Principal instabilities of large amplitude inertio-gravity waves

Abstract

We analyze the three-dimensional stability properties of monochromatic, large-amplitude, nonhydrostatic inertio-gravity waves propagating in a rotating stratified fluid with no mean shear. The ratio of Coriolis parameter f to the Brunt-Väisälä frequency N is fixed at a value f/N=0.01, chosen to be representative of the Earth’s atmosphere and upper oceans in extra-tropical latitudes. By using Floquet theory combined with suitable coordinate transformations, our analysis provides an exact representation of the spatial and temporal periodicity of the basic wave state. Growth rates are computed for several discrete phase-elevation angles 80°⩽θ⩽89.7°, spanning the frequency range in which rotational effects become important, and the basic wave state changes from fully overturned to suboverturned. For each basic wave, we identify the growth rate peaks in disturbance wavenumber space—these are the principal modes. At the lowest phase-elevation angle considered, θ=80°, the dominant principal mode is similar to its low-frequency nonrotating counterpart, with disturbance roll axes aligned in, or nearly in, the direction of the largest component of basic wave shear. As θ increases to values where the effects of rotation are more strongly manifested, the dominant rotating mode shifts to oblique orientation. At still higher values of θ, the fastest-growing modes have roll axes oriented orthogonal to the main shear component of the basic wave. At the highest values of θ considered in this study, the waves are no longer vertically overturned, but still feature minimum Richardson numbers below 14. We find that such low-frequency waves are subject to wave-scale instabilities: for the largest amplitude considered, the instability shows no preferred orientation, while at a somewhat lower amplitude, the dominant instability prefers an oblique orientation. Dominant oblique instabilities have not been reported in previous approximate stability analyses of inertio-gravity waves, although they have been found in nonlinear simulations. Our analysis resolves this discrepancy.