Hybrid Profile–Gradient Approaches for the Estimation of Surface Fluxes
Abstract
The Monin–Obukhov similarity theorybased wind speed and potential temperature profiles are inherently coupled to each other. We have developed hybrid approaches to disentangle them, and as a direct consequence, the estimation of Obukhov length (and associated turbulent fluxes) from either windspeed or temperature measurements becomes an effortless task. Additionally, our approaches give rise to two easily measurable indices of atmospheric stability. We compare these approaches with the traditional gradient and profile methods that require both windspeed and temperature profile data. Using MonteCarlotype numerical experiments we demonstrate that, if the input profiles are free of any random errors, the performance of the proposed hybrid approaches is almost equivalent to the profile method and better than the gradient method. However, the proposed hybrid approaches are less competitive in comparison to their traditional counterparts in the presence of random errors.
Keywords
Gradient method Obukhov length Profile method Similarity theory1 Introduction
“In principle, it should be possible to determine the three parameters \(z_0\) [aerodynamic roughness length], H [sensible heat flux], and \(u_*\) [friction velocity] from three good wind observations close to the ground. But Priestley (1959) has pointed out that a small error in one or more of the winds leads to a huge error in the stress, so that this technique is not practical. Priestley further suggests that temperature data be added to the wind data in order that accurate estimates of stress be made. The present note considers this possibility in some detail.”
After this influential publication, the boundarylayer community at large embraced the idea and decided to focus on the estimation of turbulent fluxes utilizing both windspeed and temperature data. The socalled gradient and profile methods (Appendix 1) were developed and refined. A few variants, using optimization techniques, were also proposed in parallel (e.g., Nieuwstadt 1978).
In contrast, only a handful of studies did not follow suit. Swinbank (1964), Klug (1967), and Lo (1979) explored the possibility of estimating turbulent fluxes using only windspeed measurements. Even though they documented reasonably good results, their fluxestimation approaches never received any serious attention in the literature. After all these years, it is difficult to pinpoint the exact reasons behind their unpopularity. It is plausible that the inherent complexities of the approaches by Klug (1967) and Lo (1979) utilizing numerical optimization techniques rendered them less desirable in practical applications. Klug’s approach also needed the aerodynamic roughness length (\(z_0\)) as an input, but accurate prescription of \(z_0\) was (and still remains) a challenging task. The algorithm of Lo (1979) did not require \(z_0\) as input, but suffered from convergence issues and possible mathematical errors (Zhang 1981). In addition, Lo (1979) did not include any error estimates of the derived variables as pointed out by Nieuwstadt and de Bruin (1981). The fluxestimation approach of Swinbank (1964) was more elegant, but was founded on the strong assumption that the surfacelayer wind profile follows an exponential shape (Appendix 2). This assumption departed significantly from the wellaccepted logarithmic form (with correction terms) for the wind profile, which likely contributed to its unpopularity.
With the advent of highresolution, highaccuracy instruments for the measurement of wind speed and temperature (e.g., sodars, lidars, distributed temperature sensors), it is worthwhile to revisit the assertions made by Panofsky (1963). The argument that we need both windspeed and temperature measurements for flux estimation may no longer be tenable. At the same time, one needs to have a more analytically tractable approach than that advocated by Lo (1979) or Klug (1967). Recently, in a short communication, we proposed such an approach, called the hybridwind approach (Basu 2018). With a few mathematical manipulations, we demonstrated that it is actually very straightforward to estimate turbulent fluxes from only windspeed measurements. Our hybrid approach is similar to Swinbank (1964). In the present study, we first extend this approach to utilize temperature data as input. Next, we compare the proposed hybrid approaches against traditional gradient and profile methods for a wide range of stability conditions. Last and most importantly, through uncertainty propagation experiments, we quantify the errors in estimated fluxes from all the aforementioned approaches.
The structure of the paper is as follows. Sect. 2 introduces the newly proposed hybrid fluxestimation approach, and as byproducts of this approach, two atmospheric stability indices are derived. Their characteristics are discussed in Sect. 3. Some caveats of the proposed hybrid approaches are touched upon in Sect. 4, and illustrative examples comparing the proposed approach and traditional fluxestimation approaches are documented in Sect. 5. The uncertainty propagation experiments and the associated results are also elaborated in this section. The concluding remarks including future directions are summarized in Sect. 6. Background information on the traditional fluxestimation approaches, Swinbank’s exponential windprofile equation, and several relevant stability correction formulations are provided in the Appendices.
