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A first-order closure model for the wind flow within and above vegetation canopies
Agricultural and Forest Meteorology 103 (2000) 301–313
A first-order closure model for the wind flow within and above vegetation canopies Pingtong Zeng∗ , Hidenori Takahashi Laboratory of Geoecology, Graduate School of Environmental Earth Science, Hokkaido University, Sapporo 060-0810, Japan Received 18 March 1999; accepted 11 January 2000
Abstract A first-order closure model that has the general utility for predicting the wind flow within and above vegetation canopies is presented. Parameterization schemes that took into account the influence of large turbulent eddies were developed for the Reynolds stress and the mixing length in the model. The results predicted by the model were compared with measured data for wind speeds within and above six types of vegetation canopy, including agricultural crops, deciduous and coniferous forests, and a rubber tree plantation during fully leafed, partially leafed and leafless periods. The predicted results agreed well with the measured data; the root-mean-square errors in the predicted wind speeds (non-dimensionalized by the friction velocity above the canopy) were about 0.2 or less for all of the canopies. The secondary wind maxima that occurred in the lower canopies were also correctly predicted. The influence of foliage density on the wind profiles within and above a vegetation canopy was successfully simulated by the model for a rubber tree plantation during fully leafed, partially leafed and leafless periods. The bulk momentum transfer coefficients (CM ) and the values of λ (which are defined by z0 =λ(h−d), where z0 is the roughness and d is the zero-displacement height of the canopy) for the vegetation canopies were also studied, and the relationships CMh =0.0618 exp(0.792CF ) and λ=0.209 exp(0.414CF ) were determined; here, CMh is the bulk momentum transfer coefficient at the canopy top; CF =Cd PAI zmax /h, where Cd is the effective drag coefficient of the canopy, PAI is the plant area index and zmax is the height at which the plant area density is maximum. The values of λ ranged from 0.22 to 0.32 for the canopies studied. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Numerical model; Wind velocity; Vegetation canopy; Reynolds stress; Mixing length; Turbulent eddy
1. Introduction Wind is an important factor for scalar fluxes (heat, water vapor, carbon dioxide, etc.) and movements of spores, pollen and particles within a vegetation canopy. Information about the wind flow within the vegetation canopy is important in meteorological, agricultural and ecological studies. A number of numerical models for predicting the vegetation wind flow have ∗ Corresponding author. Fax: +81-11-706-4867. E-mail address: [email protected] (P. Zeng)
been developed. However, accurate prediction of wind flow is difficult due to the complexity in the array of vegetation elements (leaves, branches and so on) and the complex process of air momentum transport within a vegetation canopy. Early modeling studies (e.g. Inoue, 1963; Cowan, 1968; Thom, 1971) were based on the K-theory or gradient-diffusion theory; that is, the momentum flux is equal to the product of an eddy viscosity and the local gradient of mean wind velocity. Wilson and Shaw (1977) (hereafter WS) point out that a K-theory model provides little insight into the nature of momentum transport processes within
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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 frequently 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) (hereafter 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 vegetation 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) proposed a non-local closure scheme for the total momentum 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 vegetation 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 ∂ hpi ∂τij ∂ hui i ∂ hui i + uj =− + +fFi + fVi (1) ∂t ∂xj ∂xi ∂xj with τij = −hu0i u0j i − hu00i w 00j i + ν
∂hui i ∂xj
(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, respectively. ui and xi are velocity and position vectors, respectively, t the time, p the kinematic pressure, fFi and fVi are form and viscous drag force vectors exerted on a unit mass of air within the averaging volume, respectively, τ ij the volume-averaged kinematic 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, respectively, 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 0 0 (3) hu w i + hu00 w 00 i = fx dz In this equation, hu0 w0 i is the Reynolds stress or turbulent flux of momentum, while hu00 w 00 i represents the dispersive flux that arises from the spatial correlation of regions of mean updraft or downdraft with regions, where u differs from its spatial mean, and
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fx (= fF1 + fV1 ) 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.
larger-scale eddies, which is determined by the mean wind speed differences between heights with large distance. Hence, the Reynolds stress is parameterized by −hu0 w0 i = K
2.2. Closure schemes 2.2.1. Reynolds stress In K-theory models, the Reynolds stress is parameterized as hu0 w 0 i = −KM
d hui dz
(4)
where KM 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 reliable 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 condition is often violated within vegetation canopies (Raupach and Thom, 1981; Baldocchi and Meyers, 1988b). Many measurements have shown that wind flow within and just above a plant canopy is dominated by turbulence with vertical length scales at least as large as the vegetation height (Kaimal and Finnigan, 1994). These large-scale turbulent eddies are intermittent and energetic, and they can penetrate the canopy crowns and enter the subcanopy trunk space to generate non-local turbulent transport. Most of the transport of momentum and scalar properties within the canopy are generated by these large-scale turbulent eddies (Raupach et al., 1986; Baldocchi and Meyers, 1988a). Baldocchi and Meyers (1988b) report that the Reynolds stress within a plant canopy is influenced not only by the product of KM 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
303
d hui z + Cg hur i (hur i − hui) dz h
(5)
where K is the eddy viscosity, h the vegetation height, ur the wind velocity at a reference height above the vegetation, and Cg is a coefficient. On the right-hand side of the equation, the first term (defined as Rs ), which has been formed according to the conventional K-theory, is responsible for small-eddy diffusion, and the second term (defined as Rl ) 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 hur i − hui to represent the intensity of the shear. hur 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 Cg in the equation is defined to be a function of the integrated plant area density and is expressed by Z h Cd A dz (6) Cg = C1 exp −C2 z
where A is the plant area density (m−1 ), Cd the effective drag coefficient of the plant elements, and C1 and C2 are constants that are determined by numerical experiments. The second term in Eq. (5) is an additional term that we have introduced. This term is similar to the term that represents dispersive flux in the model proposed by MLL, but instead of Cg defined in the present study, MLL used a constant coefficient. In the present study, we have not use the constant coefficient 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 experiments 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 dispersive 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
