Waves in the Earth’s core. V. inertial modes modified by magnetic effects

Fluids in a rotating frame exhibit inertial modes, normal modes with the Coriolis force as the restoring force. These modes persist in the presence of an additional background magnetic field, but are modified by the Lorentz force when the fluid is electrically conductive. We consider these modified inertial modes for an inviscid, electrically conducting fluid in a rotating full sphere. Owing to their close affinity to the inviscid, purely hydrodynamic inertial modes whose solutions are known analytically, a perturbative approach is taken to deduce the modifications introduced by magnetic effects. The perturbation in the frequency and decay rate of these modes is found to scale with the square of the background magnetic field strength. In addition, the frequency is shown to have a zeroth-order correction in magnetic diffusivity, i.e. a correction that does not vanish as magnetic diffusivity approaches zero owing to the presence of the magnetic boundary layer, rendering the diffusionless limit a singular limit. These results yield implications for modified inertial modes in stellar and planetary dynamos where magnetic diffusion dominates relative to viscous diffusion.