TOWARD A TURBULENCE CONSTITUTIVE RELATION FOR GEOPHYSICAL FLOWS

Rapidly rotating turbulent flows are frequently in approximate geostro phic balance. Single-point turbulence closures, in general, are not co nsistent with a geostrophic balance. This article addresses and resolv es the possibility of a constitutive relation for single-point second- order closures for classes of rotating and stratified flows relevant t o geophysics. Physical situations in which a geostrophic balance is at tained are described. Closely related issues of frame-indifference, ho rizontal divergence, and the Taylor-Proudman theorem are discussed. It is shown that, in the absence of vortex stretching along the axis of rotation, turbulence is frame-indifferent. Unfortunately, no turbulenc e closures are consistent with this frame-indifference that is frequen tly an important feature of rotating or quasi-geostrophic flows. A der ivation and discussion of the geostrophic constraint which ensures tha t the modeled second-moment equations are frame-invariant, in the appr opriate limit, is given. It is shown that rotating, stratified, and sh allow water flows are situations in which such a constitutive relation procedure is useful. A nonlinear nonconstant coefficient representati on for the rapid-pressure strain covariance appearing in the Reynolds stress and heat flux equations, consistent with the geostrophic balanc e, is described. The rapid-pressure strain closure features coefficien ts that are not constants determined by numerical optimization but are functions of the state of turbulence as parametrized by the Reynolds stresses and the turbulent heat fluxes ns is required by tensor repres entation theory. These issues are relevant to baroclinic and barotropi c atmospheric and oceanic flows. The planetary boundary layers in whic h there is a transition, with height or depth, from a thermally or she ar driven turbulence to a geostrophic turbulence is a classic geophysi cal example to which the considerations in this article are relevant.