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and negligible compared with the amount of turbulent 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 2 d hui (7) K=l dz where l is the mixing length. Above the vegetation canopy surface, the mixing length is expressed as l = κ(z − 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 mixing 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 coefficient 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 lh (= κ(h − d) according to Eq. (8)). Thus, the mixing length within the canopy is parameterized as κz (9) l= 1 + Cl Cd Az with l ≤ (h − d),
z
where Cl is a constant determined by numerical experiments. 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 fx is parameterized according to WS fx = Cd A hui2
(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 (Cd ) of a single leaf measured in wind tunnel changes with the leaf orientation and turbulent scales and intensity around the leaf (Raupach and Thom, 1981). However, MP reported that the value of Cd 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 hur i is twice the vegetation height. This selection is based on measurements showing that there is a roughness sublayer within which the eddy viscosity 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: C1 and C2 in Eq. (6) and Cl 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 C2 was adjusted to 2.0 in the simulations for the rubber canopy data. Variations in C2 only affect the modeled wind profile at the lower canopy. The final values of 0.01, 2.0 and 5.0 for C1 , C2 and Cl , 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 zg /h=0.001, where zg is the roughness length of the ground surface. Experimental results showed that the solution was not
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sensitive to the chosen of the value zg . The initial wind profile was assumed, and the solution was found iteratively 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 coniferous 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 averages. 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, respectivelya Vegetation canopies
h (m)
LAI
References
Corn Pine Oak–hickory Bean Wheat Aspen–maple
2.8 18.5 23 1.18 1.25 18
3 2.3 4.9 6.3 6.6 5
Wilson and Shaw (1977) Amiro (1990a) Meyers and Baldocchi (1991) Thom (1971) Legg (1975a, b) 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 successfully 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 Rs was near zero or even upwardly transported the momentum when the
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gradient of mean wind was reversed, the non-local transport Rl 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 gradient will be reversed if the non-local turbulent transport momentum is large enough in the lower portion of a vegetation canopy, i.e. 0 ∂ w 0 u0 w 0 ∂w0 ∂u > p0 + ∂z ∂z ∂x He also suggested that in the region where there is a reversal 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 profile predicted by their second-order closure model was not logarithmic immediately above the canopy surface. However, many researchers have observed logarithmic 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 almost 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 (Rl ) 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 canopies
Root-meansquare errors
Mean absolute errors
Mean errors
Corn Pine Oak–hickory Bean Wheat Aspen–maple
0.203 0.126 0.205 0.163 0.135 0.233
0.168 0.114 0.171 0.138 0.098 0.168
−0.074 0.020 −0.106 −0.010 −0.039 −0.026
displacement (d) is needed. This value is estimated by the model based on an additional input value of measured wind speed above the canopy. Experimental results 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 models (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 problem. Table 2 shows the mean, mean absolute and root-mean-square errors in predicted results calculated 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 observed 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 Table 3 Comparison of drag coefficients (Cd ) yielded by the present model and those estimated in other studies Vegetation canopies
Present study
Other studies
References
Corn Pine Oak–hickory Bean
0.20 0.20 0.18 0.04
0.20 0.1–0.25 0.15 0.03–0.04
Wilson and Shaw (1977) Amiro (1990b) Lee et al. (1994) Thom (1971)
between the modeled results and the measured data in the present study suggests that this technique is useful for numerical studies of wind flow within vegetation 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 (Cd ) for each canopy was used in the present study. The effective drag coefficient 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 determined by trial-and-error to produce the best agreement 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 Cd 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 estimated in other studies. The values of Cd used for the wheat and aspen–maple canopies were 0.12 and
307
0.15, respectively; published values of Cd for these two canopies are not available for comparison. The results of the present study also support the claim made by MP that the use of a constant effective drag coefficient 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 plantation located in the Hainan Island, China. The rubber trees were about 12 m in height and the foliage density were estimated from the leaf-fall collections and the measured solar radiations. Detailed information 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 plantation during fully leafed, partially leafed and leafless periods. The measured profiles were averages in near-neutral conditions and when the wind direction was ESE, in which the effect of the surrounding windbreaks 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 measured 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
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.
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of the rubber trees during the three periods. However, the normalized wind speed (u/u∗ ) 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 coefficient 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).
4.3. Bulk momentum transfer coefficients It can be seen in Figs. 1 and 2 that values of the normalized velocities u/u∗ (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 CM , which is defined as τ = u2∗ = CM u2 (11) ρ where τ is the shear stress, ρ the air density, u∗ the friction velocity, and u is the mean wind speed at a height above the vegetation. CM is also called drag coefficient by some researchers. Dolman (1986) showed that CM above an oak forest was about two-times larger on foliated conditions than that on defoliated conditions. On the other hand, WK showed that CM over a rice canopy was scattered in a wide range over periods with differing leaf area indices. It is probable that the variations in the CM above a vegetation canopy do not depend only on the foliage density of the canopy. Fig. 3a shows the variations in the CM at the canopy top versus Cd PAI, where PAI is the plant area index, for the nine canopies, which used in the above two sections, according to the modeled results. Although the values of CM are scattered at large values of Cd PAI,
Fig. 3. Variation in the bulk momentum transfer coefficients at the canopy top (CMh ) vs. (a) CD (=Cd PAI) and (b) CF (=Cd PAI zmax /h) for nine vegetation canopies.
