Floquet versus local stability analysis of internal gravity–inertial waves

This paper is concerned with the stability of time-harmonic plane internal gravity–inertial waves in an unbounded, uniformly stratified, rotating fluid under the Boussinesq and f f $f$ -plane approximations. The focus is on how the often-used ‘local’ analysis of short-scale instabilities, which is based on the WKB approximation, relates to the formal (‘global’) stability treatment using Floquet theory, and on whether any important physics is lost as a result of this approximation. By recasting the stability equations in Lagrangian coordinates following particle paths in the basic wave and introducing normal modes, it is demonstrated rigorously that in the limit of large disturbance wavenumber ( k right arrow normal infinity k → ∞ $k\rightarrow \infty$ ), such ‘Lagrangian’ modes translate to the localised wavepackets assumed by local stability. This reveals that: (i) local instabilities generally involve multiple frequency components (in the Eulerian frame) and thus are distinct from parametric subharmonic instability; and (ii) local stability neglects altogether the effects of dispersion on wavepackets. While the local approach assumes short-scale disturbances ( 1 divided by k right arrow 0 1 / k → 0 $1/k\rightarrow 0$ ) apparently without restricting the basic-wave amplitude epsilon ϵ $\epsilon$ , this approximation is not uniformly valid when epsilon much less than 1 ϵ ≪ 1 $\epsilon \ll 1$ . In this limit, the appropriate stability eigenvalue problem balances the script upper O left parenthesis 1 divided by k right parenthesis O ( 1 / k ) $\mathcal{O}(1/k)$ dispersion with the script upper O left parenthesis epsilon right parenthesis O ( ϵ ) $\mathcal{O}(\epsilon )$ interaction of perturbations with the basic wave. This uniformly valid approach is applied to short-scale instabilities of small-amplitude finite-width propagating wave beams,