Transpiration as the Leak in the Carbon Factory:
A Model of Self-Optimising Vegetation
Stanislaus Josef Schymanski
Diplom-Biologe
Albert-Ludwigs-University, Freiburg i. Br., Germany
This thesis is presented for the degree of
Doctor of Philosophy of
The University of Western Australia
School of Environmental Systems Engineering
June 2007
Schymanski (2007): PhD thesis
Abstract
ABSTRACT
In the most common (hydrological) viewpoint, vegetation is a “water pump”. However,
while vegetation does move vast quantities of water, it does not behave like a simple
pump – it is rather more like a “carbon factory” where transpiration becomes an
inevitable cost in the process of carbon assimilation. In this respect, the trade-off
between water loss and carbon gain emerges as a fundamental constraint on vegetation
function. While vegetation does experience other trade-offs (e.g. nutrient acquisition vs.
nutrient retention in nutrient-poor environments), the link between water and carbon
provides a convenient way to assemble a hypothetical optimal vegetation. If this could
be done, then hydrological models could include the effects of vegetation, even without
detailed knowledge of the local site.
The present study introduces a model of hypothetical optimal vegetation based on the
assumption that natural vegetation has co-evolved with its environment over a long
period of time and that natural selection has led to a species composition that is most
suited for the given environmental conditions. The trade-off between water loss and
carbon gain is formulated in terms of the costs associated with the maintenance of roots,
water transport tissues and foliage, and the benefits related to the exchange of water for
CO2 with the atmosphere, driven by photosynthesis. The model then calculates optimal
static and dynamic vegetation properties that would maximise the “Net Carbon Profit”,
i.e. the difference between carbon acquired by photosynthesis and carbon spent on
maintenance of the organs involved in its uptake.
This “Vegetation Optimality” model was tested in two stages at a savanna site near
Howard Springs (Northern Territory, Australia) by comparing the modelled fluxes and
vegetation properties with previous measurements. In the first stage, water availability
was prescribed through measured transpiration rates and the model was used to predict
the optimal leaf area index (LAI), the vertical distribution of photosynthetic capacity in
the canopy and the resulting rates of photosynthesis. This was done for two contrasting
months and the results were tested against the observations. In the second stage, water
availability was modelled using a physically-based catchment water balance model and
the vegetation was optimised dynamically to maximise its Net Carbon Profit over a
period of 30 years. This included a dynamic optimisation of roots and the foliage
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Abstract
Schymanski (2007): PhD thesis
properties of trees and grasses. The modelled canopy properties and fluxes matched the
available measurements surprisingly well, given that no calibration was undertaken and
only coarse climate and catchment data were used. Furthermore, no knowledge about
the vegetation on the site is needed to run the model.
Finally, the model was applied to a variety of catchments and vegetation types around
Australia, which span a wide range of climate properties, ranging from arid to humid.
The potential effect of atmospheric increase in CO2 concentrations was then
investigated by optimising the vegetation for different historic levels of atmospheric
CO2, which has led to interesting insights and surprising outcomes. These results
represent a significant advance in our ability to model or predict evapo-transpiration
rates in catchments with natural vegetation, in that hydrologists can treat vegetation
response in a more realistic way in their models than has been possible in the past.
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Schymanski (2007): PhD thesis
Table of Contents
TABLE OF CONTENTS
Abstract............................................................................................................................. i
Acknowledgements.......................................................................................................... 1
Publications Arising from this Thesis ........................................................................... 4
Chapter 1. General Introduction ................................................................................... 5
1.1
Motivation.......................................................................................................... 5
1.2
Outline of the Thesis .......................................................................................... 9
Chapter 2. A Test of the Optimality Approach to Modelling Canopy Gas
Exchange by Natural Vegetation ................................................................................. 12
2.1
Abstract ............................................................................................................ 12
2.2
Introduction...................................................................................................... 13
2.3
Methods............................................................................................................ 15
2.4
2.5
2.3.1
Overall Framework ......................................................................... 15
2.3.2
Vegetation Optimality Model ......................................................... 19
2.3.3
Study Site ........................................................................................ 36
2.3.4
Atmospheric Forcing....................................................................... 39
2.3.5
Conversion of Measured Fluxes ..................................................... 40
Results.............................................................................................................. 48
2.4.1
Canopy Optimisation ...................................................................... 48
2.4.2
Monthly Optimisations with Prescribed MA.................................... 54
2.4.3
Optimal Stomatal Conductivity ...................................................... 56
Discussion ........................................................................................................ 63
2.5.1
Modelled and Realistic Canopy Properties ..................................... 63
2.5.2
Canopy CO2 Uptake Rates .............................................................. 65
2.5.3
Optimal Stomatal Conductivity ...................................................... 66
2.5.4
Validity of the Approach................................................................. 69
2.6
Conclusions...................................................................................................... 71
2.7
Notation............................................................................................................ 72
Appendix 2.1. Photosynthetic Light Absorption....................................................... 76
A.2.1.1
Electron Transport Rates at Leaf Scale........................................... 76
A.2.1.2
Electron Transport Rates at Canopy Scale...................................... 80
Appendix 2.2. Photosynthetic Carboxylation Rates.................................................. 82
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Schymanski (2007): PhD thesis
Appendix 2.3. Atmospheric Forcing ......................................................................... 84
A.2.3.1 Separation of Global Irradiance into Direct and Diffuse
Components ..................................................................................................... 84
A.2.3.2
Solar Irradiance at the Top of the Atmosphere ............................... 85
Chapter 3. An Optimality-Based Model of the Dynamic Feedbacks Between
Natural Vegetation and Water Balance ...................................................................... 87
3.1
Abstract ............................................................................................................ 87
3.2
Introduction...................................................................................................... 88
3.3
Methods............................................................................................................ 90
3.4
3.5
3.3.1
Overall Framework for the Model .................................................. 90
3.3.2
Water Balance Model ..................................................................... 95
3.3.3
Vegetation Optimality Model ....................................................... 103
3.3.4
Study Site ...................................................................................... 116
3.3.5
Atmospheric Forcing .................................................................... 119
3.3.6
Conversion of Measured Fluxes ................................................... 122
3.3.7
Testing of the Below-Ground Component.................................... 124
3.3.8
Testing of the Coupled Model ...................................................... 125
Results............................................................................................................ 128
3.4.1
Test of the Below-Ground Component......................................... 128
3.4.2
Test of the Coupled Model ........................................................... 132
Discussion ...................................................................................................... 135
3.5.1
Relation of the Presented Approach to Existing OptimalityBased Models in Hydrology .......................................................................... 137
3.5.2
Need for Further Research ............................................................ 141
3.6
Conclusions.................................................................................................... 143
3.7
Notation.......................................................................................................... 144
Appendix 3.1. Some Thermodynamic Considerations about the Spontaneous
Flow of Water ................................................................................................ 150
Appendix 3.2. Equations Used in the Water Balance Model.................................. 153
A.3.2.1
Geometrical Relations................................................................... 153
A.3.2.2
Soil Water Fluxes.......................................................................... 156
A.3.2.3
Matric Suction Head and Hydraulic Conductivity........................ 157
A.3.2.4
Conservation of Mass and Changes in State Variables ................ 159
Appendix 3.3. Equations Used in the Vegetation Optimality Model...................... 162
A.3.3.1
Resistivity to Water Flow Towards Roots .................................... 162
A.3.3.2
Tissue Water Storage and Balance Pressure ................................. 164
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Table of Contents
Appendix 3.4. Conversions of Meteorological Data ............................................... 171
A.3.4.1
Diurnal Variation of Air Temperature .......................................... 171
A.3.4.2
Calculation of the Solar Angle...................................................... 173
Appendix 3.5. Implementation of the Shuffled Complex Evolution (SCE)
Algorithm ....................................................................................................... 174
Appendix 3.6. Soil Evaporation as a Function of Surface Soil Moisture and Soil
Temperature ................................................................................................... 177
Chapter 4. Possible Long-Term Effects of Increased CO2 on Vegetation and the
Hydrological Cycle ...................................................................................................... 179
4.1
Abstract .......................................................................................................... 179
4.2
Introduction.................................................................................................... 180
4.3
Methods.......................................................................................................... 182
4.3.1
Coupled Water Balance and Vegetation Optimality Model ......... 182
4.3.2
The “Budyko Curve” as a Benchmark .......................................... 193
4.3.3
Virtual Experiments ...................................................................... 194
4.3.4
Study Sites and Site-Specific Data................................................ 196
4.4
Results............................................................................................................ 199
4.5
Discussion ...................................................................................................... 202
4.6
Conclusions.................................................................................................... 208
Chapter 5. General Discussion................................................................................... 209
5.1
Achieved Progress.......................................................................................... 209
5.2
Recommendations for Further Research........................................................ 212
5.2.1
Further Testing of the Model ........................................................ 212
5.2.2
Prediction of Runoff...................................................................... 212
5.2.3
Improved Representation of Soil Physical Processes ................... 213
5.2.4
Plant Respiratory and Construction Costs .................................... 214
5.2.5
Develop Improved Plant Functional Types .................................. 215
5.2.6
Include additional feedbacks between vegetation, soils and the
atmosphere ..................................................................................................... 216
Literature ..................................................................................................................... 218
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Schymanski (2007): PhD thesis
LIST OF FIGURES
Figure 2.1: Net Carbon Profit as the difference between carbon acquired by
photosynthesis and the carbon used for the construction and
maintenance of organs necessary for its uptake. As CO2 uptake from
the atmosphere is inevitably linked to the loss of water from the
leaves, the root system as well as water transport and storage tissues
are essential to support photosynthesis. Soil water supply,
atmospheric water demand and daily radiation constitute the
environmental forcing. Within those constraints, vegetation is
assumed to optimise foliage, transport and storage tissues, roots and
stomata dynamically to maximise its Net Carbon Profit............................ 16
Figure 2.2: Model canopy as an array of Nl layers with randomly distributed
horizontal leaves with leaf area LA = 0.1 in each layer. ............................. 22
Figure 2.3: Subdivision of the total ground area of the site (outlined by solid
lines) into a vegetated fraction (MA, outlined by dashed lines) and a
non-vegetated fraction (1-MA). Variables are given per unit total
ground area (e.g. JA), per unit vegetated ground area (e.g. Ji) or per
unit leaf area (e.g. Jmax,i). ............................................................................ 24
Figure 2.4: Temperature dependence of Jmax with respect to its value at the
standard temperature of 298 K (Jmax25). For this plot, the optimal
temperature was assumed to be Topt = 300 K. ............................................ 29
Figure 2.5: Relationship between leaf mass per area and leaf life span from the
GLOPNET database (Wright et al. 2004). Lines are drawn by hand. ....... 31
Figure 2.6: Frequency distribution of leaf mass per area divided by leaf life span
(LMA/LL) for leaves from 678 species for which such data is
available from the GLOPNET database (Wright et al. 2004). The
median of the distribution is 12.8 g/m2/month........................................... 31
Figure 2.7: Relationship between Et and Ag for a fixed electron transport rate (JA)
and atmospheric vapour deficit (Dv), but variable stomatal
conductivity (Gs). The upper limit for Ag is determined by JA, while
the initial slope of the relationship is determined by Dv. ........................... 35
Figure 2.8: Climatic characteristics of the study site between July 2004 and June
2005. (a) Down-welling daily global irradiance (Sg,d) , (b) daily
rainfall, (c) half-hourly air temperature (Ta), and (d) surface soil
moisture (θ). (Measurements courtesy of Beringer, Hutley and
Tapper) ....................................................................................................... 37
Figure 2.9: Fraction of total shortwave irradiance in the photosynthetically active
region (400-700 nm). Data drawn for a 4-year period in Darwin,
from a global data set (Pinker and Laszlo 1997). Numbers on the
horizontal axis denote the year and month for each grid line (e.g.
“plc8401” refers to January 1984). ............................................................ 40
Figure 2.10: Measured daily evapo-transpiration (ET, grey lines) and modelled
daily soil evaporation (Es, black lines) at the flux tower site from
28/06/2004 to 27/06/2005. ......................................................................... 42
Figure 2.11: Comparison of soil respiration (Rs) obtained from a model adapted
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List of Figures
after Hanan et al. (1998) (shown in Equation ( 2.44 )) with soil
respiration formulated in Equations ( 2.42 ) and ( 2.43 ) for different
temperature and moisture conditions (Chen et al. 2002a). Left:
Variation in Rs with temperature (Ts) under wet conditions (θ = 0.2).
Right: Variation in Rs with volumetric soil moisture (θ) at Ts = 30
˚C. Solid lines show the empirical relationships in Equations ( 2.42 )
and ( 2.43 ), dashed lines show the modelled relationships using
Equation ( 2.44 ). ........................................................................................ 45
Figure 2.12: Modelled Rs (µmol m-2 s-1) for the investigated period. Left: halfhourly values plotted for 12 months. Right: box plot of dry season
(Jun-Sep) and wet season (Dec-Apr) distributions including 25%
and 75% quartiles and outliers. .................................................................. 45
Figure 2.13: Subdivision of measured net ecosystem CO2 uptake (FnC) into soil
respiration (Rs), woody tissue respiration (Rw) and foliage CO2
uptake (Ag). Rs and Rw are modelled based on measurements, while
Ag is taken as the difference. Note that measurements of FnC at
night-time are uncertain due to the frequent lack of wind. Top: a day
in the wet season (24/01/2005), bottom: a day in the dry season
(21/10/2004). .............................................................................................. 47
Figure 2.14: Optimal electron transport capacity (Jmax25) in each foliage layer and
net carbon profit (NCP) achieved with different numbers of layers.
NCP peaked when the canopy had 25 layers, and with the addition of
any more layers NCP subsequently decreased. The leaf area of each
foliage layer was prescribed as 0.1 m2/m2, so that the number of
layers can be translated directly into leaf area index by multiplying
with 0.1. ...................................................................................................... 49
Figure 2.15: Decrease in optimal electron transport capacity (Jmax25) and sunlit
leaf area (LAsun) from the top to the bottom of the canopy. Canopy
optimised for January 2005. The optimal canopy had 25 layers and
all subsequent layers would incur losses in terms of NCP (Figure
2.14). The continuation of the plot beyond 25 layers shows the
values of Jmax that would minimise the losses associated with the
maintenance of these layers. ....................................................................... 50
Figure 2.16: Half-hourly averages of canopy CO2 uptake in January 2005.
Modelled fluxes (black, thin lines) are based on optimised canopy
properties and inferred conductivity. Observed fluxes (grey, thick
lines) are inferred from eddy covariance measurements and
empirical models of soil- and above-ground wood respiration (Rs
and Rw). ....................................................................................................... 50
Figure 2.17: Ensemble means of measured and modelled diurnal canopy CO2
uptake rates. Ensemble means and standard deviations (error bars)
were computed for 31 days in January 2005. Error bars equate to
2 ä standard deviation. ............................................................................... 51
Figure 2.18: Comparison of observed (thick grey line) and modelled (thin, black
line) daily canopy CO2 uptake rates during daylight hours defined as
time intervals with mean Ia > 100 µmol s-1 m-2. The inset shows a
1:1 plot of the observed and modelled daily values. Mean absolute
error (MAE) = 0.089 mol m-2 s-1, Pearson’s r = 0.77. ................................ 51
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Schymanski (2007): PhD thesis
Figure 2.19: Comparison of observed (thick grey line) and modelled (thin, black
line) daily canopy CO2 uptake rates during daylight hours in
October 2004. Modelled values were obtained by complete
optimisation of canopy MA and LAI, as well as optimised Jmax25 in
each layer.................................................................................................... 52
Figure 2.20: Comparison of observed (thick grey line) and modelled (thin, black
line) daily canopy CO2 uptake rates during daylight hours in
October 2004. Modelled values were obtained by prescribing
MA = 0.3 and optimising canopy LAI, and Jmax25 in each layer. The
inset shows a 1:1 plot of the observed and modelled daily values.
Mean absolute error (MAE) = 0.045 mol m-2 s-1, Pearson’s r =-0.11. ....... 53
Figure 2.21: Decrease in optimal electron transport capacity (Jmax25, black
triangles) and sunlit leaf area (LAsun, grey squares) from the top to
the bottom of the canopy. Canopy optimised for October 2004, with
prescribed MA = 0.3.................................................................................... 54
Figure 2.22: Day-time canopy CO2 uptake rates for each month between July
2004 and June 2005. Modelled (black, thin lines) and observed
(grey, thick lines) canopy CO2 uptake rates were summed over the
daylight hours of each day (defined as time intervals with mean Ia
> 100 µmol s-1 m-2) and plotted together for each month. Modelled
values based on prescribed values of MA and optimised number of
foliage layers and electron transport capacity (Jmax25) in each layer.
Values of MA were set to either 1.0 or 0.3, which ever led to a better
fit with observed CO2 uptake rates. Leaf area index (LAI) in each
month is calculated from the prescribed value of MA and the
resulting optimal number of foliage layers (Nl). ........................................ 55
Figure 2.23: Modelled (black, thin lines) and observed (grey, thick lines) rates of
transpiration (Et) and canopy CO2 uptake (Ag) in January 2005.
Modelled Rates were obtained using constant λ = 1681 mol/mol.
Plotted rates are the sums for each day during daylight hours
(Ia > 100 µmol s-2 m-2). The insets show 1:1 plots of the observed
and modelled daily values. MAE: mean absolute error in mm day-1
and mol m-2 day-1 respectively. .................................................................. 58
Figure 2.24: Modelled (black, thin lines) and observed (grey, thick lines) rates of
transpiration (Et) and canopy CO2 uptake (Ag) in January 2005.
Modelled Rates were obtained using constant
Gs = 0.24225 mol s-1 m-2. Plotted rates are the sums for each day
during daylight hours (Ia > 100 µmol s-2 m-2). The insets show 1:1
plots of the observed and modelled daily values. MAE: mean
absolute error in mm day-1 and mol m-2 day-1 respectively. ....................... 59
Figure 2.25: Modelled (black, thin lines) and observed (grey, thick lines) rates of
transpiration (Et) and canopy CO2 uptake (Ag) in October 2004.
Modelled Rates were obtained using constant λ = 4952. Plotted rates
are the sums for each day during daylight hours
(Ia > 100 µmol s-2 m-2). The insets show 1:1 plots of the observed
and modelled daily values. MAE: mean absolute error in mm day-1
and mol m-2 day-1 respectively. .................................................................. 60
Figure 2.26: Modelled (black, thin lines) and observed (grey, thick lines) rates of
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Schymanski (2007): PhD thesis
List of Figures
transpiration (Et) and canopy CO2 uptake (Ag) in October 2004.
Modelled Rates were obtained using constant Gs = 0.11255
mol s-1 m-2. Plotted rates are the sums for each day during daylight
hours (Ia > 100 µmol s-2 m-2). The insets show 1:1 plots of the
observed and modelled daily values. MAE: mean absolute error in
mm day-1 and mol m-2 day-1 respectively. .................................................. 61
Figure 2.27: Modelled (black, thin lines) and observed (grey, thick lines) rates of
transpiration (Et) and canopy CO2 uptake (Ag) in October 2004.
Modelled Rates were obtained using two different values of constant
λ for before and after the first of two major rain events, indicated by
vertical arrows. Plotted rates are the sums for each day during
daylight hours (Ia > 100 µmol s-2 m-2). MAE: mean absolute error in
mm day-1 and mol m-2 day-1 respectively. .................................................. 62
Figure 2.28: Ensemble means of diurnal transpiration rates (Et) during daylight
hours (Ia > 100 µmol s-2 m-2). Grey lines: observed values, solid
black lines: Et modelled assuming constant λ (values given in
mol/mol); dashed lines: Et modelled assuming const. Gs (values
given in mol m-2 s-1). Means were computed for all days of the
respective month. Error bars are left out for clarity. .................................. 63
Figure 2.29: Modelled (black, thin lines) and observed (grey, thick lines) rates of
transpiration (Et) and canopy CO2 uptake (Ag) in October 2004.
Modelled Rates were obtained using constant λ = 3000. Plotted rates
are the sums for each day during daylight hours
(Ia > 100 µmol s-2 m-2). .............................................................................. 68
Figure 2.30: Hypothetical electron transport rates in four sub-areas (“A1”...”A4”)
of similar sizes with different Jmax (overlapping curves) and the total
electron transport rate of all sub-areas together (top curve, denoted
as “A”). Note that J is plotted as total J for each sub-area and hence
depends on the size of each sub-area. Although each sub-area has an
abrupt transition between non-saturated and saturated rate, the total
electron transport function for the leaf appears curved. ............................. 77
Figure 2.31: Cumulative Distribution Function of an exponential distribution of
jmax with a mean value of 160 µmol m-2 s-1. ............................................... 78
Figure 2.32: Electron transport rate as a function of Il. Comparison of the classic
formulation in Equation ( 2.48 ) (denoted Jθ=0.7), using α = 0.3,
θJ = 0.7 and Jmax = 160 µmol/m2/s, with the new formulation in
Equation ( 2.57 ) (denoted Jnew), using the same values for α and
Jmax. ............................................................................................................. 80
Figure 2.33: Jmax-limited (solid lines) and Vcmax-limited (dashed lines) CO2
assimilation rates as a function of stomatal conductivity (Gs). J was
prescribed as different proportions of Jmax, where
Jmax = 2.68 ä Vcmax. As J generally increases with irradiance between
0 and Jmax, the proportion of J to Jmax indicates the degree of lightsaturation of the photosynthetic system. Model parameters were
taken from figure 2.6 in von Caemmerer (2000):
Vcmax = 100 µmol m-2 s-2, Kc = 259 µbar, Ko = 179 mbar,
Γ* = 38.6 µbar, Rl = 1 µmol m-2 s-2, Ol = 0.2, Ca = 350ä10-6...................... 83
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Figure 3.1: Net Carbon Profit as the difference between carbon acquired by
photosynthesis and the carbon used for the construction and
maintenance of organs necessary for its uptake. As CO2 uptake from
the atmosphere is inevitably linked to loss of water from the leaves,
the root system as well as water transport and storage tissues are
essential to support photosynthesis. Soil water supply, atmospheric
water demand and daily radiation constitute the environmental
forcing. Within those constraints, vegetation is assumed to optimise
foliage, water transport and storage tissues, roots and stomata
dynamically to maximise its Net Carbon Profit. ........................................ 91
Figure 3.2: Representation of perennial (left) and seasonal (right) vegetation
components. The perennial vegetation component was assumed to
be composed of evergreen trees, while the seasonal component was
assumed to be composed of annual grasses only. Thus, perennial
vegetation was allowed to optimise root depth (yr,p) without
constraints, while the seasonal vegetation had a prescribed root
depth (yr,s) of 1 m only. At the same time, the fraction of the area
taken up by perennial vegetation (MA,p) was fixed, while the area
fraction taken up by seasonal vegetation (MA,s) was allowed to vary
from day to day. For simplicity, deciduous trees were neglected in
this study. ................................................................................................... 93
Figure 3.3: Cross-section of a simplified elementary watershed. Variables on the
left hand side denote spatial dimensions (see text), while the
variables on the right hand side denote water fluxes (precipitation
(Qrain), infiltration (Qinf), infiltration excess runoff (Qiex), evapotranspiration from the saturated zone and the unsaturated zone (ETs
and ETu respectively), flow between saturated and unsaturated layer
(Qu) and outflow across the seepage face (Qsf)). ........................................ 96
Figure 3.4: Conceptual catchment, with the unsaturated zone subdivided into
three soil layers. Soil layers are given indices (i = 1...nlayers) that
increase with increasing depth (right). The indices relating to fluxes
refer to fluxes across the bottom boundary of the respective layer
(left). ........................................................................................................... 97
Figure 3.5: Conceptual dependence of the size of the vascular system on the
rooting depth and horizontal cover. Deep rooting trees with rooting
depth yr,p need larger vascular systems per unit horizontal cover than
shallow rooting annuals. ........................................................................... 111
Figure 3.6: Seasonality of modelled soil respiration (Rs). Box plots of dry season
(Jun-Sep) and wet season (Dec-Apr) distributions include 25% and
75% quartiles and outliers of half-hourly Rs over 5 years. Modelled
values correspond well with observations between September 1998
and January 2001, when soil respiration was reported to vary
between 0.95 and 3.5 µmol m-2 s-1 in the dry season and between 3.5
and 8.4 µmol m-2 s-1 in the wet season (Chen et al. 2002b). .................... 123
Figure 3.7: Subdivision of measured net ecosystem CO2 uptake (FnC) into soil
respiration (Rs), foliage CO2 uptake (Ag) and woody tissue
respiration (Rw, not shown). Rs and Rw are modelled based on
measurements, while Ag is taken as the difference. For clarity and
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Schymanski (2007): PhD thesis
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compatibility with common practice, all fluxes have been plotted in
g carbon per day per m2 catchment area, with negative values for
carbon uptake and positive values for carbon release (note the signs
in legend). ................................................................................................. 124
Figure 3.8: Flow diagram of the coupled water balance and Vegetation
Optimality model. Input variables are at the top, while model
outputs are separated into state variables (dashed boxes) and fluxes
(along arrows). Symbols are explained in Table 3.2 on page 143 (the
subscript i denotes a vector over all soil layers). For clarity, only
selected model outputs are drawn. ........................................................... 127
Figure 3.9: Below-ground conditions 20 days after model initialisation. Vertical
soil profiles show values for each soil layer between the surface and
the variable water table. Soil saturation (su,i, bottom left) is clearly
reduced in the root zone, the distribution of root surface area (SAr,i,
top right) in the soil profile is skewed towards the deeper soil layers,
while daily root water uptake (Qr,i, top left) slowly increases with
soil depth, with a sharp incline in the lowest layer of the root zone.
The plot on the bottom right shows observed (grey line) and
modelled (black line) saturation degree in the top soil layer (su,i=1)
for 12 months, with a round dot indicating the position in time of the
other three plots. ....................................................................................... 129
Figure 3.10: Below-ground conditions during the wet season. Vertical soil
profiles show values for each soil layer between the surface and the
variable water table. The distribution of soil saturation (su,i, bottom
left) shows the propagation of multiple wetting fronts through the
soil profile, while the distributions of root surface area (SAr,i, top
right) and root water uptake (Qr,i, top left) in the soil profile are
concentrated in the top soil with a spike in the lowest layer of the
root zone. The plot on the bottom right shows observed (grey line)
and modelled (black line) saturation degree in the top soil layer
(su,i=1) for 12 months, with a round dot indicating the position in
time of the other three plots. ..................................................................... 130
Figure 3.11: Observed (grey line) and modelled (black line) saturation degree in
the top soil layer (su,i=1) using a dynamically optimised root profile. ..... 131
Figure 3.12: Observed (grey line) and modelled (black line) saturation degree in
the top soil layer (su,i=1) using a constant, homogeneous root profile. .... 131
Figure 3.13: Modelled seasonal variation in the area fraction covered by grasses
(MA,s) between 2001 and 2005.................................................................. 133
Figure 3.14: Modelled (black) and observed (grey) daily evapo-transpiration rates
(ET, top) and CO2 uptake rates (Ag,tot, bottom). Observed and
modelled total CO2 uptake during the plotted period was 629 mol
m-2 and 725 mol m-2 respectively. The dashed line shows scaled
daily averages of the validity flag values, ranging between -0.2 for a
whole day of valid measurements and -0.4 for a whole day of gapfilled data using a neural network approach. ............................................ 134
Figure 3.15: Existing optimality-based models of vegetation water use.
Ecophysiological models ask how stomata should adjust to
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Schymanski (2007): PhD thesis
maximise CO2 uptake over periods of time when water is limited,
given biochemical and structural constraints (constant leaf area and
photosynthetic capacity). Ecohydrological approaches to ecological
optimality on the other hand look at large scales and optimise longterm averages of vegetation properties, vegetation type and cover.
They do not include an explicit coupling between CO2 assimilation
and transpiration, but assume that total water use is maximised,
where the trade-off is to avoid water depletion and “stress”. These
models are not suitable for shorter time scales, because the objective
function does not differentiate when the water should be used. The
dashed box stands for the medium scale, which is the subject of the
model presented in this paper. .................................................................. 139
Figure 3.16: Schematic linear hill slope of length Lh with maximum land
elevation zm, water table at elevation zw and a channel at elevation zr. .... 154
Figure 3.17: Examples of matric suction heads in the soil profile and estimated
gradients between sub-layers. (a) Non-equilibrium situation, which
would lead to water flux out of layer 2 into layer 1 and layer 3. (b)
Equilibrium situation, where no fluxes would occur. .............................. 157
Figure 3.18: Example plots of matric pressure head (h) and relative hydraulic
conductivity (Kunsat/Ksat) using the van Genuchten (“vG”, thick grey
lines) and Brooks & Corey (“BC”, thin black lines) formulations. ......... 158
Figure 3.19: Comparison between original and modified formulation of κ as a
function of qx. The curves are virtually indistinguishable for the
majority of the range of qx. ....................................................................... 166
Figure 3.20: Relations between Pb and rWL for different values of qx. .......................... 166
Figure 3.21: Estimated and observed air temperature for 5 days in the dry season
in 2004 (left) and 5 days in the wet season in 2005 (right).
Irregularities in the estimated time series can occur at midnight,
when the model jumps from one day’s temperature characteristics to
the next day’s characteristics.................................................................... 172
Figure 3.22: Scatter plot of estimated versus observed air temperature for halfhourly measurements over 12 months (July 2004 to June 2005).
Dashed line marks the 1:1 line. ................................................................ 172
Figure 4.1: Flow diagram of the coupled water balance and Vegetation
Optimality model. Input variables are at the top, while model
outputs are separated into state variables (dashed boxes) and fluxes
(along arrows). Symbols are explained in the main text (the
subscript i denotes a vector over all soil layers). For clarity, only
selected model outputs are drawn. ........................................................... 183
Figure 4.2: Net Carbon Profit as the difference between carbon acquired by
photosynthesis and the carbon used for the construction and
maintenance of organs necessary for its uptake. As CO2 uptake from
the atmosphere is inevitably linked to the loss of water from the
leaves, the root system as well as water transport and storage tissues
are essential to support photosynthesis. Soil water supply,
atmospheric water demand and daily radiation constitute the
environmental forcing. Within those constraints, vegetation is
xii
Schymanski (2007): PhD thesis
List of Figures
assumed to optimise foliage, water transport and storage tissues,
roots and stomata dynamically to maximise its Net Carbon Profit.......... 193
Figure 4.3: Representation of perennial (left) and seasonal (right) vegetation
components. The perennial vegetation component was assumed to
be composed of evergreen trees, while the seasonal component was
assumed to be composed of annual grasses only. .................................... 193
Figure 4.4: Annual energy and water budgets for catchments modelled with an
atmospheric CO2 concentration of 350 ppm. The ratio ET,a/(In,a/λE)
represents the partitioning of annual net radiation (In,a) into evapotranspiration (ET,a) and sensible heat flux, while the ratio
Qrain,a/(In,a/λE) denotes the radiative dryness index for a year. Data
points to the left of the vertical dashed line are generally considered
to represent “water-limited” conditions, while data points to the
right of the line are considered to represent “energy-limited”
conditions. Curved lines are empirical curves fitted to observed
long-term averages in a wide range of catchments by different
authors. Each data point represents one year from a set of 30 years
per site, except for data points denoted as “Choudhury (1999)”,
which refer to long-term averages of a global variety of catchments
compiled by Choudhury (1999). .............................................................. 200
Figure 4.5: Modelled change in average annual evapo-transpiration (ET) as a
consequence of medium-term adaptation and long-term adaptation
to increased CO2 from 317 ppm to 350 ppm. The dashed line
represents the reduction in ET that would be necessary to explain the
observed increase in runoff between 1960 and 1990 (Gedney et al.
2006). See text for details. ........................................................................ 202
Figure 4.6: Examples of relationships between Et and Ag for different levels of
atmospheric CO2 concentration (top) and different values of
photosynthetic electron transport rate (Je) (bottom). Plots were
obtained by applying Equations ( 4.18 ) and ( 4.25 ) with varying
stomatal conductance Gs. Common parameter values were
Γ* = 38.6ä10-6 mol/mol, Jmax = 300 µmol m-2 s-1 and a = 1.6 and
Dv = 0.03 mol/mol. In the top plot, atmospheric CO2 was set to
Ca = 317ä10-6 mol/mol (left-most curve), Ca = 350ä10-6 mol/mol
(middle curve) and Ca = 375ä10-6 mol/mol (right-most curve). In the
bottom plot, Ca = 350ä10-6 mol/mol, while Je was increased by 10%
(middle curve) and 20% (right-most curve) from the original value
(left-most curve). Dots represent the locations where the slopes of
the curves are 2000 mol/mol. Dashed lines in the top plot represent
the decrease in Et that would have resulted from increased CO2 if Ag
was held constant...................................................................................... 206
xiii
Schymanski (2007): PhD thesis
Acknowledgements
ACKNOWLEDGEMENTS
First of all, I wish to thank my supervisors, Prof. Murugesu Sivapalan and Dr. Michael
Roderick for their continuous support and their contributions to this thesis in the form of
ideas and critical review. Siva, thanks so much for making this project possible and for
your help and encouragement to cut a rough way through the jungle before coming back
and paving it. I would have been truly lost without your ability to structure things and
expose the essence of them. Mike, thanks a thousandfold for your constant flow of ideas
and inspiration. Without your lateral thinking, I wouldn’t even see the jungle I was
trying to get through. Siva or Mike was always available when I needed feedback, even
when they were thousands of kilometres away. I truly cannot imagine a better team of
supervisors.
I would also like to thank Dr. Lindsay Hutley, Dr. Jason Beringer and Prof. Nigel
Tapper from the Tropical Savannas CRC (Cooperative Research Centre) for sharing
their eddy flux measurements with me. Moreover, Lindsay and Jason provided a wealth
of background information about the experimental site and were very helpful in
eliminating some unrealistic hypotheses. Thanks for all the extremely helpful email
discussions and for letting me join you on a field campaign.
Thanks to Assoc. Prof. Keith Smettem and Dr. Richard Harper for helping with my
supervision in the initial stages and for putting the student’s interest first. Thanks also to
Dr. Anas Ghadouani for taking on the administrative part of my supervision following
Siva’s move to the U.S. I appreciate it very much, especially considering the
administrative workload related to my PhD.
During my candidature, I came across a large number of very helpful researchers, who I
wish to thank sincerely for their generous contributions to my work. Dr. Sandra Berry
provided very helpful advice and references concerning various aspects of plant
adaptations, Dr. Erik Veneklaas helped with my understanding of root and leaf costs
through discussions and very useful references, Dr. Yoshiyuki Yokoo contributed the
initial code to compute the water balance of a catchment and Dr. Neil Viney explained
the initial code for the SCE optimisation algorithm to me. Haksu Lee and Prof. Majid
Hassanizadeh helped me understand the REW theory and model. Dr. John Evans was
extremely helpful in explaining various aspects of plant physiology to me, Dr. Tom
1
Acknowledgements
Schymanski (2007): PhD thesis
Buckley shared his ideas about optimality with me and gave me some inspiration for the
parameterisation of root costs and Dr. Susanne von Caemmerer was very helpful in
answering questions about the photosynthesis model. Further, Dr. Miko Kirschbaum
gave valuable advice in relation to testing my model, Dr. Lynda Prior helped with my
understanding of the variability in leaf properties and Dr. Richard Silberstein gave me
valuable insights into tree water storage. Many other researchers, mainly from the
University of Western Australia (UWA) and the Cooperative Research Centre for
Greenhouse Accounting (CRC-GA), freely shared their questions, opinions and advice
with me, which certainly influenced my understanding and thinking about ecology and
hydrology. Thanks to all of you!
I am also indebted to my office mates Iain Struthers, Mike Meuleners, Dyah
Kusumastuti and Jos Samuel for their frequent ad-hoc advice on both scientific and
personal matters. These acknowledgments should also be extended to the entire group
of students and staff at the Centre for Water Research (CWR) and now School of
Environmental Systems Engineering (SESE), which made my time here very enjoyable
and instructive. Big thanks also go to Ivan Kwan and Ken Summers for the contribution
of computing resources.
On an even more personal matter, I am deeply thankful to my wife Emma, her parents
and of course my parents for supporting me at all the levels they could. Their support is
irreplaceable.
I would also like to acknowledge the different institutions that funded my research and
the people that helped me with getting it all organised. The University of Western
Australia granted me a University Postgraduate Award (International Students) and the
Department for Education, Science and Training contributed an International
Postgraduate Research Scholarship. Thanks to Wendy Nadebaum and Heather Williams
for their help with the application process and administration. The Cooperative
Research Centre for Greenhouse Accounting (CRC-GA) contributed a Postgraduate
Research Grant and an additional stipend for the finalisation of my thesis. Many thanks
to Dr. Janette Lindesay and Dr. Robyn Harris for their dedication to helping students at
the CRC-GA. Being part of the SESE and the CRC-GA at the same time was a very
inspiring situation for me to do inter-disciplinary research. I learned an incredible
Schymanski (2007): PhD thesis
Acknowledgements
amount through the Annual Science Meetings and the student workshops organised by
the CRC-GA.
Finally, I would like to thank the external reviewers of this thesis, Prof. Larry Band,
Prof. Amilcare Porporato and Prof. Ian Woodward, for their encouragement and helpful
recommendations.
3
Publications Arising from this Thesis
Schymanski (2007): PhD thesis
PUBLICATIONS ARISING FROM THIS THESIS
This thesis comprises three paper manuscripts that are intended for journal publication.
Each of the manuscripts is self-contained and forms a chapter in the thesis (Chapters 2,
3 and 4). However, due to page limits applicable to journal submissions, the chapters
included in this thesis are extended versions of the manuscripts, containing details that
will be left out in the published papers. This decision was made to provide a
comprehensive description of the models used and to facilitate further application of the
new modelling approach.
An abbreviated version of Chapter 2 will be submitted for publication in Plant, Cell and
Environment as “A Test of the Optimality Approach to Modelling Canopy Gas
Exchange by Natural Vegetation” by Schymanski, S. J.; Roderick, M. L.; Sivapalan, M.;
Beringer, J. and L. B. Hutley.
Chapter 3 will be submitted in an abbreviated version for publication to Water
Resources Research as “An Optimality-Based Model of the Dynamic Feedbacks
Between Natural Vegetation and Water Balance” by Schymanski, S. J.; Sivapalan, M.;
Roderick, M. L.; Hutley, L. B. and J. Beringer.
Chapter 4 will be submitted for publication to Water Resources Research as “Possible
Long-Term Effects of Increased CO2 on Vegetation and the Hydrological Cycle” by
Schymanski, S. J.; Sivapalan, M. and M. L. Roderick.
All the theoretical and modelling work presented in this thesis has been carried out
entirely by me, with valuable advice and guidance provided by my supervisors Prof. M.
Sivapalan and Dr. Michael Roderick. The field measurements included in Chapter 2 and
Chapter 3 were kindly provided by Dr. Lindsay Hutley and Dr. Jason Beringer, who
also helped with the interpretation thereof.
Schymanski (2007): PhD thesis
Chapter 1
CHAPTER 1. GENERAL INTRODUCTION
1.1
MOTIVATION
Evapo-transpiration is often seen as a physical process (e.g. evaporation as a result of
the energy partitioning into latent and sensible heat flux), controlled by energy, vapour
pressure and turbulence. Vegetation is thereby treated as a wick that facilitates the flow
of water from the soil to the atmosphere.
In vegetated areas, however, evapo-
transpiration is often dominated by transpiration, which is a biological process
controlled by stomatal regulation and vegetation dynamics. Biologists commonly
consider transpiration as a side-effect of photosynthesis, as water vapour exits the leaf
along the same pathway as CO2 enters the leaf, and because they see the leaf as a
specialised organ to assimilate carbon, not to lose water. The biological control of
transpiration hence depends on many factors other than energy, vapour pressure and
turbulence. For example, stomatal conductance per unit leaf area is coupled to
photosynthesis and largely controlled by plant hormones, while leaf area itself varies
seasonally depending on climate, soil type, nutrient availability and vegetation type.
Current practice for modelling of transpiration in ecology and hydrology relies on long
term observations of the system in question and the derivation of correlations between
observed variables. Unfortunately, such observations are often not available, or the
correlations turn out to be highly non-linear and often complicated by hysteresis effects
(e.g. Jarvis 1976). Furthermore, establishment of correlations for a given set of
conditions does not guarantee the validity of these correlations for other conditions,
which is particularly a problem in connection with environmental change.
The prediction of change in the water, carbon and nutrient cycles due to changes in
climate or land use is currently one of the most challenging and urgent issues in science
and policy. However, in order to make predictions into the future with some certainty,
we need models that do not rely on calibration using past measurements at a given site.
A promising alternative to the conventional, correlation-based models, are models based
on optimality. The assumption that the self-organisation of biological systems is
governed by some principles of optimal adaptation has been implicit in biological
thinking for thousands of years, but it is not until recently that this assumption has been
5
1.1 Motivation
Schymanski (2007): PhD thesis
used to construct quantitative models and test their predictions (Sutherland 2005).
Optimality-based models have the advantage that they do not rely on observed
correlations as model input, but aim at predicting the correlations themselves, or in
other words:
“A considerable strength of using optimization is that once we understand why
organisms are as they are, then it should be possible to understand how they will
respond to new conditions.” (Sutherland 2005)
Models based on optimality generally involve three elements:
–
An objective function that guides the optimisation (e.g. maximisation of
photosynthesis, maximisation of water use or minimisation of “stress”)
–
Adjustable levers or “degrees of freedom” that can be adjusted by the system
(e.g. stomatal conductivity, canopy cover, rooting depth)
–
Constraints that limit the feasible range of values for the adjustable levers (e.g.
water balance, energy balance, biochemical or biophysical constraints)
Optimality principles have been employed by a number of researchers to model
transpiration. The idea to use optimality principles in eco-hydrology was promoted by
Peter Eagleson in the late 1970s and early 1980s (Eagleson 1978; Eagleson 1982).
Eagleson used two objective functions to optimise the two vegetation parameters:
–
“vegetated fraction of surface” (M) and
–
“plant coefficient (equal to potential rate of transpiration divided by potential
(soil surface) rate of evaporation [...])” (kv) (Eagleson 1978).
The two objective functions were
–
maximisation of soil moisture or equivalently, the minimisation of evapotranspiration (Eagleson 1978, page 755) and
–
the maximisation of the “index of potential biomass” (M kv) (Eagleson 1978,
page 756).
Schymanski (2007): PhD thesis
Chapter 1
The first objective function was used to establish a relationship between the parameters
M and kv, while the second one was used to find the optimal kv for a given climate and
soil type. Eagleson (1982) later extended the framework to include soil properties in the
optimisation, where he hypothesised that soils and vegetation would co-develop over
very long periods of time to approach a dynamic equilibrium (near optimal for
vegetation). Both M and kv were formulated as annual averages in Eagleson’s work, so
that intra-annual vegetation dynamics could not be modelled. Recently, Kerkhoff et al.
(2004) evaluated Eagleson’s optimality hypotheses from an ecological perspective and
pointed out several inconsistencies with current understanding of vegetation ecology.
Most importantly, the hypothesis that vegetation would aim at minimising evapotranspiration has been pointed out to be unrealistic, as this would effectively result in a
minimisation of photosynthetic activity (Kerkhoff et al. 2004).
Rodriguez-Iturbe and co-workers also assumed that vegetation would aim at minimising
“water stress”, but, in contrast to Eagleson, they defined water stress quantitatively as a
non-linear function of soil moisture. Different stress functions for grasses and trees were
defined based on empirically derived functions of evapo-transpiration in relation to soil
moisture for both vegetation types. Hypothesising that individuals in a plant community
would act together to reduce their water stress, the researchers showed numerically that
spatial interactions between woody and grassy vegetation types can lead to a more
efficient community water use and decreased global water stress, even if both
vegetation types compete for the same resource (Rodriguez-Iturbe et al. 1999a;
Rodriguez-Iturbe et al. 1999b).
In a later published framework, Rodriguez-Iturbe and co-workers linked climate
dynamics, soils and vegetation to obtain probabilistic soil moisture patterns and values
for vegetation water stress with the aim of investigating “the optimal environmental
condition for different functional types of vegetation” (Laio et al. 2001a; Laio et al.
2001b; Porporato et al. 2001; Rodriguez-Iturbe et al. 2001). As in Eagleson’s approach,
total evapo-transpiration was assumed to be proportional to productivity and the optimal
condition was defined as “somewhere” between minimum vegetation water stress and
maximum total evapo-transpiration (Porporato et al. 2001). Although this model may
allow comparison of the suitability of different climates for a given plant community, it
does not allow the reverse, a prediction of an optimal vegetation composition for a
7
1.1 Motivation
Schymanski (2007): PhD thesis
given climate, as the range of possible stress functions and the number of possible
vegetation compositions could be very large.
Optimality principles have also been used in eco-physiology, for example to make
predictions of gas exchange at leaf scale. Cowan and Farquhar (1977) assumed a priori
that plants would optimise stomatal conductivity dynamically in order to maximise total
photosynthesis for a given amount of transpiration. This assumption, together with a
quantitative theory about the non-linear coupling between transpiration and CO2
assimilation, allowed them to formulate how stomatal conductivity should vary in
response to the rate of photosynthesis and atmospheric water vapour deficit, given a
fixed amount of water available for transpiration. Later, Cowan (1982) looked at a
longer time scale and introduced competition for water by processes not under the
control of stomata (e.g. drainage, soil evaporation, water extraction by other plants).
This permitted the assessment of day to day changes in transpiration during a dry
period. Mäkelä and co-workers built a model using these concepts and tested it
successfully against field measurements at leaf scale (Berninger et al. 1996; Mäkelä et
al. 1996; Hari et al. 1999; Hari et al. 2000). However, this was based on water
availability per unit leaf area and did not allow predictions about changes in leaf area
itself, and was therefore limited to time scales at which leaf area was not expected to
change (~ days to weeks).
In summary, the eco-hydrological concept of Ecological Optimality is suited for the
prediction of long-term averages of transpiration only, due to the neglect of the nonlinearity between carbon uptake and transpiration, while the eco-physiological concept
is suited for the prediction of short-term dynamics only, due to its neglect of the longterm water balance and associated changes in leaf area.
So far, predictions at the medium scale, i.e. the month-to-month and year-to-year
variability of transpiration at canopy or catchment scale have not been investigated with
optimality-based models. Considering that this is the scale that is most relevant to
surface hydrologists and many ecologists, the present thesis is devoted to formulating
and testing an optimality-based model of transpiration at canopy to catchment scale,
representing dynamics from diurnal to inter-annual.
Schymanski (2007): PhD thesis
1.2
Chapter 1
OUTLINE OF THE THESIS
The present work is a multi-disciplinary project drawing together, and contributing to,
plant physiology, ecology and hydrology all at the same time, where the ecology links
all parts of the thesis. Chapter 2 focuses on the plant physiological aspects of
transpiration and photosynthesis, and implements them into an optimality-based model
to predict optimal foliage properties and transpiration in the short term (diurnal to
monthly scale). This chapter does not contribute directly to the hydrological science, as
it does not consider any below-ground processes. The findings from Chapter 2 are then
used to formulate a coupled vegetation and water balance model at catchment scale in
Chapter 3, which is again based on optimality. The focus of Chapter 3 is more on the
hydrological aspects of plant water use and how they influence optimal vegetation and
water use in the longer term (intra-annual to inter-annual scale). The same model is then
taken to an even larger scale in Chapter 4, where it is applied to different catchments
along an aridity gradient from arid to humid. Chapter 4 focuses mainly on the
implications of the model for our understanding of the long-term adaptations of
vegetation to climate and increased atmospheric CO2 concentrations.
The following paragraphs describe the three main thesis chapters in more detail.
Chapter 2 introduces the concept of the “Vegetation Optimality” (VO) model developed
in this thesis and tests whether its predictions are consistent with observations. The
concept is based on a direct coupling between photosynthesis and transpiration,
following the eco-physiological approach (Cowan and Farquhar 1977). The major
innovation in this concept however is the maximisation of the “Net Carbon Profit” (NCP)
as objective function, as opposed to for example the maximisation of water use,
photosynthesis, Net Primary Production (NPP), or the minimisation of “stress”, which
were proposed by other authors. The Net Carbon Profit is defined as the difference
between carbon acquired by photosynthesis and carbon spent on maintenance of the
organs involved in its uptake. This definition allows attributing generic costs in terms of
NCP to plant organs like leaves or roots, so that unrealistically high values for leaf area
or root abundance are avoided. Constraints on the optimisable parameters are thus
implicit in the costs and do not need to be defined separately. The general question
tackled by the Vegetation Optimality model is:
9
1.2 Outline of the Thesis
Schymanski (2007): PhD thesis
Given the climate, topography and soils, how should vegetation adapt, to make
the best use of the resources, in order to maximise its Net Carbon Profit?
Plants have a wealth of choices in shape, form and metabolic processes to adapt to their
environment, but in order to make the solution computationally feasible, we tried to
focus on the most relevant ones only. For the short term, for which gas exchange data
was available and could be used as a constraint on water use, we were able to simplify
the above question to:
Given the climate and measured water use, what would be the optimal canopy
structure and distribution of photosynthetic capacity within this canopy to
maximise Net Carbon Profit in each month?
Both the maintenance of a certain leaf area and of the photosynthetic capacity within
this leaf area require carbon expenditure, so plants have to decide how many leaves they
put up and how much chlorophyll and associated biochemical “machinery” they put into
these leaves in each season.
