Tree Physiology 21, 889–897
© 2001 Heron Publishing—Victoria, Canada
Tree stem diameter variations and transpiration in Scots pine: an
analysis using a dynamic sap flow model
MARTTI PERÄMÄKI,1 EERO NIKINMAA,1 SANNA SEVANTO,2 HANNU ILVESNIEMI,1
ERKKI SIIVOLA,3 PERTTI HARI1 and TIMO VESALA2
1
Department of Forest Ecology, P.O. Box 24, FIN-00014 University of Helsinki, Finland
2
Department of Physics, P.O. Box 9, FIN-00014 University of Helsinki, Finland
3
Laboratory of Applied Electronics, Helsinki University of Technology, Otakaari 5 A, FIN-02150, Espoo, Finland
Received August 18, 2000
Summary A dynamic model for simulating water flow in a
Scots pine (Pinus sylvestris L.) tree was developed. The model
is based on the cohesion theory and the assumption that fluctuating water tension driven by transpiration, together with the
elasticity of wood tissue, causes variations in the diameter of a
tree stem and branches. The change in xylem diameter can be
linked to water tension in accordance with Hooke’s law. The
model was tested against field measurements of the diurnal xylem diameter change at different heights in a 37-year-old Scots
pine at Hyytiälä, southern Finland (61°51′ N, 24°17′ E, 181 m
a.s.l.). Shoot transpiration and soil water potential were input
data for the model. The biomechanical and hydraulic properties of wood and fine root hydraulic conductance were estimated from simulated and measured stem diameter changes
during the course of 1 day. The estimated parameters attained
values similar to literature values. The ratios of estimated parameters to literature values ranged from 0.5 to 0.9. The model
predictions (stem diameters at several heights) were in close
agreement with the measurements for a period of 6 days. The
time lag between changes in transpiration rate and in sap flow
rate at the base of the tree was about half an hour. The analysis
showed that 40% of the resistance between the soil and the top
of the tree was located in the rhizosphere. Modeling the water
tension gradient and consequent woody diameter changes offer
a convenient means of studying the link between wood hydraulic conductivity and control of transpiration.
Keywords: Pinus sylvestris, water tension, xylem diameter
change.
Introduction
Water flow from the soil through plants to the atmosphere is of
interest partly because it limits carbon uptake through its influence on stomatal aperture (e.g., Hubbard et al. 1999). Moreover, it can affect the allocation of carbon compounds (e.g.,
Hari et al. 1986), and is important to an understanding of vegetation–atmosphere interactions. Since the beginning of the
20th century, water flow in trees has been explained by the co-
hesion theory (e.g., Zimmermann 1983), which holds that water is raised from soil to leaf by tension in xylem conduits
created by the evaporation of water from leaf cell walls. Water
in xylem is under tension, i.e., it has negative pressure. Moreover, at pressures below the saturation vapor pressure, liquid
water is in a metastable state and is vulnerable to transition to
the stable vapor phase, which causes xylem conduit cavitation
and termination of water flow.
The cohesion theory has been questioned because of the difficulty of proving the existence of liquid water under high tension in living trees (e.g., Zimmermann et al. 1994). The water
tension of leaves or small twigs has been measured in a pressure chamber (Scholander et al. 1965), and the stem water potential with a stem psychrometer (Dixon and Tyree 1984), a
pressure probe (Zimmermann et al. 1994) or, more recently, by
the centrifugal method (Holbrook et al. 1995, Pockman et al.
1995). All these methods are destructive and their use with intact trees is problematic. Irvine and Grace (1997) introduced a
nondestructive method of evaluating tension based on the elasticity of wood, whereby water tension is derived from changes
in xylem diameter and the elastic modulus of wood.
The dynamic modeling approach allows the study of complicated systems. With models based on real physical interactions, it is also possible to analyze phenomena that are difficult
to measure directly. A key question in modeling is the structure of the model (Bossel 1991). State variables and processes
must have counterparts in the physical world. The values of
the parameters of the model must be obtained by direct independent measurements.
Water flow in an individual tree has been modeled in previous studies (Edwards et al. 1986, Tyree 1988, Früh 1995),
where the relationship between pressure (tension) and the
amount (deficit) of water was based on the use of the capacitance analogy (the change in water content divided by the
change in tension). The problem with this approach is the difficulty in determining the capacitance (Holbrook 1995).
