The impact of heterogeneity of land surface roughness length on estimation of turbulent flux in model

Bin Chen , XiangDe Xu , YuGuo Ding , XiaoHui Shi

1. State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, Beijing 100081, China

2. Nanjing University of Information Science & Technology, Nanjing, Jiangsu 210044, China

*Correspondence to: Dr. Bin Chen, Institute of Space and Earth Information Science, The Chinese University of Hong Kong,Hong Kong, China. Tel: 010-68408656; Email: chenbin@cuhk.edu.hk

The impact of heterogeneity of land surface roughness length on estimation of turbulent flux in model

Bin Chen1*, XiangDe Xu1, YuGuo Ding2, XiaoHui Shi1

1. State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, Beijing 100081, China

2. Nanjing University of Information Science & Technology, Nanjing, Jiangsu 210044, China

*Correspondence to: Dr. Bin Chen, Institute of Space and Earth Information Science, The Chinese University of Hong Kong,Hong Kong, China. Tel: 010-68408656; Email: chenbin@cuhk.edu.hk

Based on preliminary theoretical analysis and numerical experiment, it is found that land surface heterogeneity plays an important role in the models turbulent flux calculation. In nearly neutral atmosphere conditions, variation coefficient of sub-scale roughness length, cell-average roughness, and reference height are main factors affecting the calculation of grid turbulent fluxes. The first factor has a determinant role on calculation deviation. The relative error generated by roughness heterogeneity could be more than 40% in some cases in certain areas (e.g., in vegetation-climate transition belt). Selecting a specific reference height may improve the calculation of turbulent flux. In stable or unstable atmosphere conditions, with sensible heat flux as an example, analysis shows that the discrepancy is correlated to the sub-grid distributions of mean wind velocity, potential temperature gradient between land surface and reference levels, and atmosphere stability near surface layer caused by the heterogeneity of land surface roughness.The calculation of turbulent flux is the most sensitive to stability in the above three factors. The above analysis shows that it is necessary to make a further consideration for the calculation deviation of the turbulent fluxes brought from land surface heterogeneity in the present numerical models.

land surface heterogeneity; sub-grid scale; grid turbulent flux; surface roughness length

1. Introduction

The major role of a land-surface model is to describe in detail the land-surface physical and dynamic processes, and to calculate accurately the momentum, energy fluxes, as well as other exchanges between the land surface and atmosphere, thus providing a reasonable coupling boundary condition for an atmosphere model. Because of a wide range of heterogeneities of natural land-surface characteristics (e.g., surface soil type, soil humidity, vegetation distribution, topography) and the highly nonlinearity of land-surface process response to these land-surface characteristics, atmospheric boundary-layer structure and motion state are always heterogeneous in various temporal and spatial scales. Therefore, a land-surface process model coupled with weather and climate models need to determine not only homogeneous-scale land-surface flux on single-point but also the area-averaged flux in the regions including various land-surface types. This corresponds to the area-flux parameterization of global circulation model grid scale (about 104-106km2).

Most boundary-layer parameterization schemes of land-surface process are based on the classical atmospheric boundary-layer theory with homogeneous underlying surface.As driven by homogeneous atmospheric conditions,land-surface fluxes calculation are usually based on grid-averaged parameters (Sun and Jin, 1997). Previous studies have shown that the sub-grid scale variations of land-surface physical parameters definitely affect all calculations of surface-atmosphere exchange (e.g., sensible heat, latent heat and momentum flux) in different degrees, and produce an additional sub-grid flux (Giorgi and Avissar, 1997; Giorgiet al., 2003;Zhanget al., 2006). During the Project for the Intercomparison of Land-surface Parameterization Schemes (PILPS) (Hendersonet al., 1996), it was found that the all climate model performed better in homogeneous rather than heterogeneous areas. If the sub-grid flux parameterization could be employed in the appropriate numerical model, the interaction between the land surface and atmosphere on complex surfaces may acquire a better description with the improvement of the heat and water exchange calculation for land-surface interface. Furthermore, the accuracy of the weather and climate simulation in complex terrain areas should be enhanced due to detailed descriptions of different thermal and dynamic processes near the land surface. Thus, the presentation of heterogeneous land surface processes will provide a better understanding of the mechanism of regional climate formation, change and prediction.