The use of observed transpiration for prescribing water use allowed us to neglect the
whole below-ground part of vegetation and concentrate on canopy processes in more
detail. In particular, it was possible to test whether maximisation of NCP together with
the parameterised costs for leaf area and the photosynthetic apparatus would allow the
prediction of canopy properties that were previously treated as input variables in
biological models.
In addition to the costs for foliage, the supply of water from the ground to the leaves
requires carbon expenditure (i.e. construction of a vascular system). The larger the
vegetated area within a catchment and the greater the depth to ground water, the greater
are these costs. These costs are introduced in Chapter 3, where the Vegetation
Optimality model is supplemented by a below-ground component that computes the
vertical soil moisture distribution and runoff at the sub-hourly time step and optimises
the vertical root distribution and rooting depth given the water demand determined by
the above-ground component. Due to the increased complexity brought about by the
below-ground component, the canopy had to be further simplified in order to maintain
the computational feasibility of the model. Therefore, the catchment-scale Vegetation
Optimality model, as presented in Chapter 3, does not resolve the vertical organisation
Schymanski (2007): PhD thesis
Chapter 1
of the canopy, but represents the canopy by two “big leaves”, one for perennial
vegetation (trees) and one for seasonal vegetation (grasses). The distinction between
perennial and seasonal vegetation allowed us to account for different resource use
strategies, such as the ephemeral exploitation of the surface soil moisture by grasses
versus the continuous exploitation of a deeper soil profile by trees. Although the model
in Chapter 3 works at a sub-hourly time step, only seasonal variations in fluxes and state
variables are compared with observations. This is because of the above-mentioned
simplification and because the model now optimises vegetation over 30 years, for which
only coarse climate data was available. The last four years of a 30-year simulation are
then compared with available observations of canopy evapo-transpiration and CO2
exchange in a tropical savanna.
Chapter 4 applies the coupled water balance and Vegetation Optimality model to a
range of catchments around Australia along an aridity gradient from arid to humid.
After testing whether the model gives reasonable results in terms of annual evapotranspiration in all catchments, it is used to investigate the possible medium- and longterm responses of vegetation to increased levels of atmospheric CO2 in the different
climates. This analysis takes advantage of the model’s ability to predict evapotranspiration without prescribing any vegetation properties a priori. The results put into
perspective and also shed light on the widespread opinion that increased levels of
atmospheric CO2 would lead to a general decrease in evapo-transpiration because of the
CO2 effect on stomatal closure.
11
2.1 Abstract
Schymanski (2007): PhD thesis
CHAPTER 2. A TEST OF THE OPTIMALITY APPROACH
TO MODELLING CANOPY GAS EXCHANGE BY
NATURAL VEGETATION
2.1
ABSTRACT
Natural vegetation has co-evolved with its environment over a long period of time and
natural selection has led to a species composition that is most suited for the given
conditions. Part of this adaptation is the vegetation’s water use strategy, which
determines the amount and timing of water extraction from the soil. Knowing that water
extraction by vegetation often accounts for over 90% of the annual water balance in
water-limited ecosystems, we need to understand its controls in order to model the
hydrologic cycle properly.
Water extraction by roots is driven by transpiration from the canopy, which in turn is an
inevitable consequence of CO2 uptake for photosynthesis. Photosynthesis provides
plants with their main building material, carbohydrates, and with the energy necessary
to thrive and prosper in their environment. Therefore we expect that natural vegetation
would have evolved an optimal water use strategy to maximise its “Net Carbon Profit”
(the difference between carbon acquired by photosynthesis and carbon spent on
maintenance of the organs involved in its uptake).
Based on this hypothesis and on an ecophysiological gas exchange and photosynthesis
model, we modelled the optimal vegetation for a site in sub-tropical Australia and
compared the modelled fluxes with observations on the site. The comparison gives
insights into theoretical and real controls on transpiration and photosynthesis and tests
the optimality approach to the modelling of gas exchange in natural vegetation with
unknown properties.
The main advantage of the optimality approach we adopt is that no assumptions about
the particular vegetation on a site are needed, which makes it very powerful for
predicting vegetation response to long-term climate- or land use change.
12
Schymanski (2007): PhD thesis
2.2
Chapter 2
INTRODUCTION
In the past century, plant biologists have achieved a great deal of understanding about
the biochemical processes controlling photosynthesis and the mechanisms by which
stomata open and close. With the help of boundary layer meteorologists, sophisticated
models have been developed to calculate the gas exchange between a canopy of leaves
and the atmosphere if stomatal conductivity, canopy properties and atmospheric
conditions are known. These models have proven relatively successful for crops, where
canopy properties can be monitored with handheld devices and the response of stomata
to environmental variables can be modelled empirically, based on prior observations of
the given species. Natural vegetation, on the other hand, is very difficult to parameterise
in these models, because it often consists of a range of species whose stomatal response
is largely unknown. In addition, both species composition and canopy structure of
natural vegetation can vary widely in space and time. The application of gas exchange
models that use prescribed vegetation properties becomes even more problematic if the
aim is to make predictions into the future, particularly with respect to changes in climate
or land use.
However, it may be possible to get away from simple extrapolations of past
observations if the problem is approached from a different perspective. Cowan and
Farquhar (1977) assumed a priori that plants would optimise stomatal conductivity
dynamically in order to maximise total photosynthesis for a given amount of
transpiration. This optimality assumption allowed them to formulate how stomatal
conductivity should vary in response to the rate of photosynthesis and atmospheric
water vapour deficit, given a fixed amount of water available for transpiration. Other
authors subsequently used this optimality approach in numerical models to predict gas
exchange rates at leaf level and at diurnal to daily scales without prescribing stomatal
response to environmental forcing (Berninger et al. 1996; Hari et al. 1999; Hari et al.
2000). However, the photosynthetic capacity of the leaves and the amount of water
available per unit leaf area still had to be prescribed. Hence, the application of this
approach at larger scales in time and space (e.g. canopies, days to years) requires
detailed information about canopy properties and water availability.
Biochemical properties of foliage have been observed to adapt to environmental
conditions as well, not only spatially within a canopy (Kull and Niinemets 1998;
13
2.2 Introduction
Schymanski (2007): PhD thesis
Niinemets et al. 1999a; Niinemets et al. 2004), but also seasonally (Misson et al. 2006).
Kull (2002) reviewed a range of models that derived theoretically optimal distributions
of photosynthetic capacity in the canopy and noted that all of these models overpredicted canopy photosynthesis and the slope of the nitrogen profile through the
canopy, while under-predicting leaf area index. The author blamed inappropriate merit
functions, over-simplified photosynthesis models and the lack of consideration of whole
plant processes in the acclimation of the canopy for the discrepancy between model
results and reality. Kull also stated that
“Poor results in optimum modelling are not proof that optimality fails; they
merely imply that the function to be maximised in a natural community remains
undiscovered.” (Kull 2002)
If some of the self-organising principles of vegetation could be summarised in an
appropriate merit function (or “objective function” in mathematical language), this
would greatly improve our ability to describe how vegetation will change in a changing
environment. Furthermore, the generality of such an objective function would facilitate
the construction of global models and may reduce the uncertainty associated with the
extrapolation of local observations, as is the case now. However, the appropriateness of
the objective function and the associated constraints can only be tested by comparison
of model predictions with observations in nature.
The aim of this paper is to formulate a quantitative, general concept of “Vegetation
Optimality” and to test whether it is consistent with observations. In order to test the
generality of the concept, no attempt was made to “fit” parameters in this exercise.
14
Schymanski (2007): PhD thesis
2.3
2.3.1
Chapter 2
METHODS
OVERALL FRAMEWORK
The approach adopted in this study is based on the assumption that natural vegetation
has co-evolved with its environment over a long period of time and that natural
selection has led to a species composition that is optimally adapted to the given
conditions. If this were true, the question arises, what would be the properties of such
optimal vegetation and how would it use the available resources?
The energy acquired through photosynthesis is stored in carbohydrates, which are vital
for plant fitness. Carbohydrates are both energy carriers and building materials for plant
organs. They can be used for many purposes, including seed production and the
maintenance of symbiotic relations with bacteria and fungi to mobilise nitrogen and
other nutrients from the soil or atmosphere. In addition, all living plant tissues
continuously consume energy to stay alive and require carbohydrates for their
construction. Thus, part of the carbon acquired through photosynthesis has to be reinvested into the construction and maintenance of the organs involved in its uptake.
Only what is left over, the “Net Carbon Profit” (NCP), is assumed to be useful for
increasing a plant’s fitness. Hence we postulated that the optimal resource use strategy
of plants is the one that maximises NCP.
The organs ultimately involved in carbon uptake are not just leaves, but also roots and
transport tissues, which supply the leaves with water and nutrients. For simplicity, the
costs related to nutrient uptake have been neglected in this model, as they are largely
unknown and nutrients can, to a certain extent, be recycled within plants. The chosen
optimisation problem is then to maximise NCP by adjusting foliage properties and
stomatal conductivity dynamically, while adapting roots and transport tissues to meet
the variable demand for water by the canopy (Figure 2.1).
15
2.3 Methods
Schymanski (2007): PhD thesis
Atmosphere
Water
Water Transport & Storage
Carbon
Foliage
Net Carbon
Profit
Root System
Soil
Figure 2.1: Net Carbon Profit as the difference between carbon acquired by photosynthesis and the
carbon used for the construction and maintenance of organs necessary for its uptake. As CO2
uptake from the atmosphere is inevitably linked to the loss of water from the leaves, the root system
as well as water transport and storage tissues are essential to support photosynthesis. Soil water
supply, atmospheric water demand and daily radiation constitute the environmental forcing.
Within those constraints, vegetation is assumed to optimise foliage, transport and storage tissues,
roots and stomata dynamically to maximise its Net Carbon Profit.
The simultaneous variations of the water supply from the soil, atmospheric water
demand and daily radiation, together with the number of optimisable parameters make
the solution of the optimisation problem mathematically and numerically challenging.
Furthermore, modelling the processes of canopy photosynthesis, root water uptake and
water transport is a very complex task. We believe, however, that for the purposes of
this study, the mechanistic details of plant functions are less important than their costs
and benefits in terms of the Net Carbon Profit. Hence, in order to maintain the
generality of the model while keeping the optimisation problem solvable with a standard
personal computer, a number of simplifications have been made.
In this paper, we are primarily interested in establishing whether the overall optimality
approach is feasible. To do that, we simplified the problem even further, firstly, by
ignoring costs associated with roots and water transport, and secondly, by prescribing
total transpiration, as obtained from measurements. By doing this, we were able to focus
on formulating the problem in terms of canopy properties and the optimal CO2 uptake
by the canopy. If the results were judged to be encouraging, a more complete
formulation would be pursued later. The more complete approach would couple the
plant component with a water balance model, and would then handle roots and water
transport.
16
Schymanski (2007): PhD thesis
Chapter 2
The rate of CO2 uptake by a leaf is limited by one or more of: absorbed irradiance,
biochemical capacity to carry out photosynthesis, or conductivity for CO2 between the
atmosphere and the photosynthesising sites within the leaf.
The commonly used model of leaf photosynthesis by Farquhar et al. (1980) subdivides
the biochemical capacity into the capacity to generate photosynthetic electron transport
through the absorption and processing of light and the enzymatic capacity to use the
electron transport for the carboxylation process. The latter limitation is predominantly
caused by the activity of the enzyme rubisco, which catalyses the construction of sugar
molecules out of CO2 and water molecules. In a canopy, irradiance is often limiting so
that, to a good approximation, photosynthesis is limited by electron transport in those
leaves (Farquhar and von Caemmerer 1982). However, this approximation is less
accurate in bright sunlight, when sunflecks can penetrate deep into the canopy at any
instant in time and cause the sunlit fraction of foliage to be light-saturated (de Pury and
Farquhar 1997).
Ideally, canopy photosynthesis would be computed as the sum of photosynthesis rates
over all leaves, where each leaf would absorb a certain amount of light and would have
a certain biochemical capacity and stomatal conductivity. However, such a level of
detail was not feasible for the test of the optimality-based approach here. “Big leaf”
models on the other hand, which represent the canopy as a single horizontal leaf with
homogeneous light absorption, stomatal conductivity and biochemical properties, are
computationally much more feasible, but do not permit the computation of an optimal
leaf area index or optimal biochemical capacities that could then be compared with
observations. As a compromise, the current study assumed that the canopy is composed
of leaves that are horizontal and randomly distributed in homogeneous layers of foliage
for light processing purposes, but considered the canopy as a single big leaf for gas
exchange purposes. This permitted us to include more realistic costs and benefits of
maintaining a certain leaf area index than would be possible with a “big leaf” model,
while holding the number of optimisable parameters in a computationally feasible range
(a single value for photosynthetic capacity in each layer of foliage and the number of
foliage layers with a prescribed leaf area in each, plus a single, dynamic variable to
describe canopy conductivity).
17
2.3 Methods
Schymanski (2007): PhD thesis
To test the model, a highly dynamic vegetation type in subtropical Australia was
chosen, which consists of a sparse, long-lived tree layer and a dense, short-lived grass
layer, where the latter only appears during the wet season. Thus, the vegetation changes
intra-annually from a grass-dominated one in the wet season to a tree-dominated one in
the dry season.
The resources considered were light, CO2 and water, and the optimisable controls were:
–
leaf area index (assumed constant for each month),
–
vertical distribution of photosynthetic capacity in the canopy (assumed constant for
each month), and
–
stomatal conductivity (dynamic at the diurnal scale).
The maintenance of leaf area index and photosynthetic capacity had carbon costs
associated with it, while stomatal conductivity was constrained by the observed monthly
total transpiration.
Given observed meteorological data and transpiration each month, the above parameters
were optimised to achieve a maximum in Net Carbon Profit, while the costs related to
organs involved in water uptake and transport (roots, vascular system etc.) were
neglected.
By comparing the model results with observations, we aimed at answering the following
questions:
–
Can the observed magnitude of daily canopy CO2 uptake be reproduced by the
calculated “optimal vegetation”?
–
Can Vegetation Optimality reproduce the observed absolute values of leaf area
index and photosynthetic capacity within the canopy?
–
Are the observed dynamics of canopy transpiration and CO2 uptake consistent with
optimal stomatal control?
It would be very unlikely to achieve correspondence in all three points if the vegetation
optimality assumption was fundamentally flawed or if factors other than the ones
considered were limiting plant function. Of course, if parameter values were
18
Schymanski (2007): PhD thesis
Chapter 2
“calibrated” or the model structure modified with the sole aim to improve the match of
model results and observations, affirmative results would be much more likely but a lot
less meaningful. In the present study, all parameter values were taken from the literature
and the model structure was determined based on its capability to account for the most
dominant costs and benefits of the vegetation traits under investigation.
2.3.2
VEGETATION OPTIMALITY MODEL
Energy contained in sunlight is transformed into chemical energy via photosynthesis
and is stored as carbohydrates (sugar). This process is also called “CO2 assimilation”,
because terrestrial plants take up CO2 from the atmosphere and incorporate it in sugar
molecules. The energy stored in carbohydrates can be released by the reverse process
called “CO2 respiration”, in which the carbohydrates are oxidised and transformed into
CO2 and water. The overall reaction can be summarised as:
light
CO2+H2O
HoooooI
CH2O+O2
( 2.1 )
energy
where the top arrow indicates photosynthesis or “assimilation” and the bottom arrow
“respiration”. Ultimately the energy captured from light will be degraded to heat,
through a complicated cascade of processes, which support most life on our planet.
The reactions and kinetics involved in photosynthesis are complex and part of a vast
biochemical network with a variety of energy carriers other than carbohydrates, and
frequent self-regulating feedback loops. As stated previously, the purpose of this study
is not to capture all of the known constituent processes involved with mechanistic
accuracy. Instead we aim to formulate the simplest possible model of canopy
photosynthesis that would include the key degrees of freedom by which a canopy can
adapt to a certain habitat and their energetic costs and benefits.
After identifying these “degrees of freedom” and quantifying their costs and benefits,
we searched for the optimal values of the adjustable parameters that would maximise
the Net Carbon Profit. We formulated all costs and benefits in terms of carbon, using
molar units throughout (Cowan 1977).
19
2.3 Methods
Schymanski (2007): PhD thesis
In this section, we first describe the biochemical photosynthesis model used, its
prescribed and optimisable parameters and how the model was extended from a leaf to a
canopy. Then we describe the link between photosynthesis and transpiration rates,
which introduces the role of atmospheric water deficit and stomatal conductivity. A
quantitative description of the costs involved in the maintenance of leaf area will then
enable us to formulate the mathematical optimisation problem in terms of costs and
benefits of all the optimisable parameters. The section ends with a stepwise description
of the numerical optimisation performed in this study.
2.3.2.1
Photosynthesis Model
We subdivided photosynthesis into two steps. The first step was the absorption of light
and generation of electron transport, which depends on the amount and organisation of
leaf area and the distribution of chlorophyll over the leaf area. The second step was the
assimilation of CO2 into carbohydrates, which depends primarily on the electron
transport rate generated in the first step and the supply of CO2 through the stomata.
Photosynthetic Light Absorption
The first step in photosynthesis is the generation of electron transport through
absorption of photosynthetically active irradiance (Il) and splitting of water molecules.
The electron transport rate (J) generated by this process is a non-linear function of
irradiance and the biochemical electron transport capacity (Jmax) of the leaves. In the
present study, this function is expressed as:
J
a Il y
ij
z
J
j
Jmax jj1 - ‰ max zzz
k
{
( 2.2 )
where the parameter α is the initial slope of the curve, which we set to the value 0.3
following common practice (e.g. Medlyn et al. 2002). For a derivation of this function
see Appendix A.2.1.1.
Canopy Light Absorption
Plant canopies do not form a single, closed layer of leaves without any gaps, but are
usually organised into a three-dimensional array of leaves, in which the bottom layers
20
Schymanski (2007): PhD thesis
Chapter 2
absorb some of the light that penetrates through the gaps in the top layers. Under direct
sunlight, the leaves in the upper layers cast shadows on the layers below, which wander
with the position of the sun, so that at any instant in time certain parts of the leaves in
the lower canopy layers are sunlit, while the rest remain shaded. In addition, the
intensity of radiation experienced by sunlit leaves depends on their slope angle with
respect to the solar angle (Monsi and Saeki 1953; de Pury and Farquhar 1997). Based on
the assumption of a uniform leaf angle distribution in the canopy, de Pury and Farquhar
(1997) developed a canopy radiation model for sunlit and shaded leaves in the canopy.
However, in an optimised canopy, it would be expected that not only would the electron
transport capacity of leaves follow the average light conditions experienced by each
leaf, but the leaf angles and the leaf distribution in space would also be optimised to
maximise the Net Carbon Profit. The acclimation time scale of leaf angles can be
relatively short, for example in plants that exhibit diurnal leaf movements. In fact, the
phenomenon that leaves change their angle at the time scale of changes in the solar
angle is not a rare occurrence, as a brief literature search reveals. The “Science Citation
Index Expanded (SCI-EXPANDED)” contains 1,456 citations between 1900 and 2006
dealing with leaf movements, while the first reported observations of leaf movements
reach as far back as 400 B.C., when Androsthenes described diurnal leaf movements of
the tamarind tree (McClung 2001). Darwin (1880) observed movements of leaves in
almost all of the species he investigated and linked many of the movements to stimuli
by light or the control of leaf temperature.
A quantitative theory about the costs and benefits of the acclimation of leaf angles is
presently not available. However, in the light of the above and in the framework of this
study, a canopy with horizontal leaves would appear closer to optimal than a canopy
with a uniform leaf angle distribution. Furthermore, consideration of different leaf angle
classes in the numerical optimisation would substantially increase the computational
demands. To make the numerical optimisation feasible, given the available computing
resources, a simplified canopy light absorption model was developed for the present
study. This model subdivides the canopy into horizontal layers of foliage, where all
leaves are assumed to be randomly distributed and horizontal. For computational
simplicity, each layer (i = 1, Nl) has a total leaf area per ground area (LA) of 0.1 (see also
Figure 2.2).
21
2.3 Methods
Schymanski (2007): PhD thesis
i=1
i=2
i=3
.
.
.
.
i = Nl
ground surface
Figure 2.2: Model canopy as an array of Nl layers with randomly distributed horizontal leaves with
leaf area LA = 0.1 in each layer.
Each layer of horizontal leaves can be further subdivided into sunlit and shaded
fractions. The sunlit fraction receives beam (or “direct”) and diffuse light, while the
shaded fraction only receives diffuse light. For simplicity, it was assumed that
reflectance and scattering of beam light by the foliage is negligible.
For each layer i, the radiation intensity for sunlit and shaded leaves was calculated by
assuming that the diffuse radiation is reduced from layer to layer by a factor equivalent
to the leaf area of the layer above. As all layers are assumed to have the same leaf area
(LA), we get:
Id,i
Id,i-1 H1 - LA L
( 2.3 )
where i indicates the order of the layer from the top to the bottom of the canopy (with
layer 1 being the top layer), Id,i is the diffuse irradiance in layer i and Id,i-1 is the diffuse
irradiance in the layer above. As we assumed that all leaves are horizontal, the equations
can now be formulated in terms of irradiance per unit ground area (Ia). Equation ( 2.3 )
can be simplified to:
Id,i
Id,1 H1 - L A Li-1
( 2.4 )
As shaded leaves only receive diffuse radiation, Id,i represents the total irradiance that
reaches the shaded leaf area in layer i.
The sunlit leaves in each layer receive a radiation intensity of
22
Schymanski (2007): PhD thesis
Chapter 2
Isun,i
Ib + Id,i
( 2.5 )
where Ib is the beam irradiance (i.e. direct sunlight), which is assumed to be constant
between layers.
The sunlit part of layer i was calculated by assuming that each layer above reduces the
sunlit area by its own sunlit area, and that the remaining direct radiation is intercepted
by a factor equivalent to the projected foliage cover in this layer:
LAsun,i
H1-LAsun,1-LAsun,2-...-LAsun,i-1L LA
( 2.6 )
Equation ( 2.6 ) can be simplified to (see Appendix A.2.1.2):
H 1 - L A L i- 1 L A
LAsun,i
( 2.7 )
The shaded part of layer i is the total leaf area of this layer minus the sunlit part:
LAshade,i
L A - LAsun,i
( 2.8 )
Thus, if the beam and diffuse fractions of Ia at the top of the canopy are known, the
above equations allow us to calculate the incident irradiance on the sunlit and shaded
fractions of each layer of foliage. Now, assuming that the distribution of Jmax within
layer i is exponential with a mean value of Jmax,i (see Appendix A.2.1.1), the rate of
electron transport achieved in this layer of the canopy per unit of ground area is given as
the electron transport rate achieved by shaded leaves (Jshade,i) plus the electron transport
rate achieved by sunlit leaves in this layer (Jsun,i):
Ji
Jshade,i + Jsun,i
( 2.9 )
where, following Equation ( 2.57 ) in Appendix A.2.1.1, sunlit leaves generate an
electron transport rate equivalent to:
Jsun,i
a Isun,i
ij
LAsun,i jjjj1 - ‰ Jmax,i
k
yz
zz J
zz max,i
{
( 2.10 )
and shaded leaves generate an electron transport rate equivalent to:
J shade,i
a I d,i
ij
yz
L Ashade,i jjjj1 - ‰ Jmax,i zzzz J max,i
k
{
23
( 2.11 )
2.3 Methods
Schymanski (2007): PhD thesis
The sum of all Ji would give the total electron transport rate per ground area in our
conceptual canopy if the canopy were horizontally homogeneous. However, vegetation
does not always cover all the ground area of a site. The patchiness of canopy cover will
be expressed in the parameter MA, which denotes the fraction of the ground surface that
is covered by vegetation, ranging between 0 (no vegetation) and 1 (full cover) (see
Figure 2.3). Electron transport generated by the processing of light per unit total ground
area of the site then becomes:
M A ‚ Ji
Nl
JA
( 2.12 )
i= 1
where Nl denotes the total number of foliage layers in the canopy. Note that Ji is given
per unit vegetated ground area, while JA is given per unit total ground area. In contrast,
Jmax,i is given per unit leaf area in layer i.
Figure 2.3: Subdivision of the total ground area of the site (outlined by solid lines) into a vegetated
fraction (MA, outlined by dashed lines) and a non-vegetated fraction (1-MA). Variables are given per
unit total ground area (e.g. JA), per unit vegetated ground area (e.g. Ji) or per unit leaf area (e.g.
Jmax,i).
Photosynthetic CO2-Uptake
Terrestrial plants have developed different biochemical and physiological strategies to
perform the carboxylation of sugars using electron transport and atmospheric CO2.
These are commonly grouped into three different groups, named as “C3”, “C4” and
“CAM” pathways. Intermediate pathways are possible as well. The C3 pathway is by far
the most common one, while the other pathways are usually considered to constitute
special adaptations to high temperatures (C4) or water-stress (CAM). These special
adaptations bring along their own costs and benefits, but their discussion is beyond the
scope of this study. Although most of the grasses on the study site use the C4 pathway,
24
Schymanski (2007): PhD thesis
Chapter 2
the present model is based on C3 photosynthesis. The reason for this is that C3
photosynthesis is much more common among plants than the C4 pathway, and
generality is one of the main aims of the present model. It is possible to add the C4
pathway in future, as needed.
CO2 assimilation was modelled following the description of C3 photosynthesis by von
Caemmerer (2000), with the exception that only the electron transport-limited
carboxylation rate was considered, as explained previously (Section 2.3.1). A
comparison between electron transport-limited and rubisco-limited carboxylation rates
is given in Appendix 2.2.
After obtaining the total electron-transport rate (JA) from the layered canopy model, the
nonlinear dependence of the carboxylation rate (Ac) on leaf-internal CO2 concentration
(Cl) and JA was expressed as:
JA HCl -G*L
4Cl + 8 G*
Ac
( 2.13 )
where Γ* is the CO2-compensation point, which is only dependent on temperature (see
Equation ( 2.22 )).
This and the following equations were originally derived for photosynthesis at leaf level
and all variables were defined per unit leaf area. However, as explained previously, the
current model considers the canopy as a single big leaf for gas exchange purposes, so
that the variables are now defined per unit ground area.
The rate of CO2-exchange across the leaf stomata depends on the stomatal conductivity
for CO2 (Gs) and the difference between the mole fraction of CO2 in the atmosphere
(Ca) and inside the leaf (Cl) (Cowan and Farquhar 1977):
Ag
HCa - ClL Gs
( 2.14 )
Inside the leaf, CO2 is depleted by Ac, but also replenished by leaf respiration (Rl), so
that leaf CO2 (Cl) would be at a steady-state if:
Ag
Ac - Rl
( 2.15 )
Inserting Equation ( 2.13 ) and Equation ( 2.14 ) into Equation ( 2.15 ) and solving for
Cl gives Cl as a function of Gs, JA, and Rl:
25
2.3 Methods
-
Cl
Schymanski (2007): PhD thesis
1
J-4Ca Gs + 8 G* Gs + JA - 4Rl +
8Gs
"######################################################################################################
HJA - 4Rl - 4Gs HCa - 2 G*LL2 + 16Gs H8Ca Gs + JA + 8RlL G* N
( 2.16 )
Inserting Equation ( 2.16 ) back into Equation ( 2.14 ), yields the rate of CO2-exchange
as a function of stomatal conductivity:
Ag
1
J4 Ca Gs + 8 G* Gs + Ja - 4 Rl -
"####################################################################################################################################
H-4 Ca Gs + 8 G* Gs + Ja - 4 Rl L2 + 16 Gs H8 Ca Gs + Ja + 8 Rl L G* N
8
( 2.17 )
Leaf respiration (Rl) was modelled as a linear function of photosynthetic capacity
(Amax):
Rl
A max cRl
( 2.18 )
where cRl = 0.07 following Givnish (1988), who showed for a wide range of species that
Rl is approximately 7% of Amax.
In Equation ( 2.13 ), Cl can vary between 0 and Ca, while JA can vary between 0 and the
total Jmax of all leaves in the canopy (Jmaxtot). Thus, the theoretical maximum CO2
assimilation rate or “photosynthetic capacity” of the canopy can be obtained by
replacing Cl by Ca and JA by Jmaxtot:
Amax
Jmaxtot HCa - G* L
- Rl
4 Ca + 8 G *
( 2.19 )
where Jmaxtot is given by:
LA MA ‚ Jmax,i
Nl
Jmaxtot
0.1 MA ‚ Jmax,i
Nl
i=1
( 2.20 )
i=1
Note that Jmax,i was given per unit leaf area, while Jmaxtot is formulated per unit total
ground area. Hence Jmax,i in each layer has to be multiplied by the vegetated fraction
(MA) and the leaf area per unit vegetated area (LA).
Inserting Equation ( 2.19 ) into Equation ( 2.18 ) and solving for Rl gives Rl as a
function of Jmaxtot:
26
Schymanski (2007): PhD thesis
Rl
cRl Jmaxtot HCa -G*L
4 HcRl + 1L HCa + 2 G*L
Chapter 2
( 2.21 )
The above model implicitly assumes that CO2 generated by leaf respiration during
daylight is re-cycled within the leaves, as it contributes to an increase in Cl (see
Equation ( 2.16 )).
Temperature Dependence of the Biochemical Parameters of Photosynthesis
Medlyn et al. (2002) give an overview of different formulations for the temperature
dependencies of biochemical parameters of photosynthesis. As we did not expect to
have measurements of leaf temperature, we assumed that leaf temperature is the same as
air temperature, independent of the position in the canopy. The close coupling between
leaf and air temperature for well ventilated leaves has also been shown by other authors
on the grounds of boundary layer heat transfer (Pearman et al. 1972).
For the temperature dependence of Γ* we used the empirical relationship established by
Bernacchi et al. (2001), as reformulated in Medlyn et al. (2002). After converting to
molar units, this becomes:
G*
126.946 HTa-298L
0.00004275 ‰ Rmol Ta
( 2.22 )
where Ta is air temperature in Kelvin and Rmol is the molar gas constant.
For the temperature dependence of Jmax, we used Equation 18 in Medlyn et al. (2002):
f HTaL
‰
Ha HTa -Topt L
Rmol Ta Topt
Hd kopt
Hd HTa -Topt L y
ij
zz
jj
R
T T
Hd - jj1 - ‰ mol a opt zzz Ha
k
{
( 2.23 )
where Ta is the air temperature in Kelvin, Rmol is the molar gas constant, Topt is the
“optimum temperature” of the function, kopt is the value of Jmax at temperature Topt,
while Hd and Ha are parameters that determine the slope of the function above and
below Topt.
To calculate what a given Jmax at Ta = 298 K (Jmax25) would be at an arbitrary
temperature Ta, we normalised the equation and obtained:
27
2.3 Methods
Schymanski (2007): PhD thesis
H HTopt-298L y
ijij
yz
- d
jjjj
z
298 Rmol Topt zzz
1
+
‰
H
+
H
jj
zz a
dzzz
jj
kk
{
{
Hd HTa-Topt L y
ij
jj
R
T T zz
jj-1 + ‰ mol a opt zzz Ha + Hd
k
{
Ha HTa-298L
‰ 298 Rmol Ta
Jmax
Jmax25
( 2.24 )
Medlyn et al. (2002) fitted values of Ha, Topt, Hd, and kopt to empirical curves for
different plant species. In the present study, their parameter estimates for Eucalyptus
pauciflora were used after transforming them into SI units:
Ha = 43.79ä103 J mol-1
Hd = 200ä103 J mol-1
The molar gas constant was taken as Rmol = 8.314 J mol-1 K-1, while the optimal
temperature Topt was taken as the mean daytime temperature on site during the period of
interest, to account for the possibility of acclimation to growth conditions (Niinemets et
al. 1999b; Ngugi et al. 2003; Whitehead and Beadle 2004).
Inserting the above parameter values for Ha and Hd into Equation ( 2.24 ), Jmax can be
written as a function of Jmax25, Topt and Ta only:
Jmax
Jmax25
7.40753 µ 1012 ‰
18788.8
Ta
24055.8 + 18788.8
Ta
23
+ 1.81664 µ 10 ‰ Topt
24055.8
156210 ‰ Ta
24055.8
+ 43790 ‰ Topt
The general form of this relationship is shown in Figure 2.4.
28
( 2.25 )
Schymanski (2007): PhD thesis
Chapter 2
Jmax
Jmax25
1
0.8
0.6
0.4
0.2
290
300
310
320
Ta HKL
Figure 2.4: Temperature dependence of Jmax with respect to its value at the standard temperature
of 298 K (Jmax25). For this plot, the optimal temperature was assumed to be Topt = 300 K.
While the values of Jmax,i vary with temperature according to the above equation, the
values of Jmax25,i are assumed to be controlled by the plants themselves.
The temperature dependence of Jmax causes leaf respiration (Rl) to be dependant on
temperature as well, as Rl was formulated as a function of Jmaxtot earlier (Equation
( 2.21 )).
2.3.2.2
Stomatal Conductivity and Transpiration Rate
Gas exchange across stomata was assumed to occur by diffusion and was formulated for
CO2 in Equation ( 2.14 ). The transport of water vapour across the stomata can be
written similarly, taking into account the different diffusivities of water vapour and CO2
in air. This was done by multiplying the conductivity for CO2 by a factor a, which
generally has the value 1.6 (Cowan and Farquhar 1977).
Et
a Gs HWl - WaL
( 2.26 )
The mole fraction of water vapour in air can be estimated by dividing the partial vapour
pressure (pva) by the air pressure (P). Assuming that the air space inside the leaves has
saturation vapour pressure (pvsat), we can replace the term (Wl - Wa) with the molar
vapour deficit Dv:
Wl - Wa º
pvsat - pva
P
Dv
( 2.27 )
As stomatal conductivity self-optimises in our model to achieve a maximal CO2 uptake,
we did not consider the effect of a possible canopy boundary layer. Whenever a
29
2.3 Methods
Schymanski (2007): PhD thesis
boundary layer inhibits gas exchange, stomatal conductivity would have to increase in
order to maintain a certain CO2 assimilation rate. Hence the boundary layer would only
have an effect on gas exchange if the boundary layer resistance could not be overcome
by increasing stomatal conductivity or if the gas exchange across the boundary layer
was by turbulent transfer, rather than by diffusion. The first case does not apply to our
model, as we did not define any upper limit to stomatal conductivity a priori and hence,
in this model, stomata would always be able to overcome any additional resistances to
gas exchange. A turbulent boundary layer could theoretically alter the ratio between
CO2 and water vapour exchange, as can be verified by calculating a simple mass
balance with three boxes, the inside of a leaf, the boundary layer and the atmosphere.
However, the ratio could only change significantly if the boundary layer resistance was
very high, while the transport across the boundary layer would remain turbulent. This
scenario would not be very realistic, as turbulence generally decreases the resistance to
mixing.
2.3.2.3
Foliage Turnover Costs
Wright et al. (2004) compiled an extensive database of leaf properties (GLOPNET),
which is available online. The database contains “leaf dry mass per area” (LMA) and
“leaf life span” (LL) for a global range of plant species, which we were able to use to
obtain an estimate of the costs involved in the maintenance of a certain leaf area.
Whereas both LMA and LL vary widely across species and biomes (Figure 2.5), their
ratio is confined to a range of 5-50 g/month/m2 leaf area, with a median of 12.8 (g dry
mass)/(m2 leaf area)/(month life span). The frequency distribution is given in Figure 2.6.
Assuming a construction cost equivalent to 2 g CO2 per g leaf dry matter (Givnish
2002), we formulated foliage turnover costs (Rft, given in mol carbon s-1 m-2 ground
area) as:
Rft = 2.2ä10-7 mol s-1 m-2 LAI
( 2.28 )
where LAI is the leaf area index of the site, which is obtained from the number of foliage
layers (Nl) and the vegetated cover (MA):
LAI MA Nl LA 0.1 LA MA
30
( 2.29 )
Schymanski (2007): PhD thesis
Chapter 2
900
Leafmass/area
mass/area (g/m2)
Leaf
(g/m2)
800
700
600
500
400
300
200
100
0
0
100
200
300
400
Leaf life span (m onths)
Figure 2.5: Relationship between leaf mass per area and leaf life span from the GLOPNET
database (Wright et al. 2004). Lines are drawn by hand.
180
160
Number of Observations
140
120
100
80
60
40
20
90-95
85-90
80-85
75-80
70-75
65-70
60-65
55-60
50-55
45-50
40-45
35-40
30-35
25-30
20-25
15-20
10-15
5-10
0-5
0
LMA/LL (g/m2/month)
Figure 2.6: Frequency distribution of leaf mass per area divided by leaf life span (LMA/LL) for
leaves from 678 species for which such data is available from the GLOPNET database (Wright et
al. 2004). The median of the distribution is 12.8 g/m2/month.
31
2.3 Methods
2.3.2.4
Schymanski (2007): PhD thesis
Optimisation
Optimisation of Canopy Structure
The Net Carbon Profit (NCP) of foliage was formulated as the difference between CO2
assimilation (Ag) and foliage turnover costs (Rft):
NCP
‡
tend
tstart
8 Ag @ J A HtL, Gs HtL, TaHtL, CaHtLD - RftHt, LAL< „ t
( 2.30 )
Foliage CO2 uptake per ground area (Ag) is positive if there is a sufficiently large
electron transport rate (JA) and stomatal conductivity (Gs), and negative if there is no
electron transport, e.g. at night. Hence it accounts for leaf respiration as well as CO2
assimilation. The total electron transport rate at any time (JA(t)) depends on the light
conditions and canopy properties, i.e. the number of foliage layers (Nl), vegetated
fraction (MA) and electron transport capacity in each layer (Jmax25,i). Foliage turnover
costs (Rft) are directly determined by the canopy leaf area index.
The parameters to be optimised in the model are the vegetation cover (MA), number of
layers of foliage (Nl), electron transport capacity in each layer at standard temperature
(Jmax25,i) and stomatal conductivity (Gs).
Of the above, stomatal conductivity is highly variable on a diurnal scale and does not
have a direct carbon cost, but is dependent on atmospheric water demand and water
supply by the roots. To simplify the problem, stomatal conductivity was inferred from
measured water vapour fluxes by inversion of Equation ( 2.26 ) and prescribed in the
first step. Using the prescribed stomatal conductivity, MA, Nl, and the vertical
distribution of Jmax25,i. were optimised following the below procedure:
1. Extract meteorological data and inferred Gs for one month.
2. Prescribe a low value for MA and set i=0.
3. Add a layer of foliage with LA = 0.1, increasing i and Nl by 1.
4. Optimise Jmax25,i to maximise NCP: Prescribe a positive value for Jmax25,i in this
layer, calculate JA following Equation ( 2.12 ) and compute NCP following
32
Schymanski (2007): PhD thesis
Chapter 2
Equation ( 2.30 ), using inferred Gs and meteorological data for one month.
Iterate Jmax25,i to maximise NCP and save.
5. Go to step 3 and repeat until NCP peaks.
6. Reset, increase MA, and go to step 3.
7. Keep increasing MA until NCP peaks or MA reaches 1.0.
From the resulting values of MA and Nl, we calculated the modelled “clumped leaf area
index” (LAIc = LA Nl = 0.1 Nl) and the leaf area index of the site (LAI =
LAIc MA = MA LA Nl), which were compared with the observed leaf area index on the
site. We also compared the resulting time series of foliage CO2 uptake (Ag(t)) with
measured CO2 fluxes on the site.
This procedure was repeated for every month of the investigated period to assess
temporal variability in canopy properties.
Optimisation of Stomatal Conductivity
Both water loss by transpiration (Et) and rate of photosynthesis (Ag) are linked by
stomatal conductivity, such that any increase in CO2 uptake rate under given
atmospheric conditions and foliage properties has the inevitable consequence of
increased water loss through the stomata. This relationship between Et and Ag can be
expressed by combining Equations ( 2.17 ) and ( 2.26 ) to obtain:
Et
aAg Dv Gs H4Ag - JA + 4RlL
Ca H4Ag - JA + 4RlL + H8Ag + JA + 8RlL G*
( 2.31 )
The characteristics of this function are shown in Figure 2.7. Both Ag and Et depend on
stomatal conductivity, but while Ag has an upper limit determined by JA, Et increases
indefinitely with increasing stomatal conductivity. The slope of the curve (λ) is thus
determined by the atmospheric vapour deficit (Dv), as well as irradiance and canopy
properties responsible for the magnitude of JA.
The shape of the curve changes dramatically during the day with changes in light and
atmospheric vapour pressure. Thus, the timing of water use can make a difference to the
total carbon uptake achieved with a limited amount of water available for transpiration.
33
2.3 Methods
Schymanski (2007): PhD thesis
As noted previously, Cowan and Farquhar (1977) postulated that for any given amount
of total water available for transpiration in a period of time, a leaf can achieve a
maximum in CO2 uptake if it adjusts Gs in such a way that λ is maintained at a constant
value throughout this period. This was originally derived at leaf level, but later extended
to a whole plant by postulating that the optimality criterion would require all leaves to
function at the same value of λ (Cowan 1982). The value of λ has to be adapted to the
total amount of water available during the period. λ can be expressed either as a
function of Ag or a function of Et by solving
l
∑Et
∑Ag
( 2.32 )
and
1
l
∑ Ag
∑Et
( 2.33 )
respectively.
After solving Equation ( 2.33 ) for Et, we obtained Et as a function of λ:
Et
aDv HCa HJA - 4RlL - 4 HJA + 2RlL G*L
+
4 HCa + 2 G*L2
è!!! è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
3 aDv JA G* Hl Ca - 2aDv + 2 lG*L2 Hl Ca - aDv + 2 lG*L HCa HJA - 4RlL - HJA + 8RlL G*L
4 HCa + 2 G*L2 Hl Ca - aDv + 2 lG*L
( 2.34 )
Except for the value of λ, all other variables in Equation ( 2.34 ) were taken from the
previous test. The value of λ was assumed to be constant during a month, and was
adjusted iteratively to match the monthly modelled Et with the monthly value estimated
from measurements. This value of λ was then used to compute instantaneous rates of Et
and Ag and plot these against measured values in the same time period.
To test how fluxes resulting from dynamically optimised stomatal conductivity (Gs)
would differ from fluxes modelled using a constant Gs, the above procedure was
repeated, but this time adjusting the value of constant Gs instead of λ to match the
observed monthly Et.
34
Schymanski (2007): PhD thesis
Chapter 2
Figure 2.7: Relationship between Et and Ag for a fixed electron transport rate (JA) and atmospheric
vapour deficit (Dv), but variable stomatal conductivity (Gs). The upper limit for Ag is determined by
JA, while the initial slope of the relationship is determined by Dv.
35
2.3 Methods
Schymanski (2007): PhD thesis
2.3.3
STUDY SITE
2.3.3.1
Location
The study site is located in the Northern Territory (Australia), 35 km south-east of
Darwin, near Howard Springs in the Howard River catchment.
The geographical location of the flux tower used for the measurements presented here
(July 2004 to June 2005) is 12°29'39.30"S, 131°09'8.58"E (Beringer, pers. comm.).
2.3.3.2
Climate
The climate is sub-humid, with 1750 mm mean annual rainfall and 2300 mm mean
annual class A pan evaporation. The rainfall between July 2004 and June 2005 was
2084 mm, which is above average but not extreme compared with the highest rainfall on
record of 2834 mm between September 1996 and August 1997 (Hutley et al. 2000).
Around 95% of the annual precipitation falls during the wet season (December to
March), when atmospheric water demand is low (day-time relative humidity >60%),
while the dry season (May to September) is characterised by virtually no rainfall and
high atmospheric water demand (day-time relative humidity 10-40%). Air temperatures
range between roughly 25 and 35 ˚C in the wet season and between 15 and 30 ˚C in the
dry season (Figure 2.8).
Daily shortwave radiation during the dry season is between 15 and 25 MJ/m2/d, while
during the wet season it varies between 5 and 30 MJ/m2/d (Figure 2.8). The larger dayto-day variation during the wet season is due to the higher variability in cloud cover.
The seasonality of climate is also expressed in the dynamics of volumetric soil moisture,
which varied between 0.08 in August 2004 and 0.3 after rain during the wet season
2004/2005 (Figure 2.8).
36
Schymanski (2007): PhD thesis
Chapter 2
(b)
(a)
Rain Hmm êday L
350
Sg,d HMJ êm2êdL
30
300
25
250
20
200
15
150
10
100
5
50
Month
7 8 9 10 11 12 1 2 3
Month
4 5 6
7 8 9 10 11 12 1 2 3 4 5
(c)
Ta Ho CL
40
6
(d)
θ
0.4
35
0.3
30
25
0.2
20
0.1
15
Month
7
8
9 10 11 12
1
2 3
4
5
Month
6
7
8
9 10 11 12
1
2
3
4
5
6
Figure 2.8: Climatic characteristics of the study site between July 2004 and June 2005. (a) Downwelling daily global irradiance (Sg,d) , (b) daily rainfall, (c) half-hourly air temperature (Ta), and (d)
surface soil moisture (θ). (Measurements courtesy of Beringer, Hutley and Tapper)
2.3.3.3
Topography and Soils
The terrain at the study site is very flat, with slopes <1˚ (Beringer et al. 2003). The
surface of the lowland plains, where the study site is situated, is a late Tertiary
depositional surface, with a sediment mantle that seldom reaches more than 30 to 40 m
in depth. Due to the pronounced climatic seasonality, the surface has been intensively
weathered, resulting in a standard lateritic profile, with infertile, acidic soils (RussellSmith et al. 1995). On the study site itself, the soil profile has been described as a Red
Kandosol, with sandy loams and sandy clay loams in horizons A and B respectively and
weathered laterite in the C horizon, below about 1.2 m (Kelley 2002).
2.3.3.4
Vegetation
The vegetation on the site has been described in detail elsewhere (Hutley et al. 2000;
O'Grady et al. 2000), so only a brief account is presented here.
37
2.3 Methods
Schymanski (2007): PhD thesis
The vegetation has been classified as Eucalypt open forest (Specht 1981), with a mean
canopy height of 15 m, where the overstorey has an estimated cover of 50% (Hutley et
al. 2000) and is dominated by the evergreen Eucalyptus miniata (Cunn. Ex Schauer)
and Eucalyptus tetrodonta (F. Muell.). Visual estimates of projected tree cover by
analysis of cast shadows in June 2005 suggested values closer to 30% than 50%. The
dominant tree species contribute to 60-70% of the total basal area (i.e. the ground area
covered by tree trunks) of this forest, while the remaining basal area is composed of
some brevi-, semi- and fully deciduous tree species (Williams et al. 1997; O'Grady et al.
2000).
Williams et al. (1997) give a detailed account of the seasonal variation of canopy
fullness of the different species. According to their observations, the canopy fullness of
the evergreen species in the tree layer varies only very little over the year, while the
variation increases with increasing deciduousness for the other species.
The understorey on the site comprises small individuals of the tree species, some fully
or partly deciduous shrubs and some perennial grasses during the dry season. During the
wet season it is dominated by a thick layer of annual C4 grasses of the genus Sorghum
sp.
Recent estimates give a leaf area index on the site of 0.8 in the dry season and 2.5 in the
wet season (Hutley and Beringer, unpublished data).
2.3.3.5
Measurements
The measurement techniques used on the site are described in detail elsewhere
(Beringer et al. 2003; Hutley et al. 2005) and will be only summarised here. The present
study site is described as the “moderate intensity site” in Beringer et al. (2003).
Flux measurements were conducted at the top of an 18 m tower over the 12-14 m tall
canopy, in flat terrain (slopes < 1o) with a near homogeneous fetch of more than 1 km in
all directions. The eddy covariance technique was used to infer vertical fluxes of latent
heat and CO2 from three-dimensional wind velocities, as well as turbulent fluctuations
of CO2 and water vapour in the air. Incoming shortwave radiation, air pressure and air
temperature were also measured at the top of the tower. Soil moisture was measured
using “Time Domain Reflectometry” (TDR) probes (Campbell Scientific) at 10 cm
38
Schymanski (2007): PhD thesis
Chapter 2
depth and soil temperature was obtained from soil thermocouple sensors at 2 and 6 cm
depth. All flux variables were sampled at 20 Hz and averaged over 30 minutes.
Continuous measurements have been logged since 2001. To create a continuous data
set, small gaps (less than 2 hours) were filled using linear interpolation, while larger
gaps were filled using a neural network model, fitted to the whole data set. Periods with
gap-filled data were flagged for later recognition.
The procedure of gap filling applied in conjunction with eddy-covariance measurements
has been identified to be a major contributor to uncertainty of results, followed by
uncaptured spatial heterogeneity and then instrument uncertainty (Oren et al. 2006).
Nocturnal fluxes in particular can be systematically underestimated by the eddy
covariance method, due to cold-air drainage flows and a lack of turbulence (Hutley et
al. 2005). For this reason we chose a period in the data set with only a low proportion of
gap filled values (July 2004 to June 2005) and we only considered day time fluxes for
the comparison between model results and observations.
2.3.4
ATMOSPHERIC FORCING
Air temperature and pressure, shortwave radiation, atmospheric vapour pressure deficit
and CO2 concentration were measured on site at the eddy covariance tower. To obtain
the molar vapour deficit (Dv), we divided the measured vapour pressure deficit by
atmospheric pressure. The molar fraction of CO2 in the air was determined from the
measured CO2-concentration in air (mg m-3), the molar weight of CO2 (44 g mol-1),
measured air temperature and the ideal gas law. A period of around 40 days had a
clearly lower range of measured CO2 concentrations than the rest of the data set. Due to
the sharp transition at both ends of this period we concluded that it was caused by a
calibration error and adjusted the values to match the rest of the data set.
