## 1. Introduction

The turbulent exchange process plays an important role in land–atmosphere interaction. Accurate representation of the eddy fluxes is essential for understanding the energy transfer in the land–atmosphere system and for successful numerical simulations of the atmosphere processes (Lee 1997). Generally, the momentum and heat fluxes are estimated by the Bowen ratio energy balance (BREB) method (Fritschen and Simpson 1989) and the profile method based on the Monin–Obukhov similarity theory (MOST). It is well known that the BREB, which estimates surface heat fluxes on the basis of measurements of temperature and specific humidity gradients and surface energy budget, becomes computationally unstable and results in spurious large values when the Bowen ratio is in the vicinity of −1. Meanwhile, many attempts have been made to develop and improve the profile method. Some flux-profile relationships suitable for various conditions of atmospheric stability have been well documented (Businger et al. 1971, hereinafter B71; Dyer 1974, hereinafter D74; Beljaars and Holtslag 1991, hereinafter BH91; Högström 1996, hereinafter H96; Cheng and Brutsaert 2005; Yang et al. 2008; Song et al. 2010, hereinafter S10). Despite all these efforts, the estimated fluxes are still scattered compared with observations. The disagreements mainly result from the lack of consideration of the energy budget, the limitation of the MOST over a flat homogeneous surface, and the uncertainties of the input parameters in the flux-profile relationship functions, especially under the stable stratification condition.

To make sufficient use of the observed meteorological information and the similarity law, Xu and Qiu (1997) applied the variational method to compute surface heat fluxes over a flat homogeneous surface. The variational method searches for realistic meteorological variables (e.g., the friction velocity, potential temperature, and specific humidity scale) through constructing a cost function, aiming to minimize the difference between the estimated and observed fluxes. It is found that the variational method is more reliable and useful in practical applications of flux calculations by including different physical constraints in the cost function [i.e., the heat transfer coefficient as a constraint in Ma and Daggupaty (2000), the surface energy budget in Zhang et al. (2004), and temperature variance in Cao and Ma (2005)]. Meanwhile, the applicability of the variational method has been tested over a heterogeneous surface and sea ice surface (Cao and Ma 2005, 2009). Although the results of the variational method with those physical constraints show evident improvements in flux calculations, the observed information such as surface energy flux, radiation measurements, soil temperature, and soil humidity are not included in many observational datasets. This makes it impossible to adopt the physical constraints for use in the variational method. The classical variational method, which is derived from maximum likelihood estimation theory, performs better under overdetermined circumstances (Lorenc 1986), that is, the number of observations in the cost function is greater than the number of unknown parameters. Ma and Daggupaty (2000) also pointed out that the variational method would reduce to the profile method if the number of constraints is equal to that of unknowns. Accordingly, to make use of most of the conventional observational data, it is necessary and important to introduce physical constraints into the cost function in a reasonable way. The mass conservation principle, proposed by Tajchman (1981), is used for determining the zero-plane displacement and roughness length (Molion and Moore 1983; De Bruin and Moore 1985; Lo 1990, 1995; Zhong et al. 2011). It is suggested that the mass conservation principle is a useful concept as a weak constraint for the variational method in most field experiments, where only wind, temperature, and humidity profiles are measured and no additional experimental configurations are available. In this work, on the basis of measurements of wind, temperature, and relative humidity, we intend to verify the application of the variational method with mass conservation constraint to estimate surface fluxes. The results have been compared against the direct measurements and against those estimated by the variational method without physical constraints and by the conventional flux-profile method. Section 2 introduces the measured dataset used for flux calculations and comparisons. The description of the flux-profile method and the variational method based on the mass conservation principle is given in section 3, the computational results and the sensitivity test are presented and compared with the eddy-correlation-measured fluxes in section 4, and the conclusions are given in section 5.

