We study instabilities of internal gravity waves (IGWs), Alfvén waves (AWs) and hybrid Alfvén-gravity waves (AGWs) propagating through a stably-stratified Boussinesq fluid with constant buoyancy frequency permeated by a uniform background magnetic field. This is achieved using the Boussinesq MHD equations, which admit plane wave solutions satisfying a quintic dispersion relation with three branches: a zero-frequency branch, an AW branch and an AGW branch.
We develop a theoretical framework with which to predict the growth rate of resonant instabilities of small-amplitude AGWs asymptotically. We also create a Floquet eigenvalue solver to describe the linear stability of finite-amplitude AGWs. Since the plane wave is an exact solution of the ideal nonlinear Boussinesq MHD equations, the eigenvalue solver allows us to study instabilities of waves with arbitrary amplitudes.
In the absence of a magnetic field, many instabilities of small-amplitude IGWs can be attributed to resonant triad interactions between three IGWs. We show that in the presence of a magnetic field, many instabilities of small-amplitude AGWs can be attributed to resonant triad interactions between AGWs and AWs in various combinations. The asymptotic framework for small-amplitude waves is used to classify instabilities observed using the eigenvalue solver in terms of such resonant interactions. Instabilities of these waves are not limited to those induced by resonant interactions. We derive a system of energetic equations to determine which physical processes are driving the instabilities energetically.
The model we use can be applied in a variety of astrophysical settings, including radiative zones in stellar interiors. Instabilities of IGWs play an important role in the mixing of matter, energy and angular momentum in astrophysical fluids, so it is crucial that we understand them mathematically. It is instructive to consider the interaction of IGWs with a magnetic field in many astrophysical fluids. We determine the influence of magnetic field strength and orientation on the growth rate of the instabilities we study.