Measured shortwave irradiance in the wavelengths 0.2-4.0 µm (Kshort) was converted to
photosynthetically active irradiance (IaW) in the wavelengths 0.4-0.7 µm following a
conversion coefficient obtained from Pinker and Laszlo (1997). Figure 2.9 shows the
variation of the coefficient for Darwin between 1984 and 1988. As the variation was
only relatively small, a typical value for Darwin of 0.45 (IaW = 0.45 Kshort) was used as a
constant for this study.
39
2.3 Methods
Schymanski (2007): PhD thesis
Figure 2.9: Fraction of total shortwave irradiance in the photosynthetically active region (400700 nm). Data drawn for a 4-year period in Darwin, from a global data set (Pinker and Laszlo
1997). Numbers on the horizontal axis denote the year and month for each grid line (e.g. “plc8401”
refers to January 1984).
To convert from energetic (W m-2) to molar (mol quanta s-1 m-2) units, we used a
conversion coefficient of 4.57ä10-6 mol W-1 (Thimijan and Heins 1983).
For the canopy light absorption model we needed to estimate how much of the
photosynthetically active radiation reaches the canopy as direct sunlight, and how much
is diffuse radiation, scattered by aerosols in the air. The ratio between diffuse and direct
sunlight was estimated from the relationship between global irradiance and top-of-theatmosphere irradiance. The relevant equations are given in Appendix A.2.3.1.
2.3.5
CONVERSION OF MEASURED FLUXES
The on-site eddy covariance measurements delivered half-hourly averages of latent heat
flux (in W m-2) and net CO2 flux (in mg CO2 m-2 s-1).
Latent heat flux is the result of all water vapour moving past the sensor and accounts for
transpiration as well as evaporation from the soil and wet surfaces. The transport of
liquid water (e.g. rain) is not captured by the sensor. In fact, when the sensor gets wet it
cannot resolve the water vapour concentration accurately and therefore flux
measurements during rainfall are not included.
Net CO2 flux represents a sum of all processes within the system, which either take up
or release CO2. The only significant process that leads to CO2 uptake on this site is
40
Schymanski (2007): PhD thesis
Chapter 2
photosynthesis, while CO2 release happens through the respiration of leaves, sapwood
and roots, as well as soil decomposition processes.
The optimality model used here predicts only foliage gas exchange due to leaf
photosynthesis and leaf respiration as well as transpiration through stomata. In order to
make valid comparisons between modelled and measured fluxes, we needed to extract
the relevant parts from the measured bulk fluxes.
2.3.5.1
Transpiration and Stomatal Conductivity
In order to obtain an estimate of the transpiration part of the latent heat flux, we needed
to subtract soil evaporation from the measurements.
Soil evaporation was typically between 0.1 to 0.5 mm/day in the dry season and up to
0.65 mm/day in the wet season (Hutley, pers. comm.), i.e. reaching 25% of total evapotranspiration. We estimated soil evaporation (Es) using a flux-gradient approach as
follows.
Es Gsoil HWs - WaL
( 2.35 )
where (Ws – Wa) is the difference between the mole fraction of water in the laminar
layer immediately above the soil (Ws) and the mole fraction of water in the atmosphere
(Wa), while Gsoil is the conductivity of the soil to water vapour fluxes.
Wa was obtained from measured atmospheric vapour pressure divided by air pressure,
and Ws was calculated as the vapour pressure in the laminar layer immediately above
the soil (pvs) divided by air pressure (P).
Ws
pvs
P
( 2.36 )
To get pvs, we followed a model by Lee and Pielke (1992), which was found by
Silberstein et al. (2003) to give better soil evaporation results than the use of the vapour
pressure in the soil pore spaces. The model formulates pvs as a function of the
atmospheric vapour pressure (pva), the saturation vapour pressure (pvssat) at soil
temperature Ts, actual volumetric soil moisture (θ) and volumetric soil moisture at field
capacity (θfc):
41
2.3 Methods
Schymanski (2007): PhD thesis
pvs
Ø
≤ pvssat sin4J p q N + J1 - sin4J p q NN pva q < qfc
2 qfc
2 qfc
∞
≤
q ¥ qfc
± pvssat
( 2.37 )
Atmospheric vapour pressure and as the volumetric soil moisture (θ) in the top soil was
measured on site. The value for soil moisture at field capacity (θfc) for the site was set to
0.156, equivalent to the soil moisture at a matrix pressure head of -0.01 MPa (Kelley
2002). The saturation vapour pressure just above the soil (pvssat, in Pa) was taken as the
saturation vapour pressure at soil temperature Ts (in ˚C) and calculated using the widely
used approximation (Allen et al. 1998):
17.27Ts
T
610.8 ‰ s+237.3
pvssat
( 2.38 )
The parameter Gsoil in Equation ( 2.35 ) was set to the value 0.03 mol m-2 s-1, which
resulted in a time series of soil evaporation of around 0.1 to 0.2 mm/day in the dry and
0.4 to 1.0 mm/day in the wet season.
Daily Evaporation
Edaily@mmD
(mm)
7
8
9 10 11 12
1
2
3
4
5
6
9 10 11 12 1 2
Month
Month
3
4
5
6
7
ET, d a il y
Es,d ai l y
6
5
4
3
2
1
7
8
Figure 2.10: Measured daily evapo-transpiration (ET, grey lines) and modelled daily soil
evaporation (Es, black lines) at the flux tower site from 28/06/2004 to 27/06/2005.
Transpiration (Et) was then calculated as measured evapo-transpiration (ET) minus
estimated soil evaporation (Es):
Et
ET - Es
( 2.39 )
Stomatal conductivity (Gs,meas) was inferred by inversion of Equation ( 2.26 ), using Et
from above:
42
Schymanski (2007): PhD thesis
Chapter 2
Gs,meas
2.3.5.2
Et
aDv
( 2.40 )
CO2 Fluxes
We subdivided the measured net CO2 uptake by the soil-vegetation system (FnC,
mol m-2 s-1) conceptually into net CO2 uptake by foliage (Ag), CO2 release by soil
respiration (Rs) and CO2 release by sapwood respiration (Rw).
FnC
Ag-Rs-Rw
( 2.41 )
Soil Respiration
Soil respiration is commonly measured as the release of CO2 at the soil surface and can
be seen as a result of decomposition of organic matter in the soil, as well as respiration
by plant roots. The rate of microbial decomposition depends on the amount of organic
matter, as well as soil temperature and moisture, while root respiration responds to soil
temperature, moisture and nutrient availability in the soil. Due to these non-linear
feedbacks, measured soil respiration can vary seasonally, but also in response to rapid
changes in soil moisture and soil temperature.
Chen et al. (2002a) conducted extensive soil respiration measurements using closed
chambers at the experimental site from September 1998 to January 2001 and obtained
respiration rates of 5.06 ± 1.40 µmol m–2 s–1, with a range of 3.5–8.4 µmol m–2 s–1
during the wet season (November-April), 1.53 ± 0.29 µmol m–2 s–1 during the mid-dry
season (June-July) and 1.51 ± 0.23 µmol m–2 s–1 during the late dry season (AugustSeptember). During the dry season, the values ranged from 0.95 to 3.5 µmol m–2 s–1.
The authors also derived empirical relationships between pooled daily means of soil
respiration, temperature and moisture on the site. They formulated two different linear
relationships for Rs as a function of temperature for wet conditions (θ>0.07) and dry
conditions (θ<0.07), of which only the first one was significant.
Rs HTs L
Ø
≤
≤
∞
≤
≤
±
0.352 Ts -5.26
1000000
0.04 Ts +0.78
1000000
The relation of Rs to soil moisture had the form:
43
q > 0.07
q < 0.07
( 2.42 )
2.3 Methods
Schymanski (2007): PhD thesis
RsHq L
0.00002196 q 0.71
( 2.43 )
The authors also fitted a quadratic model of Rs as a function of Ts and θ to their data set,
but the equation given in Chen et al. (2002a) does not follow the trends mentioned
above, so we assumed that there was a typographical error in the document. However,
the relationships in Equations ( 2.42 ) and ( 2.43 ) do reflect the observed trends and
were used to test another model of soil respiration.
In order to model soil respiration on a diurnal time scale, we fitted a model derived for
an African savanna (Hanan et al. 1998) to the above data. We simplified their model to:
Rs
i ‰qsÆ HTs-20L yz q - qmin
N
zJ
R0 jj
k 1 + ‰qs∞ HTs -T∞L { qmax - qmin
( 2.44 )
where qs↓= 0.507(˚C)-1, qs↑ = 0.059(˚C)-1 and T↓ = 35.8 ˚C are empirical temperature
response parameters given by Hanan et al. (1998), R0 is the “intrinsic soil respiration
rate at 20 ˚C” per m2 ground area, and θmin = 0.01 and θmax = 0.12 are “soil moisture
limits”.
We set the critical temperature (T↓) to the maximum soil temperature recorded in our
data set (T↓ = 44.95 ˚C), R0 to 1.862 µmol m-2 s-1 and left all other parameters
unchanged. Using measured soil moisture (θ) and measured soil temperature (Ts), we
obtained a time series of soil respiration that had similar characteristics to the
measurements described by Chen et al. (2002a), both in terms of sensitivity to
temperature and soil moisture (Equations ( 2.42 ) and ( 2.43 ), Figure 2.11) as well as
magnitude and range of variability in the wet and dry seasons (Figure 2.12).
44
Schymanski (2007): PhD thesis
Chapter 2
RsHµmolêm2 êsL
RsHµmolêm2êsL
7
Rs,Hanan
RsHTs L,wet
10
6
8
Rs,Hanan
RsHθL
5
4
6
3
2
4
1
25
30
35
θ
Ts
40
0.05
0.1
0.15
0.2
12
12
10
10
8
8
6
6
4
4
2
2
7
8
9 10 11 12 1 2 3
Month
4
5
6
Jun−Sep
Rs (µmol/m2/s)
Rs Hµmolêm2 êsL
Figure 2.11: Comparison of soil respiration (Rs) obtained from a model adapted after Hanan et al.
(1998) (shown in Equation ( 2.44 )) with soil respiration formulated in Equations ( 2.42 ) and ( 2.43 )
for different temperature and moisture conditions (Chen et al. 2002a). Left: Variation in Rs with
temperature (Ts) under wet conditions (θ = 0.2). Right: Variation in Rs with volumetric soil
moisture (θ) at Ts = 30 ˚C. Solid lines show the empirical relationships in Equations ( 2.42 ) and
( 2.43 ), dashed lines show the modelled relationships using Equation ( 2.44 ).
Dec−Apr
Figure 2.12: Modelled Rs (µmol m-2 s-1) for the investigated period. Left: half-hourly values plotted
for 12 months. Right: box plot of dry season (Jun-Sep) and wet season (Dec-Apr) distributions
including 25% and 75% quartiles and outliers.
Above-Ground Respiration
Total above-ground woody-tissue respiration (Rw) has been measured on site by
Cernusak et al. (2006) and estimated to be around 297 g C m−2 year-1, which is
equivalent to 0.78 µmol m-2 s-1 averaged over the whole year. No clear seasonal
variation was identified, while the diurnal variation was parameterised by the authors as
Rw
Rw25 Q10,wHTw-TrLê 10
( 2.45 )
where Rw25 is the wood respiration at reference temperature (Tr=25 ˚C), Q10,w is the
proportional increase in Rw with 10 K increase in wood temperature Tw. To estimate Tw,
45
2.3 Methods
Schymanski (2007): PhD thesis
we used the observation that the daily mid-range stem temperature at the study site was
linearly correlated with daily mid-range air temperature and that the daily temperature
amplitudes of stems and air were similar, leading to the following formulation
(Cernusak et al. 2006):
Tw 1.11 Ta-1.36
( 2.46 )
Q10,w was given by Cernusak et al. (2006) as 1.92 for all species, and we fitted the value
of Rw,r (Rw,r = 0.606 µmol CO2 m-2 s-1) to match the observed annual woody-tissue
respiration of 297 g C m-2.
Figure 2.13 shows the components of the observed FnC as calculated using the above
equations for a day in the wet season and a day in the dry season. Note that the
estimated soil- and above-ground woody tissue respiration can exceed Fc at night,
leading to apparently positive night-time Ag. This is likely due to an underestimation of
night-time fluxes by the eddy covariance technique as mentioned before and/or an overestimation of night-time respiration by the soil and wood respiration model. Due to this
uncertainty, we decided to only compare observed and modelled day-time fluxes, where
the eddy covariance technique delivers more reliable results and where the respiration
part plays a relatively smaller role.
46
Schymanski (2007): PhD thesis
1
3
5
7
9
24 ê01 ê2005
11 13 15
Chapter 2
17
19
21
23
1
19
21
23
1
CO2-flux (mol/s/m2)
25ä10-6
20ä10-6
Ag
15ä10-6
FnC
10ä10-6
Rs
5ä10-6
Rw
0
-5ä10-6
1
3
5
7
9
11
13 15
Hour
1
3
5
7
9
21 ê10 ê 2004
11 13 15
17
17
19
21
23
1
17
19
21
23
1
15ä10-6
CO2-flux (mol/s/m2)
Ag
10ä10-6
FnC
5ä10-6
Rs
Rw
0
-5ä10-6
-10ä10-6
1
3
5
7
9
11
13 15
Hour
Figure 2.13: Subdivision of measured net ecosystem CO2 uptake (FnC) into soil respiration (Rs),
woody tissue respiration (Rw) and foliage CO2 uptake (Ag). Rs and Rw are modelled based on
measurements, while Ag is taken as the difference. Note that measurements of FnC at night-time are
uncertain due to the frequent lack of wind. Top: a day in the wet season (24/01/2005), bottom: a day
in the dry season (21/10/2004).
47
2.4 Results
2.4
Schymanski (2007): PhD thesis
RESULTS
2.4.1
CANOPY OPTIMISATION
2.4.1.1
January (Wet Season)
As an example for wet season conditions, the optimal canopy properties were modelled
using inferred stomatal conductivity and meteorological data for January 2005. The
optimal canopy properties are summarised in Table 2.1. The model predicted that a
closed canopy (MA = 1.0) with a leaf area index of 2.5 would be optimal for maximising
the Net Carbon Profit. This was consistent with reported values of wet season leaf area
index of about 2.5 (Hutley and Beringer, unpubl. data). Optimal electron transport
capacity (Jmax25) decreased from the top to the bottom of the canopy, but it decreased
faster from layer to layer than the sunlit leaf area fraction (Figure 2.15). The overall
range of modelled Jmax25 was between 435.8 µmol s-1 m-2 in the top layer of foliage and
14 µmol s-1 m-2 in the lowest layer. Initially, Net Carbon Profit increased steeply with
the addition of each layer, before peaking in a very flat plateau at a leaf area index of
2.5 (Figure 2.14).
Modelled canopy CO2 uptake resembled observed canopy CO2 uptake both in
magnitude and dynamics during day-time (Figure 2.16). Ensemble averages of modelled
and measured fluxes were also very similar during day-time (Figure 2.17).
Measurements and modelled data were subsequently pooled in daily totals during
sunlight time. For this purpose, sunlight time was defined as the time intervals with
mean Ia greater than 100 µmol s-1 m-2. Absolute values and day-to-day variations of
modelled and observed daily fluxes were similar, but showed deviations on certain days
(Figure 2.18).
48
Schymanski (2007): PhD thesis
Chapter 2
Table 2.1: Optimal canopy properties for January 2005
PROPERTY
VALUE
MA (optimised)
1.0
Nl (optimised)
25
Jmax25,1 (optimised)
435.8 µmol/s/m2 leaf area
Jmax25,25 (optimised)
14.0 µmol/s/m2 leaf area
LAI (computed)
2.5
NCP (computed)
14.22 mol/m2 ground area in 31 days
14.
14
12.
12
Maximum
10
10.
300
300
Jmmaax x2 52 5
J
NCCPP
N
200
200
8
8.
6.
6
NCP (mol/m2)
Jmax25 (mol/s/m2)
400
400
4.
4
100
100
2.
2
0.
0
0
55
Top of Canopy
10
15
15
20
20
25
25
30
30
Bottom of Canopy
Foliage Layers
Figure 2.14: Optimal electron transport capacity (Jmax25) in each foliage layer and net carbon profit
(NCP) achieved with different numbers of layers. NCP peaked when the canopy had 25 layers, and
with the addition of any more layers NCP subsequently decreased. The leaf area of each foliage layer
was prescribed as 0.1 m2/m2, so that the number of layers can be translated directly into leaf area
index by multiplying with 0.1.
49
2.4 Results
Schymanski (2007): PhD thesis
300
0.08
LAsun Hm2êm2L
Jmax25 Hµmolêm2êsL
0.1
Jm a x 2 5
LA s u n
400
0.06
200
Lowest layer
100
0.04
0.02
0.
0
5
Top of Canopy
10
15
20
Foliage Layers
25
30
Bottom of Canopy
Figure 2.15: Decrease in optimal electron transport capacity (Jmax25) and sunlit leaf area (LAsun)
from the top to the bottom of the canopy. Canopy optimised for January 2005. The optimal canopy
had 25 layers and all subsequent layers would incur losses in terms of NCP (Figure 2.14). The
continuation of the plot beyond 25 layers shows the values of Jmax that would minimise the losses
associated with the maintenance of these layers.
60
Observed
Modelled
Ag Hµmolêm2êsL
40
20
0
-20
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
Day
Figure 2.16: Half-hourly averages of canopy CO2 uptake in January 2005. Modelled fluxes (black,
thin lines) are based on optimised canopy properties and inferred conductivity. Observed fluxes
(grey, thick lines) are inferred from eddy covariance measurements and empirical models of soiland above-ground wood respiration (Rs and Rw).
50
30
Chapter 2
Observed A g
Modelled Ag Hµmolêsêm2L
Observed Ag Hµmolêsêm2L
Schymanski (2007): PhD thesis
30
Modelled A g
20
20
10
10
0
0
0
5
10
15
20
25 0
5
10
Hour
15
20
-10
25
Hour
Figure 2.17: Ensemble means of measured and modelled diurnal canopy CO2 uptake rates.
Ensemble means and standard deviations (error bars) were computed for 31 days in January 2005.
Error bars equate to 2 â standard deviation.
1
Ag Hmolêm2êdayL
Observed
Observed
Modelled
Modelled
0.8
0.6
0.4
0.2
MA =1 LAI=2.5
5
10
15
Day
20
25
30
Figure 2.18: Comparison of observed (thick grey line) and modelled (thin, black line) daily canopy
CO2 uptake rates during daylight hours defined as time intervals with mean Ia > 100 µmol s-1 m-2.
The inset shows a 1:1 plot of the observed and modelled daily values. Mean absolute error
(MAE) = 0.089 mol m-2 s-1, Pearson’s r = 0.77.
2.4.1.2
October (Dry Season)
As an example for dry season conditions, canopy properties were modelled using
inferred stomatal conductivity and meteorological data for October 2004. The optimal
canopy properties are summarised in Table 2.2.
The model predicted that a closed canopy (MA = 1.0) with a leaf area index of 2.0 would
be optimal for maximising the Net Carbon Profit. This was not consistent with reported
values for leaf area index, which are about 0.7 in dry seasons on the site (Hutley and
Beringer, unpubl. data). The optimisation also led to a large over-estimation of canopy
CO2 uptake rates (Figure 2.19). However, if MA was fixed at a value of 0.3, while
allowing the number of foliage layers and Jmax25 in each layer to be optimised, the
resulting canopy had a leaf area index of 0.78 and realistic CO2 uptake rates (Table 2.3
51
2.4 Results
Schymanski (2007): PhD thesis
and Figure 2.20). The optimised values of Jmax25 per unit leaf area were very similar to
the ones in the wet season (Figure 2.21, cf. Figure 2.15).
Table 2.2: Optimal canopy properties for October 2004, if the vegetated fraction of surface (MA)
was optimised.
PROPERTY
VALUE
MA (optimised)
1.0
Nl (optimised)
20
Jmax25,1 (optimised)
459.5 µmol/s/m2 leaf area
Jmax25,25 (optimised)
21.0 µmol/s/m2 leaf area
LAI (computed)
2.0
NCP (computed)
11.66 mol/m2 ground area in 31 days
1
Observed
Modelled
mol m−2 day−1
0.8
0.6
0.4
0.2
MA = 1. LA I=2.
5
10
15
day
20
25
30
Figure 2.19: Comparison of observed (thick grey line) and modelled (thin, black line) daily canopy
CO2 uptake rates during daylight hours in October 2004. Modelled values were obtained by
complete optimisation of canopy MA and LAI, as well as optimised Jmax25 in each layer.
52
Schymanski (2007): PhD thesis
Chapter 2
Table 2.3: Optimal canopy properties for October 2004, if the vegetated fraction of surface (MA)
was prescribed as 0.3.
PROPERTY
VALUE
MA (prescribed)
0.3
Nl (optimised)
26
Jmax25,1 (optimised)
480.0 µmol/s/m2 leaf area
Jmax25,25 (optimised)
13.6 µmol/s/m2 leaf area
LAI (computed)
0.78
NCP (computed)
5.29 mol/m2 ground area in 31 days
Ag Hmolêm2êdayL
1
Observed
Observed
Modelled
Modelled
0.8
0.6
0.4
0.2
MA =0.3 LA I=0.78
5
10
15
Day
20
25
30
Figure 2.20: Comparison of observed (thick grey line) and modelled (thin, black line) daily canopy
CO2 uptake rates during daylight hours in October 2004. Modelled values were obtained by
prescribing MA = 0.3 and optimising canopy LAI, and Jmax25 in each layer. The inset shows a 1:1 plot
of the observed and modelled daily values. Mean absolute error (MAE) = 0.045 mol m-2 s-1,
Pearson’s r =-0.11.
53
2.4 Results
Schymanski (2007): PhD thesis
0.1
Jm a x 2 5
LA s u n
400
0.08
LAsun Hm2êm2L
Jmax25 Hµmolêm2êsL
500
300
0.06
200
0.04
100
0.02
0
5
Top of Canopy
10
15
Foliage Layer
0.
20
25
Bottom of Canopy
Figure 2.21: Decrease in optimal electron transport capacity (Jmax25, black triangles) and sunlit leaf
area (LAsun, grey squares) from the top to the bottom of the canopy. Canopy optimised for October
2004, with prescribed MA = 0.3.
2.4.2
MONTHLY OPTIMISATIONS WITH PRESCRIBED MA
Vegetation on the study site oscillates between a grass-dominated wet season state with
complete vegetation cover (MA = 1.0) and a dry season state, where only trees prevail
(MA = 0.3). In the following, the values of MA were prescribed for each month, and set
either to 1.0 or 0.3, whichever led to more realistic results for the month. The model was
then run for each month between July 2004 and June 2005, optimising the number of
foliage layers and electron transport capacity (Jmax25) in each layer, in order to maximise
the Net Carbon Profit (NCP) of the foliage in the given month. Modelled and observed
canopy CO2 uptake rates were summed over the sunlight time of each day and plotted
together for each month (Figure 2.22).
Modelled CO2 uptake rates matched observations best if MA was set to 0.3 between June
and October and to 1.0 between December and March. In August 2004, a bushfire led to
a sudden decrease in the observed uptake rates, but they gradually recovered and
reached the modelled values again after approximately 6 weeks. The periods between
November and December and between March and June were characterised by a
transition between the dry and wet season vegetation states.
54
Schymanski (2007): PhD thesis
Chapter 2
Ag Hmolêm2êdayL
0.6
0.4
0.2
0.8
0.6
Fire
0.4
0.2
MA =0.3 LAI=0.75
20
25
5
30
October
1
0.8
0.6
0.4
0.2
10
20
0.2
Observed
Modelled
15
Day
20
25
0.8
0.6
0.4
0.2
10
15
Day
20
25
0.4
0.2
10
Observed
Modelled
15
Day
20
25
20
25
30
0.2
Observed
Modelled
5
10
15
Day
20
25
March
0.8
0.6
0.4
0.2
Observed
Modelled
5
10
15
Day
20
25
30
June
0.8
0.6
0.4
0.2
Observed
Modelled
0.8
0.6
0.4
0.2
MA =0.3 LAI=0.72
15
Day
20
25
30
5
10
15
Day
20
25
Figure 2.22: Day-time canopy CO2 uptake rates for each month between July 2004 and June 2005.
Modelled (black, thin lines) and observed (grey, thick lines) canopy CO2 uptake rates were summed
over the daylight hours of each day (defined as time intervals with mean Ia > 100 µmol s-1 m-2) and
plotted together for each month. Modelled values based on prescribed values of MA and optimised
number of foliage layers and electron transport capacity (Jmax25) in each layer. Values of MA were
set to either 1.0 or 0.3, which ever led to a better fit with observed CO2 uptake rates. Leaf area
index (LAI) in each month is calculated from the prescribed value of MA and the resulting optimal
number of foliage layers (Nl).
55
30
1
Observed
Modelled
10
30
0.4
30
May
5
25
0.6
MA =1 LA I=2.4
MA =0.3 LAI=0.75
15
Day
20
December
1
0.6
5
15
Day
0.8
30
February
0.8
30
10
MA =1 LAI=2.4
MA =1 LA I=2.4
MA =0.3 LAI=0.78
10
5
1
0.2
1
Observed
Modelled
5
0.2
30
0.4
5
April
1
25
0.6
30
Ag Hmolêm2êdayL
0.4
10
20
Observed
Modelled
1
0.6
MA =1 LAI=2.5
Ag Hmolêm2êdayL
25
Ag Hmolêm2êdayL
Ag Hmolêm2êdayL
15
Day
0.8
5
0.4
MA =0.3 LAI=0.84
January
1
0.6
MA =0.3 LA I=0.72
15
Day
0.8
MA =0.3 LAI=0.78
5
10
November
1
Observed
Modelled
Ag Hmolêm2êdayL
Ag Hmolêm2êdayL
15
Day
Ag Hmolêm2êdayL
10
Observed
Modelled
0.8
MA =0.3 LAI=0.72
Ag Hmolêm2êdayL
5
1
Observed
Modelled
Ag Hmolêm2êdayL
Ag Hmolêm2êdayL
0.8
September
August
1
Observed
Modelled
Ag Hmolêm2êdayL
July
1
30
2.4 Results
2.4.3
Schymanski (2007): PhD thesis
OPTIMAL STOMATAL CONDUCTIVITY
To test whether observed canopy transpiration (Et) and CO2 uptake rates (Ag) are
consistent with the Cowan-Farquhar theory of optimal stomatal conductivity, the
optimised canopy structure from above was taken to calculate the slope (λ) between Et
and Ag at each time step. According to this theory, the value of λ should be fairly
constant during a period in which neither leaf properties nor soil water supply changes
significantly (Cowan and Farquhar 1977; Cowan 1978; Cowan 1982).
Based on the dynamics shown in Figure 2.22, we chose July and October 2004 as well
as January and February 2005 for the test, as these months had no obvious trends in the
CO2 uptake rates. The results for January 2005 and October 2004 will be described in
more detail as examples for wet and dry season conditions respectively.
Since the relationship between Et and Ag is determined by canopy properties and
atmospheric conditions only, prescribing a fixed value of λ allowed the computation of
Et and Ag at any time step.
The fixed value of λ was chosen such that the modelled monthly total in Et would match
the observed monthly total in canopy transpiration. The modelled and observed daily
totals of Et and Ag were then plotted and compared with each other.
Modelled monthly total Et for January 2005 matched the observed total if λ was held
constant at λ = 1681 mol/mol. At the same time, modelled daily Et followed the
observed pattern very closely, while modelled canopy CO2 uptake rates (Ag) were
generally more variable than the observed, but showed a similar pattern (Figure 2.23).
An alternative hypothesis to “constant λ” would be that day-time stomatal conductivity
Gs is constant during the period. This hypothesis was tested by choosing a constant
value for Gs, such that the modelled monthly total in Et would match the observed
monthly total in canopy transpiration. Again, the modelled and observed daily totals of
Et and Ag were then plotted and compared with each other.
When stomatal conductivity was held constant, a value of Gs = 0.24225 mol s-1 m-2 led
to the same monthly total Et as the observed. There was no obvious difference between
the daily canopy CO2 uptake rates, when modelled using constant λ or constant Gs, but
56
Schymanski (2007): PhD thesis
Chapter 2
constant Gs led to a large spike in ET on 7 January, which was not in accordance with
observations (Figure 2.24).
In October 2004, modelled monthly ET matched the observed total if λ was held constant
at λ = 4952 or Gs was held constant at Gs = 0.11255 mol s-1 m-2. Again, prescribing a
constant value for λ led to a better match of modelled daily Et with observed daily Et,
than if the value of Gs was held constant, while the daily rates of Ag were relatively
unaffected (Figure 2.25 and Figure 2.26). A much better match with the observations
could be achieved when fluxes were modelled using different values for λ before and
after the first significant rainfall in October (Figure 2.27).
In all four months, ensemble means of observed diurnal Et rates were better reproduced
by the model based on “constant λ” than the one based on “constant Gs”, the latter of
which resulted in a skew towards the afternoon (Figure 2.28).
57
2.4 Results
Schymanski (2007): PhD thesis
10
Pearson's r = 0.86
MAE = 0.51
Observed
Modelled
EtHmmêdayL
8
6
4
2
Observed
Modelled
5
Ag Hmolêm2êdayL
1
10
15
Day
20
25
30
Observed
Modelled
0.8
0.6
0.4
Observed
Modelled
0.2
Pearson's r = 0.73
MAE = 0.11
5
10
15
Day
20
25
30
Figure 2.23: Modelled (black, thin lines) and observed (grey, thick lines) rates of transpiration (Et)
and canopy CO2 uptake (Ag) in January 2005. Modelled Rates were obtained using constant
λ = 1681 mol/mol. Plotted rates are the sums for each day during daylight hours (Ia > 100
µmol s-2 m-2). The insets show 1:1 plots of the observed and modelled daily values. MAE: mean
absolute error in mm day-1 and mol m-2 day-1 respectively.
58
Schymanski (2007): PhD thesis
Chapter 2
10
Pearson's r = 0.60
MAE = 0.69
Observed
Modelled
EtHmmêdayL
8
6
4
2
Observed
Modelled
5
Ag Hmolêm2êdayL
1
10
15
Day
20
25
30
Observed
Modelled
0.8
0.6
0.4
Observed
Modelled
0.2
Pearson's r = 0.71
MAE = 0.10
5
10
15
Day
20
25
30
Figure 2.24: Modelled (black, thin lines) and observed (grey, thick lines) rates of transpiration (Et)
and canopy CO2 uptake (Ag) in January 2005. Modelled Rates were obtained using constant
Gs = 0.24225 mol s-1 m-2. Plotted rates are the sums for each day during daylight hours
(Ia > 100 µmol s-2 m-2). The insets show 1:1 plots of the observed and modelled daily values. MAE:
mean absolute error in mm day-1 and mol m-2 day-1 respectively.
59
2.4 Results
Schymanski (2007): PhD thesis
4
3.5
Et HmmêdayL
3
2.5
2
1.5
1
0.5
Observed
Modelled
Observed
Modelled
Pearson's r = 0.21
MAE =0.51
5
10
15
Day
20
25
30
0.35
Ag Hmolêm2 êdayL
0.3
0.25
0.2
0.15
0.1
0.05
Observed
Modelled
Observed
Modelled
Pearson's r = -0.36
MAE =0.036
5
10
15
Day
20
25
30
Figure 2.25: Modelled (black, thin lines) and observed (grey, thick lines) rates of transpiration (Et)
and canopy CO2 uptake (Ag) in October 2004. Modelled Rates were obtained using constant
λ = 4952. Plotted rates are the sums for each day during daylight hours (Ia > 100 µmol s-2 m-2). The
insets show 1:1 plots of the observed and modelled daily values. MAE: mean absolute error in
mm day-1 and mol m-2 day-1 respectively.
60
Schymanski (2007): PhD thesis
Chapter 2
4
3.5
Et HmmêdayL
3
2.5
2
1.5
1
0.5
Observed
Modelled
Observed
Modelled
Pearson's r = 0.23
MAE =0.58
5
10
15
Day
20
25
30
0.35
Ag Hmolêm2 êdayL
0.3
0.25
0.2
0.15
0.1
0.05
Observed
Modelled
Observed
Modelled
Pearson's r = -0.4
MAE =0.038
5
10
15
Day
20
25
30
Figure 2.26: Modelled (black, thin lines) and observed (grey, thick lines) rates of transpiration (Et)
and canopy CO2 uptake (Ag) in October 2004. Modelled Rates were obtained using constant
Gs = 0.11255 mol s-1 m-2. Plotted rates are the sums for each day during daylight hours
(Ia > 100 µmol s-2 m-2). The insets show 1:1 plots of the observed and modelled daily values. MAE:
mean absolute error in mm day-1 and mol m-2 day-1 respectively.
61
2.4 Results
Schymanski (2007): PhD thesis
λ=3000
λ=7000
4
3.5
Observed
Modelled
EtHmmêdayL
3
2.5
2
1.5
1
0.5
Observed
Modelled
Pearson's r = 0.63
MAE = 0.38
5
10
15
Day
20
25
λ=3000
λ=7000
Observed
λ=
3000
λ=7000
30
0.4
Ag Hmolêm2êdayL
Modelled
0.3
0.2
0.1
Observed
Modelled
Pearson's r = -0.26
MAE =0.035
5
10
15
Day
20
25
30
Figure 2.27: Modelled (black, thin lines) and observed (grey, thick lines) rates of transpiration (Et)
and canopy CO2 uptake (Ag) in October 2004. Modelled Rates were obtained using two different
values of constant λ for before and after the first of two major rain events, indicated by vertical
arrows. Plotted rates are the sums for each day during daylight hours (Ia > 100 µmol s-2 m-2). MAE:
mean absolute error in mm day-1 and mol m-2 day-1 respectively.
62
Schymanski (2007): PhD thesis
Chapter 2
July
October
January
February
Figure 2.28: Ensemble means of diurnal transpiration rates (Et) during daylight hours
(Ia > 100 µmol s-2 m-2). Grey lines: observed values, solid black lines: Et modelled assuming constant
λ (values given in mol/mol); dashed lines: Et modelled assuming const. Gs (values given in
mol m-2 s-1). Means were computed for all days of the respective month. Error bars are left out for
clarity.
2.5
2.5.1
DISCUSSION
MODELLED AND REALISTIC CANOPY PROPERTIES
The optimal distribution of photosynthetic capacity (expressed as Jmax25) in the modelled
canopy follows the vertical distribution of beam irradiance relatively closely, with a
slightly steeper decline (Figure 2.15 and Figure 2.21). This is in line with observations
in other vegetation types, where electron transport capacity per unit leaf area has been
shown to be closely related to the integrated light availability (Niinemets et al. 1999a;
Misson et al. 2006). The results are also in line with the finding that if the light
environment is dominated by beam irradiance, the optimal distribution of
photosynthetic capacity (often expressed as distribution of leaf nitrogen) would be
steeper than the distribution of light in the canopy (de Pury and Farquhar 1997).
The range of absolute values of optimal Jmax25 within the modelled canopy was similar
to values found in literature. Optimal Jmax25 given by the model had values between
63
2.5 Discussion
Schymanski (2007): PhD thesis
13.6 µmol s-1 m-2 at the bottom of the canopy and 480 µmol s-1 m-2 at the top of the
canopy, while values observed in a deciduous forest in Estonia ranged from 14 to 335
µmol s-1 m-2 (Niinemets et al. 1999a). A compilation of results from studies covering
109 C3 plant species contained average values for Jmax in the range between 17 and
372 µmol s-1 m-2 (Wullschleger 1993). This correspondence is particularly interesting,
as we did not impose any upper or lower limits for Jmax25 and hence the predicted range
is solely a result of the trade-off between the prescribed maintenance costs of Jmax25 and
leaf area, and the achieved carbon uptake. While it should be expected that there is a
physiological upper limit to photosynthetic capacity per unit leaf area due to, for
example, space requirements for the photosynthetic apparatus, the lower limit should
indeed be determined solely by the associated costs and benefits. Below a certain light
limit, the maintenance costs even for the thinnest leaves would exceed their carbon
uptake. Hence, if our model predicts the same lowest feasible value for Jmax25 during
periods with ample water supply as the lowest values observed in nature, we can be
confident that the parameterisation of the costs related to the maintenance of leaf area
and photosynthetic capacity in our model are realistic. This also seems to be supported
by the reasonable match between modelled and observed leaf area index on the site.
However, it should be noted that the model was based on a very simplistic
parameterisation of the canopy. The assumption that all leaves are horizontal and
randomly distributed in homogeneous layers of foliage does not apply to real canopies.
However, while non-horizontal leaf angles would increase optimal leaf area index (data
not shown), non-random distributions of leaves would probably decrease the optimal
leaf area index, so the effects of relaxing these two assumptions could partly negate
each other.
The prescribed costs and benefits of leaf area and Jmax25 led to realistic predictions
during the wet season, but were not sufficient to explain the low vegetation cover (MA)
observed during the dry season. In the dry season, the optimisation only led to realistic
results if the vegetation cover was prescribed. In fact, we could have spared ourselves
the optimisation of MA, as there were no additional costs related to maintaining a certain
value of MA, other than the costs related to the associated leaf area. As vegetation gets
the largest returns for its leaf area in the top layer of foliage, optimal MA was always 1.0
and leaf area was reduced at the bottom of the canopy if water supply was low (compare
Table 2.2 with Table 2.3). With the same amount of water use, a closed canopy could
have achieved a much higher Net Carbon Profit in the dry season than the actual open
64
Schymanski (2007): PhD thesis
Chapter 2
canopy (NCP = 11.66 mol/m2 in Table 2.2 compared with NCP = 5.29 mol/m2 in Table
2.3).
This leads to the conclusion that factors other than the amount of water must be limiting
MA. A possible explanation would be the location of water in the soil. While water is
available at the surface during the wet season, it needs to be taken up from deeper soil
layers during the dry season, and re-distributed at the surface. Observations show that
the top 3-4 m of soil are severely depleted of water during the dry season in this region
(Kelley 2002). The transport and re-distribution of water requires the construction and
maintenance of vascular infrastructure, which is obvious in tap roots, trunks and
branches of the trees prevailing during the dry season. These costs were not considered
in the present study, but were included in a follow-up study that deals with the dynamics
of soil water as well as vegetation dynamics (Schymanski et al. in prep.-c). In the
present study, the value of MA had to be prescribed for periods when water supply was
limiting.
2.5.2
CANOPY CO2 UPTAKE RATES
The relatively close match of the magnitudes of modelled and observed canopy CO2
uptake rates is, in our opinion, remarkable, particularly since no site-specific vegetation
parameters were prescribed. In fact, the model did not even consider that the dominant
CO2 uptake process during the wet season on this site is C4 photosynthesis, and hence
predicted the optimal properties of a canopy of C3 species. Our results suggest that the
inherent ratio between costs and benefits related to the maintenance of leaf area and the
photosynthetic apparatus may not change significantly between the different
photosynthetic pathways. One interpretation is that increased benefits due to special
adaptations are likely to bring along increased metabolic costs as well. The net benefit
of employing a specialist photosynthetic pathway, although significant enough to favour
certain species over others in a given environment, may be smaller at canopy scale.
During the wet season, there was also a good correspondence between modelled and
observed day-to-day variations in canopy CO2 uptake (Figure 2.22). Such a
correspondence could not be achieved during the dry season. However, it has to be
noted that the fluctuations in daily canopy CO2 uptake are naturally low during the dry
season, so that failure to capture these fluctuations may be a result of the weakness of
the signal, relative to the noise in the measurements.
65
2.5 Discussion
Schymanski (2007): PhD thesis
Two relatively static states of vegetation cover (MA) have been identified on the study
site, the wet season state with MA = 1.0 and the dry season state with MA = 0.3. By
prescribing MA = 0.3 for the period between April and November and MA = 1.0 for the
period between December and March, the model could be used to describe the duration
of the transition periods between these two states. The data presented in Figure 2.22
reveal that the transition from the dry season state to the wet season state happened
much faster (~ 6 weeks) than from the wet season state to the dry season state (~ 10
weeks). Comparison of observed and modelled fluxes also reveals the impact a bush fire
had on the fluxes, after its occurrence on 6 August 2004. The data suggest that the
recovery to the pre-bushfire CO2 uptake rates took ~ 8 weeks.
2.5.3
OPTIMAL STOMATAL CONDUCTIVITY
Cowan and Farquhar’s theory of optimal stomatal conductivity (Cowan and Farquhar
1977) suggests that diurnal variations in stomatal conductivity can be predicted by
prescribing a single constant (λ), if the functional dependency of photosynthesis and
transpiration on stomatal conductivity is known. We will call this theory the “constant λ
theory”. As in the case of empirically-based models of stomatal conductance (e. g. Ball
et al. 1987; Leuning 1990; Leuning 1995), the constant λ theory incorporates a
sensitivity of stomata to the rate of photosynthesis and the atmospheric vapour deficit
(Dv), as shown in Equations ( 2.31 ) to ( 2.34 ). However, in contrast to the empirical
models, Cowan and Farquhar’s theory gives an explanation of why stomata should be
sensitive to these factors and relates this sensitivity directly to the availability of soil
water.
The constant λ theory has been successfully used in a number of studies to model gas
exchange under field conditions at leaf level (Berninger and Hari 1993; Berninger et al.
1996; Hari et al. 1999; Koskela et al. 1999; Hari et al. 2000; Aalto et al. 2002). More
recently, the theory has also been found applicable at canopy level (Styles et al. 2002;
Wilson et al. 2003; Mercado et al. 2006). On the other hand, experiments under
controlled conditions often led to questioning of the theory (Franks et al. 1997; Buckley
et al. 1999; Thomas et al. 1999). However, this is not surprising, considering that:
“The optimisation hypothesis, relating as it does to the ideal adaptation of
stomatal behaviour to a putative selection pressure, can only be tested in the
circumstances in which we think adaptation has occurred.” (Cowan 2002)
66
Schymanski (2007): PhD thesis
Chapter 2
Cowan (1982; 2002) also pointed out that the benefit of an optimal variation of stomatal
conductivity is often very small compared with constant stomatal conductivity. This
implies that the selective pressure for the stomatal apparatus to be able to respond
rapidly to changes in environmental conditions is low, which might explain why
stomata have a relatively slow response to environmental conditions, in the order of
minutes rather than seconds.
Our results indicate that the constant λ theory is useful at canopy scale under natural
conditions, and that transpiration rates modelled using a constant value of λ reproduce
observed transpiration rates significantly better than a model based on “constant
conductivity” (Figure 2.23 to Figure 2.28). In line with the above, the modelled rates of
CO2 uptake were less affected by the assumption of “constant λ” or “constant
conductivity”.
Besides leading to more realistic results in terms of daily and diurnal transpiration rates,
the “constant λ theory” also has the advantage of generating a measure for how
conservatively vegetation uses water. As illustrated in Figure 2.7, a high value of λ
means that a large decrease in canopy transpiration (Et) would only lead to a small
decrease in canopy CO2 uptake (Ag), while a low value of λ implies that a small increase
in Et would result in a large increase in Ag. Thus, we would expect that so-called “waterstressed” vegetation would operate at a lower value of λ than vegetation with ample
water supply. In this context, it is interesting that the value of λ was much higher in the
dry season (λ = 4952) than in the wet season (λ = 1681), which suggests that the plants
were more “wasteful” with water in the dry season. In fact, if vegetation adopted a
lower value of λ in October 2005, monthly total Et could be decreased by more than
20%, while monthly total Ag would only decrease by 6%. An example is shown in
Figure 2.29. Describing the trees as “water-stressed” during the dry season does not
seem reasonable from this perspective. It is to be noted that the site received around 30
mm of rainfall on 10 October 2004, followed by some more rain between 14 and 17
October. Observed transpiration rates are fairly well reproduced by the model using
λ = 3000 before the rainfalls, and λ = 7000 after the first rain (Figure 2.27). This is
consistent with the theory that the value of λ should depend on the amount of available
soil water (Cowan 1982; Cowan 1986). Part of the observed increase in λ after rainfalls
could also be due to the evaporation of intercepted water, which has not been
considered in the present model. However, this effect is unlikely to last for more than a
67
2.5 Discussion
Schymanski (2007): PhD thesis
day after the rainfall and hence can not explain the persistently increased λ shown in
Figure 2.27.
It appears that adaptation to the more restricted water supply during the dry season is
primarily not achieved by means of stomatal closure, but by a reduction of vegetation
cover (MA), while the remaining vegetation continues transpiring at a high rate. In fact,
the reduction in MA and associated change in leaf area index was greater than the
reduction in stomatal conductivity per ground area, so that stomatal conductivity per
leaf area was on average even higher during the dry than during the wet season.
If vegetation cover MA and hence the leaf area index of the site is reduced, while
stomatal conductivity (and hence water use) per leaf area increases in the dry season,
this suggests that the costs related to the maintenance of MA must have increased more
than the costs related to water uptake. This has been explored further in a water balance
study of the site and will be presented in a follow-up paper (Schymanski et al. in prep.c).
5
EtHmmêdayL
4
3
2
Observed
Modelled
1
5
10
15
Day
20
25
30
Ag Hmolêm2êdayL
0.4
0.3
0.2
0.1
Observed
Modelled
5
10
15
Day
20
25
30
Figure 2.29: Modelled (black, thin lines) and observed (grey, thick lines) rates of transpiration (Et)
and canopy CO2 uptake (Ag) in October 2004. Modelled Rates were obtained using constant
λ = 3000. Plotted rates are the sums for each day during daylight hours (Ia > 100 µmol s-2 m-2).
68
Schymanski (2007): PhD thesis
2.5.4
Chapter 2
VALIDITY OF THE APPROACH
The optimality approach to vegetation modelling appeals to the theory of evolution and
natural selection as a justification to assume a general tendency towards optimality in
plant functioning. However, it is a great leap of faith to infer optimality of a community
of plants from this theory, as implicitly done in the present study. In fact, it has been
shown elsewhere that optimal water use by competing plants could be quite different to
optimal water use by a community of plants acting as an entity (Cowan 1982). The
question of whether natural selection favours competitive individuals or well-adapted
plant communities (“group selection”), is still the subject of considerable debate
(Wilson 1983; Wynne-Edwards 1993; Smillie 1995; Okasha 2003). Some of the
controversy about “group selection” can be related to the question of whether selection
is based on the “fitness of the group” or the “average individual fitness of its constituent
organisms” (Okasha 2003). This is not different from looking at natural selection as an
optimisation problem, where the “fitness function” constitutes the objective function for
the optimisation. Thus, the search for the appropriate “objective function” in optimality
modelling, as outlined in the introduction of this paper, is akin to the search for the
appropriate “fitness function” in evolutionary modelling. Traditionally, evolutionary
models often define “fitness” as the number of offspring, while vegetation optimality
models define the objective function as some quantity related to the use of resources. It
could be argued that Net Carbon Profit, be it for an individual or a group, is a more
general formulation of fitness than the “number of offspring”, as persistence in a given
environment can be achieved by means other than the generation of a large number of
offspring. Whatever the means to persist in the environment, the energy stored in
carbohydrates is a universal currency that can be invested in reproduction or any other
means.
The presented model contains a range of very crude simplifications, for example the
assumption that all leaves are horizontal, the subdivision of the canopy into layers with
similar leaf area and randomly distributed leaves, the neglect of the limitation of
photosynthesis by the biochemical carboxylation capacity and the treatment of the
canopy as a single leaf for carboxylation purposes. Some of these assumptions could
probably be relaxed using the same optimality considerations as we used for the existing
model. For example, the co-optimisation of electron transport capacity and
carboxylation capacity and the optimisation of leaf angles in a canopy have been
69
2.5 Discussion
Schymanski (2007): PhD thesis
applied elsewhere separately (Friend 1991; Herbert 1991; Herbert and Nilson 1991;
Herbert 1992; Niinemets et al. 2004) and could be included in the overall Vegetation
Optimality model when the available computing power improves.
70
Schymanski (2007): PhD thesis
2.6
Chapter 2
CONCLUSIONS
1. Maximisation of Net Carbon Profit is a possible principle for self-organisation
of plant communities
The objective function of maximising Net Carbon Profit of foliage led to the emergence
of vegetation properties and CO2 uptake rates in the model, which were consistent with
observations during the wet season.
2. Costs and benefits of foliage are sufficient to predict fluxes and canopy
properties if rooting and water transport costs are small
Although the study site is generally regarded as a nutrient-poor site, the leaf area index,
photosynthetic capacity and canopy photosynthesis during the wet season were as high
as the optimality model predicted without considering nutrient limitation at all.
Meteorological data, observed water use and the prescribed costs related to the
maintenance of the photosynthetic apparatus and the maintenance of leaf area were the
only constraints needed to reproduce the wet season fluxes. We conclude that other
costs, such as for example the costs involved in water uptake and water transport are
likely to have been relatively small during the wet season, so that they did not need to
be considered.
3. If rooting costs are large (deep roots needed), they have to be included in order
to predict canopy structure and fluxes
In the dry season, vegetated cover (MA) had to be prescribed in order to achieve realistic
model results. As the value of MA was not constrained by the amount of available water
alone, we conclude that costs associated with deep rooting and redistribution of water at
the surface must be included to make realistic predictions during the dry season.
Inclusion of these costs during the wet season was not necessary, as soil moisture was
sufficient in the upper soil layers and the plants that grew in the wet season could
survive with much “cheaper” root systems. In fact, we will show in a follow up study
that the low projected cover of tree crowns on the site could well be explained by the
costs related to the transport of water from deeper soil layers to the surface and the
horizontal distribution of the water over large crown areas (Schymanski et al. in prep.c).
