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
The selected reference height for the reference wind speed
hu
r
i is twice the vegetation height. This selec- tion is based on measurements showing that there is
a roughness sublayer within which the eddy viscos- ity is found to be enhanced and the semi-logarithmic
law is not obeyed over a rough surface e.g. Garrat, 1978; Raupach et al., 1980; Simpson et al., 1998.
Three new parameters are introduced in the present model: C
1
and C
2
in Eq. 6 and C
l
in Eq. 9. The values of these three parameters were firstly fitted to
be 0.01, 1.0 and 5.0, respectively, by using the corn canopy data, but C
2
was adjusted to 2.0 in the simula- tions for the rubber canopy data. Variations in C
2
only affect the modeled wind profile at the lower canopy.
The final values of 0.01, 2.0 and 5.0 for C
1
, C
2
and C
l
, respectively, were used for all the canopies in the present study. The computation grid included 60 equal
intervals from the ground surface to three times the vegetation height. All variables in the model were
non-dimensionalized by scales of h and the wind speed at 3h height. Non-dimensional wind speed was given
by 1 at the upper boundary and 0 at z
g
h =0.001, where
z
g
is the roughness length of the ground surface. Ex- perimental results showed that the solution was not
P. Zeng, H. Takahashi Agricultural and Forest Meteorology 103 2000 301–313 305
sensitive to the chosen of the value z
g
. The initial wind profile was assumed, and the solution was found iter-
atively until the differences in the computed hui were
less than the control level 10
−4
in this study.
4. Results and discussion
4.1. Wind profiles in various types of vegetation canopy
Wind speeds were predicted by the present model within and above six types of vegetation canopy, which
included agricultural crops, and deciduous and conif- erous forests with leaf area indices LAI from about
two to near seven and vegetation heights from about 1 m to more than 20 m Table 1. The predicted and
measured wind-speed profiles ranging from ground surface to twice the canopy height for three canopies
and to three times the canopy height for the other three canopies, are shown together with the distributions of
plant area density PAD in Fig. 1. The measured wind profiles in the pine and oak–hickory forests were av-
erages. The leaf area density of the pine forest was the average of pine A and pine B in Amiro 1990a,
and the tree height of 18.5 m was used in the present study, which was based on the reported tree height
of 15–20 m for the pine forest. Though the measured wind profiles within and above the canopies show large
differences according to the species and structure of the canopies, the predicted wind profiles matched them
excellently. The measured wind profiles in most of the canopies show small gradients or slight reversals
Table 1 List of vegetation canopies and the references. Oak–hickory and
aspen–maple are mixed deciduous forests with principal species of oak and hickory, and aspen and maple, respectively
a
Vegetation h m
LAI References
canopies Corn
2.8 3
Wilson and Shaw 1977 Pine
18.5 2.3
Amiro 1990a Oak–hickory
23 4.9
Meyers and Baldocchi 1991 Bean
1.18 6.3
Thom 1971 Wheat
1.25 6.6
Legg 1975a, b Aspen–maple
18 5
Neumann et al., 1989; Gao et al. 1989
a
h is the vegetation height, and LAI is the leaf area index. Fig. 1. Comparison of predicted solid lines and measured circles
wind speeds within and above six types of vegetation canopy. Vertical distributions of plant area density PAD, dashed lines are
also shown. Wind profiles ranging to 2h h is vegetation height are shown for corn, pine and oak–hickory canopies, and those ranging
to 3h are shown for wheat, bean and aspen–maple canopies. u
∗
is the friction velocity above the canopy.
in gradient in the lower portion of the canopies, and an obvious secondary maximum can be seen in the
oak–hickory canopy. K-theory models are incapable of predicting these features. The present model suc-
cessfully demonstrated these features, revealing that the non-local transport was accurately considered in
the model. It was revealed by the present model that though the small-eddy diffusion R
s
was near zero or even upwardly transported the momentum when the
306 P. Zeng, H. Takahashi Agricultural and Forest Meteorology 103 2000 301–313
gradient of mean wind was reversed, the non-local transport R
l
was large and maintained the Reynolds stress in the lower canopies see also Section 4.4.
Shaw 1977 has pointed out that the mean wind gra- dient will be reversed if the non-local turbulent trans-
port momentum is large enough in the lower portion of a vegetation canopy, i.e.
∂ w
′
u
′
w
′
∂z p
′
∂u
′
∂z +
∂w
′
∂x He also suggested that in the region where there is a re-
versal in mean wind gradient, larger scales of turbulent motion transport momentum downward, while smaller
scales transport momentum upward according to the local gradient. Very strong wind shear appears near the
canopy top in two deciduous canopies oak–hickory and aspen–maple canopies because there are very
dense foliage layers at the upper portion of these two canopies. The foliage is also very dense at the upper
portion of the bean canopy, but the wind shear is not as large as that in the deciduous canopies, because
the drag coefficient of the bean canopy is very small Table 3. The wind speed within the wheat canopy,
which has the greatest foliage density, is very low. The predicted wind profile within the pine canopy is also
very similar to those measured in other pine forests that have similar distributions of foliage densities e.g.
