An asymptotic model for resistive instability in the Earth's outer core

Numerical work indicates that resistive instability may be the dominant mod e of instability in the Earth's outer core for realistic core parameter reg imes. In this paper, we assume that the Elsasser number is large in order t o obtain an asymptotic analysis of resistive instability in an electrically conducting fluid confined to a rotating cylindrical shell of infinite exte nt in the axial direction. The dimensionless equations of motion are linear ized about an ambient magnetic field which is purely azimuthal and depends only on the cylindrical radial variable. Applying the theory of ordinary di fferential equations with a large parameter, we obtain an asymptotic approx imation to the solution. Relatively simple analytic expressions for the com plex frequencies are obtained by applying the boundary conditions for insul ating boundaries at the cylindrical sidewalls and then assuming that the am bient magnetic field vanishes at one or both of those sidewalls. The result s appear to be consistent with previous numerical work.