71
2.7 Notation
2.7
Schymanski (2007): PhD thesis
NOTATION
Table 2.4: Parameters used in this study and their units. If not stated otherwise in this table,
addition of the subscript i to a symbol in the main text denotes reference to foliage layer i (with i = 1
at the top of the canopy).
Symbol
Description
Value / Units
a
Molecular diffusion coefficient of CO2 in air, relative to
that for water vapour
1.6
Ac
Photosynthetic carboxylation rate per ground area
mol m-2 s-1
Ag
CO2 flux through stomata per ground area
mol m-2 s-1
Amax
Photosynthetic capacity per ground area
mol m-2 s-1
Av
Rubisco-limited carboxylation rate per leaf area
mol m-2 s-1
Ca
Mole fraction of CO2 outside of leaf
mol/mol
Cl
Mole fraction of CO2 in the air spaces within a leaf
mol/mol
cRl
Leaf respiration coefficient (Rl = Amax cRl)
0.07
Dv
Molar atmospheric vapour deficit (1.0-Wa)
mol/mol
Dy
Day of the year
-
Es
Evaporation from the soil surface
mol m-2 s-1
Et
Transpiration rate per unit ground area
mol m-2 s-1
ET
Evapo-transpiration; total of all evaporative processes in
the system per unit ground area (ET = Es + Et)
mol m-2 s-1
FnC
Net CO2 uptake by the soil-vegetation system per ground
area
mol m-2 s-1
ft
Equation of time
˚
Gs
Total stomatal conductivity for CO2 per ground area
mol m-2 s-1
Gs,meas
Total stomatal conductivity for CO2 per ground area,
derived from measurements
mol m-2 s-1
Gsoil
Soil conductivity to water vapour
0.03 mol m-2 s-1
h
Hour angle of the sun
˚
Ha
Rate of exponential increase of Jmax with temperature
43,790 J mol-1
Hd
Rate of decrease of Jmax with temperature above Topt
200ä103 J mol-1
I2
Useful light absorbed by Photosystem II
mol m-2 s-1
Ia
Photosynthetic active irradiance in mol quanta per
ground area per unit time
mol m-2 s-1
IaW
Photosynthetic active irradiance in Watts per ground area
W m-2
Ib
Beam component of photosynthetic active irradiance in
mol quanta per ground area per unit time
mol m-2 s-1
72
Schymanski (2007): PhD thesis
Chapter 2
Symbol
Description
Value / Units
Id
Diffuse component of photosynthetic active irradiance in
mol quanta per ground area per unit time
mol m-2 s-1
Il
Photosynthetic active irradiance in mol quanta per leaf
area per unit time
mol m-2 s-1
J
Photosynthetic electron transport rate per leaf area
mol m-2 s-1
JA
Photosynthetic electron transport rate per total ground
area
mol m-2 s-1
Ji
Photosynthetic electron transport rate in layer i per unit
vegetated area
mol m-2 s-1
Jmax
Leaf photosynthetic electron transport capacity per leaf
area
mol m-2 s-1
jmax
Local photosynthetic electron transport capacity per unit
leaf sub-area
mol m-2 s-1
Jmax25
Photosynthetic electron transport capacity per leaf area at
25˚C (standard temperature)
mol m-2 s-1
Jmaxtot
Photosynthetic electron transport capacity per ground
area
mol m-2 s-1
Jshade,i
Photosynthetic electron transport rate by shaded leaves in
layer i per vegetated ground area
mol m-2 s-1
Jsun,i
Photosynthetic electron transport rate by sunlit leaves in
layer i per vegetated ground area
mol m-2 s-1
kd
Transmission coefficient for diffuse radiation
-
kopt
Value of Jmax at optimal temperature
mol m-2 s-1
Kshort
Shortwave irradiance in the wavelengths 0.2-4.0 µm in
Watts per ground area
W s-1 m-2
LA
Leaf area per vegetated ground area in a layer of foliage
0.1 m2/m2
LAI
Leaf area index of the site
m2/m2
LAIv
Leaf area index of vegetated patches (LAIv = LAI/MA)
m2/m2
LAshade,i
Shaded leaf area in layer i (per vegetated ground area)
m2/m2
LAsun,i
Sunlit leaf area in layer i (per vegetated ground area)
m2/m2
LAunsat
Leaf area that is not light-saturated (per leaf area of leaf)
m2/m2
LD
Length of the day (sunrise to sunset)
h
Ll
Local geogr. longitude
131.15˚
Ls
Standard geogr. longitude of the time zone
142.5˚
MA
Fraction of surface covered by vegetation (projection of
outlines of all living plants)
m2/m2
NCP
Net carbon profit
mol m-2
Nl
Number of layers of foliage in canopy
-
73
2.7 Notation
Schymanski (2007): PhD thesis
Symbol
Description
Value / Units
P
Barometric air pressure
Pa
Prob[…]
Probability of…
-
pva
Atmospheric vapour pressure
Pa
pvs
Water vapour pressure in the laminar layer immediately
above the soil surface
Pa
pvsat
Saturated atmospheric vapour pressure
Pa
pvssat
Saturated water vapour pressure in the laminar layer
immediately above the soil surface
Pa
Q10,w
Temperature response parameter for woody tissue
respiration
1.92
qs↑
Temperature response parameter for soil respiration
0.059(˚C)-1
qs↓
Temperature response parameter for soil respiration
0.507(˚C)-1
R0
Intrinsic soil respiration rate per ground area at Ts=20 ˚C
and θ = θmax
1.862ä10-6
mol m-2 s-1
Rft
Foliage turnover costs per ground area
mol m-2 s-1
Rl
Leaf respiration rate per ground area
mol m-2 s-1
Rmol
Molar universal gas constant
8.31441
J mol-1 K-1
Rs
Soil respiration rate per ground area
mol m-2 s-1
Rtc
Foliage turnover costs
mol m-2 s-1
Rw
Woody tissue respiration rate per ground area
mol m-2 s-1
Rw25
Woody tissue respiration rate at 25 ˚C per ground area
0.606ä10-6
mol m-2 s-1
Sd
Diffuse solar irradiance at the top of the canopy
W m-2
Sd,d
Daily diffuse solar irradiance at the top of the canopy
J m-2 day-1
Sg,d
Daily global solar irradiance at the top of the canopy
J m-2 day-1
So
Solar irradiance at the top of the atmosphere
W m-2
So,d
Daily solar irradiance at the top of the atmosphere
J m-2 day-1
SSC
Solar constant
1367 W m-2
T↓
Temperature response parameter for soil respiration
35.8 ˚C
Ta
Air temperature
K
td
Hour of the day
h
to
Solar noon
h
Topt
Optimum temperature for electron transport
K
Tr
Reference temperature for wood respiration
25 ˚C
Ts
Soil temperature
˚C
74
Schymanski (2007): PhD thesis
Chapter 2
Symbol
Description
Value / Units
Tw
Wood temperature
˚C
Wa
Molar fraction of water vapour in the atmosphere
mol/mol
Wl
Molar fraction of water vapour in the air spaces within a
leaf
mol/mol
Ws
Molar fraction of water vapour in the laminar layer
immediately above the soil surface
mol/mol
α
Quantum yield of electron transport
0.3 mol/mol
β
Solar elevation angle
˚
Γ*
CO2-compensation point in the absence of respiration
mol/mol
Γd
Day angle
˚
δ
Solar declination angle
˚
θ
Volumetric soil moisture
m3 m-3
θfc
Volumetric soil moisture at field capacity
0.15575 m3 m-3
θmax
Soil moisture response parameter for soil respiration
0.12 m3 m-3
θmin
Soil moisture response parameter for soil respiration
0.01 m3 m-3
λ
Slope of Et(Ag) function
mol/mol
λexp
Distribution-specific parameter of exponential jmaxdistribution
m2 s mol-1
Φ
Geogr. latitude in decimal degrees
-12.4952˚
75
Appendix 2.1
Schymanski (2007): PhD thesis
APPENDIX 2.1. PHOTOSYNTHETIC LIGHT
ABSORPTION
A.2.1.1
ELECTRON TRANSPORT RATES AT LEAF SCALE
It is assumed that the electron transport rate (J) increases linearly with irradiance, until
all reaction centres get saturated and J reaches saturation (Jmax). This can be summarised
as:
J
Ø
≤
≤ Il a
∞
≤
≤ Jmax
±
Il <
Il ¥
Jmax
a
Jmax
a
( 2.47 )
where α is the relation between absorbed quanta and resulting electron transport rate.
The value of α is commonly set to around 0.3
However, observed J(Il)-curves are non-linear and have a smooth transition between the
linear and saturating parts of the curve. Traditionally, the smooth curvature is
represented by a rectangular hyperbola with an empirical curvature parameter θJ, (e.g.
Equation 2.15 in von Caemmerer (2000)), shown here as Equation ( 2.48 ):
J
I2 + Jmax -
!
è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
HI2 + JmaxL2 - 4 I2 Jmax qJ
2 qJ
( 2.48 )
where I2 = α Il and the empirical curvature parameter θJ is commonly set to a value
between 0.7 and 0.9.
The same value for θJ that applies at leaf scale, does not necessarily apply at canopy
scale (de Pury and Farquhar 1997). Therefore we sought an alternative formulation that
would not involve an empirical curvature parameter derived at leaf-scale.
When integrating over a horizontal layer of leaves, where it cannot be expected that all
leaves would have exactly the same Jmax, the question arises how the curvature would
depend on the horizontal distribution of Jmax. In fact, it can be shown that the value of θJ
in a single leaf could be explained by an inhomogeneous distribution of Jmax over the
area of that leaf.
76
Schymanski (2007): PhD thesis
Chapter 2
The following thought experiment was performed to develop an alternative approach to
modelling J as a function of Jmax and absorbed light:
The total area of a single leaf can be occupied by sub-areas with different values for
Jmax. As an example, let us assume that a total area of 1.0 m2 is occupied by four equal
sub-areas with Jmax of 100, 140, 180 and 220 µmol m-2 s-1 respectively. The average Jmax
is 160 µmol m-2 s-1, and each sub-area has a linear J(Il) function with α = 0.3, as in
Equation ( 2.47 ).
Figure 2.30 shows the electron transport rates of each sub-area as functions of light, and
the total electron transport rate.
J H µ molêm2êsL
A
150
125
100
75
50
A4
25
A1
.
.
200
400
600
800
1000
Il (µmol/m2/s)
Figure 2.30: Hypothetical electron transport rates in four sub-areas (“A1”...”A4”) of similar sizes
with different Jmax (overlapping curves) and the total electron transport rate of all sub-areas
together (top curve, denoted as “A”). Note that J is plotted as total J for each sub-area and hence
depends on the size of each sub-area. Although each sub-area has an abrupt transition between
non-saturated and saturated rate, the total electron transport function for the leaf appears curved.
It can be seen in Figure 2.30 that although the J(Il) curves are piece-wise linear within
each sub-area, the total electron transport rate of all sub-areas together appears curved,
reaching saturation when all sub-areas are light-saturated. The value of J at full
saturation coincides with the average Jmax of the whole area (160 µmol m-2 s-1).
Based on the above example, we concluded that inhomogeneity in the distribution of
Jmax can result in the observed curvature of the J(Il) curves.
To derive a new formulation of J(Il) that does not require an empirical curvature
parameter, we will assume for the moment that a total leaf area of 1.0 m2 is occupied by
many sub-areas with different values of local electron transport capacity (jmax). The
77
Appendix 2.1
Schymanski (2007): PhD thesis
distribution of jmax over the sub-areas shall be exponential, with the average jmax over the
whole leaf area written as Jmax = 160 µmol m-2 s-1.
The probability density function of the exponential distribution can be written as:
f H jmaxL
lexp ‰- jmax lexp
( 2.49 )
where 0 ≤ jmax ≤ ∞ and λexp is a distribution-specific parameter.
The expected value for an exponential distribution is E[X] = 1/λexp, so if we want to
achieve E[X] = 160ä10-6 mol m-2 s-1, we would need to insert λexp = 6250 m2 s mol-1.
The cumulative distribution function of the exponential distribution gives
Prob@ X § jmax D
- jmax lexp
1- ‰
( 2.50 )
1 - ‰-6250 jmax
( 2.51 )
Inserting λexp = 6250 m2 s mol-1, we get:
Prob@ X § jmaxD
Prob @X≤jmaxD
1
0.8
0.6
0.4
0.2
200
400
600
jmax Hµmol êm2êsL
800 1000
Figure 2.31: Cumulative Distribution Function of an exponential distribution of jmax with a mean
value of 160 µmol m-2 s-1.
A plot of the cumulative distribution with a mean value of 160 µmol m-2 s-1 is plotted in
Figure 2.31. The electron transport curve is the sum of the electron transport rates of all
sub-areas. If we assume that all sub-areas have linear light responses similar to Figure
2.30, the slope of the total curve will depend on how much of the area is saturated at a
given irradiance. The slope of the electron transport curve can thus be calculated using
the total unsaturated leaf area (leaf area with Jmax/α ¥ Il), as this is the leaf area that
contributes to an increase in J with increasing Il:
∑J ê ∑I l
a L Aunsat
78
( 2.52 )
Schymanski (2007): PhD thesis
Chapter 2
LAunsat is equivalent to “leaf area with Jmax/α ≥ Il”, or in other words “leaf area with
Jmax ≥ (α Il)”. To be able to integrate Equation ( 2.52 ) and to obtain J(Il), we need to
convert the Prob[Jmax ≤ Jmax0] function in Equation ( 2.51 ) to a curve of “leaf area with
Jmax ≥ (α Il)”. Cumulative probability functions are directly equivalent to area functions,
so we can convert the above Prob[Jmax ≤ Jmax0] to "area with Jmax > Jmax0" as a function
of Jmax0:
LAunsat
Prob@Jmax >Jmax0 D
1 - Prob@Jmax§Jmax0 D
( 2.53 )
Inserting Equation ( 2.51 ) into Equation ( 2.53 ), we get LAunsat as a function of Il:
L Aunsat
‰- 6250 a I l
( 2.54 )
After inserting ( 2.54 ) into ( 2.52 ), we can solve the differential equation for J(Il) using
the boundary condition J(0) = 0.
J HIlL
0.000048 - 0.000048 ‰-6250 a Il
a
( 2.55 )
Generalising, we could calculate J(Il) for any area with an exponential distribution of
Jmax and an average Jmax equal to the expected value of the distribution (E[Jmax] = 1/λexp)
by solving:
∑J
∑Il
-
a‰
a Il
Jmax
( 2.56 )
with the boundary condition J(0) = 0. Thus, we will write for the electron transport rate
as a function of electron transport capacity and photosynthetically active radiation:
J
a Il y
ij
z
J
j
Jmax jj1 - ‰ max zzz
k
{
( 2.57 )
Figure 2.32 shows electron transport curves formulated using the traditional equation
and Equation ( 2.57 ). As the scatter in data is usually very high at high irradiance,
either of the curves could be used to match measured data, but the new formulation does
not contain the empirical parameter θJ . Furthermore, the curvature of the electron
transport curve at leaf scale is now formulated as a result of a non-homogeneous,
exponential distribution of electron transport capacities (Jmax0) over the surface area of a
79
Appendix 2.1
Schymanski (2007): PhD thesis
leaf. This is very useful for integrating electron transport over several leaves, as nonhomogeneity is already factored in.
J Hµ molê m2ês L
150
125
100
75
Jmax
Jnew
Jθ= 0.7
50
25
1000
2000
3000
4000
(µmol/m
/s)
Ial H
µ molêm 22ê
sL
Figure 2.32: Electron transport rate as a function of Il. Comparison of the classic formulation in
Equation ( 2.48 ) (denoted Jθ=0.7), using α = 0.3, θJ = 0.7 and Jmax = 160 µmol/m2/s, with the new
formulation in Equation ( 2.57 ) (denoted Jnew), using the same values for α and Jmax.
A.2.1.2
ELECTRON TRANSPORT RATES AT CANOPY
SCALE
The sunlit part of layer i was calculated by assuming that each layer above has reduced
the sunlit area by its own sunlit area, and the remaining direct radiation is intercepted by
a factor equivalent to the projected foliage cover in this layer:
LAsun,i
H1-LAsun,1-LAsun,2-...-LAsun,i-1L L A
( 2.58 )
Similarly, for the layer above layer i we obtain:
LAsun,i-1
LA
H1-LAsun,1-LAsun,2-...-LAsun,i-2L
( 2.59 )
Therefore, the sequence in Equation ( 2.59 ) can be written as:
LAsun,i-1
- LAsun,i-1
LA
H1-LAsun,1 -LAsun,2-...-LAsun,i-1L
( 2.60 )
A comparison of Equations ( 2.60 ) and ( 2.58 ) permits to formulate LAsun,i for i ≥ 1 as a
function of LAsun,i-1 only:
LAsun,i
LA J
LAsun,i-1
- LAsun,i-1N
LA
80
( 2.61 )
Schymanski (2007): PhD thesis
Chapter 2
The above is a recurrence equation, which can be solved using the boundary condition
LAsun,1 = LA to obtain Equation ( 2.7 ) in the main text:
LAsun,i
H1 - LALi-1 LA
81
( 2.62 )
Appendix 2.2
Schymanski (2007): PhD thesis
APPENDIX 2.2. PHOTOSYNTHETIC CARBOXYLATION
RATES
It has been mentioned previously (Section 2.3.2.1), that the photosynthetic
carboxylation rate in a leaf can be limited by either photosynthetic electron transport (J)
or by the activity of the enzyme rubisco, which catalyses the carboxylation. It was
assumed that, to a good approximation, carboxylation is predominantly light-limited in
a canopy, so that Equation ( 2.17 ) was deemed adequate for modelling canopy
photosynthesis (Farquhar and von Caemmerer 1982). On the other hand, if
carboxylation was limited by rubisco, the photosynthetic carboxylation rate would take
the following form (von Caemmerer 2000):
Av
Vcmax HCl - G* L
O
Cl + Kc J l + 1N
( 2.63 )
Ko
where Vcmax, Vomax, Kc and Ko are the maximal rates and the Michaelis-Menten constants
of carboxylation and oxygenation respectively (von Caemmerer 2000) and Ol is the
molar fraction of oxygen inside the leaf. Replacing Ac by Av in Equation ( 2.15 ) and
following the same procedure from there yields the rubisco-limited big-leaf CO2
assimilation rate as a function of stomatal conductivity (Ag,v):
Ag,v
1
2Ko
JCa Gs Ko + Gs Kc Ko - Rl Ko + Vcmax Ko + Gs Kc Ol -
"#####################################################################################################################################################################################################
#
HCa Gs Ko + HRl - VcmaxL Ko - Gs Kc HKo + Ol LL2 + 4Gs Ko HKc HKo + Ol L HCa Gs + Rl L + Ko Vcmax G* L N
( 2.64 )
Optimal allocation of resources within the leaf is likely to lead to co-limitation of both
processes at the normal working conditions of the leaf (Field and Mooney 1986). This
would imply that both Vcmax and the capacity to generate electron transport (Jmax) are
adjusted in such a way that they are, on average, both equally limiting for overall
photosynthesis.
It has been shown elsewhere that Jmax and Vcmax, when standardised to a common
temperature, follow the relation Jmax = 2.68äVcmax over a wide range of plant species
(Wullschleger 1993; Leuning 1997). Using this relation, we computed Ag for different
degrees of light-saturation using Equation ( 2.17 ) and plotted the results against Ag,v
(Figure 2.33). Light-saturation degrees were simulated by setting the electron transport
82
Schymanski (2006): PhD thesis
Chapter 2
rate (JA) to a pre-defined fraction of Jmax. Light-saturated photosynthesis would mean
that the value of JA is close to Jmax.
The results show that neglecting Vcmax-limitation would lead to a significant overestimation of CO2 assimilation only when light is close to saturating and stomatal
conductivity (Gs) is in an intermediate range (Figure 2.33b, c). When the electron
transport rate is less than half the potential rate, Vcmax-limitation is not relevant at any
values of Gs (Figure 2.33a).
Ag @µmol êsD
Ag @µmol êsD
(a)
Ag @µmol êsD
(b)
(c)
30
30
30
25
25
25
20
20
15
Ag
Ag,v
10
5
J = 0.5 Jmax
0.2
0.4
0.6
0.8
1
Gs @mol êsD
20
15
Ag
Ag,v
10
5
J = 0.65 Jmax
0.2
0.4
0.6
0.8
1
Gs @mol êsD
Ag
Ag,v
15
10
5
J = 0.7 Jmax
0.2
0.4
0.6
0.8
1
Gs @mol êsD
Figure 2.33: Jmax-limited (solid lines) and Vcmax-limited (dashed lines) CO2 assimilation rates as a
function of stomatal conductivity (Gs). J was prescribed as different proportions of Jmax, where
Jmax = 2.68 â Vcmax. As J generally increases with irradiance between 0 and Jmax, the proportion of J
to Jmax indicates the degree of light-saturation of the photosynthetic system. Model parameters were
taken from figure 2.6 in von Caemmerer (2000): Vcmax = 100 µmol m-2 s-2, Kc = 259 µbar, Ko = 179
mbar, Γ* = 38.6 µbar, Rl = 1 µmol m-2 s-2, Ol = 0.2, Ca = 350â10-6
83
Appendix 2.3
Schymanski (2007): PhD thesis
APPENDIX 2.3. ATMOSPHERIC FORCING
A.2.3.1
SEPARATION OF GLOBAL IRRADIANCE INTO
DIRECT AND DIFFUSE COMPONENTS
According to Spitters et al. (1986), the instantaneous amount of diffuse solar irradiance
(Sd) reaching the canopy can be estimated from the instantaneous solar irradiance at the
top of the atmosphere (So), daily diffuse (Sd,d) and daily top of the atmosphere solar
irradiance (So,d):
Sd
So Sd,d
So,d
( 2.65 )
Obviously, the above is a very coarse approximation as atmospheric scattering can
change at smaller than daily time scales, but it is the best-guess estimate in the absence
of more detailed measurements.
We will call Sd,d/So,d the transmission coefficient for diffuse radiation (kd).
Sd kd So
( 2.66 )
Roderick (1999) derived an empirical relationship between daily global and daily
diffuse solar irradiance. The relationship can be summarised in our notation as:
Sd,d
Sg,d
Sg,d
Ø
≤
0.96
§ 0.26
≤
≤
So,d
≤
≤
≤
≤
≤ 5377.27 So,d-20681.8 Sg,d + 0.96 0.26 < Sg,d § 0.000044 †f§2 - 0.0017 †f§ + 0.8
∞
≤
I†f§2-38.6364 †f§+12272.7M So,d
So,d
≤
≤
≤
≤
Sg,d
≤
≤
> 0.000044 †f§2 - 0.0017 †f§ + 0.64
≤ 0.05
So,d
±
( 2.67 )
where Sg,d is the daily global solar irradiance and Φ is the geographic latitude in decimal
degrees.
Solving for Sd,d and dividing by So,d yields Sd,d/So,d, and the transmission coefficient for
diffuse radiation (kd):
84
Schymanski (2007): PhD thesis
kd
Chapter 2
S
Sg,d
Ø
≤
0.96 g,d
§ 0.26
≤
≤
So,d
So,d
≤
≤
≤
≤
S
≤ Sg,d ij 5377.27So,d-20681.8Sg,d
y
j 2
+ 0.96zz 0.26 < g,d § 0.000044 †f§2 - 0.0017 †f§ + 0.8 ( 2.68 )
∞
≤
So,d k I†f§ -38.6364 †f§+12272.7M So,d
So,d
≤
{
≤
≤
≤
Sg,d
Sg,d
≤
≤
> 0.000044 †f§2 - 0.0017 †f§ + 0.8
≤ 0.05 S
S
±
o,d
o,d
Sg,d was obtained by integrating measured short wave irradiance over a day, and So,d was
modelled following Spitters et al. (1986):
J‡ sinH bHtLL „ tN SSC
So,d
‡ sinH b HtLL „ t
LD
i 24 cosHdL
3600 jjjj
k
( 2.69 )
è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
y
1 - tan2 HdL tan2HfL cosHfL
+ sinHdL sinHfL LDzzzz
p
{
ij
0.397949 cosH0.0172142 D y + 0.172142L tanHfL
12 - 7.63944 sin jj
j "########################################################################################
#
k 0.920818 - 0.0791816 cosH0.0344284 D y + 0.344284L
-1j
j
( 2.70 )
yz
zz
zz
z
{
( 2.71 )
where β is the solar elevation angle, SSC is the solar constant (1367 Watt m-2), δ is the
solar declination angle, LD is the length of the day (between sunrise and sunset) and Dy
is the day of the year.
A.2.3.2
SOLAR IRRADIANCE AT THE TOP OF THE
ATMOSPHERE
The instantaneous solar irradiance at the top of the atmosphere (So) was modelled as in
the appendix of de Pury and Farquhar (1997). Solar noon was calculated according to
the local longitude (130.15˚ for Howard Springs) and the standard longitude of the time
zone (142.5˚). The relevant equations are reprinted here for convenience.
So was formulated as a function of the solar angle (β) and the solar constant (SSC):
So
sinH bL SSC
where
85
( 2.72 )
Appendix 2.3
Schymanski (2007): PhD thesis
sin(b) cos(h) cos(d) cos(f)+sin(d) sin(f)
( 2.73 )
δ is the solar declination angle:
d
- 23.45 cosJ
2
p H D y + 10LN
365
( 2.74 )
h is the hour angle of the sun:
h
1
p Htd - to L
12
( 2.75 )
td is the actual time and to is the time of solar noon, both in decimal hours:
to
1
H4 HLs - LlL - ftL + 12
60
( 2.76 )
Ll and Ls are the local longitude and the standard longitude of the time zone respectively
and ft is the equation of time, accounting for the non-linearity of solar motion
throughout the year:
ft 0.4281cosHGdL - 3.349cosH2 GdL - 17.082sinHGdL + 0.017
( 2.77 )
Γd is the day angle:
Gd
2
p Htd - 1L
365
86
( 2.78 )
Schymanski (2007): PhD thesis
Chapter 3
CHAPTER 3. AN OPTIMALITY-BASED MODEL OF THE
DYNAMIC FEEDBACKS BETWEEN NATURAL
VEGETATION AND WATER BALANCE
3.1
ABSTRACT
The partitioning of rainfall into runoff, evaporation and transpiration in a catchment is
not only a function of the energy balance, but depends largely on the structure of the
catchment, soils and vegetation. However, it is a well known fact that the vegetation
properties relevant to the partitioning can change dynamically, some of them (e.g.
stomatal conductance) within minutes. The present study discusses transpiration from a
biological view point, according to which the loss of water through stomata is the
inevitable consequence of carbon acquisition by photosynthesis. Photosynthesis
provides plants with their main building material, carbohydrates, and with the energy
necessary to thrive and prosper in their environment. At the same time, water uptake
from the soil incurs construction and maintenance costs of a root system. Therefore we
expect that natural vegetation would have evolved an optimal water use strategy to
maximise its “Net Carbon Profit” (i.e. the difference between carbon acquired by
photosynthesis and carbon spent on maintenance of the organs involved in its uptake).
This hypothesis gave rise to a novel approach of modelling the interactions between
vegetation and the catchment water balance, which does not rely on prior knowledge
about the vegetation on a particular site.
The model is based on a multi-layered physically-based catchment water balance model,
an eco-physiological gas exchange and photosynthesis model and optimisation
algorithms to find those static and dynamic vegetation properties, which would
maximise “Net Carbon Profit” under the given conditions.
The model was tested on a savanna site near Howard Springs (Northern Territory,
Australia) by comparing the modelled fluxes and vegetation properties with long-term
observations on the site. Given only catchment properties, 30 years of meteorological
data and the above-noted optimality criteria, the model was able to reproduce the
observed rates of canopy transpiration and CO2 uptake, rooting depth and dynamics in
87
3.2 Introduction
Schymanski (2007): PhD thesis
vegetation cover surprisingly well. The results suggest that the model is a potentially
powerful tool for understanding and predicting vegetation response to long-term
climate- or land use change.
3.2
INTRODUCTION
The biosphere is characterised by a complex system of interfaces between solid, liquid
and gas phases. Processes across these interfaces follow physical laws, but the
arrangement of the interfaces determines the dominant processes at larger scales. This
complex and largely unpredictable structure renders modelling of processes at larger
scales very difficult and too often we have to resort to empirical extrapolations of
observed processes. For example, the partitioning of rainfall into runoff, evaporation
and transpiration in a catchment depends not only on the energy balance, but also on the
structure of the catchment, soils and vegetation. To make things even more complicated,
the vegetation properties relevant to the partitioning can change dynamically, some of
them (e.g. stomatal conductance) within minutes.
Current practice for ecological and hydrological modelling relies on long term
observations of the system in question and the derivation of correlations between
observed variables. However, many of the correlations are highly non-linear and often
complicated by hysteresis effects (e.g. Jarvis 1976). Furthermore, establishment of
correlations for a given set of conditions does not guarantee the validity of these
correlations for other conditions, which is particularly a problem in connection with
environmental change. Especially plants are known to adapt dynamically to their
environment, which makes extrapolation of past observations very uncertain.
An attempt to account for the adaptation of vegetation to its environment has resulted in
the development of Dynamic Global Vegetation Models (DGVM). Most of these
models trace their intellectual origin to Woodward (1987) and subdivide vegetation into
different “plant functional types” (PFT) with fixed properties like for example root to
shoot ratios, photosynthetic properties, stomatal behaviour and biogeographical
distributions. The PFTs defined for each biogeographical region can then be subjected
to climate scenarios calculated using General Circulation Models (GCM) and they grow
or decline as a result of their CO2 uptake to respiration ratios. This modelling approach
allows assessing the relative suitabilities of different PFTs for a given climate, but it still
88
Schymanski (2007): PhD thesis
Chapter 3
relies on a priori assumptions about some fixed properties of PFTs in a given region.
Over long time periods, however, it is likely that the existing PFTs (if the large variety
of plant responses to their environment can be realistically discretised into such PFTs at
all) would have adapted to the new conditions, or species of different PFTs would have
migrated between biogeographical regions.
The prediction of change in the water, carbon and nutrient cycles due to changes in
climate or land use is currently one of the most challenging and urgent issues in science
and policy. However, in order to make predictions into the future with some certainty,
we need models that do not rely on calibration using past measurements at a given site.
One challenging but promising way forward has been proposed by A. Bejan, who
formulated the “Constructal Law”, under which systems self-organise to offer the
minimal resistance to flow through them (e.g. Bejan 2000). This allows prediction of
some structural features if the flows are known. Another related approach to predict the
behaviour of complex systems is the “Maximum Entropy Production” (MEP) principle,
that has been used in coarse climate predictions (e.g. Ozawa et al. 2003), or global
trends in Gross Primary Production (e.g. Kleidon 2004). These approaches can be put
into the broad category of optimality-based approaches, as they seek the optimal
properties of a system, which would maximise or minimise a specified objective
function. Their big attraction is that if the objective function is generally valid, the
results would also be valid, provided the adjustable properties are free to adjust and all
relevant constraints have been considered. Herein also lies the biggest problem of the
optimality-based approaches; to use either of the above approaches, for example, we
need to know the degrees of freedom that can be optimised in order to minimise
resistance to flow or to maximise entropy production. Another complication in the
biosphere is that living organisms could have objective functions other than minimising
resistance to flow or maximising entropy production. In fact, most ecologically
motivated approaches to optimality (e.g. “Ecological Optimality” in Eagleson 1982;
Eagleson and Tellers 1982) formulate long-term net primary production (NPP) as the
objective function (Raupach 2005).
More recently, it has been shown that “Maximisation of Net Carbon Profit” can be a
very useful objective function to predict a range of properties of natural vegetation and
its resource use (Schymanski et al. in prep.-a). As in the case of the above approaches,
89
3.3 Methods
Schymanski (2007): PhD thesis
however, this “Vegetation Optimality” principle requires knowledge of the adjustable
parameters (degrees of freedom) of the system as well as constraints imposed by the
physiology of the organisms and by the physics of the adjacent domains (soil and
atmosphere). The authors simplified the problem by prescribing transpiration rates
based on measurements, thus limiting the degrees of freedom to foliage properties and
the physical considerations to above-ground processes only. This simplification was
very useful for eliminating the uncertainty of below-ground processes and for testing the
overall feasibility of the Vegetation Optimality approach. However, it is the same
below-ground processes that are of prime interest to hydrologists, as these processes
determine the soil moisture distribution and how much water is potentially available for
runoff.
The aim of the present study is to extend the existing Vegetation Optimality model to
include below-ground processes and to test whether this would enable realistic
predictions of dynamic transpiration rates in the long term, without prescribing any
local vegetation parameters. Such a model would represent a significant advance for
hydrological modelling, as it would allow incorporation of the adaptation of natural
vegetation to environmental conditions and hence increase our confidence in long-term
predictions of hydrological models.
3.3
3.3.1
METHODS
OVERALL FRAMEWORK FOR THE MODEL
The Vegetation Optimality approach is based on the assumption that natural vegetation
has co-evolved with its environment over a long period of time and that natural
selection has led to a species composition that is optimally adapted to the given
conditions. If this were true, the question arises, what would be the properties of such
optimal vegetation and how would it use the available resources?
The energy acquired through photosynthesis is stored in carbohydrates, which are vital
for plant fitness. Carbohydrates are both energy carriers and building materials for plant
organs. They can be used for many purposes, including seed production and the
maintenance of symbiotic relations with bacteria and fungi to mobilise nitrogen and
other nutrients from the soil or atmosphere. In addition, all living plant tissues
continuously consume energy to stay alive and require carbohydrates for their
90
Schymanski (2007): PhD thesis
Chapter 3
construction. Thus, part of the carbon acquired through photosynthesis has to be reinvested into the construction and maintenance of the organs involved in its uptake.
Only what is left over, the “Net Carbon Profit” (NCP), is assumed to be useful for
increasing a plant’s fitness. Hence we defined the optimal resource use strategy as the
one that maximises NCP.
The organs ultimately involved in carbon uptake are not just leaves, but also roots and
transport tissues, which supply the leaves with water and nutrients. For simplicity, the
costs related to nutrient uptake have been neglected in this model, as they are largely
unknown and nutrients can, to a certain extent, be recycled within plants. The
optimisation problem is then to maximise NCP by adjusting foliage properties and
stomatal conductivity dynamically, while adapting roots and transport tissues to meet
the variable demand for water by the canopy (Figure 3.1).
Atmosphere
Water
Water Transport & Storage
Carbon
Foliage
Net Carbon
Profit
Root System
Soil
Figure 3.1: Net Carbon Profit as the difference between carbon acquired by photosynthesis and the
carbon used for the construction and maintenance of organs necessary for its uptake. As CO2
uptake from the atmosphere is inevitably linked to loss of water from the leaves, the root system as
well as water transport and storage tissues are essential to support photosynthesis. Soil water
supply, atmospheric water demand and daily radiation constitute the environmental forcing.
Within those constraints, vegetation is assumed to optimise foliage, water transport and storage
tissues, roots and stomata dynamically to maximise its Net Carbon Profit.
The overall approach was introduced in a previous study (Schymanski et al. in prep.-a),
where it was tested by using it to predict foliage properties if water use and vegetation
cover were prescribed. The findings suggested that the “Maximisation of Net Carbon
Profit” leads to useful predictions in natural vegetation. However, the “Vegetation
Optimality” model as formulated in that study would be of limited use for hydrological
modelling, as transpiration was prescribed and below-ground processes were neglected.
91
3.3 Methods
Schymanski (2007): PhD thesis
The present study couples the existing Vegetation Optimality model with a water
balance model and a model for root water uptake. This eliminates the necessity of
prescribing water use or vegetation cover and might, therefore, allow modelling of
water use in the long term, based on abiotic forcing only.
However, the inclusion of below-ground processes requires optimisation of root
properties as well as canopy properties. Furthermore, modelling water use in the long
term requires dynamic optimisation of these properties, as many of them cannot be
expected to be stationary from month to month (e.g. seasonality in vegetation cover and
root distribution). Dynamic optimisation in connection with stochastically varying
control and state variables (e.g. rainfall) is a mathematical challenge in itself, which has
not yet been tackled in the Ecological Optimality context (Raupach 2005).
The complexity of the problem made it necessary to further simplify the canopy part, in
order to maintain computational feasibility. This was achieved by neglecting the vertical
distribution of light and photosynthetic capacity within the canopy and effectively
representing the canopy by means of two “big leaves”. One big leaf covering an
invariant area fraction (MA,p) represented perennial vegetation (trees) and another big
leaf covering a varying area fraction (MA,s) represented seasonal vegetation (grasses)
(Figure 3.2). The division of the canopy into two leaves in the present model is an
allowance for different dynamics and water use strategies by seasonal and perennial
plants. As the big leaves were not assumed to transmit any light, no overlap between
these two leaves was allowed, so that MA,s + MA,p ≤ 1.
The seasonal component of vegetation was allowed to vary in its spatial extent (MA,s),
but had only a limited rooting depth (yr,s = 1 m), while the perennial component covered
a fixed fraction of the surface (MA,p), but was not limited in the choice of its rooting
depth (yr,p). Rooting depths were assumed to be invariant in time, but the distribution of
roots within each root zone was allowed to vary on a day-by-day basis. The
photosynthetic capacity in each “big leaf” was also allowed to vary from day to day,
while stomatal conductivity in each “big leaf” was allowed to vary on an hourly scale.
92
Schymanski (2007): PhD thesis
Chapter 3
MA,p
yr,p
yr,s
MA,s
Figure 3.2: Representation of perennial (left) and seasonal (right) vegetation components. The
perennial vegetation component was assumed to be composed of evergreen trees, while the seasonal
component was assumed to be composed of annual grasses only. Thus, perennial vegetation was
allowed to optimise root depth (yr,p) without constraints, while the seasonal vegetation had a
prescribed root depth (yr,s) of 1 m only. At the same time, the fraction of the area taken up by
perennial vegetation (MA,p) was fixed, while the area fraction taken up by seasonal vegetation (MA,s)
was allowed to vary from day to day. For simplicity, deciduous trees were neglected in this study.
93
3.3 Methods
Schymanski (2007): PhD thesis
The adjustable controls in the model can be summarised as:
– Constant area fraction covered by trees (MA,p);
– Variable area fraction covered by grasses (MA,s);
– Variable photosynthetic capacity of trees;
– Variable photosynthetic capacity of grasses;
– Variable stomatal conductivity of trees;
– Variable stomatal conductivity of grasses;
– Constant rooting depth of trees;
– Variable distribution of tree roots in the soil profile; and
– Variable distribution of grass roots in the soil profile.
The maintenance of leaf area, photosynthetic capacity and roots had carbon costs
associated with it, while stomatal conductivity was constrained by the available soil
water.
To test the model, we modelled 30 years of vegetation dynamics and fluxes (1976-2005)
for a savanna site with very dynamic vegetation and a 5-year long record of canopyscale transpiration and CO2 flux data (Beringer et al. 2003; Hutley et al. 2005;
Schymanski et al. in prep.-a). Long-term meteorological data for this site was obtained
from the Queensland Department of Natural Resources, Mines and Water (SILO Data
Drill1), while soil and catchment properties were obtained from the literature.
Given meteorology and catchment properties, the adjustable controls mentioned above
were optimised to achieve a maximum in combined Net Carbon Profit between trees
and grasses over the 30 year period. The last 5 years of the modelled fluxes were then
compared with the available measurements, and other model outputs were compared
with observations in order to address the following questions:
1
http://www.nrm.qld.gov.au/silo
94
Schymanski (2007): PhD thesis
Chapter 3
– Can the observed magnitudes of both daily canopy CO2 uptake and transpiration
be reproduced by the calculated “optimal vegetation”?
– Is the observed seasonality of fluxes consistent with “Vegetation Optimality”?
– Can “Vegetation Optimality” reproduce the observed patchiness in tree cover
and seasonal variation in grass cover?
As no site-specific information about the vegetation was used in the model, we would
not expect the results to match observations very closely. However, even an
approximate correspondence in all three points would indicate that the objective
function and the constraints are likely important principles governing the adaptation of
natural vegetation to its environment.
3.3.2
WATER BALANCE MODEL
To account for the transfer of water between the atmosphere, unsaturated zone,
saturated zone and the river channel, we initially followed the “Representative
Elementary Watershed” (REW) approach formulated by Reggiani et al. (2000), but
extended it to allow the calculation of the vertical distribution of water within the
unsaturated zone and adjusted some of the closure relations to be consistent with our
formulation. For simplicity, interactions between elementary watersheds and stream
flow routing were neglected, so that all water reaching the channel was assumed to be
instantaneous runoff.
In the REW model, an elementary watershed is subdivided into two soil layers, the
saturated and unsaturated layer, both of variable thickness. The thickness of each zone
and all fluxes are spatially averaged in the REW model, so that the model can be
summarised with just the following few variables:
– average bedrock elevation from reference datum (zs)
– average channel elevation from reference datum (zr)
– average depth of the pedosphere (Z)
– average thickness of saturated zone (ys)
95
3.3 Methods
Schymanski (2007): PhD thesis
– average thickness of unsaturated zone (yu)
– average saturation degree in unsaturated zone (su)
– unsaturated surface area fraction (ωu)
– saturated surface area fraction contributing to seepage face and overland flow
(ωo)
wu ä Lh
E
Qrain Tu
wo ä Lh
ETs
yu
Qiex
Qsf
Z
ys
Qu
Qinf
zr
Lh
Datum
zs
Figure 3.3: Cross-section of a simplified elementary watershed. Variables on the left hand side
denote spatial dimensions (see text), while the variables on the right hand side denote water fluxes
(precipitation (Qrain), infiltration (Qinf), infiltration excess runoff (Qiex), evapo-transpiration from
the saturated zone and the unsaturated zone (ETs and ETu respectively), flow between saturated and
unsaturated layer (Qu) and outflow across the seepage face (Qsf)).
For simplicity, we set the datum to coincide with the average bedrock elevation, so that
zs = 0. Any fluxes into and out of the saturated zone (Qu, Qsf or ETs) lead to changes in
the thickness of both the saturated and the unsaturated zones, and also to changes in the
unsaturated and saturated surface fractions if ys>zr. The relationships between ys, yu, ωu
and ωo depend on the geometry of the catchment and are given for an unspecified
hypothetical catchment geometry in Reggiani et al. (2000). However, the equations
given in Reggiani et al. (2000) result in yu = const., as shown in A.3.2.1, where we
derived a corrected set of geometrical relations for linear hill slopes.
The unsaturated surface fraction (wu) and the thickness of the unsaturated zone (yu) were
both calculated as a function of the thickness of the saturated zone (ys):
96
Schymanski (2007): PhD thesis
Z- ys
Ø
≤
≤ "########################
HZ- ysL HZ- zrL
∞
≤
≤
±1
wu
and
yu
Chapter 3
µ
è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
HZ - ysL HZ - zrL
Z - ys
ys > zr
( 3.1 )
ys § zr
ys > zr
ys § zr
( 3.2 )
while the saturated surface area fraction (ws) is the complement of wu. The above
equations were then used to calculate ωo, ωu and yu for any given values of Z, zr and ys
in a linear hill slope.
3.3.2.1
Subdivision of the Unsaturated Zone
In the original REW model, the unsaturated zone was treated as a lumped volume, so
that changes in soil moisture due to rain would happen very slowly if the unsaturated
zone is reasonably thick. We know, however, that vegetation can react to pulses of rain
very quickly, so the model needed to be able to account for rapid wetting and drying of
the near-surface soil, where the majority of roots are active. This was achieved by
subdividing the unsaturated zone into several sub-layers and calculating soil moisture in
each of these layers separately.
Qinf↓
Q1↕
Q2↕
Qnlayers↕
i=1
i=2
i=nlayers
Figure 3.4: Conceptual catchment, with the unsaturated zone subdivided into three soil layers. Soil
layers are given indices (i = 1...nlayers) that increase with increasing depth (right). The indices
relating to fluxes refer to fluxes across the bottom boundary of the respective layer (left).
Starting from the top of the unsaturated zone, we divided the unsaturated zone into soil
layers of prescribed thickness dyumin, until we reached the groundwater table. The soil
layer adjacent to the water table was the only layer with differing thickness, which we
set to a value between dyumin and 2 dyumin, depending on the total thickness of the
unsaturated zone (yu). The number of soil layers in the unsaturated zone (nlayers) was
generally the integer value of yu/dyumin:
97
3.3 Methods
nlayers
3.3.2.2
hh y
u
hh
d d yumin
xx
xx
t
Schymanski (2007): PhD thesis
( 3.3 )
Soil Water Fluxes
Fluxes Between Soil Layers
Water fluxes between different soil layers were calculated using a discretisation of the
Buckingham-Darcy Equation (Radcliffe and Rasmussen 2002), which is the 1-D
equivalent to Richards’ equation for steady flow. All fluxes are averaged over the
catchment area and their derivations are given in Appendix A.3.2.2.
The flux between layer i and layer i+1 was calculated as:
Qi
hi+1 - hi
i
y
-wu 0.5 H Kunsat,i + Kunsat,i+1L jj
+ 1zz
k 0.5 Hdyu,i + dyu,i+1L {
( 3.4 )
where h the “matric suction head” and Kunsat the unsaturated hydraulic conductivity. The
subscript i denotes the ith layer, Qi denotes the flux across the bottom boundary of layer
i and dyu,i denotes the thickness of layer i. In the present work, h with units of pressure
head (m), is defined as positive and increases with decreasing soil saturation. Qi is
defined as positive if water flows upwards and negative if it flows downwards. For a
discussion of the meaning of h see Appendix 3.1.
In the saturated zone, the hydraulic conductivity is Ksat and the matric suction head (h)
is assumed to be 0, so that the flux across the boundary between the unsaturated and the
saturated zone was written as:
hi yz
i
z
Qnlayers -wu 0.5 HKunsat,i + KsatL jj1 k 0.5 dyu,i {
( 3.5 )
Infiltration
During rainfall, infiltration was assumed to only occur into the unsaturated zone and the
infiltration capacity was expressed by imagining an infinitely thin layer of water above
the top soil layer and writing the infiltration capacity as Qi for i = 0 , where Kunsat,i is
replaced by Ksat, hi by 0, and dyu,i by 0 in Equation ( 3.4 ). The rate of infiltration (Qinf)
was then formulated as the lesser of infiltration capacity and rainfall intensity (Qrain):
98
Schymanski (2007): PhD thesis
Q inf
Chapter 3
ii
h1
y
y
Minjj jj
+ 1zz K sat w u , Q rain w u zz
kk 0.5 d yu,1
{
{
( 3.6 )
Runoff
Rainfall exceeding Qinf and all rainfall falling onto the saturated area fraction were
assumed to contribute to immediate runoff (Qout). In the presence of a seepage face (i.e.
when ys>zr), flow across the seepage face also contributed to runoff. We calculated the
intensity of the seepage face flow following Reggiani et al. (2000):
Qsf
Ksat H y s - zrL wo
2 cosHg 0 L L s
( 3.7 )
The parameter g0 was described as the average slope angle of the seepage face, while Ls
is a typical horizontal length scale, which depends on the length of the hill slope. In the
absence of a more rigorous treatment of this parameter, we followed the approach by
Reggiani et al. (2000) and set the parameter value to 10.0 m, irrespective of the
geometry of the catchment.
The total runoff summed to:
Q out
- Q inf + Q rain + Q sf
( 3.8 )
Evaporative Fluxes
Figure 3.3 included two other fluxes: the evapo-transpiration from the unsaturated zone
(ETu) and the evapo-transpiration from the saturated zone (ETs). For the layered model
(Figure 3.4), we have to distinguish between soil evaporation (Esu and Ess for the
unsaturated and the saturated zones respectively), which happens at the soil-air interface
only, and transpiration by vegetation, which is linked to root water uptake (Qr,i) from all
layers within the rooting zone. Root water uptake will be described in Section 3.3.3.3,
as it is part of the vegetation optimality model, so we will only formulate equations for
soil evaporation here.
The climate data set used for the simulations did not contain soil temperature
information, so soil evaporation could not be modelled as a function of soil temperature.
Instead, soil evaporation was modelled as a linear function of surface soil water
saturation (su,1), global irradiance (Ig) and irradiated soil surface fraction (1-MA). Solar
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3.3 Methods
Schymanski (2007): PhD thesis
irradiance was converted to an evaporation equivalent using the latent heat of
vaporisation (λE), taken as 2.45ä106 J kg-1 and the density of water (ρ), taken as 1000 kg
m-3. However, soil evaporation is expected to occur even if no direct radiation reaches
the soil surface, so the amount of energy available for evaporation was arbitrarily set to
Ig (1 - 0.8 (1-MA)), leading to the following equations for soil evaporation from the
unsaturated zone (Esu) and from the saturated zone (Ess):
Ig H1 - 0.8 H1 - MALL wu su,1
Esu
lE r
Ess
3.3.2.3
Ig H1 - 0.8 H1 - MALL wo
lE r
( 3.9 )
( 3.10 )
Matric Suction Head and Hydraulic Conductivity
Kunsat and h in each layer are considered to be functions of soil properties and water
saturation degree (su), but there are many, mainly empirical, formulations of these
functions in literature. The formulations by Brooks and Corey (1966) and van
Genuchten (1980) are among the most widely used for modelling. However, we found
that the Brooks and Corey formulation gave a positive matric head at full saturation,
which would cause a fully saturated soil layer to keep taking up water from its
surroundings.
Because of a more realistic representation of matric pressure near full soil saturation, we
chose the van Genuchten formulation, which is described briefly in Appendix A.3.2.3.