One of the major challenges for land-atmosphere sciences is to develop a performance land-surface process parameterization scheme for heterogeneous land surfaces, and to estimate accurately the mass and energy exchanges on land-atmosphere interface. At present, climate simulation and atmospheric boundary-layer studies are very active (Zhang, 1998; Mahrt, 2000;Giorgiet al., 2003; Liuet al., 2005), while numerous studies have reviewed previous worldwide research (Hu and Zuo, 2004;Chenet al., 2008).

Recently, to solve the problem of land-surface heterogeneity,researchers developed numerous parameterization schemes for heterogeneous land-surface process in numerical models. Three of the most popular methods are: the bulk transfer coefficient method (Avissar and Pielke, 1989; Zhonget al., 2003), Mosaic method (Desborough, 1999; Esseryet al., 2002; Molod and Salmun, 2002; Deryet al., 2004), and probability density function method (Liuet al., 2002a,b). Most numerical simulations indicate that the performance of weather and climate models using the above methods could improve to some degree. On the other hand, the effects of surface heterogeneity on the land-surface fluxes calculation also show different characteristics at different situations. Land-surface heterogeneity is different in the grid of land-surface process model coupled with climate model. For example, all the land-surface parameters almost not present obvious heterogeneity, including topography and geomorphology (altitude height, azimuth, terrain slope and orientation slope and aspect), physical and chemical parameters(land-surface heat, soil moisture, soil property and type), and biological ecological parameters (vegetation cover, roughness).Therefore, we cannot take into consideration all land surface heterogeneous characteristics at once in estimating land-surface gird flux. Several methods mentioned above mostly aim at the sub-grid variation rate at some certain underlying surface. How and what order of magnitude the land-surface sub-grid heterogeneity affects the models flux calculation are not clear so far.Therefore, at present, there are three continuous objectives to be met before the development of a new land surface parameterization scheme: (1) to determine the main sub-grid factors affecting the models grid flux estimation in specific cases, (2) to ascertain the order of magnitude of the nonlinear response of the models grid flux to these factors (quantitative description of heterogeneity), and (3) to explore the applicability and the possibility of improving the heterogeneity parameterization method. All these problems need further study since they are basic to the development of a land-surface parameterization scheme.

Numerous studies have been conducted on the effects of heterogeneity of land-surface physical parameters (e.g.,land-surface temperature, soil humidity) on grid water and heat fluxes, and have provided some parameterization methods(Chen and Avissar, 1994; Zhang, 2001a; Zenget al., 2002;Zhanget al., 2002, 2006; Giorgiet al., 2003; Liuet al., 2003;Wanget al., 2004; Chenet al., 2006). For example, Zhang(2001b) and Zhanget al. (2002) discussed the influence of meteorological elements heterogeneous distribution on the grid flux from the viewpoints of theory and numerical calculation, respectively. Chenet al.(2006) and Zhanget al. (2006) analyzed terrain heterogeneity’s effect on grid long-wave radiation flux.However, the sub-grid distributions of mean wind velocity,potential temperature gradient between land surface and reference levels and atmosphere stability near surface layer, which is associated with heterogeneous roughness, inevitably affect the calculation of grid turbulent flux. Whether or not we take into account this effect on numerical models, which is an interesting problem, it is of some significance for land surface physical scheme development. So far however, there are very few qualitative and quantitative analyses because of a lack of land-surface turbulent flux observations.

In this study, a method of combining theoretical analysis with numerical experiments is adopted to preliminarily investigate the influence of land-surface parameters (roughness length) heterogeneity on the calculation of surface turbulent flux based on observed data. The major purpose is to provide a theoretical basis for the development of heterogeneous land-surface parameterization.

