## 1. Introduction

A land surface model (LSM) is one of the components of a climate model. It computes the momentum, heat, and gas exchange between the underlying surface and the atmosphere. The roughness length (RL) and zero-plane displacement (ZPD) are the external parameters in LSMs to describe the frictional effect of the underlying surface. Sensitive studies have shown that the RL has a strong impact on the modeling results (Sud and Smith 1985; Sud et al. 1988), particularly at regional scales (Henderson-Sellers and Pitman 1992). Therefore, the development of schemes to determine RL and ZPD in the land surface models is ongoing (Zeng and Wang 2007; Zilitinkevich et al. 2008; Holland et al. 2008; Colin and Faivre 2010; Gualtieri and Secci 2011).

The roughness of real terrain is often heterogeneous over the area covered by a grid square in large-scale general circulation models (GCMs). Parameterizations of land surface heterogeneity can be generally divided into discrete or “mosaic,” and continuous or probability density function (PDF) method. In the mosaic approach, the model grid square is divided into a number of subgrid cells. Surface calculation is conducted separately in each subgrid and the land–atmosphere exchange is then reaggregated on the original coarse grid square (Avissar and Pielke 1989; Giorgi et al. 2003). In PDF methods, the heterogeneous variables are represented via analytical or empirical PDFs and relevant processes are integrated over the appropriate PDF (Entekhabi and Eagleson 1989; Avissar 1992; Giorgi 1997a,b).

On the other hand, one can estimate the effect of subgrid terrain (i.e., heterogeneous terrain) on the surface flux by introducing the so-called effective roughness length (ERL). In the LSMs, a grid square is usually composed of several land-use categories. One can determine a specific ERL representing the integrated frictional effect for those different land-use categories.

In early numerical models, RL for the model grid square was generally determined by the “dominant terrain type” method because of the lack of precise data of surface features and tested methods for calculating ERL. Since in the 1980s, the studies on ERL have attracted more and more attention (Kondo and Yamazawa 1986; Andre and Blondin 1986; Taylor 1987; Mason 1988; Claussen 1990, 1991; Vihma and Savijärvi 1991; Wood and Mason 1991; Schmid and Bünzli 1995; Hasager and Jensen 1999; Zhong et al. 2003; Bou-Zeid et al. 2004, 2007; Zeng and Wang 2007).

*z*

_{0}) inside each grid square (Taylor 1987; Zeng and Wang 2007); namely,

It has been demonstrated that the ERL calculated in Eq. (1) is valid only if the RL does not vary much in a model grid. Based on the flux conservation law, Zhong et al. (2003) concluded that the ERL calculated by (1) is approximately real when the roughness step [defined by ln(*z*_{01}/*z*_{02}), where *z*_{01} and *z*_{02} are the RLs in a grid square, respectively, and suppose *z*_{01} ≥ *z*_{02}] is less than 1.2 for the “half-and-half” case.

There are some alternative methods in determining ERL. For example, Schmid and Bünzli (1995) used a mixing-length model and a *κ*–*ɛ* model to calculate the ERL for momentum based on the concept of a blending height. The microscale aggregation model is also an effective way (Mason 1988; Hasager and Jensen 1999). However, the effect of the rough-portion zero-plane displacement (RZPD) (*d*), when the high obstacles, such as tall trees in forest, city buildings, etc., exist in the grid square, is not taken into account in estimating ERL in all methods mentioned above. Therefore, it would be of particular interest to include the effect of RZPD in ERL estimation in the case of significant topological complexity (Andre and Blondin 1986).

The present study will investigate the properties of the ERL *d*^{eff}) affected jointly by RL and RZPD for a simple two-category case within a grid square. Section 2 introduces the calculation scheme based on the flux-profile relationship and the flux and mass conservation laws. The evaluation of the new scheme is presented in section 3. Finally, the concluding remarks are given in section 4.

