Any instantaneous flow variable (e.g., s) is decomposed as s=s‾+s′, where s‾ is an “ensemble mean” quantity, and s′ is a turbulent perturbation defined as a departure from s‾. Operationally, primed variables are determined as excursions from the time-averaged state (hereafter indicated by the overbar). The atmospheric stability is quantified using the dimensionless stability parameter ζ=z/L, where z is the wall-normal distance from the surface, and L is the Obukhov length . Under stable conditions, which are the focus here, ζ > 0.

The strength of the variability in any flow variable is quantified by σs=(s′s′‾)1/2, the root mean squared value of s′, while the covariance w′s′‾ is the average net vertical kinematic scalar flux, and w′y is the vertical velocity fluctuation. From definitions, σs is related to w′s′‾ using the correlation coefficient Rws defined as 1Rws=w′s′‾σwσs.

Using Eq. (), a simplified flux model can be defined based on an empirical parameterization of Rws as a function of ζ: 2w′s′‾=Rws(ζ)σwσs. The empirical relation (Rws(ζ), a linear fit here) is, in general, non-generalizable as it may be site-specific and dependent on some other meteorological variables or surface conditions.

Again using Eq. (), one could also test another simplified model with an averaged correlation coefficient 〈Rws〉, taken as the mean over all the available observational periods, yielding 3w′s′‾=〈Rws〉σwσs. In addition to assuming that the correlation coefficient is stability-independent, the same potential drawbacks of the variable correlation coefficient formulation also apply to this model, and the results could not be extrapolated to other sites, where other factors may be present, such as heterogeneity, seasonality, and the influence of synoptic variability, to name a few.

proposed the REA method to compute turbulent scalar fluxes. The REA method is ideally appropriate to use when fast-response sensors are available for w′ (typically from sonic anemometers) but only slow-response measurements are available for the scalar concentration (slow trace gases sensors or even trace gas samples that need to be collected and analyzed subsequently). In such cases, the REA approach offers an enhanced representation of these scalar fluxes .

The basic idea is inspired by the work of , who used conditional sampling techniques to collect scalar information (along with vertical wind speed) in two electronic counters, one for upflow and another for downflow. From linear correlation analysis, the regression slope of w′/σw against s′/σs may be estimated from the correlation coefficient following 4Rws=(s+‾-s-‾)/σs(w+‾-w-‾)/σw, where s+‾ is the conditional average of scalar s instantaneously attributed to updraft events (w′>0), and s-‾ is the conditional average of scalar s instantaneously attributed to downdraft events (w′<0), Δs=s+‾-s-‾ reflects this difference in collecting scalar information from the two samples and likewise for the vertical velocity statistics. When this estimate for Rws is inserted into Eq. (), the REA expression emerges as 5w′s′‾=σw(w+‾-w-‾)σw(s+‾-s-‾). For a Gaussian-distributed w′, it can be shown that σw/(w+‾-w-‾)=2π/4 , a constant whose numerical value is 0.63.

Since the linear regression analysis to estimate Rws as featured in Eq. () is imperfect, an operational REA model is typically used and is expressed as 6w′s′‾=βsσw(s+‾-s-‾), where βs is now treated as an empirical coefficient that corrects for the abovementioned shortcomings of such a slope estimation of Rws. Many studies investigated the choice of optimal βs over a wide range of atmospheric stabilities, surfaces, and meteorological conditions . The choice of βs is still debatable, yet various studies reported a βs ≈ 0.59 , which is not far from a Gaussian prediction derived from w′ statistics (i.e., 0.63). Hence, a βs=0.59 is selected in the current study as a reference baseline in assessing the REA method.

What is less debatable is the expectation of theoretical invariance of βs with stability changes: it was recently shown that the required independence of the REA formulation in the limit of free convection from the friction velocity (u∗) is not compatible with a stability-dependent βs , and this stability invariance was in fact reported in many field observational studies. The arguments of for a stability-invariant βs can be deduced from the dimensionless form of the REA expression 71[βs]=σwu∗(s+‾-s-‾)s∗, where s∗=w′s′‾/u∗, with s+‾-s-‾∼σs. Noting that scalar flux–variance expressions of σs/s∗ exhibit opposite scaling exponents with ζ compared to σw/u∗ across all stability regimes, the dependence of βs on ζ is likely to be small as the two terms on the right-hand side of Eq. () cancel each others' stability dependence. In convective conditions, σw/u∗∼|ζ|+1/3, whereas σs/s∗∼|ζ|-1/3. For near-neutral conditions, MOST predictions suggest σw/u∗∼|ζ|0 and σs/s∗∼|ζ|0 as well, making βs also independent of stability in that limit.

