Boundary-Layer Meteorol
DOI 10.1007/s10546-013-9823-0
ARTICLE
Large-Eddy Atmosphere–Land-Surface Modelling over
Heterogeneous Surfaces: Model Development and
Comparison with Measurements
Yaping Shao · Shaofeng Liu · Jan H. Schween ·
Susanne Crewell
Received: 10 May 2012 / Accepted: 26 March 2013
© Springer Science+Business Media Dordrecht 2013
Abstract A model is developed for the large-eddy simulation (LES) of heterogeneous
atmosphere and land-surface processes. This couples a LES model with a land-surface
scheme. New developments are made to the land-surface scheme to ensure the adequate
representation of atmosphere–land-surface transfers on the large-eddy scale. These include,
(1) a multi-layer canopy scheme; (2) a method for flux estimates consistent with the large-eddy
subgrid closure; and (3) an appropriate soil-layer configuration. The model is then applied
to a heterogeneous region with 60-m horizontal resolution and the results are compared with
ground-based and airborne measurements. The simulated sensible and latent heat fluxes are
found to agree well with the eddy-correlation measurements. Good agreement is also found
in the modelled and observed net radiation, ground heat flux, soil temperature and moisture.
Based on the model results, we study the patterns of the sensible and latent heat fluxes, how
such patterns come into existence, and how large eddies propagate and destroy land-surface
signals in the atmosphere. Near the surface, the flux and land-use patterns are found to be
closely correlated. In the lower boundary layer, small eddies bearing land-surface signals
organize and develop into larger eddies, which carry the signals to considerably higher levels. As a result, the instantaneous flux patterns appear to be unrelated to the land-use patterns,
but on average, the correlation between them is significant and persistent up to about 650 m.
For a given land-surface type, the scatter of the fluxes amounts to several hundred W m−2 ,
due to (1) large-eddy randomness; (2) rapid large-eddy and surface feedback; and (3) local
advection related to surface heterogeneity.
Keywords
Atmosphere–land interaction · Heterogeneous surfaces · Large-eddy simulation
Y. Shao (B) · S. Liu · J. H. Schween · S. Crewell
Institute for Geophysics and Meteorology, University of Cologne, Cologne, Germany
e-mail: [email protected]
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Y. Shao et al.
1 Introduction
In atmospheric numerical models, land-surface processes are commonly represented using
land-surface schemes. The first generation schemes were designed to estimate surface sensible
and latent heat fluxes in climate models (Manabe 1969). In the second generation schemes, the
diurnal variations of the fluxes were considered by taking the force-restore approach to soiltemperature and moisture modelling (Bhumralker 1975; Deardorff 1978). More sophisticated
schemes (e.g. Dickinson et al. 1993) were developed in the 1980s and 1990s, with improved
treatments for plant canopy and surface soil hydrology. The third generation schemes also
include the components for plant physiology and photosynthesis, which enable the estimation
of CO2 and other trace-gas fluxes (e.g. Oleson et al. 2007).
A basic assumption made in existing land-surface schemes is that land-surface processes
are horizontally homogeneous and the transfers of the physical quantities are onedimensional. Thus, the existing framework for land-surface parametrization appears to be
fundamentally inadequate for heterogeneous land surfaces. For example, the bulk transfer
method based on the Monin–Obukhov similarity theory (MOST) (Monin and Obukhov 1954)
assumes that the atmospheric surface layer is in equilibrium with the surface and its evolution
is free from the effects of advection. Foken (2006) pointed out that the transfer mechanisms in
the heterogeneous atmospheric boundary layer can significantly deviate from MOST. Shao
et al. (2001) and Heinemann and Kerschgens (2005) have demonstrated that land-surface
heterogeneity impacts strongly on the atmosphere and land-surface exchanges.
The parametrization of heterogeneous land-surface processes has been active since the
late 1990s (e.g. Giorgi and Avissar 1997). Progress has been made in dealing with spatially distributed land-surface properties, by using the techniques of mosaic (e.g. Ament and
Simmer 2006) and parameter hierarchy (e.g. Oleson et al. 2007). It is now understood that
land-surface heterogeneity has two major effects on surface fluxes, known as the aggregation
effect and the dynamic effect. The aggregation effect occurs because the spatial variations
of the land-surface properties (albedo, hydraulic properties, etc.) result in spatial variations
of the land-surface state variables (soil moisture, soil temperature, etc.), and because the
transfer processes are non-linear, the aggregation of surface parameters does not necessarily
produce the correct aggregation of fluxes. The techniques of mosaic and parameter hierarchy
are commonly used to account for the aggregation effect. The dynamic effect occurs because
contrasts in surface conditions generate turbulence and horizontal advection that leads to
spatial variations in turbulent transfer. To date, no theoretical framework exists, equivalent
to MOST, which effectively represents the dynamic effect.
Three-dimensional atmospheric and land-surface data are required to better understand
and parametrize heterogeneous atmosphere–land-surface systems. Such data are difficult to
obtain from field or laboratory measurements, but synthetic data can be generated using largeeddy simulation (LES) and land-surface coupled models. The emphasis on large eddies is
important because they are the main contributors to the transfer processes in the atmospheric
boundary layer, and their developments are closely related to the distribution of land-surface
properties. LES models have been under development since the 1960s (Smagorinsky 1963;
Deardorff 1970; Moeng 1984), and are now widely used for atmospheric turbulent flow
simulations (Sullivan et al. 1998; Beare et al. 2004; Kleissl et al. 2006; Kumar et al. 2006).
Earlier LES models were not coupled with land-surface schemes, and the simulations mostly
had pre-specified land-surface forcing (e.g. Hechtel et al. 1990; Avissar and Schmidt 1998;
Albertson et al. 2001; Raasch and Harbusch 2001; Letzel and Raasch 2003; Huang et al.
2008; Maronga and Raasch 2013) with emphasis placed on the responses of the atmospheric
boundary layer to the forcing. More recently, LES atmosphere–land-surface coupled models
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Large-Eddy Atmosphere–Land-Surface Modelling
have been under development and have been applied to the simulation of boundary-layer flows
over synthetic (e.g. Patton et al. 2005; Huang and Margulis 2009) and natural heterogeneous
land surfaces (e.g. Huang and Margulis 2010; Brunsell et al. 2011). However, the land-surface
models used are not adequately adapted to the large-eddy scale and the model results are not
yet thoroughly validated against measurements.
