The evolution and collapse of a localised patch of turbulence in a stratified fluid is significantly influenced by the presence of internal waves. For example, a turbulent patch collapses rapidly in a stratified fluid, in part because internal waves that propagate away from the mixing region extract energy and momentum from the turbulence. Numerous laboratory experiments have been performed to quantify the energy and momentum fluxes associated with internal waves radiating from stratified turbulence, for example, in the wake behind a towed obstacle and surrounding an oscillating grid. In such experiments, an unexplained phenomena is observed. Although in theory the turbulence excites a broad frequency range of disturbances, nonetheless internal waves typically are generated within only a narrow frequency band. In mixing-box experiments, for example, the largest amplitude waves are observed to propagate at angles between 30 to 45 degrees to the vertical.
In this work, a simple argument based on weakly nonlinear theory is proposed to help explain this phenomena. Specifically, it is proposed that an internal wave is unstable if the ratio of its maximum vertical displacement to its horizontal wavelength is greater than sin(2 Theta)/(pi sqrt(8)), in which Theta is the angle of the wave characteristics to the vertical. In particular, waves that propagate at an angle of 45 degrees to the vertical are the most stable, and waves that propagate at very small angles to the vertical or horizontal are stable only if they are of infinitessimal amplitude. The predicted stability boundary is in excellent agreement with numerical simulations of horizontally periodic internal waves in uniformly stratified, stationary and non-rotating flow. However, horizontally compact waves are stable for larger amplitudes than predicted if the angle of their characteristics to the vertical is moderately smaller than 45 degrees.
In light of these results it is proposed that turbulence-excited internal waves are observed within a narrow frequency band because the waves are of such large amplitude that only a narrow frequency band is stable. To test this hypothesis experimentally, a new laboratory technique is used, in which the amplitude of internal waves can be measured continuously but non-intrusively over a two-dimensional area.
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12th Conference on Atmospheric and Oceanic Fluid Dynamics