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Conventional parameterizations of turbulent friction in atmospheric circulation models imply an energy cascade since they represent a sink for the globally integrated kinetic energy of the resolved flow. This property is, however, not sufficient for consistence with conservation laws, which hold for arbitrary control volumes. While the usual vertical momentum diffusion is conserves angular momentum, hyperdiffusion schemes or numerical filters to represent non-resolved horizontal momentum exchange do not. Also the widely used method to equate the local frictional heating by the local negative frictional rate of change of kinetic energy is flawed.
This contribution recapitulates recent developments in applying the generalized mixing-length concept of Smagorinsky for the anisotropic turbulent stress tensor in atmospheric circulation models. In particular, we will consider the proper formulation of nonlinear harmonic horizontal diffusion, as well as the finite difference analogue of the no-slip condition to discretize the shear production associated with vertical momentum diffusion. These methods prove to be essential to simulate a closed Lorenz energy cycle. The Smagorinsky-type diffusion is extended by the Richardson criterion for dynamic instability such that, for high spatial resolution, the interaction of resolved gravity waves with the mean flow in the middle atmosphere is simulated in a self-consistent fashion. Since unresolved scales in atmospheric circulation models are still large against the outer turbulent scale, their representation by turbulent diffusion is generally questionable. This becomes obvious from the global energy spectrum, which is too flat for the smallest scales when using the Smagorinsky method. The latter may be completed by a linear harmonic diffusion applied to the spectrally filtered flow in order to tune the high-wavenumber end of the energy spectrum while maintaining the conservation laws. Nonetheless, a parameterization that properly describes unresolved dynamics in atmospheric circulation models is still not at hand. |
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