A theory for drag partition over rough surfaces

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113,, doi: /2007jf000791, 2008 A theory for drag partition over rough surfaces Yaping Shao 1 and Yan Yang 2 Received 2 March 2007; revised 15 October 2007; accepted 15 November 2007; published 2 April [1] We present a theory for drag partition over rough surfaces of arbitrary roughness density. The total drag is partitioned into a pressure drag, a ground-surface drag, and a roughness-element-surface skin drag. The theory is simple but allows for the estimations of drag partition functions, friction velocity, zero-displacement height, and roughness length. The model estimates of these quantities are compared with observations and the model is found to perform well. The theory explains several known facts from observations such as the dependency of aerodynamic roughness length on roughness density. It is shown that drag partition is governed entirely by two functions f r and f s which represent the dependencies of surface drag coefficient and the roughness drag coefficient on roughness density. Under the condition of f r = f s, the Raupach (1992) model is derived without assumptions in addition to the drag laws. Citation: Shao, Y., and Y. Yang (2008), A theory for drag partition over rough surfaces, J. Geophys. Res., 113,, doi: /2007jf Problem of Drag Partition [2] A rough surface can be considered as a relatively smoother surface superposed with roughness elements. One of the quantities for characterizing the rough surface is the roughness density (or roughness frontal area index), l l ¼ na f where n is the roughness-element number density (number of roughness elements per unit area) and A f is the frontal area of a roughness element. For elements with rectangular frontal areas, l can be expressed as l = nbh with b and h being the width and height of the roughness elements, respectively. The total drag on the rough surface, t, can be expressed as t ¼ t r þ t s þ t c where t r is the pressure drag, t s is the ground-surface drag, and t c is the roughness-element-surface drag which arises due to the skin friction on the surface of the roughness element. If l is small t c can be neglected. In several previous studies [e.g., Schlichting, 1936; Wooding et al., 1973; Arya, 1975; Raupach, 1992], t is written as t ¼ t r þ t s [3] Equation (3) is probably an appropriate simplification for many natural surfaces, as t r is usually the dominating 1 Institute for Geophysics and Meteorology, University of Cologne, Cologne, Germany. 2 Canada Centre for Remote Sensing, Natural Resources Canada, Ottawa, Ontario, Canada. Copyright 2008 by the American Geophysical Union /08/2007JF ð1þ ð2þ ð3þ term for a wide range of l values. However, for surfaces with sufficiently large l, equation (3) becomes increasingly less adequate. [4] In a drag partition theory, we make two interrelated statements. The first concerns the behavior of drag partition functions, e.g., t r /t as a function of l, and the second concerns the behavior of u * /U, where u * is friction velocity and U is mean wind speed at a reference height, or equivalently a statement on the behavior of roughness length z 0. The theory of drag partition has applications to the studies of scalar transfer in urban canyons [Yang and Shao, 2005], sediment transport over Aeolian surfaces [Raupach et al., 1993; Crawley and Nickling, 2003] and land surface modeling. [5] Raupach [1992] developed a drag partition theory. The main results are t r t ¼ bl 1 þ bl t s t ¼ 1 1 þ bl ð4aþ ð4bþ where b = C r /C s with C s being the drag coefficient for the ground surface and C r the drag coefficient for the roughness element. Equation (4) is an elegant, simple and robust model and is supported by measurements. Raupach et al. [1993] applied the scheme to determining the threshold friction velocity for wind erosion, u *t and found that the ratio R t = u *ts / u *tr, with u *ts being the threshold friction velocity for the smooth surface (the surface without the roughness elements) and u *tr for the rough surface (the same underlying surface with roughness elements), can be expressed as R 2 1 t ¼ ð1 malþð1 þ mblþ ð5þ 1of9