Allen, 1968; Halldin and Lindroth, 1986.
Shaw and Pereira 1982 reported that the wind pro- file predicted by their second-order closure model was
not logarithmic immediately above the canopy sur- face. However, many researchers have observed loga-
rithmic or near-logarithmic wind profiles above many vegetation canopies e.g. Thom, 1971; Oliver, 1971;
Dolman, 1986. Fig. 1 shows that the predicted and measured wind profiles above the bean canopy are al-
most the same. The measured profiles were the profiles of D–F in Fig. 7 of the report by Thom 1971, which
were reported to be logarithmic above the canopy. The wind profile above a canopy computed by the
present model is approximately logarithmic because the non-local transfer R
l
is small above a canopy see also Section 4.4. The roughness sublayer over a plant
canopy is believed to be shallow e.g. Simpson et al., 1998 except in cases where the canopy is very sparse
e.g. Garrat, 1978. In order to predict the wind profile above the canopy, a correct value for the zero-plane
Table 2 Mean errors in predicted wind speeds non-dimensionalized by
the friction velocity above the canopy Vegetation
Root-mean- Mean absolute
Mean canopies
square errors errors
errors Corn
0.203 0.168
−0.074 Pine
0.126 0.114
0.020 Oak–hickory
0.205 0.171
−0.106 Bean
0.163 0.138
−0.010 Wheat
0.135 0.098
−0.039 Aspen–maple
0.233 0.168
−0.026
displacement d is needed. This value is estimated by the model based on an additional input value of mea-
sured wind speed above the canopy. Experimental re- sults showed that the predicted wind profile within the
canopy was not sensitive to the choice value of d if d is lower than 0.85 h.
Predicted results according to higher-closure mod- els e.g. MP; Shaw and Seginer, 1987; Meyers and
Baldocchi, 1991 often show larger errors near the canopy top, where strong shear in the flow field
occurs. The present model does not have this prob- lem. Table 2 shows the mean, mean absolute and
root-mean-square errors in predicted results calcu- lated by comparing predicted and measured values.
The largest root-mean-square error is 0.23 for the oak–hickory canopy, and other root-mean-square
errors are 0.2 or less. The mean absolute errors are smaller than 0.2, and the mean errors are almost zero
for all canopies. MP compared their predicted results by using a higher-order closure model with measured
data for six canopies in which corn and bean canopies were the same as those used in the present study.
Their root-mean-square errors were larger than 0.2 for most canopies, and the largest error was 0.54; these
values were larger than those in the present study.
In the present study, triangular distributions for the plant area density were used to approximate the ob-
served distributions for all of the canopies except the aspen–maple canopy. This technique has been used by
Pereira and Shaw 1980, who reported that the wind profile for a corn canopy the same corn canopy as that
used in the present study calculated by the triangular distribution was virtually indistinguishable from that
calculated using the observed distribution. The data of observed plant area density of a vegetation canopy
generally shows great scattering. The good agreement
P. Zeng, H. Takahashi Agricultural and Forest Meteorology 103 2000 301–313 307
Table 3 Comparison of drag coefficients C
d
yielded by the present model and those estimated in other studies
Vegetation Present
Other References
canopies study
studies Corn
0.20 0.20
Wilson and Shaw 1977 Pine
0.20 0.1–0.25
Amiro 1990b Oak–hickory
0.18 0.15
Lee et al. 1994 Bean
0.04 0.03–0.04
Thom 1971
between the modeled results and the measured data in the present study suggests that this technique is use-
ful for numerical studies of wind flow within vege- tation canopies. A beta distribution, which was used
by Meyers and Baldocchi 1991, was used for the aspen–maple canopy in the present study.
As was the case in the study by MP, a constant effective drag coefficient C
d
for each canopy was used in the present study. The effective drag coeffi-
cient for a vegetation canopy is difficult to measure due to problems of the shelter effect Thom, 1971 and
leaf orientation. In numerical studies, it is usually de- termined by trial-and-error to produce the best agree-
ment with observations. The effective drag coefficient for each canopy in the present study was determined
by trial-and-error. Table 3 shows a comparison of the values of C
d
yielded by the present model and those estimated in other studies for the same canopies. It
is shown that those yielded by the present model are almost the same as or within the range of those es-
timated in other studies. The values of C
d
used for the wheat and aspen–maple canopies were 0.12 and
Fig. 2. Simulated solid lines and measured circles wind profiles in the rubber tree plantation during a fully leafed, b partially leafed and c leafless periods. Profiles of PAD are shown by dashed lines.
