The EKE budget and the eddy diffusivity

It has recently been proposed to formulate eddy diffusivities in ocean models based on a mesoscale eddy kinetic energy (EKE) budget. Given an appropriate length scale, the mesoscale EKE can be used to estimate an eddy diffusivity based on mixing length theory. Indeed many of the commonly cited scaling arguments for the eddy diffusivity in a baroclinic flow can be derived from such a mesoscale eddy kinetic energy budget, by assuming a local balance between the extraction of available potential energy (APE) from the background flow, and a loss rate of EKE. The differences arise from different assumptions for the mixing length, as well as for the EKE loss rate.

We will revisit the problem of what controls the eddy mixing length and EKE loss rate, with the help of a series of idealized numerical simulations. The simulations use a zonally re-entrant beta-plane channel model with quadratic bottom drag. Although arguably a more realistic description of turbulent boundary layers, the quadratic drag case has received relatively little attention in the theoretical literature. The results clearly show that in the presence of any significant planetary vorticity gradient, the mixing length is governed by the Rhines scale (much more so than the eddy length scale itself). Estimating the EKE loss instead is more complicated, and all traditional scaling arguments fail. We derive a revised scaling relation for the eddy diffusivity, which holds in a broad parameter regime that is likely to be important for the real ocean. Curiously — even though the mixing length is limited by the Rhines scale — EKE loss in this regime remains dominated by frictional dissipation acting directly on EKE, with the transfer of energy into jets playing a secondary role.