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there is a clear tendency for CM to increase as Cd PAI increases. Furthermore, when the variations in CM are plotted against Cd PAI zmax /h, where zmax 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 zmax is an important factor in determining the aerodynamic roughness of a plant canopy. The value of Cd PAI zmax /h is smallest (0.146) in the case of the leafless rubber plantation canopy. It seems that Cd PAI zmax /h can accurately represent the momentum absorption ability of a vegetation canopy. Let CD ≡ Cd PAI,
(12)
zmax h
(13)
CF ≡ CD
and CMh represent the bulk momentum transfer coefficient at the canopy top, then the fitting-curve for the nine canopies in Fig. 3b is expressed as CMh = 0.0618 exp(0.792CF )
(14)
and the correlation coefficient was equal to 0.889. Using a K-theory model, WK predicted that CM increased with an increase in CD when CD <0.3, but it decreased with an increase in CD when CD increased further for a rice canopy. However, their prediction was not supported 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 z−d u∗ ln u= (15) κ z0 where z0 is the roughness of the canopy surface. Thom (1971) suggested that z0 = λ(h − 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 observation by Kondo (1971). However, the value of λ was estimated by Moore (1974) to be 0.26 according to 105 published d, z0 and h data, and it was also estimated by Shaw and Pereira (1982) to be 0.26 (when CD >0.2) on the basis of results of numerical experi-
309
ments using a higher-order closure model. From Eq. (15), we can get z0 =
1 (h − d) exp(κ(uh /u∗ ))
(17)
where uh is the mean wind speed at the canopy top. From Eqs. (11), (16) and (17) we find that λ=
1 1 = √ exp(κ(uh /u∗ )) exp(κ CMh )
(18)
Thus, λ is a function of CMh , also changes with CF (according to Eq. (14)), and is not a constant when CF >0.2. It was difficult to obtain a simple expression for the relationship between λ and CF from Eqs. (14) and (18). However, the results of least-squares analysis for the values of λ and CF for the nine canopies showed that λ = 0.209 exp(0.414CF )
(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 absorbed 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 results 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 components of the Reynolds stress, Rl and Rs , for the corn canopy. Though Rs is larger than Rl 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
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(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, Rs is more than three times larger than Rl and is the main source of the Reynolds stress, implying that the predicted mean wind profile above the canopy is approximately 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.
Fig. 4. Profile of the computed Reynolds stress (line) and the measured data (closed circles) for the corn canopy.
Rl . 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 vegetation canopies. Rl peaks at about 0.8h and decreases above and below this height, and this distribution pattern is similar to those of the measured hw 0 u0 w 0 i
Fig. 5. Profiles of two components of the Reynolds stress in the corn canopy: small-eddy diffusion Rs (solid line) and non-local transfer Rl (dash line).
5. Conclusions Taking into account the non-local turbulent transport, we have developed a first-order closure model for predicting the wind flow within and above vegetation canopies. The high accuracy and the universal utility of the model were verified by comparisons of modeled results and measured data in six types of vegetation canopy and a rubber tree plantation during fully leafed, partially leafed and leafless periods. The root-mean-square errors in the predicted wind speeds were about 0.2 or less for all of the canopies; these errors are smaller than those results from a higher-order closure model. The wind speeds in the lower canopies, which appeared to be almost constant or reversed in gradient, were also correctly predicted. In addition to its high accuracy and universal utility, the present model costs little computation time due to its simplicity. The model would be very useful for the applications for predicting vegetation wind flows or scalar fluxes (e.g. heat and water vapor) between the atmosphere and vegetated surfaces when it is coupled with other models. A simulation study on the influence of foliage density on the wind profiles within and above a vegetation canopy was performed for a rubber tree plantation during fully leafed, partially leafed and leafless periods. The simulated wind profiles within the canopy changed little in form during the three periods but that the normalized wind speed (normalized by the friction velocity above the canopy) within the canopy increased as the foliage density decreased. The slope of the wind profile above the fully leafed canopy was larger than that above the leafless canopy.
