302 P. Zeng, H. Takahashi Agricultural and Forest Meteorology 103 2000 301–313
the vegetation canopy. Moreover, Pereira and Shaw 1980 and Watanabe and Kondo 1990 hereafter
WK show that such a model can not provide accurate predictions of wind velocity in the lower portion of
a plant canopy, where a near-zero vertical gradient of mean wind velocity or a wind velocity ‘bulge’ is fre-
quently observed Shaw, 1977. In another approach, WS proposed a higher-order second-order closure
model in which turbulent kinetic energy equations and Reynolds stress equation were solved simultaneously
with that for mean momentum. However, WS reported that the calculated velocity profile was sensitive to the
parameterization scheme for the turbulent transport term in their model. In order to avoid the deficiency
in the model of WS, Meyers and Paw 1986 here- after MP developed a third-order closure model using
third-order closure principles. However, the utility of a higher-order closure model is still limited. The model
proposed by MP includes about 10 equations for only a one-dimensional problem, and the computation cost
is therefore high. In addition, their modeled results for the turbulent field have large errors MP; Meyers and
Baldocchi, 1991. The parameterization schemes for higher-order closure models used for predicting veg-
etation wind flow need to be further improved MP; Shaw and Seginer, 1987; Wilson, 1988.
As an alternative approach that does not use higher-order closure principles, Li et al. 1985 pro-
posed a non-local closure scheme for the total mo- mentum flux Reynolds stress and dispersive flux
and developed a first-order closure model that was capable of predicting the wind velocity peaks in lower
canopies. However, several problems in their model have been pointed out by Van Pul and Van Boxel
1990. To correct these problems, Miller et al. 1991 hereafter MLL made some modifications to their
model and applied it to wind flow across an alpine forest clearing. However, the utility of their model for
different types of vegetation canopy was not tested. Over the past few decades, extensive measurements
e.g. Raupach et al., 1986; Shaw et al., 1988; Gao et al., 1989 of turbulent flows within and above vege-
tation canopies have been carried out. On the basis of the results of the previous studies, the purpose of
the present study is to develop a first-order closure model, which is simple in computation and has the
general utility for predicting wind flow within and above vegetation canopies, and to investigate the in-
fluences of canopy structure and foliage density on the wind velocities within and above the canopies.
2. Theory and the model
2.1. Governing equation Following Raupach et al. 1986, under neutrally
buoyant conditions, the time- and volume-averaged equation for the mean momentum within vegetation is
∂ hu
i
i ∂t
+ u
j
∂ hu
i
i ∂x
j
= − ∂
hpi ∂x
i
+ ∂τ
ij
∂x
j
+f
F
i
+ f
V
i
1 with
τ
ij
= −hu
′ i
u
′ j
i − hu
′′ i
w
′′ j
i + ν ∂
hu
i
i ∂x
j
2 Here, i and j are index notations with values of 1–3
and Einstein’s summation is used. The overbars and single primes denote time averages and fluctuations,
while the angle brackets and double primes denote spatial volume averages and departures therefrom, re-
spectively. u
i
and x
i
are velocity and position vec- tors, respectively, t the time, p the kinematic pressure,
f
F
i
and f
V
i
are form and viscous drag force vectors exerted on a unit mass of air within the averaging
volume, respectively, τ
ij
the volume-averaged kine- matic momentum flux or stress tensor, and ν is the
kinematic viscosity. Interpretations of each term in the above equations have been described by Raupach
et al. 1986.
For a horizontal homogeneous vegetation, let u and w represent the streamwise and vertical velocity, re-
spectively, the x-coordinate in the mean streamwise direction, and the z-coordinate normal to the ground;
assuming steady-state conditions and neglecting the pressure gradient and molecular transport terms, Eq.
1 becomes
d dz
hu
′
w
′
i + hu
′′
w
′′
i = f
x
3 In this equation,
hu
′
w
′
i is the Reynolds stress or tur- bulent flux of momentum, while
hu
′′
w
′′
i represents the dispersive flux that arises from the spatial corre-
lation of regions of mean updraft or downdraft with regions, where u differs from its spatial mean, and
P. Zeng, H. Takahashi Agricultural and Forest Meteorology 103 2000 301–313 303
f
x
= f
F
1
+ f
V
1
is the total streamwise drag per unit mass of air within the averaging volume. These terms
must be parameterized in order to solve the equation and estimate the wind velocity
hui. 2.2. Closure schemes
2.2.1. Reynolds stress In K-theory models, the Reynolds stress is param-
eterized as hu
′
w
′
i = −K
M
d hui
dz 4
where K
M
is the eddy viscosity. This model shows that the turbulent flux of momentum results from the local
gradient in the mean wind velocity. Hence, it is also called a small-eddy closure technique Stull, 1988.
K-theory models have been widely used for studies in the surface layer and have been proven to be reli-
able for the inertial sublayer above a surface Raupach et al., 1980. However, Corrsin 1974 has pointed out
that the application of K-theory is limited to the place where the length scales of flux-carrying motions have
to be much smaller than the scales associated with average gradients. It is unfortunate that such a condi-
tion is often violated within vegetation canopies Rau- pach and Thom, 1981; Baldocchi and Meyers, 1988b.
