Exact two-dimensionalization of rapidly rotating large-Reynolds-number flows

TL;DR

This paper proves that rapidly rotating, high-Reynolds-number flows in bounded domains become two-dimensional over time, providing rigorous insights into the stability of 2D solutions and implications for rotating turbulence theories.

Contribution

It establishes rigorous conditions under which 3D rotating flows stabilize to 2D solutions at high Reynolds numbers, challenging existing turbulence models.

Findings

2D solutions are stable under strong rotation at high Reynolds numbers

Flow becomes 2D in the long-time limit, with no energy dissipation anomaly

Results challenge wave turbulence theory applicability in bounded rotating domains

Abstract

We consider the flow of a Newtonian fluid in a three-dimensional domain, rotating about a vertical axis and driven by a vertically invariant horizontal body-force. This system admits vertically invariant solutions that satisfy the 2D Navier-Stokes equation. At high Reynolds number and without global rotation, such solutions are usually unstable to three-dimensional perturbations. By contrast, for strong enough global rotation, we prove rigorously that the 2D (and possibly turbulent) solutions are stable to vertically dependent perturbations: the flow becomes 2D in the long-time limit. These results shed some light on several fundamental questions of rotating turbulence: for arbitrary Reynolds number and small enough Rossby number, the system is attracted towards purely 2D flow solutions, which display no energy dissipation anomaly and no cyclone-anticyclone asymmetry. Finally, these results challenge the applicability of wave turbulence theory to describe stationary rotating turbulence in bounded domains.