Experimental observation of steady inertial wave turbulence in deep rotating flows

Abstract

The theoretical framework that should be used for describing rotating turbulence 1-3 is the subject of an active debate. It was shown experimentally 4,5 and numerically 6,7 that the formalism of 2D turbulence is useful in the description of many aspects of rotating turbulence. On the other hand, theoretical and numerical work suggests that the formalism of wave turbulence 8-10 should provide a reliable description of the entire 3D flow field 11-15. The waves that are suggested as the basis for this turbulence are Coriolis-force-driven inertial waves 1. Here we present experimental results that suggest the existence of inertial wave turbulence in deep steady rotating turbulence. Our measurements show energy transfer from the injection scale to larger scales, although the energy spectra are concentrated along the dispersion relation of inertial waves. The turbulent fields are, therefore, well described as ensembles of 3D interacting inertial waves. Rotating flows are described by the rotating Navier-Stokes equation 1. For incompressible fluids this equation is characterized by two dimensionless numbers: the Reynolds number, Re = UL/v, and the Rossby number, Ro = U /2ΩL (U and L are the typical velocity and length in the flow, v the kinematic viscosity and Ω = | | the rotation rate of the system). Rotating turbulence is obtained for Re 1 and Ro < 1. Such flows, which are common in various geophysical and planetary systems, have been studied extensively 16. It was found that the horizontal component of the velocity field in such flows shares many similarities with 2D turbulence. In particular, the cascade of energy to large scales 5,7,17 , the formation of coherent structures 18 and the self-similarity of the flow field 4 appear in both types of flows. However, the 2D description cannot describe the entire 3D rotating turbulent field. In addition, it is not obtained rigorously from the rotating Navier-Stokes equation. For small enough Ro (fast rotation), the linearized (rotating) Navier-Stokes equation supports the propagation of waves, known as inertial waves 1. Each component of the velocity field, u = (u x , u y , u z), varies in space and time via the propagation of plane waves of the form u x,y,z ∝ e i(k•r+ωt) , where ω and k are the wave frequency and wavevector, respectively. These waves obey a unique dispersion relation:

Absorption of waves by large-scale winds in stratified turbulence

“…An interesting fact, that sheds more light on how these mechanisms work, is that these effects are not observed in purely rotating flows [26,43]. While these flows can also generate strong horizontal velocity fields, waves tend to travel in the vertical direction (as compared to the stratified case, where they tend to travel horizontally), and no Doppler shift develops as the waves are perpendicular to the large-scale flow.…”

“…As already mentioned, an inverse cascade is a self-similar process by which energy is transferred nonlinearly from small scales to larger scales, so energy gets accumulated in structures with larger correlation length than that of the injection mechanism. Inverse cascades have been ex-tensively studied in two dimensional turbulence [1], and have also been observed in rotating turbulence (including rotating stratified turbulence) [24][25][26]. But results for purely stratified turbulence are unclear.…”

Inertial Wave Turbulence Driven by Elliptical Instability

“…Following [31,32], we analyze how the kinetic energy is located around the dispersion relation of inertial waves given by ω ¼ AE2 cos θ, where ω is the dimensionless frequency, and θ is the angle between the wave vector and the rotation axis. This is achieved by computing the spatiotemporal Fourier transformûðk; ωÞ of the velocity field uðx; tÞ and summing contributions associated with the same angle θ.…”

“…Barker and Lithwick [19], in their local model of the elliptical instability, nonetheless showed that strong geostrophic flows emerge during the saturation and disrupt the inertial wave resonances, leading instead to growth and decay cycles. This competition between dominant geostrophic modes and inertial waves is reminiscent of the duality observed in rotating turbulence where, on one hand, geostrophic flows are widely observed [30], and on the other hand, inertial waves have been shown to survive on top of the turbulent background [31][32][33][34]. Conversely from experiments and numerical simulations of rotating turbulence, where energy is injected arbitrarily into both vortices and inertial waves through an external artificial forcing, the elliptical instability provides a natural mechanism that initially injects energy into a few inertial waves only, whose properties can be theoretically predicted.…”

“…The energy stored in the geostrophic modes is comparable to the energy in the rest of the flow. Similarly to what is observed in forced rotating turbulence [31,40] or rapidly rotating thermal convection [41][42][43], a nonlocal inverse cascade of the geostrophic modes takes place until, at t ≃ 120 rotations onwards, one single large-scale vortex, or condensate, remains [ Fig. 1(c)].…”