April 1990 T. Watanabe and J. Kondo 227. The Influence of Canopy Structure and Density. upon the Mixing Length within and above Vegetation

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1 April 1990 T. Watanabe and J. Kondo 227 The Influence of Canopy Structure and Density upon the Mixing Length within and above Vegetation By Tsutomu Watanabe and Junsei Kondo Geophysical Institute, Tohoku University, Sendai 980, Japan (Manuscript received 30 November 1989, in revised form 29 January 1990) Abstract Taking into account the intermittent gust motion within a canopy layer and the influence of both the canopy elements and the underlying ground surface on the turbulent motion within the layer, a new mixing-length model was developed to simulate the exchange of momentum, sensible heat and water vapor between the atmosphere and the vegetated surfaces. It was found that traditional models overestimate the mixing length when the canopy density is neither dense nor sparse, since these models assume that the mixing length is limited by either the canopy elements or the height from the underlying ground surface. According to the present model, for the case of a vertically uniform canopy, the mixing length within the canopy layer is approximately equal to kz (k : von Karman constant, z : height) near the underlying ground surface. It remains approximately constant far enough from the ground surface, decreasing gradually as the canopy density increases. The validity of the model was determined by comparing model results with observed data. 1. Introduction K-theory (flux-gradient theory) is often used to express the exchange of momentum, sensible heat and water vapor between the atmosphere and vegetated surfaces (e.g., Inoue, 1963; Cionco, 1965; Cowan, 1968; Seginer, 1974; Kondo and Akashi, 1976; Li et al., 1985; Kondo and Kawanaka, 1986; Massman, 1987a, b). According to Corrsin (1974), K-theory is applicable under the condition that the length scale associated with the transporting mechanism is small enough and changes slowly with height. However, flow statistics change rapidly with height within a real canopy layer, so that the above condition is often violated (Shaw, 1977). In addition, K-theory cannot explain the "secondary wind speed maxima" observed in forest canopies (Shaw, 1977; Wilson and Shaw, 1977). For these reasons, many investigators have adopted a higher-order closure model (Shaw, 1977; Wilson and Shaw, 1977; Inoue, 1981; Meyers and PawU, 1986; Shaw and Seginer, 1987). For estimating the fluxes transported through the canopy top, however, it is not necessary to precisely simulate the wind field in the entire canopy layer. For example, for the case of a forest canopy which has a dense crown, the exchange is limited to within the upper portion of the canopy layer. Therefore, it C1990, Meteorological Society of Japan is sufficient to simulate the wind field only in this portion. The turbulence in the upper portion of the canopy layer is produced by the local wind shear (Shaw, 1977; Raupach et al., 1986; Raupach, 1988), which allows K-theory to be applicable. Also, this scheme has the advantage of computational simplicity. For these reasons, it is worth while developing a canopy model based on K-theory. In order to estimate the turbulent diffusivity used in K-theory, the mixing length must be evaluated. Similarly, in higher-order closure models, the length scale is often used to parameterize the higher-order moments. Nevertheless, knowledge of the mixing length within a canopy layer has been insufficient. Several models have been developed on certain assumptions (Uchijima, 1962; Inoue, 1963; Cowan, 1968; Seginer, 1974; Kondo and Akashi, 1976; Inoue and Uchijima, 1979). The purpose of this study is to formulate the mixing length within a canopy layer on the basis of recent experimental and theoretical studies (e.g., Wilson and Shaw, 1977; Finnigan, 1979; Raupach et al., 1986; Baldocchi and Meyers, 1988a, b; Raupach, 1988). In these studies, the turbulence within a canopy layer is mainly produced by the intermittent gust motion. We are now interested in parameterizing the turbulent transport between the atmosphere and vegetated surfaces by means of a simple model. From