2 where a is the ratio of roughness-element basal area to frontal area, and m an empirical parameter. [6] Despite its success, the Raupach model encounters the following three problems. [7] 1. The validity of equations (4a) and (4b) is limited to about l 0.1. In practice, we have rough surfaces (e.g., urban area) with much larger roughness densities. [8] 2. In deriving equation (4), Raupach made two hypotheses, one on the effective sheltering area and volume associated with a roughness element and the other on the superposition of the areas and volumes. For surfaces with large l, it is not clear how the effective sheltering areas and volumes can be evaluated and how they superpose due to the complicated interactions of turbulent wakes associated with the roughness elements. Shao and Yang [2005] attempted to extend equation (4) to rough surfaces with large roughness densities by introducing an effective frontal area index, l e, but retaining the key ingredients of the Raupach model. The idea of l e is sound, but the method for determining l e requires further consideration and validation with experimental data. [9] 3. While equation (5) is supported by experimental data [e.g., Wolfe and Nickling, 1996], there are uncertainties in the values of b and m. Both b and m depend on the roughness-element aspect ratio [Crawley and Nickling, 2003]. The value of C r also requires further study, especially for surfaces with porous roughness elements [Gillies et al., 2000]. [10] In this study, we propose a theory for drag partition over rough surfaces of arbitrary l. The basic problems to be tackled in this study and those in the work of Shao and Yang [2005] are the same. In comparison to the latter paper, we propose a very different formulation of the drag partition theory. The idea of effective frontal area index, as proposed by Shao and Yang [2005], is not used. Instead, we first show that the Raupach model can be derived without assumptions in addition to the drag laws. We then show that the essence of the drag partition theory is framed by two assumptions, one on the mean wind profile and the other on the eddy diffusivity in the roughness sublayer. Formally, these assumptions appear as modification functions to C r and C s, namely, f r and f s. The theory is verified with published data. The theory presented in this study has a very simple formulation and is easily applicable to practical problems. 2. Simple Derivation of the Raupach Model [11] Raupach [1992] introduced the concept of effective sheltering area, A, and effective sheltering volume V associated with an individual roughness element, and made two hypotheses. The first hypothesis is A ¼ c 1 b h U=u * V ¼ c 2 b h 2 U=u * ð6aþ ð6bþ [12] To obtain equation (4), c 1 = c 2 is assumed, which implies V ¼ h A [13] As we shall see later, the assumption c 1 = c 2 and equation (7) have far-reaching consequences. The second ð7þ hypothesis is that the effective sheltering areas and volumes can be randomly superposed. This hypothesis is more difficult to understand. The difficulty lies in that U/u * in equation (6) is not known, so each time when a roughness element is added to the surface U/u * is changed. [14] Equation (4) shows that given l, drag partition is governed entirely by b. This indicates that it can be derived without the detailed hypotheses involving effective sheltering. For instance, we can abandon the second hypothesis of Raupach [1992]. Suppose associated with a roughness element is an effective sheltering area A, then t s ¼ rc s U 2 ð1 naþ ð8þ [15] Similarly, the pressure force exerted on the n roughness elements can be written as f ¼ n r C r b h U 2 1 n V 1 h [16] We emphasis that if equation (7) is correct, then we have t r ¼ rc r lu 2 ð1 naþ ð9þ [17] Equation (4) is merely a rearrangement of equations (3), (8), and (9). [18] Two comments are appropriate to the above derivation of equation (4). First, we have made no hypotheses apart from the drag laws t s rc s U 2 f rc r lu 2 [19] Second, equations (8) and (9) are identical to equations (13) and (17) of Raupach [1992] for small l. Taking equation (8) as example and making use of A in equation (6), we obtain t s ¼ rc s U 2 1 c 1 l U ð10þ u which takes the same form as the Taylor expansion of equation (13) of Raupach [1992] for small values of l. [20] More generally, we write t s ¼ rf s C s U 2 t r ¼ rf r C r lu 2 ð11aþ ð11bþ where f r and f s are functions of quantities related to roughness elements. Equation (4) follows from equations (3) and (11) if 3. General Formulation f s ¼ f r ð12þ [21] Equation (4) is valid for small l only. For example, as l!1, we expect t r /t! 0, but equation (4a) gives t r /t! 1. This is partially because in deriving equation (4), 2of9

3 Figure 1. A schematic illustration of three situations of a rough surface. (a) Bare surface with roughness length z 0s, (b) open canopy of roughness elements with zero-displacement height d < h and roughness length z 0, and (c) close canopy of roughness elements with zero-displacement d = h and roughness length z 0sc. we have neglected t c and partially because equation (12) cannot be correct for all l. Consider the situation illustrated in Figure 1 and suppose a surface of roughness length z 0s is superposed with roughness elements of breadth b, height h, and basal area q ( b 2 ). The roughness element aspect ratio is s = h/b. The frontal-area index l and basal-area index h (or fraction of cover) are l ¼ nbh 0 l s ð13aþ h ¼ nq 0 h 1 ð13bþ respectively [22] The maximum of l is s because the maximum of n is q 1 (b 2 ). In general, the total drag can be partitioned into three components following equation (2), which can be expressed as t r ¼ rf r C rg lu 2 ð1 hþ ð14aþ closed then the drag coefficient on the surface of the roughness element is C sgc ¼ k 2 ln 2 z w h z 0sc ð16þ where z 0sc is the roughness height of the closed canopy. It is not exactly known how the drag coefficient on the surface of the roughness element varies with l and h. However, we expect that this coefficient does not vary over a wide range. For simplicity, it is assumed to vary linearly with h from C sg and C sgc and it follows that f c ¼ 1 þ C sgc 1 h C sg [23] The drag partition functions can now be written as t r t ¼ 1 þ f r f 1 s f r fs 1 bl bl þ f c fs 1 h= ð1 hþ ð17þ ð18aþ t s ¼ rf s C sg U 2 ð1 hþ ð14bþ t s t ¼ 1 1 þ f r f 1 bl þ f c f 1 h= ð1 hþ s s ð18bþ t c ¼ rf c C sg U 2 h ð14cþ where C rg and C sg are the values of C r and C s at zero l and h and f r and f s are functions of l and h representing the modifications to C r and C s arising from the sheltering of the roughness elements. The consideration of expressing t c as equation (14c) is as follows. Because the roughness length for the underlying surface is z 0s, then C sg is C sg ¼ k 2 ln 2 z w z 0s ð15þ t c t ¼ f c fs 1 hð1 h 1 þ f r fs 1 bl þ f c fs 1 h= ð1 hþ Þ 1 ð18cþ with b = C rg /C sg which is the same as the b in equation (4). Here we merely used a slightly different notation because we wish to emphasis that in general the ratio C r /C s depends on l and h rather than a constant. For any rational choices of f r and f s, equation (18) satisfies the limits t r t ¼ 0; t s t ¼ 1; t c t ¼ 0 for l ¼ 0 where k is the von Karman constant and z w is the reference height at which U is taken. When the roughness canopy is t r t ¼ 0; t s t ¼ 0; t c t ¼ 1 for l ¼ l max 3of9

4 h would have different f r and f s ; third, both f r and f s must be zero if h = 1. On the basis of these considerations, we propose " # f r ¼ exp a rl ð1 hþ k ð21aþ " # f s ¼ exp a sl ð1 hþ k ð21bþ Figure 2. Dependency of f r and f s on l assuming s = 1 for a r =3,a s = 5, and k = 0.1. [24] Again, for f r f 1 s = 1 and h! 0, equation (18) reduces to equation (4). [25] The drag partition theory is incomplete without the specific functional forms of f r and f s because in general f r and f s are not identical and because these functions determine u * /U (or z 0 ). It follows from equations (2) and (14) that u 2 U 2 ¼ f rlð1 hþc rg þ ½f s ð1 hþþf c hšc sg ð19þ [26] Equation (19) shows that u * 2 /U 2 is a weighted average of C rg and C sg, with the weights being f r l(1 h) and f s (1 h) +f c h, respectively. It can also be written as k 2 ln 2 z w d z 0 ¼ f r lð1 h ÞC rg þ ½f s ð1 