Many measurements have shown that wind flow within and just above a plant canopy is dominated by turbu-
lence with vertical length scales at least as large as the vegetation height Kaimal and Finnigan, 1994. These
large-scale turbulent eddies are intermittent and ener- getic, and they can penetrate the canopy crowns and
enter the subcanopy trunk space to generate non-local turbulent transport. Most of the transport of momen-
tum and scalar properties within the canopy are gen- erated by these large-scale turbulent eddies Raupach
et al., 1986; Baldocchi and Meyers, 1988a. Baldoc- chi and Meyers 1988b report that the Reynolds stress
within a plant canopy is influenced not only by the product of K
M
and the local vertical gradient in the mean wind velocity but also by the non-local turbulent
transport through the activities of large-scale eddies. Based on the results obtained from previous studies,
the turbulent momentum flux is divided into two parts in the present study: one diffused by the smaller-scale
eddies, which depends on the local gradient of the mean wind velocity; and the other transported by the
larger-scale eddies, which is determined by the mean wind speed differences between heights with large dis-
tance. Hence, the Reynolds stress is parameterized by
−hu
′
w
′
i = K d
hui dz
+ C
g
hu
r
i hu
r
i − hui z
h 5
where K is the eddy viscosity, h the vegetation height, u
r
the wind velocity at a reference height above the vegetation, and C
g
is a coefficient. On the right-hand side of the equation, the first term defined as R
s
, which has been formed according to the conventional
K-theory, is responsible for small-eddy diffusion, and the second term defined as R
l
is responsible for non-local transport. Since non-local transport is
caused by the shear between the wind flows above and within the canopy, we used
hu
r
i − hui to represent the intensity of the shear.
hu
r
i outside of the parentheses was used to account for the intensity of the wind flow
above the canopy. As it is easier for turbulent eddies to penetrate into a sparse canopy than into a dense
canopy Shaw et al., 1988, the coefficient C
g
in the equation is defined to be a function of the integrated
plant area density and is expressed by C
g
= C
1
exp −C
2
Z
h z
C
d
A dz 6
where A is the plant area density m
−1
, C
d
the ef- fective drag coefficient of the plant elements, and C
1
and C
2
are constants that are determined by numerical experiments. The second term in Eq. 5 is an addi-
tional term that we have introduced. This term is sim- ilar to the term that represents dispersive flux in the
model proposed by MLL, but instead of C
g
defined in the present study, MLL used a constant coefficient. In
the present study, we have not use the constant coef- ficient proposed by MLL because we believe that the
momentum transported by large eddies is modified by the structure of a canopy. Results of the numerical ex-
periments revealed that wind velocities predicted by MLL’s model are very sensitive to the selected values
of the coefficient, and we failed to find a ‘universal constant’ that can produce correct predictions for the
wind velocities in canopies with different structures. Although there is little information about the disper-
sive flux in a real vegetation canopy, a wind-tunnel study using a model canopy Raupach et al., 1986 has
shown that the amount of dispersive flux is very small
304 P. Zeng, H. Takahashi Agricultural and Forest Meteorology 103 2000 301–313
and negligible compared with the amount of turbu- lent flux. In addition, wind velocities within vegetation
canopies have been successfully predicted by WS and MP, who used higher-order closure models in which
a term of dispersive flux is not included. Therefore, dispersive flux is considered to be negligible and was
omitted in the present study.
2.2.2. Mixing length The eddy viscosity K is parameterized according to
the Prandtl–von Kármán mixing-length theory K
= l
2
d hui
dz 7
where l is the mixing length. Above the vegetation canopy surface, the mixing length is expressed as
l = κz − d,
z ≥ h
8 where κ is the von Kármán’s constant equal to 0.4,
and d is the zero-plane displacement m. The mix- ing length within a canopy is complicated due to the
effects of canopy elements. In conventional K-theory models, Inoue 1963 suggested that l is constant
throughout the canopy layer, while Seginer 1974, Kondo and Akashi 1976 and WK considered l to
be constrained by the ground surface and the local internal structure of the canopy and defined it as a
function of the local plant area density, drag coeffi- cient and height. As the conventional K-theory model
is used in the present model to represent only the small-eddy diffusion within the canopy, the definition
of mixing length within the canopy in the present model is different from that in conventional K-theory
models. However, the effects of ground surface and canopy structure pointed out by Seginer 1974 and
other researchers will be qualitatively the same. In addition, it is considered that the value of l within
the canopy can not be larger than that at the canopy top l
h
= κh − d according to Eq. 8. Thus, the mixing length within the canopy is parameterized as
l =
κz 1
+ C
l
C
d
Az 9
with l
≤ h − d, z h
where C
l
is a constant determined by numerical ex- periments. This model is different from that proposed
by MLL, which is a function of z and the total leaf area in the portion below d and does not reflect the
effect of local canopy structure.
2.2.3. Drag force The drag of plant elements f
x
is parameterized ac- cording to WS
f
x
= C
d
A hui
2
10 This parameterization scheme has been widely used
in studies on wind flow within a vegetation canopy e.g. MP; WK; Wang and Takle, 1996. The effective
drag coefficient C
d
of a single leaf measured in wind tunnel changes with the leaf orientation and turbu-
lent scales and intensity around the leaf Raupach and Thom, 1981. However, MP reported that the value of
C
d
of a vegetation canopy is a constant, i.e. it does not depend on wind speed and the position within the
canopy.
3. Numerical aspects