In this study, a large-eddy simulation atmosphere–land-surface model, LES-ALM, is
developed, which couples a LES model with a new land-surface model adapted to the LES
requirements. In the new model, as detailed in Sect. 2, a multi-layer canopy scheme is
employed and the surface-flux computation is consistently formulated with the subgrid turbulence closure scheme and no longer requires MOST. The soil-layer configuration used
herein differs from that used in earlier studies because, as later shown, very thin soil layers need to be used to capture the rapid exchanges between the land surface and the large
eddies. LES-ALM is then applied to a heterogeneous area and its performance tested against
observations. The comparison confirms that LES-ALM is an adequate and powerful tool for
studying heterogeneous atmosphere–land-surface systems. Using the model simulations, we
examine the relation of large eddies to surface heterogeneity and the persistency of surface
heterogeneity in the atmosphere. The emphasis is placed on the development and validation
of the large-eddy model.
2 Large-Eddy Simulation Atmosphere–Land-Surface Model
2.1 Large-Eddy Simulation Model and Radiation Scheme
The LES-ALM model integrates a LES model with a radiation scheme and a land-surface
scheme. The LES model is the Weather Research and Forecast (WRF) model in its largeeddy mode, but improved by the inclusion of a vegetation canopy scheme. The flow is
assumed to be compressible and non-hydrostatic. The model separates the turbulent flow
into a grid-resolved component and a subgrid component. Several subgrid closures can be
selected (e.g. Smagorinsky√1963), but the k–l closure (Deardorff 1980) is used here. A subgrid
scaling velocity u ∗s = Ck e/κ is defined, with e being the subgrid turbulent kinetic energy
(TKE) determined by solving the TKE equation (Skamarock et al. 2008), Ck is an empirical
parameter of about 0.15, and κ is the von Karman constant. The subgrid eddy viscosity can
be expressed as:
K m,sg = κu ∗s l,
(1a)
where l is a mixing length that differs for horizontal and vertical directions. Equation 1a is
identical to
√
K m,sg = Ck l e.
(1b)
Suppose the horizontal and vertical grid resolutions are x and z, respectively. Then, we
set l x = x and l z = z. The subgrid eddy diffusivity for a scalar (e.g. heat), K h,sg , can be
expressed as
K h,sg = K m,sg Pr−1
(2)
with Pr being the Prandtl number of about 0.3. For radiation estimates, the RRTMG scheme
implemented in the WRF model is adapted. This is a corrected k-distribution model for both
shortwave and longwave broadband radiation transfers (Mlawer et al. 1997). More details
can be found in Clough et al. (2005) and Iacono et al. (2008).
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Y. Shao et al.
2.2 Large-Eddy Land-Surface Scheme
Numerous land-surface schemes have been proposed for numerical weather prediction and
climate models (e.g. Dickinson et al. 1993; Irannejad and Shao 1998; Ek et al. 2003; Oleson
et al. 2007), but substantial modifications are necessary to suit the purpose of large-eddy
modelling. In this study, a new land-surface scheme is proposed, which is based on the Noah
community land-surface model (Chen and Dudhia 2001), but with three new key features
that will be described below.
2.2.1 Explicit Multi-Layer Canopy Scheme
Due to the high resolution used for LES (x ∼ 10 m, z ∼ 1 m), a vertically resolved
canopy layer is necessary (Fig. 1). The effects of vegetation on the flow are treated as sinks
(sources) in the conservation equations for momentum, heat and moisture. The treatment of
vegetation as a momentum sink is not new (e.g. Shaw and Schumann 1992), but heat and
moisture sources in the context of land-surface modelling are critical for coupling a largeeddy flow model with a land-surface scheme. The drag induced by vegetation on the flow is
explicitly treated as a momentum sink in the equations of motion,
∂ ū i ū j
∂τi j
∂ ū i
1 ∂ p̄
∂ 2 ū i
+
= −δi3 g + εi j3 f ū j −
−
+ ν 2 + SMi ,
∂t
∂x j
ρ ∂ xi
∂x j
∂x j
(3)
where ū i is the gird-resolved flow velocity, g is the acceleration due to gravity, f is the
Coriolis parameter, ρ is the air density, p̄ is pressure, τi j is the subgrid stress, ν is the
kinematic viscosity, δi3 is the Kronecker operator and εi j3 is the alternating operator. SMi is
the canopy drag in the xi direction, given by
SMi = −αf Cd V ū i ,
(4)
where Cd is a dimensionless drag coefficient of 0.15 (Shaw et al. 1988), and depends on
canopy porosity and sheltering effects (Raupach 1992; Shao and Yang 2008; but not considered in this study), V is the local wind speed, and αf is the vegetation frontal area density
(frontal area per unit air volume, m2 m−3 ). For a given location (x, y), the vegetation frontal
area index (FAI, vegetation frontal area per unit land surface, m2 m−2 ) can be estimated from
the leaf area index (LAI) for given plant configuration (e.g. FAI = LAI for plants of spherical
shape). For a given vegetation type, a shape function (Fig. 1) for the vertical distribution of
the frontal area, f v (z) (m−1 ) is specified, and αf can be estimated as
αf (x, y, z) = F AI (x, y) f v (z),
(5)
with f v satisfying
∞
f v (z)dz = 1.
(6)
0
Likewise, vegetation also acts as sources (sinks) for heat, moisture and carbon dioxide. These
sources and sinks can be treated similarly as for momentum. The source term in the equation
for air temperature can be expressed as
ST = −αt CT V (T̄ − Tc ),
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(7)
Large-Eddy Atmosphere–Land-Surface Modelling
where αt is the leaf area density (total leaf area per unit volume, m2 m−3 ), CT is a dimensionless exchange coefficient analogous to Cd , T̄ is the air temperature in the canopy and
Tc is the canopy temperature. The source term in the equation for specific humidity can be
expressed as
Sq = −(1 − f wet )αq Cq V [q̄ − qs (Tc )] − f wet αq Cd V [q̄ − qs (Tc )] ,
(8)
where f wet is the fraction of wet vegetation, q̄ is the specific humidity, αq = αt/2 and qs (Tc )
is the saturation specific humidity at canopy temperature, Tc . Taking the vegetation stomatal
resistance into account, Cq can be expressed as
Cq =
Cd
,
C d V rb + 1
where the bulk vegetation resistance is given by
rst
,
rb =
αq l
(9)
(10)
where l is a unit length. Many schemes for rst exist, and we follow the method described
in Noilhan and Planton (1989), which relates rst to several environmental control factors,
including the root zone soil moisture.