hþþ f c hšc sg ð20þ [27] Equation (20) shows that f r and f s also determine the roughness length, z 0, for given zero-displacement height, d. [28] It is not necessary to derive universally correct and exact expressions of f r and f s on the basis of detailed fluid dynamic processes. A kinematical approach is simpler. We argue that in general, f r and f s must depend on roughness Reynolds number and parameters such as l and h. With fixed wind speed, an increase in l would result in enhanced interferences among the turbulent wakes generated by the roughness elements and thus a decrease in pressure drag on an individual roughness element. Further, the increase in roughness density would lead to increased sheltering and reduce the surface drag. Thus we expect both f r and f s to decay with l. Several considerations can be made on the choices of f r and f s. First, f s must decay faster with l than f r because the sheltering effect on the surface is stronger than that on the roughness elements; second, both decay rates must be affected by the aspect ratios of the roughness elements, i.e., two surfaces with the same l but different where a r, a s, and k are parameters to be determined by fitting equation (21) to data. As there is a lack of observations for determining these parameters, we have carried out large eddy simulations of flow over rough surfaces with various roughness densities. On the basis of the numerical simulations, we propose to select a r =3,a s = 5, and k = 0.1. The inclusion of the (1 h) k term ensure f r = f s = 1 for l = 0 and f r = f s = 0 for h = 1, at the same time as k is set to a small value, the exponential decays of f r and f s are not substantially affected. Also, a s is larger than a r,sof s decays more rapidly than f r. Figure 2 is a plot of f r and f s against l for s = 1. Additional experimental dada are required to better estimate f r and f s. 4. Comparison With Data [29] The predicted u * /U using equation (19) is shown in Figure 3. For small l, the predictions fit well to the data of Raupach et al. [1980] and O Loughlin [1965] and are almost identical to the predictions using the Raupach [1992] model. For l > 0.2, Raupach [1992] argued that u * /U approaches a constant. Our model is different and predicts that u * /U reaches a maximum at around l = 0.2 and then decays with l. At l = l max, (u * /U) 2 must satisfy equation (16). The maximum of u * /U at around l = 0.2 is understandable from equation (19), because at this l the Figure 3. Variation of u*/u with l, calculated using equation (19) with z 0s /h and z 0sc /h set to , z w /h = 1.5, and b = 150. The data of Raupach et al. [1980], RTE1980, and O Loughlin [1965], OL1965, are used for comparison. The estimates of Raupach [1992], R92, are shown for comparison. 4of9

5 Figure 4. Comparison of pressure drag partition (r r = t r /t) calculated using equation (18a) with the data of Marshall [1971] and the Raupach [1992] model; (b) as in Figure 4a but for surface drag partition (r s = t s /t) calculated using equation (18b); and (c) dependencies of r c, r, and r c = t c /t on l. The symbols are the estimates from a large eddy simulation. In the calculations, b = 150 and s = 1 are assumed. first term on the right hand side of the equation carries more weight, i.e. the pressure drag dominates the momentum transfer and hence for given U, u * is the largest. At other l values, ground surface and roughness-element-surface drags become more important and hence, for given U, u * is smaller. [30] Figure 4a shows the comparison of t r /t calculated using equation (18a), with f r and f s calculated using equations (21a) and (21b), with the data of Marshall [1971]. As expected, the estimates agree with the measurements for small l. Our model is almost identical to the Raupach model for l < 0.1, as can be seen from Figure 4a and 4b. Differences between the two models occur for larger l. Our scheme requires t r /t! 0asl! l max, accompanied by t s / t! 0 and t c /t! 1, as shown in Figure 4c. We are not aware of observed data for verifying these scheme estimates for large l. Again, we have conducted large eddy simulations of flow