Wave turbulence in a rotating channel

Abstract

AbstractThis paper describes wave-turbulence closure and its consequences for rapidly rotating (i.e. small Rossby number) turbulence confined by two infinite, parallel walls perpendicular to the rotation axis. Expressing the flow as a combination of inertial waveguide modes leads to a spectral matrix, whose diagonal elements express the distribution of energy over modes and whose off-diagonal elements represent correlations between modes of different orders. In preparation for wave-turbulence closure, the flow is decomposed into two-dimensional and wave components. The former is found to evolve as if it were a classical, two-dimensional, non-rotating flow, but with wall friction due to Ekman pumping by the boundary layers. Evolution equations for the wave-component elements of the spectral matrix are derived using a wave-turbulence approach. Detailed analysis of these equations shows that, surprisingly, the two-dimensional component has no effect on wave-component energetics. As expected for wave turbulence, energy transfer between wave modes is via resonant triads and takes place at times $O(\varepsilon ^{-2})$ multiples of the rotational period, where $\varepsilon $ is the Rossby number. Despite playing no role in wave-mode energetics, the two-dimensional component produces decay of the off-diagonal elements of the spectral matrix on a time scale that is small compared with $O(\varepsilon ^{-2})$ rotation periods. There are thus three asymptotically distinct stages in the evolution of the turbulence in the limit of small Rossby number: the two-dimensional flow begins to evolve at the usual large-eddy turnover time scale ($O(\varepsilon ^{-1})$ multiples of the rotation period) and continues to develop thereafter. This is followed by decorrelation of different wave orders and finally evolution of the wave energy spectra due to resonant interactions.