As Eqs. 7 and 8 show, a scheme for canopy temperature, Tc , is necessary. Such a scheme
involves the transfer of (shortwave and longwave) radiation, which in detail depends on the
absorbance, reflectance, distribution and orientation of the leaves, solar view angle, etc. In
this study, a simple canopy temperature model is used, with which Tc can be estimated by
solving the following diagnostic equation,
αt εσ Tc4 = λs (Rs↓ + Rs↑ ) + λl (Rl↓ + Rl↑ ) − ρcp ST − ρ L Sq ,
(11)
where ε is vegetation emissivity, σ is the Stefan–Boltzmann constant, λs and λl are the
extinction coefficients for shortwave and longwave radiation, Rs↓ and Rs↑ are the downward
and upward shortwave fluxes, while Rl↓ and Rl↑ are the downward and upward longwave
fluxes, respectively; ρ is air density, cp is the air specific heat capacity at constant pressure
and L is the latent heat of vaporization. Details of the canopy temperature scheme, because
of its complexity, will be fully described in a separate paper, but an outline is given in the
Appendix.
2.2.2 Flux Formulation
The atmospheric surface layer is commonly divided into an inertial layer and a roughness
sublayer. In conventional land-surface models, fluxes are computed for the inertial layer using
the bulk transfer method that can be formulated as follows.
Suppose the surface layer is homogeneous and stationary. Then, the profiles of mean
wind, ū, potential temperature, θ̄ , and specific humidity, q̄, are almost logarithmic and the
momentum flux, τ , sensible heat flux, H , and latent heat flux, LE, are constant in the vertical.
The flux-gradient relationship for H (and similarly for L E) is expressed as
H = −ρcp K h
∂ T̄
.
∂z
The eddy diffusivity, K h , is derived from MOST as
κu ∗ z
Kh =
,
ϕh
(12)
(13)
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Y. Shao et al.
Fig. 1 LES model configuration. The vegetation canopy is vertically resolved in multiple layers, and thin soil
layers are used to allow the land surface to respond on the large-eddy time scale
with ϕh being the MOST stability function and u ∗ is the friction velocity. Using the bulk
formulation, we have
H = −ρcp
(T̄a − T̄0 )
,
rh
(14)
K h−1 dz
(15)
where rh is the aerodynamic resistance,
z
rh =
z0
and T̄a and T̄0 respectively are the reference-level air temperature and surface temperature.
Using MOST, we find
z
z
1
rh =
ln
− ψh
,
(16)
κu ∗
z 0h
Lo
z
where L o is the Obukhov length, ψh = z 0h (1 − ϕh )d ln z, and z 0h is the roughness length for
heat. Although this type of flux formulation has been used in recent large-eddy atmosphere
and land-surface coupled simulations, its applicability must be questioned for the following
reasons:
(1) The derivation of MOST assumes horizontal homogeneity with the effect of advection
being negligible. These assumptions do not hold on the scale of atmospheric large eddies.
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Large-Eddy Atmosphere–Land-Surface Modelling
(2) The MOST similarity functions are empirically derived using averaged (e.g. over 15–
30 min or over several km) boundary-layer measurements. We are not aware of MOST
similarity functions derived from large-eddy scale measurements.
(3) In the framework of a LES model, the MOST-based diffusivity and viscosity estimates
near the surface are inconsistent with the model closure-based diffusivity and viscosity
estimates, causing contradictions between model closure and boundary condition.
(4) Even if MOST were applicable, the similarity parameters could not be specified with
confidence (e.g. roughness length), especially in areas of land-surface heterogeneity,
causing large uncertainties in flux estimates.
In contrast, in a LES model, the main fractions of the fluxes are computed by resolving
the energy-containing eddies, and only the inertial sub-range parts of the fluxes need to be
parametrized. We express a flux (e.g. H and L E) as the sum of a grid-resolved flux (Hg and
L E g ) and a subgrid flux (Hsg and L E sg )
H = Hg + Hsg ,
L E = L E g + L E sg ,
(17a)
(17b)
where Hg and L E g are computed from the grid-resolved vertical velocity, w̃, air temperature,
T̃ , etc., namely,
Hg = ρcp w̃ T̃ ,
L E g = ρ L w̃q̃.
(17c)
(17d)
H and L E calculated using Eqs. 17a, b are then included in the surface energy and water
balance equations, i.e.
Rn − (Hg + Hsg ) − (L E g + L E sg ) − G = 0,
(17e)
P − (E g + E sg ) − I − Ro = 0,
(17f)
where Rn is net radiation, G is ground heat flux, P is precipitation, I is infiltration and
Ro is surface runoff. Equation 17a–f represents an essential difference between large-eddy
atmosphere–land-surface modelling and the conventional land-surface modelling. In the latter
case, Hg and E g are zero.
Hsg and L E sg can be expressed as
Hsg = −ρcp
(T̃a − T̃0 )
,
rh,sg
L E sg = −ρ Lβ
(q̃a − qs (T̃0 ))
,
rq,sg
(18)
(19)
where T̃a and q̃a are the air temperature and specific humidity at the lowest model level, T̃0
is the surface skin temperature and qs (T̃0 ) is the saturation specific humidity at T̃0 . Various
formulations exist for the β parameter, which is usually assumed to be a linear function of
the soil moisture in the top soil layer (Irannejad and Shao 1998).
For simplicity, we assume rh,sg = rq,sg , but these resistances can no longer be determined
from MOST. Instead,
z 1
rh,sg =
−1
K h,sg
(z)dz,
(20)
z 0s
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Y. Shao et al.
where z 0s is a roughness length depending on local aerodynamic characteristics of the landsurface, and z 1 is the lowest model level height. Suppose
n
z
(21)
K h,sg (z) = K h,sg (z 1 )
z1
with K h,sg (z 1 ) being estimated from the subgrid closure scheme (i.e. Eq. 2) for the first model
level height, z 1 . For n = 1, Eqs. 20 and 21 give
z1
z1
rh,sg =
ln
.
(22a)
K h,sg (z 1 )
z 0s
For other n values, they give
rh,sg
n−1 z1
z1
=
.
1−
(1 − n)K h,sg (z 1 )
z 0s
(22b)
It is sensible to use Eq. 22a for its simplicity, rather than Eq. 22b, which implies that the
shape of K h,sg (z) also affects rh,sg . In summary, the flux computation in a LES model differs
from that of a conventional land-surface scheme in, (a) the main components of the fluxes
are grid resolved; (b) the subgrid components are parametrized in consistency with the flow
subgrid closure; and (c) the flux computation does not rely on MOST, although the validity
of Eq. 21 also requires further scrutiny.
2.2.3 Soil Temperature and Soil Moisture
Soil temperature obeys the heat diffusion equation
∂ Ts
∂ Ts
∂
=
νG
+ sT ,
∂t
∂z
∂z
(23)
where Ts is soil temperature, νG is the soil thermal diffusivity and sT is a temperature source.
