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) Ns¯ of a specified atmospheric constituent s to be measured with the constraint of conservation of any other atmospheric constituent (e.g. N2, argon, or dry air). It is shown that if the condition Ns¯≫χs¯NCO2¯ 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:Ns¯=cd¯w′χ′s¯h+∫0hcd¯(z)∂χs∂t¯ dz+∫0h[χs¯(z)−χs¯(h)]∂cd∂t¯ dz≈cd¯(h)w′χ′s¯h+∫0h∂χs∂t¯ dz.Here w is vertical velocity, c molar density, t time, h eddy flux measurement height, z vertical distance and χs = cs/cd 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 (WPL; 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 cd¯w′χ′s¯ but requires the change in storage to be calculated as the ‘effective change in storage’ as follows:∫0h∂cs∂t¯ dz−χs¯(h)∫0h∂cd∂t¯ dz=∫0hcd¯(z)∂χs∂t¯ dz+∫0h[χs¯(z)−χs¯(h)]∂cd∂t¯ dz≈cd¯(h)∫0h∂χs∂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.When the EC instrumentation measures densities rather than mixing ratios, it is necessary to use:Ns¯≈w′c′s¯h+χs¯w′c′v¯+c¯w′T′¯T¯h+cd¯(h)∫0h∂χs∂t¯ dz.Here T is temperature and cv and c are the molar densities of water vapor and moist air, respectively.For some atmospheric gas species such as N2 and O2, the condition Ns¯≫χs¯NCO2¯ is not satisfied and additional information is needed in order to apply the EC technique with the constraint of conservation of dry air.

► The fundamental equation of eddy covariance is derived. ► Misunderstandings about the eddy covariance theory are clarified. ► The steady-state theory is shown to cause substantial biases. ► The proper application of the non-steady state theory is presented.