Toward a turbulence constitutive relation for rotating ows

In rapidly rotating turbulent ows the largest scales of the motion are in approximate geostrophic balance. Single-point turbulence closures, in general, cannot attain a geostrophic balance. This article addresses and resolves the possibility of constitutive relation procedures for single-point second order closures for a speciic class of rotating or stratiied ows. Physical situations in which the geostrophic balance is attained are described. Closely related issues of frame-indiierence, horizontal nondivergence, Taylor-Proudman theorem and two-dimensionality are, in the context of both the instantaneous and averaged equations, discussed. It is shown, in the absence of vortex stretching along the axis of rotation, that turbulence is frame-indiierent. A derivation and discussion of a geostrophic constraint which the prognostic equations for second-order statistics must satisfy for turbulence approaching a frame-indiierent limit is given. These ow situations, which include rotating and nonrotating stratiied ows, are slowly evolving ows in which the constitutive relation procedures are useful. A nonlinear non-constant coeecient representation for the rapid-pressure strain covariance appearing in the Reynolds stress and heat ux equations consistent with the geostrophic balance is described. The rapid-pressure strain model coeecients are not constants determined by numerical optimization but are functions of the state of the turbulence as param-eterized by the Reynolds stresses and the turbulent heat uxes. The functions are valid for all states of the turbulence attaining their limiting values only when a limit state is achieved. These issues are relevant to strongly vortical ows as well as ows such as the planetary boundary layers, in which there is a transition from a three-dimensional shear driven turbulence to a geostrophic or horizontal turbulence.