Short-wavelength instabilities of waves in rotating stratified fluids

Standing and propagating plane waves in a rigidly rotating stably stratified fluid are considered. Based on a WKB analysis, it is shown numerically that almost all of them are unstable with respect to three-dimensional short-wavelength perturbations. The waves whose wavenumber vector is parallel to the axis of rotation and stratification are stabilized as the Brunt-Väisälä frequency N (normalized by the angular velocity of rigid rotation) increases, whereas the inclined waves are unstable at any N. These instabilities are caused by parametric resonances of low order (first and second) and the instability growth rate is determined theoretically, when the wave amplitude is small.