On plane internal waves: their amplification, and potential to break in a rotating stratified quiescent fluid

Abstract

ABSTRACTIn this article, we assess the potential of plane internal waves to break in the interior of a rotating but otherwise quiescent stratified fluid. We show that wave energy density of a monochromatic wave is uniform in space and time, and calculate two heuristic stability measures: the isopycnal steepness (for overturning instability) and the Froude number (for shear instability) as functions of the wave energy density and phase to determine the regions where fast perturbation growth and subsequent wave breaking are likely to occur. For purely inertial waves, the Froude number field is uniform, and wave breaking is equally probable at any point. For non-inertial waves the locations of the minimal convective and shear stability always coincide, but the onset of shear instability almost always occurs at lower energy densities than the onset of overturning for all wave frequencies. In a rotating fluid, the separation between the two critical energy densities is most pronounced at low frequencies. At high frequencies, the critical energies for the instabilities nearly coalesce. In a non-rotating fluid, in contrast, the two critical energies coalesce at all frequencies. When a wave travels through a jump from weaker to stronger stratification, it amplifies in both steepness and Froude number. For the transmitted wave, the amplification factors for both stability measures are a monotonic function of the ratio of the two buoyancy frequencies. The Froude number amplification factor is a nonlinear function of the wave energy density: waves entering a region with a stronger stratification with larger initial energy amplify to a larger Froude number. The amplification upon refraction is stronger in both steepness and Froude number when nonhydrostatic terms are retained in the governing equations. In contrast to the transmitted wave, the superposition of the incident and the reflected waves on the other side of the stratification jump can become destabilised for both scenarios (the incident wave entering either stronger or weaker stratification). When the sufficiently energetic incident wave enters stronger stratification, the flow is destabilised simultaneously on both sides of the interface. When the incident wave enters weaker stratification, only the superposition of the incident and reflected waves can become destabilised, but not the transmitted wave.