Evolution and Stability of Two-Dimensional Anelastic Internal Gravity Wave Packets

Abstract

The weakly nonlinear evolution, stability, and overturning of horizontally and vertically localized internal gravity wave packets is examined for a nonrotating, anelastic atmosphere that is stationary in the absence of waves. The weakly nonlinear evolution is examined through the derivation of their wave-induced mean flow, which is used to formulate a nonlinear Schrödinger equation. The induced flow is manifest as a long, hydrostatic, bow wake-like disturbance, whose flow direction transitions from positive on the leading flank of the wave packet to negative on the trailing flank of the wave packet. As such, two-dimensional wave packets are always modulationally unstable. This instability results in enhanced amplitude growth confined to either the leading or trailing flank. Hence, when combined with anelastic growth predicted by linear theory, we anticipate two-dimensional waves will overturn either somewhat below or just above the heights predicted by linear theory. Numerical solutions of the Schrödinger equation are compared with the results of fully nonlinear simulations to establish the validity of the weakly nonlinear theory. Actual wave overturning heights are determined quantitatively from a range of fully nonlinear simulations.

Flows induced by Coriolis-influenced vertically propagating two-dimensional internal gravity wave packets

“…Because the waves also grow in amplitude due to anelastic effects, the induced flow may have significant impact upon the evolution and consequent breaking level of the waves. This work has focused upon waves of such small amplitude that the induced flow has negligible weakly nonlinear influence upon the evolution of the waves through Doppler-shifting that can lead to enhanced amplitude growth through (modulational instability) causing the waves to overturn at altitudes lower than predicted by linear theory, or can enhance vertical dispersion (modulational stability) leading to breaking at relatively higher altitudes 21,23,24 . Consequently, the change in the structure of the mean flow predicted by our theory as it depends upon |m/k|, |f |/N and kσ z /µ 0 suggests that this should be an important consideration in predicting the ultimate level of breaking and momentum deposition.…”

“…This is because the negative flow on the trailing flank would increase the extrinsic frequency of the waves causing their vertical group velocity to increase. Hence the amplitude envelope in the lee of the wavepacket would narrow and grow in amplitude, as has been investigated in studies of horizontally periodic (one-dimensional) wavepackets [21][22][23] and non-rotating two-dimensional wavepackets 24 . However, if the waves dominately induce an evanescent disturbance, then the flow is negative over the center of the wavepacket.…”

Self‐Acceleration and Instability of Gravity Wave Packets: 2. Two‐Dimensional Packet Propagation, Instability Dynamics, and Transient Flow Responses

“…A recent analysis by Gervais et al () addressed 2‐D GW packet evolution in an anelastic atmosphere. They inferred that modulational instability occurs for all 2‐D GW packets but did not consider the influences of rapid 3‐D instabilities that likely obviate, or reduce, the importance of modulational instability in 2‐D and 3‐D packet evolutions.…”

Propagation and overturning of three-dimensional Boussinesq wave packets with rotation

“…Two-dimensional wave packets, unlike their one-dimensional counterparts, are always modulationally unstable, as the induced mean flow (long waves) is positive above and negative below the center of the wave packet [13]. Sutherland [14] heuristically formulated a prediction for the amplitude at which Boussinesq wave packets ultimately overturn due to self-acceleration, according to the wave-induced mean flow exceeding the streamwise group velocity.…”