From a Two-Compartment to a Single Compartment Relaxed Eddy Accumulation Method

Background: The relaxed eddy accumulation (REA) method is now routinely used to estimate turbulent fluxes of scalar quantities when high frequency turbulent vertical velocity but not scalar concentration measurements are available. The premise behind estimating turbulent fluxes using conditional sampling was first conceived in the mid to late 1970s but the REA was derived in its current form in the early 1990s as an alternative to the eddy covariance (EC) techniques. Since then, the operational version of the REA method requires the accumulation of scalar concentrations in two separate compartments that conditionally sample updrafts and downdraft events. Several prototypes have been proposed and built for terrestrial ecosystems and air-sea exchange studies, but the methodology for computing turbulent fluxes remains the same across all of them. The REA has been and continues to be used to measure turbulent fluxes of compounds such as ammonia, butene, carbonyl sulfide, carbon disulfide, ethene, halocarbon, HONO, isoprene, methane, nitrous oxides, ozone, terpenoid, and even dimethyl sulfide in air-sea exchanges. The REA concept has been extended to measure aerosols and ultrafine particle fluxes and was recently used for estimating sediment-water exchanges in aquatic systems.

What is Proposed: It is here demonstrated that the assumptions behind the conventional REA approach allow for one-compartment sampling (1-C-REA) instead of 2 compartments thereby expanding the operational utility of REA. The presentation here offers a proof of concept regarding the utility of a 1-C-REA and comparisons with the conventional or 2-compartment REA and the eddy-covariance method.

What is Accomplished: The one-compartment sampling is tested using high frequency time series measurements of scalars and momentum across many land cover types and atmospheric stability conditions. The land cover types considered span an ice sheet, a lake, a grassland in a forest clearing, and coniferous and deciduous forests. The ‘formulation’ testing considered key assumptions invoked in the conventional REA and the 1-C-REA. These assumptions pertain to the structure of the joint probability density function (JPDF) of the vertical velocity (w’) and the scalar concentration (c’) fluctuations. It is shown that when the JPDF(w’,c’) is Gaussian characterized by a large correlation coefficient (Rwc), the conventional or two-compartment REA and 1-C-REA recover exactly the turbulent flux <w’c’>, where <.> is time-averaging. Using a new information entropy measure derived for the empirical association between w’ and c’, it is shown that deviations from Gaussian JPDF(w’,c’) are ubiquitous for many scalars and momentum (i.e. c’=u’, the longitudinal velocity fluctuations) in the roughness sublayer (RSL) and in the atmospheric surface layer (ASL) across a wide range of atmospheric stability conditions. When such deviations are approximated using a fourth order cumulant expansion that retains third and fourth moment combinations beyond a joint Gaussian JPDF, the much-discussed REA coefficient beta that links the turbulent flux to the vertical velocity standard deviation and mean scalar concentration difference between updrafts and downdrafts events can be explicitly derived. Its numerical value is shown to depend on the imbalance between two high-order moments: M40 and M31/M11, where M40=<w’w’w’w’>/[<w’^2>^2], M31 = <w’w’w’c’>/[<w’w’>^(3/2) <c’c’>^(1/2)], and M11=Rwc. When |Rwc|>0.15, these high-moments tend to act in opposing manner, roughly canceling their own contributions to beta and leading to an apparent near-constant value theoretically predicted to be 0.56. Interestingly, this theoretical beta is lower than beta predicted from a Gaussian distributed w’ alone (beta = 0.63) and agrees with many prior empirical experiments (beta = 0.55-0.58). It is also shown that when |Rwc|>0.15, a linear regression approximation in the w’-c’ phase space enables the use of both conventional and 1-C-REA approximations with no tunable parameters. The large relative errors in REA turbulent flux determination can be attributed to a small |Rwc|. For such a small |Rwc| (usually but not necessarily associated with small fluxes), contributions from inward and outward interaction quadrants in the w’-c’ phase space are significant. Those quadrants act to weaken the ejection-sweep contributions to turbulent fluxes and are the main culprit in the breakdown of linear regression theory used in REA to estimate Rwc from updraft (w’>0) - downdraft (w’<0) scalar contributions. Such conditions can be isolated and studied in detail when c’=T’ for near-neutral conditions, where T’ is the temperature fluctuations. Near-neutral runs are characterized by small sensible heat fluxes, large temperature variances, and high mechanical production of turbulent kinetic energy. On a positive note, the work here also shows that the ensemble-averaged turbulent fluxes across all the small |Rwc| cases remain reasonably approximated by both conventional and 1-C-REA formulations though run-to-run scatter can be substantial. This scatter has been empirically reported as large excursions in the numerical value of beta from a constant in prior studies.

Conclusions: The one compartment REA can provide turbulent flux measurements commensurate with those determined from conventional 2-compartment REA. This finding enables rapid expansion and practical utility of REA in studies of surface-atmosphere exchanges, interactions and feedbacks.