On the parametric instabilities of internal gravity waves in a density-stratified fluid

The parametric instability of a finite-amplitude, internal gravity wave is a widely studied process in atmospheric and oceanic fluid dynamics, and has been extensively investigated through experiments and direct numerical simulations. The mathematical approach of the Floquet-Fourier method leads to a linear algebraic computation of Floquet exponents (stability eigenvalues) as a function of disturbance wavenumbers. The number of numerical eigenvalues is determined by the truncation of the Fourier series in the Floquet solution of the linearized Boussinesq equations. Yet, the physical linear dispersion relation for the frequency eigenvalues is only a double-valued function of wavenumbers. We investigate this ambiguity in the eigenvalue count through the development of resonant-mode perturbation analyses that identify the physically relevant instabilities. Our choice of Floquet exponents is interpreted as branches of a Riemann surface from the complex analysis of Floquet spectral theory.