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Based on the modeled wind speeds using the present model, we were able to clarify the effects of canopy density, structure and effective drag coefficient on the bulk momentum transfer coefficient and the coefficient λ, and we were also able to determine the correlations between CM and CF and between λ and CF . The Reynolds stress in the present model was parameterized by two terms: one representing the small-eddy diffusion, and one representing non-local transport through large-scale turbulent eddies. The modeled results showed that the non-local transfer component was small above the canopy but large, and the main source of the Reynolds stress, in the lower portion of the canopy. The model can also account for counter-gradient momentum transport occurs in the lower portion of a vegetation canopy. A parameterization scheme was developed for the mixing length within a canopy. 6. Nomenclature A plant area density (m2 m−3 ) C1 , C2 , Cl constants equal to 0.01, 2 and 5, respectively a coefficient equal to Cd PAI CD canopy effective drag coefficient Cd a coefficient defined as Cd PAI zmax /h CF coefficient of non-local momentum Cg transport bulk momentum transfer coefficient CM bulk momentum transfer coefficient at CMh the canopy top d zero plane displacement height (m) total (form and viscous) streamwise drag fx force (N) form drag force (N) fF viscous drag force (N) fV h mean vegetation height (m) i, j index notations, with values of 1, 2 or 3 K momentum eddy viscosity (defined in present study) (m2 s−1 ) conventional momentum eddy viscosity KM (m2 s−1 ) l mixing length (m) LAI leaf area index (m2 m−2 ) lh mixing-length at vegetation canopy top (m)
p PAD PAI Rl Rs t T u ui ur u∗ hu0 w0 i V w xi z z0 zmax
311
kinematic pressure (Pa) plant area density (m2 m−3 ) plant area index (m2 m−2 ) non-local momentum transport (m2 s−2 ) small-eddy diffusion (m2 s−2 ) time (s) time average interval (s) streamwise velocity (m s−1 ) wind velocity vector (m s−1 ) wind velocity at a reference height (m s−1 ) friction velocity above vegetation canopy (m s−1 ) Reynolds stress (m2 s−2 ) volume (m3 ) vertical velocity (m s−1 ) position vector (m) height (m) roughness of vegetation surface (m) the height at which the plant area density is maximum (m)
Greek letters ν kinematic viscosity of the air (m2 s−1 ) κ von K´arm´an’s constant (equal to 0.4) λ the coefficient between z0 and λ ρ air density (kg m−3 ) τ shear stress (m2 s−2 ) kinematic momentum flux (m2 s−2 ) τ ij φ a scalar or vector variable Other symbols overbar ( – ) prime ( 0 ) angle brackets h i double prime ( 00 )
time average deviation from time average volume average deviation from volume average
Acknowledgements Micrometeorological measurements in a rubber plantation were conducted as a part of the project ‘A Study on the Geoecological Factors of Land Use and Vegetation in the Tropics and Subtropics in the Peoples Republic of China,’ which was supported by grants-in-aid from the Japanese Ministry of Education, Science and Culture (leader: Masatoshi Yoshino,
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Nos. 61041012 and 62043011). We are grateful to Prof. T. Maitani (Okayama University), Mr. T. Sato (Hokkaido Head Office, Japan Weather Association) and Mr. Natsume (Hokkaido Institute of Environmental Science) for their helpful comments. Mr. Natsume also checked our computing programs. References Allen, L.H. Jr., 1968. Turbulence and wind speed spectra within a Japanese larch plantation. J. Appl. Meteorol. 7, 73–78. Amiro, B.D., 1990a. Comparison of turbulence statistics within three boreal forest canopies. Boundary-Layer Meteorol. 51, 99–120. Amiro, B.D., 1990b. Drag coefficients and turbulence spectra within three boreal forest canopies. Boundary-Layer Meteorol. 52, 227–246. Baldocchi, D.D., Meyers, T.P., 1988a. Turbulence structure in a deciduous forest. Boundary-Layer Meteorol. 43, 345–364. Baldocchi, D.D., Meyers, T.P., 1988b. A spectral and lag-correlation analysis of turbulence in a deciduous forest canopy. Boundary-Layer Meteorol. 45, 31–58. Bruin, H.A.R., Moore, C.J., 1985. Zero-plane displacement and roughness length for tall vegetation, derived from a simple mass conservation hypothesis. Boundary-Layer Meteorol. 31, 39–49. Corrsin, S., 1974. Limitations of gradient transport model in random walks and in turbulence. Adv. Geophys. 18A, 25–60. Cowan, I.R., 1968. Mass, heat and momentum exchange between stands of plants and their atmospheric environment. Quart. J. R. Meteorol. Soc. 94, 523–544. Dolman, A.J., 1986. Estimates of roughness length and zero plane displacement for a foliated and non-foliated oak canopy. Agric. For. Meteorol. 36, 241–248. Gao, W., Shaw, R.H., Paw, U.K.T., 1989. Observation of organized structure in turbulent flow within and above a forest canopy. Boundary-Layer Meteorol. 47, 349–377. Garrat, J.R., 1978. Flux-profile relations above tall vegetation. Quart. J. R. Meteorol. Soc. 104, 199–211. Halldin, S., Lindroth, A., 1986. Pine forest microclimate simulation using different diffusivities. Boundary-Layer Meteorol. 35, 103–123. Inoue, E., 1963. On the turbulent structure of air flow within crop canopies. J. Meteorol. Soc. Jpn. 41, 317–325. Kaimal, J.C., Finnigan, J.J., 1994. Atmospheric Boundary Layer Flows: Their Structure and Measurement. Oxford University Press, New York, p. 289. Kondo, J., 1971. Relationship between the roughness coefficient and other aerodynamic parameters. J. Meteorol. Soc. Jpn. 49, 121–124. Kondo, J., Akashi, S., 1976. Numerical studies on the two-dimensional flow in horizontally homogeneous canopy layers. Boundary-Layer Meteorol. 10, 255–272. Lee, X., Shaw, R.H., Black, T.A., 1994. Modelling the effect of mean pressure gradient on the mean flow within forests. Agric. For. Meteorol. 68, 201–212.