The soil moisture θ obeys the Richards equation
∂w
∂ (ψw + z)
∂
(24)
=
KW
+ sw ,
∂t
∂z
∂z
where ψw is the hydraulic suction head, K W is the hydraulic conductivity and sw is a moisture source. The simplifications of the soil temperature and soil moisture equations to the
one-dimensional Eqs. 23 and 24 are justifiable from scaling analysis even for large-eddy
simulation.
In land-surface models used for weather prediction, it is appropriate to select the thicknesses of the soil layers, for example, as 0.1, 0.3, 0.6, 1.0 m. However, for LES, much thinner
layers must be selected to allow the land surface to respond to the effects of large eddies.
Suppose the typical time scale of the atmospheric system is tA . Then, because the soil thermal
diffusivity, νG , is of the order of 10−6 to 10−7 m2 s−1 , the corresponding thickness of the soil
layer, s, must satisfy
√
s ∼ νG tA .
(25)
For tA = 1 day, s is about 0.2 m; for tA = 10 min, s is about 0.01 m. Thus, we set the
thickness of the soil column as 0.2 m to allow for soil response to the diurnal variation of the
atmosphere, and set the thickness of the first soil layer as 0.01 m to allow for soil response
to the large-eddy fluctuations. The soil-layer configuration used in LES-ALM is as shown in
Fig. 1.
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Large-Eddy Atmosphere–Land-Surface Modelling
Fig. 2 Land-use map of the simulation area. For each of the nine land-use types, a set of vegetation parameters
is assigned, i.e. leaf area index (LAI), vegetation height and vegetation cover fraction. The Selhausen and
Merken measurement sites are marked by the black dots. A selected subset of the aircraft flight paths (white
lines) is projected onto the map. All flights were between 120 and 200 m above ground. The numbers stand
for the start and end time of the selected flights. The black line represents the cross-section to be shown in
Fig. 10
3 Model Application Site and Simulation Set-up
3.1 Field Measurements
The model has been set-up to simulate a case from the FLUXPAT experiments conducted within the German research collaborative SFB/TR 32 “Patterns in Soil-VegetationAtmosphere-Systems: Monitoring, Modelling, and Data Assimilation” (Vereecken et al.
2010). The FLUXPAT experiments include measurements of soil state parameters and efflux,
plant structural parameters such as LAI, plant physiological status, micrometeorological quantities from surface (tower-based) to higher levels (by airplane). Measurements took place in
the Rur river catchment between Aachen and Cologne in western Germany, close to the Jülich
Research Centre (50◦ 53 , 6◦ 27 , see Fig. 2). The area is arable land dominated by field crops
of sugar beet and two grain species (winter wheat and winter barley), which amount to about
75 % of the cultivated plants in the area. Typical field sizes of the region are in the range of
one to a few hectares or a typical length scale of some 100 m. Measurements focused on
a flat area of roughly 10 × 10 km2 about 100 m above mean sea level. Height differences
within the terrain are small with the wider bed of river Rur lying about 10 m lower than the
surrounding and a slight downhill slope of 20 m from south to north. Two main sites near
Selhausen and Merken were established.
In this study, we use micrometeorological measurements from two field plots with sugar
beet and harvested winter wheat, respectively. These measurements included standard air
temperature and humidity and turbulent fluxes using the eddy-covariance method. To characterize the development of the boundary layer, radiosondes were released every hour. A
research airplane flew at 120 m and 200 m above ground in a polygonal pattern and measured meteorological parameters at 10 Hz to derive fluxes based on the eddy-covariance
method (see Schmitgen et al. 2004). A part of the flight pattern can be seen in Fig. 2. Flights
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Y. Shao et al.
Table 1 Settings of model simulation
Characteristic
Exp 1
Exp 2
Exp 3
Land-surface scheme
New scheme with
improvements
based on Noah
LSM: an explicit
multilayer
canopy scheme;
much finer
soil-layer
configuration as
shown in Fig. 1
New scheme as
described by
Eqs. 20–22
7.5 × 6.0 × 2.2 km3
Noah LSM: bulk
canopy; 4 soil
layers: 0.1, 0.3,
0.6, 1 m
As Exp 1, but with
conventional
soil-layer
configuration
MOST
As Exp 1
As Exp 1
As Exp 1
Surface layer scheme
Domain size
Grid
Horizontal: x =
y = 60 m;
Vertical:
logarithmically
stretched with
z varying from
2 m near the
surface to 24 m
for z ≥ 80 m
125 × 100 × 100
Time step
0.2 s
Simulation period
0800–2000 UTC, 5 Aug 2009
Spatial resolution
Lateral BCs
Periodic
Upper BCs
Constant pressure
with zero
vertical velocity
were made under fair weather conditions, such that the influences of synoptic variations and
clouds were minimized. The FLUXPAT campaigns covered different stages of plant growth
and produced a comprehensive data set for model comparison. In this study, we use data from
August 5, 2009, a day with weak south-easterly winds of 3–4 m s−1 in the entire boundary
layer and no clouds except for some thin cirrus in the afternoon.
The land-use data used for the modelling are derived from the Advanced Spaceborne
Thermal Emission and Reflection Radiometer (Waldhoff 2010) data. The land-use map with
a resolution of 15 m is shown in Fig. 2.
3.2 Model Initialization and Parameter Setting
The model domain covers a 7.5 ×6 km2 flat area with different land-use types. The upper
boundary of the domain is 2.2 km above the ground. The model domain (7.5 ×6 × 2.2 km3 )
is covered with 125 ×100 × 100 grid points, with x = y = 60 m and z stretched
from 2 m near the surface to 24 m for z ≥ 80 m. Periodic boundary conditions are used for
the horizontal boundaries. The upper boundary is assumed to have a constant pressure and
zero vertical wind. The layer between 1.8 and 2.2 km is assumed to be a damping layer.
For comparison, three numerical experiments are carried out: one with the new land-surface
scheme (Exp 1), one with the original Noah land-surface scheme using MOST-based flux
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Large-Eddy Atmosphere–Land-Surface Modelling
Fig. 3 Vertical profiles of potential temperature (a), humidity mixing ratio (b) and wind speed (c) obtained
through radiosounding (with minor smoothing) between 0758 and 0810 UTC, 5 August 2009. The data are
used to initialize the model at 0800 UTC, 5 August 2009
Table 2 Soil temperature, soil moisture and surface albedo for different land types used to initialize the
land-surface model
Soil temperature (◦ C)
Soil moisture (m3 m−3 )
Layer 1
Layers 2–4
Layers 1–4
Bare soil
20
18
0.20
Settlement
20
18
0.20
0.15
Bog
–
–
–
0.08
Water
–
–
–
0.08
Rapeseed
20
18
0.20
0.33
Beet
20
18
0.24
0.22
Land-use type
Surface albedo
0.33
Grain
20
18
0.20
0.33
Pasture
20
18
0.24
0.19
Forest
20
18
0.24
0.14
formulations (Exp 2), and one with the new land-surface scheme but with the conventional
(thick) soil-layer configuration (Exp 3). A summary of the simulation settings is given in
Table 1. Exp 1 is used as the reference simulation for comparison.
