## Introduction

The surface energy and radiation balance system (SERBS) measures near-surface wind, vertical gradients of temperature and water vapor pressure, and surface energy budget. Using these observations, surface fluxes of sensible and latent heat can be computed by the Bowen ratio energy balance (BREB) method (Fritschen and Simpson 1989). However, the BREB method becomes computationally unstable and produces spurious large values in the computed fluxes when the Bowen ratio is in the vicinity of −1. In addition, the BREB method does not make full use of the information provided by the similarity law for turbulent flow in the surface layer. On the other hand, the profile method uses the equations of the similarity law to compute surface fluxes of sensible and latent heat from the atmospheric measurements in the surface layer (Panofsky and Dutton 1984), but it does not use the information provided by the surface energy budget measurements. Since neither of the above methods uses the complete information provided by the SERBS measurements and the similarity law, a variational method is developed in this paper to make full use of the information. The new method is applied to the SERBS data collected during 10–19 July 1994 at the Oklahoma Atmospheric Radiation Measurement Cloud and Radiation Testbed (ARM-CART) central station (Stokes and Schwartz 1994). The SERBS data are described in the next section with a brief review of the BREB method and the profile method. The variational method is introduced in section 3. The method is tested with the SERBS data in section 4, and the results are compared with those computed by the BREB method and the profile method. Conclusions follow in section 4.

## Data and conventional methods

### Data

In this paper, we use the data measured by the surface energy and radiation balance system during 10–19 July 1994 at the Oklahoma ARM-CART central station (36.61°N, 97.49°W, altitude 315 m). The ground surface around the station is flat and covered with short grasses. The data were collected every 30 min, so there are 480 time levels of observations for the selected 10-day period. The SERBS makes the following measurements: *u* is the horizontal wind speed at 3.4 m AGL, Δ*θ*_{1} is the temperature difference between 0.96 and 1.96 m (inside the temperature sensor chamber), Δ*θ*_{2} is the temperature difference between 0.96 and 1.96 m (inside the humidity sensor chamber), Δ*e* is the water vapor pressure difference between 0.96 and 1.96 m, *G* is the soil heat flux at the surface, and *R* is the net radiative flux (near the surface). Here, Δ*θ*_{1} is measured inside the temperature sensor chamber, Δ*θ*_{2} is measured together with the relative humidity inside the humidity sensor chamber, and the water vapor pressure difference Δ*e* is calculated from the relative humidity and temperature measurements inside the humidity sensor chamber (at 0.96 and 1.96 m) together with the measurement of the atmospheric pressure *p* at the surface level. The measurement resolutions are ±0.1 m s^{−1} for *u,* ±0.01 K for Δ*θ*_{1} and Δ*θ*_{2}, ±0.01 mb for Δ*e* (or ±0.6 × 10^{−5} for Δ*q*), and ±0.1 W m^{−2} for *G* and *R.* In this paper, the water vapor pressure difference is converted into the specific humidity difference by Δ*q* ≈ 0.622Δ*e*/*p.*

### Profile method

*u*

_{*}

*u*

^{2}

_{*}

*τ*/

*ρ*in association with the wind stress

*τ*and air density

*ρ*;

*z*

_{0}is the surface roughness length;

*κ*≈ 0.4 is the von Kármán constant;

*θ*

_{*}

*q*

_{*}

_{M}, Ψ

_{H}, and Ψ

_{Q}are the stability functions [see (6)–(9)];

*L*=

*u*

^{2}

_{*}

*T*/

*κ*

*g*

*θ*

_{*}

*T*= [

*θ*(

*z*

_{1}) +

*θ*(

*z*

_{2})]/2; and

*g*is the acceleration of gravity. The flux temperature scale is related to the sensible heat flux by

*c*

_{P}is the specific heat at constant pressure. The flux specific humidity scale

*q*

_{*}

*λ*is the latent heat of evaporation.

*θ*

_{*}

*L*< 0), and the formulations are given by

*θ*

_{*}> 0 or

*L*> 0), we use the stability functions suggested by Holtslag and DeBruin (1988) and Beljaars and Holtslag (1991); that is,

*a*= 1,

*b*= 0.667,

*c*= 5, and

*d*= 0.35. In (7) and (9), Ψ

_{H}= Ψ

_{Q}is assumed.

On the left-hand side of (1)–(3), the horizontal wind speed, temperature difference, and specific humidity difference can be directly determined by the SERBS measurements. On the right-hand side of (1)–(3), the roughness length can be estimated as an external parameter (see section 4a), so there are three unknowns in (1)–(3): the friction velocity *u*_{*}*θ*_{*}*q*_{*}*G* and *R* in section 2a). Due to the approximations used in (1)–(9) and limited accuracy in the observational data, the total flux of *H* + *λ**E* computed by the profile method could be significantly different from the observed value of *G* + *R* and thus violate the surface energy balance [see (10)].