2 228 Journal of the Meteorological Society of Japan Vol. 68, No. 2 recent experience, it has been known that the most difficult case in parameterizing the exchange occurs when the canopy density has a moderate value. This is a preliminary study to remove this difficulty. The present results will be applied to the evapotranspiration model in our successive study. and the boundary condition 2. Several mixing-length models K-theory relates the shear stress * to the local shear of the mean wind speed u as where * is the air density, KM is the turbulent diffusivity and z is the height. KM is conventionally described by the mixing length l, i.e., Here, zg is the aerodynamic roughness of the underlying ground surface and li is the intrinsic mixing length-the mixing length within a vertically homogeneous canopy that has the same internal structure as the level under consideration. Kondo and Akashi (1976) and Kondo and Kawanaka (1986) adopted this model. According to this model, l within a vertically homogeneous canopy is expressed as,* The budget of the shear stress within and above the canopy layer is where cd is the effective drag coefficient of each leaf element, a is the leaf area density (one-sided), and h is the canopy height. If l can be expressed by the known parameters cd and a(z), profiles of the mean wind speed and the shear stress within and above the canopy layer can be derived from Eqs. (1) to (3) numerically. Uchijima (1962) proposed that l is proportional to the height within a canopy layer, and formulated l such as where d is the zero-plane displacement and k is the von Karman constant. This formulation is applicable to a sparse canopy, but fails for a dense canopy, since the turbulent motion in a dense canopy layer is strongly affected by the canopy elements rather than by the ground surface. On the other hand, Inoue (1963) suggested that is constant throughout the canopy layer. This l model, however, cannot be applied to a sparse canopy; even when the canopy is dense, it cannot be applied near the ground surface. In these cases, the turbulent motion is limited by the ground surface, so that l may depend upon z. Some field experiments support this dependence (e.g., Uchijima and Wright, 1964; Cionco, 1965). Also, within a vertically inhomogeneous canopy, l may depend upon z. In order to overcome the difficulty of these models, Seginer (1974) supposed that the value of l is the maximum possible under the two constraints This formula implies that the mixing length in a sparse canopy is as large as that in the atmospheric surface layer, even though the canopy elements modify the turbulent motion. Consequently, the fluxes of momentum, sensible heat and water vapor are overestimated, especially when the canopy density has a moderate value. In the present study, a new model is suggested (Section 4), which can be applied to all casessparse, moderate and dense canopies. 3. Dimensional analysis The properties of the mixing length are preliminarily investigated by means of dimensional analysis. In this section, only a vertically homogeneous canopy will be considered. a) Surface layer without a canopy The turbulent motion is limited by the ground surface only, and thus the characteristic length scale consists only of the height z. Dimensional analysis predicts that In fact, many experimental data have supported the result that b) Tall and dense canopy-the case of the intrinsic mixing length In this case, the turbulent motion is modified by the canopy elements through the action of the form drag. Thus, z is no longer important. However, the term (cda0)-1 where a0 is the leaf area density, is the significant length scale. From dimensional analysis, it is predicted that

3 April 1990 T. Watanabe and J. Kondo 229 where * is a constant that must be determined experimentally. Kondo and Akashi (1976) applied von Karman's similarity hypotheses to the exponential wind profile (Inoue, 1963), and obtained a value of *=2k3 (=0.128) for a general canopy. Wilson and Shaw (1977) found that a value of *=0.06 was needed in order to obtain the best fit between simulated and observed profiles within and above a corn canopy. c) Vertically homogeneous canopy-the general case Turbulent motion is limited by the ground surface as well as the canopy elements, so that both z and li are significant. In this case, the mixing length must be expressed as, where f is a universal function and * is the nondimensional height defined as, As * increases, the limitation due to the canopy elements becomes dominant, and the mixing length approaches its intrinsic value. As * decreases, on the other hand, the limitation due to the ground surface becomes dominant, and the mixing length approaches the value of the atmospheric surface layer. These are expressed by where dlocal is the local displacement height which is dependent on the canopy-density profile. Above the canopy layer, the local displacement height corresponds to the zero-plane displacement, i. e., The