a. Flux-profile method

On the basis of the similarity hypothesis of Monin and Obukhov, the profiles of wind, potential temperature, and specific humidity for turbulent flows in a horizontally homogeneous surface layer are expressed as follows (BH91):

and

where is the friction velocity; and are the potential temperature and humidity scale, respectively; , , and are the roughness lengths for momentum, heat, and humidity, respectively ( and ; see Lo 1996); is the von Kármán constant; is the zero-plane displacement associated with the upward displacement in the mean airflow trajectory; is the turbulent Prandtl number; , , and are the stability functions determined experimentally in various fields; and is the Monin–Obukhov length given by , where is gravitational acceleration. The momentum flux, namely, the wind stress, is related to the friction velocity by

where is the air density. The sensible heat flux is associated with the temperature scale by

where is the specific heat of air at constant pressure. The latent heat flux is given by

where is the latent heat of evaporation. The negative in Eqs. (6) and (7) represents the upward flux.

Various analytical expressions for stability functions have been given in previous works. No consensus, however, has been reached so far on the values of the experimentally determined constants or functions. Guilloteau (1998) suggested that under unstable stratification condition, the schemes of H96 give good approximations to the measurements. H96 is proportional to powers of z/L and can be expressed as

and

where and . For the stable condition, the universal functions of BH91 are reliable across a wider range of stable stratification. The functions of BH91 are expressed as

and

where , , , and .

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

The variational method makes full use of the observed meteorological data through a searching procedure for , , and to minimize the cost function 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 by

where is the mass flow, is the height, and is air density.

The hypothesis of Marunich (1971) (referenced in Tajchman 1981) assumes that the rate of mass flow between ground level and level over the smooth surface is equal to the rate of mass flow between ground level and level above the rough surface. This is considered to be the “mass conservation hypothesis” and is expressed as

Marunich further hypothesized that if levels and are high enough, then the velocities at and are nearly constant and equal. On the basis of this hypothesis, he suggested the distance between and is the zero-plane displacement. De Bruin and Moore (1985) extended the Marunich's approach without the need of a hypothetical smooth surface, taking into account the existence of the transition layer above the surface. They assumed the condition of mass conservation could be imposed on the logarithmic wind profiles, such that the logarithmic wind profile, extrapolated to , transports the same amount of mass flow as the actual wind profile. Although the method to impose a condition of conservation of mass flow on the logarithmic wind profile is primarily developed for practical reasons, theoretically, it also bears a physical meaning when parameters and are linked to mass flow. This condition can be written as

where is calculated by Eq. (2) according to different stability cases and is the measured velocity. In this study, we assume that the airflow in the surface layer is incompressible and that air density is constant with height; hence, the density is negligible. The left-hand side of Eq. (14) represents the estimated mass flow, and the right‐hand side of Eq. (14) represents the mass flow transported by the measured velocity profile, which can be regarded as the area enclosed by the measured wind profile. For simplicity, we employ the trapezoidal rule to compute it directly, and the calculation can be expressed as

where means the measured wind speed at ith level of and in the dataset.

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 and the measurements are obtained by Eq. (15) through measured wind profiles. It can be seen in Fig. 1a that within the entire range of , the estimations are consistent with the measurements, especially under near-neutral and stable conditions. The estimations are somewhat larger than the measurements in unstable stratifications. Further analysis reveals that the discrepancy can be attributed to the uncertainties in idealistic flux-profile functions and the inaccuracy in measured and , as well as large samples under unstable conditions. Nevertheless, the difference between them is not significant, with the absolute error closing to zero and the relative error less than about 10% (Figs. 1b,c). In general, the mass conservation hypothesis in this study can be applied for non-neutral conditions, agreeing with the studies of Lo (1990, 1995) and Zhong et al. (2011).

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(a) The variations of the measured (dashed line) and estimated (solid line) mass flow vs . The dots represent the mean estimated values at a 0.15 stability interval. The error bars indicate the standard deviation of estimated mass flow in stability interval. (b) The difference of mass flow between the measurements and the estimations. (c) The relative error of mass flow between the measurements and the estimations.