over regular arrays of roughness elements with various roughness densities and then estimated drag partitions from the numerical results. The procedure for the large eddy simulation has been described in the Appendix of Shao and Yang [2005]. As Figure 4c shows the numerical results support our drag partition scheme. [31] It has been shown by Shao and Yang [2005] that the zero-displacement height can be estimated as d h ¼ t r pffiffi t c h þ t t ð22þ Figure 5. Comparison of d/h estimated using equation (22) with the wind tunnel data (symbols) of Hall et al. [1996], with open circles representing the square array case and full dots representing the staggered array case. [32] In Figure 5, d/h estimated by using equation (22), with t r /t, t s /t, f r and f s estimated using equations (18a), (18b), (21a), and (21b), respectively, is compared with the wind tunnel data of Hall et al. [1996]. The comparison is good. [33] As Equation (20) shows that z 0 can be estimated as z 0 h ¼ z w h d exp ku h u ð23þ 5of9

6 elements are cubes of 0.01 m in height. For both the staggered array case and the square array case, z w =2h, z 0s = h and z 0sc = h are set. For the staggered array case (case 1), b = 120 is used, while for the square array case (case 2), b = 80 is used. A comment on the choice of z 0s is warranted. The roughness Reynolds number R r R r ¼ u * n z 0s can be used to define whether a surface is smooth or rough (n is the kinematic viscosity of air). If R r < 0.11, the surface can be considered to be smooth and the relationship between z 0s and u * is approximately z 0s ¼ 0:11 n u * Figure 6. Comparison of z 0 /h estimated using equation (23) with the wind tunnel data of Hall et al. [1996], HMWS1996, for the square-array and staggered-array cases of cubic roughness elements of 0.01 m. (a) For both the staggered and square array cases, z w =2h, z 0s = h and z 0sc = h are set. For case 1 and case 2, b = 120 and b = 80 are used, respectively. The data from Liedtke [1992], L1992, Raupach et al. [1980], RTE1980, and O Loughlin and MacDonald [1964], OM1964, are also shown for comparison. (b) For both the staggered and square array cases, b = 100, z 0s = h and z 0sc = h are set. For case 3 and case 4, z w =2.5h and z w = 1.5h are used, respectively. The estimates of z 0 /h using equation (23a) and the model of Macdonald et al. [1998], MGH1998, are also shown. For the staggered and the square array case, a = 1 and a = 0.55 are used, respectively. with the u * /U relationship given by equation (19). In principle, z 0 should not depend on the choice of the reference height, z w, as long as z w is within the log layer. However, as both C sg and C rg vary with z w, b is related to z w. Although z w appears in equation (23), if C sg, C rg, and b are consistently calculated, the choice of z w should not affect the calculation of z 0. Observations indicate that the height of the log-layer varies from case to case. According to Cheng and Castro [2002], when the wind profile is spatially averaged the log-layer extends down to the level of h. In practice, z w may be chosen between h and 4h. We suggest to set z w =2h. In Figure 6a, we use the wind tunnel data of Hall et al. [1996] together with several other published data sets to verify equation (23). The wind tunnel experiments of Hall et al. were conducted for square and staggered arrays of roughness elements. The roughness [34] This result can be incorporated into our model for determining z 0s if the underlying surface is indeed smooth. However, it is not necessary to assume the underlying surface is smooth in this sense; rather it is advantageous to treat it as one composed of another underlying surface superposed with smaller rough elements (e.g., low buildings in a city with high risers or shrubs among forest trees). We thus consider z 0s as a tunable parameter which ensures that z 0 approaches the right limit as l approaches 0. The same argument applies to z 0sc which is chosen to ensure that z 0 approaches the right limit as l approached l max (in our case one). Figure 6a shows that the behavior of z 0, as observed in the wind tunnel, is reproduced by the