Fluctuating hydrodynamics and turbulence in a rotating fluid: Universal properties

TL;DR

This paper develops a theoretical framework to analyze the statistical properties of 3D turbulence in rotating fluids, revealing anisotropic scaling behaviors and deviations from pure 2D turbulence, with results applicable to experiments and simulations.

Contribution

It introduces a generating functional and hierarchical equations for rotating turbulence, providing new insights into anisotropic scaling and the nature of flows at large rotation.

Findings

Velocity correlations parallel to rotation scale as q^{-5/3} (Kolmogorov scaling).

Perpendicular velocity correlations scale as q^{-3}.

Large rotation induces anisotropic turbulence behavior.

Abstract

We analyze the statistical properties of three-dimensional ($3d$) turbulence in a rotating fluid. To this end we introduce a generating functional to study the statistical properties of the velocity field $\bf v$. We obtain the master equation from the Navier-Stokes equation in a rotating frame and thence a set of exact hierarchical equations for the velocity structure functions for arbitrary angular velocity $\mathbf \Omega$. In particular we obtain the {\em differential forms} for the analogs of the well-known von Karman-Howarth relation for $3d$ fluid turbulence. We examine their behavior in the limit of large rotation. Our results clearly suggest dissimilar statistical behavior and scaling along directions parallel and perpendicular to $\mathbf \Omega$. The hierarchical relations yield strong evidence that the nature of the flows for large rotation is not identical to pure two-dimensional flows. To complement these results, by using an effective model in the small-$\Omega$ limit, within a one-loop approximation, we show that the equal-time correlation of the velocity components parallel to $\mathbf \Omega$ displays Kolmogorov scaling $q^{-5/3}$, where as for all other components, the equal-time correlators scale as $q^{-3}$ in the inertial range where $\bf q$ is a wavevector in $3d$. Our results are generally testable in experiments and/or direct numerical simulations of the Navier-Stokes equation in a rotating frame.