The objectives of this study were to: (a) describe sap flow as
a physical process and derive a dynamic model that simulates
sap flow and stem diameter changes in a single tree; and (b)
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PERÄMÄKI ET AL.
evaluate how well field observations can be explained by this
method. We follow the approach introduced by Irvine and
Grace (1997) of combining water tension and stem diameter
change and use it for the first time, as far as we know, in sap
flow modeling for a whole tree. In addition, we simultaneously monitor transpiration and soil water potential in a tree
stand at the SMEAR II Station (Hyytiälä, southern Finland)
used as input data for the model. As a result, we have a tool for
analyzing coupling of stem diameter variation and transpiration rate.
Three model parameters (the radial elasticity of wood, the
specific permeability of wood and the fine root conductivity)
were fitted against water relations measurements for one
representative day. Simulations with these measurements as
input variables were performed for a longer period. The results
were compared with observations of diurnal variation in stem
diameter, and reasons for discrepancies are discussed. Finally,
the performance of the model, which couples soil, tree stem
and crown by means of the physics of sap ascent and wood
elasticity, is evaluated.
Materials and methods
Figure 1. Schematic illustration of the modeled tree.
tension between two consecutive elements, according to
Darcy’s law (Siau 1984), and may be expressed as:
Qin, i =
ks, i Pi −1 − Pi − Ph
Ai ρ ,
l
η
(2)
Modeling
Model assumptions Water flow in trees is explained by the
cohesion theory. The cohesion forces keep the water column
continuous and adhesion binds it to the cell walls of the conducting elements. Transpiration in the foliage creates a water
tension gradient that initiates sap flow. Because of wood elasticity, wood volume decreases under tension. The water tension can be derived from the diameter decrease in the stem and
branches using Hooke’s law (Irvine and Grace 1997). Ignoring
the effect of cavitation of water columns within the xylem, the
change in wood volume equals the net change in the amount of
water.
Description of the model A modeled Scots pine (Pinus sylvestris L.) is described as a combination of stem, branches,
needles and fine roots (Figure 1). The amount of needles on a
branch is proportional to the sapwood area of the branch (e.g.,
Berninger and Nikinmaa 1994). Needles are located at the outermost end of a branch. Branches in one whorl are identical and
horizontal. Water-absorbing fine roots are directly connected
to the base of the tree.
The tree is divided into a number of stem and branch elements of constant length. State variables of each element are
wood element dimensions, water mass and tension. The water
mass balance equation for each element is:
dm w, i
= Qin, i − Qout, i ,
dt
(1)
where mw,i (kg) is the mass of water of an element i, Qin,i (kg
s –1) is the inward mass flow rate and Qout,i is the outward mass
flow rate. The inward mass flow is caused by the difference in
where ks,i (m 2) is the wood specific conductivity or permeability, η (Pa s) is the dynamic viscosity of water, Pi (Pa) is the water tension of the element i, Pi–1 is the water tension of the
element below i, l (m) is the length of an element, Ai (m 2) is the
basal area of sapwood of an element and ρ (kg m –3) is the density of water. In stem elements, where the sap flow is vertical,
the effect of gravitation is included under Ph = ρgl, i.e., hydrostatic pressure, where g (m s –2) is the acceleration due to gravity. From the geometry of the tree, the change in the amount of
water and change in diameter of an element are related by:
dd i
2 dm w, i
=
,
dt
ld i πρ dt
(3)
where di (m) is the xylem diameter of an element. Water tension causes a reversible change in the diameter of an element.
The relation between water tension and diameter change is described by Hooke’s law:
d Pi
1 d di
= Er
,
dt
d sw, i dt
(4)
where Er (Pa) is the elastic modulus of wood in the radial direction and dsw is the diameter of the sapwood. Tension
changes the diameter of water-conducting sapwood only
(Irvine and Grace 1997). Outflow from an element is equal to
the inflow of the upper element plus the inflow to the
branches:
Qout , i = Qin , i + 1 + ∑ Qin, branch .
TREE PHYSIOLOGY VOLUME 21, 2001
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TREE STEM DIAMETER VARIATIONS AND TRANSPIRATION IN SCOTS PINE
A schematic picture of a stem element is presented in Figure 2.