model, i.e., z 0 first increases with l to a maximum at around l = 0.2 and then decreases to z 0sc as l further increases. More data showing similar behavior of z 0 can be found in the work of Wooding et al. [1973], Raupach et al. [1980, 1991], and Hatfield [1989]. The wind tunnel observations further show that z 0 varies from case to case, e.g., z 0 for the staggered array case is considerably larger than for the square array case of roughness elements. This is caused by the fact that the staggered array case generates a deeper mixing layer, and therefore, by choosing different b values, the variations for both cases can be well represented. [35] We can fix b and vary z w to achieve the same goodness of fitting equation (23) to the data. In the tests shown in Figure 6b, we have set b = 100, z 0s = h and z 0sc = h for both staggered and square array cases. For the staggered array case (case 3), z w =2.5h is assumed and for the square array case (case 4), z w =1.5h. A comparison of Figures 6a and 6b confirms that the agreement between the model and the observation is equally as good. [36] Several empirical relationships between z 0 and l have been proposed (see Macdonald et al. [1998] and Shao and Yang [2005] for a summary). By accounting for the displacement height in the z 0 equation derived by Lettau [1969], Macdonald et al. [1998] found that z 0 h ¼ 1 d ( exp 0:5a C d h k 2 1 d ) 0:5 l h ð23aþ 6of9

7 b is C rg /C sg and b is not a constant but variable depending on the characteristics of the roughness elements as well as z 0s and h. The studies of Raupach et al. [1993], Wolfe and Nickling [1996], and Wyatt and Nickling [1997] suggest that b broadly varies between 90 and 200. Part of this variation is attributed to the dependency of C rg on the type of roughness elements. For example, for cylinders with Reynolds number in the range of 10 3 to 10 5 C rg is a constant of about 0.25 to 0.3 [Taylor, 1988], while for vegetation C rg is in the range of 0.3 to 0.9 [Grant and Nickling, 1998]. [39] Further, b must also depend on the ratio of z 0s /h. To illustrate this, we consider a roughness element of height h located on a surface with roughness length z 0s. Suppose wind is logarithmic. Then C rg can be interpreted as C rg U 2 r ¼ 1 h Z h z 0s C d U 2 ðþdz z ð24þ Figure 7. Sensitivity of drag partition to roughness element aspect ratio, s. (a, b, and c) z 0 /h, u * /U, and the drag partition functions given by equation (18) are plotted, respectively. where C d is drag coefficient, k is the von Karman constant, and a is an empirical constant of about 0.55, which accounts for the variable obstacle shapes and flow conditions. The displacement height, d, is given by d h ¼ 1 þ M l ðl 1Þ ð22aþ [37] The best fit to the wind tunnel data of Hall et al. [1996] is achieved by setting M = 4.43 for the staggered arrays and M = 3.59 for the square arrays of roughness elements. On average, M is approximately 4. As shown in Figure 6b, equation (23a) also quite well fits to the data of Hall et al. [1996], except probably for the region of l > 0.7. It is interesting to note that equations (23) and (23a), although derived through different approaches, are indeed quite similar. 5. Discussions 5.1. Parameter b [38] As can be directly seen from equations (4) and (18), the impact of b on drag partition is significant. By definition where U r is the mean wind at the reference level and C d is the drag coefficient for the same roughness element in a uniform flows (which depends on roughness-element Reynolds number). Assuming z 0s h, U(z) is logarithmic, C d is constant (if the roughness-element Reynolds number is sufficiently large) and C sg = u* 2 /U r 2, by evaluating the integral on the right hand of equation (24), we obtain ( b ¼ C ) d h 2 ln 1 þ1 k 2 z 0s ð25þ [40] Thus, b must be a function of z 0s /h. It follows that for fixed b, b depends on s. We now examine the drag partition for the three cases listed in Table 1. The results of the sensitivity tests are shown in Figure 7. The model shows that the dependencies of the drag partition functions, z 0 / h and u * /U on l can be very different for different values of s. These