The Primitive Equations in the Stochastic Theory of Adiabatic Stratified Turbulence

Abstract

The stochastic theory of compressible turbulent fluid transport recently developed by Dukowicz and Smith is applied to the ensemble-mean primitive equations (PEs) for adiabatic stratified flow. The theory predicts a generalized Gent?McWilliams form for the bolus velocity and a single symmetric positive-definite diffusivity tensor for along-isopycnal Fickian diffusion of layer thickness and tracer distributions. When the theory is applied to the active tracer potential vorticity it provides constraints on the form of the Reynolds correlation in the momentum equation, and the turbulence closure problem is reduced to the determination of one 2 ? 2 symmetric diffusivity tensor and one scalar field related to the eddy kinetic energy. The role of the rotational eddy fluxes of thickness, tracers, and potential vorticity is investigated, and a key feature of the closure is that the mean PEs do not depend on the gauge field associated with the rotational component of thickness flux, thereby eliminating the need to parameterize it. The relationship between this closure and closure schemes proposed by others in the quasigeostrophic regime is discussed. It is shown that the eddy-induced transport velocity can be parameterized as diffusion of either thickness or potential vorticity, and the resulting closure schemes are equivalent in the quasigeostrophic regime. The implications of the theory for energy and enstrophy balances are also discussed.