0.15, respectively; published values of C
d
for these two canopies are not available for comparison. The re-
sults of the present study also support the claim made by MP that the use of a constant effective drag coef-
ficient is appropriate for a plant canopy.
4.2. Influence of foliage density on the wind profile Influence of foliage density on the wind profile was
investigated using the data measured in a rubber plan- tation located in the Hainan Island, China. The rub-
ber trees were about 12 m in height and the foliage density were estimated from the leaf-fall collections
and the measured solar radiations. Detailed informa- tion of the instrumentation and site has been described
by Takahashi et al. 1986 and Yoshino et al. 1988. Fig. 2 shows the simulated using the present model
and measured wind profiles in the rubber tree plan- tation during fully leafed, partially leafed and leaf-
less periods. The measured profiles were averages in near-neutral conditions and when the wind direction
was ESE, in which the effect of the surrounding wind- breaks was smallest and the fetch was the longest. The
plant area indices PAIs during the three periods were about 5, 3 and 1, respectively.
Again, the simulated results agreed with the mea- sured data excellently; the root-mean-square errors
were less than 0.2, and the mean errors were almost zero for the canopy during the three periods. As can
be seen in the figure, the wind profiles change little in form during the three periods, because there was
little change in the distribution patterns of the PAD
308 P. Zeng, H. Takahashi Agricultural and Forest Meteorology 103 2000 301–313
of the rubber trees during the three periods. How- ever, the normalized wind speed uu
∗
within the canopy increases as the foliage density decreases,
because it is easier for momentum to penetrate the canopy from above the canopy if the canopy is thin.
Although the PAI decreased about five-fold from the fully leafed to leafless periods, the normalized wind
speed within the leafless canopy, except that at the very upper portion of the canopy, increased less than
two-fold. This discrepancy can be explained by the effect of the effective drag coefficient. Seginer et al.
1976 has shown, from the results of wind-tunnel experiments, that the effective drag coefficient of a
dense canopy is smaller than that of a thin canopy. The effective drag coefficients used by the model for
the fully leafed and leafless canopies were 0.15 and 0.36, respectively, showing that the effective drag co-
efficient of tree branches in a fully leafed canopy is smaller than that in a leafless canopy. The same fact
was also found by Sato personal communication in his studies on wind flow through windbreaks. Each
of the three wind profiles showed a weak secondary wind maximum in the lower portion of the canopy
because the trunk space of the rubber plantation was very open. The wind profiles are near logarithmic
above the three canopies, and the slope of the wind profile during fully leafed period is larger than that
during leafless period. These findings are consistent with those observed above a Japanese larch forest
Allen, 1968 and an oak forest Dolman, 1986.
Fig. 3. Variation in the bulk momentum transfer coefficients at the canopy top C
Mh
vs. a C
D
=C
d
PAI and b C
F
=C
d
PAI z
max
h for nine vegetation canopies.
4.3. Bulk momentum transfer coefficients It can be seen in Figs. 1 and 2 that values of the
normalized velocities uu
∗
u
∗
is the friction velocity above the canopy at the top of all the canopy tops are
very similar, but changes of them according to the type of canopy are discernible. The same phenomenon was
also shown by MP. If the normalized velocity above the canopy is inverted and squared, it becomes the bulk
momentum transfer coefficient C
M
, which is defined as
τ ρ
= u
2 ∗
= C
M
u
2
11 where τ is the shear stress, ρ the air density, u
∗
the fric- tion velocity, and u is the mean wind speed at a height
above the vegetation. C
M
is also called drag coeffi- cient by some researchers. Dolman 1986 showed that
C
M
above an oak forest was about two-times larger on foliated conditions than that on defoliated condi-
tions. On the other hand, WK showed that C
M
over a rice canopy was scattered in a wide range over peri-
ods with differing leaf area indices. It is probable that the variations in the C
M
above a vegetation canopy do not depend only on the foliage density of the canopy.