Under stable conditions, similar plausibility arguments for a stability-independent βs can be made based on the observations of that σs/s∗∼|ζ|0 under very stable conditions. This result was explained by the author based on arguments first presented by that under very stable conditions the active eddy size scales with L rather than z, and thus turbulence statistics should become independent of ζ. This would then also apply to σw/u∗ and by extension to β. However, as later shown in the present paper, (i) the practical application of REA uses devices with finite mechanical response time to physically separate the accumulation of the trace gas in updrafts and downdrafts; (ii) a “dead band” is introduced at small w′, where the concentrations are counted neither towards s+‾ nor towards s-‾; and (iii) the slow response of the scalar sensors may all induce an indirect stability dependence.

Despite its inherent assumptions and some remaining challenges, REA is at present the main tool for estimating fluxes with slow-response gas sensors. However, the device it requires is not simple to design and construct and is not yet available off the shelf. Therefore, we propose and test a version where the separation of air streams into downdrafts and updrafts is done virtually, without a mechanical device. The resulting virtual eddy accumulation (VEA) uses the same REA expression in Eq. (); however, it assumes that the scalar concentrations are measured using a slow-response gas sensor. In this paper, we simulate this slow signal based on the actual high-frequency scalar concentration measurements using a digital filter; the details will be provided in Sect. . This method offers the alternative of using cheaper and simpler slow-response scalar sensors instead of setting up an REA apparatus. Sensors operating at measurement speeds slower than the 10 Hz required for eddy covariance but still somewhat rapid (e.g., 0.5 to 1 Hz) are now more widely available and less expensive and cover more gas species . Furthermore, slower gas sampling speeds reduce the needs for massive pumps in closed-path sensors, decreasing overall power usage and bulkiness, thus easing field deployability, especially in remote areas . Slower-response sensors also tend to have higher precision, other factors being held equal, as white noise decreases with increasing averaging times . In laser-based gas analyzers operating at rates slower than 10 Hz, spectroscopic and temperature corrections can be done in real time, simplifying post-processing and reducing frequency-correction errors . All these advantages motivate the investigation of such a virtual counterpart to REA that can work with moderate speed sensors.

Another advantage of the VEA is that, in theory, it can be applied over averaging periods comparable to eddy covariance (e.g., as short as 15 min), overcoming some of the challenges of conventional REA or of disjunct eddy covariance (DEC) (another technique for measuring fluxes with a slow gas sensor) that usually require longer averaging time periods of 30 to 60 min to compute fluxes. Such long sampling durations, while improving statistical convergence, might be problematic in the stable ABL, especially when turbulence is intermittent, and thus non-stationarity becomes a concern.

A close analogue of the classic eddy-covariance technique can be obtained when the EC calculations are applied as usual, but here the scalar sensor has a slow physical response time. We refer to this calculation as the large-eddy-covariance (LEC) approach. The sensor may still be sampled at a high rate, equal to that of w′, to compute the LEC flux as w′s′‾, but the method should be cognizant of the inherent physical filtering of the fluxes carried by fast eddies that the slow scalar sensor cannot resolve.

Recently proposed models for momentum and heat fluxes based on mixing length analogies that outperformed MOST during stable periods marked with intermittent turbulence dynamics are also tested here. These models were initially formulated using an eddy diffusion representation of fluxes: 8w′s′‾=-Ks∂s‾∂z=-(σwLmix)∂s‾∂z. The eddy diffusivity (Ks) was then defined as the product of a characteristic velocity scale (Uchar) and a mixing length scale (Lmix): Ks=UcharLmix. Here, Lmix will be defined differently for momentum (Ku) and heat (KTv, where Tv is the virtual temperature), but both use the standard deviation of the vertical velocity (σw) as the characteristic velocity scale (similar to REA); i.e., Ks=σwLmix.

For momentum, Lmix=Lu, and Lu is defined as a harmonic average between two competing shear length scales (Lu1, a local turbulent shear scale, and Lu2, the classic bulk shear scale) as follows 9aLu1=(1-αu)σw∂u‾∂z-1,9bLu2=αuu‾∂u‾∂z-1,9cLu=1Lu1+1Lu2-1.

In this model, αu is an empirical constant; its value is determined as αu=αTv = 0.35 (same value found to be also adequate for the heat flux model described next). The mean wind speed at the measurement height is given by u‾.

