The Stability of Short Symmetric Internal Waves on Sloping Fronts: Beyond the Traditional Approximation

The interaction of internal waves with geostrophic flows is found to be strongly dependent upon the background stratification. Under the traditional approximation of neglecting the horizontal component of the earth's rotation vector, the well-known inertial and symmetric instabilities highlight the asymmetry between positive and negative vertical components of relative vorticity (horizontal shear) of the mean flow, the former being stable. This is a strong stratification limit but, if it becomes too low, the traditional approximation cannot be made and the Coriolis terms caused by the earth's rotation vector must be kept in full. A new asymmetry then appears between positive and negative horizontal components of relative vorticity (vertical shear) of the mean flow, the latter becoming more unstable. Particularly conspicuous at low latitudes, this new asymmetry does not require vanishing stratification to occur as it operates readily for rotation/stratification ratios 2 Omega/N as small as 0.25 (the stratification still dominates over rotation) for realistic vertical shears. Given that such ratios are easily found in ocean atmosphere boundary layers or in the deep ocean, such ageostrophic instabilities may be important for the routes to dissipation of the energy of the large-scale motions. The energetics show that, depending on the orientation of the internal wave crests with respect to the mean isopycnal surfaces, the unstable motions can draw their energy either from the kinetic energy or from the available potential energy of the mean flow. The kinetic energy source is usually the leading contribution when the growth rates reach their maxima.