Nonlinear Eddy-Viscosity Turbulence Parameterization in Anelastic Three-Dimensional Flow: Some Mathematical Aspects

Abstract

This paper considers some mathematical aspects of the nonlinear eddy-viscosity turbulence parameterization with quasi-isotropic partitioning of turbulent kinetic energy in three-dimensional anelastic flow. The parameterization, which several investigators have used with the Boussinesq approximation in simulating boundary layer flow and shallow cumulus convection, is generalized in a reasonable way, modifying the formulations of subgrid velocity variances and mean flow deformation to take nonzero three-dimensional divergence into account. It is shown that while strict dissipation of domain-integrated mean kinetic energy due to turbulence is insured for Boussinesq flow, a source/sink term analogous to the pressure-divergence term is also formally present for anelastic flow. Also, in seeking a straightforward parameterization of turbulent kinetic energy so as to insure mathematical realizability for the turbulent velocity variance and covariances, it is found that the formulation of Lilly (1967) meets these requirements provided that his proportionality factor is suitably restricted. In turn, this places a restriction on the relative magnitudes of two universal constants that arise in a more recent formulation of Deardorff (1973). (1973).