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The Monin‐Obukhov Similarity Theory (MOST) links turbulent statistics to surface fluxes through universal functions. Here, we investigate its performance over a large lake, where none of its assumptions (flat homogeneous surface) are obviously violated. We probe the connection between the variance budget terms and departure from the nondimensional flux‐variance function for CO_{2}, water vapor, and temperature. Our results indicate that both the variance storage and its vertical transport affect MOST, and these terms are most significant when small fluxes and near neutral conditions were prevalent. Such conditions are common over lakes and oceans, especially for CO_{2}, underlining the limitation of using any MOST‐based methods to compute small fluxes. We further show that the relaxed eddy accumulation (REA) method is more robust and less sensitive to storage and transport, adequately reproducing the eddy‐covariance fluxes even for the smallest flux magnitudes. Therefore, we recommend REA over MOST methods for trace‐gas flux estimation.

The Monin‐Obukhov Similarity Theory (MOST) is a cornerstone of Earth observations and models. It defines a set of functions that are commonly used to estimate the exchange of heat, water vapor, and other trace gases between the surface and the atmosphere. MOST was originally formulated for “ideal” conditions that rarely prevail in the real world (e.g., fair steady weather and large flat surfaces with little spatial variation in surface cover or moisture conditions). Here, we show that the theory may be equally inaccurate under supposedly ideal conditions where its assumptions are met. We investigate the performance of MOST above a large lake, identifying additional constraints on its performance related to the behavior of turbulence in the atmosphere. One consequential finding is that the theory misrepresents the exchange of small fluxes, which are prevalent over lakes and oceans. In our analyses, it predicted high CO_{2} fluxes from the lake when in reality the fluxes were very small, a problem that can hinder the understanding of the global carbon cycle. We show that the relaxed eddy accumulation method, an alternative to MOST, not only outperforms, but is also more robust to deviations from ideal conditions, representing small fluxes more accurately.

Equilibrium between variance production and dissipation is a requirement for good performance of Monin‐Obukhov Similarity Theory (MOST)

Fluxes estimated by MOST are more uncertain when the flux magnitudes are small and when variance transport or storage are high

The relaxed eddy accumulation method can still yield robust flux estimates even when MOST models underperform

Since its inception almost 70 years ago (Monin & Obukhov, 1954), the validity of the Monin‐Obukhov Similarity Theory (MOST) has been investigated across a range of sites, with mixed results, above both flat and complex terrains (Asanuma & Brutsaert, 1999; Barskov et al., 2019; Cancelli et al., 2012; Chor et al., 2017; Dellwik & Jensen, 2005; Denmead & Bradley, 1985; Detto et al., 2008; Dyer, 1974; Dyer & Hicks, 1970; Hogstrom, 1988; Hsieh et al., 2008; Katul et al., 1995; Katul, Hsieh, et al., 1996; Li et al., 2012; Raupach, 1979; Waterman et al., 2022; Weaver, 1990; Williams et al., 2007; Zahn, Dias, et al., 2016). Nonetheless, despite its known limitations, MOST functions—in particular the flux‐gradient relation—have been directly or indirectly implemented to estimate scalar fluxes in field measurements or as surface parameterization schemes in large scale models. The variance function has also been used in closure parameterizations in Earth System Models that implement high‐order schemes (Waterman et al., 2022). Furthermore, the variance method has been used in gap‐filling methods (Cava et al., 2008; Dias et al., 2009). Alternatives and adjustments to MOST have been proposed, but none seems to be able to compete with the elegant simplicity and breadth of applicability of the original theory. As a consequence, studies focused on advancing MOST implementation, delineating its skill limits, or proposing alternative formulations continue to be of great relevance to the various geoscientific research communities.