Following this formulation, the soil saturation degree (su) is written as a function of
volumetric water content (θ) and the empirical soil properties θr and θs:
su
q -qr
qs -qr
( 3.11 )
The matric suction head h was then written as a function of su and the empirical soil
properties αvG, nvG and mvG:
1
1 ijj - mvG yzz nvG
- 1zz
jsu
avG j
k
{
1
h
100
( 3.12 )
Schymanski (2007): PhD thesis
Chapter 3
where nvG and mvG are assumed to follow the relation:
mvG
1-
1
n vG
( 3.13 )
Then, the unsaturated hydraulic conductivity (Kunsat) was computed as a function of the
saturated hydraulic conductivity (Ksat) and the above values:
K unsat
1 mvGy2
mvG yzz
z
è!!!!! ijj ijj
Ksat su jj1 - jj1 - su zz zzz
j j
z z
k k
{ {
( 3.14 )
To parameterise the above equations, values of the parameters Ksat, θr, θs, αvG, and mvG
have to be obtained from soil physical measurements, or “typical” values for the
prevalent soil type can be obtained from literature.
3.3.2.4
Conservation of Mass and Changes in State Variables
Changes in the state variables su,i, dyu,i, ωo, ωu, ys and yu due to water fluxes must satisfy
conservation of mass. Ignoring density variations due to changes in temperature, we
replaced the mass of water per square meter of area with the volume of water per square
meter of area, so that the units of mass of water per unit area become m3/m2 = m, which
is consistent with water flux units of m3 m-2 s-1 = m/s.
The derivations of the following equations can be found in Appendix A.3.2.4.
Downwards flux of water into the saturated zone would cause it to expand into the
unsaturated zone, while an upward flux would cause it to contract and leave an
unsaturated volume behind. The change in the thickness of the saturated zone was thus
expressed as a function of the fluxes in and out of the saturated zone and the average
saturation of the unsaturated zone:
∑ ys Ht L
∑t
EssHt L + QnlayersHt L + Qsf Ht L
¶ HsuHt L - 1L
( 3.15 )
where Ess is the soil evaporation rate from the saturated zone, Qnlayers is the flux across
the bottom boundary of the unsaturated zone, Qsf is outflow across the seepage face and
su is the average saturation degree in the unsaturated zone.
The change in soil moisture was written for the top soil layer as:
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3.3 Methods
Schymanski (2007): PhD thesis
∑ su,1 HtL
-EsuHtL + Q1HtL + Qinf HtL - Qr,1 HtL
∑t
¶ wu dyu,1
( 3.16 )
and for the soil layers between the top and the bottom layer:
∑ s u,i H tL
- Q i- 1 H tL + Q iH tL - Q r,1 H tL
¶ w u d yu,i
∑t
( 3.17 )
To calculate the change in the state variables for a finite time step from t1 to t2, the
above equations were solved for t1 and then multiplied by the length of the time step.
This gave ys and su,i at time t2 for all layers apart from the bottom layer of the
unsaturated zone. The saturation degree in the bottom layer at time t2 was then
calculated by difference, from the fluxes in and out of the whole soil domain and the
change in water storage in all other compartments.
The water storage in each soil layer was written as
ws,i
¶ wu su,i dyu,i
( 3.18 )
and the sum of fluxes in and out of the soil domain between time t1 and time t2 was
written as
layers
ij
yz
j
Ht2 - t1L jjQinf Ht1L - ‚ HQr,i Ht1LL - EssHt1L - EsuHt1L - Qsf Ht1Lzzz
i=1
k
{
n
dwc
( 3.19 )
where dwc denotes the change in the total water store of the soil domain per unit
catchment area.
The water storage in the bottom soil layer at time t2 (ws,nlayers(t2)) was calculated as
ws,nlayersHt2L
dwc + ‚ ws,i Ht1L - ‚ ws,i Ht2L + ¶ ysHt1L - ¶ ysHt2L
nlayers
i=1
nlayers-1
( 3.20 )
i=1
The value of su,nlayers at time t2 was then obtained from ws,nlayers:
su,nlayers Ht2 L
ws,nlayers Ht2 L
¶ w u Ht2 L d yu,nlayers Ht2 L
102
( 3.21 )
Schymanski (2007): PhD thesis
3.3.3
Chapter 3
VEGETATION OPTIMALITY MODEL
Vegetation was subdivided into a perennial component (assumed to consist of trees) and
a seasonal component (assumed to consist of annual grasses). Both were treated
identically in terms of physiology (both were treated as C3 species), but had different
structural degrees of freedom. The rooting depth of grasses was fixed at 1 m, while the
rooting depth of trees was an optimised constant that could take on any value. On the
other hand, the projected cover of grasses was optimised dynamically from day to day,
while the projected cover of trees was optimised but constant over time.
In this section, we will explain the physiological model that couples light absorption,
CO2 uptake and transpiration, before introducing a model for root water uptake that
imposes soil moisture constraints on water use. Root water uptake by trees was linked to
a tree water storage model, which also allowed hydraulic redistribution in the model.
After quantifying the carbon costs of roots, foliage and water transport tissues, we will
explain how the optimisation of the vegetation was performed.
3.3.3.1
Photosynthesis Model
CO2-assimilation was calculated following a physiological canopy gas exchange model
described elsewhere (Schymanski et al. in prep.-a), simplifying the canopy as two “big
leaves” with one “big leaf” covering the area fraction MA,p for perennial vegetation
(trees) and the second “big leaf” covering the area fraction MA,s for seasonal vegetation
(grasses). We assumed no light transmission through the big leaves, so that the two big
leaves were not allowed to overlap, and the combined MA of trees and grasses could not
be larger than 1.0. The gas exchange of trees and grasses was then modelled
independently as a function of the electron transport rates and stomatal conductivities of
trees and grasses respectively.
The functional dependence of CO2 uptake rate (Ag) on photosynthetic active irradiance
(Ia), electron transport capacity (Jmax), stomatal conductivity (Gs), air temperature (Ta)
and the mole fraction of CO2 in the air (Ca, in mol/mol) has been formulated elsewhere
(Schymanski et al. in prep.-a) and will only be summarised here:
103
3.3 Methods
Ag
Schymanski (2007): PhD thesis
1
J4 Ca Gs + 8 G* Gs + Je - 4 Rl +
8
"###########################################################################################################
#
HJe - 4 Rl - 4 Gs HCa - 2 G*LL2 + 16 Gs H8 Ca Gs + Je + 8 Rl L G* N
( 3.22 )
The electron transport rate (Je) is a function of irradiance (Ia), the big leaf’s electron
transport capacity (Jmax) and the size of the leaf (MA):
Je
a Ia y
ij
jj1 - ‰ Jmax zzz Jmax M A
j
z
k
{
( 3.23 )
while the leaf respiration rate (Rl) is determined by Jmax:
Rl
cRl Jmax HCa - G* L
8 HCa + 2 G* L
( 3.24 )
The parameter cRl is a constant that has been given the value of 0.07, following an
empirical relationship between photosynthetic capacity and leaf respiration (Givnish
1988; Schymanski et al. in prep.-a).
The CO2 compensation point (Γ*, in mol/mol) is fairly constant between most plant
species, but varies with temperature following an Arrhenius-type function:
G*
126.946 H Ta-25L
0.00004275 ‰ Rmol H Ta +273L
( 3.25 )
where Rmol is the universal gas constant, and Ta is air temperature, which we assumed to
represent leaf temperature.
Jmax,p also varies with temperature, following an equation proposed by Medlyn et al.
(2002), which has been altered to express Jmax,p at a given temperature as a function of
Jmax at the reference temperature of 25˚ C (Jmax25,p) (Schymanski et al. in prep.-a):
ijij - Hd HTopt-298L
yz
yz
jjjj 298 R mol Topt
zz
z
- 1zz H a + H dzzz
jj ‰
jj
kk
{
{
H
H
T
T
L
ij d a opt
yz
jj R mol Ta Topt
z
- 1zzz H a + H d
jj‰
k
{
Ha HTa - 298L
‰ 298 R mol Ta
Jmax
Jmax25
( 3.26 )
where we used parameter values derived for Eucalyptus pauciflora, as presented in
Medlyn et al. (2002).
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Schymanski (2007): PhD thesis
Chapter 3
Equations ( 3.22 ) to ( 3.26 ) were equally applied to grasses and trees, with different
values of MA, Jmax25, and Gs. The total vegetation CO2 uptake rate (Ag,tot) was calculated
as the sum of CO2 uptake by trees (Ag,p) and by grasses (Ag,s):
Ag, p + Ag,s
Ag,tot
3.3.3.2
( 3.27 )
Stomatal Conductivity and Transpiration Rate
Transpiration (Et) was modelled as a diffusive process, where the stomatal conductivity
for CO2 (Gs) was multiplied by a constant (a = 1.6) to account for the different
diffusivity of water vapour and CO2 in air (Cowan and Farquhar 1977):
Et
a Gs HWl - WaL
( 3.28 )
Wl and Wa denote the mole fraction of water vapour in air inside the leaf and in the
atmosphere respectively.
The mole fraction of water vapour in air can be estimated by dividing the partial vapour
pressure (pva) by air pressure (Pa). Assuming that the air space inside the leaves is
saturated (pvsat) at air temperature (Ta), we can replace the term (Wl - Wa) by the molar
vapour deficit Dv:
Wl - Wa º
3.3.3.3
pvsat - pva
P
Dv
( 3.29 )
Root Water Uptake
Root water uptake was modelled using an electrical circuit analogy, where radial root
resistivity and soil resistivity are in series in each soil sub-layer (Hunt et al. 1991).
Water uptake per unit root surface area in a soil layer (Jr,i) was thus written as:
Jr ,i
hr ,i - hi
Wr + Ws ,i
( 3.30 )
where Ωr is root resistivity to water uptake per unit root surface area (assumed to have
the same value in all soil layers), and Ωs,i is the resistivity to water flow towards the
roots in the soil. The driving force for water uptake by roots is the difference between
the forces holding the water in the soil (hi) and the forces holding the water in the roots
(hr,i). Defining SAr,i as the root surface area per m2 ground area in layer i, we can write
the root water uptake rate in layer i as:
105
3.3 Methods
Schymanski (2007): PhD thesis
Qr,i
i hr,i - hi yz
SAr,i j
k Wr + Ws,i {
( 3.31 )
The resistivity to water flow towards the roots in the soil (Ωs,i) was formulated as a
function of unsaturated hydraulic conductivity (Kunsat,i), root radius (rr) and root surface
area density in soil layer i (SAdr,i):
Ws,i
1
Kunsat,i
$%%%%%%%%%%%%%%%%%%%
p rr
2SAdr,i
( 3.32 )
Equation ( 3.32 ) has the desired properties, that Ωs,i decreases with increasing Kunsat,i
and decreasing distance between roots (represented by the second term). For a
derivation of Equation ( 3.32 ) see Appendix A.3.3.1.
3.3.3.4
Water Storage and Tissue Balance Pressure
In the above, root water uptake was modelled as a function of root- and soil properties
and the suction head gradient between the soil and the inside of the roots. The suction
head inside the roots is often considered to be linked to the suction head in leaves,
which is caused by adhesive forces and is driven by transpiration. Thus, water transport
from the soil to the leaves could happen passively, without the expenditure of energy
other than the maintenance of the plant tissues involved. In Appendix A.3.3.2, we
developed a model to quantify the forces involved in such a passive process. The model
was based on measurements of tissue balance pressure (Pb, the pressure that has to be
applied in order to force water out of the tissue) as a function of tissue water content
(Mq):
i 750 Md
1 yz
z
Pb HMqx - MqL jj
+
k HMd + MqxL2 Mqx {
( 3.33 )
where Pb is the tissue balance pressure in bars, Mqx and Mq are the potential and actual
amount of water stored in plant tissues per unit catchment area respectively, and Md is
the total mass of dry matter associated with living tissues per unit catchment area.
If the tissue balance pressure is assumed to represent the suction force exerted by the
tissue, Equation ( 3.33 ) implies that the suction force increases as the tissue water
content decreases. However, Pb can only increase until Mq reaches a value of 0.9 Mqx,
106
Schymanski (2007): PhD thesis
Chapter 3
because any further decrease in water content is assumed to lead to tissue damage (see
Appendix A.3.3.2).
In order to use the balance pressure in plant organs above ground (Pb) to drive passive
water uptake by roots, Pb was translated into the root suction head hr,i by taking into
account the hydrostatic head between roots and trunks:
cPbm Pb - hh,i
hr,i
( 3.34 )
where hh,i is the hydrostatic head difference between the soil surface and the depth of
layer i, while cPbm = 10.2 m bar-1 is a conversion coefficient to convert from units of Pb
(bar) to units of hr,i (m). The height of the canopy was not considered in the calculation
of hh,i, as the model did not include any information about tree heights.
Equation ( 3.33 ) was applied to compute Pb for the perennial part of vegetation, which
made it necessary to compute Mq in trees as a function of time, as described below.
Grasses were not expected to have any considerable water storage capacity (Mq), so Pb
was set to a constant 15 bars in Equation ( 3.34 ) to enable the calculation of root
suction hr,i.
Water Storage and Limits on Transpiration
While it is clear from the above that the value of Mq is important for calculating water
uptake rates by roots in the present model, it was also postulated that Mq should not
decrease below 90% of its maximum value, Mqx. The rate of change in Mq can be
written as a function of root water uptake and transpiration rate:
∑ Mq Ht L
∑t
‚ Qr, p,i Ht L - Et, pHt L
i r, p
( 3.35 )
i =1
where Qr,p,i is the water uptake rate by tree roots in soil layer i and ir,p is the deepest soil
layer accessed by tree roots. This layer is obtained by dividing rooting depth (yr,p) by the
minimum thickness of the soil layers (δyumin) and rounding to the next higher integer:
ir,p
y p
h̀h r,p xx
hh dyumin xx
( 3.36 )
To hold Mq above 0.9 Mqx, Et,p could not exceed root water uptake when Mq = 0.9 Mqx.
This was set as a condition for root-induced stomatal closure in the model. However, if
107
3.3 Methods
Schymanski (2007): PhD thesis
hr,i < hi, as could be the case for very dry soil, Qr,i could be negative (Equation ( 3.31 )),
and Mq could decrease even if stomata are fully closed. If such a situation caused Mq to
decrease below 0.9 Mqx in the model, the modelled vegetation was deemed unsuited for
the given environment and the computation was aborted.
A large tree water storage capacity (Mqx) can act as a buffer for meeting peak foliage
water demands that exceed root water uptake rates during the day in the present model.
In contrast, the water storage capacity of grasses is generally negligible, so that grass
transpiration rates Et,s were not allowed to exceed total grass root water uptake at any
instant in time.
Cernusak et al. (2006) measured specific leaf area at the study site as 5.5 m2/kg, so with
a leaf area index of 0.7 in the dry season, this would result in 0.127 kg of dry matter in
the foliage per m2 catchment area. The same authors also estimated the total aboveground volume of sapwood on the site to be 0.0032 m3 per m2 catchment area and
observed mean values of sapwood density ranging between species from 0.81 to 0.94 g
cm-3. In rough terms, this would give an estimate of 3000 g sapwood dry matter per m2
catchment area, or, using the estimate that 30% of the area was covered by trees, the
amount of sapwood dry matter per m2 area covered by trees was 10,000 g. Based on
these considerations, we expressed sapwood dry matter per m2 catchment area (Md) as a
function of the area fraction covered by trees (MA,p):
Md = 10 kg m-2 MA,p
( 3.37 )
Knowing that the vegetation on the site experiences prolonged drought periods, we set
qx to 0.5, which would support the largest values of Pb in the present model. This
implies that Mq = Md.
3.3.3.5
Carbon Costs Related to Plant Functions
The leaf respiration rate (Rl) was formulated earlier as a function of the leaf’s
photosynthetic capacity (Equation ( 3.24 )). This formulation implies that the
maintenance of infrastructure to perform photosynthesis has an associated cost, which
can be expressed in the expenditure of carbon for respiration. More generally,
respiration can be seen as a result of active processes, which include the synthesis of
proteins and structural dry matter, but also nutrient conversions, ion transport or phloem
108
Schymanski (2007): PhD thesis
Chapter 3
loading (Dewar 2000). What is generally referred to as “maintenance respiration” could
be thought of as the respiration resulting from the active processes necessary to make up
for natural decay. Extending this further, we will formulate “maintenance costs” as the
carbon that has to be invested to negate the decay rates of certain structures. These can
also include the carbon that is for example lost due to litter fall.
Foliage Turnover Costs
It is obvious from Equation ( 3.23 ) that a larger vegetated fraction (MA) leads to larger
electron transport rates (Je), which would result in larger CO2 uptake rates. However,
leaves have limited life times, so that the plants need to invest carbon into the
replacement of fallen leaves. It has been shown that these turnover costs can be
estimated to be equivalent to an average carbon investment of 0.22 µmol s-1 per m2 leaf
area (Schymanski et al. in prep.-a). To relate MA to leaf area, we assumed that the
clumped leaf area index within vegetated patches is 2.5 (Schymanski et al. in prep.-a),
so the carbon costs related to MA were approximated as:
Rf = 5.5ä10-7 mol s-1 m-2 MA
( 3.38 )
In other words, if MA is seen as the leaf area of the “big leaf”, the maintenance of this
leaf area is assumed to be 2.5 times more expensive than the maintenance of small
leaves with the same cumulative leaf area.
Root Maintenance Costs
As in the case of leaves, root longevity seems to be generally related to tissue density.
However, the small range of data currently available does not allow us to model root life
span as a function of particular root properties (Eissenstat and Yanai 1997; Eissenstat et
al. 2000).
We assumed that there would be a relation between the vegetation’s capacity to extract
water from the soil and the amount of carbon that has to be invested in the root system.
However, such a general relationship was difficult to obtain from literature, as
respiration rates, root hydraulic properties and turnover rates are rarely measured on the
same plants, while all of these are highly variable, not only between species, but even
within the same root over time (Steudle 2000).
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3.3 Methods
Schymanski (2007): PhD thesis
In the absence of a general relationship between root costs and their water uptake
capacity, we used published measurements on citrus roots, for which observations of
both respiration rates and hydraulic properties were available in the literature. Huang
and Eissenstat (2000) measured radial conductivity in different citrus species and found
hydraulic conductivities per m2 root area of 1 to 3 µm/s/MPa/m2 in first-order lateral
roots (0.34 to 0.44 mm diameter). In second-order lateral roots (0.58 to 0.87 mm
diameter), they found values of 0.2 to 0.75 µm/s/MPa/m2. Using 1 µm/s/MPa/m2 as a
typical value for radial root conductivity per root area, we set the root resistivity to
water uptake to:
1.01937 µ 108 s
Wr
( 3.39 )
Bryla et al. (2001) gave values of root respiration for a single citrus fine root on a dry
weight (DW) basis in the order of 10 nmol/(g DW)/s. The average fine root diameter of
the measured roots was 0.6 mm. Eissenstat (1991) gave values for the dry mass to
volume relationships of different citrus roots between 0.15 and 0.2 g/cm3. Taking
0.17 g/cm3 as a typical value, 1 m3 root volume should have 0.17ä106 g dry weight.
Consequently the respiration rate for 1 m3 of fine roots would be 0.0017 mol/s.
Assuming cylindrical roots, we get root respiration (Rr) as a function of root radius (rr)
and root surface area (SAr):
Rr
cRrJ
rr
SAr N
2
( 3.40 )
where, following the above, cRr = 0.0017 mol s-1 m-3 and rr = 0.3ä10-3 m
In any single root layer, the root respiration rate per unit catchment area was modelled
as a function of root surface area density in that layer (SAdr,i) and the volume of the soil
layer i per unit catchment area:
Rr,i
cRrJ
rr
SAdr,i dyu,i wuN
2
( 3.41 )
Water Transport Costs
Water that is taken up by fine roots needs to be transported to leaves where it gets
transpired. The vascular system required for this transport is most obvious in the stems
and branches of trees. An equivalent structure is needed below ground, so that we
expect that deeper root systems and larger crown areas require larger vascular systems
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Chapter 3
(Figure 3.5). These structures are expected to have specific decay rates, but unlike in the
case of leaf area, a relationship between size and carbon costs could not be derived from
the literature. Therefore, we simply assumed that the carbon costs related to the
maintenance of the vascular system are a linear function of rooting depth and the
horizontal extent of the vegetation:
Rv = crv MA yr
( 3.42 )
where crv is an unknown proportionality constant, which, at this stage, was set to an
arbitrary value that would lead to realistic model results.
yr,p
Figure 3.5: Conceptual dependence of the size of the vascular system on the rooting depth and
horizontal cover. Deep rooting trees with rooting depth yr,p need larger vascular systems per unit
horizontal cover than shallow rooting annuals.
3.3.3.6
Degrees of Freedom (Adjustable Variables)
Vegetation has a large number of degrees of freedom to enable it to adapt to the
environment (e.g. leaf area index, canopy structure, photosynthetic capacity, stomatal
conductance, species combination, root properties and distribution, etc.). The time
scales of adaptation range between minutes (stomatal conductance) to decades (species
composition). As mentioned earlier, we subdivided vegetation into two components:
perennial trees and seasonal grasses. The above-ground part of each component was
represented by a big leaf of adjustable horizontal extent (MA), with adjustable electron
transport capacity (Jmax25) and stomatal conductivity (Gs). The below-ground part of
each component was represented by a vertical distribution of root surface area into the
different soil layers, as defined in the water balance model. As root abundance in the
soil can be very dynamic and can follow the distribution of soil water (Schenk 2005),
root surface area in each soil layer was allowed to vary from day to day. Similarly, the
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Schymanski (2007): PhD thesis
horizontal extent of the seasonal vegetation (MA,s) was allowed to vary from day to day,
while the horizontal extent (MA,p) and average thickness of the root zone (yr,p) of the
perennial vegetation were assumed to be constant over the period of investigation
(30 years). The adjustable variables and their time scales of variation are summarised in
Table 3.1.
Table 3.1: Degrees of freedom (adjustable variables) and time scales of variation
ADJUSTABLE VARIABLE
TIME SCALE OF VARIATION
Fraction of area covered by perennial vegetation
(MA,p)
constant over 30 years
Fraction of area covered by seasonal vegetation
(MA,s)
varying on daily scale
Electron transport capacity of perennial vegetation
(Jmax25,p)
varying on daily scale
Electron transport capacity of seasonal vegetation
(Jmax25,s)
varying on daily scale
Stomatal conductance of perennial vegetation (Gs,p)
varying on hourly scale
Stomatal conductance of seasonal vegetation (Gs,s)
varying on hourly scale
Thickness of root zone of perennial vegetation (yr,p)
constant over 30 years
Root surface area density of perennial vegetation in
each soil layer (SAdr,i,p)
varying on daily scale
Root surface area density of seasonal vegetation in
each soil layer (SAdr,i,s)
varying on daily scale
3.3.3.7
Objective Function
Following the concept of Vegetation Optimality, as described in Section 3.3.1, the
objective function for the optimisation was taken as the maximisation of “Net Carbon
Profit” (NCP) of the total vegetation. The Net Carbon Profit was defined as total CO2uptake of trees and grasses over the entire period, minus all identified maintenance costs
of the organs assisting photosynthesis, including foliage, roots and water transport
tissues.
NCP
‡
tend
tstart
H Ag,totHtL - Rf HtL - RrHtL - RvHtLL „t
( 3.43 )
where Ag,tot is the combined CO2 uptake by trees and grasses (given in Equation
( 3.27 )), Rf stands for the foliage costs of grasses and trees combined, Rr are the root
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Chapter 3
costs of trees and grasses summed over all soil layers, and Rv are the costs associated
with the vascular systems of trees and grasses combined.
3.3.3.8
Optimisation Strategy
In this study, it was assumed that the adaptation of vegetation to the environment was
driven by the maximisation of “Net Carbon Profit” (NCP). The optimisable vegetation
properties (or “degrees of freedom”) were divided into properties that are adapted in the
short-term and respond to the day-to-day changes in environmental conditions, and
those properties that are assumed to be adapted to the long-term environmental
conditions on the site. Examples for short-term adaptation are the biochemical foliage
properties of trees and grasses, grass projected cover and root distributions of trees and
grasses. These were optimised each day, based on the conditions of the previous day
(e.g. increase in Jmax if a higher value would have led to a larger daily NCP on the
previous day). Examples for long-term adaptation are the projected cover and rooting
depth of trees, and the water use strategies of trees and grasses.
The long-term adaptation was simulated by running the model over a period of 30 years
and searching for the optimal vegetation properties that would maximise NCP over that
period. In order to make the optimisation problem computationally feasible, the
vegetation properties to be optimised over the long-term were represented by a small
number of invariant parameters. The number of optimisable parameters had to be kept
small, because with each degree of freedom the complexity of the problem and
associated computing time increases. We used the Shuffled Complex Evolution (SCE)
algorithm (Duan et al. 1993; Duan et al. 1994) to find the optimal values for the
invariant parameters that would maximise NCP over 30 years. The algorithm is based on
a synthesis of several concepts developed for global optimisation, and basically
performs a search of the parameter space for the optimal parameter values that would
satisfy the objective function. The algorithm and its implementation are described in
more detail in Appendix 3.5.
Below, we will describe which vegetation properties were optimised for the long-term
and which were optimised day-by-day.
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Schymanski (2007): PhD thesis
Long-Term Optimisation: MA,p, yr,p and Water Use
As mentioned previously, the area fraction covered by trees (MA,p) and the rooting depth
(yr,p) of the trees were considered constant over 30 years, and their values were
optimised as part of the SCE optimisation.
Water use (transpiration) can be limited either by stomatal conductance or by root water
uptake. However, for achieving maximum Net Carbon Profit with a limited amount of
water, transpiration should be controlled by stomata in such a way that the slope
between CO2 uptake and transpiration is maintained as constant during a day (Cowan
and Farquhar 1977; Cowan 1982; Cowan 1986; Schymanski et al. in prep.-a). This
slope will be called λs and λp for grasses and trees respectively. Over longer time
periods, the parameters λs and λp should be sensitive to the availability of soil water and
this sensitivity could be seen as a plant physiological response shaped by evolution to
suit a given environment
(Cowan and Farquhar 1977). In the present study, the
sensitivity of λs and λp to soil water has been parameterised as:
ls
and
lp
ij ir,s yzcle,s
clf,s jjj‚ hi zzz
k i=1 {
ij i r,p yzcle,p
clf,p jjj‚ hi zzz
j
z
k i =1 {
( 3.44 )
( 3.45 )
where ir,s and ir,p denote the deepest soil layer accessed by roots of grasses and trees
respectively. The calculation of ir,p was given in Equation ( 3.36 ) and ir,s can be
calculated similarly by dividing the rooting depth (yr,s) by the minimum thickness of the
soil layers (δyumin) and rounding to the next higher integer:
ir,s
y p
h̀h r,s xx
hh dyumin xx
( 3.46 )
Equations ( 3.44 ) and ( 3.45 ) are aimed at allowing the largest possible flexibility for
the relationship between soil water and λs or λp while using the least possible number of
adjustable parameters. The above formulations include only two adjustable parameters
for each relationship (cλf and cλe) and allow for relationships ranging from constant λ
(cλe = 0) through to linear relationships (cλe = 1.0) and to highly non-linear relationships
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Chapter 3
(cλe >> 1.0 or cλe << 0). Soil water was represented by the suction head in the rooting
zone, because we expected that plants would more likely be able to “sense” the suction
head than the total amount of water available in their rooting zones.
In summary, long-term adaptation of vegetation to the environment was modelled by
the optimisation of six parameters: MA,p, yr,p, cλf,p, cλe,p, cλf,s and cλe,s.
Daily Optimisation: MA,s, Jmax25, and Root Distributions
The fraction occupied by seasonal vegetation (MA,s) and the electron transport capacities
of seasonal and perennial vegetation (Jmax25,s, Jmax25,p) can change dynamically and have
a direct impact on photosynthetic rates (Ag,s, Ag,p) and daily Net Carbon Profit (NCPd).
Their dynamic adaptation to the environment was modelled on a daily scale, by
computing NCPd for each day using their actual values and alternative values taken as a
specified increment higher and lower than the actual values. The values for MA,s, Jmax25,s
and Jmax25,p on the subsequent day were then set to the combination of values that would
have led to the maximum NCPd on the previous day. The daily increment had to be small
enough to prevent oscillation between two highly non-optimal states under stable
environmental conditions and large enough to allow a quick enough adaptation to
seasonal changes in environmental conditions. This was achieved by setting the daily
increment for MA,s to 0.02, while the daily increment for Jmax25,s and Jmax25,p was set to
1% of their actual values.
The values of λs and λp, in combination with the above parameters and meteorological
conditions, determine the canopy water demands by grasses and trees respectively.
These water demands have to be met by root water uptake, so that the optimisation
problem for the root system is the minimisation of costs while meeting the water
demand by the canopy. The optimisation of the root systems was performed on a daily
scale and involved two steps. The first step was to determine whether the actual root
surface area was more or less than adequate to meet the water demand during the
previous day. For trees, this was performed by recording the minimum value of the
tissue water store (Mq) during the previous day (Mqmin), which was then used to compute
a coefficient of change for the root system (kr,p):
kr, p
0.95 Mqx- Mqmin
0.05 Mqx
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( 3.47 )
3.3 Methods
Schymanski (2007): PhD thesis
Mqmin can range between 0.9 Mqx if root water uptake did not meet the canopy water
demand and Mqx if canopy water demand did not deplete Mq at all, while the values of
kr,p range between 1 for the first case and -1 for the latter case. For grasses, the value of
kr,s was simply set to 1 if Et,s was limited by root water uptake at least once during the
previous day or to -1 if it was not limited by root water uptake.
The second step was to determine the relative effectiveness of roots in different soil
layers (kreff,p,i) on the previous day. This was performed for trees by dividing the daily
water uptake per unit root surface area in each soil layer (Jrdaily,p,i) by the maximum
daily water uptake per unit root surface area in the whole soil profile.
kreff , p,i
0.5 Jrdaily, p,i
MaxH Jrdaily, p,i L
( 3.48 )
This step was omitted for grass roots, as their root systems were only shallow. The
change in root surface area density (SAdr) for grass roots from day to day was calculated
as:
dSAdr,s,i
SAdr,s,i H0.9 + 0.2 kr,s L
( 3.49 )
For tree roots, the change in root surface area in each soil layer from day to day was
computed as a function of kr,p and kreff,p,i:
dSAdr, p,i
SAr, p,i k reff , p,i k r, p Grmax
( 3.50 )
where Grmax is the maximum daily increment in root surface area in a soil layer, set to
10%.
3.3.4
STUDY SITE
The model presented in this study optimises vegetation for given environmental
conditions, which consist of long-term meteorological data (solar irradiance, vapour
pressure, temperature, rainfall) and catchment properties (catchment geometry and soil
type). To compare model results with observations, we chose a test-site for which longterm observations of vegetation and canopy-scale water and CO2 fluxes were available.
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3.3.4.1
Chapter 3
Location
The site chosen for the present study is located in the Northern Territory of Australia, 35
km South-East of Darwin, near Howard Springs in the Howard River catchment. A flux
tower recording meteorological data and fluxes of CO2 and water vapour was located at
12°29'39.30"S, 131° 9'8.58"E.
3.3.4.2
Climate
Due to the site’s proximity to the Northern coast of Australia, the climate is sub-humid
on an annual basis (1750 mm mean annual rainfall vs. 2300 mm mean annual class-A
pan evaporation), but with a very strong monsoonal seasonality. 95% of the 1750 mm
mean annual rainfall is restricted to the wet season (December to March, inclusive),
while the dry season (May to September) is literally dry (Hutley et al. 2000). Thus, the
availability of water is highest when atmospheric water demand is low (day-time
relative humidity >60% during the wet season), and lowest when the atmospheric water
demand is high (day-time relative humidity 10-40% during the dry season). Air
temperatures range between roughly 25 and 35 ˚C in the wet season and between 15 and
30 ˚C in the dry season.
Daily shortwave radiation during the dry season is between 15 and 25 MJ/m2/d, while
during the wet season it varies between 5 and 30 MJ/m2/d. The larger day-to-day
variation during the wet season is due to the higher variability in cloud cover.
3.3.4.3
Topography and Soils
The terrain at the study site is very flat, with slopes <1˚ (Beringer et al. 2003). The
surface of the lowland plains, where the study site is situated, is a late Tertiary
depositional surface, with a sediment mantle that seldom reaches more than 30 to 40 m
in depth. Due to the pronounced climatic seasonality, the surface has been intensively
weathered, resulting in a lateritic profile, with infertile, acidic soils (Russell-Smith et al.
1995). On the study site itself, the soil profile has been described as a red kandosol, with
sandy loams and sandy clay loams in horizons A and B respectively and weathered
laterite in the C horizon, below about 1.2 m (Kelley 2002).
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Schymanski (2007): PhD thesis
The site is situated between the Howard River, which is at an elevation of 20 m (per
AHD, Australian Height datum) and around 4.5 km to the West of the site, and a
smaller river channel, which is around 0.5 km to the East of the site, at around 30 m
AHD. The maximum elevation between these two channels is roughly 40 m AHD. In
terms of the catchment conceptualisation in Figure 3.3, we interpreted the catchment as
having an average depth of the pedosphere (Z) of 15 m, and an average channel
elevation (zr) of 10 m from the reference datum (zs). The reference datum was set to
10 m AHD to coincide with the assumed average bedrock elevation.
3.3.4.4
Vegetation
The vegetation on the site has been described in detail elsewhere (Hutley et al. 2000;
O'Grady et al. 2000), so we will only give a brief account.
The vegetation has been classified as a Eucalypt open forest (Specht 1981), with a mean
canopy height of 15 m, where the overstorey has an estimated cover of 50% (Hutley et
al. 2000) and is dominated by the evergreen Eucalyptus miniata and Eucalyptus
tetrodonta. Visual estimates of projected tree cover by analysis of cast shadows in June
2005 suggested values closer to 30% than 50%. The dominant tree species contribute to
60-70% of the total basal area (i.e. the ground area covered by tree trunks) of this forest
and are accompanied by some brevi-, semi- and fully deciduous tree species (O'Grady et
al. 2000).
The understorey on the site is very dynamic. During the dry season it is composed of
small individuals of the tree species, some fully or partly deciduous shrubs and some
perennial grasses, while during the wet season it is dominated by a thick layer of annual
C4 grasses of the genus Sorghum sp.
Recent estimates give a total leaf area index on the site of 0.8 in the dry season and 2.5
in the wet season (Hutley and Beringer, unpubl. data).
The root system of the vegetation on the site is mainly limited to the top 4-5 m of soil,
with single roots observed at depths of up to 9 m, but not in significant quantities
(Kelley 2002; Hutley, unpubl. data).
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3.3.4.5
Chapter 3
Measurements
The measurement techniques used on the site are described in detail elsewhere
(Beringer et al. 2003; Hutley et al. 2005) and will be only summarised here. The present
study site is described as the “moderate intensity site” in Beringer et al. (2003).
Flux measurements were conducted at the top of an 18 m tower over the 12-14 m tall
canopy, in a flat terrain (slopes < 1o) with a near homogeneous fetch of more than 1 km
in all directions. The eddy covariance technique was used to infer vertical fluxes of
latent heat and CO2 from three dimensional wind velocities and turbulent fluctuations of
CO2 and H2O in the air. Soil moisture was measured using Time Domain Reflectometry
(TDR) probes (Campbell Scientific) at 10 cm depth and soil temperature was obtained
from soil thermocouple sensors at 2 and 6 cm depth. All flux variables were sampled at
20 Hz and averaged over 30 minutes. Continuous measurements have been collected
since 2001. To ensure a continuous data set, small gaps (less than 2 hours) were filled
using linear interpolation, while larger gaps were filled using a neural network model
fitted to the whole data set (Hutley and Beringer, unpubl.). Periods with gap-filled data
were flagged for later recognition.
3.3.4.6
Disturbances
The most significant disturbances affecting vegetation at the site are probably severe
cyclones and fires. In fact, the population and size structure of the trees on the site
indicates that the vegetation may still be recovering from destruction caused by Cyclone
Tracy in 1974 (O'Grady et al. 2000). Bush fires occur more frequently on the study site,
with generally less permanent damage than severe cyclones. Since the establishment of
the eddy flux tower in 2001, the site has been burnt every year around July or August.
However, these fires were of low intensity and, for two of these fires, it has been shown
that vegetation recovered within a few weeks (Beringer et al. 2003; Schymanski et al. in
prep.-a).
3.3.5
ATMOSPHERIC FORCING
Although local meteorological and flux measurements were only available for the
period 2001 to 2005, the model was run for 30 years, between 1976 and 2005, in order
to model acclimatisation of the vegetation to the long-term environmental conditions.
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Schymanski (2007): PhD thesis
Meteorological data for this period was obtained from the Queensland Department of
Natural Resources, Mines and Water (SILO Data Drill2). The original data set
contained, among others, daily totals of global radiation, precipitation, and class A pan
evaporation, daily maxima and minima of air temperature and daily values of
atmospheric vapour pressure, all of which were obtained by interpolation of data from
the nearest measurement stations and/or estimation based on proxy data. The
methodology used for the compilation of the data set is described in Jeffrey et al.
(2001). The data was interpolated to the location 12°30'S, 131°09'E and was not
expected to be an accurate representation of the weather on the site on any given day,
but sufficient to capture the general trends and cross-correlations between different
meteorological variables on the site.
To run the Vegetation Optimality model, the daily data had to be transformed into
diurnal data, especially solar irradiance, temperature and atmospheric water vapour
deficit. The mole fraction of CO2 in the air was assumed to be invariant at
0.00035 mol/mol and daily rainfall was distributed evenly over 24 hours.
3.3.5.1
Diurnal Variation in Global Irradiance
The diurnal variation in global irradiance (Ig) was estimated from the solar elevation
angle (β) and daily global irradiance (Ig,d), after Spitters et al. (1986):
Ig
Ig,d J‡ sinH bHthLL „ th N sinH bHth LL
( 3.51 )
where th is the hour of the day. The solution of this equation is given in Appendix
A.3.4.2.
3.3.5.2
Diurnal Variation in Air Temperature
Based on the review of temperature models by Bilbao et al. (2002), we chose “Erbs’
Model” of diurnal variation in air temperature, which requires only information about
the mean and range of temperature. We replaced the respective monthly parameters in
2
http://www.nrm.qld.gov.au/silo
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Chapter 3
the original model with daily mean air temperature and daily temperature range and
wrote:
Ta
Ta,m + Ta,r H0.4632 cos Hch-3.805L + 0.0984 cos H2 ch - 0.360L +
0.0168 cos H3 ch - 0.822L + 0.0138 cos H4 ch - 3.513LL
( 3.52 )
where Ta,m is the daily mean temperature and Ta,r is the daily temperature range, all in
Kelvins. The parameter ch changes with the hour of the day (th):
ch
1
p Hth - 1L
12
( 3.53 )
A test of the applicability of this temperature model to the study site is presented in
Appendix A.3.4.1.
3.3.5.3
Diurnal Variation in Atmospheric Vapour Deficit
Atmospheric vapour deficit (Dv) was defined in Equation ( 3.29 ) as the difference
between the partial pressure of water vapour in air (pva) and saturation vapour pressure
(pvsat), divided by air pressure (Pa):
Dv
pvsat - pva
Pa
( 3.54 )
The absolute vapour pressure (pva) is often fairly constant during the day (Woodward
1987; Goudriaan and Van Laar 1994), so that the diurnal variation in Dv is largely a
result of the diurnal variation in saturation vapour pressure (pvsat), which can be
calculated using the common approximation (Allen et al. 1998):
pvsat
17.27 HTa-273L
610.8 ‰ Ta-35.7
( 3.55 )
where pvsat has units of Pa and Ta is given in units of K.
The diurnal variation in Dv was thus estimated using the diurnal variation of air
temperature (Ta) from the previous section, and daily values of pva from the
meteorological database. Barometric air pressure (Pa) was not included in the
meteorological database and was set to a constant of 101325 Pa.
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3.3.6
Schymanski (2007): PhD thesis
CONVERSION OF MEASURED FLUXES
The on-site eddy covariance measurements delivered half-hourly averages of latent heat
flux (in W m-2) and net CO2 flux (in mg CO2 m-2 s-1). Latent heat flux is the result of all
water vapour moving past the sensors and accounts for transpiration, evaporation from
the soil and wet surfaces and potential sinks due to the formation of dew. Net CO2 flux
represents a sum of all processes within the system, which either take up or release CO2.
The only significant process that leads to CO2 uptake on this site is photosynthesis,
while CO2 release occurs through the respiration of leaves, sapwood and roots, as well
as soil decomposition processes.
The optimality model used here predicts only foliage gas exchange due to leaf
photosynthesis and leaf respiration (lumped in the variable Ag,tot), transpiration through
stomata (Et) and soil evaporation (Esu and Ess). In order to make valid comparisons
between modelled and measured fluxes, we needed to extract the relevant parts from the
measured bulk fluxes.
We assumed that modelled transpiration and soil evaporation combined would be
equivalent to the latent heat flux estimated using the eddy covariance technique, and
lumped both into “evapo-transpiration” (ET):
ET
Ess + Esu + Et
( 3.56 )
The measured CO2 uptake by the soil-vegetation system (FnC, mol m-2 s-1) was
subdivided conceptually into net CO2 uptake by foliage (Ag,tot), CO2 release by soil
respiration (Rs) and CO2 release by sapwood respiration (Rw).
FnC
Ag,tot - Rs - Rw
( 3.57 )
Following earlier work (Schymanski et al. in prep.-a), we fitted a model derived for an
African savanna (Hanan et al. 1998) to reported observations of soil respiration on our
study site (Chen et al. 2002b):
Rs
i ‰qsÆ HTs-20L yz q - qmin
R0jj
N
zJ
k 1 + ‰qs∞ HTs-T∞ L { qmax - qmin
( 3.58 )
where qs↓= 0.507(˚C)-1, qs↑ = 0.059(˚C)-1 and T↓ are temperature response parameters, R0
is the “intrinsic soil respiration rate at 20 ˚C” per m2 ground area and θmin (= 0.01 here)
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Chapter 3
and θmax (= 0.12 here) are “soil moisture limits”. We set the critical temperature (T↓) to
the maximum soil temperature recorded in our data set (T↓ = 44.95 ˚C), R0 to
1.862 µmol m-2 s-1 and left all other parameters unchanged. Using measured soil
moisture (θ) and measured soil temperature (Ts), we obtained a time series of soil
respiration that had similar characteristics to the measurements described by Chen et al.
(2002b), both in terms of sensitivity to temperature and soil moisture (shown in
Schymanski et al. in prep.-a) and seasonality (Figure 3.6).
Above-ground woody-tissue respiration (Rw) has been measured on site by Cernusak et
al. (2006) and estimated to be around 297 g C m−2 year-1, which is equivalent to 0.78
µmol m-2 s-1 or 0.8 g C m−2 day-1, averaged over the whole year. No clear seasonal
variation was identified, so we took this value as a constant over the whole period.
The relative magnitudes of the different components of the net carbon flux are shown in
Figure 3.7, with reversed signs for FnC and Ag,tot to achieve greater clarity.
Rs (µmol/s/m2)
12
10
8
6
4
2
Jun −Sep
Dec −Apr
Figure 3.6: Seasonality of modelled soil respiration (Rs). Box plots of dry season (Jun-Sep) and wet
season (Dec-Apr) distributions include 25% and 75% quartiles and outliers of half-hourly Rs over 5
years. Modelled values correspond well with observations between September 1998 and January
2001, when soil respiration was reported to vary between 0.95 and 3.5 µmol m-2 s-1 in the dry season
and between 3.5 and 8.4 µmol m-2 s-1 in the wet season (Chen et al. 2002b).
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Schymanski (2007): PhD thesis
2002
2003
2004
2005
Carbon flux HgCêdayêm2L
10
5
0
-5
-10
−Ag,tot
−FnC
Rs
-15
2002
2003
Year
2004
2005
Figure 3.7: Subdivision of measured net ecosystem CO2 uptake (FnC) into soil respiration (Rs),
foliage CO2 uptake (Ag) and woody tissue respiration (Rw, not shown). Rs and Rw are modelled based
on measurements, while Ag is taken as the difference. For clarity and compatibility with common
practice, all fluxes have been plotted in g carbon per day per m2 catchment area, with negative
values for carbon uptake and positive values for carbon release (note the signs in legend).
3.3.7
TESTING OF THE BELOW-GROUND COMPONENT
To test the below-ground component of the model separately from the above-ground
optimisation, we used observed transpiration rates between July 2004 and June 2005
(Schymanski et al. in prep.-a) to drive the root water uptake and root optimisation. For
simplicity, we did not distinguish between tree and grass water use, but modelled a
single rooting system with a rooting depth of 4.0 m, while root water uptake and
transpiration were linked to an above-ground plant storage compartment with a storage
capacity of Mq = Md = 3 kg/m2.
Z and zr were set to 15 m and 10 m respectively, following Section 3.3.4.3, while Λs was
set to 10 m, for reasons explained in Section 3.3.2.2. The vertical resolution of the soil
profile was set to δyumin = 0.1 m. Based on comments found in the literature, we
concluded that sandy loams were the most dominant soils on the site, and used the same
van Genuchten soil parameters (avG and nvG) and hydraulic conductivity (Ksat) as given
124
Schymanski (2007): PhD thesis
Chapter 3
by the Hydrus 1-D Software package (Simunek et al. 2005) for sandy loam
(avG = 7.5 m-1, nvG = 1.89, Ksat = 1.23ä10-5 m s-1, θs = 0.41 and θr = 0.065).
Soil evaporation was computed as a function of modelled surface soil saturation,
observed atmospheric vapour pressure and soil temperature, as described in Appendix
3.6. All meteorological input variables (rainfall, soil temperature and atmospheric
vapour pressure) were taken from half-hourly observations on the site between July
2004 and June 2005.
The initial conditions for the optimisation were set to a homogeneous root distribution
of SAdr,i = 0.3 m2/m3 in each layer within the root zone, while the initial position of the
water table was set to coincide with the channel depth (ys = yr = 10.0 m) and the initial
soil moisture profile was set to equilibrium conditions (i.e. no spontaneous flow
between soil layers). The root system was then optimised dynamically from day to day,
as described in Section 3.3.3.8 and the modelled surface saturation (su,1) was compared
with observations of surface soil moisture during the same period. Observed soil
moisture (θ) in the top 10 cm of soil was converted to units of relative saturation (su)
using Equation ( 3.76 ), where θs was replaced by the maximum observed soil moisture
(0.307), while θr was kept at a value of 0.065 m3/m3 as estimated previously.
As a control, the model was also run without root optimisation, keeping the root surface
area distribution constant at the initial conditions.
3.3.8
TESTING OF THE COUPLED MODEL
The fully integrated catchment water balance and Vegetation Optimality model was
tested by running the model for 30 years between 1976 and 2005 and optimising the
vegetation properties to maximise Net Carbon Profit (NCP) for the whole period. The
model results were then compared with observations on the site.
The catchment and soil properties used for the model were given in the previous section
(Section 3.3.7). However, to speed up the computations, the vertical resolution of the
soil profile was decreased to δyumin = 0.5 m. Spatial heterogeneity in soil properties was
not considered in this study, but could be included in future studies. Rainfall (Qrain),
atmospheric vapour deficit (Dv), air temperature (Ta) and irradiance (Ia) were derived
125
3.3 Methods
Schymanski (2007): PhD thesis
from the meteorological database (Data drill), as described in Section 3.3.5. No local
vegetation parameters were used for the model.
Figure 3.8 gives an overview of site-specific input variables needed for the combined
water balance and Vegetation Optimality model, and how the different components are
coupled. The water balance and vegetation component are coupled by root water uptake
(Qr,i) and soil evaporation (Esu and Ess), while the “Shuffled Complex Evolution” (SCE
algorithm) is coupled with the vegetation model by the Net Carbon Profit (NCP) and the
constant parameters optimised by the SCE algorithm. The maximum time step length of
the combined model was set to 1 hour, and the individual time steps were reduced as
necessary to prevent any state variables from changing by more than 10% in a single
time step. The model was run with a climate data set of 30 years, which resulted in a
single value for NCP over that period. After each run of 30 years the SCE algorithm was
invoked to generate a new set of input parameters (cλf,s, cλe,s, cλf,p, cλe,p, yr,p and MA,p), and
the model was re-run with this new parameter set, generating a new value for NCP and
so on. For a description of the implementation of the SCE algorithm, please refer to
Appendix 3.5.
Many of the model’s state variables and fluxes could be potentially compared against
observations, but in this study we only used time series of evapo-transpiration
(ET = Et,p + Et,s + Esu + Ess), CO2 assimilation (Agtot), and the area fraction covered by
trees (MA,p) for the validation of the model.
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Schymanski (2007): PhD thesis
Z, zr, Λs
Ksat, avG, nvG
Chapter 3
Qrain
Ia
SCE model
Dv, Ta
cλf,s, cλf,p, cλe,s
cλe,p, yr,p, MA,p
NCP
Water balance
model
Vegetation Optimality
model
su,i, hi
Qsf
Qiex
Qr,i
Ag
Mq, Pb
Et
Jmax25
Gs,
ys, yu,wu,wo,
SAr,i
Esu
Rr
yr
Ess
MA
Rl
Rv
Figure 3.8: Flow diagram of the coupled water balance and Vegetation Optimality model. Input
variables are at the top, while model outputs are separated into state variables (dashed boxes) and
fluxes (along arrows). Symbols are explained in Table 3.2 on page 144 (the subscript i denotes a
vector over all soil layers). For clarity, only selected model outputs are drawn.