2. The effect of land-surface heterogeneity on grid turbulent flux in model and parameterization

One approach to calculate the grid area flux is the so-called distributed method in which the large-scale mesh is divided into many finer sub-grid cells. The higher resolution provided in this method is assumed to have homogeneous land surface characteristics and every sub-grid cell flux is calculated separately. In adopting an average method (area-weight average, for example),the averaged flux in a large-scale mesh is obtained as the sum of all sub-grids. Another approach is the bulk calculation scheme.Using the area average land surface physical parameters, which is derived from the separated sub-grid tiles (e.g., arithmetic average value), the area grid flux is derived directly, while the large-scale grid is supposed to be homogeneous in geographical distribution, with derived physical parameters.

2.1. The calculation deviation of grid turbulent flux caused by heterogeneity

The vertical component of turbulent flux for variableφ(e.g.,temperature, humidity, or wind speed) can be written as

whereρis air density,δis turbulent flux parameters,CDis bulk transfer coefficient,uis the velocity at reference height,φ1andφ2denote variable values in reference height and land surface respectively, and Δφrepresents the difference (φ1－φ2) betweenφ1andφ2.

Given a model mesh and an initial time, land-surface physical parameter at every geographical coordinate point is the sum of averaged value and disturbance due to heterogeneity. Becauseρa(bǔ)ndδvary little in sub-grid scale, they are assumed as constants. Thus from formula(1), it is derived as whereWdandWare the area-averaged turbulent flux derived from distributed computation and bulk method, respectively,andR1is the discrepancy between them.

It should be noted thatW= ＜Δφ＞·＜u＞·＜CD＞ = ＜Δφ＞·＜u＞·CDis in formula(2).CDis the nonlinear function of both land-surface roughness (Z0) and Richardson number (Ri) representing atmospheric stability,i.e.,CD=f(Z0,Ri). During the actual computation process in most numerical models, ＜CD＞ is derived from grid averaged roughness () and averaged Richardson number (). Therefore, the real land-surface turbulent flux is calculated according to belowformulas

From expression(6), generally, it is found that there is usually a grid turbulent flux deviation (R1+R2) between the numerical model result (Wreal) and the "real value" (Wd), which is caused by land-surface sub-grid scale variation rate.R1is the term related to land-surface sub-grid scale bulk transfer coefficient and surface temperature and humidity mixed disturbance related items,R2is the disturbance term introduced by the nonlinear correlation between surface roughness and stability during calculating the bulk transfer coefficient. Thus, it is first necessary to estimate properlyR1andR2values so as to accurately calculate surface grid turbulent flux, and to parameterize the sub-grid flux process.

For the dynamic equations of atmosphere, a variable (temperature, humidity or wind speed) at reference level in a large-scale mesh is assumed as a constant, which is reasonable in some degree. Thus, the first and the third terms in equation(2)could be omitted. The relative errors of land-surface grid turbulence flux caused by grid heterogeneity are defined as,

According to above derivation, there are

2.2. The effect of land-surface characteristics heterogeneity under different atmospheric conditions

2.2.1 Neutral or nearly neutral atmospheric conditions

The bulk transfer coefficient in neutral or nearly neutral atmospheric conditions is written as

wherekis Karman constant and is usually 0.4;Zris reference height;Z0is land-surface roughness length. Obviously, the bulk transfer coefficient is only the function of roughness and reference height. The evolutions ofCDand other physical parameters(e.g., temperature, humidity or wind speed) do not present in(out of) phase correlation. From equation(7),Er1≈0 exists.Combine(5)and(8),Er2under nearly neutral condition is derived as

whereCVZ0is the sub-grid scale variation coefficient of grid land-surface roughness. Therefore, it is exhibited as the effect of sub-grid variation rate of surface roughness on the calculation of grid turbulent flux. The turbulent relative error (Er2) is associated with grid-scale averaged roughness, the sub-grid variation coefficient of roughness, and the selected reference height.