Similarly for heat, Lmix=LTv, and LTv is defined as a harmonic average between two competing length scales. The first is LTv1, the Ellison length scale and the second is LTv2, the buoyancy length scale . These scales are formulated as 10aLTv1=αTvσTv∂Tv‾∂z-1=αTv2TPENBV,10bLTv2=1-αTvσwNBV,10cLTv=1LTv1+1LTv2-1. Here, 11NBV=gTv‾∂Tv‾∂z is the Brunt–Väisälä frequency; g is the gravitational acceleration; and TPE is the turbulent potential energy, which is related to NBV as shown in LTv1 .

Based on dimensional analysis, formulated flux–gradient relations that are still used widely in weather prediction and climate models. MOST has inherent limitations as it applies to planar homogeneous conditions and stationary flows at very high Reynolds number in the absence of subsidence and requires turbulent kinetic energy (TKE) production to be balanced by the TKE dissipation rate. Nevertheless, MOST still serves as a reference for idealized conditions . MOST fluxes could still capture the observed fluxes under weakly stable conditions where turbulence is continuously sustained and not intermittently suppressed . MOST fluxes here are computed using the Businger–Dyer relations as those relations remain pervasively in use today. Such relations are expressed by non-dimensional gradient (diabatic) functions, Ψs(ζ), relating the scalar concentration surface scale s∗=w′s′‾/u∗ to the gradient following 12Ψs(ζ)=∂s‾∂zκzs∗, where κ is the von Kármán constant (i.e., 0.4).

All introduced models pertinent to high-frequency measurements are now tested at the Utqiaġvik site because it has the multiple levels that are required for testing the A22 and MOST models. The middle panel subplots of Fig. , corresponding to the REA and A22 models, depict the strongest correlation between modeled and observed EC fluxes for both momentum and heat. In addition, the ACC model, which incorporates stability information (Eq. ), performs slightly better than the constant ACC model (Eq. ). Although these models at first may appear to be better approximations of the eddy-covariance fluxes than REA, the REA model performance is in practice superior, benefiting from the cancellation of the effect of stability in the model coefficient βs, as detailed previously. This agrees with prior findings for unstable conditions. All proposed models outperform MOST (Fig. c–f). With their superior performance established, the REA and A22 models will be the only ones retained in the subsequent analyses. When filtering is applied, the VEA model replaces the REA model.

To address the limited bandwidth of many trace gas sensors, we simulate the output of a real slow sensor s̃ measuring a variable s as the numerical solution to the first-order ODE in Eq. (). Here, s would be the “fast” turbulent sensor, and Δ‾ is the timescale of the filter width (the response timescale of the slow sensor), which we selected to vary in the range [0–5 s] with increments of 1 s. Thus, solving the following equation numerically (using explicit forward Euler time advancement scheme) converts the s time series from 10 Hz to a lower frequency down to 0.2 Hz when Δ‾ = 5 s: 13ds̃dt+1Δ‾s̃=1Δ‾s. The observed filtered EC fluxes computed with the filtered scalar signal s̃ are referred to as the LEC fluxes. We also tested another filter type, a low-pass Gaussian filter, and the obtained results were not sensitive to the filter type, but the ODE solution is a more accurate model for a first-order slow-sensing system. The filtering is not applied to the vertical velocity (w) as high-frequency anemometers are readily available. Hence, a tilde denotes the filtered virtual temperature (Tṽ), scalar concentration (s̃), and streamwise velocity (ũ). All three-dimensional velocity components (u, v, and w) are available in high-frequency output of the sonic anemometer, but only u is filtered (ũ) to compare its kinematic momentum fluxes to those of scalars.

In theory, an REA system should not suffer from the slow response of the trace gas sensor since it only requires the mean measurements of s+‾ and s-‾. However, this requires a mechanism to separate the gas streams that has a 10 Hz response time as well. While such systems have been constructed and used, they are typically operated using a certain threshold value that defines a dead-band velocity w0 below which the system does not switch intake or keeps both intakes closed. This is designed to avoid an excessive number of movements and to guarantee larger individual air samples. In addition, since these systems are typically custom made, there is still a possibility of some latency in the response of the mechanics of some models. In the present implementation, a dynamic dead band that is linked to the turbulence conditions in each period is adopted such that similar amounts of air are sampled in updrafts and downdrafts. We used the empirical finding of , as depicted in Eq. (), which yielded roughly similar amounts of air in the updraft and downdraft reservoirs. Sampling is only activated if the vertical wind exceeds this threshold value in w0, i.e., |w|>w0. 14w0=σw3.5 We should underline the fact that the computations of s+‾ and s-‾ for the REA are first done with the unfiltered s signal (mimicking a flawless and very fast mechanical system), and the dead band of vertical velocity defined above is then the only effective filter that applies to the REA computations. However, when exploring the VEA approach that we will detail later, we will compute the VEA fluxes with s+‾ and s-‾ from the filtered signal produced by Eq. ().