Legg, B.J., 1975a. Turbulent diffusion within a wheat canopy: I. Measurement using nitrous oxide. Quart. J. R. Meteorol. Soc. 101, 591–610. Legg, B.J., 1975b. Turbulent diffusion within a wheat canopy: II. Results and interpretation. Quart. J. R. Meteorol. Soc. 101, 611–628. Li, Z.J., Miller, R., Lin, J.D., 1985. A first-order closure scheme to describe counter-gradient momentum transport in plant canopies. Boundary-Layer Meteorol. 33, 77–83. Meyers, T.P., Baldocchi, D.D., 1991. The budgets of turbulent kinetic energy and Reynolds stress within and above a deciduous forest. Agric. For. Meteorol. 53, 207–222. Meyers, T.P., Paw, U.K.T., 1986. Testing of a higher-order closure model for airflow within and above plant canopies. Boundary-Layer Meteorol. 37, 297–311. Miller, D.R., Lin, J.D., Lu, Z.N., 1991. Air flow across an alpine forest clearing: a model and field measurements. Agric. For. Meteorol. 56, 209–225. Moore, C.J., 1974. A comparative study of forest and grassland micrometeorology. Ph.D. Thesis, The Flinders, University of South Australia, p. 237. Neumann, H.H., Den Hartog, G., Shaw, R.H., 1989. Leaf area measurements based on hemispheric photographs and leaf-letter collection in a deciduous forest during autumn leaf-fall. Agric. For. Meteorol. 45, 325–345. Oliver, H.R., 1971. Wind profiles in and above a forest canopy. Quart. J. R. Meteorol. Soc. 97, 548–553. Pereira, A.R., Shaw, R.H., 1980. A numerical experiment on the mean wind structure inside canopies of vegetation. Agric. Meteorol. 22, 303–318. Raupach, M.R., Thom, A.S., 1981. Turbulence in and above plant canopies. Ann. Rev. Fluid Mech. 13, 97–129. Raupach, M.R., Coppin, P.A., Legg, B.J., 1986. Experiments on scalar dispersion within a model plant canopy: Part I. The turbulence structure. Boundary-Layer Meteorol. 35, 21– 52. Raupach, M.R., Thom, A.S., Edwards, I., 1980. A wind-tunnel study of turbulent flow close to regularly arrayed rough surfaces. Boundary-Layer Meteorol. 18, 373–397. Seginer, J., 1974. Aerodynamic roughness of vegetated surfaces. Boundary-Layer Meteorol. 5, 383–393. Seginer, J., Mulhearn, P.J., Bradley, E.R., Finnigan, J.J., 1976. Turbulent flow in a model plant canopy. Boundary-Layer Meteorol. 10, 423–453. Shaw, R.H., 1977. Secondary wind speed maxima inside plant canopies. J. Appl. Meteorol. 16, 514–521. Shaw, R.H., Pereira, A.R., 1982. Aerodynamic roughness of a plant canopy: a numerical experiment. Agric. Meteorol. 26, 51–65. Shaw, R.H., Seginer, I., 1987. Calculation of velocity skewness in real and artificial plant canopies. Boundary-Layer Meteorol. 39, 315–332. Shaw, R.H., Den Hartog, G., Neumann, H.H., 1988. Influence of foliar density and thermal stability on profiles of Reynolds stress and turbulence intensity in a deciduous forest. Boundary-Layer Meteorol. 45, 391–409. Simpson, I.J., Thurtell, G.W., Neumann, H.H., Den Hartog, G., Edwards, G.C., 1998. The validity of similarity theory in the
P. Zeng, H. Takahashi / Agricultural and Forest Meteorology 103 (2000) 301–313 roughness sublayer above forests. Boundary-Layer Meteorol. 87, 69–99. Stull, B.S., 1988. An Introduction to Boundary Layer Meteorology. Kluwer Academic Publishers, Boston, p. 666. Takahashi, H., Makita, H., Nakagawa, K., Hayashi, Y., Hao, Y., Chin, L., Wang, P., Wei, F., Yao, M., 1986. Micrometeorology in a rubber plantation during cold wave weather conditions. In: Yoshino, M. (Ed.), Climate, Geoecology and Agriculture in South China, Vol. 1. Climatological notes, Institute of Geoscience, University of Tsukuba, Tsukuba, Japan, pp. 37–58. Thom, A.S., 1971. Momentum absorption by vegetation. Quart. J. R. Meteorol. Soc. 97, 414–428. Van Pul, A., Van Boxel, J.H., 1990. Comment on a first-order closure scheme to describe counter-gradient momentum transport in plant canopies by Z.J. Li, D.R. Miller and J.D. Lin. Boundary-Layer Meteorol. 51, 313–315.
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Wang, H., Takle, E.S., 1996. On three-dimensionality of shelterbelt structure and its influences on shelter effects. Boundary-Layer Meteorol. 79, 83–108. Watanabe, T., Kondo, J., 1990. The influence of canopy structure and density upon the mixing length within and above vegetation. J. Meteorol. Soc. Jpn. 68, 227–235. Wilson, J.D., 1988. A secondary-order closure model for flow through vegetation. Boundary-Layer Meteorol. 42, 371–392. Wilson, N.R., Shaw, R.H., 1977. A higher order closure model for canopy flow. J. Appl. Meteorol. 1197–1205. Yoshino, M., Jiang, A., Hao, Y., Takahashi, H., Nakagawa, K., Yamakawa, S., Makita, H., Chujo, H., Xie, L., Wang, P., 1988. System of micrometeorological observation in and out of a rubber plantation in Hainan Island. In: Yoshino, M. (Ed.), Climate, Geoecology and Agriculture in South China, Vol. 2. Climatological notes, Institute of Geoscience, University of Tsukuba, Tsukuba, Japan, pp. 5–16.