Boundary conditions The outflow from the outermost branch
elements equals the transpiration rate (i.e., the transpiration
rate per unit needle area multiplied by the needle area of the
branch). Inflow to the base of the tree is caused by the difference between the soil water potential and the water tension at
the base of the tree, and is directly proportional to the root surface area:
Qin,0 = Lpr (Ps − P0 ) Ar ρ ,
(6)
where Lpr (m s –1 Pa –1) is root hydraulic conductivity (root surface area basis), Ps is soil water tension, P0 is the water tension
at the base of the tree and Ar (m 2) is the effective fine root area,
i.e., the total surface area of water-absorbing fine roots.
Modeling canopy PAR attenuation and transpiration Transpiration was measured with two cuvettes at the top of the
crown (see field measurements section). The mean value of
these measurements was used as a reference value. The water
vapor concentration and air temperature inside the canopy
were measured at several heights (Vesala et al. 1998) and no
significant vertical variation was found in VPD (data not
shown) as a result of daytime turbulent mixing. The needle
temperature was assumed to be close to that of the ambient air
and thus transpiration inside the canopy was assumed to be dependent on stomatal conductance only. The stomatal conductance was assumed to be a hyperbolic function of photosynthetic active radiation (PAR) as in Hari et al. (1999a). The
proportion of direct and diffuse PAR was estimated by calculating the theoretical maximum of direct and diffuse PAR under a clear sky using the formulas of Campbell (1981):
I dh = S 0 τ m sin(θ)
(7)
I sh = 0.5( 0.91 S 0 sin(θ) − I d h ),
(8)
Figure 2. Schematic illustration of the stem element and inherent
physics. Parameter mw,i is the mass of water in stem element i (elements presented in Figure 1), dmw,i /dt is the change of the mass of water, Qin,i and Qout,i are inward and outward water flows, respectively.
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where Idh is direct PAR, Ish is diffuse PAR, S0 is the solar constant and equals 2700 µmol m –2 s –1, τ is atmospheric transmittance and equals 0.7, m is 1/sin(θ) and θ is the angle of elevation of the sun. Radiation inside the canopy was assumed to
attenuate as a function of cumulative needle mass according to
Lambert-Beer’s law (see Gates 1980):
 LAI ( z )

k dh 
−

sin( θ )
I dh ( z) = pdh Im e
,
(9)
for direct radiation and:
I sh ( z) = psh Im e(− LAI (z ) k sh ) ,
(10)
for diffuse radiation, respectively, where pdh = Idh /(Idh+ Ish),
psh = Ish /(Idh+ Ish), Im is the total PAR measured above the canopy, LAI is the cumulative leaf area index, kdh is the attenuation coefficient for direct radiation and ksh is the attenuation
coefficient for diffuse radiation. The values of kdh and ksh were
determined from the multipoint PAR measuring system
(Vesala et al. 1999). The values of kdh and ksh were 0.24 and
0.19, respectively (T. Markkanen, University of Helsinki,
Finland, unpublished data).
Total PAR inside the canopy, Ih(z), is the sum of direct and
diffuse components and, given the dependence of the stomatal
conductance on PAR, the transpiration inside the crown was
estimated to be:
Iz
Iz + γ
E ( z) =
ET ,
Im
Im + γ
(11)
where parameter γ equals 700 µmol m –2 s –1, and ET is the reference transpiration measured by the chambers at the canopy
top.
This approach made the stomatal conductance decrease almost linearly down to 25% of the top value at the crown base.
This is consistent with literature values. According to Beadle
et al. (1985), the stomatal conductance of the lower shoots of
Scots pine at Thetford, UK, is 25 to 50% of that of the upper
shoots.
Implementation and parameterization The model was implemented on a personal computer with the Delphi Version 4
(Inprise/Borland Software Corp., Scotts Valley, CA) programming tool. The tree was described as a linked list of a number of
stem and branch elements of constant length (0.2 m) (see Figure 1). Equations 1, 3 and 4 were solved numerically for each
element by the fourth-order Runge-Kutta method (Press et al.
1988). The time step was 0.1 s. The model simulates water relations from the vertical transpiration profile and soil water tension as input data. During a time step, water relations were
simulated for each stem and branch element starting from the
top of the tree. Initial values of the state variables are presented
in Table 1.
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PERÄMÄKI ET AL.