Again focusing on Utqiaġvik data set with its multilevel measurements, the models are now tested using inputs from a slow sensor. The top and middle panel subplots of Fig.  show that the LEC and VEA methods significantly, and in similar fashion, underestimate the observed (unfiltered) heat and momentum EC fluxes based on filtered quantities Tṽ and ũ, respectively, as Δ‾ increases. This underestimation, however, is not surprising because under stable conditions small eddies carry a significant proportion of the fluxes, especially when the background flow is laminarizing and intermittent, as shown elsewhere . These small eddies are filtered appreciably by the sensor's slow response. We should underline here that an REA system with fast-response mechanical valves will give results equivalent to the unfiltered VEA (VEA recovers an REA system for Δ‾ = 0) and will be in good agreement with the actual EC fluxes. However, any latency in the mechanical response of the REA device will introduce some type of filtering to the signal that depends on the device design; this underlines the importance of the mechanical design of these systems. Remarkably, the bottom-panel subplots of Fig. c and f show that the A22 model's performance is not sensitive to the signal filtering and provides good estimates of the observed (unfiltered) heat and momentum EC fluxes, even when filtered quantities Tṽ and ũ are used, and up to the highest filter width Δ‾ = 5 s. The A22 model relies on multilevel means and variances in computing the fluxes, which tend to be carried by larger scales than the actual fluxes; this may explain the independence of the model performance from sensor response.

Given the results above, a followup question is whether the VEA model estimates with a filtered signal actually correspond to the fluxes that would be computed using eddy covariances of the filtered scalar signal, i.e., the LEC fluxes. Figure a and b indeed show that the VEA model is still a reliable method to capture these filtered observed heat and momentum LEC fluxes at Utqiaġvik as Δ‾ increases. All quantities here, including the fluxes, are computed based on filtered series Tṽ and ũ. This implies that VEA flux estimates are broadly comparable to the filtering of the LEC fluxes by slow-response scalar sensors.

This match between VEA and LEC fluxes is also observed at the Wendell site for heat and momentum, as well as for the other scalars available at that site (CO2, NH3, and H2O). As in Utqiaġvik's site, Figs. b, d and b, d, and f show that VEA performs better when evaluated against the LEC fluxes, compared to Figs. a, c and a, c, and e that compare it to total EC fluxes. Note that since only one level of measurements at Wendell is available, gradients of first-order moments could not be computed, which precludes testing of MOST or A22 models.

These similar VEA findings among the two different sites indicate that, under stable conditions, the VEA captures the fluxes of the “resolved” eddies. Any slow-response filtering will cause the method to significantly underestimate the needed high-frequency, full observed fluxes. An important question (addressed in the next subsection) that follows is whether the VEA fluxes can be “corrected” under stable conditions to recover the missed fluxes. The proposed modifications, outlined below for the VEA methodological framework, offer a correction for VEA flux estimates to recover their corresponding total EC fluxes. It is to be noted that under unstable conditions at the Wendell site, where flux carrying eddies are of much larger scales than under stable conditions, the VEA method is found to be almost insensitive to the considered filter widths (Δ‾'s) (refer to Appendix ). Therefore, VEA performs well and captures the observed (high- and low-frequency) fluxes under unstable convective regimes, and hence biases in predicting scalar fluxes are expected to be minimal (Figs.  and ). This also implies that for conventional REA, the mechanical response speed is not as critical under unstable conditions. In the next subsection, we aim to develop an effective correction for the under-resolved fluxes under stable conditions that will provide a method to obtain continuous accurate fluxes using VEA over the whole diurnal cycle.

Figure  reveals that the scatter between VEA and EC or LEC results is larger when the fluxes are small, which would translate into larger errors and scatter in the values of βs. Analyses not shown here also reveal that βs values calculated for each period converge well towards the 0.59 value when the correlation Rws increases; i.e., Rws > 0.2, indicating larger fluxes, with more scatter for lower correlation values. However this scatter is randomly distributed around the 1:1 line, indicating some error cancellation when the fluxes are integrated over longer periods of time. Discussions on these random variations in βs have linked them to the effect of height above canopies and to the energy content influence of the associated eddy motions , among others, and are not a focus of the present paper.

Figures a and d and a and c, on the other hand, reveal that the chosen value of βs = 0.59 in the modeled fluxes is a good estimate in recovering the observed EC fluxes when the signal is not filtered (Δ‾ = 0 s). Missing smaller eddies (when the signal is filtered) that contribute significantly to fluxes under stable, but not unstable, conditions thus requires a larger βs to predict the correct fluxes using VEA. Such underestimation was attributed to the filtering operation, dictated here by the choice of Δ‾. This motivates a model development for βs that incorporates the effect of filtering of the fast eddies, which we will express as βv.