A first-order closure model for the wind flow within and above vegetation canopies Pingtong Zeng∗ , Hidenori Takahashi Laboratory of Geoecology, Graduate School of Environmental Earth Science, Hokkaido University, Sapporo 060-0810, Japan Received 18 March 1999; accepted 11 January 2000
Abstract A first-order closure model that has the general utility for predicting the wind flow within and above vegetation canopies is presented. Parameterization schemes that took into account the influence of large turbulent eddies were developed for the Reynolds stress and the mixing length in the model. The results predicted by the model were compared with measured data for wind speeds within and above six types of vegetation canopy, including agricultural crops, deciduous and coniferous forests, and a rubber tree plantation during fully leafed, partially leafed and leafless periods. The predicted results agreed well with the measured data; the root-mean-square errors in the predicted wind speeds (non-dimensionalized by the friction velocity above the canopy) were about 0.2 or less for all of the canopies. The secondary wind maxima that occurred in the lower canopies were also correctly predicted. The influence of foliage density on the wind profiles within and above a vegetation canopy was successfully simulated by the model for a rubber tree plantation during fully leafed, partially leafed and leafless periods. The bulk momentum transfer coefficients (CM ) and the values of λ (which are defined by z0 =λ(h−d), where z0 is the roughness and d is the zero-displacement height of the canopy) for the vegetation canopies were also studied, and the relationships CMh =0.0618 exp(0.792CF ) and λ=0.209 exp(0.414CF ) were determined; here, CMh is the bulk momentum transfer coefficient at the canopy top; CF =Cd PAI zmax /h, where Cd is the effective drag coefficient of the canopy, PAI is the plant area index and zmax is the height at which the plant area density is maximum. The values of λ ranged from 0.22 to 0.32 for the canopies studied. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Numerical model; Wind velocity; Vegetation canopy; Reynolds stress; Mixing length; Turbulent eddy
1. Introduction Wind is an important factor for scalar fluxes (heat, water vapor, carbon dioxide, etc.) and movements of spores, pollen and particles within a vegetation canopy. Information about the wind flow within the vegetation canopy is important in meteorological, agricultural and ecological studies. A number of numerical models for predicting the vegetation wind flow have ∗ Corresponding author. Fax: +81-11-706-4867. E-mail address: [email protected] (P. Zeng)
been developed. However, accurate prediction of wind flow is difficult due to the complexity in the array of vegetation elements (leaves, branches and so on) and the complex process of air momentum transport within a vegetation canopy. Early modeling studies (e.g. Inoue, 1963; Cowan, 1968; Thom, 1971) were based on the K-theory or gradient-diffusion theory; that is, the momentum flux is equal to the product of an eddy viscosity and the local gradient of mean wind velocity. Wilson and Shaw (1977) (hereafter WS) point out that a K-theory model provides little insight into the nature of momentum transport processes within
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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 frequently 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) (hereafter 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 vegetation 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) proposed a non-local closure scheme for the total momentum 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 vegetation 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 ∂ hpi ∂τij ∂ hui i ∂ hui i + uj =− + +fFi + fVi (1) ∂t ∂xj ∂xi ∂xj with τij = −hu0i u0j i − hu00i w 00j i + ν
∂hui i ∂xj
(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, respectively. ui and xi are velocity and position vectors, respectively, t the time, p the kinematic pressure, fFi and fVi are form and viscous drag force vectors exerted on a unit mass of air within the averaging volume, respectively, τ ij the volume-averaged kinematic 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, respectively, 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 0 0 (3) hu w i + hu00 w 00 i = fx dz In this equation, hu0 w0 i is the Reynolds stress or turbulent flux of momentum, while hu00 w 00 i represents the dispersive flux that arises from the spatial correlation of regions of mean updraft or downdraft with regions, where u differs from its spatial mean, and
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fx (= fF1 + fV1 ) 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.
larger-scale eddies, which is determined by the mean wind speed differences between heights with large distance. Hence, the Reynolds stress is parameterized by −hu0 w0 i = K
2.2. Closure schemes 2.2.1. Reynolds stress In K-theory models, the Reynolds stress is parameterized as hu0 w 0 i = −KM
d hui dz
(4)
where KM 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 reliable 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 condition is often violated within vegetation canopies (Raupach and Thom, 1981; Baldocchi and Meyers, 1988b). Many measurements have shown that wind flow within and just above a plant canopy is dominated by turbulence with vertical length scales at least as large as the vegetation height (Kaimal and Finnigan, 1994). These large-scale turbulent eddies are intermittent and energetic, and they can penetrate the canopy crowns and enter the subcanopy trunk space to generate non-local turbulent transport. Most of the transport of momentum and scalar properties within the canopy are generated by these large-scale turbulent eddies (Raupach et al., 1986; Baldocchi and Meyers, 1988a). Baldocchi and Meyers (1988b) report that the Reynolds stress within a plant canopy is influenced not only by the product of KM 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
303
d hui z + Cg hur i (hur i − hui) dz h
(5)
where K is the eddy viscosity, h the vegetation height, ur the wind velocity at a reference height above the vegetation, and Cg is a coefficient. On the right-hand side of the equation, the first term (defined as Rs ), which has been formed according to the conventional K-theory, is responsible for small-eddy diffusion, and the second term (defined as Rl ) 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 hur i − hui to represent the intensity of the shear. hur 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 Cg in the equation is defined to be a function of the integrated plant area density and is expressed by Z h Cd A dz (6) Cg = C1 exp −C2 z
where A is the plant area density (m−1 ), Cd the effective drag coefficient of the plant elements, and C1 and C2 are constants that are determined by numerical experiments. The second term in Eq. (5) is an additional term that we have introduced. This term is similar to the term that represents dispersive flux in the model proposed by MLL, but instead of Cg defined in the present study, MLL used a constant coefficient. In the present study, we have not use the constant coefficient 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 experiments 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 dispersive 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
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and negligible compared with the amount of turbulent 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 2 d hui (7) K=l dz where l is the mixing length. Above the vegetation canopy surface, the mixing length is expressed as l = κ(z − 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 mixing 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 coefficient 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 lh (= κ(h − d) according to Eq. (8)). Thus, the mixing length within the canopy is parameterized as κz (9) l= 1 + Cl Cd Az with l ≤ (h − d),
z
where Cl is a constant determined by numerical experiments. 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 fx is parameterized according to WS fx = Cd A hui2
(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 (Cd ) of a single leaf measured in wind tunnel changes with the leaf orientation and turbulent scales and intensity around the leaf (Raupach and Thom, 1981). However, MP reported that the value of Cd 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 hur i is twice the vegetation height. This selection is based on measurements showing that there is a roughness sublayer within which the eddy viscosity 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: C1 and C2 in Eq. (6) and Cl 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 C2 was adjusted to 2.0 in the simulations for the rubber canopy data. Variations in C2 only affect the modeled wind profile at the lower canopy. The final values of 0.01, 2.0 and 5.0 for C1 , C2 and Cl , 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 zg /h=0.001, where zg is the roughness length of the ground surface. Experimental results showed that the solution was not