According to Foken (2006), the ideal conditions for implementation of MOST are (a) measurement heights *z* limited to the surface layer (*z* ⪅ 3 h, where *h* is the canopy height), (b) stability range |*ζ*| ≤ 1–2 (where *ζ* = *z*/*L* is the MOST stability parameter and *L* is the Obukhov length), and (c) homogeneous surfaces. Under typical waves conditions (no swell or breaking waves), lakes are surfaces that satisfy most of these requirements, in particular the homogeneity of sources of water vapor, temperature, and CO_{2}. However, as will be discussed in our study, even under these conditions, deviations from MOST are observed. To date, few studies have focused on such assessments of MOST, or alternative flux models such as the relaxed eddy accumulation (REA) method, over water surfaces (Armani et al., 2020; Assouline et al., 2008; Cancelli et al., 2012; Dias & Vissotto, 2017), while even fewer investigated the performance of MOST functions to compute CO_{2} fluxes above freshwater bodies (Xiao et al., 2014; Zhao et al., 2019) or oceans (Iwata et al., 2004; Lammert & Ament, 2015). Nevertheless, these micrometeorological methods offer a direct approach to estimate air‐sea gas exchange that circumvents the need for estimating a transfer velocity (Zemmelink et al., 2004).

To help bridge these knowledge gaps, this study explores the turbulence dynamics responsible for modulating MOST performance. We first investigate the performance of the variance method for water vapor, temperature, and CO_{2} at four levels above lake Geneva, in Switzerland. The departure from MOST is then further explored based on the different terms of the variance budget, and implications to other similarity functions are discussed. Multiple measurement levels are rarely available in such experiments, but are needed to directly compute the budget terms we rely on in the present analyses. Given the limitations of MOST, we then investigate whether the REA method—which is similar in formulation to the variance method but does not rely on an empirical MOST function or stability parameter—is more robust to the estimation of low fluxes. The questions that motivate this study are the following

What physical processes, and corresponding terms in the variance budget, explain deviations from MOST?

Does the REA method suffer from the same limitations as the variance and flux‐gradient methods?

To answer these questions, we demonstrate the impacts of the variance budget terms, with emphasis on vertical transport and storage of variance, on the deviation from MOST. We then conclude with a comparison of fluxes for all three scalars above the lake using eddy‐covariance, variance function, and the REA methods, explaining why the REA performance is superior to MOST‐based approaches.

Assuming horizontal homogeneity, the variance budget for a scalar *s* can be written as (Stull, 1988):*μ*, *μ*′ is the turbulent fluctuation from the mean *t* is time; and *z* and *w* are the vertical coordinate and velocity component, respectively. The different budget terms are *S*_{s}, the local storage of variance; *P*_{s}, the gradient production term; *T*_{s}, the vertical turbulent transport; and *D*_{s}, the molecular dissipation of variance. The radiative term in the variance budget for temperature was neglected. Note that the negative signs were already included in the definition of each term. All terms are further nondimensionalized by an “estimate” of production *κ* is the von Karman constant, *s*. Our analyses will thus investigate the importance of each term with respect to production.

MOST postulates the existence of nondimensional universal functions valid for any scalar (Foken, 2017). According to the theory, these functions are solely governed by the Obukhov stability parameter *ζ*,*g* is gravity and *T*_{v} is the virtual temperature. By definition, *ζ* > 0 for stable conditions, *ζ* ≈ 0 for near neutral conditions, and *ζ* < 0 for unstable conditions.

Two well explored MOST non‐dimensional functions are the gradient function Ψ_{s}(*ζ*) and the variance function *ϕ*_{s}(*ζ*), defined as*s*′. The most commonly used empirical forms of these functions are (Hogstrom, 1988; Kaimal & Finnigan, 1994).*C*_{1}, *C*_{2}, *C*_{3}, and *C*_{4} are empirical parameters obtained from field (Businger et al., 1971; De Bruin et al., 1993; Hogstrom, 1988; Kaimal & Finnigan, 1994; Liu et al., 1998; Wyngaard et al., 1971) or numerical experiments (Maronga & Reuder, 2017). These functions connect the gradient or the variance of a scalar to its fluxes (represented by *s*_{*}). If the universality hypothesized by MOST is valid, Ψ_{s} (and *ϕ*_{s}) should be equal for any scalar (Hill, 1989). This is the most important feature of the theory since it provides a tool to compute fluxes of a scalar solely from its gradient or variance, which can be obtained using slower or cheaper/simpler sensors (e.g., thermocouples or capacitance hygrometers) or from fast gas sensors without a sonic anemometer. This relaxes the need for fast velocity measurements or careful sensor position or distortion corrections (Katul et al., 1994; Katul, Finkelstein, et al., 1996).