Citation: Journal of Applied Meteorology and Climatology 53, 1; 10.1175/JAMC-D-13-0121.1

The cost function can be written as

where , , and are derived from Eqs. (2)–(4); the subscript letters obs stand for observations; and 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, and in the cost function represent the differences between the above four levels, and the lowest level instead of the height .

The residual error in the mass conservation is given by

The variables , , , and are dimensional weights, which can be specified empirically to be inversely proportional to the respective observation error. In this study, the weights are set to be , , , and . Weight determination is examined in detail in the following section.

To obtain optimal estimates of , , and , it is necessary to minimize 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 in the cost function. To avoid this problem, a scanning procedure instead of a searching one is applied to find the optimal solution. The scanning scales are set up approximately: a maximum of 1 m s⁻¹ for the friction velocity, from −1.5 to 1 K for the potential temperature scale, and from to for the specific humidity scale. Thus, an extension of the limited-memory, quasi-Newton Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm (L-BFGS) for unconstrained optimization, which we call the L-BFGS-B method, is employed instead of the L-BFGS (Liu and Nocedal 1989) used in Xu and Qiu (1997). The main improvement is that L-BFGS-B is able to deal with large nonlinear optimization problems subject to simple bounds on the variables (Zhu et al. 1997). In the algorithm, the scanning direction, defined as the vector leading from the current iterate to this approximate minimizer, is determined by the gradient projection method (Bertsekas 1982; Conn et al. 1988), and the step size is computed by a procedure using the line search program of Moré and Thuente (1994). The iterative procedure is run to compute the cost function and its gradients until the convergence criterion of is satisfied. The procedure consists of the following steps.

1. Suppose the initial guessed values of the unknowns are , , and ; can be computed.
2. Calculate , , and from Eqs. (2)–(4) and from Eq. (17) based on the estimated .
3. 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.
4. 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.
5. 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.

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 are small. This means that an underprediction appears for strongly stable conditions because of the uncertainty of stability functions. Still, the measured and the computed results using the three methods agree well and their correlation coefficients are similar, as shown in Fig. 2a. In contrast, under the unstable conditions, obvious discrepancies in estimated from the three methods are found in Fig. 2d. Many points of the VOM results are scattered below the diagonal line, indicating significant underestimations. On the contrary, most points of the BHH results are distributed above the diagonal line, suggesting distinct overestimations. The above results clearly show that neither VOM nor BHH can provide satisfactory fluxes calculation. The points of the VWM results, however, are distributed along the diagonal line and correspond to the highest correlation coefficient. This result suggests that among the three methods discussed above, VWM is the most effective one to calculate momentum fluxes, although several underestimations cannot be neglected. Sensible heat fluxes and latent heat fluxes computed by the three methods are compared with observations under different stratifications in Figs. 2b–f. The points representing the BHH results are sparsely distributed, showing large vertical deviations from the diagonal line, particularly when the sensible heat flux is largely negative. Similar patterns are found in latent flux calculations under stable conditions by VOM. The results demonstrate evident underestimations. Note that the BHH overestimates latent heat flux in all cases, and some of these average errors are quite significant. For example, the average from YRSRORS observations is 19.7 W m⁻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 , of 27.190 for , and of 81.606 for , whereas the RMSEs for the same quantities are 0.036, 19.584, and 66.245 for the VOM method and 0.02, 13.443, and 40.654 for the VWM method. These results suggest that the VWM method gives the most realistic estimation of momentum and heat fluxes and performs best among the three methods.

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For (top) stable and (bottom) unstable conditions, the (a),(c) momentum fluxes, (b),(d) sensible heat fluxes, and (c),(e) latent heat fluxes of the observed vs the values computed by the flux-profile method (dots), the variational method without mass conservation constraint (circles), and the VWM constraint (plus signs).

Citation: Journal of Applied Meteorology and Climatology 53, 1; 10.1175/JAMC-D-13-0121.1

Table 1.