### BREB method

*H*

*λ*

*E*

*R*

*G,*

*G*is the soil heat flux and

*R*is the net radiative flux (see section 2a). By introducing the Bowen ratio

*B*=

*H*/

*λ*

*E,*this balance equation can be rewritten as

_{H}= Ψ

_{Q}[as assumed in (7) and (9)], we obtain

*θ*(either Δ

*θ*

_{1}or Δ

*θ*

_{2}), Δ

*q*= 0.622Δ

*e*/

*p, R,*and

*G*are all directly measured, so we can estimate the Bowen ratio from (12) and then compute the latent heat flux from (11) and obtain the sensible heat flux from

*H*=

*λ*

*EB.*

The BREB method uses all the SERBS measurements listed in section 2a and has generally been considered the most conservative and reliable all-weather technique for flux computations (Priestley and Taylor 1972; Fritschen and Simpson 1989). However, it is also well known that the BREB method will become computationally unstable when the Bowen ratio is in the vicinity of −1. In this case, as we can see from (11), a small observational error in *R* + *G* or *B* will cause very large errors in the computed fluxes. To eliminate this problem, a variational method is developed in the next section.

## Variational method

*u*

_{*}

*θ*

_{*}

*q*

_{*}

*u,*Δ

*θ*, and Δ

*q*are functions of (

*u*

_{*}

*θ*

_{*}

*q*

_{*}

*u*

^{ob}, Δ

*θ*

^{ ob}

_{i}

*i*= 1, 2), and Δ

*q*

^{ob}are the observed wind, temperature difference, and specific humidity difference, respectively (see section 2a); and

*δ*

*R*

*G*

*H*

*λ*

*E.*

*R*and

*G*are given observationally (see section 2a), and

*H*and

*λ*

*E*are computed as functions of (

*u*

_{*}

*θ*

_{*}

*q*

_{*}

*δ*measures the mismatch between the computed

*H*+

*λ*

*E*and observed

*R*+

*G*under the constraint of the surface energy balance equation (10). The first three terms on the right-hand side of (13) measure the mismatches between the computed and observed winds, temperature differences, and humidity differences, respectively. All these terms should be inversely weighted by their respective observation error variances, so the minimum point of

*J*gives the optimal estimates of (

*u*

_{*}

*θ*

_{*}

*q*

_{*}

*w*

_{u}= 10 m

^{−2}s

^{2},

*w*

_{1}= 100 K

^{−2},

*w*

_{2}=

*w*

_{1}/4,

*w*

_{q}= 10

^{8}, and

*w*

_{r}= 10

^{−4}W

^{−2}m

^{4}. Here, by choosing

*w*

_{2}=

*w*

_{l}/4,Δ

*θ*

_{2}is used as a supplemental data source that is independent of Δ

*θ*

_{1}, although Δ

*θ*

_{2}is measured in nearly the same way as Δ

*θ*

_{1}and only Δ

*θ*

_{1}is used by the BREB method to compute the Bowen ration in (12). If

*w*

_{2}= 0 and

*w*

_{r}= 0 are chosen, then the number of constraints used in (13) reduces from 5 to 3. In this case, the number of constraints (or equations) equals the number of unknowns, and the variational method reduces to the profile method.

*J*, the three gradient components of

*J*with respect to the unknown variables (

*u*

_{*}

*θ*

_{*}

*q*

_{*}

*J.*The iterative procedure consists of the following basic steps.

Start from initial guesses of the unknowns, say,

*u* = 0.1 m s_{*}^{−1},*θ* = 0, and_{*}*q* = 0._{*}Compute

*u,*Δ*θ*, and Δ*q*from (1)–(3), and compute*δ*from (14) by using (4) and (5).Calculate the cost function

*J*in (13) and its three gradient components by using (A1)–(A3).If the convergence criterion is satisfied (|

**∇***J*| ≤ 10^{−4}), then the expected estimates (*u* ,_{*}*θ* ,_{*}*q* ) are reached at the minimum of the cost function. Otherwise, determine a search direction based on the gradients and the search directions of previous iterations._{*}Determine the search step size and find the minimum of

*J*along the search direction, obtain the new estimates of (*u* ,_{*}*θ* ,_{*}*q* ), and return to step 2._{*}

In the algorithm the step size is approximately determined by fitting a cubic curve through a number of points along the line of the search. To improve the convergence rate of iteration, (*u*_{*}*θ*_{*}*q*_{*}*u*_{*}*θ*_{*}*q*_{*}^{−1}, 0.5 K, 0.5 × 10^{−3}). The convergence criterion is |**∇***J*| ≤ 10^{−4}, where **∇***J* represents the gradient of *J* with respect to the scaled (*u*_{*}*θ*_{*}*q*_{*}

## Results and discussions

The two conventional (BREB and profile) methods and the new variational method are tested with the SERBS data collected every 30 min at the Oklahoma ARM-CART central station during 10–19 July 1994. The detailed results are examined and intercompared in the following subsections.