## 2. Data

^{−1}at 12 m s

^{−1}for wind speed, ±0.2°C for temperature, and ±2% at 20°C for relative humidity. The absolute limit test and the abrupt change test have been employed to exclude abnormal observations caused by various reasons, for example, the operation problems of the observing system, the thunderstorm weather, and the noise effects. The absolute limit test detects the unrealistic values in the time series by defining the threshold for different observation variables. In the abrupt change test, we specify two thresholds: ±3 times the standard deviations from the mean of the entire observed height record and from the mean of the continuous 2.5-h measurements. When the data at a certain height or time are outside the thresholds, they are excluded from the analysis. Furthermore, the momentum and sensible and latent heat fluxes used to verify the validity of the new scheme are measured by the eddy-correlation technique at a sampling rate of 10 Hz in the dataset. All the data are collected by block averaging over 30 min, so there are 672 time periods of observations. Since the hypothesis of eddy-covariance technique is that the atmospheric turbulence is in steady state and homogeneous, the steady state test and the similarity test of turbulent covariance can be used as the standard for the data quality analysis and control (Foken and Wichura 1996). In this study, the flux data are classified into three categories, that is, 0 (good), 1 (average), and 2 (bad), according to the standard from Foken et al. (2004). Only those best samples (category 0) were utilized. After the quality control and classification (Yu et al. 2006; Foken et al. 2004), the number of data samples is reduced to 347. In addition, in an attempt to take anemometer stalling errors into account, the actual wind speed

## 3. Descriptions of methods

### a. Flux-profile method

*z*/

*L*and can be expressed asandwhere

Since the wind speed, temperature, and specific humidity gradients are measured directly, the Monin–Obukhov length can be computed by an iterative procedure through substituting Eqs. (7)–(11) into Eqs. (2)–(4). The three unknowns, that is, the friction velocity, temperature, and humidity scale, can be obtained to calculate momentum and sensible and latent heat fluxes simultaneously by Eqs. (5)–(7). While only the vertical profiles are needed in the flux-profile method, it is evident that the disagreement between the estimated and observed fluxes can be attributed to the uncertainty of universal stability functions and the inadequacy of observational data.

### b. Variational method with mass conservation constraint

*J*, which is defined as the summation of differences in wind speed, potential temperature, and specific humidity between the computed and observed values. In the study, the mass conservation hypothesis is included in the cost function to serve as a weak physical constraint. In previous papers (Molion and Moore 1983; De Bruin and Moore 1985), it is assumed that the mass flow is given bywhere

*i*th level of

On the basis of the study of De Bruin and Moore (1985), the mass conservation assumption could be used as a reasonable working hypothesis in neutral condition. Lo (1990, 1995) applied the mass conservation hypothesis to determine zero-plane displacement height and roughness length. In their study, the general forms of the logarithmic wind and temperature profiles, including diabatic influence functions, are introduced. Zhong et al. (2011) have extended the hypothesis to non-neutral conditions to estimate the effective roughness length. Figure 1 shows the comparison between the measured and estimated mass flow during 11–25 August 2010, where the estimations are computed by Eqs. (2) and (14) using the measured

*i*and

*j*denote the number of observation levels for wind speed and temperature, respectively. Note that the surface temperature and humidity are not included in the dataset,

*J*. Namely, the three gradients of

*J*relative to the unknowns should be zero:The analytical expressions of the gradient components are given in the appendix. The “non parameter” problem arises as a result of the confusion of unknowns when the unknowns in the study are related to each other (Schaudt 1998), for example, the Monin–Obukhov length

^{−1}for the friction velocity, from −1.5 to 1 K for the potential temperature scale, and from

- Suppose the initial guessed values of the unknowns are
, , and ; can be computed. - Calculate
, , and from Eqs. (2)–(4) and from Eq. (17) based on the estimated . - Calculate the cost function
*J*in Eq. (16) and its gradients with respect to the three unknowns. If the convergence criterion is reached, the expected estimates (, , ) are obtained; otherwise, the scanning direction and the step size can be determined in terms of the gradients and the scanning direction of the previous iteration. - Along the scanning direction, a line search is performed within predetermined bounds to find the minimum
*J*, and new estimates are obtained at each iteration. - Repeat steps 2–4 until the procedure converges.

The variational method explores all conventional observed data of wind speed, temperature, and relative humidity. It is suitable for the variational approach whereby an overdetermined system including three unknowns and four constraints is formed when additional information of mass conservation is introduced.