2 Methodology
We have named our fluxestimation methodology a ‘hybrid’ profile–gradient approach because it borrows ideas from both the traditional profile and gradient methods. Via mathematical manipulations, it disentangles the original MOST equations, which has not been feasible in the traditional methods. Hereafter, we make a further distinction and refer to the proposed approach as ‘hybridW’ or ‘hybridT’ depending on whether windspeed or temperature data are being utilized as inputs.
3 Characteristics of \(R_W\) and \(R_T\)
In summary, for the selected stability correction functions, \(R_W\) and \(R_T\) are singlevalued functions of L. Thus, it should be straightforward to estimate L given measured value of either \(R_W\) and \(R_T\). In this regard, any suitable rootfinding algorithm (e.g., Newton–Raphson approach) can be utilized; we make use of the wellknown Levenberg–Marquardt algorithm. Once L is estimated, one can estimate \(u_*\) from Eqs. 2a and 2b. Since there are two equations and only one unknown, the conventional linear regression approach with ordinary least squares can be employed. Having determined both L and \(u_*\), one can then estimate \(\overline{w\theta }\) from the definition of Obukhov length. A similar strategy can be followed in conjunction with \(R_T\) as input. Of course, in this case, one solves for \(\theta _*\) instead of \(u_*\), and from the definition of L, one deduces \(u_*\), and subsequently, \(\overline{w\theta }\).
4 Limitations of the Proposed Hybrid Approaches
Before delving into the results, we would like to mention a few issues that may limit the applications of the proposed hybrid approaches.
4.1 Validity of MOST
Both the hybridW and hybridT approaches are deeply rooted in MOST. Hence, they are only applicable when and where MOST is applicable. We would like to remind the readers that MOST is strictly valid in a horizontally homogeneous surface layer. In the surface layer (aka constant flux layer), the turbulent fluxes are assumed to be invariant with height. Thus, all the sensor heights (i.e., \(z_1\) , \(z_2\), \(z_3\)) should be within the surface layer to avoid violation of MOST. For strongly stratified conditions, the surface layer may be only a few metres deep; the proposed hybrid approaches should be avoided under that scenario.
4.2 Monotonicity of Input Mean Profiles
The hybridW approach implicitly assumes that wind speeds monotonically increase with height. Similarly, in the case of the hybridT approach, the potential temperature is expected to monotonically increase (decrease) with height for stable (unstable) conditions. If such monotonic conditions are not met, the proposed approaches should not be used.
4.3 Similarity of Footprints
The footprints for scalars and fluxes should be similar in order to estimate fluxes accurately via MOST; under homogeneous surface conditions, this restriction is not that important. However, for heterogeneous cases, the mismatch of footprints could pose a serious limitation. Of course, any application of MOST for these cases will also be questionable.
4.4 Multivalued Functions
4.5 Turbulent Prandtl Number
In the MOST relation for the potential temperature profile, Eq. 1b, we implicitly assume that the turbulent Prandtl number (\(Pr_T\)) is equal to one. Since the estimation of L only depends on the ratio \(R_T\), this assumption is not relevant. However, its influence on the estimations of \(\theta _*\) and \(u_*\) via the hybridT approach cannot be disregarded. Note that the hybridW approach does not involve any information about \(Pr_T\).
4.6 Effects of Moisture
Throughout this paper, we have only considered dry atmospheric conditions in the surface layer. It is, however, straightforward to extend the hybrid approaches for moist conditions (e.g., offshore environments). In these cases, one must utilize virtual kinematic heat flux and the virtual potential temperature in the definition of Obukhov length (L) and in Eq. 1b. The stability parameter (z / L) can even be partitioned to account for sensible heat flux and latent heat flux separately. For further details, see Barthelmie et al. (2010) and the references therein.
5 InterComparison of Different FluxEstimation Approaches
 (i)
To encompass a widerange of stability conditions, we assume \(u_* \in \left[ 0.1~2\right] \) m s\(^{1}\) and \(\theta _* \in \left[ \,1~0.2\right] \) K. From these sets, we randomly (with uniform probability) select a \(\left( u_*, \theta _*\right) \) pair.