## 2. Calculation scheme

*u*

_{*}is the friction velocity,

*θ*

_{*}is the scaling temperature,

*κ*= 0.4 is the von Kármán constant,

*P*

_{r}_{0}is the turbulent Prandtl number, and

*ψ*and

_{m}*ψ*are the integrated diabatic influence function for momentum and heat, respectively. Here

_{h}*ζ*is the stability parameter, which can be written asWith Eqs. (2) and (3), the bulk Richardson number (

*R*) can be represented byIf the functional forms for the flux-profile relationships of Businger et al. (1971) are used, then

_{ib}*ψ*and

_{m}*ψ*can be integrated to give the following forms: for the stable case (

_{h}*R*≥ 0),and for the unstable case (

_{ib}*R*< 0),where

_{ib}*ζ*= (

*z*−

*d*)/

*L*,

*ζ*

_{0}=

*z*

_{0}/

*L*, and

*β*= 4.67,

_{m}*β*= 6.35,

_{h}*γ*= 15,

_{m}*γ*= 9, and

_{h}*P*

_{r}_{0}=

*β*/

_{m}*β*= 0.74 (Businger et al. 1971). Though different values of these empirical constants have been recommended by different authors (Wieringa 1980; Högström 1988; Oncley et al. 1996), the comparison study conducted by Lo (1996) has demonstrated that the features of

_{h}*ζ*and

*ζ*

_{0}, as well as

*ψ*and

_{m}*ψ*, are very similar with different values of the empirical constants.

_{h}*R*and

_{ib}*L*, originally derived by Byun (1990) and later revised by Lo (1993), the stability parameter

*ζ*can be expressed as the following: for the stable case (

*R*≥ 0),and for the unstable case (

_{ib}*R*< 0),for

_{ib}*R*≤ −0.2098 or −0.024 97 ≤

_{ib}*R*≤ 0,and for −0.2098 ≤

_{ib}*R*≤ −0.024 97,where

_{ib}*d*

^{eff}should allow the total turbulent momentum/heat/vapor fluxes between the underlying surface and the atmosphere to be equal to the sum of those from all individual land-use categories within a grid square, which can be expressed by the approximate relationship between the effective drag coefficientswhere

*n*is the number of the terrain categories in the grid square, and

*r*,

_{i}*z*

_{0i}, and

*d*are the fraction, RL, and ZPD for the

_{i}*i*th category, respectively. The flux conservation law presented by Eq. (13) is identical to that proposed by Wooding et al. (1973), Van Dop (1983), and Vihma and Savijärvi (1991), but the reference level height (

*z*) of drag coefficient in their calculations was taken lower than 20 m. Whereas in accordance with general usage, this may be true at about 60 m height, as suggested by Wieringa (1986), for Eq. (13) will be valid with high precision at this level, especially with high obstacles within the grid square.

_{m}The mass conservation law is a useful concept in determining RL and ZPD for tall vegetation (Tajchman 1981; DeBruin and Moore 1985; Lo 1990, 1995). In this work, it will be employed to estimate the ERL and EZPD in addition to the flux conservation one.

*R*≥ 0),and for the unstable case (

_{ib}*R*< 0),where

_{ib}Eqs. (13) and (15) lead to an algebraic expression for *d*^{eff}. This two-equation system can be solved for *d*^{eff} via numerical iteration method from the fraction of all land-use categories within a grid square and the RLs and ZPDs. The resulting parameters (*d*^{eff}) show the integrated dynamics of all those land-use categories in the model grid square and allow the calculated turbulent fluxes and mass to be equal to the sum of those from all individual land-use categories inside the grid square.