length {z - dlocal (z)} can be regarded as the mean distance through which an air parcel can move freely without being limited by the canopy elements and/or the ground surface. This distance can be evaluated as follows. The limitation imposed by the canopy elements is proportional to the product of the drag coefficient of each canopy element cd and the canopy density a(z). Therefore, the probability that an air parcel will be limited by the canopy elements when it moves downward from the level z to z - dr, is expressed as Acda(z)dr, where A is a constant. Next, Pz(r) is assigned to be the probability that an air parcel can move downward through the distance r from the level z without being limited. Then Pz (r + dr), the probability that an air parcel can reach the level z-(r+dr) without limitations, is less than Pz(r) by a factor of the probability that the air parcel will be affected by the canopy elements between the level z - r and z - (r + dr), i.e.*, e., where and The form of f (*) will be determined in the next section. 4. The new mixing-length model According to recent experimental and theoretical studies (e.g., Wilson and Shaw, 1977; Finnigan, 1979; Raupach et al., 1986; Baldocchi and Meyers, 1988a, b; Raupach, 1988), the turbulence within a canopy layer is produced through the following process. First, large scale turbulence is produced by the mean wind shear near the top of the canopy layer. This turbulence is transported into the canopy layer by the intermittent gust motion. The transported turbulence is then modified by the canopy elements through the action of the form drag, and limited by the ground surface. On the basis of this modification process, it is assumed that the mixing length within and above a canopy layer can be expressed as follows, Since the quantity on the left-hand side Pz (r + dr) can be rewritten as Pz(r) + (dpz/dr)dr, Eq. (15) yields, After integrating, the following is obtained and since Pz(0)=1, On the other hand, the probability of an air parcel that started from level z may encounter limits between z - r and z - (r + dr) for the first time is Pz(r)µ*(z - r)dr. Also, the probability that an air parcel may be limited by the ground surface without being limited by the canopy elements is Pz(z). As a result, z - dlocal (z) is expressed in the following, using the definition of the mean value;

4 230 Journal of the Meteorological Society of Japan Vol. 68, No. 2 <case a> Vertically homogeneous canopy If the canopy density has a constant value of a0, then the mixing length within this canopy layer can be expressed by Eqs. (13) and (18) as where * is the non-dimensional height (see Eq. (11)). To satisfy the demand on the mixing length Eq. (12a), it is necessary that Combining Eq. (9) and Eq. (20) yields Fig. 1. Profiles of the mixing length within and above an ideal canopy, in which leaves exist only in the upper half portion (shaded area), for six different values of cda0. Substituting Eq. (21) into Eq. (19) results in Table 1. Best fit value of the effective drag coefficient cd and the source of the observed data for each canopy This equation satisfies the demands of Eqs. (12b), (12c) and (12d). <case b> General canopy For the general profile of a(z), the mixing length can be calculated from Eqs. (13), (14), (18) and (21). But in a canopy with a dense crown, such as that of a forest, the calculated mixing length increases quickly with depth, near the bottom of the crown. To avoid this, the following restriction, such as that found in Seginer (1974), is imposed ; Note that the calculation of the l(z)-profile must originate at the level z = h, since the turbulence within a canopy layer results from the large scale eddy produced near the top of the canopy layer. If the calculation begins from level z = 0, an incorrect result will be obtained. Figure 1 shows the l(z)-profiles calculated from Eqs. (13), (14), (18), (21) and (23) for an ideal canopy with a crown, of which a(z)-profile is expressed by 5. Testing of the model 5.1 Profiles within a canopy layer The simulated profiles of the mean wind speed and the shear stress within and above a canopy layer (Eqs. (1) to (3)) are compared with observed data from a corn field, a deciduous forest and a pine forest. The sources of these data are indicated in Table 1. Each profile of the leaf area density was approximated by a smooth function and incorporated into the model (Fig. 2a-c). The effective drag coefficient cd was assumed to be constant throughout each canopy layer, and was determined so as to fit the simulated wind and shear-stress profiles to those observed (Table 1).

