A uniform asymptotic approximation for unstable hydromagnetic waves in a rotating sphere

Abstract

Abstract The Earth's outer core is modelled by the interior of a rotating sphere containing an incompressible, inviscid, perfectly conducting fluid of constant density. Solutions are sought as small perturbations to the governing equations in the cylindrical coordinates (r, θ, z), linearized about a state of no flow and azimuthal magnetic field varying with r. A multiple scale perturbation technique is used to approximate slow hydromagnetic waves. This scheme assumes a short wavelength scale (i.e., much less than the radius of the sphere) in the r direction while the wave number in that direction varies slowly (on the scale of the spherical radius) with r. The length scales in the θ and z directions are long and the amplitude varies on the long length scale in both r and z. The waves are also assumed to be in geostrophic balance. Application of a standard ansatz yields an approximation which becomes singular in the neighborhood of the equator. Application of a revised ansatz yields a uniform approximation...