2.2.2 Non-neutral atmospheric condition

The main objective of this study is to estimate the effect of sub-grid roughness variation on flux calculation. The influence of spatial variation of bulk transfer coefficient on the calculation of grid turbulent flux has been demonstrated by roughness heterogeneity under neutral conditions. In a non-neutral case, the sub-grid variation rate ofRionly has modification effect on the bulk transfer coefficient. To simplify the problem, the order of magnitude of calculation deviation termEr1derived only by surface heterogeneity is considered.

Under non-neutral atmospheric conditions, the parameterization scheme for the bulk transfer coefficient in the numerical model from the European Centre for Medium-Range Weather Forecasts (ECMWF) is adopted.Acco'rding to equations(10)-(13), for sensible heat flux,CD' and Δφ(temperature variation) are out of phase. Surface heterogeneity will lead to a negative ad'ditional sub-grid flux, while for latent heat flux,CD' and Δφ(humidity variation) has not shown obviously direct correlation. It is difficult to discussEr1characteristics, therefore it is not considered in this study. Thus,what affects the calculation of grid turbulent flux in non-neutral cases is mainly the sub-grid variation of surface temperature but not the selected reference height.

3. The numerical experiments and results analyses

3.1. The effect of the sub-grid variation of land-surface roughness under neutral atmospheric condition

In equation(11)Richardson number is defined asRi=g.z2r.ΔT/.|u|), wheregis acceleration of gravity (9.8 m/s); ΔTis the difference between potential temperature at reference height in numerical model (T) and surface potential temperature;f(Ri) is the correction function. For the turbulent heat flux,

From equation(7), as the sub-grid variation ofCDand Δ are in phase, there isEr1＞0. That is to say, surface heterogeneity produces positive additional sub-grid flux, and vice versa.

It can be seen thatCVZ0, ＜Z0＞ and reference heightZrall together determine the magnitude of calculation error (Figure 1).The absolute value of relative calculation error (|Er2|) introduced by heterogeneous sub-grid distribution enhances with the increase of the variation coefficient of grid roughness. As the reference height is specified,Er2varies with grid averaged roughness length. There is a critical value of ＜Z0＞ to makeEr2=0. As＜Z0＞ is larger than the critical value, the sub-grid heterogeneity of roughness produces a positive additional sub-grid flux, and it decreases with the increase of ＜Z0＞. As ＜Z0＞ is smaller than the critical value, a negative sub-grid flux is produced and increases with the increase of ＜Z0＞. From figures 1a and 1b, both reference height and grid averaged roughness length together determines the magnitude of critical value. However,CVZ0has the decisive role on the calculation deviation. As the variation coefficient is small (e.g., ＜0.2),Er2will not be larger than 5% even if the averaged roughness length reaches 2.0 m.Er2may reach more than 50% if the variation coefficient is large (e.g., ＞1.0)with the same roughness length.

Figure 1 The distributions of relative errors Er2 with the variation coefficient of surface roughness CVZ0 and averaged roughness ＜Z0＞ at reference heights Zr=10.0 m (a) and Zr=5.0 m (b) based on equation (10)

To validate the above results, simulation experiments are performed. Figure 2 is the simulated sub-grid distribution of surface roughness with 60×60 sub-grid numbers produced by random number generator. The variation coefficient and average value of grid roughness length are 0.431 and 1.659. If taking 10 m as the reference height,there isEr2=9.81%. Figure 3 is the relative error calculated by the distributed and bulk methods by repeating the above simulated experiments 16,800 times with the same reference height and different variation coefficients and average values of roughness.

Comparing figure 3 with figure 1, the calculated results from equation(9)is extremely close to simulation, both have the same order of magnitude and distribution characteristics. The result manifests calculated results from equation(9)are completely capable of depicting the magnitude of calculation error introduced by surface heterogeneity under nearly neutral atmospheric conditions.