 (ii)
Furthermore, we assume \(z_0 = z_{0 T} = 0.1\ \hbox {m}\) and \(\Theta _s = \Theta _0 = 300\ \hbox {K}\).
 (iii)
Using these selected inputs, we first estimate L, and then in turn, predict \(U\left( z\right) \) and \(\Theta \left( z\right) \) via Eqs. 1a and 1b in conjunctions with the Businger–Dyer stability correction functions [i.e., Eqs. 13a–13c].
 (iv)
In ‘noisefree input data’ cases, we skip this specific step. Otherwise, we add random noise on the U(z) and \(\Theta (z)\) profiles. More details on the characteristics of additive noise are provided later.
 (v)
If the estimated \(z/L < 1\) and mean wind speed \(> 1 \ \hbox {m s}^{1}\), then we proceed to the following step. Otherwise, we discard the selected \(\left( u_*, \theta _*\right) \) pair and go back to the first step. In the ‘noisy input data’ cases, we enforce a few more additional exclusion criteria which will be discussed later.
 (vi)
Next, we attempt to do the following inverse computation: given the predicted mean windspeed and/or temperature profiles, can we accurately estimate the surface fluxes? In hybridW (hybridT) approach, we estimate the surface fluxes by only using windspeed (potential temperature) data from \(z = 5\), 10, and 20 m.
 (vii)
In order to have a direct comparison, we also estimate fluxes using the traditional gradient and profile methods (Appendix 1). In this case, both wind and temperature data from the lowest two levels are utilized.
 (viii)
For all the fluxestimation approaches, we quantify the relative errors in the estimations of \(u_*\) and \(\theta _*\).
 (ix)
We repeat all the previous steps until we get \(10^5\) admissible samples for all the scenarios.
5.1 NoiseFree Input Data
Relative errors (%) in the estimations of \(u_*\) and \(\theta _*\)
Min  \(p_1\)  \(p_{25}\)  \(p_{50}\)  \(p_{75}\)  \(p_{99}\)  Max  

Estimation of \({u_*}\)  
HybridW  − 2.1 \(\times 10^{3}\)  0  0  0  0  0  3.0 \(\times 10^{4}\) 
HybridT  − 90.1  0  0  0  0  0  9.3 \(\times 10^{4}\) 
Gradient  4.0  4.0  4.0  4.0  4.1  4.5  4.5 
Profile  0  0  0  0  0  0  0 
Estimation of \({\theta _*}\)  
HybridW  0  0  0  0  0  0  5100 
HybridT  0  0  0  0  0  0  0 
Gradient  0  4.0  4.0  4.1  4.4  5.1  8.3 
Profile  0  0  0  0  0  0  0 
Clearly, for both \(u_*\) and \(\theta _*\), the performance of the traditional profile method is the best among all the approaches as it leads to null errors. In contrast, the traditional gradient method seems to suffer from a systematic error of \(O(4\%)\). This error stems from finitedifference approximations, as discussed by Arya (1991).
For both hybrid approaches, the relative errors equal zero for percentiles ranging from 1 to 99. In the case of the hybridW approach, negligible errors can occur in the estimation of \(u_*\) due to roundoff errors during the optimization process. In the case of \(\theta _*\), only 17 samples (out of \(10^5\)) exceeded errors > 1%. Most of these cases had true \(\theta _*\) values close to zero and the division by a small number led to very large relative errors. The performance of the hybridT approach was perfect for the estimation of \(\theta _*\). In the case of \(u_*\) estimation, 16 samples (out of \(10^5\)) exceeded absolute relative error of 1%. In summary, for the noisefree cases, the overall performance of the proposed hybrid approaches is almost at par with the traditional profile method. In the following subsection, we investigate if this conclusion holds in the presence of random errors in input mean profiles.