differences are quite obvious for l > 0.1. Thus, it is insufficient in general to assume that drag partition only depends on l with b being an empirical constant. Rather, drag partition must depend both on l and h with b estimated from equation (25) Functions f r and f s [41] It is now clear that drag partition is entirely determined by C r and C s, which we have expressed formally as C r ¼ C rg ðh; z w Þf r ðl; hþ ð26aþ C s ¼ C sg ðh; z w Þf s ðl; hþ ð26bþ Table 1. Parameters Used for Sensitivity Tests of Drag Partition to Roughness Element Aspect Ratio a Case 1 Case 2 Case 3 z 0s /h z w /h s b a For all three cases, h = 0.01 m is assumed; b is calculated from equation (25) by assuming C d /k 2 = (this value is selected so that b = 150 for case 1). 7of9

8 [42] Assuming z w =2h, we obtain C r ¼ C rg ðþ h f r ðl; hþ ð26a 0 Þ C s ¼ C sg ðþf h s ðl; hþ ð26b 0 Þ [43] Thus, the key to drag partition is to determine f r and f s as functions of l and h. On the other hand, we can express C r and C s as C r ¼ 1 Ur 2 1 h Z h 0 C d U 2 ðþdz z where the overline represents the horizontal average, and C s ¼ z Ur 2 K j z¼0 where K 0 is the eddy diffusivity close to the ground surface. The mean wind, U is in general not logarithmic in the roughness sublayer. According to Raupach et al. [1991], it is fairly well approximated by the empirical exponential wind profile UðÞ z h Uh ð Þ ¼ exp a r 1 z i h where the coefficient a r tends overall to increase with l, but with considerable scatter. So ultimately, embedded in f r and f s are the information of how the mean wind profile and eddy diffusivity vary with roughness density. This observation is not surprising but not necessarily obvious in the first instance. The circle is now complete; namely, to be able to do drag partition, we must know the profiles of mean wind and turbulence for a given roughness configuration. 6. Conclusions [44] We have presented a model for drag partition over rough surfaces of any roughness density. In our model, the total drag is partitioned into three components, namely, the pressure drag, t r, ground surface drag, t s, and roughnesselement surface drag, t c. The model allows the calculations of dimensionless quantities, t r /t, t s /t and t c /t via equation (18), u * /U via equation (19), d/h via equation (22), and z 0 /h via equation (23). The model predictions are verified with data, and the model performance is good. The new model explains several facts known from observations, e.g., z 0 first increases with l from z 0s (for ground surface) to a maximum at around l = 0.2 and then decreases with l to z 0sc (for closed canopy of roughness elements). [45] Drag partition is governed entirely by the ratio of C r /C s and the variations of C r and C s with l and h. We have expressed C r and C s using equation (26), in which C rg and C sg are functions of the reference height z w (which is proportional to h) and f r and f s are exponential functions given by equation (21). The behavior of f r and f s boil down to those of mean wind and eddy diffusivity in the roughness sublayer. In general f s decays faster than f r with roughness density. The Raupach [1992] model can be derived without assumptions in addition to the drag laws stated in equation (11) under the condition f r = f s. Notation A effective shelter area, L 2. b width of roughness element, L. C r drag coefficient of roughness element. C rg value of C r at l =0. C s drag coefficient of surface. C sg value of C s at l =0. d zero-plane displacement height, L. h height of roughness element, L. K 0 eddy diffusivity, L 2 T 1. n roughness element number density. q basal area of roughness element. U wind speed at reference level, LT 1. u * friction velocity, LT 1. u *ts threshold friction velocity for smooth surface, LT 1. u *tr threshold friction velocity for rough surface, LT 1. V effective shelter volume, L 3. z w reference height, L. z 0 aerodynamic roughness length, L. z 0s roughness length for underlying surface, L. z 0sc roughness length for closed canyon, L. a ratio of roughness-element basal area to frontal area. k von Karman constant. l frontal area index. l e effective