Simulation and stability of two-dimensional internal gravity waves in a stratified shear flow

Abstract

Abstract Two-dimensional interaction of internal waves with critical layers is studied by a combination of numerical simulation and linear stability analysis. Results show that density overturns occur early without a Rayleigh-Taylor instability developing. Instead, Kelvin-Helmholtz billows develop later. Because the instability occurs when profiles vary slowly in a convected frame, linear stability analysis can be applied locally. The analysis shows that two classes of unstable modes should be present; they are both of mixed type. The shear is reduced to explore the properties of oblique modes. It is found that a Rayleigh-Taylor instability in the spanwise direction should be the most unstable mode. The shear-dominated mixed instability arises in the numerical simulation because the spanwise Rayleigh-Taylor instability is not allowed in a two-dimensional simulation. The two-dimensional case does not permit ‘wave breaking’ in the usual sense; instead, an instability develops.