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:
We experimentally study the characteristics of an inverse energy cascade in deep rotating turbulence. Experiments were performed in a rotating cylindrical water tank with energy injection at a small scale. The steady state energy spectrum of the horizontal velocity field at scales larger than the injection scale is very well described by the 2D inverse energy cascade spectrum. Transient energy spectra evolve via 2D-like inverse cascade, with energy transfer from small to large scales. The measured energy transfer rates are in good quantitative agreement with the predictions for 2D turbulence.
We measure and quantify the spatial transport of energy within steady rotating hydrodynamic turbulence. A steady turbulent field in a rotating tank is perturbed by a short and abrupt increase of energy injection at a well-defined plane. Initially, a wave packet of inertial modes is generated within the background turbulent field, propagating according to the dispersion relation of inertial waves, much like wave propagation within a static fluid. At this stage, the background turbulence is only weakly affected by the pulse passage. Only at longer times, which are determined by the pulse initial amplitude and spectrum, energy is efficiently transferred from the wave packet to the background flow. The energy, which is physically injected at the bottom of the tank, is, therefore, effectively injected at a higher plane. The height of maximum energy transfer efficiency is estimated and varied via manipulations of experimental parameters.
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