*d*

^{eff}, the effective drag coefficient for momentum can be evaluated as follows:

## 3. Preliminary evaluation of the new scheme

In this section, we will give a preliminary evaluation of our new scheme for a two-category system. As pointed out by Taylor (1987), the area-weighted logarithmic average roughness length (*z*_{0m}) is a good approximation for the ERL with a small roughness step. To illustrate the difference between our ERL scheme and the approximation (*z*_{0m}), we first compared the new scheme for the simple half-and-half case. Assuming that one had *z*_{01} = 1 m and 0 ≤ *d*_{1} ≤ 30 m, *z*_{02} = 0.1 m and *d*_{2} = 0 m for simplicity. The ZPD is insignificant for a smooth surface but significant for a rough one, which is usually true in reality. The proportional coefficient used was *r*_{1} = *r*_{2} = 0.5, with *z*_{1} = 60 m as suggested by Wieringa (1986). In this case, one can have *z*_{0m} = 0.316 m. We conducted calculations and obtained results in terms of the bulk Richardson number (*R _{ib}*) in the range of −1.2 ≤

*R*≤ 0.2, which was identical to that proposed by Lo (1996), and the drag coefficient approached zero when

_{ib}*R*≥ 0.2.

_{ib}To investigate the effects of RZPD (*d*_{1}) on *d*^{eff}, we calculated the ERL and EZPD for different values of *d*_{1} and *R _{ib}.* Figure 1 shows the variation of

*R*for

_{ib}*d*

_{1}= 0, 5, 10, 15, 20, 25, and 30 m. It can be seen that the

*z*

_{0m}(=0.316 m) for all cases. The minimum ERL is approximately 0.45 m for

*d*

_{1}= 0 m with

*R*= 0.2 and the maximum is approximately 0.6 m for

_{ib}*d*

_{1}= 30 m with

*R*= −1.2. The effect of RZPD (

_{ib}*d*

_{1}) on ERL is also shown in Fig. 1. It is found that the ERL increases with

*d*

_{1}and the increase becomes more significant at higher

*d*

_{1}. It is necessary to point out that the calculated ERL shows a slight dependence on the stability parameter

*R*with a small increase with instability and shows a significant variation under near-neutral conditions. Moreover, the dependence of ERL on atmospheric stability becomes more significant with increasing

_{ib}*d*

_{1}. The maximum change in ERL under different stability conditions is approximately 0.125 m when

*d*

_{1}= 0 m; however, it becomes 0.215 m for

*d*

_{1}= 30 m.

Corresponding to the ERL estimation, the effect of RZPD (*d*_{1}) on EZPD is shown in Fig. 2. In contrast to the increase of *d*_{1} for ERL, the increase of EZPD seems to be independent of *d*_{1} because it increases approximately by 2.25 m with a 5-m increase of *d*_{1}. The estimated EZPD is also independent of atmospheric stability under the unstable conditions as seen from Fig. 2; therefore, we can have an approximate relationship *d*^{eff} ≈ 0.45*d*_{1} for the half-and-half case when *d*_{2} = 0. This approximate value of EZPD is slightly smaller than the simple area-weighted average value (*d _{m}* = 0.5

*d*

_{1}) proposed by Zeng et al. (2005) and Zeng and Wang (2007), in which the weights may relate to the sum of leaf and stem area indices. The EZPD becomes increasingly dependent on the atmospheric stability under near-neutral conditions and only slightly increasing with stability in stable cases. Note that the EZPD is no longer zero during

*d*

_{1}= 0 m for

*R*> −0.1. A numerical calculation error may be the reason behind why the calculated EZPD shows this feature for

_{ib}*R*> −0.1. Comparing with Fig. 1, it also suggests that the ERL and EZPD can adjust one another to satisfy the two-equation system (13) and (15), assuring the momentum/heat/vapor fluxes and mass conservation.

_{ib}Generally, the RL and the ZPD vary seasonally in weather and climate models, and they are assumed to be independent of atmospheric stability. However, the effect of atmospheric stratification stability on the RL and the ZPD was demonstrated in more recent studies (Zilitinkevich et al. 2008). Therefore, it is important to illustrate the impact of the stability-dependent ERL and EZPD calculated with our scheme on the effective drag coefficient, which is the ultimate representation of surface friction associated with these two parameters (Monteith and Unsworth 2007).

