# Berlin 2014 – wissenschaftliches Programm

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# UP: Fachverband Umweltphysik

## UP 13: Klimamodellierung

### UP 13.3: Vortrag

### Donnerstag, 20. März 2014, 14:00–14:15, MAG 100

**A Scale Invariance Criterion for LES Parametrizations** — •Urs Schaefer-Rolffs — Institut für Atmosphärenphysik Kühlungsborn

The role of scale invariance (SI) is mandatory to explain a turbulent kinetic energy cascade in different fluid dynamical systems, including rotating and stratified flows. However, the representation of subgrid-scales in large eddy simulations do not necessarily fulfill this constraint. So far, SI has been considered in the context of isotropic, incompressible, and three-dimensional turbulence. In this presentation, 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 involved are obtained by demanding that the mathematical structure of the equations does not change.

The method is validated by the breakdown of SI occurring in the conventional Smagorinsky model, but not in the Dynamic Smagorinsky Model. Further, the criterion proves that the compressible continuity equation is intrinsically scale invariant, while for the hydrostatic approximation horizontal and vertical scales exhibit different scaling behaviour. The criterion also shows that a simplified kinetic energy equation can be scale invariant. Finally, in models of turbulent transport of a tracer, a constant mixing length must be avoided to allow for SI.