RMSE (W m⁻²) between the observed momentum fluxes (), sensible heat fluxes (), and latent heat fluxes () and those computed by the three methods BHH, VWM, and VOM. The terms Uns, Sta, and Ent stand for unstable stratification, stable stratification, and entire stratification, respectively.

Figure 3 shows the time evolution of the relative errors (relative differences between measured and calculated variables) of , , and during a period of 10 consecutive days (from 11 to 21 August 2008). The weather condition during the period was relatively fair, with mean winds blowing mostly from the south or the southwest. There were no recorded rainfalls and the soil experienced gradual drying. After the quality control procedure, 253 wind, temperature, and humidity profiles, of which 179 are for unstable stratification and 74 are for stable stratification, are selected. Simultaneous flux measurements are obtained also. Figure 3c shows that VWM can give a reasonable result, with relative uncertainty generally within 40% for and , and 50% for . Results of the other two methods, however, have much larger biases than that of VWM. Overestimation of sensible and latent heat fluxes by BHH is evident (Fig. 3a) with relative error up to about 170%. The overestimation is less severe in results of the VOM, but the relative error of friction velocity can reach about 100%. Figure 3d shows the variation of mean relative difference of and for the consecutive days. A slightly negative diurnal cycle tends to appear, especially in the first, fourth, and tenth days. During the daytime, the mean relative error becomes smaller but increases in magnitude during the nighttime. Although the diurnal cycle looks weak, it provides evidence to support our previous results that the VWM performs best under unstable stratification conditions (i.e., daytime). Table 2 lists the mean relative difference between the observed and computed heat fluxes by the VWM during the daytime (0800–1800 LST) and the nighttime (2000–0600 LST), respectively. The biases in the daytime are significantly less than that in the nighttime, indicating that the estimated results for the daytime are more consistent with observations. This can be attributed to the sufficient number of samples and reliable stability functions under unstable and convective conditions. In general, the VWM shows substantial improvements in the flux calculations relative to the VOM and the flux-profile method.

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Time series of relative differences between observed and computed friction velocity (circles), sensible heat flux (dots), and latent heat flux (plus signs) obtained by the (a) BHH, (b) VOM, and (c) VWM methods. (d) The variation of mean relative difference of estimated heat fluxes by the VWM method or continuous days. Time starts at 0000 LST 11 Aug 2010.

Citation: Journal of Applied Meteorology and Climatology 53, 1; 10.1175/JAMC-D-13-0121.1

Table 2.

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

Since there is no clear way to select the weights in the cost function 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 and examine the sensitivity of the estimated fluxes to the weights, we compute the RMSE of sensible heat fluxes between the observed values and those estimated by VWM with different weight groups. First, suppose is proportional to the measurement resolution in specific humidity of a constant , is determined within the range of 0–100, and and are changing from 1 to 100; thus, there are 100 × 100 × 101 experiments in all. Next, the sensitivity of the estimated heat fluxes to is tested. As shown in Fig. 4, when is assumed to be 0 (i.e., the VOM), the RMSE ranges from 13.7 to 32, with different combinations of and . It is clear that with larger values of , the variation of RMSEs is significant. On the contrary, when the mass conservation constraints are constructed in the cost function, it can be seen from Fig. 5 that with different weight groups RMSEs vary from 12.65 to 13.85, which is not distinct at all. This result further proves the advantage of VWM and indicates its weaker sensitivity to weights compared to the VOM. Figure 5 also shows that the RMSEs are barely dependent on when and are fixed. It should be mentioned, however, that as the assumed increases, the relatively small value strip in RMSE (shaded with slanted lines in each panel) shifts to the larger . It seems that there probably exists a comprehensible relationship between and the averaged denoted by the dashed line over the shaded area, which may provide a basis for setting . To ascertain whether the relationship exists, we choose the weight combinations with which the RMSEs of estimated sensible heat fluxes are within the range of , and compute the respective correlation coefficients of selected values with and . The represents the minimum RMSE regarding all weights experiments. On the basis of the calculation, the correlation coefficient of with is equal to 0.9153 () at the statistically significant level of 99.9%. This indicates a reasonable linear relationship between them. Meanwhile, the correlation coefficient of with is 0.1453, suggesting a poor linear relationship between them. As a result, can be determined through regression by minimizing the RMSE. The linear regression shown in Fig. 6 is valid for sufficiently large samples and the optimally fitted function for relative to is given by

