Rotating magnetic shallow water waves and instabilities in a sphere

Waves in a thin layer on a rotating sphere are studied. The effect of a toroidal magnetic field is considered, using the shallow water ideal MHD equations. The work is motivated by suggestions that there is a stably stratified layer below the Earth's core mantle boundary, and the existence of stable layers in stellar tachoclines. With an azimuthal background field known as the Malkus field, B-phi = B(0)sin theta, theta being the co-latitude, a non-diffusive instability is found with azimuthal wavenumber m = 1. A necessary condition for instability is that the Alfven speed exceeds Omega R-0(0) where Omega(0) is the rotation rate and R-0 the sphere radius. Magneto-inertial gravity waves propagating westward and eastward occur, and become equatorially trapped when the field is strong. Magneto-Kelvin waves propagate eastward at low field strength, but a new westward propagating Kelvin wave is found when the field is strong. Fastmagnetic Rossby waves travelwestward, whilst the slowmagnetic Rossby waves generally travel eastward, except for some m = 1 modes at large field strength. An exceptional very slow westward m = 1 magnetic Rossby wave mode occurs at all field strengths. The current-driven instability occurs for m = 1 when the slow and fast magnetic Rossby waves interact. With strong field the magnetic Rossby waves become trapped at the pole. An asymptotic analysis giving the wave speed and wave form in terms of elementary functions is possible both in polar trapped and equatorially trapped cases.