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sensitive to the chosen of the value zg . The initial wind profile was assumed, and the solution was found iteratively 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 coniferous 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 averages. 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, respectivelya Vegetation canopies
h (m)
LAI
References
Corn Pine Oak–hickory Bean Wheat Aspen–maple
2.8 18.5 23 1.18 1.25 18
3 2.3 4.9 6.3 6.6 5
Wilson and Shaw (1977) Amiro (1990a) Meyers and Baldocchi (1991) Thom (1971) Legg (1975a, b) 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 successfully 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 Rs was near zero or even upwardly transported the momentum when the
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gradient of mean wind was reversed, the non-local transport Rl 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 gradient will be reversed if the non-local turbulent transport momentum is large enough in the lower portion of a vegetation canopy, i.e. 0 ∂ w 0 u0 w 0 ∂w0 ∂u > p0 + ∂z ∂z ∂x He also suggested that in the region where there is a reversal 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 profile predicted by their second-order closure model was not logarithmic immediately above the canopy surface. However, many researchers have observed logarithmic 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 almost 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 (Rl ) 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 canopies
Root-meansquare errors
Mean absolute errors
Mean errors
Corn Pine Oak–hickory Bean Wheat Aspen–maple
0.203 0.126 0.205 0.163 0.135 0.233
0.168 0.114 0.171 0.138 0.098 0.168
−0.074 0.020 −0.106 −0.010 −0.039 −0.026
displacement (d) is needed. This value is estimated by the model based on an additional input value of measured wind speed above the canopy. Experimental results 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 models (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 problem. Table 2 shows the mean, mean absolute and root-mean-square errors in predicted results calculated 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 observed 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 Table 3 Comparison of drag coefficients (Cd ) yielded by the present model and those estimated in other studies Vegetation canopies
Present study
Other studies
References
Corn Pine Oak–hickory Bean
0.20 0.20 0.18 0.04
0.20 0.1–0.25 0.15 0.03–0.04
Wilson and Shaw (1977) Amiro (1990b) Lee et al. (1994) Thom (1971)
between the modeled results and the measured data in the present study suggests that this technique is useful for numerical studies of wind flow within vegetation 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 (Cd ) for each canopy was used in the present study. The effective drag coefficient 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 determined by trial-and-error to produce the best agreement 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 Cd 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 estimated in other studies. The values of Cd used for the wheat and aspen–maple canopies were 0.12 and
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0.15, respectively; published values of Cd for these two canopies are not available for comparison. The results of the present study also support the claim made by MP that the use of a constant effective drag coefficient 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 plantation located in the Hainan Island, China. The rubber trees were about 12 m in height and the foliage density were estimated from the leaf-fall collections and the measured solar radiations. Detailed information 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 plantation during fully leafed, partially leafed and leafless periods. The measured profiles were averages in near-neutral conditions and when the wind direction was ESE, in which the effect of the surrounding windbreaks 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 measured 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
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.
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of the rubber trees during the three periods. However, the normalized wind speed (u/u∗ ) 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 coefficient 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).
4.3. Bulk momentum transfer coefficients It can be seen in Figs. 1 and 2 that values of the normalized velocities u/u∗ (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 CM , which is defined as τ = u2∗ = CM u2 (11) ρ where τ is the shear stress, ρ the air density, u∗ the friction velocity, and u is the mean wind speed at a height above the vegetation. CM is also called drag coefficient by some researchers. Dolman (1986) showed that CM above an oak forest was about two-times larger on foliated conditions than that on defoliated conditions. On the other hand, WK showed that CM over a rice canopy was scattered in a wide range over periods with differing leaf area indices. It is probable that the variations in the CM above a vegetation canopy do not depend only on the foliage density of the canopy. Fig. 3a shows the variations in the CM at the canopy top versus Cd PAI, where PAI is the plant area index, for the nine canopies, which used in the above two sections, according to the modeled results. Although the values of CM are scattered at large values of Cd PAI,
Fig. 3. Variation in the bulk momentum transfer coefficients at the canopy top (CMh ) vs. (a) CD (=Cd PAI) and (b) CF (=Cd PAI zmax /h) for nine vegetation canopies.
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there is a clear tendency for CM to increase as Cd PAI increases. Furthermore, when the variations in CM are plotted against Cd PAI zmax /h, where zmax 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 zmax is an important factor in determining the aerodynamic roughness of a plant canopy. The value of Cd PAI zmax /h is smallest (0.146) in the case of the leafless rubber plantation canopy. It seems that Cd PAI zmax /h can accurately represent the momentum absorption ability of a vegetation canopy. Let CD ≡ Cd PAI,
(12)
zmax h
(13)
CF ≡ CD
and CMh represent the bulk momentum transfer coefficient at the canopy top, then the fitting-curve for the nine canopies in Fig. 3b is expressed as CMh = 0.0618 exp(0.792CF )
(14)
and the correlation coefficient was equal to 0.889. Using a K-theory model, WK predicted that CM increased with an increase in CD when CD <0.3, but it decreased with an increase in CD when CD increased further for a rice canopy. However, their prediction was not supported 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 z−d u∗ ln u= (15) κ z0 where z0 is the roughness of the canopy surface. Thom (1971) suggested that z0 = λ(h − 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 observation by Kondo (1971). However, the value of λ was estimated by Moore (1974) to be 0.26 according to 105 published d, z0 and h data, and it was also estimated by Shaw and Pereira (1982) to be 0.26 (when CD >0.2) on the basis of results of numerical experi-
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ments using a higher-order closure model. From Eq. (15), we can get z0 =
1 (h − d) exp(κ(uh /u∗ ))
(17)
where uh is the mean wind speed at the canopy top. From Eqs. (11), (16) and (17) we find that λ=
1 1 = √ exp(κ(uh /u∗ )) exp(κ CMh )
(18)
Thus, λ is a function of CMh , also changes with CF (according to Eq. (14)), and is not a constant when CF >0.2. It was difficult to obtain a simple expression for the relationship between λ and CF from Eqs. (14) and (18). However, the results of least-squares analysis for the values of λ and CF for the nine canopies showed that λ = 0.209 exp(0.414CF )
(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 absorbed 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 results 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 components of the Reynolds stress, Rl and Rs , for the corn canopy. Though Rs is larger than Rl 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
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(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, Rs is more than three times larger than Rl and is the main source of the Reynolds stress, implying that the predicted mean wind profile above the canopy is approximately 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.