127
3.4 Results
3.4
3.4.1
Schymanski (2007): PhD thesis
RESULTS
TEST OF THE BELOW-GROUND COMPONENT
Starting with an initially uniform root distribution in the soil profile, the root system
self-optimised to assume a distribution of root surface area (SAr,i) that was skewed
towards the deeper soil after 40 days (Figure 3.9). During the wet season, however, the
root distribution was concentrated in the top soil layers, with a spike remaining in the
bottom root layer (Figure 3.10). In summary, the root distribution seems to reflect the
general direction of water flow. During the dry season, when the root zone can only be
recharged from the moisture soil layers below, the roots concentrate at the bottom
boundary of the root zone, while in the wet season, when most of the recharge happens
through rainfall, the roots concentrate at the top of the root zone, while still maintaining
a high abundance at the bottom boundary, where water supply is lower, but steady. The
top layers of the soil profile can dry down intermittently even during the wet season.
When the distribution of soil moisture was very heterogeneous in the soil profile, the
model predicted temporal release of water by roots in the driest soil layers (not shown),
but this effect was not observed at the daily time scale. Over 24 hours, water uptake by
roots was always greater than their water release in all soil layers.
Modelled and observed surface soil saturation (su,1) were very similar both in magnitude
as well as dynamics (Figure 3.11). If we assumed a static root distribution in the soil
profile, we were not able to achieve a similar correspondence between modelled and
observed soil moisture (Figure 3.12).
128
Chapter 3
0
0
-1
-1
Depth HmL
Depth HmL
Schymanski (2007): PhD thesis
-2
-3
-2
-3
-4
-4
-5
-5
0
1
2
3
4
Qr,i HmmêdayL
5
0.2
0.3
SAr,i Hm2êm2L
0.1
0
su,i=1
1
0.4
Modelled
Observed
-1
Depth HmL
0.8
-2
0.6
-3
0.4
-4
0.2
-5
0.2
0.4
0.6
0.8
Day
1
50
su,i
100
150
200
250
300
350
Figure 3.9: Below-ground conditions 20 days after model initialisation. Vertical soil profiles show
values for each soil layer between the surface and the variable water table. Soil saturation (su,i,
bottom left) is clearly reduced in the root zone, the distribution of root surface area (SAr,i, top right)
in the soil profile is skewed towards the deeper soil layers, while daily root water uptake (Qr,i, top
left) slowly increases with soil depth, with a sharp incline in the lowest layer of the root zone. The
plot on the bottom right shows observed (grey line) and modelled (black line) saturation degree in
the top soil layer (su,i=1) for 12 months, with a round dot indicating the position in time of the other
three plots.
129
Schymanski (2007): PhD thesis
0
0
-1
-1
Depth HmL
Depth HmL
3.4 Results
-2
-3
-2
-3
-4
-4
-5
-5
0
1
2
3
4
Qr,i HmmêdayL
0.2
0.3
SAr,i Hm2êm2L
0.1
5
0
su,i=1
1
0.4
Modelled
Observed
-1
Depth HmL
0.8
-2
0.6
-3
0.4
-4
0.2
-5
0.2
0.4
0.6
0.8
Day
1
50
su,i
100
150
200
250
300
350
Figure 3.10: Below-ground conditions during the wet season. Vertical soil profiles show values for
each soil layer between the surface and the variable water table. The distribution of soil saturation
(su,i, bottom left) shows the propagation of multiple wetting fronts through the soil profile, while the
distributions of root surface area (SAr,i, top right) and root water uptake (Qr,i, top left) in the soil
profile are concentrated in the top soil with a spike in the lowest layer of the root zone. The plot on
the bottom right shows observed (grey line) and modelled (black line) saturation degree in the top
soil layer (su,i=1) for 12 months, with a round dot indicating the position in time of the other three
plots.
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Schymanski (2007): PhD thesis
su,i=1
1
Chapter 3
Modelled
Observed
0.8
0.6
0.4
0.2
Day
50
100
150
200
250
300
350
Figure 3.11: Observed (grey line) and modelled (black line) saturation degree in the top soil layer
(su,i=1) using a dynamically optimised root profile.
su,i=1
1
Modelled
Observed
0.8
0.6
0.4
0.2
Day
50
100
150
200
250
300
350
Figure 3.12: Observed (grey line) and modelled (black line) saturation degree in the top soil layer
(su,i=1) using a constant, homogeneous root profile.
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3.4 Results
3.4.2
Schymanski (2007): PhD thesis
TEST OF THE COUPLED MODEL
To test the fully integrated catchment water balance and Vegetation Optimality model,
we ran the model for 30 years between 1976 and 2005 and optimised the vegetation
properties to maximise Net Carbon Profit (NCP) for the whole period. As mentioned in
the Methods section, the value of the cost parameter crv, which relates the water
transport costs (Rv) to rooting depth (yr) and the area fraction covered by vegetation
(MA) in Equation ( 3.42 ), could not be derived from the literature and had to be fitted so
that the model would give realistic results. We found that a value of crv = 1.2 µmol m-3
gave the most realistic results, leading to an optimal tree cover (MA,p) of 0.33 and an
optimal rooting depth of trees (yr,p) of 4.0 m, while the grass cover (MA,s) varied
between 0 in the dry season and 0.67 in the wet season, adding up to a total vegetation
cover of 1 (MA,s + MA,p = 1.0) in the wet season (Figure 3.13). At the same time,
modelled and observed rates of evapo-transpiration (ET) and CO2 uptake (Ag,tot) were
very similar in the period for which observations were available. These comparisons of
fluxes are shown in Figure 3.14.
During the period for which the comparison was carried out, total observed ET was
4672 mm, and total modelled ET was 4878 mm. The model captured the magnitude and
seasonal dynamics of ET fairly accurately, except that some spikes in observed ET were
not captured and transpiration on the declining part of the year was over-estimated by
the model in two years (Figure 3.14). The same statements can be made with respect to
modelled and observed CO2 uptake rates, only that the differences between modelled
and observed values are more pronounced, leading to an apparent over-prediction of
fluxes during the plotted period, with 629 mol m-2 and 725 mol m-2 for observed and
modelled total CO2 uptake respectively (Figure 3.14) .
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Schymanski (2007): PhD thesis
Chapter 3
2002
2003
2002
2003
2004
2005
2004
2005
1
0.8
MA,s
0.6
0.4
0.2
Year
Figure 3.13: Modelled seasonal variation in the area fraction covered by grasses (MA,s) between
2001 and 2005.
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3.4 Results
Schymanski (2007): PhD thesis
2002
2003
2004
ET HmmEêTda
yL
(mm/day)
10
2005
Observed
Observed
Modelled
Modelled
Flag
8
6
4
2
0
2002
2003
2004
2005
1.5
Ag,tot Hmolêm2êdayL
Observed
Observed
Modelled
Modelled
Flag
Flag
1
0.5
0
-0.5
2002
2003
2004
2005
Year
Figure 3.14: Modelled (black) and observed (grey) daily evapo-transpiration rates (ET, top) and
CO2 uptake rates (Ag,tot, bottom). Observed and modelled total CO2 uptake during the plotted
period was 629 mol m-2 and 725 mol m-2 respectively. The dashed line shows scaled daily averages
of the validity flag values, ranging between -0.2 for a whole day of valid measurements and -0.4 for
a whole day of gap-filled data using a neural network approach.
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3.5
Chapter 3
DISCUSSION
The aim of this study was to construct and test a model of transpiration by natural
vegetation that would not require any knowledge about the local vegetation on a site or
any parameter fitting to match the results with observations. Given that the value for the
cost parameter crv (Equation ( 3.42 )) had to be “guess-timated” to give reasonable
results, the reader might find that the authors have not succeeded. However, it is
remarkable that the “right” choice of only this one parameter value resulted in
reasonable time series of evapo-transpiration (ET), CO2 uptake (Ag,tot) and grass cover
(MA,s), as well as reasonable values for tree cover (MA,p) and rooting depth (yr,p). The
carbon costs of the vascular system (Rv) of trees, resulting from the choice of this
parameter, would be equivalent to a constant carbon loss of 1.6 µmol s-1 m-2. Cernusak
et al. (2006) measured sapwood respiration rates on the site and estimated the total
respiration by above-ground sapwood to be 297 g C m-2 ground area year-1, which
would be equivalent to a constant respiration rate of 0.79 µmol s-1 m-2. Considering a
1:1 partitioning between above-ground and below-ground carbon allocation as a first
estimate, the total carbon costs of the vascular system given by the model seem very
reasonable. The remaining question is whether the chosen value of crv is a universal
constant valid for a wide range of conditions and vegetation types, or whether it is likely
to depend on, for example soil type, temperature or other environmental conditions.
This has partly been tested in a follow-up study, where the model was applied to a range
of catchments in different climates, using the same value of crv throughout (Schymanski
et al. in prep.-b).
The present model suggests that most of the vegetation dynamics on the site can be
captured by sub-dividing the vegetation into only two components, a perennial
component, which has to be adapted to withstand the severest conditions experienced on
the site, and a seasonal component, which is very flexible and makes use of the best
conditions on the site. This is in line with investigations by Eamus et al. (2000), who
found that dry season conditions determine the wet season water use by trees. However,
the assumption that the seasonal vegetation component is only composed of grasses
with shallow root systems is probably not applicable to vegetation types where
deciduous trees play a significant role. Further differentiation of different vegetation
components might be necessary for broader application of the model. For example,
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3.5 Discussion
Schymanski (2007): PhD thesis
deciduous trees could be represented by a third “big leaf” with optimised invariant
rooting depth and area fraction, but variable leaf area.
A part of the success of the model relies on a realistic representation of below-ground
processes and their associated costs. The time series of surface soil moisture obtained
from a test run of the below-ground model component alone corresponded well with
observed surface soil moisture if the root distribution in the soil profile was dynamically
optimised to meet canopy demands estimated from measurements (figures on pages 129
to 131). Although these model results appear very promising, we wish to point out that
the roots were only optimised for the uptake of water in the present model, but if
nutrient uptake was made part of the objective function for their optimisation, the
optimal root distributions would likely be different. In addition, variability in root
resistivity to water uptake was neglected and the parameterisation of the root costs was
very simplistic and based on observations in citrus roots only, although it is known that
roots can be very versatile. For instance in Lotus japonicus, radial root hydraulic
conductivity has been shown to vary 4-fold between 8.0ä10-9 m s-1 MPa-1
-8
-1
-1
4.7ä10 m s MPa
and
on a diurnal basis, due its control by aquaporins (Henzler et al.
1999). Aquaporins can be thought of as another degree of freedom available to plants
for the regulation of their water uptake. For example, they could open when tissue
“suction” would lead to water uptake (during the day) and close to reduce the reverse
effect at night. Or, they could open in soil patches with high concentrations of particular
nutrients and close in nutrient-poor patches, to use the transpiration stream for the
selective uptake of nutrients. They could even discriminate salty water against fresher
water in salinity-affected soils. However, despite their functional similarity to
“Maxwell’s demon” (Maxwell 1871), which was a theoretical construct used to defy the
second law of thermodynamics, aquaporin regulation requires energy expenditure by
plant cells, but quantitative data on these costs are not yet available. More research is
needed to understand all of the plants’ degrees of freedom related to water uptake and
their associated costs.
The present model also allows investigation of processes like the uptake of water by
roots in wet soil and simultaneous exudation of water by roots in dry soil layers
(“hydraulic redistribution”) to a certain extent. Hydraulic redistribution (HR) has been
widely observed and could be seen as a passive process, which depends on the soil
suction head and the root distribution within the soil column. Species not showing
136
Schymanski (2007): PhD thesis
Chapter 3
hydraulic lift are often those whose roots do not survive in dry soils and hence avoid the
loss of water from roots (Espeleta et al. 2004). Hydraulic redistribution could have the
advantage of improving the uptake of nutrients from the surface soil, which would
otherwise be inhibited by dryness. On the other hand, HR could be an undesired “leak in
the system”, a view which is supported by the observation that root resistance to water
release seems to be generally higher than root resistance to water uptake (Hunt et al.
1991). Although root resistance was assumed to be the same for water movement in
both directions in the present study, the predicted water release by roots, when it
occurred, was very small and not apparent at the daily scale. This is in line with field
observations elsewhere, which showed that tree roots take up more water from shallow
soil than they exude via hydraulic lift (Ludwig et al. 2004).
3.5.1
RELATION OF THE PRESENTED APPROACH TO EXISTING
OPTIMALITY-BASED MODELS IN HYDROLOGY
Optimality principles have been used previously by a number of researchers for
modelling vegetation-environment interactions.
The initial idea for using Optimality principles in eco-hydrology was taken from the
work by Peter Eagleson in the late 1970s and early 1980s (Eagleson 1978; Eagleson
1982). Eagleson used two objective functions to optimise the two vegetation
parameters:
– “vegetated fraction of surface” (M) and
– “plant coefficient (equal to potential rate of transpiration divided by potential
(soil surface) rate of evaporation [...])” (kv) (Eagleson 1978).
The two objective functions were
– maximisation of soil moisture or equivalently, the minimisation of evapotranspiration (Eagleson 1978, page 755) and
– the maximisation of the “index of potential biomass” (M kv) (Eagleson 1978,
page 756).
The first objective function was used to establish a relationship between the parameters
M and kv, while the second one was used to find the optimal kv for a given climate and
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3.5 Discussion
Schymanski (2007): PhD thesis
soil type. Eagleson (1982) later extended the framework to include soil properties in the
optimisation, where he hypothesised that soils and vegetation would co-develop over
very long periods of time to achieve an equilibrium state (near optimal for vegetation).
Both M and kv were formulated as annual averages in Eagleson’s work, so that intraannual vegetation dynamics could not be predicted. Recently, Kerkhoff et al. (2004)
evaluated Eagleson’s optimality hypotheses from an ecological perspective and pointed
out several inconsistencies with current understanding of vegetation ecology. Most of
all, the hypothesis that vegetation would aim at minimising evapo-transpiration has been
pointed out to be unrealistic, as this would effectively result in a minimisation of
photosynthetic activity (Kerkhoff et al. 2004).
Rodriguez-Iturbe and co-workers also assumed that vegetation would aim at minimising
“water stress”, but, in contrast to Eagleson, they defined water stress quantitatively as a
non-linear function of soil moisture. Different stress functions for grasses and trees were
defined based on empirically derived functions of evapo-transpiration in relation to soil
moisture for both vegetation types. Hypothesizing that individuals in a plant community
would act together to reduce their water stress, the researchers showed numerically that
spatial interactions between woody and grassy vegetation types can lead to a more
efficient community water use and decrease global water stress, even if both vegetation
types compete for the same resource (Rodriguez-Iturbe et al. 1999a; Rodriguez-Iturbe et
al. 1999b).
In a later published framework, Rodriguez-Iturbe and co-workers linked climate
dynamics, soils and vegetation to obtain probabilistic soil moisture patterns and values
for vegetation water stress with the aim to investigate “the optimal environmental
condition for different functional types of vegetation” (Laio et al. 2001a; Laio et al.
2001b; Porporato et al. 2001; Rodriguez-Iturbe et al. 2001). As in Eagleson’s approach,
total evapo-transpiration was assumed to be proportional to productivity and the optimal
condition was defined as “somewhere” between minimum vegetation water stress and
maximum total evapo-transpiration (Porporato et al. 2001). Although this model may
allow comparison of the suitability of different climates for a given plant community, it
does not allow the reverse, a prediction of an optimal vegetation composition for a
given climate, as the range of possible stress functions and the number of possible
vegetation compositions could be very large.
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Schymanski (2007): PhD thesis
Chapter 3
Ecophysiology
Ecohydrology
(Cowan, Farquhar, Mäkelä)
(Eagleson, Rodriguez-Iturbe)
ale
al sc
n
o
i
Reg
cale
Leaf-s
Optimise Vegetation
type and cover
Canopy/Catchment scale?
Optimise stomata
Maximise water use,
Minimise stress
Maximise CO2 uptake
Biochemical/Structural/Water
constraints
Days-years?
Energy/Water constraints
Annual averages
Minutes-days
Figure 3.15: Existing optimality-based models of vegetation water use. Ecophysiological models ask
how stomata should adjust to maximise CO2 uptake over periods of time when water is limited,
given biochemical and structural constraints (constant leaf area and photosynthetic capacity).
Ecohydrological approaches to ecological optimality on the other hand look at large scales and
optimise long-term averages of vegetation properties, vegetation type and cover. They do not
include an explicit coupling between CO2 assimilation and transpiration, but assume that total
water use is maximised, where the trade-off is to avoid water depletion and “stress”. These models
are not suitable for shorter time scales, because the objective function does not differentiate when
the water should be used. The dashed box stands for the medium scale, which is the subject of the
model presented in this paper.
Optimality principles have also been used in eco-physiology, for example to make
predictions about gas exchange at leaf scale. Cowan and Farquhar (1977) assumed a
priori that plants would optimise stomatal conductivity dynamically in order to
maximise total photosynthesis for a given amount of transpiration. This assumption,
together with a quantitative theory about the non-linear coupling between transpiration
and CO2 assimilation, allowed them to derive how stomatal conductivity should vary in
response to the rate of photosynthesis and atmospheric water vapour deficit, given a
fixed amount of water available for transpiration. Later, Cowan (1982) looked at a
longer time scale and introduced competition for water by processes not under the
control of stomata (e.g. drainage, soil evaporation, water extraction by other plants).
This allowed assessing day to day changes in transpiration during a dry period. Mäkelä
and co-workers built a model using these concepts and tested it successfully against
field measurements at leaf scale (Berninger et al. 1996; Mäkelä et al. 1996; Hari et al.
1999; Hari et al. 2000). However, this was based on water availability per unit leaf area
139
3.5 Discussion
Schymanski (2007): PhD thesis
and did not allow predictions about changes in leaf area itself, and was therefore limited
to time scales at which leaf area was not expected to change.
In summary, the eco-hydrological concept of Ecological Optimality is suited for
prediction of long-term averages of transpiration only, due to the neglect of the nonlinearity between carbon uptake and transpiration, while the eco-physiological concept
is suited for the prediction of short-term dynamics only, due to its neglect of the longterm water balance and associated changes in leaf area (Figure 3.15).
The concept presented here combines the eco-physiological approach to optimality with
the eco-hydrological one. It accounts explicitly for the non-linear coupling between CO2
uptake and transpiration, adopting a biochemical model of photosynthesis. At the same
time, the presented concept considers the dynamics of soil water and explicit carbon
costs of maintaining roots and water transport tissues. This allows modelling gas
exchange at smaller scales than the eco-hydrological optimality models, and at larger
scales than the eco-physiological models mentioned above.
Further, the present concept is based on the assumption that “Net Carbon Profit” (NCP,
as defined in Equation ( 3.43 )) is maximised by vegetation. This is in contrast to the
commonly assumed maximisation of “Net Primary Production” (NPP) in other
optimality-based models (Raupach 2005). Neither NPP nor NCP are easily observable in
nature, as the carbon gained is not necessarily all invested into the build-up of biomass
(Roxburgh et al. 2005). NPP refers to “the rate at which solar energy is stored by
plants as organic matter” (Roxburgh et al. 2005), irrespective of whether this energy is
subsequently available to the plants or not. In fact, most of the “observable” part of NPP
corresponds to energy that is locked up in cellulose and lignin and hence not retrievable
by the plants. We believe that maximisation of NCP is a more appropriate objective
function than the maximisation of NPP, because NCP only refers to the energy that is
available to the plants for increasing their “biological fitness” (e.g. production of seeds,
maintenance of defence mechanisms against pests and herbivores or maintenance of
symbiotic relationships to improve nutrient uptake). While the growth of new leaves
itself could be interpreted as a strategy to increase a plant's “fitness”, here it is merely
considered a means to increase the carbon gain in the current framework. We have to
accept that the magnitude of NCP is not measurable itself, but the magnitudes of the
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Schymanski (2007): PhD thesis
Chapter 3
fluxes leading up to its calculation are the ones of interest and can be tested against
observations, as demonstrated in this study.
3.5.2
NEED FOR FURTHER RESEARCH
The present study represents only one major step towards a better understanding of the
interplay between vegetation and hydrology. The idea behind it was that vegetation
optimises itself to make best use of the available water, light and CO2. To test this idea
against available observations, we constructed a minimalist model that would include
only the key degrees of freedom for which we were able to estimate the associated costs
and benefits in terms of the Net Carbon Profit. The results obtained using this relatively
simple model were very encouraging and, in our opinion, warrant a more detailed
investigation of the optimality hypothesis, especially as computing power increases and
our understanding of hydrology and plant physiology improves. However, for the
further advance of the presented framework, experimental and modelling approaches
have to be conducted side-by-side and to complement each other.
From the view point of Vegetation Optimality, it is unfortunate that the majority of
studies on plants are performed on individuals removed from their natural habitat or
cultivated under experimental conditions with little or no reference to the importance of
their observed behaviour for survival in their natural environment. Often, great
importance is placed on differences between species or genera of plants, but no analysis
of the ecological distribution of these taxa is provided. However, this might lead to little
insight anyway, as the same plant species can often be found in very different
environments, where it finds an ecological niche suitable for its success. What would be
more interesting from a vegetation optimality perspective, are studies of the functional
properties of whole plant communities, which are more likely to converge with
environmental conditions than those of individuals or species. The “functional trait
approach” (McGill et al. 2006; Westoby and Wright 2006) constitutes a step in this
direction.
On the other hand, experimental setups where plants are removed from their natural
environment and investigated under controlled conditions could deliver valuable
insights into generic cost-benefit relationships of different plant organs. These are
crucial for understanding, let alone predicting patterns like vertical root distributions or
canopy structures. Unfortunately, not many researchers focus on the link between
141
3.5 Discussion
Schymanski (2007): PhD thesis
energy expenditure and the amount of useful work a plant organ performs, but most
concentrate on one side of the equation only and relate either respiration rates or plant
functioning to environmental variables, not both to each other.
The present study demonstrated how important a realistic parameterisation of costbenefit relationships of different plant adaptations can be for the prediction of optimal
vegetation properties and resulting fluxes. The authors hope that the promising results
of this study will motivate future research into the links between the costs and benefits
of different plant functions in terms of the Net Carbon Profit. Some important questions
are:
– Is there a generic relationship between the total costs related to water uptake
from the soil, soil moisture and soil properties?
– Is there a generic relationship between the construction and maintenance costs of
the vascular system and its ability to conduct and distribute water?
– How can the costs and benefits of nutrient uptake and processing be
incorporated into the optimality framework?
– What are the costs and benefits of active water uptake and -transport as opposed
to the passive processes modelled in the present study (e.g. Dewar 2000)?
The hydrological part of the presented model also needs improvement. Currently,
drainage and discharge are modelled in a very simplistic way, assuming that all water
that reaches the river channel is instantly lost and the only input of water to the system
is rainfall. A more sophisticated hydrological model could account for routing and
backlogging of water in the channel, as well as surface or sub-surface contribution of
water from adjacent domains. This would then allow comparison of hydrographs
generated by the Optimality-based model with readily available stream flow
hydrographs. It would also allow investigation of spatial heterogeneity in soil moisture
and vegetation properties. The only prerequisite for the hydrology-part is to provide
output that is useful for computing the root water uptake and root costs necessary to
meet the water demand by the canopy.
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Schymanski (2007): PhD thesis
Chapter 3
We are confident that more scientific insight into these and related questions will further
improve our ability to analyse possible responses of eco-hydrological systems to
environmental change.
3.6
CONCLUSIONS
1. Maximisation of the Net Carbon Profit is a possible principle for selforganisation of plant communities
The objective function of maximising Net Carbon Profit of the whole vegetation on the
site led to the emergence of vegetation properties and CO2 uptake rates in the model,
which were consistent with observations.
2. Costs associated with water transport may be responsible for the limitation of
the vegetated surface fraction
Realistic predictions of the seasonality in grass cover and the total surface area fraction
covered by the deeper rooting trees were only achieved if the carbon costs related to
water transport tissues were parameterised appropriately. Without consideration of these
costs, optimal vegetation cover would be 100% throughout the year (Schymanski et al.
in prep.-a).
3. If rooting costs are derived from the feedback between vegetation water use and
the catchment water balance, Vegetation Optimality allows modelling of water
and CO2 fluxes and some key vegetation properties from day to day and from
year to year without a priori assumptions about the vegetation on a particular
site.
The coupling of a physical water balance model with the Vegetation Optimality model
allowed realistic predictions of the magnitude and seasonality of canopy water use at
daily scale over many years. As the model did not require any input about the vegetation
on the site, it is potentially useful for the prediction of the long-term response of natural
vegetation to environmental change. In addition, the model could potentially be used to
predict the “Potential Natural Vegetation” on sites where this cannot be achieved using
observations of remnant vegetation. Predictions about the magnitude and seasonality of
transpiration obtained from the model could further be used as input to more
143
3.7 Notation
Schymanski (2007): PhD thesis
sophisticated runoff models that suffer from uncertainty in the estimation of the
transpiration component.
3.7
NOTATION
Table 3.2: Parameters used in this study and their units. If not stated otherwise in this table,
addition of the subscript i, 1, or nlayers to a symbol in the main text denotes reference to soil layer i,
the top soil layer or the bottom soil layer of the unsaturated zone respectively. Addition of the
subscript p or s refers to the perennial or seasonal vegetation component respectively.
Symbol
Description
Value / Units
g0
Average slope angle of the seepage face
2º
Ls
Typical horizontal length scale for seepage face flow
10 m
dyu
Thickness of a soil layer
m
dyumin
Minimum soil layer thickness in unsaturated zone
m
a
Molecular diffusion coefficient of CO2 in air, relative
to that for water vapour
1.6
Ag
CO2 uptake rate per unit ground area
mol m-2 s-1
Ag,tot
CO2 uptake rate by all vegetation per unit ground
area
mol m-2 s-1
Ca
Mole fraction of CO2 in the air
mol/mol
ch
Parameter for the calculation of the diurnal variation
in air temperature
-
cPbm
Conversion coefficient to convert from pressure (bar)
to hydraulic head (m).
10.2 m bar-1
cRl
Leaf respiration coefficient
0.07
cRr
Root respiration rate per volume of fine roots
0.0017 mol m-3 s-1
crv
Proportionality constant for water transport carbon
costs
mol m-3
cλe,p
Constant of the relationship between λp and soil water -
cλe,s
Constant of the relationship between λs and soil water
cλf,p
Constant of the relationship between λp and soil water -
cλf,s
Constant of the relationship between λs and soil water
-
Dl
Day length (time between sunrise and sunset in
hours)
h
dSAdr
Daily increment in root surface area density
m2 m-3 d-1
Dv
Molar atmospheric vapour deficit (1.0-Wa)
mol/mol
dx
Increment in distance
m
Dy
Day of the year
-
144
-
Schymanski (2007): PhD thesis
Chapter 3
Symbol
Description
Value / Units
Ess
Soil evaporation from the saturated zone
m/s
Esu
Soil evaporation from the unsaturated zone
m/s
Et
Transpiration rate per unit catchment area
m/s
ETs
Evapo-transpiration rate per unit horizontal area from
saturated zone
m/s
ETu
Evapo-transpiration rate per unit horizontal area from
unsaturated zone
m/s
FnC
CO2 uptake rate by the soil-vegetation system per
unit catchment area
mol m-2 s-1
Grmax
Maximum daily increment in root surface area per
unit ground area
10%
Gs
Stomatal conductivity per unit ground area
mol m-2 s-1
h
Matric suction head
m
Ha
Rate of exponential increase of Jmax with temperature
43,790 J mol-1
Hd
Rate of decrease of Jmax with temperature above Topt
200ä103 J mol-1
hh,i
Hydrostatic head difference between soil surface and
layer i
m
hi
Matric suction head in layer i
m
hr,i
Root suction head in layer i
m
Ia
Photosynthetic active irradiance in mol quanta per
horizontal area
mol m-2 s-1
Ig
Global irradiance per unit horizontal area
W/m2
Ig,d
Daily global irradiance per unit horizontal area
J m-2 day-1
ir
Deepest soil layer accessed by roots
-
Je
Photosynthetic electron transport rate per leaf area
mol m-2 s-1
Jmax
Photosynthetic electron transport capacity per leaf
area
mol m-2 s-1
Jmax25
Photosynthetic electron transport capacity per leaf
area at 25˚C (standard temperature)
mol m-2 s-1
Jr,daily
Daily root water uptake per unit root surface area
m/day
Jr,i
Water uptake per unit root surface area in a soil layer
m/s
Jw
Rate of vertical water flow per unit cross-sectional
area in unsaturated zone (positive for upwards and
negative for downwards flow)
m/s
kr,p
Coefficient of root optimisation
-
kreff,p,i
Relative effectiveness of tree roots in layer i
-
Ksat
Saturated hydraulic conductivity
m/s
145
3.7 Notation
Schymanski (2007): PhD thesis
Symbol
Description
Value / Units
Kunsat
Unsaturated hydraulic conductivity
m/s
Lh
Length of hill slope
m
lr
Total root length
m
m
Mass of a plant organ (e.g. leaf)
kg
MA
Fraction of catchment area covered by vegetation or
“leaf area of big leaf per unit catchment area”
-
md
Mass of dry matter within a plant organ
kg
Md
Mass of dry matter in living plant tissues per unit
ground area
kg m-2
mq
Mass of liquid matter within a plant organ
kg
Mq
Mass of liquid matter in living plant tissues per unit
ground area
kg m-2
Mqmin
Daily minimum of liquid matter in living plant tissues kg m-2
per unit ground area
mqx
Mass of liquid matter within a plant organ at
saturation
kg
Mqx
Water storage capacity of living plant tissues per unit
ground area
kg m-2
mvG
Empirical parameter of the van Genuchten water
retention model
-
mx
Mass of a plant organ at saturation (in equilibrium
with 100% relative humidity)
kg
NCP
Net Carbon Profit of all vegetation on the site per unit mol/m2
ground area
NCPd
Daily Net Carbon Profit, computed by setting tstart
and tend to the start and the end of a day in Equation
( 3.43 )
mol/m2
nlayers
Number of soil layers in the unsaturated zone
-
nvG
Empirical parameter of the van Genuchten water
retention model
-
Pa
Barometric air pressure
Pa
Pb
Balance pressure of a plant organ, measured by the
pressure bomb
bar
Pbx
Balance pressure that cannot be exceeded without
damage to a plant organ
bar
pva
Partial vapour pressure of water vapour in
atmospheric air
Pa
pvsat
Saturation vapour pressure as a function of air
temperature
Pa
146
Schymanski (2007): PhD thesis
Chapter 3
Symbol
Description
Value / Units
q
Relative water content of a plant organ (q = mq/m)
kg/kg
Qi
Flow rate across the bottom boundary of layer i, per
unit horizontal catchment area (positive in upwards
direction)
m/s
Qiex
Infiltration excess runoff rate per unit horizontal
catchment area
m/s
Qinf
Infiltration rate per unit horizontal catchment area
m/s
Qnlayers
Flow rate across the bottom boundary of the bottom
m/s
of the unsaturated zone, per unit horizontal catchment
area (positive in upwards direction)
Qout
Immediate runoff per unit horizontal catchment area
m/s
Qr
Root water uptake per unit horizontal catchment area
m/s
Qrain
Precipitation rate per unit horizontal catchment area
m/s
qs↑
Temperature response parameter for soil respiration
0.059(˚C)-1
qs↓
Temperature response parameter for soil respiration
0.507(˚C)-1
Qsf
Flow rate across the seepage face per unit horizontal
catchment area
m/s
Qu
Flow rate between saturated and unsaturated zone per
unit horizontal catchment area (= Qnlayers)
m/s
qx
Relative water content of a plant organ at saturation
(qx = mqx/mx)
kg/kg
R0
Intrinsic soil respiration rate per ground area at Ts=20 1.862ä10-6
˚C and θ=θmax
mol m-2 s-1
Rf
Carbon costs related to the maintenance of foliage per mol m-2 s-1
unit ground area
Rl
Leaf respiration rate per ground area
mol m-2 s-1
Rmol
Molar universal gas constant
8.31441
J mol-1 K-1
Rr
Root respiration rate per unit ground area
mol m-2 s-1
rr
Mean radius of fine roots
0.0003 m
Rs
Soil respiration rate per unit ground area
mol m-2 s-1
Rv
Carbon costs related to water transport tissues per
unit ground area
mol m-2 s-1
Rw
Above-ground woody tissue respiration rate per unit
catchment area
mol m-2 s-1
rWL
Relative water loss from a plant organ
(rWL = (mx - m)/mx)
kg/kg
rWLx
Relative water loss that cannot be exceeded without
damage to a plant organ
kg/kg
147
3.7 Notation
Schymanski (2007): PhD thesis
Symbol
Description
Value / Units
SAdr
Root surface area density (root surface area per unit
soil volume)
m2/m3
SAr
Root surface area in a given soil volume
m2
su
Average saturation degree in the unsaturated zone
(range 0 to 1)
-
T↓
Temperature response parameter for soil respiration
35.8 ˚C
Ta
Air temperature
K
Ta,m
Daily mean air temperature
K
Ta,r
Daily temperature range
K
th
Hour of day (0-24)
Topt
Optimum temperature for electron transport
305 K
Ts
Soil temperature
˚C
Ui,j
Deviation of a degree of freedom from its mean in the population, relative to its feasible range
Vs
Volume of soil containing roots
m3
Wa
Mole fraction of water vapour in atmospheric air
mol/mol
wc
Total amount of water stored in the soil domain per
unit catchment area
m
Wl
Mole fraction of water vapour in the intercellular air
spaces of leaves
mol/mol
ws,i
Amount of water stored in soil layer i per unit
catchment area
m
yd
Maximum thickness of the unsaturated zone
m
yr
Average thickness of the root zone
m
ys
Average thickness of the saturated zone
m
yu
Average thickness of the unsaturated zone
m
Z
Average depth of the pedosphere
15.0 m
z
Distance to the soil surface
m
zm
Maximum land elevation
m
zr
Average channel elevation from reference datum
10.0 m
zs
Average bedrock elevation from reference datum
0.0 m
zw
Average water table elevation from reference datum
m
α
Quantum yield of electron transport
0.3 mol/mol
αvG
Empirical parameter of the van Genuchten water
retention model
m-1
β
Solar elevation angle
º
148
Schymanski (2007): PhD thesis
Chapter 3
Symbol
Description
Value / Units
βw
Proportionality constant for geometrical relations
-
Γ*
CO2-compensation point in the absence of respiration
mol/mol
δ
Solar declination angle
˚
δyu,i
Thickness of soil layer i
m
δyumin
Minimum thickness of soil layers
m
ε
Soil porosity (ε = θs – θr)
m3/m3
θ
Volume of water per unit soil volume
m3/m3
θmax
Soil moisture response parameter for soil respiration
0.12 m3 m-3
θmin
Soil moisture response parameter for soil respiration
0.01 m3 m-3
θr
Residual water content
m3/m3
θs
Volume of water per unit soil volume in saturated
soil
m3/m3
κ
Proportionality constant for balance pressure
relationship
bar
λE
Latent heat of vaporisation at 20 oC
2.45ä106 J kg-1
λp
Targeted slope between Et,p and Ag,p
mol/mol
λs
Targeted slope between Et,s and Ag,s
mol/mol
π
Trigonometrical constant (3.1415…..)
-
ρ
Density of water
1000 kg m-3
φ
Geogr. latitude in decimal degrees
-12.4952˚
ωo
Saturated surface area fraction contributing to
seepage face and overland flow (range 0 to 1)
-
Ωr
Root resistivity to water uptake per unit root surface
area
s
Ωs
Resistivity to water flow towards the roots in the soil
s
ωu
Unsaturated surface area fraction (range 0 to 1)
-
149
Appendix 3.1
Schymanski (2007): PhD thesis
APPENDIX 3.1. SOME THERMODYNAMIC
CONSIDERATIONS ABOUT THE SPONTANEOUS FLOW
OF WATER
The term “suction head” used in soil water relations can be confusing, as the variable
refers to a force rather than a pressure. It is caused by adhesive and capillary forces that
hold the water in the soil (Roderick 2001). The “suction head” refers to the force per
unit cross-sectional area that has to be overcome by another force in order to extract
water from the soil. Being a force per unit area, it is commonly expressed as a pressure
head (with units of length), while keeping the directional component of a force. This can
lead to the confusing concept of “negative pressure”.
Common descriptions of water flow in unsaturated soil or in plant tissues are derived
from the electrical circuit analogue and consider various forms of driving forces and
resistances (e.g. “water potential gradient”). More confusion arises from the use of
different units and different reference systems (Hunt et al. 1991). More recently, it has
been observed that the use of thermodynamic expressions for the description of water
flow in plants and soil is not consistent with thermodynamic literature (Roderick 2001).
In particular, the common practice of neglecting the role of the gravitational potential,
temperature gradients and the importance of interfaces between different phases has
been criticised. Neglecting these could lead to interesting conclusions, as noted by
Roderick (2001):
“That is, if water did actually spontaneously move from a region of high
chemical potential to low chemical potential, as is commonly stated in plant-water
relations texts, then it would spontaneously flow up a liquid column because the
chemical potential is higher at the bottom than the top of the column.”
Kondepudi and Prigogine (1998) include external fields in the definition of chemical
potential, so that differences in chemical potential can be used to predict the direction of
flow under any conditions.
Roderick (2001) used Gibbs energy to calculate a system’s capacity for work and noted
that whereas the chemical potential of water can be the same in two systems, their
ability to perform work can be very different depending on the amount of water in each
150
Schymanski (2007): PhD thesis
Chapter 3
of them. This could be interpreted in the following way: the difference in chemical
potentials between states or subsystems is called the “thermodynamic force”, so that the
chemical potential of water in an isolated system in thermodynamic equilibrium
determines the thermodynamic force this system could apply if connected to another
system. The amount of water on the other hand determines how long this force can be
maintained for, if an exchange of energy or matter happens. Mechanical work is the
product of force and distance, so the product of chemical potential and the amount of
water is the thermodynamic equivalent of the amount of mechanical work that can be
done by the system. Note that thermodynamic force has the units of J mol-1, while the
units of mechanical force are equivalent to J m-1. The product of the chemical potential
and the number of molecules is the Gibbs energy (Equation 20a in Roderick 2001) and
matches the units of mechanical work.
Let us think of a horizontal soil column, containing a mixture of liquid water and gas,
where the liquid water is assumed to be in thermodynamic equilibrium with its vapour
in the gas phase. We can ask how to determine the chemical potential of water in this
column. Common methods consist of connecting the column to a reservoir of water with
a standard chemical potential of water and measuring the forces between the reservoir
and the soil column. In general, these forces are expressed in units of pressure, which
are the same as energy/volume. However, the units of chemical potential are
energy/amount of substance, so to convert from one to the other we need to know the
partial molar volume of water. This is straightforward for liquid water, where the
density does not change much with temperature or pressure, but if the system contains
water vapour, the conversion becomes difficult. For this reason, the “water potential”,
expressed in units of pressure, cannot be seen as equivalent to the “chemical potential”
of water in a system containing water in liquid and gas form.
Roderick (2001) also criticised the use of the electric circuit analogue for the description
of bulk flow, mainly because the driving force and resistance would not be simple
functions of pressure head, gravitational head, velocity head or frictional losses.
According to current thermodynamic literature, flow and driving force are only linearly
related when a system is “close to thermodynamic equilibrium”. A system is “close to
thermodynamic equilibrium” or “in the linear regime” if the difference in chemical
potentials between its compartments is small compared with (Rmol T), i.e.
(µ1-µ2) << Rmol T (Kondepudi and Prigogine 1998, page 409). Thus, at least in theory,
151
Appendix 3.1
Schymanski (2007): PhD thesis
flow could be modelled using the linearity assumption if subsystems are chosen such
that the conditions within each subsystem can be assumed to be in thermodynamic
equilibrium, while the differences in chemical potential of water between the
subsystems are small enough for the linearity condition to hold (“local equilibrium
hypothesis”). In this context, a living plant can hardly be considered an appropriate
sub-system, as life and thermodynamic equilibrium are mutually exclusive. As long as a
plant is alive, ongoing energy conversion keeps the plant as a whole far away from
thermodynamic equilibrium. Yet equilibrium considerations can be used to assess
potential flows of water that could happen spontaneously, without additional
expenditure of energy by the plant. If water does flow spontaneously from roots to
leaves due to a gradient in the chemical potential of water between soil and atmosphere,
there would be no need for the plant to invest in the maintenance of ionic concentration
gradients to support the flow. This “cheap” flow was the one modelled in the present
study.
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Schymanski (2007): PhD thesis
Chapter 3
APPENDIX 3.2. EQUATIONS USED IN THE WATER
BALANCE MODEL
A.3.2.1
GEOMETRICAL RELATIONS
The equations for the geometrical relations given in Reggiani et al. (2000) result in yu =
const., as shown below:
Equation B1 in Reggiani et al. (2000):
y s + y u wu
Z
( 3.59 )
Equation B3 in Reggiani et al. (2000):
H ys - zr + zsL bw
wo
( 3.60 )
Equation B4 in Reggiani et al. (2000):
bw
1
Z - zr + zs
( 3.61 )
As ωu and ωo are complementary (i.e. ωu = 1-ωo), we could insert ( 3.60 ) and ( 3.61 )
into ( 3.59 ) and solve for yu, which gives
yu
Z - zr + zs
( 3.62 )
This would imply that the average thickness of the unsaturated zone (yu) is constant and
independent of the thickness of the saturated zone (ys). However, we could not come up
with any hill slope geometry, for which a horizontal water table could change position
without changing the value of yu.
Assuming linear slopes as in Figure 3.16, we found a more coherent form for ωo by
explicitly calculating the average thickness of each zone as a function of the position of
a horizontal water table.
153
Appendix 3.2
Schymanski (2007): PhD thesis
wu ä Lh
A
yd
C
G
F
zm
wo ä Lh
zw
Datum
B
D
E
zr
Lh
Figure 3.16: Schematic linear hill slope of length Lh with maximum land elevation zm, water table at
elevation zw and a channel at elevation zr.
It is obvious from Figure 3.16 that the parameter pairs (yd, zw) and (ωo, ωu) are
complementary, so that we can write:
yd
zm - zw
( 3.63 )
wo
1 - wu
( 3.64 )
and
The average thickness of the pedosphere for this linear hill slope can be calculated as:
Z
1
H zm + zrL
2
( 3.65 )
The average thickness of the saturated zone (ys) in Figure 3.16 can be formulated as:
ys
Lh zw - 0.5 Lh H zw - zrL wo
Lh
( 3.66 )
The average thickness of the unsaturated zone (yu) can similarly be calculated as the
area of the rectangle ABDG minus the area of triangle ABE plus the area of triangle
CDE, all divided by the length of line GC:
yu
H zm- zwL Lh-0.5 Lh H zm- zrL+0.5 wo Lh H zw- zrL
Lh wu
154
( 3.67 )
Schymanski (2007): PhD thesis
Chapter 3
If we solve Equation ( 3.66 ) for zw, insert it into Equation ( 3.67 ) and then replace zm
by zm = 2 Z - zr from Equation ( 3.65 ), we could prove that Equation ( 3.59 ) is still
valid.
As the hillslope surface is linear, we can write the saturated area fraction (ωo) as a linear
function of water table elevation (zw):
wo
zw-zr
Ø
≤ zm-zr
∞
≤
±0
zw > zr
( 3.68 )
zw § zr
After inserting Equation ( 3.68 ) into ( 3.66 ) we solved for zw to obtain zw as a function
of ys:
zw
è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
zm - H zm - zrL H-2 ys + zm + zrL
µ
ys
ys > zr
ys § zr
( 3.69 )
Inserting this back into Equation ( 3.68 ) yields ωo as a function of ys:
wo
Ø
≤
≤ 1∞
≤
≤
±0
è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Hzm -zr L H-2 ys +zm+zr L
zm-zr
ys > zr
ys § zr
( 3.70 )
Using Equation ( 3.65 ), we can eliminate zm in all of the above equations and use the
average pedosphere depth (Z) instead:
wo
ys-Z
Ø
+1
≤
≤ "########################
HZ- ysL HZ- zrL
∞
≤
≤
±0
ys > zr
( 3.71 )
ys § zr
The complement of ωo, ωu = 1- ωo is then:
wu
Z- ys
Ø
≤
≤ "########################
HZ- ysL HZ- zrL
∞
≤
≤
±1
ys > zr
( 3.72 )
ys § zr
Inserting Equation ( 3.72 ) into Equation ( 3.59 ) gives a convenient formula for yu:
yu
è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
HZ - ysL HZ - zrL
µ
Z - ys
ys > zr
ys § zr
( 3.73 )
With this set of equations, we can calculate ωo, ωu and yu for any given values of Z, zr
and ys in a linear hill slope. However, for strongly non-linear hill slopes, the use of these
equations would lead to a bias, particularly in the surface fractions of saturated and
155
Appendix 3.2
Schymanski (2007): PhD thesis
unsaturated zones. Note that the length of the hillslope (Lh) is not needed, as it cancelled
out.
A.3.2.2
SOIL WATER FLUXES
Steady, vertical flow in a uniform soil in the absence of osmotic potential gradients can
be expressed by the Buckingham-Darcy Equation (Radcliffe and Rasmussen 2002):
Jw
i ∑ h H zL
y
- Kunsat j
+ 1z
k ∑z
{
( 3.74 )
In the above, Jw is the flow rate per unit area, h the “matric suction head”, z the distance
to the soil surface and Kunsat the unsaturated hydraulic conductivity. The “matric suction
head” refers to the force per unit cross-sectional area that has to be overcome by another
force in order to extract water from the soil matrix. It is commonly measured by
inserting a porous cup into the soil layer of interest, filled with pure water and coupled
with a manometer at atmospheric pressure. The change in pressure caused by
equilibration with the soil is then the “suction head”. In the present work, h with units of
pressure head (m), is defined as positive and increases with decreasing soil saturation.
Jw is defined as positive if water flows upwards and negative if it flows downwards.
To calculate the flow rate across the interfaces between different soil layers, the head
gradient (∂h/∂z) and Kunsat at these interfaces are needed. For simplicity, the head was
calculated using the average saturation degree in each layer. Then, the average heads in
all layers were connected with straight lines to estimate the gradients. This procedure is
shown in Figure 3.17, with an example of a non-equilibrium situation in (a), which
would result in an upward flux between layer 1 and 2, and a downward flux from layer
2 and layer 3. An example for an equilibrium situation is given in (b), with ∑h/∑z=-1
and hence no fluxes occurring between the layers.
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Schymanski (2007): PhD thesis
a)
Chapter 3
b)
h
0
h
0
h1
h1
h2
z
h2
h3
z
h3
Figure 3.17: Examples of matric suction heads in the soil profile and estimated gradients between
sub-layers. (a) Non-equilibrium situation, which would lead to water flux out of layer 2 into layer 1
and layer 3. (b) Equilibrium situation, where no fluxes would occur.
Following Equation ( 3.74 ), we expressed the fluxes between layer i and layer i+1 as:
Qi
A.3.2.3
wu Jw,i
i
h i+1 - h i
y
- wu 0.5 H K unsat,i+ K unsat,i+1 L jj
+ 1zz
k 0.5 Hdyu,i+ dyu,i+1 L
{
( 3.75 )
MATRIC SUCTION HEAD AND HYDRAULIC
CONDUCTIVITY
The formulations by Brooks and Corey (1966) (“BC”) and van Genuchten (1980)
(“vG”) are among the most widely used for modelling matric suction head and
hydraulic conductivity as a function of soil moisture. The BC formulation is simpler
than the vG formulation, but does not reproduce the decline of hydraulic head towards
zero near full saturation (Figure 3.18), which would lead to the undesired effect in the
model that saturated soil could still take up water from its surroundings. For a more
detailed comparison between the two formulations and possible parameter conversions,
please refer to Morel-Seytoux et al. (1996).
Because of a more realistic representation of matric pressure near full soil saturation, we
chose the vG formulation, which will be briefly described here.
157
Appendix 3.2
Schymanski (2007): PhD thesis
1
1
vG
BC
0.8
KunsatêKsat
h @mD
0.8
0.6
0.4
0.6
0.4
vG
BC
0.2
0.2
0
0
0
0.2
0.4
su
0.6
0.8
0
1
0.2
0.4
su
0.6
0.8
1
Figure 3.18: Example plots of matric pressure head (h) and relative hydraulic conductivity
(Kunsat/Ksat) using the van Genuchten (“vG”, thick grey lines) and Brooks & Corey (“BC”, thin black
lines) formulations.
The absolute volume of water per unit soil volume (θ) can take values within a range
specific to the particular soil type. The maximum value (θs) depends on the soil
porosity, i.e. the volume of voids in the soil that can be taken up by water. The
minimum value (θr), called the “residual water content”, is not defined precisely but is
often determined as the water content at some large value of the pressure head, where a
further increase in pressure head does not significantly decrease the soil moisture any
further. In this context, the saturation degree (su) of the soil can be defined as:
q -qr
qs -qr
su
( 3.76 )
The relation between matric head (h) and saturation (su) was given as (van Genuchten
1980):
su
1
ij
yzmvG
j
z
k Hh avGLnvG + 1 {
This can be inverted to
h
1
avG
ij - m1
y nvG
jjsu vG - 1zzz
j
z
k
{
( 3.77 )
1
( 3.78 )
where αvG, nvG and mvG are empirical parameters, fitted to water retention curves of a
specific soil type. The author also showed that if nvG and mvG are not fitted
independently, but satisfy the relation
mvG
1-
1
n vG
158
( 3.79 )
Schymanski (2007): PhD thesis
Chapter 3
then the unsaturated hydraulic conductivity can be expressed as:
K unsat
1 mvGy2
z
mvG yzz
è!!!!! ijj ijj
Ksat su jj1 - jj1 - su zz zzz
j j
z z
k k
{ {
( 3.80 )
The above equations, in conjunction with appropriate values for the parameters αvG, nvG
and Ksat, allow us to calculate matric suction head and unsaturated hydraulic
conductivity as a function of saturation degree (su). These are, however, very
approximate estimates as we did not consider hysteresis effects between wetting and
drying of soil (Pham et al. 2005), nor the spatial heterogeneity of the soil parameters.
