The scale invariance of geophysical fluids is investigated in terms of a scale invariance criterion. It was developed by Schaefer-Rolffs et al. (2015) based on the implication that each scale invariant subrange shall have its own criterion. Two particular cases are considered, namely the synoptic scales where the Coriolis term is significant, and the smaller scales where the anelastic approximation is valid. The first case is characterized by a constant enstrophy cascade rather than an energy cascade, leading to a significant restatement of the invariance criterion. In the second case, small-scale fluctuations of density, pressure, and temperature are taken into account that have to be included into the known scale invariance criterion as specific scaling relations. It is further shown that the scaling of Billant and Chomaz (2001) is a special case of the present anelastic scaling relations.
In both cases, the respective scale invariance criteria are applied to the turbulent nonlinear harmonic, the k-model, and the passive tracer diffusion. It is demonstrated that, in analogy to previous work, only the dynamic approaches are scale invariant. In addition, it is shown that for the anelastic approximation the scaling ratio of stratified turbulence has to be applied to the vertical length scale instead of the vertical mixing length.
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