The profiles of the atmospheric variables (wind speed, potential temperature, humidity,
etc.) are idealized from the radiosonde for 0800 UTC, 5 August 2009 and used for initializing
the LES model (Fig. 3). As seen, an inversion existed at z = 1,500–1,700 m, with a lapse
rate of about 8 K km−1 . A near surface inversion existed between 100 and 300 m and the
bulk of the boundary layer was weakly stable. The humidity mixing ratio was between 5.5
and 7 g kg−1 , decreasing with height, and from 5.5 to 0.5 g kg−1 across the capping inversion
layer. The wind speed was about 3.6 m s−1 from the north-east.
There are nine land-use types in the simulation domain, each corresponding to a set of
vegetation parameters, including LAI, vegetation height and cover fraction. The soil type is
assumed to be loam and uniform in space. The initial temperature of the top soil layer is set to
20 ◦ C based on observation, decreasing to 18 ◦ C for the remaining layers. Soil moisture was
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Y. Shao et al.
Fig. 4 a Profiles of H , Hg and Hsg averaged over the model domain and the time period of 1300–1400
UTC. The black full dots are the aircraft measurements of H along different flight paths and the red dot is
the corresponding average. Also shown is the profile of potential temperature, θ , and corresponding aircraft
measurements (blue dots); b as (a), but for modelled L E, L E g and L E sg , aircraft measured L E and specific
humidity, q; c Profiles of H , Hg and Hsg for the lower 200 m; d as (c), but for L E, L E g and L E sg
measured for grain und sugar beet surfaces, but not for the other land-use types. The initial
soil moistures for the latter cases were set empirically with reference to the measurements
for the grain and sugar beet surfaces, e.g. the forested area was assumed to be slightly wetter
than the sugar beet. The surface albedo varies between 0.08 and 0.33 according to field
observation. A summary of the initializations is given in Table 2.
4 Comparison of Model Results with Observations
4.1 Large-Eddy Simulated Flux Patterns
We first examine the general features of the model simulated fluxes. Figure 4a shows the
profiles of the time- and domain-averaged sensible heat fluxes, H, Hg and Hsg ,and potential
temperature, θ , and Fig. 4b shows those of latent heat fluxes, L E, L E g and L E sg , and specific
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Large-Eddy Atmosphere–Land-Surface Modelling
humidity, q. The time averages are for the 1-h period of 1300–1400 UTC (all time averages
refer to this period, unless otherwise stated). For both Exp 1 and Exp 2, the profiles of H
are typical for a convective boundary layer, i.e., they linearly decrease with height until the
inversion level (Fig. 4a). In the bulk of the boundary layer, H is mainly due to Hg , and Hsg
is negligible. Close to the surface, as the scale of turbulence is fine, Hsg dominates (Fig.
4c). Below about the 40-m level, H increases slightly with height. This is probably due to
an underestimate of Hg near the surface. The latent heat flux profiles (Fig. 4b, d) show that
L E increases with height to the inversion level. This result is consistent with the finding
of Deardorff (1974) who conducted a large-eddy simulation of a convective boundary layer
without a land-surface scheme. This profile of L E is caused by the entrainment of dry air
from aloft, which contributes to a positive latent heat flux, but we do not have measurements
to verify the correctness of the simulated latent heat flux profile. With respect to Exp 1, H
from Exp 2 is generally lower due to its smaller value at the surface. As a consequence of the
weaker thermal instability, L E from Exp 2 is also smaller in the upper part of the atmospheric
boundary layer due to the weaker entrainment of dry air from aloft. Our comparison shows
that the formulation of the land-surface scheme can have a significant quantitative effect on
the flux profiles of the atmospheric boundary layer.
Also shown in Fig. 4a, b is the aircraft measurements of sensible heat flux, potential
temperature, latent heat flux and specific humidity. These values are averaged along the flight
paths shown in Fig. 2. The simulated (Exp 1 and Exp 2 are almost the same) and observed
potential temperatures are in good agreement, while the simulated specific humidity (Exp
1 and Exp 2 are almost the same) is slightly lower than the observed. This is similar to
the results of Zacharias et al. (2012) and probably due to advection of moist air that is
not included in the model. Since the model is run with no data assimilation, and given the
strong humidity decrease across the inversion layer and the mixing in the boundary layer, the
slight decrease of specific humidity in time is expected. Therefore, the degree of discrepancy
between the modelled and aircraft-observed mixing ratio can be explained and is acceptable.
The aircraft H and L E measurements show a large scatter between 50 and 400 W m−2 . The
scatter is not surprising, as we shall also see from the model simulation (e.g. Fig. 8), the
instantaneous fluxes can vary greatly in space and time because large eddies move around
with considerable randomness. However, the averages of H and L E over all aircraft flight
paths are in reasonable agreement with the model estimates.
The time-averaged sensible and latent heat fluxes (Fig. 5) show that the near-surface
patterns of H and L E are closely correlated with the land-surface properties. For example
(Fig. 5a), the highest H values are found over the settlement areas, in excess of 400 W m−2 ,
while the values of H over the forest, pasture and sugar beet surfaces are much smaller, at
around 120 W m−2 . Over the lake surface, H is negative. The impacts of the land surface on
the patterns of LE are also visible (Fig. 5d): the largest L E values occur over the pasture and
forest areas along the river, where the surface is moist and rough. Figure 5a, d also shows
that, while the features of land-surface properties are clearly reflected in the patterns of H
and L E, they quickly become fuzzy as a consequence of large-eddy mixing (Fig. 5b, e), and
only the larger land-surface features are identifiable.
At Merken, sensible and latent heat fluxes were measured using the eddy-correlation technique over the (harvested) wheat and sugar beet surfaces. The fluxes of Exp 1 are calculated
using Eq. 17a, b and then averaged over 30-min intervals for comparison with the measurements. Figure 6a shows that, for the wheat surface, H of Exp 2 is underestimated by as much
as 100 W m−2 from the late morning to the early afternoon. The agreement between the
simulated H of Exp 1 and the observation is much better. For the sugar beet surface (Fig. 6b),
L E is overestimated in Exp 2, but the overestimate is reduced in Exp 1. In both Exp 1 and
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Y. Shao et al.