The effective drag coefficients varying with atmospheric stability for different *d*_{1} are shown in Fig. 3a. It shows that the effective drag coefficient increases with *d*_{1}, especially in the unstable cases. The effect of *d*_{1} becomes less apparent as *R _{ib}* approaches 0.2. Figure 3b presents the percentage errors between the effective drag coefficients and those calculated with the mean ERL and EZPD under different stability conditions shown in Figs. 1 and 2. The absolute value of the maximum percentage error is less than 2% under stable conditions and less than 1.5% under highly unstable conditions. Therefore, it can be concluded that the mean ERL and EZPD calculated with our scheme can be used as constants in LSMs under all stability conditions. Using the proposed method will introduce a small but insignificant error in drag coefficient calculation. If

*R*= −0.3 is used for the drag coefficient, the error will be minimized for the half-and-half case as shown in Fig. 3b.

_{ib}To investigate the influence of the fractional area of smooth/rough surface, the ERL and EZPD were calculated using a range of rough area fraction (*r*_{1}) values. Figure 4 displays the distributions of *d*^{eff} as a function of *r*_{1} and *d*_{1} for the cases where *z*_{01} = 1 m and *z*_{02} = 0.1 m, respectively, and *R _{ib}* = −0.3, as described in the previous section. Numerical calculation errors and the mutual adjustment of ERL and EZPD prevent the ERL from being 0.1 or 1.0 exactly when only a smooth or a rough surface exists in the grid square (Fig. 4a). It clearly shows that the effect of rough area fraction on ERL is larger than that of the RZPD since the ERL increases with

*d*

_{1}only slightly. This is especially true for the case where there is a smaller rough area fraction. Furthermore, the ERL results increase linearly with respect to the rough area fraction. The effect of the rough area fraction on the EZPD, however, is nonlinear. The EZPD increases with both the rough area fraction and the RZPD significantly (Fig. 4b). Again, the EZPD is not exactly 0 or

*d*

_{1}for a rough area fraction of 0 or 1, respectively, because of numerical calculation errors and the mutual adjustment of ERL and EZPD.

The RZPD has an enhancement effect on the ERL as shown in the previous section. It has been demonstrated that the roughness step is also a factor in magnifying the ERL (Taylor 1987; Zhong et al. 2003). For a better illustration of the enhancement effect of the roughness step on ERL, we present the ratio of ERL to the area-weighted logarithmic average *r _{a}* (=

*z*

_{01}/

*z*

_{02}) of 10, 100, and 1000, with

*z*

_{01}= 1 m and

*z*

_{02}= 0.1, 0.01, and 0.001 m, respectively (Fig. 5). This clearly shows that the roughness ratio [or the roughness step ln(

*z*

_{01}/

*z*

_{02})] has a considerable contribution to the ERL. This contribution is caused by the RZPD. The larger the roughness ratio is, the larger the enhancement effect on the ERL, which increases with ZPD significantly. In addition, as shown in previous studies (Schmid and Bünzli 1995; Hasager and Jensen 1999; Zhong et al. 2003), the maximum enhancement does not occur for the half-and-half case. It appears when the rough area fraction is 0.35–0.4 for the given roughness ratio of 10–1000; this shows that the effect of the rough area on the ERL becomes more dominant as the roughness ratio increases. From Fig. 5, it can be estimated that the maximum ERL is approximately 1.5, 3.5, and 13 times of the area-weighted logarithmic average for the roughness ratios of 10, 100, and 1000, respectively, even though the enhancement effect of RZPD is not considered (

*d*

_{1}= 0 m). However, these estimated values increase when the RZPD is taken into account (

*d*

_{1}> 0 m). In addition, the maximum ratio of

*r*is approximately 3 and 10 times when the roughness ratio is 10 times (Figs. 5a,b) and 100 times (Figs. 5a,c) for the large RZPD, respectively. These results illustrate the nonlinear dependence of

_{e}*r*on the roughness ratio.

