Nonlinear transfer timescales, steady-state spectral regimes, and wave-mediated energy transport in the buildup of forced rotating turbulence

An experiment by Kolvin, Cohen, Vardi and Sharon (Phys. Rev. Lett. 102, 014503, 2009) showed that when turbulence builds up in a rotating tank from an abruptly applied forcing, energy transport separates into two time scales: a linear one, $\tau_1(k) = hk/2\Omega$, on which inertial waves (restored by the Coriolis force) carry the injected energy through the volume at the linear group velocity, and a longer nonlinear one, $\tau_2(k)$, on which energy is transferred between modes toward large scales. The paper left open four linked questions: the exact functional dependence of $\tau_2(k)$ on wave number and velocity; the much longer relaxation time on which the full spectrum reaches steady state; the parameter conditions under which the steady spectra reported for rotating turbulence ($k^{-3}$ or $k^{-5/3}$) develop; and how much energy linear inertial waves can transport before nonlinear interactions destroy or significantly modify them. Later work, largely by the same laboratory and by independent groups, resolved parts of these questions in restricted regimes: at small Rossby number the nonlinear time extracted from spectral widths scales as $(Uk)^{-1}$, and the steady state decomposes into a quasi-two-dimensional inverse-cascade component plus a three-dimensional inertial-wave component that follows weak-turbulence scaling within a measured validity window. What remains open is the regime of Rossby number of order one, which is the regime of the original buildup experiment and of many geophysical flows: the functional form and multiplicity of the nonlinear transfer times, whether the inverse cascade remains mediated by propagating waves as it is in the strongly driven small-Rossby-number experiments ($0.006 < \mathrm{Ro} < 0.02$) of Shaltiel et al. (2024), the interplay between quasi-2D and wave components (including high-frequency modes beyond the weak-turbulence window), the relaxation dynamics to steady state under steady and intermittent forcing, and a quantitative parameter map of the steady spectral regimes across rotation rate, injected power, Reynolds and Rossby numbers, and domain geometry.

Background

Rotating turbulence and its two competing paradigms. A turbulent flow in a system rotating at angular velocity $\Omega$ is characterized, besides the Reynolds number $\mathrm{Re} = UL/\nu$ (the ratio of inertial to viscous forces, with $U$, $L$ characteristic velocity and length and $\nu$ the kinematic viscosity), by the Rossby number $\mathrm{Ro} = U/(2\Omega L)$, the ratio of inertial to Coriolis forces. When $\mathrm{Ro} \ll 1$ the Coriolis force dominates: the flow becomes columnar along the rotation axis (quasi two-dimensional) and supports inertial waves, the linear oscillations of a uniformly rotating incompressible fluid in which the Coriolis force acts as a restoring force. A plane inertial wave with wave vector $k$ obeys the anisotropic dispersion relation $\omega = 2\Omega\cos\theta$ ($\theta$ the angle between $k$ and the rotation axis) and its group velocity (the velocity at which wave energy travels) has magnitude $|v_g| = 2\Omega/k$, so longer waves carry energy faster. Two idealized descriptions of rotating turbulence compete. In the quasi-2D picture the flow behaves like two-dimensional turbulence, in which energy cascades to large scales (an inverse energy cascade, contrasting with the forward cascade of three-dimensional turbulence) with a horizontal spectrum (in the horizontal wave number $k_\perp$) close to $k_\perp^{-5/3}$; resonant interactions of inertial waves transferring energy to two-dimensional modes have been proposed to build instead a $k^{-3}$ spectrum at large scales (Smith and Waleffe 1999), and the crossover between two- and three-dimensional behavior with increasing rotation was demonstrated numerically by Smith, Chasnov and Waleffe (1996). In the wave-turbulence picture, rotating turbulence is a weakly nonlinear ensemble of inertial waves: weak-turbulence theory (Galtier 2003) predicts an anisotropic forward energy cascade with a definite scaling of the energy spectrum with wave number, frequency, $\Omega$, and the energy flux. Experiments and simulations support aspects of both pictures, and their coexistence and mutual interference are a central modern issue (Campagne et al. 2014, 2015; Alexakis 2023; van Kan 2024).

The source experiment. Kolvin, Cohen, Vardi and Sharon studied the transition from rest to turbulence in a rotating water cylinder (80 cm diameter, 90 cm tall) forced abruptly at the bottom at $t = 0$, measuring horizontal velocity fields by particle image velocimetry (PIV, an optical technique that reconstructs velocity fields from images of tracer particles) at height $h$ above the injector, over rotation rates $6.3 < \Omega < 14$ rad/s and injected powers $1.5 < P < 45$ W, reaching $\mathrm{Re} \sim 10^3$-$10^4$ at $\mathrm{Ro} \sim 1$. In the time evolution of the energy spectrum $E(k,t)$, two fronts appear. The first, arriving at $\tau_1(k) \propto hk/2\Omega$, marks planar inertial waves traveling at the linear group velocity from the injector to the measurement plane; $\tau_1$ is independent of $P$ over a decade of injected power, showing that even high-amplitude waves propagate linearly in this regime. The second, at $\tau_2(k)$, marks nonlinear energy transfer from small to large scales; $\tau_2$ decreases with $k$ and with $P$, and is expected to scale as the eddy-turnover time $\tau_2 \sim (vk)^{-1}$ (the time for a flow structure of size $1/k$ to be overturned by a velocity $v$). Where $\tau_1(k) < \tau_2(k)$, i.e. for $k < (\Omega/(hv))^{1/2}$ (an order-of-magnitude crossover obtained by equating the scaling forms of $\tau_1$ and $\tau_2$, as quoted in the source), inertial waves are the sole energy-transport mechanism, so temporal fluctuations of the energy source are carried through the medium by waves before nonlinear interactions modify them. The paper explicitly leaves open: the exact functional dependence of $\tau_2(k)$ on $k$ and $v$; the much longer time scale on which the entire spectrum reaches steady state; the conditions under which the steady spectra reported in the literature ($k^{-3}$ or $k^{-5/3}$) develop, in particular whether a $k^{-5/3}$ spectrum always develops in driven rotating turbulence; and how and how much energy linear inertial waves can transfer to the whole volume before nonlinearities destroy them.

