Energy and entropy evolution of interacting internal gravity waves and turbulence

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Abstract
The energy and entropy evolutions of two-dimensional interacting internal gravity waves and turbulence in the (x, z) plane are studied in inviscid, viscous decay and forced-dissipative numerical simulation experiments using a spectral model. In each case the entropy evolution is compared with that predicted by the eddy-damped quasinormal Markovian (EDQNM) closure formulated by Carnevale and Frederiksen (1983) for two-dimensional internal waves. Although there is no spectral gap between the internal waves and the turbulence in the experiments, we define a transition wavenumber k 1 such that for wave-numbers k ≪ k 1 wave motion predominates and for wavenumbers k≫k 1 turbulence is dominant. In both experiments in which wave motion is generated from turbulence and in which turbulence is generated from wave instability, it is found that the entropy production and energy transfers are inhibited for increased, Brunt-Väisälä frequencies or transition wavenumbers. In experiments in which a single wave (k x=2, k z= 1) decays to smaller scale disturbances, it is found that there is preferential entropy and energy production of smaller scales at the peak of the potential energy cycle of the wave (2,1). The behaviour is similar to that of the Mathieu equation describing linear internal wave instability, although nonlinear effects are shown to be important at least during some parts of the cycle. The relaxation back to statistical steady state of a forced-dissipative internal wave system displaced from that state by the addition of a wave component (2,1) having substantial amplitude is studied. The initial energy spectrum with an approximate k −3 power law for k ≳ 7 first becomes flatter as the wave (2,1) loses energy to neighbouring wavenumber bands and to the smaller scales and then eventually the spectrum returns to the approximate k −3 power law as viscous effects remove the excess small scales. The initial increase and subsequent decrease of the entropy as the system returns to statistical steady state is again consistent with that predicted by the EDQNM closure. There is a tendency for the energy of the large scales to preferentiaily populate modes which are elongated in the x-direction.

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