Similarly, in Earth System or numerical prediction models, Equation 6 is used to estimate the flux exchange between the surface and atmosphere. Explicitly, turbulent fluxes from the flux‐gradient method (FG) and from the flux‐variance method can be computed as follows:

Both equations can be further manipulated by combination with expressions 3, 6, and 7. Depending on the available data, the fluxes might be found using an iterative process (such as when *u*_{*} is not available) or directly when *u*_{*} is measured or when measurements for “proxy” scalars, assumed to be similar to the scalar of interest, are available.

The REA method (Businger & Oncley, 1990) can be derived directly from the eddy‐covariance expression, after assuming linearity between *w*′ and *s*′ (Katul, Finkelstein, et al., 1996; Katul et al., 2018). The derivation of REA is shown Section S1 in Supporting Information S1, where the final flux expression is

*β*_{s} is an empirical coefficient, *σ*_{w} is the standard deviation of the vertical velocity, and *s* for updrafts (*s* when *w*′ > 0) and downdrafts (*w*′ < 0).

While the coefficient *β*_{s} was originally proposed to be similar across scalars but a function of the stability parameter, many studies found no clear variability with *ζ* (Businger & Oncley, 1990; Katul, Finkelstein, et al., 1996; Zahn, Dias, et al., 2016). In fact, various studies show that *β*_{s} is quite steady with an average of ≈0.6 (Bowling et al., 1998; Katul, Finkelstein, et al., 1996) for different scalars, suggesting that the coefficient computed for a “proxy” scalar can be extrapolated to other scalars.

Field implementations of REA usually consist of two separate containers sampling air for updrafts and downdrafts in order to compute _{2} fluxes over the ocean. According to the authors, EC sensors might lack the sensitivity to measure the concentrations when CO_{2} fluxes are very small. REA, on the other hand, allows measuring the concentration from larger air samples and more accurate analysis with high‐precision techniques. Thus, the REA method is a very popular alternative to measure trace gases when measurements by eddy‐covariance systems are expensive or not possible.

The data set used in our analyses is from the Lake‐Atmosphere Turbulent Exchange field campaign (August–October 2006) over Lake Geneva, Switzerland (details of the experiment can be found in Bou‐Zeid et al. (2008) and Vercauteren et al. (2008)). Eddy‐covariance data was sampled at four levels above the water surface (*z* = 1.65, 2.30, 2.95, and 3.60 m) at 20 Hz. All levels were equipped with a sonic anemometer (Campbell Scientific CSAT3) and open‐path gas analyzer (Licor‐7500), which simultaneously collected time series of CO_{2} (*c*), water vapor (*q*), atmospheric pressure (*P*), sonic temperature (*T*_{s}), and the velocity components in the streamwise (*u*), cross‐stream (*v*), and vertical (*w*) directions. Only unstable conditions were considered.

Quality control of the raw high‐frequency data included removal of periods identified by various sensors flags, detection and removal of outliers, and gap‐filling (Zahn, Chor, & Dias, 2016). Pre‐processing consisted of density corrections for *c* and *q* (Detto & Katul, 2007) and double rotation of the velocity components. The turbulent fluctuations were computed around the 30‐min average. In addition, a stationarity statistic for variances was computed based on the test described in Foken (2017),

who suggested that steady‐state conditions can be assumed if *ξ*_{s} < 30%. We did not discard data based on this criterion; instead, *ξ*_{s} is used solely for the purpose of investigating how nonstationarity impacts the variance function. Nonetheless, we restricted our data according to the wind direction, selecting only periods when the angle between the mean wind and the sensor axis was smaller than 135°.

