Discreteness and resolution effects in rapidly rotating turbulence

Rotating turbulence is characterized by the nondimensional Rossby number Ro, which is a measure of the strength of the Coriolis term relative to that of the nonlinear term. For rapid rotation (Ro -> 0), nonlinear interactions between inertial waves are weak, and the theoretical approaches used for other weak (wave) turbulence problems can be applied. The important interactions in rotating turbulence at small Ro become those between modes satisfying the resonant and near-resonant conditions. Often, discussions comparing theoretical results and numerical simulations are questioned because of a speculated problem regarding the discreteness of the modes in finite numerical domains versus continuous modes in unbounded continuous theoretical domains. This argument finds its origin in a previous study of capillary waves, for which resonant interactions have a very particular property that is not shared by inertial waves. This possible restriction on numerical simulations of rotating turbulence to moderate Ro has never been quantified. In this paper, we inquire whether the discreteness effects observed in capillary wave turbulence are also present in inertial wave turbulence at small Ro. We investigate how the discreteness effects can affect the setup and interpretation of studies of rapidly rotating turbulence in finite domains. In addition, we investigate how the resolution of finite numerical domains can affect the different types of nonlinear interactions relevant for rotating inertial wave turbulence theories. We focus on Rossby numbers ranging from 0 to 1 and on periodic domains due to their relevance to direct numerical simulations of turbulence. We find that discreteness effects are present for the system of inertial waves for Rossby numbers comfortably smaller than those used in the most recent numerical simulations of rotating turbulence. We use a kinematic model of the cascade of energy via selected types of resonant and near-resonant interactions to determine the threshold of Ro below which discreteness effects become important enough to render an energy cascade impossible.