This helps to improve the choice of weights in the cost function. After is obtained, the sensitivity of the estimated heat fluxes to is examined. We assume that , , and , derived from Eq. (19), and eight different values for are listed in Table 3. Note that the magnitude of the selected is presumably the inverse square of the observational error in specific humidity. The RMSEs between the computed heat fluxes and the measurements are also listed in Table 3 for each value. In general, the variation of RMSEs for sensible heat fluxes is smaller than that for latent heat fluxes in eight tests. As shown, the RMSEs of sensible heat fluxes vary from 12.8 to 13.1 and RMSEs of the latent heat fluxes vary from 41.5 to 42.3, indicating that the computed heat fluxes do not correspond clearly to the change in , which is mainly caused by the small difference in specific humidity.

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The RMSEs of sensible heat fluxes between the measured values and values estimated by the VOM method, namely, the weight of the mass conservation term () in the cost function is equal to zero. The x axis and the y axis represent the weight of differences in the potential temperature term () and that of differences in the wind speed term (), ranging from 1 to 100, respectively.

Citation: Journal of Applied Meteorology and Climatology 53, 1; 10.1175/JAMC-D-13-0121.1

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The RMSEs of sensible heat fluxes between the measured values and values estimated by the VWM with constraint for different weights: (a) , (b) , (c) , (d) , (e) , and (f) . Contours with shaded with enclosed slant lines show the relatively small values within , where indicates the minimum RMSE for each specified.

Citation: Journal of Applied Meteorology and Climatology 53, 1; 10.1175/JAMC-D-13-0121.1

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The scatterplot of selected values of vs with which the RMSEs of sensible heat fluxes estimated by VWM method are less than 1.005 times the minimum RMSE relevant to all weights experiments. The solid line represents the mean relative to , where the linear regression relationship is shown by a dashed line.

Citation: Journal of Applied Meteorology and Climatology 53, 1; 10.1175/JAMC-D-13-0121.1

Table 3.

List of experiments, parameters of assumed , and the RMSEs (W m⁻²) in sensible and latent heat fluxes between the observations and those estimated by experiments for different values.

As mentioned above, in this study the cost function 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 by

where and are the ith (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.

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The sensible heat fluxes estimated by (a) VWM and (b) profile methods using different flux-profile functions.

Citation: Journal of Applied Meteorology and Climatology 53, 1; 10.1175/JAMC-D-13-0121.1

Table 4.

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 , and (about 2% relative humidity) are added to the wind at 4.2 m, temperature and specific humidity at 7.17 m, respectively, for the time levels. Eight combinations of errors and the MAE of fluxes estimated by each method with and without the errors are listed in Table 5. Because the mass flow is defined by the wind profile [Eq. (12)], the mass conservation constraint is certainly affected by the wind error. This explains why the MAE generated by VWM is slightly larger than that by the VOM (see the fourth and fifth rows in Table 5) when wind error is imposed. The VWM method, however, exhibits the lowest sensitivities to temperature and specific humidity errors since it uses the observational information and extra physical constraint based on wind speed. For the flux-profile method, the estimated latent heat fluxes are most sensitive to specific humidity error and moderately sensitive to temperature error. The estimated sensible heat fluxes are highly sensitive to temperature error. Note that the specific humidity is not used in sensible heat flux calculations by the profile method, so error of specific humidity exerts no influence on the results. Relative to the flux-profile method, the VOM method is much less sensitive to errors of wind and specific humidity. But the VOM is sensitive to temperature errors, as discussed by Zhang et al. (2004). The above results clearly show that among all three methods used in this study, the VWM is the least sensitive to observation errors, except for speed error, and the profile method is the most sensitive.

Table 5.

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.