We verified the existence of a time‐varying sensor mean bias for scalars in our data set, which prevented the accurate computation of mean scalar gradients. For this reason, the flux‐gradient expression 6 was here estimated from the residual of the variance budget equation (*P*_{s} = *S*_{s} − *T*_{s} − *D*_{s}) assuming that the effects of horizontal advection are small, which was confirmed by Assouline et al. (2008) for this data set. Note that the transport term is solely based on turbulent fluctuations—not being affected by mean biases—and was thus directly computed.

The storage term in Equation 1 was computed from the slope of a linear fit of the scalar variance *z* = 3 m. The dissipation of variance was estimated from the second‐order structure function *D*_{ss} following Chamecki and Dias (2004) (more details in Section S2 in Supporting Information S1). Finally, while there are multiple variants of REA, depending on sensor response time and approaches to accumulate the fluxes (Foken, 2017), here we will only test that basic formulation where the *s*^{+} and *s*^{−} are computed from high frequency data to simulate an actual REA system. We thus compute the term Δ*s* by conditionally sampling *s* based on the sign of *w*′ with no lower threshold on |*w*′|.

We first compute the variance function for all three scalars as shown in Figures 1a–1c, where we include the best fit for the empirical form as defined by Equation 7 (the color scheme represents the stationarity statistic from Equation 11). In the curve fitting process, we only considered data points where production and dissipation were in near balance to guarantee optimal MOST applicability as we will discuss in Section 4.1. Thus, by visual inspection of Figure 2, we selected only periods when transport and storage are negligible using the following criteria: _{2}, and

Empirical curves obtained over a homogeneous surface by Liu et al. (1998) are shown in the figure, indicating a close agreement with the curves found over our lake.

Clearly, CO_{2} has the most noticeable scatter, i.e., deviation from the MOST variance function even after filtering out periods most affected by storage and transport. Water vapor and temperature, on the other hand, are more similar (with closer fitting coefficients) when only “good” periods are retained (i.e., steady periods with small transport terms). In terms of stability, more deviation from MOST is observed for mildly unstable conditions (*ζ* ≈ −0.1). Notice that steady periods (*ξ*_{s} < 30%) usually correspond to points close to the empirical curves, meaning that they are more likely to follow the scaling. Nonetheless, the presence of stationary periods that deviate from the expected values indicate that steadiness alone does not guarantee agreement with MOST. To further identify other causes for the deviation from MOST, we will now explore its connection to the variance budget.

To better compare departures from MOST across scalars and conditions and the potential role of various variance budget terms, we first define the metric *χ*_{s},

As a reference, we chose *ϕ*_{s} obtained for water vapor over the lake, shown in Figure 1b, given its closer proximity to results obtained by Liu et al. (1998). In addition, because we computed the transport term at *z* = 3 m, we focus subsequent analyses on *χ*_{s} as measured by the third sensor (at *z* = 2.95 m).

The impact of the transport and storage terms on MOST is depicted in Figures 2a–2c, where the color scheme represents *χ*_{s}. Their dependence on stability is shown in Figure S2 in Supporting Information S1. Clearly, large transport values are associated with large deviation from MOST. In addition, large storage values also seem to cause large deviations from the theory. Note that transport and storage are slightly correlated (more obvious for CO_{2}), where large transport can be associated with large storage. Overall, the smallest deviations are observed when both terms are small compared to production. This result reveals that transport and storage are important in practice for the applicability of MOST, and they can independently cause large deviations from the theory. These findings corroborate the hypothesis proposed by Cancelli et al. (2012) that both terms could play an important role in the failure of MOST.