5 April 1990 T. Watanabe and J. Kondo 231 Fig. 2. Profiles of the leaf area density, assumed (curves) and measured (histogram or plotted points), for the corn field (a), deciduous forest (b), and pine forest (c). Fig. 3. Calculated profiles of the mixing length (solid lines) and measured Eulerian length scale (open circle: Lu, closed circle: Lw), for the corn field (a) and deciduous forest (b). Uchijima and Wright (1964) calculated a value of cd= from data observed in a corn field, while Grant (1983) measured cd= for a coniferous twig in a wind tunnel. The results indicated in Table 1 fall within the observed range. For the deciduous forest, an experimentally determined cd was not available for comparison. The profiles of the mixing length calculated by means of the present scheme for the corn field and the deciduous forest are shown in Fig. 3. In this calculation, a value of * = 2k3 (Kondo and Akashi, 1976) was adopted for Eq. (21). The plotted points denote observed values of the Eulerian length scale; Lu, Lw, which were computed by Here, *w is the standard deviation of the vertical velocity, and Tw is the Eulerian time scale, defined as the integral of the autocovariance function, i. e., where Rww (*) is the autocovariance of the vertical velocity with a time lag *. The calculated profile of the mixing length has a maximum within these canopies. This represents that the turbulence within

6 232 Journal of the Meteorological Society of Japan Vol. 68, No. 2 Fig. 4. Calculated profiles (solid lines) and measured values (plotted points) of the normalized mean wind speed, for the corn field (a), deciduous forest (b), and pine forest (c). uh denotes the mean wind speed at the top of the canopy layer and u* the friction velocity (see Section 5). these canopies is maintained by the downward transport of the large-scale turbulent kinetic energy. In the atmospheric surface layer, the mixing length is often considered to coincide with the Eulerian length scale (e.g., Tennekes and Lumley, 1972). However, as can be seen in Fig. 3, both do not coincide within the canopy layer. Similarly, Saito et al. (1970) in a corn canopy and Seginer et al. (1976) in a model canopy, found that the mixing length and the Eulerian length scale were not coincident. Figure 4 shows a comparison between the calculated and measured mean wind speed. Good agreement is found for the corn canopy (Fig. 4a), which supports the notion that K-theory can be applied to a low canopy, such as cereal crops. On the other hand, for a canopy which has a dense crown such as forests, this scheme cannot describe the secondary wind-speed maximum (Fig. 4b, c). However, it is sufficient for the present purpose to simulate the upper part of the wind profile within a canopy layer, as was mentioned in Section 1. The calculated shear-stress profile is shown in Fig. 5, along with experimental data for comparison. The agreement is good in each case, except for data at two points above the deciduous forest. Regarding this discrepancy, it should be noted that the observers commented that the instrument errors and the inhomogeneity of the topography and the treeheight affected these data. 5.2 Zero-plane displacement and the bulk momentum transfer coefficient The calculation of the zero-plane displacement and the bulk momentum transfer coefficient for a rice paddy field are compared with the present experimental data. The experiments were performed during the summers of , at the atmospheric boundarylayer observatory of Tohoku University, located in Kitaura, Miyagi Prefecture, Japan. A horizontally homogeneous rice paddy field surrounds this observatory. Sensitive cup-anemometers and ventilated psychrometers were mounted at six different levels on a 10 m-tall tower, and profiles of wind speed, air temperature and specific humidity above the canopy layer were obtained. Simultaneously, the downward shortwave and longwave radiation, along with the infrared radiative temperature of the canopy layer were measured. Fitting the profile function described by Kondo (1975) to these data, the fluxes of momentum, sensible heat and water vapor, the zero-plane displacement d and aerodynamic roughness z0 were obtained. The bulk momentum transfer coefficient CM is defined as where u* is the friction velocity, and ua is the mean wind speed at a reference level (za =10h, h : canopy height ). Above a canopy layer, the logarithmic wind

