Anelastic Internal Wave Packet Evolution and Stability

Abstract

Abstract As upward-propagating anelastic internal gravity wave packets grow in amplitude, nonlinear effects develop as a result of interactions with the horizontal mean flow that they induce. This qualitatively alters the structure of the wave packet. The weakly nonlinear dynamics are well captured by the nonlinear Schrödinger equation, which is derived here for anelastic waves. In particular, this predicts that strongly nonhydrostatic waves are modulationally unstable and so the wave packet narrows and grows more rapidly in amplitude than the exponential anelastic growth rate. More hydrostatic waves are modulationally stable and so their amplitude grows less rapidly. The marginal case between stability and instability occurs for waves propagating at the fastest vertical group velocity. Extrapolating these results to waves propagating to higher altitudes (hence attaining larger amplitudes), it is anticipated that modulationally unstable waves should break at lower altitudes and modulationally stable waves should break at higher altitudes than predicted by linear theory. This prediction is borne out by fully nonlinear numerical simulations of the anelastic equations. A range of simulations is performed to quantify where overturning actually occurs.