A.3.2.4
CONSERVATION OF MASS AND CHANGES IN
STATE VARIABLES
Saturated Zone
Downwards flux of water into the saturated zone would cause it to expand into the
unsaturated zone, while an upward flux would cause it to contract and leave an
unsaturated volume behind. A change in the thickness of the saturated zone (ys) changes
the total volume of the unsaturated zone(ωu ä yu), which can be expressed by taking the
derivative over time of Equation ( 3.59 ), where only ωu, ys and yu are functions of time:
wu HtL
∑ yu HtL
∑ wu HtL
+ yuHtL
∑t
∑t
-
∑ ys HtL
∑t
( 3.81 )
The saturation degree in the unsaturated soil layers, on the other hand, is an intensive
variable and should not change with a change in volume. However, if the volume of the
unsaturated zone changes, but the saturation degree stays constant, the total water stored
in the unsaturated zone would change. This would result in a “loss” or “creation” of
water in the model if this water was not explicitly accounted for elsewhere. The amount
of water to be accounted for can be calculated from the change in volume of the
unsaturated zone (see Equation ( 3.81 )) and the average saturation degree in the
unsaturated zone (su):
Water loss
¶ su Ht L
∑ ys Ht L
∑t
159
( 3.82 )
Appendix 3.2
Schymanski (2007): PhD thesis
Considering that the water “lost” from the unsaturated zone due to a decrease in volume
of this zone has been incorporated by the expanding saturated zone, it has been added to
the fluxes into the saturated zone, so that the mass balance of the saturated zone was
expressed as:
¶
∑ ys HtL
∑t
-EssHtL -QnlayersHtL - Qsf HtL + ¶ suHtL
∑ ys HtL
∑t
( 3.83 )
where Qnlayers denotes Qi with i = nlayers. The left hand side denotes the change in
storage, while the right hand side is the sum of the fluxes in and out of the saturated
zone. After solving for the change in ys, we obtained:
EssHtL + QnlayersHtL + Qsf HtL
¶ HsuHtL - 1L
∑ ys Ht L
∑t
( 3.84 )
Unsaturated Zone
As stated previously, changes in the saturation degree (su) are independent of changes in
volume of each layer. The mass balance for each layer was therefore calculated at
constant volume, so that ωu and δyu,i were not written as functions of time in the below
equations.
The mass balance of the top soil layer was formulated as:
¶ wu dyu,1
∑ su,1 HtL
- EsuHtL + Q1HtL + QinfHtL - Qr,1HtL
∑t
( 3.85 )
The mass balance of the layers between the top and the bottom layers was calculated as:
¶ wu dyu,i
∑ su,i HtL
∑t
- Qi-1HtL + Qi HtL - Qr,1HtL
( 3.86 )
Equations ( 3.85 ) and ( 3.86 ) can be solved for the rate of change in soil moisture in
each layer, yielding:
∑ su,1 HtL
-EsuHtL + Q1HtL + Qinf HtL - Qr,1 HtL
∑t
¶ wu dyu,1
∑ su,i HtL
- Qi-1 HtL + QiHtL - Qr,1 HtL
∑t
¶ wu dyu,i
160
( 3.87 )
( 3.88 )
Schymanski (2007): PhD thesis
Chapter 3
The mass balance for the bottom layer of the unsaturated zone was then calculated as a
function of the change in total water store (wc) and the net flux in or out of the soil
domain. Total water store per unit catchment area is the sum of water stored in the
saturated zone (= ε ys) and the water stored in each layer of the unsaturated zone (ws,i).
This can be written as
¶ wu su,i dyu,i
( 3.89 )
wc ¶ ys + ‚ ws,i
( 3.90 )
ws,i
and
nlayers
i=1
For a finite time step between t1 and t2, we approximated the change in water store (dwc)
by summing the fluxes in and out of the soil domain at time t1 and multiplying them by
the length of the time step:
layers
yz
ij
j
Ht2 - t1L jjQinf Ht1L - ‚ HQr,i Ht1LL - EssHt1L - EsuHt1L - Qsf Ht1Lzzz
i=1
{
k
n
dwc
( 3.91 )
The soil water stored in the bottom layer at time t2 (ws,nlayers(t2)) was then approximated
as the difference between dwc and the change in storage in all other compartments:
ws,nlayersHt2L
dwc + ‚ ws,i Ht1L - ‚ ws,i Ht2L + ¶ ysHt1L - ¶ ysHt2L
nlayers
i =1
nlayers-1
( 3.92 )
i =1
The value of su,nlayers at time t2 was then obtained from ws,nlayers:
su,nlayersHt2L
ws,nlayersHt2L
¶ wuHt2L dyu,nlayersHt2L
( 3.93 )
This procedure ensured that the mass balance was closed at every time step and the
model did not “create” or “lose” any water.
161
Appendix 3.3
Schymanski (2007): PhD thesis
APPENDIX 3.3. EQUATIONS USED IN THE
VEGETATION OPTIMALITY MODEL
A.3.3.1
RESISTIVITY TO WATER FLOW TOWARDS ROOTS
The flow of water towards roots can be inhibited by the soil’s resistivity to the
movement of water. This resistivity has been included in the expression for root water
uptake (Equation ( 3.30 )) as Ωs,i. Hunt et al. (1991) mentioned that Ωs,i is small
compared with Ωr under normal rooting conditions and can probably be neglected.
However, we encountered numerical problems when neglecting Ωs,i as root water uptake
could then still occur at a high rate, even in very dry soil layers. Numerical stability had
to be maintained even with unrealistic vegetation parameters, as the optimisation
algorithm included a random search of the parameter space (see Section 3.3.3.8).
Ωs,i can be expressed as a function of the unsaturated hydraulic conductivity (Kunsat,i) if
one takes into account that Ωs,i is used to calculate flow as a function of a head
difference (∆h, see also Equation ( 3.30 )), while Kunsat,i is used to calculate flow as a
function of a head gradient (dh/dx, see also Equation ( 3.74)). Head difference is
generally used if dx is fixed, as in the case of roots with a certain diameter, where dx is
already factored into Ωr. In the soil, horizontal flow between two points, a distance dx
apart, with hydraulic heads h1 and h2, can thus be expressed using Ωs,i, which depends
on dx, or using Kunsat,i:
Jw
h1 - h2
Ws ,i
i h1 - h2 yz
z
k dx {
K unsat,i jj
( 3.94 )
Solving for Ωs,i, gives the relation between Ωs,i and Kunsa,it:
Ws
dx
K unsat,i
( 3.95 )
If soil conductivity is low, water flow towards the root would be slow and hi in the
vicinity of the root would decline, which would feed back onto the root water uptake
rate. An increase in root density can partly overcome low soil conductivity by
shortening the average distance water has to flow through the soil towards the roots.
162
Schymanski (2007): PhD thesis
Chapter 3
These effects were parameterised by incorporating the average distance between roots
into the expression for Ωs,i.
The value of dx was defined as half of the average distance between two roots and can
be calculated from the root surface area and root radius if we assume that all roots are
straight lines at equal distances from each other. If n lines have length l and are a
distance 2ädx apart, the volume of soil they would take up is given as:
4 dx2 l n
Vs
( 3.96 )
Now, län is the total root length (lr) in the given volume of soil, so that we can write:
è!!!!!
è!!!!
Vs
dx
( 3.97 )
2 lr
The surface area (SAr) of roots with a total root length lr and a radius rr each can be
written as:
SAr
2 p lr rr
( 3.98 )
Solving Equation ( 3.98 ) for lr and inserting into Equation ( 3.97 ) yields dx as a
function of soil volume, root radius and root surface area:
p rr Vs
$%%%%%%%%%%%%%%
2 SAr
dx
( 3.99 )
To calculate the soil resistance component for layer i, we defined SAdr,i as the root
surface area density (root surface area per unit soil volume) in that layer:
SAdr,i
SAr
( 3.100 )
Vs
Inserting Equation ( 3.100 ) into ( 3.99 ) and then into Equation ( 3.95 ) gives soil
resistivity as a function of the unsaturated hydraulic conductivity (Kunsat,i), root radius
(rr) and root surface area density in soil layer i (SAdr,i):
Ws ,i
1
K unsat,i
$ %%%%%%%%%%%%%%%%%%%
p rr
2 SAdr,i
163
( 3.101 )
Appendix 3.3
Schymanski (2007): PhD thesis
In the above, the subscripts i indicate that the symbols refer to soil layer i. Fine root
radius rr was assumed to be the same in all soil layers and was set to rr = 0.3ä10-3 m, a
typical value for citrus fine roots following Bryla et al. (2001).
A.3.3.2
TISSUE WATER STORAGE AND BALANCE
PRESSURE
A common method for estimating the forces acting on the liquid phase in a plant organ
is the use of a pressure bomb (“Scholander bomb”). The procedure has been described
as follows:
“A cut leaf (or other plant part) is placed inside a sealed chamber with the cut
proximal end protruding from the chamber into the atmosphere. Gas is forced
into the sealed chamber, and when liquid appears in the xylem at the cut end the
pressure in the chamber is denoted as the balance pressure (Pb)”(Roderick and
Canny 2005)
The common assumption is that if the liquid inside the intact plant organ was under
tension, cutting it would lead to air entry at the cut surface and relaxation of the tension
by retraction of the liquid. This retraction is assumed to be reversed by applying an
external pressure to the intact bit of the cut plant organ until the liquid re-appears at the
cut surface. The pressure applied is then assumed to be equivalent to the suction that
was present in the intact plant organ. The balance pressure measured in this way has
been frequently used to quantify a plant’s water status, assuming that the balance
pressure would increase as the plant’s water content decreases.
Roderick and Canny (2005) formulated some theoretical relationships between the
balance pressure measured by the “Scholander bomb” and the relative saturation of the
leaf, based on a simple “plastic-bottle analogy”.
First of all, they divided the mass of a leaf into its dry matter (md) and its liquid matter
(mq), so that the total mass of a leaf (md) at any instant in time was written as:
m
md + mq
( 3.102 )
When a leaf is saturated (in equilibrium with 100% relative humidity), the liquid mass is
at its maximum (mqx) so that the maximum mass of a leaf (mx) is given by:
164
Schymanski (2007): PhD thesis
Chapter 3
md + mqx
mx
( 3.103 )
The leaf water status can be represented by its relative water content (q), i.e. the mass of
its liquid matter divided by the total leaf mass:
q
mq
m
( 3.104 )
mqx
mx
( 3.105 )
The saturated leaf water status is then:
qx
Another useful representation of leaf water status is the “relative water loss” (rWL),
defined as:
r WL
mx - m
mx
mqx - mq
md + mqx
( 3.106 )
Based on theoretical grounds, Roderick and Canny (2005) predicted that the balance
pressure as measured by the pressure bomb should increase linearly with increasing rWL:
Pb
k rWL
( 3.107 )
The proportionality constant κ was predicted to vary with the type of leaf, from
hygrophytes to xerophytes. The rationale was that if leaves behave like plastic bottles
that are partly filled with a liquid, the application of a certain external pressure would
squeeze the bottle enough to lead to the appearance of the liquid at the opening of the
bottle. The amount of pressure necessary for the squeeze should depend not only on the
amount of liquid in the bottle, but also on the thickness of the bottle walls. Bottles with
thicker walls (low values of qx) would generally require more pressure to achieve a
certain amount of squeeze than thin-walled bottles (high values of qx). Thus, the value
of κ should increase with decreasing value of qx. At qx = 1, the value of κ should be 0, as
no solid matter would be present to oppose the squeeze in this case. Roderick and
Canny (2005) predicted that the relationship between κ and qx should be generally linear
for the natural range, but tend towards infinity for very small values of qx, as no
pressure can squeeze out water where there is none present. From measurements on a
wide range of leaves, they derived the following empirical relationship, if κ is expressed
in bars:
165
Appendix 3.3
Schymanski (2007): PhD thesis
k
753 - 763 qx
( 3.108 )
To make the relationship consistent with κ → ∞ as qx → ∞, we modified it to:
k
750-750 qx +
1
qx
( 3.109 )
A comparison between the original and the modified relationship is shown in Figure
3.19.
κ HbarL
750 − 750 qx + 1êqx
753 − 763 qx
1750
1500
1250
1000
750
500
250
0.2
0.4
0.6
0.8
1
qx
Figure 3.19: Comparison between original and modified formulation of κ as a function of qx. The
curves are virtually indistinguishable for the majority of the range of qx.
Combining Equations ( 3.109 ) and ( 3.107 ) gives the balance pressure as a function of
qx and rWL:
Pb
1y
ij
j- 750qx + 750 + zz rWL
qx {
k
( 3.110 )
Pb HbarL
600
500
qx=0.2
400
qx=0.4
300
200
qx=0.6
100
qx=0.8
rWL
0.2
0.4
0.6
0.8
Figure 3.20: Relations between Pb and rWL for different values of qx.
166
1
Schymanski (2007): PhD thesis
Chapter 3
Theoretical relations between Pb and rWL for different values of qx are shown in Figure
3.20. Roderick and Canny (2005) presented measurements of these properties for leaves
with qx ranging between 0.52 and 0.85.
Balance pressure, turgor loss, cavitation and plasmolysis
Zweifel et al. (2000) described water loss from tree trunks during air-drying with the
following words:
"The air-drying period can be roughly divided into two phases: a first phase,
during which the amount of water lost from the stem segments is linearly
correlated to stem radius, and a second phase, where the relation is not linear,
which means that the evaporated water will not be expressed as a volume change.
The transition between the two phases can be assigned to a range of a, between
-2.0 and -2.6 MPa."
The authors measured stem diameter, weight and Pb in the top segments of young trees
as a function of available water in the bark while the samples were progressively airdried. Pb changed linearly with available water in the bark at balance pressures of less
than 20 bar and available water between 2.7 and 3.0 g. As available water decreased
past 2.7 g, and Pb increased past 20 bar, the relationship became non-linear and Pb
stayed close to constant between 25 and 30 bar, even if available water in the bark
decreased as far as 0.5 g. Thus, the linear relationship between Pb and stored water only
held for the first 10% of relative water loss. Within this linear range, the decrease in
water content was explained by a decrease in volume of the stem. This is in accordance
with Roderick and Canny’s (Roderick and Canny 2005) “plastic-bottle analogy”, as it
confirms that the pressure-induced extraction of water from the tissue is indeed due to a
change in volume. As water gets depleted past the linear stage, stem volume remains
constant, while water is removed from the xylem of the stem and replaced by air
(Zweifel et al. 2000).
It is remarkable that most of the data sets used in Roderick and Canny (2005) also only
span over less than 10% of the water storage capacity, and one of those that span over a
range of 20% shows a distinct change in slope past 10% (Figure 6 in Roderick and
Canny 2005). The same threshold of 10% for a change in slope of Pb versus relative
water content is shown in Brodribb and Holbrook (2003) for leaves of a leguminous tree
167
Appendix 3.3
Schymanski (2007): PhD thesis
(Gliricidia sepium). Myers et al. (1997) documented incipient plasmolysis at a relative
water content of 85% to 94% in a range of trees from fully deciduous to fully evergreen
in a savanna close to Darwin, Australia.
The above observations led us to the following hypothesis:
A reduction in water content of living tissues by 10% from the saturated value is
reversible, while further reduction in water content may lead to permanent cell
damage.
If this is the case, the maximum value of relative water loss (rWLx) that living tissues can
be subjected to without damage can be calculated by combining Equations ( 3.102 ) and
( 3.106 ), and replacing mq with 0.9ämqx:
r WLx
0.1 m qx
m d + m qx
( 3.111 )
This can also be written as a function of qx, if we combine Equations ( 3.103 ) with
( 3.105 ) and insert into Equation ( 3.111 ) to obtain:
rWLx
0.1 qx
( 3.112 )
Inserting the above into Equation ( 3.107 ) yields the maximum value of Pb that can be
achieved before permanent damage occurs (Pbx):
P bx
- 75 q2x + 75 qx + 0.1
( 3.113 )
Equation ( 3.113 ) suggests that the maximum supportable balance pressure (Pbx) of a
living tissue depends on its intrinsic property qx. It is a negative quadratic function, with
a maximum of Pbx = 18.85 bar at qx = 0.5. The water storage capacity per unit dry mass
increases with qx, which suggests that there is a trade-off between water storage
capacity and maximum supportable balance pressure for qx > 0.5. For qx < 0.5, both
water storage capacity per unit dry mass and maximum supportable balance pressure
decrease. Hence, if these relations are true, we would not expect to find any living plant
tissues with qx < 0.5. Considering that the wide variety of leaves investigated by
Roderick and Canny (2005) had values of qx ranging between 0.52 and 0.85, this
statement seems to be supported by observations.
168
Schymanski (2007): PhD thesis
Chapter 3
According to Equation ( 3.113 ), the maximum balance pressure that can be supported
by living tissues should not exceed 18.85 bar (or 1.885 MPa) for tissues with any values
of qx. Lo Gullo et al. (1998) observed plasmolysis in Olea oleaster leaves at 1.89 MPa
balance pressure, Brodribb and Holbrook (2003) determined the turgor loss point at 1.5
MPa and the aforementioned example from Roderick and Canny (2005) shows a change
in slope at 15 bar. Koch et al. (2004) reported the lowest xylem pressure in the tallest
trees as 1.84 MPa. All of this is consistent with the theory presented here, based on the
“plastic-bottle analogy” by Roderick and Canny (2005).
Balance pressure and root suction head
Equation ( 3.110 ) combined with the condition that rWL ≤ 0.1 qx, as stated in Equation
( 3.112 ), provides a model of the “balance pressure” as a function of relative water loss
and tissue properties, which gives realistic values in terms of observed values of qx and
observed ranges of balance pressure.
In the following, we will use the above equations to relate the total amount of water
stored in living plant organs per unit catchment area to the suction head within these
plant organs. This will allow modelling of root water uptake as a function of vegetation
water status.
Stepwise insertion of Equations ( 3.106 ), ( 3.105 ) and ( 3.103 ) into Equation ( 3.110 )
results in the balance pressure (Pb) as a function of the extensive variables mqx, md and
mq:
Pb
i 750md
1 yz
z
Hmqx - mqL jj
+
k Hmd + mqxL2 mqx {
( 3.114 )
In the above expression, md is mainly related to the amount of carbon that has been
invested into cell walls, while the relation between md and mqx is a plant morphological
feature. These parameters are not expected to change very much from day to day, while
the amount of water stored in the plants (mq) is expected to change on a diurnal scale,
affected by water loss through stomata and water uptake by the roots. This diurnal cycle
in stored water is often expressed in the swelling and shrinking of tree trunks.
Equation ( 3.114 ) and was used to compute the root suction head in each soil layer as a
function of variable vegetation water storage (mq), for given values of mqx and md. For
169
Appendix 3.3
Schymanski (2007): PhD thesis
consistency with the water balance equations, mqx, md, and mq were replaced by their
respective values per m2 ground area (Mqx, Md, and Mq):
i 750 Md
1 yz
z
Pb HMqx - MqL jj
+
k HMd + MqxL2 Mqx {
170
( 3.115 )
Schymanski (2007): PhD thesis
Chapter 3
APPENDIX 3.4. CONVERSIONS OF METEOROLOGICAL
DATA
A.3.4.1
DIURNAL VARIATION OF AIR TEMPERATURE
The long-term meteorological data set contained daily minimum and maximum
temperatures only. This section describes how we inferred the diurnal variation in
temperature from daily data and compares inferred temperature with temperature
measured on the site over a shorter time period.
Based on the review of temperature models by Bilbao et al. (2002), we chose “Erbs’
Model” of diurnal variation in air temperature, which requires only information about
the mean and range of temperature. We replaced the respective monthly parameters in
the original model by the daily mean air temperature (Ta,m, in K) and daily temperature
range (Ta,r, in K) and wrote:
Ta
Ta,m + Ta,r H0.4632 cos Hch-3.805L + 0.0984 cos H2 ch - 0.360L +
0.0168 cos H3 ch - 0.822L + 0.0138 cos H4 ch - 3.513LL
( 3.116 )
The parameter ch changes with the hour of the day (th):
ch
1
pHth - 1L
12
( 3.117 )
To test whether this model was applicable to our study site, we used half-hourly air
temperature measurements for 2004/2005 (Schymanski et al. in prep.-a), computed the
mean air temperature and temperature range for each day and used these to simulate the
diurnal temperature variation for each day using the above equations (Figure 3.21 and
Figure 3.22).
171
Appendix 3.4
Schymanski (2007): PhD thesis
219
220
221
222
Air temperature HoCL
224
Modelled
Observed
35
30
25
20
219
33
220
221
222
Day of year
34
35
223
36
37
224
38
Modelled
Observed
35
Air temperature HoCL
223
32.5
30
27.5
25
22.5
33
34
35
36
Day of year
37
38
Figure 3.21: Estimated and observed air temperature for 5 days in the dry season in 2004 (left) and
5 days in the wet season in 2005 (right). Irregularities in the estimated time series can occur at
midnight, when the model jumps from one day’s temperature characteristics to the next day’s
characteristics.
Modelled Ta HoCL
35
30
25
20
15
15
20
25
Observed Ta HoCL
30
35
Figure 3.22: Scatter plot of estimated versus observed air temperature for half-hourly
measurements over 12 months (July 2004 to June 2005). Dashed line marks the 1:1 line.
172
Schymanski (2007): PhD thesis
A.3.4.2
Chapter 3
CALCULATION OF THE SOLAR ANGLE
The solar angle (β) was calculated after Spitters et al. (1986).
The sine of the solar angle was written as:
sinH bL
cosHdL cosHfL cosH15 Hth - 12LL + sinHdL sinHfL
( 3.118 )
where φ is the geographical latitude and all angles are given in decimal degrees. The
solar declination (δ) was modelled using the following equation:
sinHdL
-0.397949 cosH360HDy+10Lê365L
( 3.119 )
where Dy is the day of the year.
The integral of sin(β) over the day was written as:
‡ sinH bHthLL „ th
3600 J
24
"###############################
cosHdL 1 - tan2HdL tan2HfL cosHfL + sinHdL sinHfL DlN
p
( 3.120 )
where Dl is the day length, defined as the time between sunrise and sunset:
Dl
24
sin-1HtanHdL tanHfLL + 12
180
( 3.121 )
Note that Equation ( 3.121 ) was printed with a “-” between the bracketed expression in
Spitters et al. (1986), but this appeared to be a typographical error.
173
Appendix 3.5
Schymanski (2007): PhD thesis
APPENDIX 3.5. IMPLEMENTATION OF THE SHUFFLED
COMPLEX EVOLUTION (SCE) ALGORITHM
The Shuffled Complex Evolution (SCE) algorithm (Duan et al. 1993; Duan et al. 1994)
is an algorithm to find the optimal values of n optimisable parameters (degrees of
freedom) that would maximise or minimise a function of these parameters. In the case
of the present study, n = 6 and the parameters are MA,p, yr,p, cλf,p, cλe,p, cλf,s and cλe,s. The
objective function is the maximisation of the Net Carbon Profit (NCP). We will call the
computation of a value of the objective function (NCP) for one parameter set, i.e. the
completion of 30 years of simulation in this case, a “run”. Many runs with different sets
of values for the optimisable parameters led to a “population” of parameter sets and
their respective values of NCP.
The SCE algorithm subdivides a population of parameter sets into a pre-defined number
of complexes, and then applies a combination of the local search simplex procedure
(Nelder and Mead 1965), a systematic complex evolution and competitive mixing
between complexes. The completion of all of these procedures marks the completion of
one “loop” and ends with a new population, which gets sorted according to the values of
the objective function (NCP), and cropped to remove the worst sets before a new loop is
started.
At the end of each loop, the SCE algorithm assesses whether any convergence
conditions have been reached, in which case the algorithm would stop and the parameter
set with the best objective function would be considered the optimal one. Common
convergence conditions are for example the achievement of a certain value for the
objective function or the convergence of the parameter sets to an arbitrarily small part of
the feasible parameter space. In the present study, the real value of maximal NCP was
not even known approximately, so that convergence of the parameter sets was the only
convergence condition that was applied. The optimisation was assumed successful, if
maximal relative parameter range explored by the population was smaller than 0.0001:
MaxHUi, jL < 0.0001
where
174
( 3.122 )
Schymanski (2007): PhD thesis
Ui, j
†ci, j - ci,mean§
, i = 1, ... n; j = 1, ..., s
cmax,i - cmin,i
Chapter 3
( 3.123 )
The symbol cj,k denotes the value of an optimisable parameter cj in set k, cj,mean denotes
the average value of parameter cj over all sets, the value n is the number of optimisable
parameters (degrees of freedom) and s is the number of parameter sets in the population.
The feasible range of ach parameter cj was predefined as the range between cmin,j and
cmax,j.
However, upon repetition of the optimisation with the same parameter values, we
experienced that the model sometimes converged prematurely, before reaching even a
local maximum. To break such premature convergence, we included a perturbation of
the parameters in the code, which was applied every time when convergence occurred.
The added code perturbed each parameter of the “optimal” set by 0.1%, 1%, 10% and
100% of its feasible range (cmax,j – cmin,j), one parameter at a time, and assessed whether
the objective function was improved by any of the perturbations. If so, the newly
created parameter sets were included in the population and the optimisation was
continued with the extended population of parameter sets. If none of the perturbations
led to an improvement of the objective function, it was assumed that the SCE algorithm
has found a local or the global maximum.
The convergence into local maxima was reduced by seeding the model with a
systematically located initial population (Muttil and Liong 2004) that explored the
feasible parameter space more evenly than a randomly generated population would.
However, none of the methods employed could guarantee that the global maximum in
NCP has been found, so the procedure was frequently repeated to confirm or disprove
previous results. We tried a number of different parameterisations of the SCE model and
found that the parameterisation given in Table 3.3 yielded reasonably robust results with
an acceptable number of runs (in the order of 3000-5000 runs for the optimisation of 6
degrees of freedom).
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Appendix 3.5
Schymanski (2007): PhD thesis
Table 3.3: Parameterisation of the SCE model used in the present study
PARAMETER DESCRIPTION
VALUE
Number of complexes (parameter p in Duan et al. 1993)
2
Number of optimisations per complex and run (parameter α in Duan
et al. 1993)
3
Convergence criterion
MaxHUi,jL < 0.0001
Feasible range of MA,p
0.0 to 1.0
Feasible range of yr,p
1.0 to 9.0
Feasible range of cλf,p
0.0 to 104
Feasible range of cλe,p
-3.0 to 1.0
Feasible range of cλf,s
0.0 to 104
Feasible range of cλe,s
-3.0 to 1.0
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Schymanski (2007): PhD thesis
Chapter 3
APPENDIX 3.6. SOIL EVAPORATION AS A FUNCTION
OF SURFACE SOIL MOISTURE AND SOIL
TEMPERATURE
Observed soil evaporation on the study site was typically between 0.1 to 0.5 mm/day in
the dry season and up to 0.65 mm/day in wet season (Hutley and Beringer, unpubl.
data), i.e. up to 25% of total evapo-transpiration. In the presence of observed soil
temperature data, soil evaporation (Es) was modelled using a flux-gradient approach as
follows.
Es
Gsoil HWs - WaL
( 3.124 )
where (Ws – Wa) is the difference between the mole fraction of water in the laminar
layer immediately above the soil (Ws) and the mole fraction of water in the atmosphere
(Wa), while Gsoil is the conductivity of the soil to water vapour fluxes.
Wa was obtained from measured atmospheric vapour pressure divided by air pressure,
and Ws was calculated as the vapour pressure in the laminar layer immediately above
the soil (pvs) divided by air pressure (P).
Ws
pvs
P
( 3.125 )
To get pvs, we followed a model by Lee and Pielke (1992), which has been found by
Silberstein et al. (2003) to give better soil evaporation results than the use of the vapour
pressure in the soil pore spaces. The model formulates pvs as a function of the
atmospheric vapour pressure (pva), the saturation vapour pressure (pvssat) at soil
temperature Ts, actual volumetric soil moisture (θ) and volumetric soil moisture at field
capacity (θfc):
pvs
Ø
≤ pvssat sin4J p q N + J1 - sin4J p q NN pva q < qfc
2 qfc
2 qfc
∞
≤
q ¥ qfc
± pvssat
( 3.126 )
Atmospheric vapour pressure, as well as the volumetric soil moisture (θ) in the top soil
was measured on site and the value for soil moisture at field capacity (θfc) for the site
was set to 0.15575, equivalent to the soil moisture at a matrix pressure head of -0.01
MPa (Kelley 2002). The saturation vapour pressure just above the soil (pvssat, in Pa) was
177
Appendix 3.6
Schymanski (2007): PhD thesis
taken as the saturation vapour pressure at soil temperature Ts (in ˚C) and calculated
using the commonly used approximation (Allen et al. 1998):
pvssat
17.27Ts
T
610.8 ‰ s+237.3
( 3.127 )
The parameter Gsoil in Equation ( 3.124 ) was set to the value 0.03 mol m-2 s-1, which
resulted in a time series of soil evaporation of around 0.1 to 0.2 mm/day in the dry and
0.4 to 1.0 mm/day in the wet season if observed values of θ were used. To model Es
from modelled surface soil saturation (su,1), su,1 was converted to θ using Equation
( 3.76 ).
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Schymanski (2007): PhD thesis
Chapter 4
CHAPTER 4. POSSIBLE LONG-TERM EFFECTS OF
INCREASED CO2 ON VEGETATION AND THE
HYDROLOGICAL CYCLE
4.1
ABSTRACT
The continuous increase in atmospheric CO2 concentrations has the potential to
significantly influence hydrological cycles. Besides having an effect on global
temperatures and climate, increased CO2 is now thought to lead to a global decrease in
transpiration by reducing stomatal aperture in plants. The effect of CO2-induced
stomatal closure is progressively being incorporated in global climate predictions and
has even been attributed to be responsible for the observed increase in global
continental runoff during the past century. However, stomatal closure might not be the
only means by which vegetation responds to an increase in atmospheric CO2
concentrations. If, for example, stomatal closure is offset by an increase in vegetation
cover in the long term, the effect of increased CO2 on global transpiration could, in fact,
be reversed.
In the present study we use a coupled water balance and vegetation model to investigate
the possible long-term response of vegetation to increased levels of atmospheric CO2. In
contrast to conventional models, the model used here not only allows for the adaptation
of stomatal conductance, but also for adjustments in biochemical foliage properties,
vegetation cover and rooting depth. Short-term responses to increased CO2 are separated
from long-term responses by distinguishing which vegetation properties can adapt in the
long term from those capable of adaptation in the short term. The model is applied to a
number of catchments in different climates and reveals that, in certain climates, the
long-term effect of elevated CO2 on transpiration can be opposite to the short-term
effect. The ramifications for global climate predictions and the dangers in the
extrapolation of short-term observations to long-term environmental change are
discussed.
179
4.2 Introduction
4.2
Schymanski (2007): PhD thesis
INTRODUCTION
Almost 30 years ago, Ian R. Cowan concluded a chapter on water use in higher plants
with a vision about the greenhouse effect:
“There is certainly no precedent in agricultural history, perhaps none in biological
history, for such a rapid relative change in atmospheric CO2 concentration. How will
the standing biomass of plants alter, how will competition between plant species
(particularly between C3 and C4 species) be affected, what is the nature of the genetic
variance which plant breeders have at their disposal to take advantage of the change
and so on? All of these questions have implications in the study of plant-water relations.
For example, if plants do not or cannot be made to grow faster it may be that they will,
or can be made to grow at the same rate, with a smaller usage of water. Such
speculations emphasize the importance of understanding the interplay of hydrology,
meteorology, and carbon fixation in the biosphere.” (Cowan 1978)
This interplay of hydrology, meteorology and carbon fixation in the biosphere has only
recently been put into the spotlight again, when a team of researchers attributed the
observed increase in global continental river runoff to the effect of increased CO2 on
global vegetation water use (Gedney et al. 2006). Their results were based on a
mechanistic model that incorporates a sensitivity of stomatal conductance to
atmospheric CO2 concentration. In this model, the observed increase in atmospheric
CO2 concentrations since 1960 led to stomatal closure, which resulted in a decrease in
transpiration approximately equivalent to the observed increase in continental runoff
since 1960. The authors concluded that the direct CO2 effect on transpiration may
increase freshwater availability and even suggested the use of long term records of river
runoff to infer historical CO2 concentrations in the atmosphere.
The perception that the continuous increase in atmospheric CO2 concentrations may
increase runoff and the amount of available fresh water in the long term is widespread
in the climate change community. To some extent, it may somehow alleviate the
otherwise grim future scenarios associated with the greenhouse effect (Oki and Kanae
2006). However, the evidence that increased CO2 will lead to decreased transpiration in
the long term is rather weak. Large scale free-air CO2 enrichment (FACE) experiments
have shown different responses of plants to elevated CO2 than the more traditional
chamber experiments, probably because they focus on different vegetation types
180
Schymanski (2007): PhD thesis
Chapter 4
(Ainsworth and Long 2005; Körner 2006). As opposed to small-scale chamber
experiments, FACE experiments allow observations on whole canopies of crops and
relatively homogeneous forest plantations, but they have only been conducted since the
1990s and are mainly limited to temperate vegetation (Ainsworth and Long 2005;
Körner 2006). Even if the leaf area index of existing plants is not significantly affected
by increased CO2, as suggested by many of the FACE experiments so far, it is possible
that the number of plants could increase in natural systems where it is currently limited
by the availability of water. An increase in the number of plants could lead to more
efficient water extraction by roots, which could eventually even result in increased
overall transpiration.
Longer-term FACE observations and experiments encompassing a wider range of
vegetation types are needed to forecast future food and water supply empirically
(Ainsworth and Long 2005). In the meantime, ecological and hydrological modelling
may give us hints about the direction of future changes. If the aim is to forecast longterm response of the biosphere to climate change and increased atmospheric CO2, we
cannot rely on extrapolations of past trends into future. A more thorough understanding
of the reasons behind adaptation and acclimatisation, the degrees of freedom that can be
adjusted and their limitations is needed.
The present study investigates how increased CO2 might affect vegetation and the water
balance in the medium and long term, using a previously tested model of the dynamic
feedbacks between natural vegetation and the water balance (Schymanski et al. in prep.c). Rather than prescribing vegetation response to environmental change, the model is
based on the assumption that vegetation self-optimises to maximise its “Net Carbon
Profit” (i.e. maximising the difference between carbon acquired by photosynthesis and
carbon spent on maintenance of the organs involved in its uptake) and finds the
“optimal” vegetation for given environmental conditions. The “optimal” vegetation
represents a long-term adaptation, allowing for changes in vegetation cover, rooting
depths and species composition. Some dynamic vegetation parameters, on the other
hand, are optimised in the short to medium term and follow the seasonality and year-toyear variability of climate.
Using catchment properties and meteorological data for five contrasting catchments
around Australia, the present study aims at answering the following questions:
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4.3 Methods
Schymanski (2007): PhD thesis
1. Does the model give reasonable estimates of evapo-transpiration in all five
catchments without any information about the local vegetation?
2. What would be the difference in predicted annual transpiration rates if only
dynamic vegetation properties were allowed to adapt to increased CO2 (mediumterm adaptation)?
3. What would be the difference in predicted annual transpiration rates if all
vegetation properties were allowed to adapt to increased CO2 (long-term
adaptation)?
4. Does an increase in atmospheric CO2 have similar effects on transpiration in all
five catchments and climates?
4.3
4.3.1
METHODS
COUPLED WATER BALANCE AND VEGETATION
OPTIMALITY MODEL
The model used in this study is a coupled water balance and vegetation dynamics
model, which does not rely on any input of site-specific vegetation properties or past
observations of vegetation response to environmental forcing. This model has been
described elsewhere in detail (Schymanski et al. in prep.-c) and will only be
summarised here. Figure 4.1 gives an overview of site-specific input variables needed
for the combined water balance and Vegetation Optimality model, and how the different
components are coupled. The parameters featuring in the model and their computation
will be briefly described below.
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Schymanski (2007): PhD thesis
Z, zr, Λs
Ksat, avG, nvG
Chapter 4
Qrain
Ia
SCE model
Dv, Ta
cλf,s, cλf,p, cλe,s
cλe,p, yr,p, MA,p
NCP
Water balance
model
Vegetation Optimality
model
su,i, hi
Qsf
Qiex
Qr,i
Ag
Mq, Pb
Et
Jmax25
Gs,
ys, yu,wu,wo,
SAr,i
Esu
Rr
yr
Ess
MA
Rl
Rv
Figure 4.1: Flow diagram of the coupled water balance and Vegetation Optimality model. Input
variables are at the top, while model outputs are separated into state variables (dashed boxes) and
fluxes (along arrows). Symbols are explained in the main text (the subscript i denotes a vector over
all soil layers). For clarity, only selected model outputs are drawn.
4.3.1.1
Water Balance
The water balance part of the model is spatially lumped (based on the “Representative
Elementary Watershed” (REW) concept by Reggiani et al. 2000), but has an added
vertical resolution of the unsaturated zone into layers of 0.5 m thickness to account for
the importance of surface wetting and drying for plant water use (Schymanski et al. in
prep.-c).
The geometrical catchment properties needed for the model were the average depth of
the pedosphere (Z), the average elevation of drainage channels with respect to the
average bedrock elevation (zr) and the average slope angle of the seepage face (g0).
Estimates of these parameters were obtained by studying the topography of each
catchment and then used to compute the area fraction covered by the unsaturated zone
(wu) and the thickness of the unsaturated zone (yu) as a function of the thickness of the
saturated zone (ys) at each time step:
wu
Z- ys
Ø
≤
≤ "########################
HZ- ysL HZ- zrL
∞
≤
≤
±1
and
183
ys > zr
ys § zr
( 4.1 )
4.3 Methods
yu
è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
HZ - ysL HZ - zrL
µ
Z - ys
Schymanski (2007): PhD thesis
ys >
ys §
zr
zr
( 4.2 )
The unsaturated zone was then sub-divided into soil layers of thickness δyu,i each and the
water fluxes between different soil layers were calculated based on a one-dimensional
representation of Richards’ equation.
The flux between layer i and layer i+1 was calculated as:
Qi
hi+1 - hi
i
y
-wu 0.5 H Kunsat,i + Kunsat,i+1L jj
+ 1zz
k 0.5 Hdyu,i +dyu,i+1L {
( 4.3 )
where h is the “matric suction head” and Kunsat the unsaturated hydraulic conductivity.
The subscript i denotes the ith layer (i = 1 at the top), Qi denotes the flux across the
bottom boundary of layer i and dyu,i denotes the thickness of layer i. In the present work,
h with units of pressure head (m), is defined as positive and increases with decreasing
soil saturation. Qi is defined as positive if water flows upwards and negative if it flows
downwards.
In the saturated zone, the hydraulic conductivity is Ksat and the matric suction head (h)
is assumed to be 0, so that the flux across the boundary between the unsaturated and the
saturated zone was written as:
hi yz
i
z
Qnlayers -wu 0.5 HKunsat,i + KsatL jj1 k 0.5 dyu,i {
( 4.4 )
During rainfall, infiltration was assumed to only occur into the unsaturated zone and the
infiltration capacity was expressed by imagining an infinitely thin layer of water above
the top soil layer and writing the infiltration capacity as Qi for i = 0 , where Kunsat,i is
replaced by Ksat, hi by 0, and dyu,i by 0 in Equation ( 4.3 ). The rate of infiltration (Qinf)
was then formulated as the lesser of infiltration capacity and rainfall intensity (Qrain):
Q inf
ii
h1
y
y
Minjj jj
+ 1zz K sat w u , Q rain w u zz
kk 0.5 d yu,1
{
{
( 4.5 )
The above equations require the calculation of matric suction head (h) and unsaturated
hydraulic conductivity (Kunsat) in each soil layer at each time step. These were obtained
as a function of the soil saturation degree (su), from the widely used water retention
model by van Genuchten (1980):
184
Schymanski (2007): PhD thesis
Chapter 4
1
avG
h
ij - m1
y nvG
jjsu vG - 1zzz
j
z
k
{
1
( 4.6 )
where nvG and mvG are assumed to follow the relation:
mvG
1-
1
n vG
( 4.7 )
The unsaturated hydraulic conductivity (Kunsat) was expressed as (van Genuchten 1980):
1 mvGy2
mvG yzz
z
è!!!!! ijj ijj
Ksat su jj1 - jj1 - su zz zzz
j j
z z
k k
{ {
K unsat
( 4.8 )
The parameter values of Ksat, αvg and nvG were taken from the literature as typical values
given for the dominant soil type on each site.
Groundwater discharge (Qsf) was expressed as a function of the average thickness of the
saturated zone (ys), and the surface area fraction covered by seepage faces (wo = 1 - wu):
Qsf
Ksat H y s - zrL wo
2 cosHg 0 L L s
( 4.9 )
where the value of the “typical horizontal length scale for seepage face flow”(Ls) was
set to a value of 10.0 m, following the approach by Reggiani et al. (2000).
Soil evaporation (Esu and Ess for the unsaturated and saturated surface area fractions
respectively) was assumed to affect the top soil layer only and was written as a function
of the saturation degree in that soil layer, and the amount of global radiation (Ig) that
reaches the soil:
Esu
and
Ess
Ig H1 - 0.8 H1 - MALL wu su,1
lr
Ig H1 - 0.8 H1 - MALL wo
lr
( 4.10 )
( 4.11 )
where MA is the vegetated fraction of the surface, λE is the latent heat of vaporisation,
taken as 2.45ä106 J kg-1 and ρ is the density of liquid water.
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4.3 Methods
Schymanski (2007): PhD thesis
The final component affecting the water balance is root water uptake (Qr,i). Root water
uptake was assumed to withdraw water from the unsaturated soil layers only and was
modelled using an electrical circuit analogy, where the driving force is the difference
between root suction head and matric suction head in each layer (hr,i and hi
respectively). Radial root resistivity (Ωr) and soil resistivity (Ωs,i) were in series in each
soil layer:
Qr,i
i hr,i - hi yz
SAr,i j
k Wr + Ws,i {
( 4.12 )
where SAr,i is the root surface area per unit catchment area in soil layer i. The resistivity
to water flow towards the roots in the soil (Ωs,i) was formulated as a function of
unsaturated hydraulic conductivity (Kunsat,i), root radius (rr) and root surface area density
in soil layer i (SAdr,i):
Ws ,i
1
Kunsat,i
$%%%%%%%%%%%%%%%%%%%
p rr
2SAdr,i
( 4.13 )
The root surface area density (SAdr,i) refers to the root surface area per unit soil volume
in layer i.
Root suction head (hr,i) was expressed as a function of tissue balance pressure above
ground (Pb, bar) and the hydrostatic head between the soil surface and soil layer i (hh,i):
hr,i
cPbm Pb - hh,i
( 4.14 )
where cPbm = 10.2 m bar-1 is a conversion coefficient to convert from units of Pb (bar) to
units of hr,i (m). The height of the canopy was not considered in the calculation of hh,i,
as it was not modelled.
For grasses, Pb,s was set to a constant value of 15 bars, while for trees, Pb,p was
modelled as a function of above-ground dry matter in living tissues (Md), water storage
capacity of living tissues (Mqx) and the variable tree water store (Mq) per unit catchment
area:
Pb,p
1 yz
i 750 Md
z
+
H Mqx - MqL jj
2
Mqx {
k H Md + MqxL
( 4.15 )
where Md was assumed to be a linear function of the area fraction covered by perennial
vegetation (MA,p):
186
Schymanski (2007): PhD thesis
Chapter 4
Md = 10 kg m-2 MA,p
( 4.16 )
and Mqx was proportional to Md:
Mqx = cMd Md
( 4.17 )
The proportionality constant cMd is typically greater than or equal to 1.0 (Schymanski et
al. in prep.-c) and was set to 1.0 in this study. Changes in tree water storage (Mq) at each
time step were calculated as the difference between total root water uptake and
transpiration by trees. The transpiration model will be described in the next section.
Given the above fluxes, changes in the state variables of the water balance model (su,i,
dyu,i, ωo, ωu, ys and yu) were computed using the balance of mass, with a soil water
storage capacity determined by the soil porosity, which was obtained from the
difference between saturated water content (θs) and residual water content (θr). The
values of θs and θr were specific for each soil type and obtained as part of the
parameterisation of the van Genuchten water retention model (see above).
4.3.1.2
Vegetation Optimality
The Vegetation Optimality approach is based on the assumption that natural vegetation
has co-evolved with its environment over a long period of time and that natural
selection has led to a species composition that is optimally adapted to the given
conditions. If this were true, the question arises, what would be the properties of such
optimal vegetation and how would it use the available resources?
The energy acquired through photosynthesis is stored in carbohydrates, which are vital
for plant fitness. Carbohydrates are both energy carriers and building materials for plant
organs. They can be used for many purposes, including seed production and the
maintenance of symbiotic relations with bacteria and fungi to mobilise nitrogen and
other nutrients from the soil or atmosphere. In addition, all living plant tissues
continuously consume energy to stay alive and require carbohydrates for their
construction. Thus, part of the carbon acquired through photosynthesis has to be reinvested into the construction and maintenance of the organs involved in its uptake.
Only what is left over, the “Net Carbon Profit” (NCP), is assumed to be useful for
187
4.3 Methods
Schymanski (2007): PhD thesis
increasing a plant’s fitness. Hence the optimal resource use strategy was defined as the
one that maximises NCP.
The organs ultimately involved in carbon uptake are not just leaves, but also roots and
transport tissues, which supply the leaves with water and nutrients. For simplicity, the
costs related to nutrient uptake have been neglected in this model, as they are largely
unknown and nutrients can, to a certain extent, be recycled within plants. The
optimisation problem is then to maximise NCP by adjusting foliage properties and
stomatal conductivity dynamically, while adapting roots and transport tissues to meet
the variable demand for water by the canopy (Figure 4.2).
The canopy was represented by two “big leaves”. One big leaf of invariant size (MA,p)
represented perennial vegetation (trees) and another big leaf of varying size (MA,s)
represented seasonal vegetation (grasses). As the big leaves were not assumed to
transmit any light, no overlap between these two leaves was allowed, so that
MA,s + MA,p ≤ 1 (Figure 4.3). The seasonal vegetation was allowed to vary in its spatial
extent (MA,s), but had only a limited rooting depth (yr,s = 1 m), while the perennial
component covered a fixed fraction of the surface (MA,p) but had an unlimited rooting
depth (yr,p). Rooting depths were assumed to be invariant in time, but the distribution of
roots within each root zone was allowed to vary on a day-by-day basis. The
photosynthetic capacity in each “big leaf” was also allowed to vary from day to day,
while stomatal conductivity in each “big leaf” was allowed to vary on an hourly scale.
In the following, we will summarise the costs and benefits in terms of Net Carbon Profit
that were associated with the optimised parameters.
The computation of leaf CO2 uptake (Ag) was based on a biochemical model of
photosynthesis (von Caemmerer 2000) and expressed as (Schymanski et al. in prep.-a;
Schymanski et al. in prep.-c):
Ag
1
J4Ca Gs + 8 G* Gs + Je - 4Rl +
8
"####################################################################################################
#
HJe - 4Rl - 4Gs HCa - 2 G*LL2 + 16Gs H8Ca Gs + Je + 8RlL G* N
( 4.18 )
where Ca is the mole fraction of CO2 in the atmosphere, Gs is the stomatal conductivity,
Je is the photosynthetic electron transport rate, Rl is leaf respiration and Γ* is the CO2
compensation point (in mol/mol).
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Schymanski (2007): PhD thesis
Chapter 4
The electron transport rate (Je) is a function of photosynthetic active irradiance (Ia), the
big leaf’s electron transport capacity (Jmax) and the size of the leaf (MA):
a Ia y
ij
jj1 - ‰ Jmax zzz Jmax M A
j
z
k
{
Je
( 4.19 )
where α is the initial slope of the curve and is commonly given the value 0.3.
The leaf respiration rate (Rl) is also determined by Jmax:
Rl
cRl Jmax HCa - G* L
8 H Ca + 2 G * L
( 4.20 )
The parameter cRl is a constant that has been given the value of 0.07, following an
empirical relationship between photosynthetic capacity and leaf respiration (Givnish
1988; Schymanski et al. in prep.-a).
The CO2 compensation point (Γ*, in mol/mol) is fairly constant between most plant
species, but varies with temperature following an Arrhenius-type function (Bernacchi et
al. 2001; Medlyn et al. 2002):
G*
126.946 H Ta-25L
0.00004275 ‰ Rmol H Ta +273L
( 4.21 )
where Rmol is the universal gas constant, and Ta is air temperature, which we assumed to
represent leaf temperature.
Jmax,p also varies with temperature, following an equation proposed by Medlyn et al.
(2002), which has been altered to express Jmax,p at a given temperature as a function of
Jmax at the reference temperature of 25˚ C (Jmax25,p) (Schymanski et al. in prep.-a):
ijij - Hd HTopt-298L
yz
yz
jjjj 298 R mol Topt
zz
z
- 1zz H a + Hdzzz
jj‰
jj
kk
{
{
H
H
T
T
L
a
opt
d
ij
yz
jj R mol Ta Topt
zz
‰
1
jj
zz H a + H d
k
{
Ha HTa -298L
‰ 298 R mol Ta
Jmax
Jmax25
( 4.22 )
where we used parameter values derived for Eucalyptus pauciflora, as presented in
Medlyn et al. (2002).