_{e}The difference between the EZPD and the area-weighted average ZPD is shown in Fig. 6. It can be seen that the calculated EZPD is smaller than the area-weighted average one for all cases. As expected, the variability increases because of the effects of the roughness ratio and the RZPD. The maximum discrepancies are of −1.5, −2.3, and −2.8 m for the roughness ratios of 10, 100, and 1000, respectively. It also shows that the maximum discrepancy of EZPD appears when the smooth area is slightly larger than the rough one, relative to that of the maximum ratio of *r _{e}*. However, the asymmetric feature for maximum discrepancies of ERL and EZPD are independent of the RZPD.

To verify the features of the new scheme, the variations of *r _{e}* with the rough area fraction

*r*

_{1}were compared with the other models (Schmid and Bünzli 1995; Mason 1988; Hasager and Jensen 1999), in which the roughness ratio is of 100 and the effect of RZPD is not considered. As seen in Fig. 7a, all the models clearly show the dominant effect of the rough area on ERL. Among these models, the mixing-length model gives the largest ratio and the

*κ*–

*ɛ*closure model (Schmid and Bünzli 1995) the smallest. The ratio of the microscale aggregation model (Hasager and Jensen 1999) and the flux conservation model (Zhong et al. 2003) is between that of the mixing-length model and the

*κ*–

*ɛ*closure model. The corresponding variation of these ratios, when the effect of RZPD is taken into account (for a roughness ratio of 100), is shown in Fig. 7b. As expected, the ratios are larger because of the enhanced effect of a RZPD on ERL. The maximum ratio consistently occurs when the ratio of rough area to smooth area (

*r*

_{1}/

*r*

_{2}) is approximately

It is well known that the large-eddy simulation (LES) is one of the popular tools in checking the credibility of schemes for determining effective aerodynamic parameters (Glendening and Lin 2002; Bou-Zeid et al. 2004, 2007). Based on a Lagrangian scale-dependent dynamic subgrid model, a new-generation LES was applied to neutral atmospheric flow over heterogeneous land surface with a range of characteristic lengths and surface roughness values, and the ERLs were obtained with a least squares error fit using LES data for equal- (Bou-Zeid et al. 2004) and unequal-size stripes (Bou-Zeid et al. 2007). For simplicity, we will validate our scheme using the LES data from Bou-Zeid et al. (2004) and Bou-Zeid et al. (2007) directly. The simulation configuration with equal-size stripes and the ERLs from LES data (Bou-Zeid et al. 2004), as well as from our scheme, for the neutral case and area-weighted logarithmic average are listed in Table 1. It clearly shows that the ERLs of our scheme agree with the mean value of LES at a percentage error less than 15%, whereas the area-weighted logarithmic average values are much less than that of LES at a percentage error more than 20% and 65% with roughness ratios of 10 and 100, respectively. In contrast to the simulations for equal-size stripes, the LES configuration for two unequal-size stripes (Bou-Zeid et al. 2007) and the ERLs is listed in Table 2. It also shows that the ERLs from our scheme are much closer to that obtained from LES data at a percentage error less than 10% for all cases; however, the percentage error of area-weighted logarithmic average scheme is larger than 10% for most cases and the maximum percentage error (25%) occurs for simulation S67R33, which is approximately consistent with the fraction ratio

Simulation configuration and the ERLs from LES results (Bou-Zeid et al. 2004) for equal-size stripes (*z*_{0e-LES}), the scheme proposed in this paper *z*_{0m}) for the half-and-half case (*z*_{0e-LES} for four simulations with different stripe size and patch number, and *L _{p}*

_{1}and

*L*

_{p}_{2}are the patch lengths of rough and smooth surfaces; the number in the parentheses presents the percentage error of

*z*

_{0m}relative to

*z*

_{0e-LES}, respectively).

As in Table 1, but for the unequal-size stripes from Bou-Zeid et al. (2007).

The comparisons with the other models (Schmid and Bünzli 1995; Hasager and Jensen 1999; Zhong et al. 2003) and LES (Bou-Zeid et al. 2004, 2007) suggest that our scheme exhibits the same features as other models, and it can describe the enhancement effect of the roughness ratio and RZPD on ERL. Therefore, it could be an alternative scheme in the determination of ERL. Moreover, the EZPD can also be estimated synchronously, though there is no previous work for verifying the credibility of the effect of RZPD on ERL, as well as the roughness ratio on the EZPD.