Fig. 5 a Patterns of sensible heat flux H (W m−2 ) at the 10-m level, averaged over the time period 1300–
1400 UTC for Exp 1; b as (a), but for Hg ; (c) as (a), but for Hsg ; d–f as (a–c), but for L E, L E g and L E sg ,
respectively
Fig. 6 a Comparison of the simulated and observed sensible heat fluxes (red) and latent heat fluxes (blue)
for the harvested wheat surface. b As (a), but for the sugar beet surface
Exp 2, H is overestimated. The overestimation may be due to a variety of reasons, but the
most likely is that the specified albedo for the sugar beet surface (0.22) is too low. Both the
simulation and the observations show a clear dependence of the partitioning of net radiation
on the land-use type. Over the wheat surface, H dominates over L E, while over the sugar
beet surface, L E dominates over H . For both surfaces, Exp 1 correctly reproduced the diurnal
cycles and the magnitudes of the dominating flux component. There are some discrepancies,
but the improvements as a result of the new land-surface scheme are significant.
Figure 7 shows the comparison of the other quantities for the wheat surface, including
net radiation, Rn , soil heat flux, G, soil temperature in the top 50-mm soil layer and air
temperature at 2 m, as well as soil moisture at 50-mm depth. The agreement of Rn between
Exp 1 and the observations is reasonable and is evidently better than that for Exp 2 (Fig. 7a).
The simulated G at 100-mm depth and the observed G at 80 mm are in reasonable agreement,
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Large-Eddy Atmosphere–Land-Surface Modelling
Fig. 7 a Comparisons of the simulated and observed net radiation, Rn ; b simulated soil heat flux at 100 mm,
and the measured soil heat flux at 80 mm below the surface; c simulated and observed soil temperature at 50
mm below the surface (black) and air temperature at 2 m above the ground surface (blue); d Simulated and
observed soil moisture for the top soil layer; for the harvested wheat surface whose surface energy fluxes are
compared in Fig. 6a
but G of Exp 1 is up to 15 W m−2 lower than the observed values (at around 1400 UTC),
while G of Exp 2 is more than 10 W m−2 higher than the observed values after 1400 UTC. It
is generally the case that G in deeper soil is smaller. This implies that the simulation of Exp 1
using the new model is more consistent with the observation. Soil temperature and moisture
are also better simulated in Exp 1 (Fig. 7c, d). The soil temperature of Exp 2 is very much
overestimated, up to 5 ◦ C higher than that observed for the time period after 1400 UTC. The
higher soil temperature is a consequence of the low H , which is, in Exp 2, underestimated by
about 30 % for most of the daytime (Fig. 6a). This implies that the conductance used for the
flux calculation in the old land-surface scheme is too small, and consequently, in comparison
to Exp 1, H is much lower even though the surface–air temperature difference is much larger.
Both simulated air temperatures at the 2-m level are up to 2 ◦ C higher than those observed
for the time period 1000–1700 UTC. The higher air temperature and lower soil heat flux are
attributed to the use of the periodic boundary condition for the model run, while cool air
advection might have occurred in reality. At noon time, a higher air temperature implies a
lower ground heat flux and hence a lower soil heat flux. Based on the comparison with the
airborne and ground-based observations, as well as the comparison between the experiments,
we conclude that, despite some discrepancies, the model simulation of Exp 1 is successful.
Figure 8 shows the composite time-averaged profiles of H and LE, together with their
standard deviations, for four different land-use types. The differences among the profiles
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Y. Shao et al.
Fig. 8 Time-averaged profiles of turbulent fluxes for different land-use types (solid) plus, minus their standard
deviation (dashed) for Exp 1. Top the whole BL, bottom the lower 200 m, left sensible heat flux, right latent
heat flux
Fig. 9 a Simulated instantaneous surface L E versus H for all points in the simulation domain at 1300 UTC
for Exp 1. b As (a), but for Exp 3
above different land-use types are substantial, e.g. for forest and settlement areas. The differences are largest close to the surface at about 20 m, but remain quite significant to fairly high
levels. The standard deviation shows that the scatter among the fluxes for a given surface type
is substantial. For H , the largest scatter occurs near the surface, while for L E, the largest
scatter occurs at the inversion level.
Figure 9 is a scatter plot of the instantaneous L E and H for different land-surface types.
Distinct Bowen ratios (H /L E) for different land surfaces are identifiable. For example, the
Bowen ratio is close to 0.4 for sugar beet, but considerably higher for bare soil and settlement
(Fig. 9a). Figure 9a also shows that the scatter of H + L E for a given land-use type is large.
This implies that the net radiation and ground heat flux also possess strong temporal and
spatial variations as a consequence of the atmosphere–land-surface interactions. Because of
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Large-Eddy Atmosphere–Land-Surface Modelling
Fig. 10 a Snap shot of a cross section of potential temperature and surface sensible heat fluxes at y = 2.5
km (black straight line in Fig. 2) for Exp 1. b As (a), but for specific humidity and latent heat flux. A large
updraft eddy at x = 2 km carries heat and moisture upwards, and a large downdraft eddy at x = 4 km carries
heat downwards and moisture upwards
the fine soil-layer configuration we used (Fig. 1), the temperature and moisture in the first soil
layer vary rapidly with time as a consequence of the large-eddy fluxes and local advections,
details of which will be discussed in a companion paper (Liu and Shao 2013, submitted).
For comparison, the scatter of L E and H is plotted in Fig. 9b for a model run with the
conventional soil layer configuration (Exp 3). In Exp 1 (Fig. 9a), the Bowen ratios are in
general larger, especially for the bare soil and settlement. This is due to the supply limitation
of soil moisture in the thin top soil layer. In the thin soil layer configuration, the limiting effect
of soil hydraulic conductivity to evaporation from bare soil surfaces is much more obvious.
In Exp 3 (Fig. 9b), the scatter of H + L E for a given land-use type is much smaller, implying
that this quantity has weaker temporal and spatial variations due to the weaker coupling
between the land surface and the atmosphere and the weaker limiting effect of soil hydraulic
conductivity on evaporation (as a consequence of the thicker soil-layer configuration).
