A scale invariance criterion for LES parametrizations

Turbulent kinetic energy cascades in fluid dynamical systems are usually characterized by scale invariance. However, subgrid-scale (SGS) parametrizations of in large eddy simulations do not necessarily fulfill this constraint. Up to now, scale invariance has been considered only in the context of isotropic, incompressible, and three-dimensional turbulence. Here we extend the theory to anisotropic turbulence in compressible flows that obey the hydrostatic approximation. We present a criterion to check if the symmetries of the governing equations are correctly translated into the equations used in a numerical model including the corresponding SGS parametrizations (model equations).

We validate the criterion by recovering the breakdown of scale invariance in the classical Smagorinsky model and by confirming scale invariance for the Dynamic Smagorinsky Model. We further apply the criterion to the primitive equations completed by horizontal and vertical diffusion as used in a GCM. Our assumption is that the numerical resolution extends into the macroturbulent inertial range of the mesoscales, which is governed by a forward energy cascade. The aforementioned criterion then allows us to formulate both the horizontal and vertical mixing lengths for the free atmosphere in accordance with scale invariance. High-resolution runs with the Kühlungsborn Mechanistic General Circulation Model (KMCM) using triangular spectral truncation at wavenumber 330 are presented, being the first simulations of a -5/3 slope of the kinetic energy spectrum in the upper troposphere and lower stratosphere without numerical dissipation or hyperdiffusion. In particular, a dynamic vertical mixing length leads to a steepening of the spectrum in the synoptic scales and a shallowing in the mesoscales.