Figures 1d–1f shows the performance of the flux‐gradient function. As expected, more scatter is observed for CO_{2} and temperature. Better agreement with the empirical fit is seen for water vapor, where points that deviate more are periods featuring large transport or storage of variance. Overall, we notice a similarity between the variance function and the gradient‐flux methods in terms of how they are affected by variance storage and transport. This similarity can be explained by the variance budget itself. By combining a standard dissipation model *C*_{ϵ} is a similarity constant and *τ* a variance cascade time scale related to eddy turnover time) with the variance budget equation and expressions 2, 4, and 5, and ignoring the transport and storage terms, we can show that

This equation shows that the flux‐gradient and flux‐variance methods are directly related. Furthermore, since *κz*/*u*_{*} is itself a reasonable estimate for the eddy turnover time, we expect (*κz*)/(*u*_{*}*τ*) ≃ 1. Thus, any disturbances to the variance function caused by storage or transport (or horizontal advection over other surfaces) also affect the FG (and vice‐versa). Indeed, we found high correlation between the variance transport and flux transport (*w*′*w*′*s*′, not shown here), which is aligned with earlier work documenting the role of the flux transport term in the breakdown of MOST (Raupach, 1979; Simpson et al., 1998).

The fluxes for all three scalars computed from the MOST variance‐function method are compared to the eddy‐covariance fluxes in Figures 3a–3c. As expected, fluxes computed from the theory are closer to the real measurements only when the combined transport and storage terms are small. In addition, the greatest errors tend to occur when the measured fluxes are small and are then widely overpredicted by MOST, which is associated with a weak local production term. On the other hand, stronger measured fluxes lead to a strong local production, which is then balanced by dissipation. These results are in agreement with the result of Cancelli et al. (2012) and Dias and Vissotto (2017) for fluxes over a lake in Brazil.

The poor results found for CO_{2} are a consequence of the weak local production caused by the small fluxes from the lake. This result has direct implications for the exchanges of CO_{2} and other trace gases between water and the atmosphere: if their fluxes are small, such that the equilibrium between production and dissipation of variance are disturbed, estimates obtained by MOST are less reliable (as also documented by De Bruin et al. (1999)). Therefore, these results indicate that alternative approaches to estimate fluxes in such situations are needed.

As an alternative method to compute turbulent fluxes, we now investigate the REA method. We first computed the similarity coefficient *β*_{s} by inverting Equation 10. Negative coefficients were discarded, as well as *β*_{s} > 1. For all three scalars, we found the respective median and median absolute deviation to be 0.58 ± 0.10 (CO_{2}), 0.60 ± 0.03 (water vapor), and 0.58 ± 0.05 (temperature), in the range found by other authors (Businger & Oncley, 1990; Katul, Finkelstein, et al., 1996; Sakabe et al., 2014; Vogl et al., 2021; Zahn, Dias, et al., 2016). This variability in *β*_{c} of around 20% from its median (Figures S3 and S4 in Supporting Information S1) is much smaller compared to the scatter seen for the variance function (Figure 1a), which can be up to three orders of magnitude above the value predicted by MOST.

To investigate the weak dependence of *β*_{s} on stability, we rewrite the REA equation as*η*_{s} = Δ*s*/*s*_{*} is a function of stability. The bin averages of *ϕ*_{w} and *η*_{q} are shown in Figure 3g, where we can see a clear dependence on *ζ* (detailed results for all three scalars are included in Section S4 in Supporting Information S1). The same figure, on the other hand, shows that *β*_{q} is constant. But rather than being a fortuitous or empirical result, the figure and Equation 17 demonstrate that the independence of *β*_{q} from stability is the result of the exact cancellation of the *ζ* effects in the multiplication between *ϕ*_{w} ∝ *ζ*^{1/3} and *η*_{q} ∝ *ζ*^{−1/3}. Since both of these exponents are derived to ensure independence from *u*_{*} in the free convection limit, the scaling *β* ∝ *ζ*^{0} is itself a requirement if REA is to recover that free convection scaling. This result thus explains the weak or non existent dependence of *β*_{s} on *ζ*—at least under unstable conditions—found but not explained in previous studies (Baker et al., 1992; Katul, Finkelstein, et al., 1996; Zahn, Dias, et al., 2016), and confirms Foken's hypothesis who suggested that the weak dependence was likely caused by the canceling effects of stability (Foken, 2017, p. 184). We thus find it practical and reasonable to set a constant *β*_{s} = 0.59 for all scalars, which also better mimics a field experiment where no proxy scalars are available.