7 April 1990 T. Watanabe and J. Kondo 233 Fig. 6. Profile of the leaf area density in the rice paddy field, normalized by LAI (leaf area index). Fig. 5. Calculated profiles (solid lines) and measured values (plotted points) of the normalized shear stress, for the corn field (a) and deciduous forest (b). Fig. 7. Calculated the paddy field, friction compared velocity with u* above measure- ments. profile by formed in neutral stability conditions is given In the model simulation, Eq. (29) was fitted to the calculated wind profile, and u*, d and z0 were evaluated. CM under neutral stability conditions is derived from Eqs. (28) and (29) as Figure 6 illustrates the profile of leaf area density in the rice paddy field, which approximates the measurements by photographic techniques. This profile was incorporated into the model. The effective drag coefficient (Cd=0.18) was determined to minimize the error between the measured and calculated values of u* (Fig. 7). This value of cd is similar to the value of cd=0.21, obtained by Inoue and Uchijima (1979) also from a rice paddy field. The normalized zero-plane displacement and the bulk momentum transfer coefficientare plotted versus cd LAI (LAI : leaf area index) in Fig. 8 and Fig. 9 respectively. The solid line indicates the present model, while the dashed line indicates the previous model (Eqs. (5) and (9)). Even though the observed values are scattered in both figures, due to the difficulty in determining a precise value of d from the profile data, the present model simulates the obser-

8 234 Journal of the Meteorological Society of Japan Vol. 68, No. 2 Fig. 8. Normalized zero-plane displacement d as function of cd LAI (solid line : present mixing-length model, dashed line : Eqs. (5) and (9), plotted points: measurements). The mixing-length model within a canopy layer was presented, which can be adopted for all casesfrom a low sparse canopy to a tall dense canopy. This model is based on the fact that the turbulence within a canopy layer is modified by both the canopy elements and the underlying ground surface. For a vertically homogeneous canopy, the present model predicts that l is approximately equal to kz near the ground surface and is fairly constant at distances far enough away from the ground surface. Also, l decreases gradually as the canopy density increases. The present model was tested with field data of the wind speed and shear-stress profiles from three different canopies, along with the zero-plane displacement and the bulk momentum transfer coefficient from a rice paddy field. Good agreement between simulated and measured wind and shearstress profiles was found for all canopies, especially in the upper portion of the canopy layer. Also, some improvement was found in simulation of the zero-plane displacement and the bulk momentum transfer coefficient, in comparison with the previous model. These results support the validity of using this model to express the exchange between the atmosphere and vegetated surfaces. In the successive report, the exchange of sensible heat and water vapor will be parameterized by means of the present model. Acknowledgments We would like to thank Dr. T. Sato of Shinjyo Branch of Snow and Ice Studies, National Research Center for Disaster Prevention and our colleagues of Tohoku University for their assistance in the field observations. References Fig. 9. Same as Fig. 8, except for the bulk momentum transfer coefficient CM. vations relatively well. Especially, some improvement is apparent for the moderate leaf area density (Cd LAI=10-1 to 100). 