A generalized Osborn-Cox relation

The generalized temporal residual mean (TRM-G) framework is reviewed and Illustrated using a numerical Simulation of vertical shear instability. It is shown how TRM-G reveals the physically relevant amount of diapycnal eddy fluxes and implied diapycnal mixing, and how TRM-G relates to the Osborn-Cox relation, which is often used to obtain observational estimates of the diapycnal diffusivity. An exact expression for the diapycnal diffusivity in the TRM-G is given in the presence of molecular diffusion, based on acknowledging and summing Lip an entire hierarchy of eddy buoyancy moments. In this revised form of the Osborn-Cox relation, diapycnal diffusivity is related only to irreversible mixing of buoyancy, since all advective and molecular flux terms are converted to dissipation of variance and higher order moments. An approximate but closed analytical expression can be given for the revised Osborn-Cox relation with the caveat that this closed expression implies unphysical cross-boundary rotational fluxes. It is demonstrated that the original Osborn-Cox relation, in which advective and molecular flux terms are simply neglected, is an approximation to the full form valid to first order. In the numerical simulation the original Osborn-Cox relation holds to a surprisingly good approximation despite large advective fluxes of variance and large lateral inhomogeneity in the turbulent mixing.