Waves drive large-scale 2D flows in rotating turbulence and cause their demise

TL;DR

This study addresses how rotating 3D turbulence sustains large-scale 2D flows despite forcing that excites 3D inertial waves. It combines direct numerical simulations with a quasi-linear wave-mean-flow kinetic framework to show that near-resonant, scale-separated interactions impose a sign-definite helicity conservation for each wave chirality, directing energy into the large-scale 2D condensate. As rotation increases, resonant coupling weakens and 3D–2D energy transfer vanishes in the fast-rotation limit, producing a transition from 2D-dominated to 3D-dominated wave turbulence. The authors derive analytical expressions for the condensate amplitude and the 3D–2D energy transfer as functions of rotation, Reynolds number, and domain geometry, which align with DNS results. Overall, the work elucidates a fundamental mechanism for two-dimensionalization in rotating turbulence and illustrates how wave-bearing nonlinear systems can self-organize into zero-frequency, anisotropic structures.

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. However, when subject to rotation, 3D turbulence is profoundly altered: rotation produces 3D inertial waves, while also sustaining emergent two-dimensional structures and favoring domain-scale flows called condensates. This interplay raises a fundamental question: why and when are 2D flows sustained even when only 3D waves are excited? Using extensive numerical simulations of the rotating 3D Navier-Stokes equations together with a quasi-linear wave-kinetic theory, 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. However, as rotation increases, resonance conditions become more restrictive and the energy transfer from the waves to the 2D flow progressively vanishes, leading to a transition between distinct classes of turbulence, from 2D-dominated to 3D-dominated wave turbulence. We derive analytical expressions for this 3D-2D energy transfer as a function of rotation, Reynolds number and domain geometry, which show a good agreement with numerical simulations. Together, these results establish a mechanism underlying two-dimensionalization in rotating turbulence, and, more broadly, illustrate how non-linear systems sustaining waves can self-organize into anisotropic, zero-frequency structures.

Figures (9)

• Figure 1: Formation of large-scale 2D condensates in rotating 3D NSE. (a) Temporal evolution of the total energy for $Ro=0.011$ and $Re= 9.3$, plotted in viscous time scale. (b) Condensate energy (normalized by $(\epsilon L_y/2\pi)^{2/3}$) as a function of $Ro$ and $Re$. (b-d) Flow visualizations of vertical vorticity at various values of $Ro$ and fixed $Re=9.3$. At low rotation $Ro \geq 0.5$ (c), the flow is not rotationally-constrained and only exhibits a turbulence of 3D eddies. With decreased $Ro$, the flow becomes $z$-invariant and takes the form of box-filling jets (d). At very low $Ro$ for this $Re$ (e), the two-dimensionalization stops ($\langle U^2 \rangle=0$), and the flow consists of 3D inertial waves (circles in (a), which show the total energy).
• Figure 2: (a) Energy fluxes across scales, measured from the DNS at $Ro= 0.011$, $Re=23$. (b) Interaction types and (c) time-scale hierarchy in rotating 3D NSE sustaining a condensate of shear-rate ${U'}$. The 3D-forced flow is dominated by 3D-2D interactions occurring over a time scale $\sim 1/{U'} \ll \tau_{nl}$, the characteristic time scale of 3D-3D interactions. Under fast rotation ${U'} \ll 2\Omega$, 3D modes consist in inertial waves and all interactions are restricted by wave resonances.
• Figure 3: (a) Rescaled condensate mean shear rate ${U'}/\Omega$ from Navier-Stokes simulations (squares), illustrating the collapse with $Ro_{\epsilon}=\sqrt{\epsilon/\nu}/(2\Omega) = Ro Re^{1/2}$ (same colors as Fig. \ref{['fig1']}). (b) Energy fluxes to the condensate due to 3D waves of various chiralities. At high rotation, the condensate-wave interactions do not mix wave chiralities, leading to the conservation of sign-definite helicity $H^{\pm}$ for the waves. As a consequence, waves energize the condensate ($\Pi^{s,s}<0$). The 2D flow eventually decouples from the waves when $Ro_{\epsilon}\to0$ following scaling \ref{['eq:U_O_w2d']} (black lines in panel (a)). The red line in (a) shows the numerical closure of \ref{['eq:global_bal']} with $T_{\rm 3D-2D}$ in \ref{['eq:dec_scalings2']}. Inset: comparison with the numerical QL solution obtained with a Lorentzian approximation of the oscillating factor (blue line).
• Figure 4: Energy transfer from the 3D waves to the 2D flow, $T_{\rm 3D-2D}$, as a function of the mean shear rate $U'/\Omega$, as measured from the DNS (solid points) and predicted from the QL theory. In the theory, near-resonances are treated either with a Heaviside (red line) or Lorentzian function of $U'/\Omega$ (blue line).
• Figure 5: (a) Energy transfer from the 3D waves to the 2D flow, $T_{\rm 3D-2D}$, measured in the DNS of rotating 3DNSE as a function of $Ro_{\epsilon} = Ro \times Re^{\frac{1}{2}}$. As waves progressively decouple from the 2D flow, $T_{\rm 3D-2D} \underset{Ro_{\epsilon} \to 0}{\to}0$. White points correspond to values of $T_{\rm 3D-2D}$ below statistical error. Solid lines: predictions from the QL theory. Red: numerical solution with the exact function $T_{\rm 3D-2D}({U'}/\Omega)$, predicted with a Heaviside treatment of near-resonances; black: approximate expressions derived in \ref{['eq:eps32_leading_main']}. Below $Ro_{\epsilon} \sim \frac{l_f^2}{L_y^2} \left(\frac{l_f}{L_z}\right)^{\frac{1}{2}}$, the modes closest to $p_z= 0$ decouple and $T_{\rm 3D-2D} = 0$. Blue: numerical solution with a Lorentzian treatment of near-resonances. (b) Predictions for the 3D-2D energy transfer with the Heaviside approximation, shown in the ($Ro_{\epsilon}, L_y/l_f$) plane.