Horizontal Momentum Diffusion in GCMs Using the Dynamic Smagorinsky Model

Abstract

Abstract A dynamic version of Smagorinsky’s diffusion scheme is presented that is applicable for large-eddy simulations (LES) of the atmospheric dynamics. The approach is motivated (i) by the incompatibility of conventional hyperdiffusion schemes with the conservation laws, and (ii) because the conventional Smagorinsky model (which fulfills the conservation laws) does not maintain scale invariance, which is mandatory for a correct simulation of the macroturbulent kinetic energy spectrum. The authors derive a two-dimensional (horizontal) formulation of the dynamic Smagorinsky model (DSM) and present three solutions of the so-called Germano identity: the method of least squares, a solution without invariance of the Smagorinsky parameter, and a tensor-norm solution. The applicability of the tensor-norm approach is confirmed in simulations with the Kühlungsborn mechanistic general circulation model (KMCM). The standard spectral dynamical core of the model facilitates the implementation of the test filter procedure of the DSM. Various energy spectra simulated with the DSM and the conventional Smagorinsky scheme are presented. In particular, the results show that only the DSM allows for a reasonable spectrum at all scales. Latitude–height cross sections of zonal-mean fluid variables are given and show that the DSM preserves the main features of the atmospheric dynamics. The best ratio for the test-filter scale to the resolution scale is found to be 1.33, resulting in dynamically determined Smagorinsky parameters cS from 0.10 to 0.22 in the troposphere. This result is very similar to other values of cS found in previous three-dimensional applications of the DSM.