189
4.3 Methods
Schymanski (2007): PhD thesis
Equations ( 4.18 ) to ( 4.22 ) were equally applied to grasses and trees, with different
values of MA, Jmax25, and Gs. The total vegetation CO2 uptake rate (Ag,tot) was calculated
as the sum of CO2 uptake by trees (Ag,p) and by grasses (Ag,s):
Ag,tot
Ag,p + Ag,s
( 4.23 )
The costs and benefits of choosing a certain value for Jmax25 are implicit in the above
equations, as with increasing Jmax25, both Je and Rl increase at different rates in Equation
( 4.18 ), so that for given values of Ia, Gs, Ca and Ta, there would be an optimal value of
Jmax25 that would maximise Ag. It is also obvious from Equation ( 4.19 ) that a larger
vegetated fraction (MA) can lead to larger electron transport rates (Je), which would
result in larger CO2 uptake rates. However, as leaves have limited life times, plants need
to invest carbon into the replacement of fallen leaves, and the carbon costs related to MA
were approximated as (Schymanski et al. in prep.-c):
Rf = 5.5ä10-7 mol s-1 m-2 MA
( 4.24 )
This still leaves the costs of stomatal conductivity (Gs) in Equation ( 4.18 ) unaccounted
for. Stomatal conductivity does not have any direct carbon costs, but results in water
loss from the leaves (Et):
Et
a Gs HWl - WaL
( 4.25 )
where Wl and Wa denote the mole fraction of water vapour in air inside the leaf and in
the atmosphere respectively. The stomatal conductivity for CO2 (Gs) was multiplied by
a constant (a = 1.6) to account for the different diffusivity of water vapour and CO2 in
air (Cowan and Farquhar 1977). The mole fraction of water vapour in air was estimated
by dividing the partial vapour pressure (pva) by air pressure (Pa). Assuming that the air
space inside the leaves has saturation vapour pressure (pvsat), we can replace the term
(Wl - Wa) by the molar vapour deficit Dv:
Wl - Wa º
pvsat - pva
P
Dv
( 4.26 )
In the longer term, transpiration and stomatal conductivity are constrained by the
availability of water in the soil. To maximise Ag with a limited amount of water,
transpiration should be controlled by stomata in such a way that the slope between CO2
uptake and transpiration is maintained as constant during a day (Cowan and Farquhar
1977; Cowan 1982; Cowan 1986; Schymanski et al. in prep.-a). This slope will be
190
Schymanski (2007): PhD thesis
Chapter 4
called λs and λp for grasses and trees respectively. Over longer time periods, the
parameters λs and λp should be sensitive to the availability of soil water and this
sensitivity could be seen as a plant physiological response shaped by evolution to suit a
given environment (Cowan and Farquhar 1977). In the present model, the sensitivity of
λs and λp to soil water was parameterised as:
ij ir,s yz
clf,s jjj‚ hi zzz
k i=1 {
cle,s
ls
and
lp
ij i r, p yz
clf , p jjj‚ hi zzz
k i =1 {
( 4.27 )
cle, p
( 4.28 )
where ir,s and ir,p denote the deepest soil layer accessed by roots of grasses and trees
respectively. The parameters cλf,s, cλe,s, cλf,p and cλe,p are assumed to represent the longterm adaptation of a plant community to its environment and are likely to be influenced
by the species composition of the community.
Given that Equations ( 4.18 ) and ( 4.25 ) have Gs in common, they were combined to
express Et as a function of Ag. Thus, for given values of λs or λp the corresponding
values of Et, Ag and Gs were derived by inverse methods. Transpiration rates determined
in this way had to be matched by root water uptake, so that higher values of λs or λp
required more extensive root systems. The model assumed that the rooting depth of
grasses is fixed at 1 m, while the tree rooting depth (yr,p) is a result of the long-term
adaptation to a given environment. The carbon costs associated with a given rooting
depth (yr) were formulated in terms of the vascular system necessary to transport water
from deep roots to the surface and to re-distribute the water over the vegetated area (MA)
at the surface:
Rv
crv MA yr
( 4.29 )
where crv was a proportionality constant that was set to 1.2ä10-6 mol s-1 m-3 for all sites
(Schymanski et al. in prep.-c).
The surface area of fine roots (SAr) in each soil layer was optimised daily to allow
adequate root water uptake with the lowest possible total root surface area. The carbon
191
4.3 Methods
Schymanski (2007): PhD thesis
costs associated with a given root surface area of fine roots were expressed as
(Schymanski et al. in prep.-c):
Rr
cRrJ
rr
SArN
2
( 4.30 )
where rr is the average fine root radius (set to 0.6 mm) and cRr are the carbon costs per
unit fine root volume (set to 0.0017 mol s-1 m-3).
The Net Carbon Profit (NCP) was defined as total CO2-uptake of trees and grasses over
the entire period, minus all identified maintenance costs of organs assisting
photosynthesis, including foliage, roots and water transport tissues, as described above:
NCP
‡
tend
tstart
8 Ag,tot HtL - R f HtL - Rr HtL - Rv HtL< „ t
( 4.31 )
where Ag,tot is the combined CO2 uptake by trees and grasses (given in Equation
( 4.23 )), Rf stands for the foliage costs of grasses and trees together, Rr are the root
costs of trees and grasses summed over all soil layers, and Rv are the costs associated
with the vascular systems of trees and grasses combined.
Using the above equations and meteorological data over 30 years, long-term adaptation
of vegetation to the environment was modelled by the optimisation of six parameters:
MA,p, yr,p, cλf,p, cλe,p, cλf,s and cλe,s. The optimisation was performed using the Shuffled
Complex Evolution (Duan et al. 1993; Duan et al. 1994; Muttil and Liong 2004), which
searches the parameter space for the global optimum by re-running the 30-year
simulation repeatedly. During each run, electron transport capacity of grasses (Jmax25,s)
and trees (Jmax25,p), vegetated surface area covered by grasses (MA,s) and the root surface
areas of trees and grasses (SAr,p and SAr,s respectively) were optimised dynamically on a
day-by-day basis. For a more detailed description of the optimisation algorithms see
Schymanski et al. (in prep.-c).
192
Schymanski (2007): PhD thesis
Chapter 4
Atmosphere
Water
Water Transport & Storage
Carbon
Foliage
Net Carbon
Profit
Root System
Soil
Figure 4.2: Net Carbon Profit as the difference between carbon acquired by photosynthesis and the
carbon used for the construction and maintenance of organs necessary for its uptake. As CO2
uptake from the atmosphere is inevitably linked to the loss of water from the leaves, the root system
as well as water transport and storage tissues are essential to support photosynthesis. Soil water
supply, atmospheric water demand and daily radiation constitute the environmental forcing.
Within those constraints, vegetation is assumed to optimise foliage, water transport and storage
tissues, roots and stomata dynamically to maximise its Net Carbon Profit.
MA,p
yr,p
yr,s
MA,s
Figure 4.3: Representation of perennial (left) and seasonal (right) vegetation components. The
perennial vegetation component was assumed to be composed of evergreen trees, while the seasonal
component was assumed to be composed of annual grasses only.
4.3.2
THE “BUDYKO CURVE” AS A BENCHMARK
Over annual and longer time periods, evapo-transpiration at catchment scale has been
shown to be related to precipitation and net radiation in a systematic way (Budyko
1974). Early observations of this relationship include the ones by Schreiber (1904) and
Ol’dekop (1911), who formulated different expressions for annual evapo-transpiration
(ET,a), as inferred from precipitation and runoff data, as a function of annual
193
4.3 Methods
Schymanski (2007): PhD thesis
precipitation (Qrain,a) and annual potential evaporation (Ep,a). These relationships were
given by Budyko (1974) as:
ET,a
and
ET,a
E p,a
ij
y
Q
jj1 - ‰ rain,a zzz Q
j
z rain,a
k
{
i Qrain,a yz
z
Ep,a tanhjj
k Ep,a {
( 4.32 )
( 4.33 )
where Ep,a was approximated by annual net radiation (In,a) divided by the latent heat of
evaporation (λE):
E p,a
In,a
lE
( 4.34 )
The geometrical mean of the right-hand sides of Equations ( 4.32 ) and ( 4.33 ) was then
written as (Budyko 1974):
ET,a
i Qrain,a yz 0.5
i Ep,a yz
i Ep,a yz
F Ep,a Qrain,a tanhjj
+ sinh j
:B1 - cosh j
z>
k Ep,a {
k Qrain,a {
k Qrain,a {
( 4.35 )
Equation ( 4.35 ) can be used to calculate the ratio of ET,a/Ep,a as a function of the
“dryness index”, Qrain,a/Ep,a, yielding the “Budyko curve”, which has been used
previously as a benchmark for water balance models (e.g. Milly and Shmakin 2002;
Porporato et al. 2004).
In this study, annual net radiation (In,a) was estimated from daily global radiation (Ig,d)
and climate data following the procedure by Allen et al. (1998), where we assumed a
constant albedo of 0.18 (except for the tropical rainforest, where albedo was set to 0.12).
Then, modelled ET,a/Ep,a was computed for each year of the data set and plotted against
Qrain,a/Ep,a to check whether modelled values of evapo-transpiration are consistent with
the previously established empirical relationships.
4.3.3
VIRTUAL EXPERIMENTS
The model used in this study requires only information about the environmental
conditions on a site (meteorology, atmospheric CO2, soil and topography) and computes
the hypothetically optimal vegetation for the given conditions. Along with the optimal
194
Schymanski (2007): PhD thesis
Chapter 4
vegetation properties, the model also computes hourly transpiration- and CO2 uptake
rates and surface runoff that result from the interaction between climate, vegetation and
the water balance. In the present study, our primary interest was how different levels of
atmospheric CO2 would affect the modelled optimal vegetation and particularly how
they would affect transpiration rates in the long term. As described above, the model
distinguished between vegetation properties that are assumed to have evolved over a
long period of time (tens to hundreds of years) and vegetation properties that adapt
dynamically to the environment (day to day). The first category will be called “longterm properties” and includes the area fraction covered by trees (MA,p) and their rooting
depth (yr,p), and the functional dependence of water use on soil water by trees and
grasses. The second category will be called “medium-term properties” and includes the
dynamic area fraction covered by grasses (MA,s), photosynthetic electron transport
capacity of trees and grasses (Jmax25,p and Jmax25,s respectively) and root surface area of
trees and grasses (SAr,p and SAr,s respectively).
The numerical experiments were carried out with two different concentrations of
atmospheric CO2, 317 ppm and 350 ppm, approximately representing the levels of 1960
and 1990 respectively. To distinguish between medium- and long-term adaptations to
increased CO2, the numerical experiments were done in three stages:
1. Optimise all vegetation parameters using 30 years of meteorological data and a
constant CO2 level of 317 ppm.
2. Keep the long-term properties fixed and re-run the model with the same
meteorological data but a constant CO2 level of 350 ppm.
3. Optimise all vegetation parameters using the same meteorological data and a
constant CO2 level of 350 ppm.
We expected that the effect of increased CO2 on vegetation could be different under
different climates, so we chose five catchments that span a wide climatic gradient from
semi-arid to humid.
195
4.3 Methods
4.3.4
Schymanski (2007): PhD thesis
STUDY SITES AND SITE-SPECIFIC DATA
The study sites chosen were four sites that are part of the OzFlux network3 and an
additional site that has been subject to intensive investigations in the past. The sites span
a climatic gradient from semi-arid to humid. The OzFlux sites are long-term monitoring
sites for canopy-scale CO2- and water vapour exchange and are likely to deliver
valuable data that can be used in conjunction with the model results of this study in
future. These sites were Virginia Park (VIR) and Cape Tribulation (CT) in Queensland,
Tumbarumba (TUM) in New South Wales and Howard Springs (HS) in the Northern
Territory. The additional site was the Salmon catchment (SAL), which is part of the
Collie Catchment in the Southwest of Western Australia. The geographic locations,
vegetation types and coarse climate properties of the different sites are summarised in
Table 4.1.
Meteorological data for the sites was obtained from the Queensland Department of
Natural Resources, Mines and Water (SILO Data Drill4). The data set contained, among
others, daily totals of global radiation, precipitation, and class A pan evaporation, daily
maxima and minima of air temperature and daily values for atmospheric vapour
pressure, all of which were obtained by interpolation of data from the nearest
measurement stations and/or estimated based on proxy data. The methodology used for
the compilation of the data set is described in Jeffrey et al. (2001). Daily rainfall was
distributed evenly over 24 hours, while global irradiance (Ig), air temperature (Ta) were
transformed into hourly values by adding diurnal variation as described below.
The diurnal variation in global irradiance (Ig) was estimated from the solar elevation
angle (β) and daily global irradiance (Ig,d), after Spitters et al. (1986):
Ig
I g,d J‡ sinH bHth LL „ th N sinH bHth LL
( 4.36 )
where th is the hour of the day.
3
Ozflux is the Australian and New Zealand Flux Research and Monitoring Network
(http://www.dar.csiro.au/lai/ozflux/index.html), which is part of a global network
coordinating regional and global analysis of observations from micrometeorological
tower sites (Fluxnet, http://www.fluxnet.ornl.gov/fluxnet/index.cfm)
4
http://www.nrm.qld.gov.au/silo
196
Schymanski (2007): PhD thesis
Chapter 4
The diurnal variation in air temperature (Ta) was expressed as a function of daily mean
temperature (Ta,m) and daily temperature range (Ta,r), which was taken from Bilbao et al.
(2002):
Ta
Ta,m + Ta,r H0.4632 cos Hch-3.805L + 0.0984 cos H2 ch - 0.360L +
0.0168 cos H3 ch - 0.822L + 0.0138 cos H4 ch - 3.513LL
( 4.37 )
where ch changes with the hour of the day (th):
ch
1
pHth - 1L
12
( 4.38 )
Atmospheric vapour deficit (Dv) was defined as the difference between the partial
pressure of water vapour in air (pva) and saturation vapour pressure (pvsat), divided by air
pressure (Pa, set to 101325 Pa):
Dv
pvsat - pva
Pa
( 4.39 )
The vapour pressure (pva) given in the meteorological database was assumed to be
constant during the day, so that the diurnal variation in Dv was expressed as a result of
the diurnal variation in saturation vapour pressure (pvsat), which was calculated using the
common approximation (Allen et al. 1998):
pvsat
17.27 HTa-273L
610.8 ‰ Ta-35.7
( 4.40 )
where pvsat has units of Pa and Ta is given in units of K.
Catchment and soil properties that were used for modelling the different sites are given
in Table 4.2.
197
4.3 Methods
Schymanski (2007): PhD thesis
Table 4.1: Study sites and their average climate properties between 1970 and 2000, as calculated
from daily meteorological data. Annual Ep is the evaporation-equivalent of annual net radiation
(In,a).
SITE
Name
Geogr.
Location
Vegetation
Annual
Qrain
Annual
Ep=In,a/λE
VIR
Virginia Park
19º54'S
146º33'E
Open woodland
savanna
580 mm
1810 mm
SAL
Salmon River
33º25'S
115º59'E
Jarrah forest
1051 mm
1268 mm
HS
Howard Springs
12º30'S
131º09'E
Open forest
savanna
1719 mm
1876 mm
TUM
Tumbarumba
35º39'S
148º09'E
Wet sclerophyll
forest
1288 mm
1155 mm
CT
Cape Tribulation
16º06'S
145º27'E
Tropical rain forest
4097 mm
2085 mm
Table 4.2: Site-specific catchment and soil parameters. Geometrical catchment properties (Z, zr and
g0) were estimated visually from digital elevation models, while soil parameters were taken from
the software package Hydrus 1-D (Simunek et al. 2005) as the typical parameters for “Sandy
Loam” (VIR, SAL, HS), “Loam” (TUM) and “Sandy Clay Loam” (CT).
Z
(m)
zr
(m)
g0
(rad)
Ksat
(m-1)
θr
θs
nvG
αvG
(m-1)
VIR
15.0
5.0
0.033
1.23ä10-5
0.065
0.41
1.89
7.5
SAL
20.0
5.0
0.2
1.23ä10-5
0.065
0.41
1.89
7.5
-5
0.065
0.41
1.89
7.5
-6
SITE
HS
15.0
10.0
0.033
1.23ä10
TUM
30.0
5.0
0.2
2.89ä10
0.078
0.43
1.56
3.6
CT
15.0
5.0
0.033
3.64ä10-6
0.1
0.39
1.48
5.9
198
Schymanski (2007): PhD thesis
4.4
Chapter 4
RESULTS
The predicted annual evapo-transpiration rates of all sites are consistent with
empirically derived relationships between evapo-transpiration and the annual water and
energy budget (Figure 4.4). The figure also shows that the modelled sites span over the
whole range of global environments collated by Choudhury (1999) in terms of their
radiative dryness index (Qrain,a/Ep,a). According to this index, Virginia Park (VP) and
Salmon (SAL) fit into the “water-limited” range, while Cape Tribulation (CT) is far in
the “energy-limited” range. Tumbarumba (TUM) and Howard Springs (HS) would be
considered as transitional between “water-limited” and “energy-limited” according to
the radiative dryness index.
An increase in atmospheric CO2 concentrations from 317 ppm to 350 ppm had different
effects on evapo-transpiration in the model, depending on whether all vegetation
properties were allowed to adapt (“long-term adaptation”) or whether only the
“medium-term properties” were allowed to adapt to the new conditions (“medium-term
adaptation”). The response also differed between sites. In the two intermediate
environments (TUM and HS), increased CO2 led to a decrease in ET in the medium
term, but to increased ET in the long term (Figure 4.5). In the other catchments,
increased CO2 either resulted in decreased ET or in no significant change in ET. On
average, the medium-term response was a decrease of ET by 14 mm/yr (i.e.
~0.47 mm/yr2 decrease on average over 30 years), while the long-term response was an
increase in ET by 36 mm/yr (i.e. 1.2 mm/yr2 increase on average over 30 years) (Table
4.3). Applying the observed runoff trend of +0.45 mm/yr2 (Fig. 3 in Gedney et al. 2006)
to a period of 30 years, we estimated that a decrease in ET by 13.5 mm/yr would be
sufficient to explain the observed increase in runoff if no other factors were involved.
This value is shown as a dashed line in Figure 4.5. The most dramatic change in ET was
predicted for the Howard Springs (HS) site when all vegetation properties were allowed
to adapt. The increase in ET was accompanied by a significant increase in rooting depth
(yr,p) and tree cover (MA,p) (Table 4.3). Such changes in MA,p or yr,p were not predicted
for the other sites. However, time-averaged root surface area (Av(SAr)) increased in the
medium term in response to increased CO2 on all sites, even if mean annual
transpiration rates went down.
199
4.4 Results
Schymanski (2007): PhD thesis
1.20
Water limited
Energy limited
ET,a/(I
ET/Ep
n,a/λE)
1.00
0.80
Schreiber (1904)
Ol'Dekop (1911)
Budyko (1974)
Choudhury (1999)
VIR
TUM
HS
CT
SAL
0.60
0.40
0.20
0.00
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
Qr/Ep
Qrain,a/(I
n,a/λE)
Figure 4.4: Annual energy and water budgets for catchments modelled with an atmospheric CO2
concentration of 350 ppm. The ratio ET,a/(In,a/λE) represents the partitioning of annual net radiation
(In,a) into evapo-transpiration (ET,a) and sensible heat flux, while the ratio Qrain,a/(In,a/λE) denotes the
radiative dryness index for a year. Data points to the left of the vertical dashed line are generally
considered to represent “water-limited” conditions, while data points to the right of the line are
considered to represent “energy-limited” conditions. Curved lines are empirical curves fitted to
observed long-term averages in a wide range of catchments by different authors. Each data point
represents one year from a set of 30 years per site, except for data points denoted as “Choudhury
(1999)”, which refer to long-term averages of a global variety of catchments compiled by
Choudhury (1999).
200
Schymanski (2007): PhD thesis
Chapter 4
Table 4.3: Model runs for different catchments and different levels of atmospheric CO2. The leftmost column gives the catchment code, followed by the CO2 concentration (in ppm) for which the
vegetation was optimised in the long term and then the CO2 concentration with which the shortterm optimisation and the calculation of fluxes were performed. The following columns give values
of optimal tree cover (MA,p) and rooting depth (yr,p), average of optimal grass cover (Av(MA,s)),
average of root surface area per unit catchment area (Av(SAr)), average annual transpiration
(Av(Et)), average annual evapo-transpiration (Av(ET)) and Net Carbon Profit over 30 years (NCP).
Grey fields refer to changes of the average values between the 317 ppm scenario (317_317) and the
two 350 ppm scenarios (“medium-term”: 317_350 - 317_317; “long-term”: 350_350 - 317_317).
Site, run
MA,p
yr,p
(m)
Av(MA,s)
Av(SAr)
Av(Et)
(mm/yr)
Av(ET)
(mm/yr)
NCP
(mol/m2)
VIR317_317
0.16
2.0
0.54
0.69
354
503
1639
0.54
0.89
361
504
1930
0.58
0.78
350
496
1951
Diff. medium-term
+7
+1
Diff. long-term
-4
-7
VIR317_350
VIR350_350
SAL_317_317
0.14
0.21
2.5
2.5
0.59
0.96
508
653
3218
0.60
1.05
498
642
3516
0.59
0.93
479
624
3529
Diff. medium-term
- 10
- 11
Diff. long-term
- 29
- 30
SAL_317_350
SAL_350_350
TUM317_317
0.23
0.26
2.5
1.5
0.73
0.54
653
879
6166
0.73
0.55
633
862
6527
0.74
0.59
658
883
6534
Diff. medium-term
- 20
- 17
Diff. long-term
+5
+4
TUM317_350
TUM350_350
HS317_317
0.25
0.14
1.5
3.0
HS317_350
HS350_350
0.33
4.0
0.50
0.73
892
1104
2275
0.50
0.86
883
1093
2564
0.43
0.73
1169
1337
2652
-9
- 11
+ 277
+ 233
Diff. medium-term
Diff. long-term
CT317_317
0.20
2.0
0.77
0.65
1557
1892
5522
0.77
0.69
1527
1861
6059
0.75
0.70
1541
1875
6064
Diff. medium-term
- 29
- 30
Diff. long-term
- 15
- 17
Average diff. in medium-term
- 12
- 14
Average diff. in long-term
+ 47
+ 36
CT317_350
CT350_350
0.22
2.0
201
4.5 Discussion
Schymanski (2007): PhD thesis
Change in Et (mm/yr)
300
Medium-term adaptation
250
Long-term adaptation
200
150
100
50
0
-50
VIR
SAL
TUM
HS
CT
AVERAGE
Figure 4.5: Modelled change in average annual evapo-transpiration (ET) as a consequence of
medium-term adaptation and long-term adaptation to increased CO2 from 317 ppm to 350 ppm.
The dashed line represents the reduction in ET that would be necessary to explain the observed
increase in runoff between 1960 and 1990 (Gedney et al. 2006). See text for details.
4.5
DISCUSSION
The model used in the present study simulated transpiration rates by hypothetical
natural vegetation, based only on the trade-offs between CO2 uptake by photosynthesis
and CO2 loss related to the maintenance of foliage and water uptake and -transport
tissues. The model has been tested previously on the open forest savanna site (HS), and
demonstrated its capability to reproduce observed inter- and intra-annual dynamics of
transpiration, CO2 uptake and vegetation properties (Schymanski et al. in prep.-c). The
model was thought to be valid for water-limited environments, where energy constraints
on evapo-transpiration and nutrient constraints on growth would be superposed by the
constrained availability of water. The general perception is that in environments where
water is not “limiting”, transpiration would be limited by the energy balance or by limits
to leaf area imposed by nutrient deficiency. It has been shown elsewhere that the
“Budyko curve” can be reproduced very well if the water- and energy limitations to
evapo-transpiration are factored into a water balance model and the soil water holding
capacity (or rooting depth) is prescribed relative to the rainfall depth (Milly 1994;
Porporato et al. 2004). In the present model, however, neither nutrient limitation nor the
energy balance was considered. Hence, it is remarkable that transpiration rates did not
get grossly over-estimated for the tropical rainforest site (CT in Figure 4.4). In fact,
modelled annual ET,a/(In,a/λE) follows the “Budyko curve” very closely, throughout the
202
Schymanski (2007): PhD thesis
Chapter 4
range from strongly water-limited (VIR) to strongly “energy-limited” (CT) climates
(Figure 4.4). This can be explained by the costs related to water uptake. Even in very
wet soils, water uptake still requires a certain fine root surface area, and the higher the
transpiration, the more water has to be taken up by the roots. At the same time, returns
on investments into roots in terms of CO2 uptake by foliage are limited by the available
radiation (included in the variable In,a). Hence the amount of modelled transpiration can
be seen as a result of the trade-off between water uptake costs and CO2 uptake benefits
achieved through subsequent transpiration. This trade-off alone led to realistic results,
without explicit consideration of the energy balance or nutrient limitation.
The correspondence of model results with the “Budyko curve” is only a very coarse
indicator for the performance of the model. Much better indicators are direct
observations in the modelled catchments. The correspondence of the model results of
the HS site with four years of observations has been shown elsewhere (Schymanski et
al. in prep.-c). Similar observations are available for the sites TUM and VIR, where
annual ET has been reported in the ranges of 600 - 800 mm and 350 - 550 mm
respectively (Leuning et al. 2005). These observations are reasonably consistent with
the model results presented in Table 4.3. The measurements from the Ozflux station at
CT are not available yet5, but hydrologists have a fairly good understanding about the
water balance in the Jarrah forest (SAL). Annual evapo-transpiration should account for
90-99% of annual precipitation, and tree roots have been observed at 35 m depth
(Silberstein et al. 2001). This is in contrast with the model results obtained in the
present study, where annual ET only accounts for around 65% of annual precipitation
5
A recently published report suggests transpiration rates as low as 522 mm/year
(McJannet, D., J. Wallace, P. Fitch, M. Disher and P. Reddell (2006). "Water balance
measurements in Australia's Wet Tropics: Sites, methods and results." CSIRO Land and
Water Science Report 19/06.). These were obtained from sap flow measurements on
trees and it remains to be seen whether such low values will be confirmed by the eddy
covariance technique. Considering that the figure of 522 mm/year constitutes one third
of the modelled transpiration rates on that site (Table 4.3), this would mean that the site
would fall well below the Budyko curve in Figure 4.4.
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4.5 Discussion
Schymanski (2007): PhD thesis
and average tree rooting depth is only 2.5 m. The most likely reason for this discrepancy
between model results and observations is an over-estimation of the value of the
proportionality constant for water transport costs (crv) in Equation ( 4.29 ). Its value was
originally unknown and was adapted to yield realistic results at HS (Schymanski et al.
in prep.-c). It is possible that the value of crv changes from site to site depending on the
below-ground conditions, or that below-ground and above-ground water transport costs
have to be modelled separately.
The results obtained from the virtual experiment with different CO2 concentrations
suggest that the response of vegetation to increased CO2 can be very complex and
dependent on both climate and time scale. In the medium term, when species
composition, tree cover, tree rooting depth and water use strategies have not adapted
yet, stomatal closure is likely to be the dominant mechanism behind the modelled
decrease in transpiration (Et). Figure 4.6 shows examples of how the relation between
transpiration rates (Et) and CO2 uptake rates (Ag) is affected by changes in atmospheric
CO2 (Ca) or photosynthetic electron transport rate (Je). As Ca increases, the same Ag can
be achieved with less Et. If the stomata adjust in order to keep the slope of the curve
(e.g. λp) constant (as assumed in the present model), transpiration rates would decrease
slightly as a consequence of increased Ca. If, however, stomata were assumed to adjust
in order to keep Ag constant, the reduction in Et would be expected to be much stronger
(dashed lines in the plot). The bottom plot in Figure 4.6 shows how the relation between
transpiration rates (Et) and CO2 uptake rates (Ag) would be affected by changes in
electron transport rate (Je), either due to an increase in irradiance (Ia), photosynthetic
electron transport capacity (Jmax) or vegetation cover (MA) in Equation ( 4.19 ). In this
case, maintenance of a constant slope would lead to an increase in Et. This is what
happened in the long-term adaptation scenario at HS, where MA,p increased significantly
in response to increased Ca and resulted in a dramatic increase in Et.
Some of the changes in Et can be partly offset by changes in soil evaporation (Es) in the
reverse direction. The open woodland savanna (VIR) did not express any significant
response in ET in the medium term, although transpiration (Et) increased markedly
(Table 4.3). The increase in Et coincided with a significant increase in average root
surface area (Av(SAr)) and indicates that the increased Et was facilitated by a higher root
abundance, while soil evaporation went down due to the stronger competition by root
water uptake. The fact that VIR was the only site in the study that was not predicted to
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Chapter 4
respond with a decrease in Et to elevated CO2 is remarkable in itself, as VIR was the site
with the highest proportion of grasses to trees. It is thus consistent with the suggestion
that trees are generally more sensitive to changes in atmospheric CO2 than grasses
(Ainsworth and Long 2005; Körner 2006). This has possibly to do with the fact that
grasses exploit ephemeral resource pools and probably never reach an “optimal” steadystate, while trees explore more stable resource pools and therefore have a clearer
response to changes in steady-state conditions.
It is also interesting to note that the modelled medium-term response to increased CO2
included an increased average root surface area (Av(SAr)) in all study sites. This could be
seen as a result of the effect that elevated CO2 leads to higher carbon returns per unit of
water use especially during peak times when irradiance is high. As root surface area in
the model is predominantly determined by the daily peak water demand, it responds
more sensitively to the diurnal dynamics in transpiration than to the long-term average
transpiration rates.
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Et (mol/m2/s)
0.05
0.04
0.03
0.02
0.01
0
0.00001 0.00002 0.00003 0.00004
Ag (mol/m2/s)
0
0.00001 0.00002 0.00003 0.00004
Ag (mol/m2/s)
Et (mol/m2/s)
0.05
0.04
0.03
0.02
0.01
Figure 4.6: Examples of relationships between Et and Ag for different levels of atmospheric CO2
concentration (top) and different values of photosynthetic electron transport rate (Je) (bottom).
Plots were obtained by applying Equations ( 4.18 ) and ( 4.25 ) with varying stomatal conductance
Gs. Common parameter values were Γ* = 38.6â10-6 mol/mol, Jmax = 300 µmol m-2 s-1 and a = 1.6 and
Dv = 0.03 mol/mol. In the top plot, atmospheric CO2 was set to Ca = 317â10-6 mol/mol (left-most
curve), Ca = 350â10-6 mol/mol (middle curve) and Ca = 375â10-6 mol/mol (right-most curve). In the
bottom plot, Ca = 350â10-6 mol/mol, while Je was increased by 10% (middle curve) and 20% (rightmost curve) from the original value (left-most curve). Dots represent the locations where the slopes
of the curves are 2000 mol/mol. Dashed lines in the top plot represent the decrease in Et that would
have resulted from increased CO2 if Ag was held constant.
The average of the predicted medium-term responses to elevated CO2 was a decrease in
ET by 14 mm/yr. The prescribed increase in CO2 from 317 ppm to 350 ppm represents
roughly the observed increase between 1960 and 1990. Gedney et al. (2006) identified a
runoff trend for this period of around +0.45 mm/yr2. This trend, summed over 30 years,
would have led to an average runoff that is higher by 13.5 mm/yr in the 1990s compared
with the 1960s. Around 0.35 mm/yr2 of this trend was attributed to the observed change
in atmospheric CO2 (Gedney et al. 2006, Fig 3), which is equivalent to a decrease in ET
by 10.5 mm/yr over 30 years. This is consistent in magnitude with the medium-term
response obtained from the virtual experiment in the present study.
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Chapter 4
However, the average long-term effect suggested by the virtual experiment was an
increase in ET by 36 mm. This was mainly caused by an increase in optimal tree cover
and rooting depth in the open forest savanna (HS). Although such an extreme response
to elevated CO2 has only been predicted for this one site out of five, it cannot be
dismissed as a “freak result”. The increase in tree and shrub density, also referred to as
“vegetation thickening” is a world-wide phenomenon, not only reported from savannas
but also from many temperate forests and even tropical rainforests (Gifford and Howden
2001). Elevated CO2 may not be the main reason behind the observed thickening, but it
has certainly the potential to contribute to it under certain conditions, as implied by the
results presented here. Why did increased CO2 not have a similar effect on optimal tree
cover at the open woodland savanna site (VIR), then? This site is significantly more arid
and rainfalls are much more irregular than at the HS site. This pulse-driven water
availability seems to favour grasses, which, in this model, have a limited growth rate
that is not as sensitive to atmospheric CO2. Thus, elevated CO2 did not lead to increased
tree cover because of the long dry periods, while the limited growth rate of grasses
prevented a more complete use of the water pulses even if more returns in terms of CO2
uptake could be achieved under elevated CO2.
Currently observed trends in vegetation response to elevated CO2 using FACE
experiments can only represent short- to medium-term adaptation, as experiments have
not been conducted for long enough yet (Ainsworth and Long 2005). It has been
observed that the response of vegetation to elevated CO2 can be affected by the
availability of nitrogen, with the result that photosynthetic capacity gets down-regulated
on low-nitrogen sites (Ainsworth and Long 2005). Another interesting observation was
that stimulation of plant growth by elevated CO2 was more pronounced in the third
growing season than in the first two growing seasons in the data compiled by Ainsworth
and Long (2005). This is consistent with the hypothesis that changes in vegetation cover
would affect water use in the long-term only, while stomatal responses and downregulation of the photosynthetic apparatus could represent short- to medium-term
adaptations. The possible long-term effect of atmospheric CO2 concentrations on the
tree-grass ratio in savannas has also been pointed out elsewhere (Bond et al. 2003).
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4.6 Conclusions
4.6
Schymanski (2007): PhD thesis
CONCLUSIONS
The present study highlights the importance of considering all major degrees of freedom
that vegetation has for its adaptation to elevated CO2, if we want to assess the possible
long-term effects of our burning of fossil fuels on the water cycle. We showed that
medium-term and long-term responses to elevated CO2 could have opposite effects on
the water cycle under certain conditions. Furthermore, the effects could have a
threshold-like behaviour, leading to large impacts in some areas, while leaving other
areas unaffected. This suggests that observations made in the recent past cannot easily
be extrapolated into future and used for long-term predictions. Much more research into
the cause and effect of ecosystem adaptation to increasing CO2 concentrations will be
needed before we can make such long-term predictions with confidence.
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Chapter 5
CHAPTER 5. GENERAL DISCUSSION
5.1
ACHIEVED PROGRESS
The main aims of the present thesis were:
1. to demonstrate the general feasibility of the optimality approach for making
predictions in natural vegetation,
2. to construct a model of transpiration based on assumptions about vegetation
optimality, and
3. to test the model against observations.
The general feasibility of the optimality approach for making predictions about physical
and biochemical canopy properties has been demonstrated in Chapter 2. Using a largely
simplified model of canopy photosynthesis and transpiration, we were able to show that
the optimisation of some biochemical and physical foliage properties led to
photosynthetic rates that were consistent with observations in a tropical savanna in the
wet season. The only constraints on the model were climate, observed water use and
general cost factors for the optimised parameters. The cost factors were obtained from
the literature as typical values for a wide range of C3 plants6. The same model led to an
over-estimation of photosynthesis in the dry season. Only after including the observed
vegetation cover as a constraint on the optimisation of the canopy in the dry season, the
model was able to capture the observed photosynthetic rates again. This can be
explained by the fact that in the dry season, when the top soil layers are depleted of
water, each plant has to achieve enough CO2-uptake to cover the enormous costs of
accessing water from the deeper soil layers. Hence only a few trees can survive, which
use a relatively large amount of water per unit ground cover, whilst most of the surface
area is bare soil. During the wet season in contrast, when the top soil is very wet, the
6
‘C3 plants’ refers to the vast majority of terrestrial plant species that follow the
photosynthetic C3 pathway. A minority of species follow other photosynthetic
pathways, like ‘CAM’, ‘C4’ or ‘intermediate C3 and C4’ pathways. See von Caemmerer,
S. (2000). Biochemical Models of Leaf Photosynthesis. Collingwood, CSIRO
Publishing.
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water access costs are much lower, and hence many more plants can survive on a lower
water use per unit ground cover. Whereas the light conditions largely determine the
optimal amount of foliage and the distribution of photosynthetic capacity in the vertical
dimension, the location of water in the soil profile largely determines the optimal
horizontal vegetation cover.
Chapter 2 further demonstrated that dynamically optimised stomatal conductivity
(“constant lambda hypothesis”, Cowan and Farquhar 1977),
led to better
correspondence of modelled transpiration rates with the observed than constant stomatal
conductivity. To our knowledge, such a test of the “constant lambda hypothesis” has not
been performed at canopy scale in the past.
Chapter 2 mainly contributes to plant physiology and ecology, in that it uses a
minimalist biochemical canopy photosynthesis model together with optimality
assumptions to predict photosynthetic rates as well as canopy properties that were
hitherto treated as input variables for more complicated models. The implication is that
better knowledge about the costs and benefits of different plant adaptations in terms of
carbon profit could help in the understanding of why they occur in certain environments
and predicting their occurrence in plant communities quantitatively.
Based on the optimality approach presented in Chapter 2, a coupled vegetation
dynamics and water balance model was constructed in Chapter 3 to predict transpiration
rates without any use of site-specific vegetation data as input. The model optimised
static and dynamic vegetation properties over 30 years using only climate, soil,
topography and general cost factors for the optimised properties as constraints.
Comparison of the last 4 years of the simulation with observations of transpiration and
CO2 uptake rates in a tropical savanna revealed a very close resemblance between
modelled and observed fluxes. This constitutes a major advance in our understanding of
vegetation functioning. For the first time, it was possible to predict day-to-day water use
and CO2 uptake as well as seasonally varying vegetation properties from first principles,
without prescribing vegetation type or using prior observations of the modelled
parameters. It has to be admitted, however, that the model was not entirely free of
calibration, as the cost factor relating carbon costs of the vascular system to rooting
depth and vegetation cover was unknown and had to be adjusted to get realistic results.
Whereas it has been shown that the value chosen for this parameter was consistent with
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Chapter 5
the value that could be deduced from measurements of sapwood respiration on the site,
there is a possibility that the value might vary from site to site. This has partly been
tested in Chapter 4, where the same value of this parameter was used for several
different catchments ranging from semi-arid to humid climatic conditions and with
various soil types. The results revealed that whilst the model seemed appropriate for
most catchments, it was not able to reproduce transpiration accurately when very deep
roots are involved. More research will be needed to appropriately parameterise the costs
for deep rooting.
Overall, the test of the model presented in Chapter 4 turned out positive. Particularly the
fact that the model did not over-predict annual transpiration in the humid catchments is
remarkable. Given that energy was used in the model as a constraint for photosynthesis
but not for evapo-transpiration, the results suggest that transpiration can be modelled as
a result of the trade-off between the costs related to water uptake and the benefits from
its exchange for CO2 by the leaves. This is consistent with the perception that leaves are
“carbon factories” and not just wet surfaces, as implied by models that use the canopy
energy balance to compute transpiration.
The biggest advantage of the model presented in this thesis is that it is able to account
for a variety of degrees of freedom that vegetation has to adapt to its environment. This
is used in Chapter 4, where the possible long-term effects of increased atmospheric CO2
on natural vegetation are investigated. In the long-term, not only stomatal conductivity
and leaf area, but also the vegetated ground cover, the tree to grass ratio and the root
systems can change. Furthermore, changes in species compositions are likely to take
place and hence it is not feasible to make predictions based on current observations
only. For studying the possible long-term effects of increased CO2, it is not even
possible to use the “place for time substitution”, i.e. use observations from other places
that are already adapted to the conditions that are likely to occur on a particular site in
future. This is because there are not many sites on Earth that are adapted to the CO2
levels expected in 20 years time. In fact, Chapter 4 suggests that there might not even be
many places that are adapted to our current CO2 levels. If Vegetation Optimality is
anything to go by, as suggested by this thesis, we would expect very complex responses
to increased CO2 in different catchments, reaching from a clear decrease to a dramatic
increase in transpiration rates. Such responses are likely to be accompanied by changes
in both above- and below-ground vegetation structures.
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5.2
Schymanski (2007): PhD thesis
RECOMMENDATIONS FOR FURTHER RESEARCH
The present thesis represents only the initial steps towards a new generation of
hydrological and ecological models that can make predictions about vegetation
properties and the water balance from first principles and without calibration. These
predictions can then be tested by comparison with observations on a particular site.
5.2.1
FURTHER TESTING OF THE MODEL
For the further advancement of the presented framework, we would recommend to first
compile a set of well investigated “benchmark” sites that cover the climatic and edaphic
range over which we want to make predictions of transpiration. A minimum
requirement for these sites would be information about the topography, soils and longterm climate (e.g. 30 years of shortwave radiation, rainfall, vapour pressure and
temperature), as well as observations of canopy-scale transpiration that could be
compared with model outputs. The record of observed transpiration rates should be long
enough to capture the intra-annual as well as some of the inter-annual variability. Any
additional information about vegetation properties or the CO2 uptake rates on the site
would make it even more useful for testing the model and for identifying weaknesses.
Canopy-scale observations of the exchange of CO2 and water vapour are being
conducted on numerous sites as part of the FLUXNET project7. Those FLUXNET sites,
for which several years of observations and several decades of climate data is available,
could form the core of the “benchmark” sites.
5.2.2
PREDICTION OF RUNOFF
One aspect that has not been tested in the presented studies is the reproduction of
surface runoff. This was mainly because the simple water balance model did not contain
a routing component that would allow relating the averaged catchment runoff to a
hydrograph at the outlet. In catchments where no direct observations of transpiration are
available, the vegetation optimality model could be used in conjunction with a more
sophisticated water balance model to reproduce the intra- and inter-annual variability in
surface runoff. Water balance models for reproducing the dynamics of surface runoff
7
http://www-eosdis.ornl.gov/FLUXNET/
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Chapter 5
generally require calibration, as not all input parameters can be measured at the scale of
interest. The necessary calibration of the runoff parameters could then be performed in
alternation with the optimisation of the vegetation parameters, which could improve
both the runoff and transpiration parts of the model. Calibration would be reserved to
those parameters of the model that cannot be modelled using optimality principles,
either because they cannot be assumed to be “optimal” or because their costs and
benefits in terms of the objective function are unknown. A similar strategy has been
followed by Schymanski et al. (in prep.-c), where the parameter determining the water
transport costs was calibrated, while all other parameter values where either taken from
literature or optimised within the optimality framework. Further scientific progress
could be measured by the decrease in model components that need calibration or that
rely on specific observations.
5.2.3
IMPROVED REPRESENTATION OF SOIL PHYSICAL
PROCESSES
A step in this direction would be for example to include soil properties in the
optimisation instead of deriving them from observations on the site. This would follow
Eagleson’s hypothesis that vegetation influences soil conditions over very long time
scales in its own favour (Eagleson 1982). It remains to be tested whether “optimal” soil
properties for maximising Net Carbon Profit in a given climate can be found in theory,
and whether these coincide with observed soil properties. For such a test, a more
physically-based representation of soil water retention, matric suction and hydraulic
conductivity would be useful. The water retention curves and hydraulic conductivities
used in the present thesis were derived empirically for different soil types and do not
incorporate any knowledge about feasible ranges and cross-dependencies of the
different parameters. To our knowledge, there are no good alternatives to the empirical
representation available yet. Therefore we would recommend further research into the
soil physical processes that are relevant for water availability and root water uptake.
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5.2.4
PLANT RESPIRATORY AND CONSTRUCTION COSTS
5.2.4.1
Vascular System
The present thesis identified the costs related to the transport of water as one of the most
important areas requiring further research. The construction and maintenance of the
plants’ vascular systems requires carbon expenditure, which is expected to depend,
among others, on the size of the vascular systems necessary to connect the sources with
the sinks of water (e.g. rooting depth and foliage cover). Other factors could be the
required maximum water transport rates, soil- and air temperatures and soil properties.
In addition to the infrastructure connecting the sources and sinks of water, active
processes involved in water transport also require the expenditure of carbon.
Simultaneous measurements of respiration rates of sapwood and water transport rates
under different conditions could help separating the costs related to active water
transport from those related to the maintenance of the infrastructure. Such
measurements are not only needed above-ground, but also in the below-ground
sapwood, where both anaerobic and aerobic respiration has to be quantified. Anaerobic
respiration is much less energy-efficient than aerobic respiration and the need for
anaerobic respiration is likely to change with the aeration degree of the soil and hence
with soil type and water content. Therefore it is important to quantify the below-ground
costs for different soil conditions. Another aspect of the water transport costs is related
to wood anatomy. There are likely trade-offs between the investment of dry matter into
the sapwood, the bulk hydraulic conductivity of the sapwood, its susceptibility to
embolisms and the mechanical stability of stems. If these trade-offs could be captured
quantitatively and incorporated into the optimisation, it would be possible to model the
optimal xylem resistance or branching patterns of plants, instead of prescribing these
properties a priori as in current practice.
5.2.4.2
Leaves
In order to understand the costs and benefits of the different plant organs with respect to
the Net Carbon Profit, it is necessary to put their respiration rates as well as their
structural carbon into perspective with the functions they fulfil in assisting
photosynthesis. This has been done for leaves, where the structural carbon was linked to
leaf area, while respiration rates were linked to the amount of photosynthetic
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Chapter 5
“machinery” in the leaves (Schymanski et al. in prep.-a; Schymanski et al. in prep.-c).
However, there is a large variety in the observed relations between leaf area, leaf life
span and leaf carbon and it is yet to be investigated in terms of how far this variety
relates to the adaptation to different environments.
5.2.4.3
Fine Roots
The variability in observed costs and benefits of plant organs is even larger for fine
roots. Firstly, the ratios between root dry mass and root length or surface area as well as
the radial and axial hydraulic conductivities are highly variable, and secondly, it is very
hard to distinguish their energy expenditure for water uptake from the one for nutrient
uptake. Again, simultaneous measurements of water- and nutrient uptake together with
fine root respiration rates under different soil conditions could be helpful for assessing
the costs and benefits of fine roots.
5.2.4.4
Theoretical Approach
Another possible approach to quantifying the maintenance costs of plant organs in
relation to their functions is a more theoretical one. The theory of nonlinear
thermodynamics could potentially be helpful for predicting the rates of energy
dissipation necessary to maintain certain living structures. If there was a quantitative
relationship between the ability of a far-from-equilibrium system to perform work and
the rate of energy dissipation necessary for its maintenance, then the costs of different
plant functions could possibly be derived from first principles. This would be a major
scientific breakthrough, as it would eliminate the need for a large range of empirical
cost parameters in the optimality model.
5.2.5
DEVELOP IMPROVED PLANT FUNCTIONAL TYPES
In the present thesis, only deep-rooted evergreen and shallow-rooted seasonal
vegetation was conceptualised and referred to as “trees” and “grasses” respectively. In
reality, tall trees can be shallow-rooted and have seasonal foliage, while grasses can be
relatively deep-rooted and perennial. The costs and benefits of these strategies in
different environments need further investigation.
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Schymanski (2007): PhD thesis
More research into the costs and benefits of different plant adaptations would
significantly improve the models on the biological side, and better physical
representations of the catchment-scale below-ground hydraulic conditions and runoff
would greatly benefit both the predictions about vegetation properties and about the
water balance. Currently, our computing power is a constraint on how realistically
different processes can be represented and how many degrees of freedom can be
optimised, but increasing computing power means that this constraint is likely to be
relaxed more and more in the future.
5.2.6
INCLUDE ADDITIONAL FEEDBACKS BETWEEN
VEGETATION, SOILS AND THE ATMOSPHERE
In Section 5.2.4.3, it was mentioned that nutrient uptake by fine roots also requires
energy expenditure. This energy expenditure is likely to be dependent on the amount of
available nutrients in the soil. Nutrient availability, in turn, depends on the quality of
litter entering the soil, the soil microfauna and soil moisture. All of these factors can be
influenced by the plants themselves, by varying their leaf and root properties and
turnover rates, or by investing into hydraulic lift and exudates. For example, where
nitrogen is not available in adequate amounts, energy (e.g. assimilates invested into
nitrogen fixing symbionts) can be invested in order to fix it from the air. The
mobilisation of other nutrients from the soil can also be increased by the expense of
assimilates (e.g. exudation of chelates), so the soil nutrient status can not be seen as an
independent forcing for plant growth. It remains to be seen under what conditions the
neglect of such costs will lead to wrong predictions of the water cycle. In most of the
catchments investigated in this thesis, satisfactory results could be achieved without
explicit inclusion of nutrient limitations or uptake costs.
Given the complexity of soil processes, it would appear very tempting to apply
optimality principles to these processes, too. Again, the main challenges lie in
determining the costs and benefits of the vegetation’s interaction with the soil domain
and the objective function of the system. While the plants’ interaction is likely to be
motivated by the maximisation of their net carbon profit, the soil biota could follow
other objective functions. However, to test different optimality hypotheses, more
information is needed about the energetics of plant-soil interactions.
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Chapter 5
Another possible feedback that has not been considered in the present study is the
feedback between transpiration, air humidity and rainfall. This is likely to become more
important at larger scales, particularly if the vegetation optimality model was going to
be coupled to a general circulation model (GCM) and used to investigate global change
scenarios. In this case, increased transpiration could lead to increased recycling of
moisture, which would alter the costs and benefits of water use altogether. Such an
application, though challenging, might give us new insights about the possible
responses of the biosphere to global change.
217
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