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  • Open Access

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

Sébastien Gomé and Anna Frishman*

  • *Contact author: frishman@technion.ac.il
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Phys. Rev. X 16, 031013 – Published 21 July, 2026

DOI: https://doi.org/10.1103/rjxp-pcy2

Abstract

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.

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Popular Summary

Three-dimensional turbulence usually transfers energy toward ever-smaller motions. Rotation, however, profoundly changes this behavior. Rotation generates wavelike motions throughout the fluid while simultaneously promoting the emergence of large-scale, two-dimensional flows that can span the entire system. These sorts of structures play a central role in the dynamics of planetary atmospheres and oceans. We explain how these large-scale flows can be sustained even when energy is injected only into 3D waves. We show that interactions between waves and large-scale motions impose an unexpected conservation law that redirects energy toward large-scale 2D structures. Building on this realization, we develop a theory that analytically predicts the magnitude of this energy transfer as a function of rotation, Reynolds number, and domain geometry that agrees well with numerical simulations. We show how 2D flows become weaker at high rotation, as their coupling to the waves progressively vanishes. The theory solves a major puzzle in how energy can be transferred in geophysical and astrophysical flows, with possible applications in plasma physics, stratified turbulence, and chiral fluids.

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Phys. Rev. X 16, 031013– Published 21 July, 2026
  • Received 22 September 2025
  • Accepted 2 June 2026
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