2014, Schaefer-Rolffs, U., Knöpfel, R., Becker, E.

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 scaleinvariant 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.

2018, Schaefer-Rolffs, U.

A discussion of different approaches and solutions of the basic tensor equation within the Dynamic Smagorinsky Model (DSM) suitable for General Circulation Models (GCM) is presented. Particular interest is dedicated to the relationship between various approaches (i.e., the specific formulation of the tensor equation), namely a least-square approach, a time lag approach, and a simple tensor contraction approach, and the impact of the specific solution (i.e., how to solve the equation) on the Smagorinsky parameter c2S . In addition to the standard solutions, clipped solutions, absolute solutions, and tensor norm solutions are examined. The numerical results are based on calculations from a general circulation model, where the different approaches are applied to provide the turbulent horizontal momentum diffusion. Here, they are examined with focus on two issues: 1) At the beginning of the simulations, the different choices for the tensor equation result in different values for the locally distributed and zonally averaged values of the Smagorinsky parameter. These values show that for the standard solutions almost half of the values of c2S are negative, in accordance with known results from isotropic turbulence and leads to unstable simulations. In addition, the tensor norm is related to the absolute solution via the Cauchy-Schwarz inequality. 2) As the simulations proceed, the differences of the Smagorinsky parameter values diminish except for the tensor norm solutions while evolving to a stationary state in a process of self-organization such that they form a group with values comparable to isotropic three-dimensional simulations. In summary, the least-squares and time lag approaches provide reasonable results, while the simple contraction approach fluctuates more. For the solutions, it is discussed whether the clipped or the tensor norm solution is more reasonable. © 2018 The authors.