Fig. 3a shows the variations in the C
M
at the canopy top versus C
d
PAI, where PAI is the plant area index, for the nine canopies, which used in the above two sec-
tions, according to the modeled results. Although the values of C
M
are scattered at large values of C
d
PAI,
P. Zeng, H. Takahashi Agricultural and Forest Meteorology 103 2000 301–313 309
there is a clear tendency for C
M
to increase as C
d
PAI increases. Furthermore, when the variations in C
M
are plotted against C
d
PAI z
max
h, where z
max
is the height at which the plant area density is maximum, a good
correlation between them is revealed Fig. 3b. Shaw and Pereira 1982 have shown that z
max
is an impor- tant factor in determining the aerodynamic roughness
of a plant canopy. The value of C
d
PAI z
max
h is small- est 0.146 in the case of the leafless rubber planta-
tion canopy. It seems that C
d
PAI z
max
h can accurately represent the momentum absorption ability of a vege-
tation canopy. Let C
D
≡ C
d
PAI, 12
C
F
≡ C
D
z
max
h 13
and C
Mh
represent the bulk momentum transfer coef- ficient at the canopy top, then the fitting-curve for the
nine canopies in Fig. 3b is expressed as C
Mh
= 0.0618 exp0.792C
F
14 and the correlation coefficient was equal to 0.889. Us-
ing a K-theory model, WK predicted that C
M
increased with an increase in C
D
when C
D
0.3, but it decreased with an increase in C
D
when C
D
increased further for a rice canopy. However, their prediction was not sup-
ported well by the measured data. If the logarithmic law is obeyed above a canopy
surface, the wind profile above the canopy surface is expressed as
u =
u
∗
κ ln
z − d z
15 where z
is the roughness of the canopy surface. Thom 1971 suggested that
z = λh − d
16 where h is the canopy height, and the value of λ was
estimated to be 0.36 for an artificial crop. Seginer 1974 estimated the value of λ to be 0.37 on the basis
of the canopy wind model of Inoue 1963 and an ob- servation by Kondo 1971. However, the value of λ
was estimated by Moore 1974 to be 0.26 according to 105 published d, z
and h data, and it was also es- timated by Shaw and Pereira 1982 to be 0.26 when
C
D
0.2 on the basis of results of numerical experi- ments using a higher-order closure model. From Eq.
15, we can get z
= 1
expκu
h
u
∗
h − d
17 where u
h
is the mean wind speed at the canopy top. From Eqs. 11, 16 and 17 we find that
λ =
1 expκu
h
u
∗
= 1
expκ √
C
Mh
18 Thus, λ is a function of C
Mh
, also changes with C
F
according to Eq. 14, and is not a constant when C
F
0.2. It was difficult to obtain a simple expression for the relationship between λ and C
F
from Eqs. 14 and 18. However, the results of least-squares analysis
for the values of λ and C
F
for the nine canopies showed that
λ = 0.209 exp0.414C
F
19 with the correlation coefficient equaling 0.892. The
values of λ for the nine canopies ranged from 0.22 the bean canopy to 0.32 the oak–hickory canopy, and
the average for all of the canopies was 0.26, which is the same value as that obtained by Moore 1974 and
Shaw and Pereira 1982. For comparison, Bruin and Moore 1985 reported λ
=0.22 for a pine forest the Thetford Forest.
4.4. Reynolds stress The computed Reynolds stress for the corn canopy
showed that more than 80 of the momentum was ab- sorbed in the upper half of the canopy Fig. 4, which
was the same as that shown by the measured data. The computed Reynolds stress decreased to almost zero
near the ground. Though there was no measured data for comparison in the lower canopy, the modeled re-
sults were almost the same as those showed by WS, who computed using a higher-order closure model.
Fig. 5 shows the computed profiles of two com- ponents of the Reynolds stress, R
l
and R
s
, for the corn canopy. Though R
s
is larger than R
l
above the canopy and in about the upper 70 of the canopy,
it decreases rapidly with height within the canopy, and it is almost zero in the lower 40 canopy; the
Reynolds stress in the lower portion of the canopy is maintained by the non-local transport momentum
310 P. Zeng, H. Takahashi Agricultural and Forest Meteorology 103 2000 301–313
Fig. 4. Profile of the computed Reynolds stress line and the measured data closed circles for the corn canopy.
R
l
. Meyers and Baldocchi 1991 have pointed out that the shear production, which is generated by the
interaction between the turbulent field and the mean velocity gradient, is small and turbulence imported
from above the canopy is a strong source for the turbulent kinetic energy in the lower canopy. Thus,
the present model can account for the phenomena of counter-gradient momentum transport and secondary
wind maxima that occurs in the lower portions of veg- etation canopies. R
l
peaks at about 0.8h and decreases above and below this height, and this distribution
pattern is similar to those of the measured hw
′
u
′
w
′
i
Fig. 5. Profiles of two components of the Reynolds stress in the corn canopy: small-eddy diffusion R
s
solid line and non-local transfer R
l
dash line.
non-local transport of the Reynolds stress in many canopies e.g. Shaw and Seginer, 1987; Baldocchi and
Meyers, 1988a, b; Amiro, 1990a. Above the canopy, R
s
is more than three times larger than R
l
and is the main source of the Reynolds stress, implying that the
predicted mean wind profile above the canopy is ap- proximately logarithmic. For the layer above 2h, the
non-local transfer momentum is parameterized to be zero in the model. Though only the profiles for the
corn canopy are shown, those for the other canopies are qualitatively the same.
5. Conclusions