## 4. Results and discussion

### a. Estimations of the momentum, sensible, and latent heat fluxes

To evaluate the performance of the variational method with mass conservation (hereinafter VWM) in this study, we compare the fluxes computed by the new scheme and by the conventional flux-profile method with those measured by eddy correlation. In addition, the variational method without the mass conservation constraint (hereinafter VOM) is designed following the same approach of Xu and Qiu (1997), which simply inputs multiple-level observed data into the cost function and constructs the overdetermined system. The results estimated by VOM are also examined. Note that for the flux-profile method, the schemes of H96 for unstable cases and BH91 for stable cases are used and are denoted as BHH. The results are plotted as points in their respective correlation diagrams (Fig. 2). In the stable stratification (Figs. 2a–c), the correlation diagram contains 101 points corresponding to the 101 samples of observations during the selected period. For unstable stratification, there are 246 samples (Figs. 2d–f). It is seen in Fig. 2a that the correlation points of the three methods are distributed near the diagonal line in most cases. Note that they are above the diagonal line when the momentum fluxes ^{−2} for stable cases and 209.9 W m^{−2} for unstable cases, whereas from BHH results it is 39.6 W m^{−2} and 284.8 W m^{−2}, representing a nearly twofold increase in magnitude under stable conditions. In comparison with results of the other methods, sensible and latent heat fluxes estimated by VWM show relatively weaker fluctuation and agree better with observations. Meanwhile, the correlation coefficients between VWM results and measurements are the highest among all three methods. It is recognized that the validity of the mass conservation approach can be established in all stability classes. On the basis of its definition, the mass conservation constraint is used to adjust the wind profile, mainly manifesting in the friction velocity, so as to make the rate of mass flow within the logarithmic wind profile equal to that within the actual wind profile. From Eq. (5), the momentum flux is only affected by the friction velocity; therefore, the difference of results under stable and unstable condition is insignificant. The correlation coefficients between VWM results and measurements reach 87% under both conditions. Estimation of the heat fluxes, however, is determined not only by the friction velocity, but also by the potential temperature and humidity scale. As shown in Figs. 2b, 2c, 2e, and 2f, the calculated results by VWM show better agreement with the measurements under unstable condition than that under stable condition, which is consistent with results from the other two methods. This behavior may possibly be related to the stability functions. In unstable stratification, stability functions are reliable and supposed to be more important because of the larger turbulent fluxes under that condition. The stability functions and turbulence under stably stratified condition, however, have remained less understood and controversial (Cheng and Brutsaert 2005). The strongly stable boundary layer is characterized by large temperature gradients, weak momentum fluxes, light wind, and small mass flow. Therefore, the constraint derived from mass conservation becomes weak. As a result, the VWM generates more realistic estimation of heat fluxes under unstable conditions than in stable conditions. We further calculate the root-mean-square error (RMSE) to quantitatively evaluate the performance of the three methods. As demonstrated in Table 1, in all cases the BHH has an RMSE of 0.038 for

RMSE (W m^{−2}) between the observed momentum fluxes (

Figure 3 shows the time evolution of the relative errors (relative differences between measured and calculated variables) of

The mean relative difference between the observed and computed heat fluxes by the VWM with constraint during the daytime (0800–1800 LST) and nighttime (2000–0600 LST).

### b. Sensitivity tests

*J*, it can be related to the observational errors empirically and even arbitrarily (Daley 1996; Cao et al. 2006). Moreover, although the mass conservation is derived from the vertical wind profile, this does not mean that the weight of the mass conservation term depends solely on the accuracy of wind speed measurements. Thus, to search for

List of experiments, parameters of assumed ^{−2}) in sensible and latent heat fluxes between the observations and those estimated by experiments for different

*J*is based on the stability functions derived from BHH. These stability functions are supposed to be reliable. On the basis of different in situ experiments, however, various empirical coefficients and functions have been developed, leading to large differences in flux calculations, especially in the stable stratification condition. To verify the impact of empirical functions on results of the VWM, four empirical functions resulting from B71, D74, S10, and BHH are employed and are also incorporated into the cost function. The sensible heat fluxes estimated by profile methods and VWM with different flux-profile functions are shown in Fig. 7. It is clear that, for all kinds of stratification (unstable, neutral, and stable) and all flux variables, the biases estimated by VWM using different functions are pretty small (Fig. 7a), corresponding to the similar tendency. For profile methods, as seen in Fig. 7b, the discrepancies between results with different empirical functions are not evident and are similar to those of the VWM under unstable conditions. This is ascribed to the analogous −¼ law, which is adopted as the empirical functions with different coefficients in various experiments. In contrast, results with profile methods under stable stratification remain largely scattered because of the uncertainties of the applicable stability interval and various forms of flux-profile functions. The mean absolute error (MAE) and correlation coefficients (

