Scaling laws for homogeneous turbulent shear flows in a rotating frame

The scaling properties of plane homogeneous turbulent shear flows in a rotating frame are examined mathematically by a direct analysis of the Navier–Stokes equations. It is proved that two such shear flows are dynamically similar if and only if their initial dimensionless energy spectrum E*(k*,0), initial dimensionless shear rate SK0/e0, initial Reynolds number K20/νe0, and the ratio of the rotation rate to the shear rate Ω/S are identical. Consequently, if universal equilibrium states exist at high Reynolds numbers, they will only depend on the single parameter Ω/S. The commonly assumed dependence of such equilibrium states on Ω/S through the Richardson number Ri=−2(Ω/S)(1−2Ω/S) is proved to be inconsistent with the full Navier–Stokes equations and to constitute no more than a weak approximation. To be more specific, Richardson number similarity is shown to only rigorously apply to certain low‐order truncations of the Navier–Stokes equations (i.e., to certain second‐order closure models) wherein closure ...