What is already established. In the same and similar apparatus it is now established that steady rotating turbulence decomposes into a dominant quasi-2D component and a 3D inertial-wave component whose spectrum follows Galtier's weak-turbulence scaling within a measured validity window (Shaltiel, Gat and Sharon 2025); that a nonlinear time scale extracted from the width of the frequency spectrum scales as $(Uk)^{-1}$ at small Rossby number (Yarom, Salhov and Sharon 2017); that abruptly injected energy propagates as inertial-wave packets whose energy is transferred to the background flow only after a competition between propagation time and nonlinear interaction time (Salhov, Yarom and Sharon 2019; Shaltiel et al. 2024); and that in strongly driven steady turbulence at small Rossby number ($0.006 < \mathrm{Ro} < 0.02$, $1500 < \mathrm{Re} < 2100$) the inverse cascade itself appears to be mediated by propagating waves (Shaltiel et al. 2024). Quasi-linear wave-kinetic theories now describe the transfer of energy from 3D waves to large-scale 2D flows as a function of $\Omega$, $\mathrm{Re}$, and domain geometry in idealized settings (Gomé and Frishman 2026), and the linear stability and nonlinear breakdown of finite-amplitude plane inertial waves has been characterized (Skoutnev, Astoul and Barker 2026).

Problem Statement

Determine, experimentally and/or theoretically, the quantitative laws governing energy transfer in forced turbulence in a rotating system, during its buildup and in its steady states, completing the program posed by Kolvin et al. (2009). The problem has four linked components, referring to flows with $\mathrm{Re} \gg 1$ and in particular to the regime around $\mathrm{Ro} \sim 1$ of the original buildup experiment, which is not covered by existing weak-turbulence results:

(1) Nonlinear transfer time. Establish the functional dependence, and possible multiplicity, of the nonlinear mode-interaction time $\tau_2$ on wave number $k$, characteristic velocity $v$ (equivalently injected power $P$ or $\mathrm{Re}$), rotation rate $\Omega$, and system geometry: does the eddy-turnover form $\tau_2 \sim (vk)^{-1}$ hold throughout, or does the $\mathrm{Ro} \sim 1$ regime split into distinct time scales such as the rapid spectral flattening and slow inverse cascade noted in strongly driven small-Rossby-number experiments? The answer must be consistent with the established small-Rossby-number result $\tau_{nl} \propto (Uk)^{-1}$ and with the amplitude-independent linear propagation time $\tau_1 = hk/2\Omega$.

(2) Relaxation to steady state. Measure or predict the time scale on which the full energy spectrum relaxes to statistical steady state after forcing is applied or changed, as a function of $\Omega$, $P$, and geometry, including intermittently driven (energy-burst) forcing, which is the case of geophysical interest emphasized in the source.

(3) Regime map of steady spectra. Determine the regions of parameter space ($\Omega$, $P$, $\mathrm{Re}$, $\mathrm{Ro}$, domain aspect ratio, forcing spectrum and intermittency) in which each steady spectral regime develops — quasi-2D inverse cascade with $k_\perp^{-5/3}$, the $k^{-3}$ large-scale spectrum built by resonant inertial-wave interactions (Smith and Waleffe 1999), weak-turbulence (Galtier) scaling within its measured validity window, or hybrid strongly driven states in which the quasi-2D and wave components coexist — covering every spectrum named as open in the research-objective paragraph, together with a separation of the quasi-2D and wave components that is well defined (a stated, reproducible criterion) and domain-independent (unchanged by the domain aspect ratio and the finite measurement window).

(4) Wave-transport energy budget. Quantify how much energy linear inertial waves transport through the volume before nonlinear interactions redistribute it: the parameter-dependent thresholds (the crossover $k < (\Omega/(hv))^{1/2}$, understood at the scaling level, and its generalizations) and the efficiency of transfer from waves to the quasi-2D flow and to small scales.

A complete answer must be consistent with the wave-mediated inverse cascade observed in strongly driven steady turbulence at small Rossby number ($0.006 < \mathrm{Ro} < 0.02$; Shaltiel et al. 2024), a wave-mediated upscale transfer that idealized weak-turbulence theory does not reproduce and that quasi-linear wave-kinetic theory describes so far only in its scale-separated idealizations (Gomé and Frishman 2026), not in this strongly driven regime, and with the verified weak-turbulence window at small $\mathrm{Ro}$.

References

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