The fundamental equation of eddy covariance and its application in flux measurements

A fundamental equation of eddy covariance (FQEC) is derived that allows the net ecosystem exchange (NEE) (N-s) over bar of a specified atmospheric constituent s to be measured with the constraint of conservation of any other atmospheric constituent (e.g. N-2, argon, or dry air). It is shown that if the condition |(N-s) over bar| >> |(chi(s)) over bar||(N-CO2) over bar| is true, the conservation of mass can be applied with the assumption of no net ecosystem source or sink of dry air and the FQEC is reduced to the following equation and its approximation for horizontally homogeneous mass fluxes: (N-s) over bar = (C-d) over bar(w'chi(s)') over bar|(h) + integral(h)(0)(C-d) over bar (z)(partial derivative chi(s)) over bar/partial derivative tdz + integral(h)(0)[(chi(s)) over bar (z) - (chi(s)) over bar (h)](partial derivative C-d) over bar/partial derivative d dz approximate to (C-d) over bar (h) {(w'chi(s)') over bar|(h) + integral(h)(0) (partial derivative chi(s)) over bar/partial derivative t dz}. Here w is vertical velocity, c molar density, t time, h eddy flux measurement height, z vertical distance and chi(s) C-s/C-d molar mixing ratio relative to dry air. Subscripts s, d and CO2 are for the specified constituent, dry air and carbon dioxide, respectively. Primes and overbars refer to turbulent fluctuations and time averages, respectively. This equation and its approximation are derived for non-steady state conditions that build on the steady-state theory of Webb, Pearman and Leuning (WPI.; Webb et al., 1980. Quart. J. R. Meteorol. Soc. 106,85-100), theory that is widely used to calculate the eddy fluxes of CO2 and other trace gases. The original WPL constraint of no vertical flux of dry air across the EC measurement plane, which is valid only for steady-state conditions, is replaced with the requirement of no net ecosystem source or sink of dry air for non-steady state conditions. This replacement does not affect the 'eddy flux' term (C-d) over bar(w'chi(s)') over bar but requires the change in storage to be calculated as the 'effective change in storage' as follows: integral 0h (partial derivative C-s) over bar/partial derivative t dz - (chi(s)) over bar (h) integral(h)(0) (partial derivative C-d) over bar/partial derivative t dz = integral(h)(0) (C-d) over bar (z)(partial derivative chi(s)) over bar/partial derivative t dz + integral 0h [(chi(s)) over bar (h)](partial derivative C-d) over bar/partial derivative t dz approximate to (C-d) over bar (h) integral(h)(0)(partial derivative chi(s)) over bar/partial derivative t dz. Without doing so, significant diurnal and seasonal biases may occur. We demonstrate that the effective change in storage can be estimated accurately with a properly designed profile of mixing ratio measurements made at multiple heights. However further simplification by using a single measurement at the EC instrumentation height is shown to produce substantial biases. It is emphasized that an adequately designed profile system for measuring the effective change in storage in proper units is as important as the eddy flux term for determining NEE. (C) 2011 Elsevier B.V. All rights reserved. When the EC instrumentation measures densities rather than mixing ratios, it is necessary to use: (N-s) over bar approximate to (w'C-s') over bar|(h) + (chi(s)) over bar[(w'C-v') over bar+(c) over bar(w'T') over bar/(T) over bar](h) + (C-d) over bar (h) integral(h)(0)(partial derivative chi(s)) over bar/partial derivative t dz. Here T is temperature and C-v and c are the molar densities of water vapor and moist air, respectively. For some atmospheric gas species such as N-2 and O-2, the condition |(N-s) over bar >> |(chi(s)) over bar||(N-CO2) over bar is not satisfied and additional information is needed in order to apply the EC technique with the constraint of conservation of dry air. (C) 2011 Elsevier B.V. All rights reserved.