Table 1. Initial values of the state variables of the model
State variable
Unit
Description
Initial value
mw,i
Pi
kg
MPa
Mass of water in an element
Water tension in an element
dx,i
Qm,i
m
kg s –1
Diameter of an element
Water flow into an element
0.5ρVi, Vi is the volume of an element
Ps,t = 0 – ρghi.
Ps,t = 0 is the soil water tension at the beginning of the simulation period
hi is the distance of an element from the ground
Measured from the tree
0
Field measurements
Monitoring water relations in field conditions We simultaneously measured shoot transpiration, diurnal shrinking and
swelling of xylem and soil water tension in the tree stand at the
Helsinki University SMEAR II Station in Hyytiälä, southern
Finland (61°51′ N, 24°17′ E, 181 m a.s.l.). The site is of medium fertility and supports an even-aged Scots pine (Pinus
sylvestris L.) stand established from seed in 1962. The mean
height and diameter of the trees was 12 m and 13 cm, respectively, and the number of stems was 2500 h –1. Leaf area index
(LAI) was 8. The vertical distribution of the needle mass (area)
was also determined (Vesala et al. 1999). We selected one dominant tree from the stand for more detailed analysis. We measured the basic structural information on the tree including the
diameter of the stem below each branch whorl, positions of
branch whorls and the number and diameter of branches in
each whorl. The number of needles on each branch was estimated from the regression between the branch basal sapwood
area and the needle mass of the branch, as determined from
other trees of the same stand (Palmroth et al. 1997). The diameter of nonconducting heartwood was estimated by taking core
samples of the tree stem. The main characteristics of the test
tree are: height = 13.2 m, dbh = 0.198 m, height of crown base =
6 m, estimated needle area = 50.8 m and fine root area =
96.1 m 2.
Transpiration at the top of the tree was measured using two,
cylindrical trap-type chambers enclosing a shoot installed
through the chamber base. The measurement was based on the
detection of the water vapor concentration change in the
cuvette while it was closed for 70 s. The gas concentration was
recorded every 5 s. The cuvette was open for the remaining
time, thus providing an environment close to ambient conditions for the shoots. When the cuvette was closed, the airflow
to the gas analyzer (URAS 4, Mannesman, Hartmann and
Braun, Berlin, Germany) was compensated by the influx of
ambient air. The air in the cuvette was kept well-mixed by a
small fan. Measurements were taken every 20 min. The chamber system is described by Aalto (1998) and Hari et al.
(1999b). Dimensions of the needles in the cuvette were measured manually and needle areas were calculated by Tirén’s
formula (Tirén 1926):
A=
π1

 w + t  l + wl ,

22
(12)
where A is total surface area, w is needle width, t is needle
thickness and l is needle length. Transpiration per needle area
was estimated from the chamber measurements from the mass
balance equation presented in Hari et al. (1999b).
The microvariation in stem diameter was measured every
minute at four heights (2.5, 6.5, 9 and 12 m) by industrial diameter transducers (Sylvac SA, Crisser, Switzerland; Solartron Inc., West Sussex, U.K.) connected to a rigid steel frame
mounted around the tree (Figure 3). A small aluminum plate
was fixed on the wood after removing the bark on both sides of
the stem so that the frame and the sensitive diameter transducer attached to the frame rested on the plates on opposite
sides. The diameter transducers measured the distance between the plates, i.e., the effect of cambium and phloem was
eliminated. The accuracy of the instrument was 0.1 µm. The
effect of thermal expansion of the frame was eliminated by
correcting the results on the basis of measured temperatures
and the linear coefficient of thermal expansion for steel.
Soil water tension and fine roots The volumetric soil water
content was monitored by the Time-Domain Reflectometry
(TDR) method (Tektronix TDR 1502, Tektronix Inc., Beaverton, OR) in six places in the research site. The quantity of water
was transformed to soil water tension by separately determined
soil water retention curves. The mean value of several TDR
outputs from different soil layers, augmented by the vertical
fine root distribution, was taken as the soil water tension. For
determining the quantity of fine roots, 20 evenly distributed
soil samples were taken from the research site with a 50-cm
Figure 3. Schematic illustration of the xylem-diameter measuring
system.
TREE PHYSIOLOGY VOLUME 21, 2001
TREE STEM DIAMETER VARIATIONS AND TRANSPIRATION IN SCOTS PINE
long auger with an inside diameter of 46 mm (Westman 1995).
