Stratified Turbulence in the Atmosphere and Oceans: A New Subgrid Model

Abstract

Turbulence in a stratified medium is studied with emphasis on stable stratification, as it occurs in the atmosphere and oceans, and on the construction of a subgrid model (SGS) for use in large eddy simulation (LES). The two basic assumptions of all SGS models are 1) that the unresolved scales are isotropic and 2) that they can be described by a Kolmogorov spectrum and are no longer valid in a stably stratified medium. Generation of gravity waves invalidates the second assumption, while the damping of vertical motion induces a degree of anisotropy considerably higher than in unstably stratified flows. First, Weinstock's model is used to find that the energy dissipation rate ? decreases with stability. By contrast, the dissipation rate ?? of temperature variance increases with stability. The effect of shear on the subgrid scales is neglected. Second, because of the higher anisotropy of stably stratified flows, even the most complete SGS model presently in use must be enlarged to include new higher-order terms. A new second-order closure model is proposed in which the three components of the flux ui? can be obtained by inverting a 3 ? 3 matrix and uiuj can be obtained by inverting a 6 ? 6 matrix. An approximate procedure is suggested, however, to avoid the 6 ? 6 matrix inversion and yet account for anisotropic production. The kinetic energy e is a solution of a differential equation. It is also shown that in a deep LES, where the buoyancy scales are fully resolved, the standard models for ? and u? are probably adequate, whereas in a shallow LES, where the buoyancy range may not be fully resolved, the above effects on ? and u? must be accounted for. It would be of interest to perform both a shallow and a deep LES so as to cheek the predictions of the model proposed here. Preliminary results indicate that the (total) kinetic energy dissipation length scale increase with stability, in accordance with LES results but in disagreement with Deardorff's model that suggested a decrease of all dissipation scales in presence of stratification.