Fig. 4. Profile of the computed Reynolds stress (line) and the measured data (closed circles) for the corn canopy.
Rl . 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 vegetation canopies. Rl peaks at about 0.8h and decreases above and below this height, and this distribution pattern is similar to those of the measured hw 0 u0 w 0 i
Fig. 5. Profiles of two components of the Reynolds stress in the corn canopy: small-eddy diffusion Rs (solid line) and non-local transfer Rl (dash line).
5. Conclusions Taking into account the non-local turbulent transport, we have developed a first-order closure model for predicting the wind flow within and above vegetation canopies. The high accuracy and the universal utility of the model were verified by comparisons of modeled results and measured data in six types of vegetation canopy and a rubber tree plantation during fully leafed, partially leafed and leafless periods. The root-mean-square errors in the predicted wind speeds were about 0.2 or less for all of the canopies; these errors are smaller than those results from a higher-order closure model. The wind speeds in the lower canopies, which appeared to be almost constant or reversed in gradient, were also correctly predicted. In addition to its high accuracy and universal utility, the present model costs little computation time due to its simplicity. The model would be very useful for the applications for predicting vegetation wind flows or scalar fluxes (e.g. heat and water vapor) between the atmosphere and vegetated surfaces when it is coupled with other models. A simulation study on the influence of foliage density on the wind profiles within and above a vegetation canopy was performed for a rubber tree plantation during fully leafed, partially leafed and leafless periods. The simulated wind profiles within the canopy changed little in form during the three periods but that the normalized wind speed (normalized by the friction velocity above the canopy) within the canopy increased as the foliage density decreased. The slope of the wind profile above the fully leafed canopy was larger than that above the leafless canopy.
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Based on the modeled wind speeds using the present model, we were able to clarify the effects of canopy density, structure and effective drag coefficient on the bulk momentum transfer coefficient and the coefficient λ, and we were also able to determine the correlations between CM and CF and between λ and CF . The Reynolds stress in the present model was parameterized by two terms: one representing the small-eddy diffusion, and one representing non-local transport through large-scale turbulent eddies. The modeled results showed that the non-local transfer component was small above the canopy but large, and the main source of the Reynolds stress, in the lower portion of the canopy. The model can also account for counter-gradient momentum transport occurs in the lower portion of a vegetation canopy. A parameterization scheme was developed for the mixing length within a canopy. 6. Nomenclature A plant area density (m2 m−3 ) C1 , C2 , Cl constants equal to 0.01, 2 and 5, respectively a coefficient equal to Cd PAI CD canopy effective drag coefficient Cd a coefficient defined as Cd PAI zmax /h CF coefficient of non-local momentum Cg transport bulk momentum transfer coefficient CM bulk momentum transfer coefficient at CMh the canopy top d zero plane displacement height (m) total (form and viscous) streamwise drag fx force (N) form drag force (N) fF viscous drag force (N) fV h mean vegetation height (m) i, j index notations, with values of 1, 2 or 3 K momentum eddy viscosity (defined in present study) (m2 s−1 ) conventional momentum eddy viscosity KM (m2 s−1 ) l mixing length (m) LAI leaf area index (m2 m−2 ) lh mixing-length at vegetation canopy top (m)
p PAD PAI Rl Rs t T u ui ur u∗ hu0 w0 i V w xi z z0 zmax
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kinematic pressure (Pa) plant area density (m2 m−3 ) plant area index (m2 m−2 ) non-local momentum transport (m2 s−2 ) small-eddy diffusion (m2 s−2 ) time (s) time average interval (s) streamwise velocity (m s−1 ) wind velocity vector (m s−1 ) wind velocity at a reference height (m s−1 ) friction velocity above vegetation canopy (m s−1 ) Reynolds stress (m2 s−2 ) volume (m3 ) vertical velocity (m s−1 ) position vector (m) height (m) roughness of vegetation surface (m) the height at which the plant area density is maximum (m)
Greek letters ν kinematic viscosity of the air (m2 s−1 ) κ von K´arm´an’s constant (equal to 0.4) λ the coefficient between z0 and λ ρ air density (kg m−3 ) τ shear stress (m2 s−2 ) kinematic momentum flux (m2 s−2 ) τ ij φ a scalar or vector variable Other symbols overbar ( – ) prime ( 0 ) angle brackets h i double prime ( 00 )
time average deviation from time average volume average deviation from volume average
Acknowledgements Micrometeorological measurements in a rubber plantation were conducted as a part of the project ‘A Study on the Geoecological Factors of Land Use and Vegetation in the Tropics and Subtropics in the Peoples Republic of China,’ which was supported by grants-in-aid from the Japanese Ministry of Education, Science and Culture (leader: Masatoshi Yoshino,
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