frontal area index. l max maximum frontal area index. r air density, ML 3. h basal area index. b ratio of C r /C s. t total drag, ML 1 T 2. t c roughness-element surface drag, ML 1 T 2. t r pressure drag, ML 1 T 2. t s ground surface drag, ML 1 T 2. s roughness element aspect ratio. References Arya, S. P. S. (1975), A drag partition theory for determining the large-scale roughness parameter and wind stress on the Arctic Pack Ice, J. Geophys. Res., 80, Cheng, H., and I. P. Castro (2002), Near wall flow over an unban-like roughness, Boundary Layer Meteorol., 104, Crawley, D. M., and W. G. Nickling (2003), Drag partition for regularlyarrayed rough surfaces, Boundary Layer Meteorol., 107, Gillies, J. A., N. Lancaster, W. G. Nickling, and D. M. Crawley (2000), Field determination of drag forces and shear stress partitioning effects for a desert shrub (Sarcobatus vermiculatus, Greasewood), J. Geophys. Res., 105, 24,871 24,880. Grant, P. F., and W. G. Nickling (1998), Direct field measurement of wind drag on vegetation for application to windbreak design and modelling, Land Degrad. Develop., 9, Hall, D. J., R. Macdonald, S. Walker, and A. M. Spanton (1996), Measurements of dispersion within simulated urban arrays - A small scale wind tunnel study, BRE Client Rep. CR 178/96, Building Res. Estab., Garston, Watford, UK. Hatfield, J. L. (1989), Aerodynamic properties of partial canopies, Forest Meteorol., 46, Lettau, H. (1969), Note on aerodynamic roughness parameter estimation on the basis of roughness element description, J. Appl. Meteorol., 8, Liedtke, J. (1992), Experimentelle Untersuchung des Ausbreitungsverhaltens eines gasformigen Stoffes in neutral und instabil geschichteten turbu- 8of9

9 lenten Gleichdruckgrenzschichten bei unterschiedlicher Bodenrauhigkeit, Ph.D. thesis, Univ. der Bundeswehr, München, Germany. Macdonald, R. W., R. F. Griffiths, and D. J. Hall (1998), An improved method for the estimation of surface roughness of obstacle arrays, Atmos. Environ., 32, Marshall, J. K. (1971), Drag measurements in roughness arrays of varying density and distribution, Agric. Meteorol., 8, O Loughlin, E. M. (1965), Resistance to flow over boundaries with small roughness concentrations, Ph.D. dissertation, Univ. of Iowa, Iowa City. O Loughlin, E. M., and E. G. MacDonald (1964), Some roughnessconcentration effects on boundary resistance, Houille Blanche, 7, Raupach, M. R. (1992), Drag and drag partition on rough surfaces, Boundary Layer Meteorol., 60, Raupach, M. R., A. S. Thom, and I. Edwards (1980), A wind tunnel study of turbulent flow close to regularly arrayed rough surfaces, Boundary Layer Meteorol., 18, Raupach, M. R., R. A. Antonia, and S. Rajagopalan (1991), Rough-wall turbulent boundary layers, Appl. Mech. Rev., 44, Raupach, M. R., D. A. Gillette, and J. F. Leys (1993), The effect of roughness elements on wind erosion thresholds, J. Geophys. Res., 98, Schlichting, H. (1936), A wind tunnel study of turbulent flow close to regularly arrayed rough surfaces, NACA Tech. Memo., 823, Shao, Y., and Y. Yang (2005), A scheme for drag partition over rough surfaces, Atmos. Environ., 39, , doi: /j.atmosenv Taylor, P. A. (1988), Turbulent wakes in the atmospheric boundary layer, in Flow and Transport in the Natural Environment: Advances and Applications, edited by W. L. Steffen, and O. T. Denmead, pp , Springer, Berlin. Wolfe, S. A., and W. G. Nickling (1996), Shear stress partitioning in sparsely vegetated desert canopies, Earth Surf. Processes Landforms, 21, Wooding, R. A., E. F. Bradley, and J. K. Marshall (1973), Drag due to regular arrays of roughness elements of varying geometry, Boundary Layer Meteorol., 5, Wyatt, V. E., and W. G. Nickling (1997), Drag and shear stress partitioning in sparse desert creosote communities, Can. J. Earth Sci., 34, Yang, Y., and Y. Shao (2005), A scheme for scalar exchange in urban boundary layers, Boundary Layer Meteorol., 120, , doi: /s Y. Shao, Institute for Geophysics and Meteorology, University of Cologne, D Cologne, Germany. (yshao@meteo.uni-koeln.de) Y. Yang, Canada Centre for Remote Sensing, Natural Resources Canada, 588 Booth Street, Ottawa, ON, Canada K1A 0Y7. 9of9