Original paper
A scale invariance criterion for LES parametrizations
Schaefer-Rolffs, Urs; Knöpfel, Rahel; Becker, Erich
Meteorologische Zeitschrift Vol. 24 No. 1 (2015), p. 3 - 13
43 references
published: Mar 13, 2015
published online: Jan 13, 2015
manuscript accepted: Nov 4, 2014
manuscript revision received: Nov 4, 2014
manuscript revision requested: Oct 16, 2014
manuscript received: Jun 6, 2014
Open Access (paper may be downloaded free of charge)
Abstract
Turbulent kinetic energy cascades in fluid dynamical systems are usually characterized by scale invariance. However, representations of subgrid scales in large eddy simulations do not necessarily fulfill this constraint. So far, scale invariance has been considered in the context of isotropic, incompressible, and three-dimensional turbulence. In the present paper, the theory is extended to compressible flows that obey the hydrostatic approximation, as well as to corresponding subgrid-scale parametrizations. A criterion is presented to check if the symmetries of the governing equations are correctly translated into the equations used in numerical models. By applying scaling transformations to the model equations, relations between the scaling factors are obtained by demanding that the mathematical structure of the equations does not change.The criterion is validated by recovering the breakdown of scale invariance in the classical Smagorinsky model and confirming scale invariance for the Dynamic Smagorinsky Model. The criterion also shows that the compressible continuity equation is intrinsically scale-invariant. The criterion also proves that a scale-invariant turbulent kinetic energy equation or a scale-invariant equation of motion for a passive tracer is obtained only with a dynamic mixing length. For large-scale atmospheric flows governed by the hydrostatic balance the energy cascade is due to horizontal advection and the vertical length scale exhibits a scaling behaviour that is different from that derived for horizontal length scales.
Keywords
General fluid dynamics • Atmospheric physics; Scale invariance • GCMs • LES parametrizations