4.2 Persistency of Land-Surface Heterogeneity
We now study how land-surface heterogeneity propagates and diminishes in the atmosphere
and what role large eddies play in these processes. The model simulation allows a visual
examination of the large-eddy structures and the associated fluxes. As an example, Fig. 10
shows snap shots of the x–z cross-sections of potential temperature and specific humidity,
together with the surface sensible and latent heat fluxes. It is seen that the instantaneous
sensible and latent heat fluxes vary over a wide range between zero and 1,000 W m−2 . Convergence lines and divergence areas of horizontal flow often occur, accompanied by strong
updrafts and relatively weaker downdrafts as seen in Fig. 10a, b. The updrafts near the surface merge to build larger updrafts that carry heat and moisture to the upper boundary layer,
as seen at x = 2 km of Fig. 10a, b. At the same time, downdrafts from the inversion level
entrain warm (higher potential temperature) and dry air into the boundary layer and produce
negative sensible heat fluxes and positive latent heat fluxes, as seen at x = 4 km of Fig. 10a,
b. These examples show that large eddies build efficient transport pathways from the surface
to the upper atmospheric boundary layer. Thus, the mechanism of large-eddy transport in the
atmospheric boundary layer differs profoundly from the mechanism of small-eddy diffusion
commonly described using K-theory (Holtslag and Moeng 1991). This difference is particularly evident in latent heat fluxes, which in the bulk of the boundary layer are not determined
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Y. Shao et al.
Fig. 11 Propagation of the effects of land-surface heterogeneity in the atmosphere for Exp 1. a Cross-sections
of sensible heat fluxes at various times and levels. b H and LE averaged over 1300–1400 UTC
by the local vertical gradient of specific humidity, but the bottom-up transport of moist air
and the top-down entrainment of dry air and the mixing of the convective thermals.
Figure 10 suggests that large-eddy transfer is determined both by the macroscopic structure
of the boundary layer and the patterns of land-surface properties. As small eddies emerge
from the surface, they bear the signals of land-surface heterogeneity, but as the small eddies
organize to build larger eddies, a process governed by the thermal-dynamic instability of the
boundary layer, these land-surface signals weaken as a consequence of mixing. Then, if so,
how persistent are the land-surface signals? In Fig. 11a, the cross-sections of instantaneous
H are shown for z = 2, 10, 40, 160 and 640 m as a hierarchy. Near the surface (e.g. at z = 2 m),
the patterns of the fluxes and the land types are closely correlated for every instant. However,
the footprints of the land surface become more difficult to recognize at higher levels (e.g. at z
= 40 m) for a given instant. At the same time, the low-level small eddies feed to the formation
of larger eddies at higher levels. A hierarchic structure of turbulence is identifiable and, as
a consequence, the patterns of the averaged fluxes do show a close link to the patterns of
land-use (Fig. 11b).
In Fig. 11b, hourly averages of H and L E for five different heights are shown. The
correlations between the patterns of the fluxes and the land-use are clearly identifiable for
levels below 160 m. Even at the 640-m level, the influence of land-surface heterogeneity is
visible. Thus, on average, the land-surface signals are surprisingly persistent. This implies
that certain large-eddy types (e.g. warm updrafts) prevail over certain land-surface types (e.g.
settlement), such that the land-surface heterogeneity persists on average in the atmospheric
flow structure. For the simulation presented here this means that, if a blending height exists, it
would be well beyond the surface-layer depth. Figure 11b shows that, while the details of landsurface heterogeneity gradually diminish with height, the large features remain prominent.
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Large-Eddy Atmosphere–Land-Surface Modelling
5 Summary
This paper is the first part of our study on atmospheric–land-surface interactions over heterogeneous surfaces. We have presented a large-eddy simulation atmosphere–land-surface
model (LES-ALM) that is indispensable for the investigation of the specific scientific questions related to land-surface heterogeneity, e.g., how do the effects of land-surface heterogeneity propagate and persist in the atmospheric boundary layer. We have argued, based on
numerous numerical tests, that substantial improvements are necessary to the “conventional”
land-surface schemes to ensure their adequacy for large-eddy simulation of atmosphere–
land-surface processes. The improvements we have made in this study include, (1) an explicit
multi-layer treatment of the canopy layer; (2) a method for computing surface fluxes that is
consistent with the flow model subgrid closure; and (3) a thin soil-layer configuration. The
model is then applied to the Selhausen–Merken experimental site in Germany, a heterogeneous area with several land-use types (bare soil, forest, wheat, sugar beet, settlement etc.).
A 12-h daytime simulation is made to evaluate the model performance, and the model results
are compared with ground-based and airborne measurements. The model simulations of sensible and latent heat fluxes are found to be in good agreement with the observed fluxes. Good
agreement is also found between the modelled and observed net radiation, ground heat flux,
soil temperature and soil moisture. The discrepancies that still exist between the modelled
results and the observations (e.g. 2-m air temperature) can be explained.
A number of sensitivity tests have been made and the results of representative ones (Table
1) have been presented to assess the performance of the new land-surface scheme (Exp 1)
with respect to the Noah land-surface scheme (Exp 2) and to assess the effect of soil-layer
configuration (Exp 3). Exp 1 and 2 produced qualitatively similar H and L E profiles in
the atmospheric boundary layer, but significant quantitative differences between them are
found: Exp 2 produced smaller H , weaker thermal instability and smaller LE in the upper
part of the boundary layer due to the related weaker entrainment. The comparisons with the
near-surface measurements have shown that the overall performance of the new land-surface
scheme is more satisfactory in terms of sensible heat flux, H , and latent heat flux, L E, net
radiation, Rn , soil heat flux, G, as well as top layer soil temperature and soil moisture. While
the improvements may be due to a combination of the modifications we have made to the
land-surface scheme, the refinement of the soil-layer configuration appears to have had a
significant impact because the very thin top soil layer limits the availability of soil moisture
for evaporation and thereby profoundly affects the partitioning of net radiation into sensible
and latent heat fluxes (Exp 3).
An even more substantial difference between the present work and earlier studies is that
LES-ALM allows the atmosphere and the land surface to interact at the scale of large eddies,
as seen from the fluctuations and the probabilistic distributions of the near-surface energy
fluxes. A multi-layer canopy scheme has been proposed, but we have not thoroughly evaluated
its performance.
Using LES-ALM, we have made preliminary investigations on large-eddy scale
atmospheric land-surface exchanges. While many questions remain to be clarified, we have
focused on the patterns of sensible and latent heat fluxes, how such patterns come into
existence, how large eddies are related to land-surface heterogeneity and how the effects
of land-surface heterogeneity persist in the atmosphere. The model simulation revealed a
complex image of flux patterns in the atmospheric boundary layer. Near the surface (e.g.
below 10 m), the flux patterns are closely correlated with the land-use patterns, and while
this correlation rapidly diminishes with height, it remains identifiable to a level of over
60 m.
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Y. Shao et al.
