Hydromagnetic waves in rapidly rotating spherical shells generated by magnetic toroidal decay modes

Abstract

Instabilities in the form of slow azimuthally travelling hydrodynamic waves in a rapidly rotating, stress-free, electrically conducting spherical fluid shell are investigated. The instabilities are generated by the toroidal decay mode of the lowest order or a combination of toroidal decay modes. It is found that the Elsasser number Λ_c at the onset of instability is always Λ_c = O(10) for various profiles of the basic magnetic field. It is also found that the hydromagnetic waves of the preferred instability propagate eastward [i.e. for solutions proportional to exp i(mφ + ωt), ω < 0] and are characterized by nearly two-dimensional columnar fluid motions attempting to satisfy the Proudman-Taylor theorem, indicating that the most rapidly growing magnetic disturbance arranges itself in such a way that the corresponding magnetic forces balance only the ageostrophic component of the Coriolis force. Except for the Stewartson-type velocity boundary layer at the equator of the inner core, the velocity and magnetic field of the most unstable mode are always smooth, with length scale comparable with the shell width. By studying the same system with and without fluid inertia, we show that fluid inertia cannot introduce any new instability of physical relevance or significantly change the main features of the instability when T _m > 10⁵. T _m being the magnetic Taylor number. By studying the detailed dependence of the instabilities in the whole range 0 < T _m ≤ ∞, we are able to demonstrate that the existence of the Stewartson-type layer is unlikely to affect the primary properties of the instabilities. In the case of the quadruple symmetry mode, there is no indication of a Stewartson-type boundary layer; a result of the symmetry properties. For sufficiently large T _m . the most rapidly growing instability always has dipole symmetry (which allows the formation of two-dimensional fluid motions). We have compared our resuls (in the limit T _m → ∞) with those found using the magnetostrophic approximation. Calculations using the latter do not impose stress-free boundary conditions and slightly lower values of the critical Elsasser number result. Otherwise, the solutions found are broadly similar.

Key Words:

• MHD waves
• rotation
• spherical shells
• toroidal magnetic field