On the role of optimal perturbations in the instability of monochromatic gravity waves

Abstract

Motivated by the useful new insights from optimal-perturbation theory into the onset of turbulence in other fields singular vectors (SVs) in stable and unstable gravity waves have been determined within the framework of the Boussinesq equations on an f plane. The difference between the dynamics of normal modes (NMs) and SV is characterized by a time invariance in the comparative role of the various possible exchange processes between NM and basic wave, while SV can have a highly time-dependent structure, allowing a more efficient energy exchange over a finite time. Both inertia-gravity waves (IGWs) and high-frequency gravity waves (HGWs) have been considered. At Reynolds numbers typical for the middle to upper mesosphere IGW admit rapid nonmodal growth even when no unstable NMs exist. SV energy growth within one Brunt-Vaisala period can cover two orders of magnitude, suggesting the possibility of turbulence onset under conditions where this would not be predicted by a NM analysis. HGWs show a dependence of short-term optimal growth on the direction of propagation of the perturbation with respect to the wave which is, at weak to moderate wave amplitudes, quite different from that of NM but reproduced in ensemble integrations from random initial perturbations. Their SVs are sharply peaked pulses with negligible group velocity which are repeatedly excited as the rapidly propagating wave passes over them. The transition of these to the leading NM, which is not moving with respect to the wave and which is typically broader in structure, is very slow, so that in many cases the turbulence onset via local perturbations of a gravity wave might be more appropriately described using optimal-perturbation theory. This might contribute to a better understanding of the often observed occurrence of thin turbulent layers in the middle atmosphere.