Multiple scales kinetic theory for rotationally constrained slow inertial waves and anisotropic dynamics

Abstract

Wave kinetic theory for rapidly rotating flows is developed in this paper using a rigorous application of multiple scales perturbation theory. The governing equations are an asymptotically reduced set of equations that are derived from the incompressible Navier-Stokes equations. These equations are applicable for rapidly rotating flow regimes and are best suited to describe anisotropic dynamics of rotating flows. The independent variables of these equations inherently reside in a helical wave basis that is the most suitable basis for inertial waves. A coupled system of equations for the two global invariants: \textit{energy} and \textit{helicity}, is derived by extending a simpler symmetrical system to the more general non-symmetrical helical case. This approach of deriving the kinetic equations for helicity follows naturally by exploiting the symmetries in the system and is different from the derivations presented in earlier work of \citet{Galtier03, Galtier14} that uses multiple correlation functions to account for the asymmetry due to helicity. Stationary solutions, including Kolmogorov solutions, for the flow invariants are obtained as a scaling law of the anisotropic wave numbers. The scaling law solutions compare affirmatively with results from recent experimental and simulation data. The theory developed in this paper pertains to the wave dynamics supported by an asymptotically reduced set of hydrodynamic equations and therefore encompasses a different dynamical regime compared to the weak turbulence theory presented in the work of \citet{Galtier03, Galtier14}.