The soil samples were divided into humus, eluvial, illuvial and
ground soil samples in the laboratory and the depth of each
layer was measured. The samples were gently washed in a
sieve to remove the soil and all living Scots pine roots were
transferred to clear acetate sheets. The WinRhizo computer
program Version 3.0.2 (Regent Instruments Inc., Québec City,
Canada) and the Tennant calculation algorithm were used to
measure surface area of the roots. The effective fine root density Ar of the research site was 10.4 m 2 m –2 soil. The total quantity of fine roots of the test tree was estimated by assuming that
this quantity correlates with the stem base area at breast height
(Marklund 1988).
Estimation of radial elasticity and specific permeability of
xylem and fine root conductivity Three parameters were not
directly measured: the radial elastic modulus (Er), the wood
specific conductivity (ks) and the fine root hydraulic conductivity (Lpr). These parameters were estimated simultaneously by
using measured and simulated stem diameter changes at the
four heights (2.5, 6.5, 9 and 12 m) for one typical, partly cloudy
day (June 16, 1999) and minimizing the sum of residual
squares. The radial elastic modulus and specific conductivity
of wood were assumed to be a function of cambial age (Mencuccini et al. 1997).
Results
Parameter values and model sensitivity
The values of fitted parameters are presented in Table 2. Root
hydraulic conductivity deviated most from the literature value
(Rüdinger et al. 1994). However, this is acceptable because the
literature value was for Norway spruce growing in a different
environment. The specific permeability of sapwood parameter
also yielded a value smaller than that obtained by Mencuccini
et al. (1997), who used a standard method for determining permeability in which wood samples were saturated with water.
Water content has a considerable effect on the permeability of
wood (e.g., Edwards et al. 1994). Nikinmaa et al. (1997) reported a considerable wood water content variation in Scots
pine during summer and also demonstrated its effect on conductivity. Thus the value obtained, although smaller than in the
literature, is reasonable.
The sensitivity of the model to estimated parameters was
tested by performing several model runs and changing the
value of one parameter at a time ± 30%. The model was most
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sensitive to changes in elasticity and fine root conductivity. A
decrease of 30% in fine root conductivity caused an increase
of about 30% in the amplitude of the stem diameter in the
whole stem area. The effect of a change in elasticity was of the
same order of magnitude but opposite to that of fine root conductivity. The sensitivity of the model to a 30% change in specific permeability was negligible.
Short time-scale phenomena
The simulated and measured diameter changes for June 16th,
1999 are presented in Figure 4. The model was able to accurately predict the general pattern and minor changes in stem
diameter at all heights. The biggest discrepancy was at the
height of 6.5 m, where the model underestimated the diameter
change by approximately 20%. The existing discrepancies
may result partly from inaccuracies in xylem elasticity and
permeability measurements. The empirical functions (Mencuccini et al. 1997) that describe cambial age dependence are not
necessarily valid for the test tree studied. Furthermore, we
transformed the longitudinal elasticity to radial elasticity with
a constant value reported by Koponen et al. (1991). Because
the woody structure varies in a radial direction as a function of
cambial age, a constant ratio between the longitudinal and radial elastic modulus may not be valid for the entire stem.
The simulated and observed diameter changes at all heights
followed the transpiration changes quite closely. Figure 5
shows transpiration and simulated diameter change at the
2.5 m height. This suggests that tension (i.e., diameter change)
propagates quickly along the stem. The speed of propagation
of tension means that there is no significant water storage between the evaporating needles and the base of the stem. Note
that no water storage in the stem, apart from that resulting
from the tree diameter changes, was assumed in the model. If
large quantities of water came from the stores in the woody
segments, this should have resulted in considerable delays in
propagation of transpiration-induced tension down the stem.
The insignificance of the water storage in the stem is also revealed in the small time lag between simulated sap flows at the
height of 6.5 m and at the base of the stem (Figure 6). The
mean time lag for the whole day was calculated by making a
cross-correlation analysis between sap flows at these two
heights. The sap flow at 6.5 m was repeatedly shifted later in
time at a time step of 1 min. The maximum correlation (r 2 =
0.999) was found when the time shift was 7 min.