Figure 2 An example of sub-grid distribution of roughness length among the simulations with total sub-grid numbers equal 60×60 (units in m)

Figure 3 The same as figure 1(a), but the relative errors Er2 is the difference between the calculated by the distributed method and by the lumped method based on simulations results

In reality, the results derived from theory analysis and simulated experiment in figures 1 and 3 do not show the order of magnitude and characteristics ofEr2. Therefore, we determine the distribution of surface roughness (figure is omitted) according to vegetation type data with 1km×1km resolution (80.00417°E-119.9958°E, 20.00417°N-59.99583°N) from the University of Maryland, then calculate the variation coefficient (Figure 4) and average value (not shown) of surface roughness with a 100km×100km mesh comparable with the present atmospheric circulation numerical model. The calculated results show that the maximum variation coefficient and maximum averaged value of surface roughness are 2.9968 and 2.194 m. There is no correlation between two maximum-value areas. The strong signals of variation coefficient of surface roughness are mainly located in transition belts of various vegetations and the edge of mega relief,e.g., the transition between temperate and frigid zones of 45°N-50°N, and the edge of Tibetan plateau. The strong signals of averaged roughness mainly lie in forest areas with flourishing vegetation, such as the tropical rainforest in the south slope of Tibetan plateau, and the boreal forest of the Siberian plateau.

Figure 4 The distribution of variation coefficient of surface roughness length with grid area about 100km×100km

Figure 5 shows the distribution of the turbulent flux relative error derived from equation(10)(taking 5 m, 10 m, and 20 m as reference heights in figures 5a, 5b, and 5c, respectively). It is found that the sub-grid distribution of surface heterogeneity introduces larger calculation errors into grid turbulent flux. As reference height equals 5 m, sub-grid heterogeneity leads to a positive additional sub-grid flux. The maximum calculation relative error may reach up to and above 40%. Its distribution is related to the surface roughness and its average value, and is mainly located over the south slope of Tibetan plateau. If the reference height is 20 m, a negative sub-grid flux presents with the extremum of calculated relative error up to -20%. This indicates that reference height and surface sub-grid heterogeneity together determines the magnitude of calculated error, but its extreme mainly lies in the transition belt and the edge of different climatic zones.

Figure 5 The distribution of relative errors of turbulent fluxes calculated by real roughness length based on equation (10)(a, Zr=5.0 m; b, Zr=10.0 m; c, Zr=20.0 m)

3.2. The effect of the sub-grid variation of land-surface parameters under non-neutral atmospheric condition

By reference to the above analyses, the effect of surface grid heterogeneity on grid sensible heat flux under non-neutral atmospheric conditions is analyzed in this section.TakingZr=20.0 m andT=290K, the Richardson number varies with various ΔTandVvalues as shown in figure 6.

From equations(10)-(12)and the distribution of Richardson number, the bulk transfer coefficient is the single-valued function of ΔTif the surface roughness and reference height are known. Figure 7 shows the variation of the bulk transfer coefficient with ΔTas averaged surface roughness and reference heights are 0.3 m and 30 m.CD' and ΔT'are out of phase. The sub-grid variations of surface temperature and transfer coefficient produce a negative additional sub-grid flux.

Figure 6 The variation of corresponding Ri number under different combinations of ΔT (temperature difference between reference height and surface)and wind speed. (Taking Zr=20.0 m, T=290 K)

Figure 7 The variation of bulk transfer coefficient with ΔT under different wind velocities (unit in m/s) (Taking ＜Z0＞=0.3 m, Zr=30.0 m)

It is very difficult to determine the magnitude ofEr1in equation(7)because of the scarcity of data. Adopting the same numerical simulation method with the above analyses,given the maximum and minimum value of ΔT, a variety of random values will be generated for ΔTwhich represents the sub-grid distribution corresponding bulk transfer coefficient is calculated. ThenEr1is calculated in different backgrounds with various wind speeds. TakingZr=20.0 m,T=290K,＜Z0＞=0.3 m, the variation of |Er1| with ΔTand wind speeduare presented in figure 8a-c.