Furthermore, the agreement between the LEC and VEA predictions when the scalar signal is filtered also opens the possibility of applying the VEA method as a surrogate for REA, without an actual device that separates air streams from downdrafts and updrafts. If a slow-response sensor (open- or closed-path) is available, the VEA method that uses the REA equation (Eq. ) can be applied, with s+‾ and s-‾ computed using conditional averaging of the scalar time series based on the sign of w – similar to what is done to the simulated REA measurements – but without the dead band given in Eq. ().

For this purpose, we first compute each period's optimal βv (i.e., by inverting the expression in Eq. ) that causes the REA-predicted fluxes, equivalent to VEA computed with the raw signal without filtering, to match the exact observed fluxes when Δ‾ = 0 s (no filtering) across the two contrasting sites. The top-panel subplots, Fig. a for heat, Fig. c for momentum, and Fig. e for all three passive trace gases (CO2, NH3, and H2O), show a scatter plot of these exact βv's relative to the integral timescale τint(s) for each period's co-spectrum, where the data points are colored with the MOST stability parameter. The integral timescale (τint) is determined by integrating the autocorrelation function (ρw′s′(τ)) up to the first zero-crossing from the measured w′s′ instantaneous time series. We observe βv values that depart from βs = 0.59 under all stabilities, but in general there is no clear βv dependence on τint as depicted here.

For Δ‾≠0 s, the corresponding bottom-panel subplots, Fig. b, d, and f, show these same βv's relative to Δ‾/τa for different Δ‾'s, where τa is an advective timescale. After experimenting with various choices of timescales for normalizing the filter scale Δ‾, an advective timescale formed by z and u‾, hereafter labeled as τa=(κz/u‾), appears to provide the best scaling for the variations of βv with the filter size (κ is the von Kármán constant). This converges with the work of , who formulated corrections to estimate the attenuation of scalar flux measurements by the slow response of sensors (akin to LEC). They used the sensor response frequency and a normalized frequency formulated based on τa, which is the frequency of the peak of the logarithmic cospectrum fm=nmκ/τa, to correct for the missed fluxes. In that work, nm is the dimensionless frequency at the cospectral maximum, where it is estimated from observations of its behavior as a function of atmospheric stability ζ. The advective timescale was similarly found to be a plausible choice in describing the drift and non-linear diffusion terms of a proposed non-linear Langevin equation to model the turbulent kinetic energy in stably stratified ABL . This characteristic timescale, τa, measures the advection time of the attached eddies, of size κz, past a fixed sensor.

As depicted in Fig. , an empirical fit that relates βv to Δ‾/τa i.e., βv=f(Δ‾/τa), is proposed here to recover the real observed (unfiltered) EC fluxes using the VEA method with (s̃) measurements. This relation is best described using a power-law model, βv=a(Δ‾/τa)b+c, and Table  summarizes these empirical coefficients (a,b) for momentum, heat (active scalar), and passive scalar (CO2, NH3, and H2O) fluxes at both sites, where c=βv(Δ‾=0)=βs. The reported b exponent for all fluxes, which describes how βv scales with Δ‾/τa, varies between active and passive scalars. Note that if the dead-band criterion is removed, b ≈ 0.7 and does not vary much among all scalars as shown in Table , hinting at the possible universality of such dependence for a VEA model that removes the need for a complex mechanical REA system. Nevertheless, further exploration at disparate sites and analyses of observational data for different scalars across wider stability ranges and different non-ideal surfaces are needed to have increased statistical confidence in the reported values of (a,b) and their generalizability, especially for the Utqiaġvik site, which does not have measurements of the passive scalars.

A common feature for all fluxes, as depicted in the bottom panel of Fig. , is that the proposed model becomes less certain as Δ‾ increases (as expected). Therefore, such a model becomes less reliable with very slow sensors that cannot resolve most of the scales of turbulence and is ideally suited for moderately slow sensors that can still resolve the larger eddies. Otherwise, to compute βv, the only needed inputs are (i) Δ‾ (ideally provided by the slow-sensor manufacturer) and (ii) τa (computed from mean wind measurements). The reported fitting parameters in Table  are obtained using the least absolute residual (LAR) method, so that extreme values, which occur less frequently and may be related to unusual conditions or measurement errors, have a lesser influence on the fit. Given the variability in the optimal values of βv, the proposed models would be mostly suited for quantification of long-term aggregates (net averaged fluxes) of the scalars cycle.