Waves Maintain Large-Scale 2D Flows in Rotating Turbulence and Cause Their Demise

Turbulence follows a few well-known organizational principles, rooted in conservation laws. One such principle states that a system conserving two sign-definite invariants self-organizes into large-scale structures. Ordinary three-dimensional turbulence does not fall within this paradigm but is profoundly altered when subject to rotation. In rotating turbulence, 3D inertial waves coexist alongside emergent two-dimensional structures, which tend to take the form of domain-scale flows called condensates. This interplay raises a fundamental question: Why and when are 2D flows sustained if only 3D waves are excited? We develop a quasilinear wave-kinetic theory to answer this question. We show that near-resonant interactions between 3D waves and a large-scale 2D flow impose an additional conservation law: Waves must conserve their helicity separately for each helicity sign. This emergent sign-definite invariant constrains the waves to transfer their energy to large-scale 2D motions, which maintains the latter in statistical steady state. We derive analytical expressions for the 3D-2D energy transfer as a function of rotation, Reynolds number, and domain geometry in a rotation-dominated regime and compare them with extensive numerical simulations of the rotating 3D Navier-Stokes equations. As rotation increases, the energy transfer from the waves to the 2D flow progressively vanishes as the two decouple, leading to a transition between distinct classes of turbulence: from 2D-dominated to 3D-dominated wave turbulence. Our theory shows that this gradual transition is caused by a depletion of modes satisfying the resonance conditions and exhibits good agreement with numerical simulations when the number of near-resonant modes is not too small. We discuss such limitations of our theory, as well as the validity range of its underlying assumptions. Together, these results suggest a mechanism underlying two-dimensionalization in rotating turbulence and, more broadly, illustrate how nonlinear systems sustaining waves can self-organize into anisotropic, zero-frequency structures.