5.2 Noisy Input Data
Different scenarios for the noise terms
Scenario  \(\eta _U\)  \(\eta _\Theta \)  

\(\sigma \) (\(\hbox {m s}^{1}\))  \(\rho \)  \(\sigma \) (K)  \(\rho \)  
1  0.01  0.9  –  – 
2  0.01  0.5  –  – 
3  0.05  0.9  –  – 
4  0.05  0.5  –  – 
5  0.05  0.5  0.01  0.9 
6  0.05  0.5  0.05  0.5 
We consider several noise scenarios which are listed in Table 2. Specifically, we consider two noise levels (with appropriate units): 0.01 (low) and 0.05 (high). In addition, two values of \(\rho \) are considered: 0.9 (high) and 0.5 (low). Since the hybridW approach only requires windspeed data, please note that the scenarios 4, 5, and 6 are all the same for this approach.

HybridW: for scenarios 1 and 2, the error in \(u_*\) estimation is less than 10%. However, the errors increase substantially for scenarios 3 and 4. For low \(u_*\) values, the errors can range from 10 to 100%; however, for high \(u_*\) values, they are mostly less than 10%. The performance of this approach for \(\theta _*\) estimation is somewhat poorer. For stable conditions, the median absolute error values are largely on the order of 10–20%. For unstable conditions, they are higher and seem to be independent of \(\theta _*\) values. For nearneutral conditions, large errors can occur due to the division by small numbers.

HybridT: the estimation of \(\theta _*\) is far better than \(u_*\) for both scenarios 5 and 6. For unstable conditions, the median error values in \(\theta _*\) are largely less than 20%. Marginally higher errors are noticeable for stable conditions.

Gradient: for scenarios 5 and 6, for low \(u_*\) values, the errors could be on the order of 10–100%. Otherwise, for high \(u_*\) values, they are much lower than 10%. For all conditions (with the exception of nearneutral), \(\theta _*\) errors are less than 10%.

Profile: similar to the noisefree cases, this approach outperforms others in both the scenarios 5 and 6. Qualitatively, the errors in \(u_*\) estimation follow a similar trend as the hybridW approach. However, the magnitude of the errors are much smaller. The errors in the estimation of \(\theta _*\) also barely exceed 10–20% (other than the nearneutral conditions).
6 Concluding Remarks
We have developed new approaches to estimate surface fluxes utilizing either windspeed or temperature profile data. We have compared our approaches against traditional gradient and profile methods that require both windspeed and temperature profile data. For noisefree input data, the hybrid approaches perform as well as the traditional profile method. However, in the presence of random errors in input data, the proposed approaches lead to somewhat more fluxestimation errors than the traditional ones.
Given the unique onetoone relationships between the ratio of windspeed differences (or the ratio of potential temperature differences) with the Obukhov length, we propose that either of these ratios could be utilized as a proxy for atmospheric stability. In Basu (2018), we demonstrated that the ratio of windspeed differences was able to categorize observational data in a physically meaningful way. However, further direct verifications are needed.
We believe that the hybridW approach is ideally suited for sodar and lidarbased windspeed measurements owing to their high vertical resolution in the surface layer. Similarly, the distributed temperature sensingbased highresolution temperature profiles can be utilized as inputs for the hybridT approach. In our future work, observational datasets from various field campaigns will be utilized to make an indepth assessment of the proposed hybridW and hybridT approaches. Of course, we will pay close attention to the issues of nonstationarity and heterogeneity, as under such circumstances, the usage of the proposed hybrid approaches (and MOST in general) is not appropriate.
7 Appendix 1: Traditional Gradient and Profile Methods
Application of the profile method typically requires the following variables as input: wind speed at one level, temperature at two levels, and aerodynamic roughness length (Berkowicz and Prahm 1982). In a slightly modified version, one uses windspeed from an additional level instead of the roughness length. One then utilizes the MOSTbased profile equations and solves for the unknown fluxes. Brotzge and Crawford (2000) utilized this modified profile approach to estimate fluxes from the Oklahoma mesonet.
8 Appendix 2: Swinbank’s Exponential Wind Profile
9 Appendix 3: Stability Correction Functions
Over the years, numerous stability correction functions have been proposed in the literature. A few of them are listed below:
Footnotes
 1.
The text within the parentheses, [ ], are included by Basu, S. to enhance readability.
Notes
Acknowledgements
The author is grateful to Fred Bosveld, Stephan de Roode, Bert Holtslag, Harm Jonker, Branko Kosović, Peggy LeMone, Larry Mahrt, Pier Siebesma, and Bas van de Wiel for their constructive feedback on this work.
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