Toward a Unified Theory of Gravity Wave Stability

Abstract

This paper reveals relationships among linear instabilities of internal gravity waves often supposed to be independent. Using a Floquet analysis of a monochromatic wave propagating in a uniformly stratified background without shear, which accounts for finite wave amplitude, spatial and temporal periodicity, tilted phase planes, and 3D disturbances, it is demonstrated that the dominant instabilities of overturning waves are the large-wave-amplitude manifestations of resonant and slantwise instabilities of small amplitude waves, and that they possess no threshold amplitudes. An energy budget analysis examines the relation of parametric instabilities at large wave amplitude to vertical dynamic and static instabilities; however, the instability characteristics for propagating waves are very different from those inferred by analogy to Kelvin–Helmholtz instability and Bénard convection in simpler backgrounds. At small amplitudes, resonant instabilities rely on horizontal or slantwise gradients of wave properties; in particular, parametric subharmonic instability is related to slantwise static instability. Three-dimensional instabilities with a preferred oblique orientation are found: these are related to wave-shear-aligned instabilities at large amplitude and to higher-order resonances at small wave amplitude. A simplified classification of gravity wave instability is proposed.