On magnetostrophic dynamos in annular cores

ABSTRACT Magnetostrophic dynamos are studied in an annular core, adapting the seminal work of Taylor [The magnetohydrodynamics of a rotating fluid and the Earth's dynamo problem. Proc. R. Soc. London A. 1963, 274, 274] for a fluid-filled core. The model consists of an inviscid fluid core and a concentric solid inner core. The fluid is supposed to obey the Boussinesq equations of motion and is driven into motion by a flow-forcing function consisting of the buoyancy force of an adverse radial temperature gradient, opposed by the Lorentz force of the self-sustained magnetic field. Coriolis forces act but inertial and viscous forces are ignored. Taylor (1963) showed how such a “magnetostrophic dynamo” can be found when there is no solid inner core, but his ideas have to be non-trivially generalised when an inner core is present. That is undertaken in this paper. In the 1993, CUP book, “Theory of Solar and Planetary Dynamos”, Hollerbach and Proctor gave examples in which the zonal flow created by a specified flow-forcing function may be singular on the “tangent cylinder”, an imaginary cylinder tangential to the inner core and parallel to the polar axis. It is shown here how this singularity is related to the flow-forcing function, and how discontinuities of other components of the fluid velocity on the tangent cylinder are determined by that function. In appendix A, an identity is established between the leading terms in the Fourier expansion of two of the cylindrical components of an arbitrary vector field. In appendix B, eight examples are given relevant to annular dynamos. In appendix C, equatorial symmetry is considered.