It is shown that large-eddy transfer is determined both by the macroscopic structure of the
boundary layer and patterns of land-surface properties. As small eddies emerge from near the
surface, they bear land-surface signals, but as they organize and develop into larger eddies, a
process governed by the thermal-dynamic instability of the boundary layer, the land-surface
signals weaken due to turbulent mixing. As a result, the instant flux patterns (unless very close
to the surface) appear to be unrelated to the land-use patterns, but on average, the correlation
between the flux and land-use pattern is quite strong and persistent in at least the lower half of
the atmospheric boundary layer. The relationship between the sensible and latent heat fluxes
in the bulk of the boundary layer is rather interesting because the downward entrainment of
warm and dry air from aloft results in a negative sensible heat flux but a positive latent heat
flux in the upper part of the boundary layer. On occasions, the downward entrainment of the
dry air can even make a significant difference to the latent heat fluxes near the surface.
Although the near-surface flux patterns and land-use patterns are closely correlated, the
scatter of the fluxes for a given land-use type is substantial. Three sources for the scatter can
be identified: (1) large eddies generate randomness in the fluxes; (2) rapid feedback exists
between the land-surface and large eddies; (3) surface heterogeneity causes local advection
between the grid cells. For example, warm and dry air from upstream dry areas can influence
the evaporation and heat exchange over the vegetated wet areas downstream. Owing to the
advection from the dry areas, evaporation in the wet areas is fostered, while heat exchange
suppressed.
Due to the very thin soil layers that must be used for the large-eddy simulation, the
requirement for reliable land-surface input parameters and initial conditions is difficult to
meet. Thus, in general, the value of a large-eddy simulation atmospheric–land-surface model
does not lie in the quantitative accuracy of the model simulation for specific cases, but in
its application to generate understanding of atmospheric–land-surface interactions that are
difficult to observe through experiments, and to support the interpretations of the observations.
Acknowledgments This work is supported by the DFG Transregional Cooperative Research Centre 32
“Patterns in Soil-Vegetation-Atmosphere-Systems: Monitoring, Modelling and Data Assimilation”. We thank
Bruno Neininger (MetAir) for performing and processing of the aircraft measurements, Heiner Geiss (Juelich
Research Center), Martin Lennefer, Dirk Schüttemeyer, Stefan Kollet (University Bonn) who supported the
micrometeorological measurements, Gerritt Maschwitz for launching the radiosondes.
6 Appendix: Canopy Temperature Scheme
The equation for canopy temperature, Tc , can be written as
cvg
∂ Tc
· R − αt εσ Tc4 − ρcp ST − ρ L Sq
= −∇
∂t
(26)
where cvg is the volumetric vegetation heat capacity (J m−3 s−1 ), i.e., the energy required to
increase the temperature of vegetation per unit (air) volume, αt is the vegetation area density
(total area per unit volume), ε is vegetation emissivity, ρ is air density, cp is air specific heat
at constant pressure, L is the latent heat of vaporization of water, ST and Sq are as given
in Eqs. 7 and 8, R is net radiation flux. Suppose net radiation is horizontally homogeneous,
then, Eq. 26 becomes
cvg
123
∂ Tc
∂ Rn
=−
− αt εσ Tc4 − ρcp ST − ρ L Sq ,
∂t
∂z
(27)
Large-Eddy Atmosphere–Land-Surface Modelling
Fig. 12 Schematic illustration of radiation transfer through vegetation canopy
where Rn is the vertical component of the net radiation. For simplicity, we divide the radiation
spectrum into the shortwave (solar) and longwave (terrestrial) bands. Then, as illustrated in
Fig. 12, Rn for any given level can be expressed as
Rn = (Rs↑ − Rs↓ ) + (Rl↑ − Rl↓ ).
(28)
In general, radiation passing through a vegetation layer of thickness, ds, is scattered and
absorbed by leaves. The dependence of R on s can be expressed as
dR = −k Rds,
(29)
where k is the canopy extinction coefficient. It therefore follows that
−
∂ Rn
= ks (Rs↑ + Rs↓ ) + kl (Rl↑ + Rl↓ ),
∂z
(30)
noting that dR↓ = −k R↓ ds, ds = −dz, and therefore,
∂ R↑
= −k R↑ ,
∂z
∂ R↓
= k R↓ .
∂z
(31a)
(31b)
In Eq. 30, ks and kl are respectively the extinction coefficients for shortwave and longwave
radiation. It follows that Eq. 27 becomes
cvg
∂ Tc
= ks (Rs↑ + Rs↓ ) + kl (Rl↑ + Rl↓ ) − αt εσ Tc4 − ρcp ST − ρ L Sq .
∂t
(32)
Suppose cvg is small, then the canopy temperature can be determined from the following
diagnostic equation
αt εσ Tc4 = ks (Rs↑ + Rs↓ ) + kl (Rl↑ + Rl↓ ) − ρcp ST − ρ L Sq .
(33)
The treatment of the radiation fluxes is straightforward. Suppose the shortwave flux at the
top of the canopy, h, is Rsh . Then, the fraction of the shortwave radiation entering the canopy
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Y. Shao et al.
is (1 − avg )Rsh and the fraction reaching the surface is
⎛ h
⎞
Rs0 = (1 − avg )Rsh exp ⎝− ks dz ⎠ .
(34)
0
where avg is vegetation albedo. Thus, for a level z,
⎛
⎛ h
⎞
⎞
z
Rs↑ + Rs↓ = a0 (1 − avg )Rsh exp ⎝− ks dz ⎠ · exp ⎝− ks dz ⎠
⎛
+ (1 − avg )Rsh exp ⎝−
0
h
0
⎞
ks dz ⎠
(35)
z
or
⎛ h
⎞
⎞⎤
h
z
= (1 − avg )Rsh ⎣a0 exp ⎝− ks dz − ks dz ⎠ + exp ⎝− ks dz ⎠⎦ . (36)
⎡
Rs↑ + Rs↓
⎛
0
z
0
where a0 is surface albedo. Suppose the atmospheric longwave radiation at the top of the
canopy is Rlh and the ground surface temperature is T0 . Further, suppose the canopy layer
between z and h is divided into Ia layers, and the vegetation layer between 0 and z is divided
into Ib layers, each of δz thick (Fig. 12). Then
⎛
⎞
⎛ h
⎞
z
Rs↑ + Rs↓ = εσ T04 exp ⎝− kl dz ⎠ + Rlh exp ⎝− kl dz ⎠
0
+
Ib
⎛
r (z i ) exp ⎝−
i=1
z
zi
⎞
kl dz ⎠ +
z
Ia
i=1
⎛
r (z i ) exp ⎝−
zi
⎞
kl dz ⎠
(37)
z
with
1
αt (z i )εσ Tc4 (z i )δz
2
where αt (z i )is the vegetation area density at level z i .
r (z i ) =
(38)
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