Figure 8 The variation of |Er1| with ΔT and wind speed u. (a, stable atmosphere; b, unstable atmosphere; c, the blending of stable and unstable atmospheres; (Taking Zr=20.0 m, =290K, ＜Z0＞=0.3 m)

From the calculated results, the magnitude of |Er1| presents negative correlation with wind speed at reference height (u) but positive correlation with ΔT(the temperature difference between reference height and surface). What most evidently affect calculation deviation are the sub-grid distribution characteristics of atmospheric stratification stability. In the case of stable atmospheric-stratification, the calculation error will be no larger than 10%. However, the calculation error may reach up to 90% in static-wind conditions. For the blending regions of stable and unstable atmospheres, incredible calculation errors (about 2-6 times)could be present. Note here that the calculation is from the viewpoint of numerical simulation. The real sub-grid distribution characteristics of land-surface temperature and atmospheric stability still need to be validated using observed data.

4. Conclusion and discussion

(1) The effect of sub-grid physical parameters heterogeneity on grid surface turbulent flux calculation is investigated in view of theoretical analyses and numerical simulation. This shows that land-surface sub-grid heterogeneity introduces calculation deviation of the turbulent flux calculated by grid averaged land-surface parameters (e.g., roughness length, land-surface temperature) compared to the 'real'value. This deviation is composed of two parts. One is mixed disturbance term generated by sub-grid bulk transfer coefficient associated with land-surface temperature and humidity heterogeneity. The other is caused by the nonlinear correlation between land-surface roughness and the stability as calculating bulk transfer coefficient.

(2) The simulation experiments and observed data analyses shows that the sub-grid variation rate of land-surface roughness is the main factor leading to calculation error of grid turbulent flux under neutral conditions. The magnitude of calculation deviation is associated with the variation coefficient of sub-grid roughness, the grid averaged roughness, and the selected reference height. The first factor has dominant effect. In some special regions (e.g., the transition zone of different climatic belts), the calculation error can reach up to 40%, which should be taken into account in the numerical model. Selecting specific reference heights could improve the calculation of grid flux.

(3) Under non-neutral atmospheric conditions, all the sub-grid heterogeneous distributions of potential temperature difference between reference height and surface, wind speed, and atmospheric stability near ground layer may affect the calculation of grid turbulent flux (sensible heat flux).Although in different hypothesis conditions, the effect of heterogeneity presents nonlinear variations with varying land-surface parameters, the calculation error could nearly be neglected. For the blending mesh regions of stable and unstable atmospheres, incredible calculation errors could be present.

The above results show that, for regions without large changes of atmospheric stability, the effect of the sub-grid variation of land-surface roughness length is the main factor that influences the calculation of grid turbulent flux in numerical models. However, for areas with a sharp change of atmospheric stability (such as ocean-land interface, shared areas of various land-surface vegetation types), the effects of the sub-grid variation of land-surface roughness and other parameters (e.g., land-surface temperature and humidity)should be taken into account in calculating land-surface turbulent flux and devising atmospheric numerical models.

Although the above analyses improved the calculation of land-surface grid turbulent flux and perfected the land-surface numerical model, the studies in this paper are only preliminary. Only the quantitative effect of the sub-grid variation of land-surface roughness on the estimation of real turbulent flux is investigated here. Moreover, for the influences such as land-surface temperature and atmospheric stability, the possible relative contributions are shown from the viewpoint of numerical simulation. Their quantitative analyses of the real effects on the estimation of turbulent flux still requires further study by utilizing observed data or numerical simulations (Gao and Lü, 2001; Yanet al., 2001).

This research was jointly funded by the International Sci.-Tech. Cooperative Project (2007DFB20210) funded by the Ministry of Science and Technology of the People’s Republic of China, the Key Project of Basic Scientific Research and Operation fund of Chinese Academy of Meteorological Sciences (2008Z006), the Independent Research Project of LaSW (2008LASWZI04, 2009LASWZF02).

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10.3724/SP.J.1226.2011.00031

6 April 2010 Accepted: 25 June 2010