Because an increase in transpiration is directly reflected in a
decrease in diameter of the stem, there must be a considerable
Table 2. Estimated parameter values of E r, ks and Lpr. Literature values of E r,L and ks,L are functions of cambial age (Mencuccini et al. 1997), Lpr,L
is root hydraulic conductivity of Norway spruce, Picea abies (L.) Karst. (Rüdinger et al. 1994). The radial elastic modulus was estimated by applying the constant ratio (1/17.5) between radial and longitudinal elastic modulus (Koponen et al. 1991).
Parameter
Unit
Description
Value
Value versus literature value
Er
ks
Lpr
GPa
m2
m s –1 MPa –1
Radial elasticity
Specific permeability
Root conductivity
0.03–0.27
1 × 10 –12 –4 × 10 –12
3.2 × 10 –8
0.6 × Er,L
0.9 × ks,L
0.5 × Lpr,L
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PERÄMÄKI ET AL.
Figure 6. Simulated water flow at the crown base (6.5 m) (thick line)
and stem base (thin line) on June 16, 1999 at 0600–1800 h (a).
stem area (Figure 7). Diurnal variation in simulated water tension was 0.07 MPa at the base of the stem and 0.17 MPa at the
top, indicating that up to 40% of hydraulic resistance between
the soil and the top of the tree is located in the rhizosphere and
the remaining 60% is in the stem. This is in reasonable agreement with the literature (Running 1980, Irvine and Grace
1997).
Daily-scale phenomena
Figure 4. Measured (thick line) and simulated (thin line) xylem diameter change on June 16, 1999 at heights of 12 m (a), 9 m (b), 6.5 m (c)
and 2.5 m (d). The results are taken from the longer simulation period
June 12 to 17, 1999.
resistance to water flow at the root–soil interface. Uptake
seems to be restricted by low root permeability. Restricted water uptake causes fluctuation in water tension over the whole
Figure 5. Measured transpiration (thick line) and simulated diameter
change (thin line) at a height of 2.5 m on June 16. Note that on the
y-axis at right the values are in reverse order.
Model performance was tested by a longer simulation (June
12–17, 1999) with the parameter values fitted for June 16. Figure 8 shows the measured input data (PAR, air temperature,
reference transpiration, soil water tension and VPD). This period was selected because it included various radiation conditions and one rainy day (June 15), with 11 mm of precipitation.
Consequently, soil water tension decreased substantially on
June 15 (Figure 8d). These factors have a significant impact on
transpiration, sap flow and stem diameter.
The measured and simulated stem diameters are presented
in Figure 9 (note that the curves are matched with June 16).
The model is able to predict the pattern of stem diameter
changes at all heights quite well. At the beginning of the simulation the water tension at the base of the tree was set to equal
the measured soil water tension. According to the model, the
basal tension is driven to equilibrium with the tension in the
soil at night and hence the overall decreasing diameter trend
Figure 7. Simulated water tensions at a height of 12 m (thick line), at
the crown base (medium line) and at the base of the stem (thin line) on
June 16, 1999.
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Figure 8. PAR (a), air temperature (b), transpiration per unit needle
area (c), soil water tension (d), and VPD (e) during the simulation period from June 12 to 17, 1999.
(except the rainy day) follows the same pattern as the soil water tension (Figure 8d). Measurements, however, showed an
increasing trend in stem diameter for the whole period. The
contradictory trends are most apparent as different measured
and predicted diameters at the beginning of the simulation period, and at night between June 14 and 15. The daily amplitudes, however, are in agreement. The minimum nighttime
temperatures (Figure 8b) show an increasing trend similar to
the increase in maximum diameters at night. The effect of temperature on the diameter of the tree cannot, however, explain
this increase, because the coefficient of thermal expansion of
fresh wood is negative (Irvine and Grace 1997, Salmén 1990)
and its contribution is one order of magnitude smaller than the
observed swelling. The increasing trend suggests that the initial assumption of the water status equilibrium between the
stem base and soil is not strictly valid, and despite decreasing
soil water tension, water uptake is enhanced to the extent that
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Figure 9. Measured (thick line) and simulated (thin line) xylem diameter changes at heights of 12 m (a), 9 m (b), 6.5 m (c) and at 2.5 m (d)
during the simulation period from June 12 to 17, 1999.
rain does not affect the stem diameter, as it does in the case of
the model. The improved water uptake may be the result of increased root activity caused by the increase in soil temperature. There was an increasing trend in soil temperature (data
not shown) during the simulation period. Fine root biomass
has been reported to fluctuate considerably without a distinct
and clear pattern during the growing season (Makkonen and
Helmisaari 1997). The possible increase in the quantity of fine
roots may also have enhanced water uptake.