## 4. Concluding remarks

The present study proposes an approach to determine the ERL and EZPD when there are high obstacles in the grid square of the LSMs. The approach is based on similarity theory of the atmospheric surface layer and on the flux conservation and mass conservation laws. This approach is an improvement on the approaches proposed in previous studies on ERL estimation because in those studies the enhancement effect of the RZPD was not taken into consideration.

By using the derived two-equation system for determining ERL and EZPD, we first calculated the ERL and EZPD for a simple half-and-half case. The results show that the ERL is larger than the area-weighted logarithmic average. The ERL also shows a slight dependence on the stability parameter, especially in the near-neutral cases. However, the EZPD is smaller than the area-weighted average and is dependent on stability except under near-neutral conditions. Because the stability dependence of the ERL and the EZPD has no significant impact on the effective drag coefficient with percentage errors less than 2%, they can be used as constants in the initialization process and kept unchanged throughout the integration period of the numerical models, unless seasonal variation is considered.

In addition to the enhanced effect of roughness ratio on the ERL and the EZPD, the impact of the RZPD on ERL and EZPD is also presented. The larger ZPD of the rough portion in the grid square is accompanied by an enhanced effect on ERL and EZPD. It also suggests that the integrated dynamic and thermodynamic effects of the underlying surface could be presented jointly by ERL and EZPD for homogeneous or heterogeneous terrains.

The rough area has a more dominant effect on ERL and EZPD. The maximum ratio of ERL to the area-weighted logarithmic average value, which appears when the smooth area is larger than the rough one, is only proportional to the roughness ratio (or the roughness step) in the grid square, showing an asymmetric feature of the half-and-half case. It approaches 0.3 when the roughness ratio is larger than 100 for the two-category case. The maximum discrepancy of EZPD occurs when the smooth area is slightly larger than the rough one. However, the asymmetric feature for maximum discrepancies of ERL and EZPD are independent of the RZPD.

A recent study by Gualtieri and Secci (2011) has shown that the RL for the specified underlying surface is not only determined by surface characteristics, but also dependent on the wind direction. They exhibited an example for three coastal cities in Italy where the RL varies with wind direction significantly besides their monthly and diurnal variations. It should be mentioned that the effect of wind direction on the ERL and EZPD is not considered directly in our scheme. The results suggest that the proposed method can give ERL and EZPD independently of wind direction. On the other hand, the wind direction effect is considered implicitly in the scheme when it is used in the land surface models because the mean wind direction is represented in a grid square of the model, and if the RL for each patch in a grid square is related to the wind direction, then the ERL as well as EZPD is also dependent on the wind direction.

It should also be noted that one limitation of our scheme is no consideration of the effect of patch scale and the texture of surface variability, which can strongly modulate the ERL as being of a “second-order roughness” relative to the regular “first-order roughness” (Schmid and Bünzli 1995)—it means that ERL and EZPD are related to the total area of each patch regardless of the patch scale in a grid square. Another limitation of the scheme comes from the assumption of the same roughness length for heat and momentum [Eq. (3)], which would result in an inevitable error because of the different stability parameter (*ζ*) for such an assumption (Lo 1996). However, the effect of thermal roughness can be taken into account using the analytical solution of flux-profile relationships for the atmospheric surface layer, distinguishing thermal roughness and momentum roughness (Lo 1996).

The authors give thanks to Dr. Elie Bou-Zeid, Princeton University, for his suggestion and for providing the LES configuration table in the revision of the manuscript, which is very important for the verification of the scheme proposed in this paper. This work is sponsored by the National Program on Key Basic Research Project of China under Grants 2010CB428505 and 2011CB952002 and the Research and Development Special Fund for Public Welfare Industry (Meteorology) of China under Grant 201206039.

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