Two-dimensional internal gravity wave beam instability. Linear theory and subcritical instability

Abstract

The instability of propagating internal gravity waves (IGWs) is of long-standing interest in geophysical fluid dynamics since breaking IGWs exchange energy and momentum with the large-scale flow and hence they support the large-scale circulation. In this study a low-order IGW beam model is used to delineate both linear and so called non-modal transient instability. In the first part of the study, linear normal mode instability of a wave beam consisting of two finite-amplitude plane monochromatic IGWs with the same frequency and parallel wave vectors of different magnitude is investigated using the Galerkin method. It is concluded that the wave beam is linearly more unstable than its constituent plane waves, taken separately. The degree of instability increases with the separation of the constituent waves in the wave number space, that is, with the wave beam concentration in the physical space. The narrower a wave beam is, the more linearly unstable it is. In its turn, transient instability typically occurs for linearly stable flows or before linear instability can set in (subcritical instability) if the governing system matrix is non-normal. In the second part of the paper, first the non-normality of the linear system matrix of the wave beam model is examined by computing measures like the Henrici number, the pseudospectrum, and the range of the matrix. Subsequently, the robustness of the transient growth is studied when the initial condition for optimal growth is randomly perturbed. It is concluded that for full randomisation, in particular, shallow wave beams can show subcritical growth when entering a turbulent background field. Such growing and eventually breaking wave beams might add turbulence to existing background turbulence that originates from other sources of instability. However, the robustness of transient growth for wave beam perturbations depends strongly on the strength of randomisation of the initial conditions, the beam angle and the perturbation wavelength.