The model often overestimated evening and nighttime recovery of stem diameter, especially at heights of 2.5 and 6.5 m
(Figures 9c and 9d). According to the model, the water tension
at the base of the tree was driven to equilibrium with the soil
water tension. The measurements, however, showed that the
tree did not completely fill up until transpiration started; for
example, early in the morning on June 14. This is shown by the
diameter change curves. The most probable explanation of this
discrepancy is a reduction in soil hydraulic conductivity in the
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896
PERÄMÄKI ET AL.
vicinity of the fine roots during the day. This reduction is not
seen with robust soil water status measurements.
The simulated sap flow at the base of the stem is shown in
Figure 10. The flow follows transpiration quite closely. Crosscorrelation analysis between transpiration and the simulated
sap flow at a height of 2.5 m gave the maximum correlation
(0.975) when the time lag was 26 min for the whole period. For
testing the relationship between water tension and time lag we
divided the simulation period into high-tension (0600– 1800
h) and low tension (1800–0600 h) periods. The time lag varied
between 17 and 31 min (mean 22 min) during the high-tension
period and between 20 and 39 min (mean 30 min) during the
low-tension period. The higher the tension, the smaller the diameter of the stem, so the change in diameter corresponding to
a certain decrease in the amount of water is larger. To obtain a
larger change in diameter a greater tension is required, which
results in a more rapid flow and a smaller time lag. In living
trees, the increased tension often causes cavitation (e.g., Grace
1993) which decreases permeability. Decreased permeability
slows sap flow, complicating the time lag analysis. Note that in
our model, water tension has no effect on permeability.
In this study, the time lag between transpiration and sap flow
is shorter than literature values; time lag values of 1–2 between transpiration and sap flow at the base of the stem are often reported for taller trees (e.g., Herzog et al. 1995). Phillips
et al. (1997) reported a time lag of about 0.5 h for 12-year-old
loblolly pine (7 m).
Discussion
Water transport within a tree provides a key to understanding
the formation of its woody structure (Hari et al. 1986). The application of simple physical principles to the woody structure
of a tree allows a relatively straightforward construction of
water flow models, such as are to be found in the literature
(e.g., Edwards et al. 1986, Tyree 1988, Früh 1995). Most of
these models use the capacitance analogy, i.e., the relationship
between water content and water tension; an approach which
provides a reasonable description of water flow in the wood,
but involves a fairly laborious determination of wood water capacitance.
The idea that tension is reflected in the xylem diameter is
Figure 10. Transpiration (thick line) and simulated sap flow at the
base of the stem (thin line) during the simulation period from June 12
to 17, 1999.
not often considered in sap flow studies. This is analogous to
the treatment of a string: applied tension causes a force on a
stem surface element directed toward the center of the stem
and the tracheal structure resists the movement of the surface
element. Because the changes in the diameter are small, the
linear Hooke’s law can be used. The treatment of wood as an
elastic material is also a simplification, but the model predictions, which were reasonably close to the daily measured values, suggest that this simplification is justified. Because the diameter change method is accurate and reliable, water flow
models incorporating tension-induced diameter changes can
be readily evaluated. Variation in stem diameter is easily measured by the elegant technique introduced by Irvine and Grace
(1997).
The model presented here describes the daily pattern of
transpiration-induced diameter changes well. The close coupling between transpiration and diameter variation using reasonable parameter values indicates that water storage variation
in such conditions has a minor role. Tension-induced diameter
changes offer means of evaluating model performance. Model
predictions linked with observations can be used to investigate
the relationship between wood hydraulic conductivity and
control of transpiration. The larger time lag observed in large
trees in tension propagation down the stem and some discrepancies observed in this study may indicate the necessity of
considering cavitation. This model structure readily allows for
such extensions. Simulation of water flow in the soil in the vicinity of the fine roots is also a necessary improvement in research concerning water transport in the soil–tree–atmosphere
continuum.
Acknowledgments
We thank Dr. Toivo Pohja who contributed to designing the measurement facilities and Mrs Susanna Kaukinen who helped measure fine
root density. Financial support was provided by Tekes Technology
Development Centre, Finland (project 40373/98).
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