*R*) between the empirical functions are listed in Table 4. The MAE can be evaluated bywhere

*i*th (the number of samples) results computed by the different empirical functions. As shown in Table 4, for the unstable stratification there is insignificant difference between the two methods. For the stable cases, however, the MAE between results of VWM using different functions is only about half of that generated by the profile methods. This indicates that the VWM places weak dependence on the empirical equations.

The MAEs (W m^{−2}) and correlation coefficients (COR) of sensible heat fluxes estimated between different flux-profile empirical functions: B71, D74, S10, and BHH, employed by the VWM with constraint and profile methods (PFM). Here U and S stand for the unstable stratification and stable stratification, respectively.

To investigate the sensitivities of the estimated fluxes to the observational errors in wind, temperature, and specific humidity, eight sensitivity experiments were conducted. Errors of ±0.5 m s^{−1}, ±0.2

Sensitivities of estimated fluxes to observational errors. Here BHH, VOM, and VWM denote the flux-profile method and variational method without and with mass conservation constraint, respectively.

## 5. Conclusions

In this study, the mass conservation principle, which is based on wind profiles and has clear physical meanings, is chosen as a weak constraint in the cost function in the variational method for turbulent flux calculations. With the mass conservation constraint, the number of observations in the cost function is greater than the number of unknown parameters, namely, the overdetermined situation, and only the conventional meteorological data are taken into consideration. By using the limited memory quasi-Newton algorithm for large-scale bound-constrained optimization (L-BFGS-B), the optical friction velocity, potential temperature, and specific humidity scales are scanned within predefined scales for estimation of momentum and sensible and latent heat fluxes.

Data collected by YRSRORS during 11–25 August 2010 at the Maqu observation site are used to test the variational method with mass conservation constraint (VWM). Briefly, the computational results of momentum and sensible and latent heat fluxes are consistent with the eddy-correlation measurements. In addition, it is shown that the VWM results are highly correlated with the measured fluxes. Biases of fluxes calculated by the VWM method are much smaller than that by the VOM method and by the conventional flux-profile method. Note that the heat fluxes estimated by the VWM agree better with the measurements under unstable conditions because of the sufficiently large number of samples and reliable stability functions.

Sensitivity tests of the weights in the cost function have been carried out for the VWM method. The results show that compared to the VOM, the effect of changing weights is insignificant in estimation of sensible heat flux. Although the estimated fluxes varied in a limited range, an evident linear relationship is found between the weight of the mass conservation term and that of the temperature difference term, resulting in smaller RMSEs. Moreover, the computed heat fluxes by the VWM have similar values when the weight of the specific humidity difference term is changed within certain scales. Relative to the VOM and the flux-profile method, the VWM method shows less sensitivity to errors in meteorological data. The sensible heat fluxes' sensitivity to errors in temperature and the latent heat fluxes' sensitivity to errors in specific humidity are greatly reduced in VWM than in VOM and BHH, whereas the heat fluxes' sensitivity to wind speed errors is intermediate. In addition, since no consensus has been reached on the stability functions experimentally, in the paper the sensible heat fluxes estimated by the VWM using different empirical functions are examined. The results show good agreement, especially for the stable cases, and the mean absolute errors of computed fluxes using different functions are only about half of those by the profile methods. This suggests that the stability function has no substantial impact on the VWM results. Hence, introducing the mass conservation into the cost function is a reliable and simple way to carry out flux calculations. It makes full use of the existing conventional observational information in an effective approach. It is noteworthy that the VWM can be applied for flux estimations over a heterogeneous underlying surface because the mass conservation applies to significantly heterogeneous terrain complexity (Andre and Blondin 1986).

This work was supported by National Key Basic Research and Development Project of China under Grants 2010CB428505 and 2011CB952002 and by the R&D Special Fund for Public Welfare Industry (Meteorology) under Grant GYHY201306025.

# APPENDIX

## Analytical Expressions of the Gradient Components

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