6. Conclusions Baldocchi, D.D. and T.P. Meyers, 1988a: Thrbulence structure in a deciduous forest. Boundary-Layer Meteor., 43, Baldocchi, D.D. and T.P. Meyers, 1988b: A spectral and lag-correlation analysis of turbulence in a deciduous forest canopy. Boundary-Layer Meteor., 45, Cionco, R.M., 1965: A mathematical model for air flow in a vegetative canopy. J. Appl. Meteor., 4, Corrsin, S., 1974: Limitations of gradient transport models in random walks and in turbulence. Adv. Geophys., 18A, Cowan, I.R., 1968: Mass, heat and momentum exchange between stands of plants and their atmospheric environment. Quart. J. Roy. Meteor. Soc., 94, Finnigan, J.J., 1979: Turbulence in waving wheat. II: Structure of momentum transfer. Boundary-Layer Meteor., 16, Grant, R.H., 1983: The scaling of flow in vegetative structures. Boundary-Layer Meteor., 27, Halldin, S. and A. Lindroth, 1986: Pine forest microclimate simulation using different diffusivities. Boundary-Layer Meteor., 35, Inoue, E., 1963: On the turbulent structure of airflow within crop canopies. J. Meteor. Soc. Japan, 41, Inoue, K., 1981: A model study of microstructure of wind turbulence of plant canopy flow. Bull. Natl. Inst. Agric. Sci., Ser. A27, Inoue, K. and Z. Uchijima, 1979: Experimental study of microstructure of wind turbulence in rice and maize canopies. Bull. Natl. Inst. Agric. Sci., Ser. A26, Kondo, J., 1975: Air-sea bulk transfer coefficients in diabatic conditions. Boundary-Layer Meteor., 9, Kondo, J. and S. Akashi, 1976: Numerical studies on the two-dimensional flow in horizontally homogeneous canopy layers. Boundary-Layer Meteor., 10, Kondo, J. and A. Kawanaka, 1986: Numerical study of the bulk heat transfer coefficient for a variety of vege-

9 April 1990 T. Watanabe and J. Kondo 235 tation types and densities. Boundary-Layer Meteor., 37, Li, Z.J., D.R. Miller and J.D. Lin, 1985: A first-order closure scheme to describe counter-gradient momentum transport in plant canopies. Boundary-Layer Meteor., 33, Massman, W., 1987a: A comparative study of some mathematical models of the mean wind structure and aerodynamic drag of plant canopies. Boundary-Layer Meteor., 40, Massman, W., 1987b: Heat transfer to and from vegetated surfaces: an analytical method for the bulk exchange coefficients. Boundary-Layer Meteor., 40, Meyers, T. and K.T. PawU, 1986: Testing of a higherorder closure model for modeling airflow within and above plant canopies. Boundary-Layer Meteor., 37, Raupach, M.R., 1988: Canopy transport processes. Flow and transport in the natural environment : advances and applications, W.L. Steffen and O.T. Denmead (eds.), Springer-Verlag. Berlin, Raupach, MR., P.A. Coppin and B.J. Legg, 1986: Experiments on scalar dispersion within a model plant canopy. part I: The turbulence structure. Boundary- Layer Meteor., 35, Saito, T., Y. Nagai, S. Isobe and Y. Horibe, 1970: An investigation of turbulence within a crop canopy. J. Agr. Meteor. (Japan), 25, (in Japanese, with English summary.) Seginer, I., 1974: Aerodynamic roughness of vegetated surfaces. Boundary-Layer Meteor., 5, Seginer, I., P.J. Mulhearn, E.F. Bradley and J.J. Finnigan, 1976: Turbulent flow in a model plant canopy. Boundary-Layer Meteor., 10, Shaw, R.H., 1977: Secondary wind speed maxima inside plant canopies. J. Appl. Meteor., 16, Shaw, R.H. and I. Seginer, 1987: Calculation of velocity skewness in real and artificial plant canopies. Boundary-Layer Meteor., 39, Tennekes, H. and J.L. Lumley,1972: A first course in turbulence. The MIT Press. Cambridge, Mass., 300pp. Uchijima, Z., 1962: Studies on the micro-climate within plant communities. (2) The scale of turbulence and the momentum transfer within plant layers. J. Agr. Meteor. (Japan), 18, (in Japanese, with English summary). Uchijima, Z. and J.L. Wright, 1964: An experimental study of air flow in a corn plant-air layer. Bull. Natl. Inst. Agric. Sci., Ser. A11, Wilson, N.R. and R.H. Shaw, 1977: A higher order closure model for canopy flow. J. Appl. Meteor., 16, Wilson, J.D., D.P. Ward, G.W. Thurtell and G.E. Kidd, 1982: Statistics of atmospheric turbulence within and above a corn canopy. Boundary-Layer Meteor., 24,