Figures 3d–3f compares the eddy‐covariance fluxes to the fluxes computed by the REA for all scalars. Noticeably, a much better agreement is obtained, in particular for small flux magnitudes. Good agreement between REA and EC estimates has previously been observed for CO_{2} over a soybean field (Pattey et al., 1993), wheat and barley crops (Gallagher et al., 2000), a coniferous forest (Hamotani et al., 1996), and wetlands (Tsai et al., 2012), as well as for isoprene over a mixed forest (Bowling et al., 1998). Thus, we further confirm the advantages of REA as a flux estimation method over MOST‐based models. In particular, the REA is more robust to the influence of variance storage and transport. This results from the main assumption underlying its derivation, which only requires linearity between *w*′ and *s*′. Thus, as long as their linearity is not strongly impacted by transport and storage, fluxes estimated by the REA should be very close to their eddy‐covariance counterparts (Katul et al., 2018). Further discussion of the robustness of the REA method is included Sections S1 (see Figure S1 in Supporting Information S1), S4 and S5 (see Figure S5 in Supporting Information S1).

The insensitivity of REA to the production‐dissipation equilibrium, which is the exception rather than the rule in many field experiment conditions, thus emerges as a substantial practical advantage. Therefore, our findings support the implementation of the REA method, as opposed to MOST, to estimate small flux magnitudes. Nonetheless, we note that there are still practical challenges concerning accurate measurements of trace gases and implementation of the REA. This has led to modifications of the method aiming at removing noisy measurements by restricting the sampling of updrafts and downdrafts based on a minimum magnitude of the vertical velocity (Businger & Oncley, 1990; Pattey et al., 1993; Vogl et al., 2021). In this paper, however, we mainly focused on the theoretical basis of the method, ignoring other practical aspects related to sensor sensitivity and accumulation device design, among others.

Our findings confirm that MOST's performance hinges on the assumption of local equilibrium between variance production and dissipation. In other words, any disruption of this equilibrium, such as storage and transport, might lead to departure from the theory and flux overestimation. Small flux magnitudes are usually in non‐equilibrium, and thus are more likely to be misrepresented by MOST. As a consequence, MOST's application is somewhat limited to strong flux conditions (note that the definition of “weak” and “strong” fluxes depend on the scalar in question). However, since MOST overestimates the eddy‐covariance measured fluxes, it may not be possible in practice to identify low flux period using MOST itself to avoid large errors.

On the other hand, REA's performance does not rely on local equilibrium, being capable of estimating reliable fluxes even under non‐ideal conditions. Therefore, REA is a valuable tool for trace‐gas flux estimation that should be more broadly adopted in field experiments when possible, particularly that our study demonstrates why the REA coefficient *β* is required to be independent of stability. However, an alternative to MOST in the land‐surface parameterizations of Earth‐System models, where REA is not a suitable alternative, remains elusive.

EZ and EBZ are supported by the US National Science Foundation under award number AGS 2128345 and EAR 2126206 and by the Cooperative Institute for Modeling the Earth System at Princeton University under Award NA18OAR4320123 from the National Oceanic and Atmospheric Administration. This material is based upon work supported by the High Meadows Environmental Institute at Princeton University through the Walbridge Fund Graduate Award for Environmental Research. The authors would also like to thank Gabriel G. Katul for his insight on the derivations in Section S1 in Supporting Information S1.

The raw eddy‐covariance data used for analyses in the study are available at