The Temporal-Residual-Mean Velocity. Part I: Derivation and the Scalar Conservation Equations

Abstract

The time-averaged density conservation equation in z coordinates contains a forcing term that is the divergence of the transient eddy fluxes. These fluxes are due to the temporal correlation between the instantaneous velocity and density fields. Even when the instantaneous motion is adiabatic and the flow is statistically steady, the divergence of these eddy fluxes is nonzero, thereby causing the time-averaged flow to have an apparently diabatic component. That is, the time-averaged velocity has a component through the time-averaged density contours. Here a modified time-averaged velocity is derived that has a diabatic component only when there are genuine diabatic processes occurring or when the flow is statistically unsteady. This modified velocity is the sum of the usual Eulerian time-averaged velocity and an extra advection due to transient eddies. It is analogous to the residual-mean velocity defined for zonally averaged flows and is therefore termed the temporal-residual-mean (TRM) velocity. The authors also derive the time-averaged conservation equation for a simple tracer, which is a function only of density. In the absence of diabatic mixing processes and if the flow is statistically steady, the TRM circulation is shown to advect the tracer value averaged along density surfaces, not the tracer value averaged at constant height. This result has implications for the way in which datasets or numerical model output should be averaged and analyzed. The results of this paper apply to both the atmosphere and the ocean or, indeed, to any turbulent stratified fluid.