### Estimate of surface roughness length

The surface roughness length *z*_{0} is an important parameter for the profile method and variational method. The local roughness length (over an area of 1–10 km^{2}) is related to the roughness characteristics of the surface. Since the terrain around the Oklahoma ARM-CART central station is flat and covered by short grasses, a crude estimate of *z*_{0} can be made in the range from 0.005 to 0.03 m (Wieringa 1981). This estimated range of *z*_{0} can be improved by using the information provided by the surface energy budget measurements. The details are as follows.

Assume that *z*_{0} does not change during the selected period (10 days). For a selected value of *z*_{0}, the sensible and latent heat fluxes can be computed by the profile method. Substituting these computed values of *H* and *λ**E,* together with the observed values of *R* and *G,* into (14) gives the residual value of *δ*. For total *N* = 480 time levels of observations during the selected period (10–19 July 1994), the rms value of *δ* is computed by ∥*δ*∥ = (*N*^{−1} ^{N}_{n=1}*δ*^{2}_{n}^{½}, and the result is plotted as a function of *z*_{0} in Fig. 1. As shown, ∥*δ*∥ has a minimum around *z*_{0} = 0.01 m, suggesting that *z*_{0} = 0.01 m should be used in order to closely satisfy the surface energy budget. This estimated value is used in this paper, which is the same as the value obtained by Beljaars (1988) for short grass from the data collected during the Cabauw field experiment.

### Comparisons of different methods

Because direct measurements of fluxes are not available, the computed heat fluxes cannot be precisely evaluated. A qualitative evaluation, however, can be made by intercomparisons of the results obtained by the three methods. The computed sensible and latent heat fluxes are shown (for 15–16 July only) in Figs. 2 and 3, respectively. Two problems are seen for the BREB method. 1) The computed fluxes jump and cause spurious spikes when the Bowen ratio becomes close to −1 (in the nighttime between 2000 LT 15 July and 0600 LT 16 July). 2) The computed sensible (or latent) heat flux sometimes (for example, during 0000–0430 LT 15 July) has a sign opposite to the observed vertical gradient of temperature (or humidity). These problems are fairly well known for the BREB method. The profile method and the variational method do not have these problems.

In the daytime, the fluxes computed by the profile method show relatively large fluctuations and do not closely match the observed value of *R* + *G* as required by the surface energy balance equation (10). The mismatch between the computed total flux *H* + *λ**E* and the observed value of *R* + *G* can be measured by ∥*δ*∥. The indicated mismatch is ∥*δ*∥ ≥ 56 W m^{−2} for the profile method (see Fig. 1) and reduces to ∥*δ*∥ = 15 W m^{−2} for the variational method (not shown). During most of the 10-day period, the fluxes computed by the variational method are between those computed by the BREB method and by the profile method.

To show the overall comparisons, each pair of flux values computed by two different methods is plotted as a point in their respective correlation diagrams (Figs. 4–7). Each correlation diagram contains 480 points corresponding to the *N* = 480 time levels of observations during the selected period (10–19 July 1994). In Fig. 4 the profile method is compared with the variational method. As shown, the correlation points are mostly distributed above the diagonal line when the sensible heat flux is positive and large. Thus, in comparison with the variational method, the profile method tends to overestimate sensible heat flux when the stratification is unstable. The BREB method is compared with the variational method in Fig. 5, where a number of points show large vertical deviations from the diagonal line. These large deviations are caused by the spurious large fluxes computed by the BREB method for those cases in which the stratification is very stable and the Bowen ratio is close to −1. Clearly, when the Bowen ratio becomes close to −1, the BREB method fails but the variational method still works well. Similar results are obtained for the computed latent heat fluxes in Figs. 6 and 7.

### Sensitivity experiments

The relative advantages and disadvantages of the three methods can be also evaluated by examining the sensitivities of their computed fluxes with respect to data errors. For this purpose, additional “data errors” of 0.5 m s^{−1}, 0.05 K, 0.1 × 10^{−4}, and 5.0 W m^{−2} are added to the observed wind at 3.4 m, temperature at 0.96 m, specific humidity at 0.96 m, and the observed values of *R* + *G,* respectively (for the *N* = 480 time levels). Five different combinations of these data errors are listed in Table 1. Fluxes are computed by each method with and without the data errors. The rms differences between the computed fluxes with and without the data errors are listed in Table 1 for each combination and each method. As shown, among the three methods, the variational method is most insensitive to data errors, while the BREB method is most sensitive to data errors.

The high sensitivity of the BREB method is largely caused by its computational instability when the Bowen ratio becomes close to −1, or, say, when |*B* + 1| becomes close to zero. As shown in Table 2, the sensitivity of the BREB method can be reduced if the computations are restricted by |*B* + 1| > *ϵ*, or, say, if the computations are performed only for those cases in which |*B* + 1| > *ϵ*. However, the sensitivity of the BREB method cannot be reduced to the level of the variational method even when the computations are restricted by |*B* + 1| > 0.25 (with 8.5% cases excluded). Thus, although some alternate methods were previously proposed to substitute the BREB method in case of |*B* + 1| < *ϵ* ≪ 1 [see, for example, section 6 of Fritschen and Simpson (1989)], there is not a clear way to select the threshold value *ϵ* for an alternate method to be used. The variational method proposed in this paper provides a rational resolution to the problem caused by small |*B* + 1| in the BREB method.

## Conclusions

In this paper, two conventional methods, the profile method and the Bowen ratio energy balance method, are reviewed for their relative advantages and disadvantages in computing surface fluxes of sensible heat and latent heat. Briefly, the profile method uses the full information provided by the similarity law for turbulent flow in the surface layer, but it does not use the surface energy budget measurements, while the BREB method does the opposite. Since neither of the conventional methods uses the complete information provided by the measurements and the similarity law, a variational method is developed in this paper to combine the advantages of the two conventional methods.

Tested with the data collected by the surface energy and radiation balance systems during 10–19 July 1994 at the ARM central station, the variational method is found to be more reliable and less sensitive to data errors than the two conventional methods. In particular, the results show that the BREB method becomes computationally unstable and causes spurious spikes in the computed fluxes when the Bowen ratio is in the vicinity of −1. This problem is fairly well known and is now solved by the variational method in a more rational way than the previously suggested alternative methods. The variational method is also more accurate than the profile method in terms of satisfying the surface energy balance and matching the observed soil heat flux and net radiative flux.

In this paper, the variational method is evaluated only in a relative sense with respect to the conventional BREB and profile methods. In most cases, the variational method gives fluxes somewhere in between those of the two conventional methods, and in case of a clear breakdown of any of the conventional methods, the variational method gives more weight to the information that is really useful for estimating the fluxes. The variational method will be further evaluated when direct eddy-correlation measurements of fluxes become available from the ARM data in the near future.

Although the variational method is tested with the SERBS data for 10 days under a variety of dry and wet weather conditions characterized by a very wide range of Bowen ratios (from *B* ≪ −1 to *B* ≫ 1), the method may become relatively inaccurate when the temperature and humidity gradients are very small or measured by the Oklahoma Mesonet (with relatively large errors). To solve this problem, the current variational method needs to be extended. In particular, the model’s atmospheric layer should be coupled with a soil/vegetation layer, so that the extended method can utilize not only the atmospheric measurements but also the soil temperature and moisture measurements provided by the ARM data or the Oklahoma Mesonet data. This problem is under our investigation, and the preliminary results are very encouraging (Xu and Zhou 1996).

## Acknowledgments

We are thankful to Michael Splitt for the ARM data support, and to Bibin Zhou and anonymous reviewers for their comments and suggestions, which improved the presentation of the results. This work is supported by the National Oceanic and Atmospheric Administration Contract NA37RJ0203, the U.S. Air Force Grant F49620-95-1-0320, and the Department of Energy ARM Program through Battelle PNL Contract 144880-A-Q1 to CIMMS, University of Oklahoma.

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## APPENDIX

### Formulations for the Gradient Components

*J,*defined in (13) with respect to (

*u*

_{*}

*θ*

_{*}

*q*

_{*}

_{M}, Ψ

_{H}, and Ψ

_{Q}are given by (6) and (7) for an unstable stratification and by (8) and (9) for a stable stratification. Accordingly, the gradient components of the stability functions are given, for an unstable stratification, by

*x*= (1 − 16

*z*/

*L*)

^{1/4}= (1 − 16

*zκgθ*

_{*}/

*Tu*

^{ 2}

_{*}

^{1/4}is the same as defined in (6) and (7). For a stable stratification, the gradient components of the stability functions are given by

*x*=

*z*/

*L*=

*zκgθ*

_{*}/

*Tu*

^{ 2}

_{*}

*a*= 1,

*b*= 0.667,

*c*= 5, and

*d*= 0.35, as in (8) and (9).

Sensitivities of computed fluxes with respect to data errors. Here, B, P, and V denote the BREB and the profile and variational methods, respectively.

Percentage of cases restricted by |*B* + 1| > *ϵ* and related rms errors in fluxes computed by the BREB method for restricted cases. The data errors are the same as in the last line of Table 1.