On a dynamo driven topographically by longitudinal libration

Variation in the angular velocity Ω of a planetary body is called libration or longitudinal libration when the Ω-axis is fixed in direction. This motion of the body's solid mantle drives motions in its fluid core, either by viscous coupling across the core-mantle interface S, or topographically when S is asymmetric with respect to the Ω-axis, the only case considered in this article. A significant topographically-driven flow is identified having uniform vorticity within S and no component parallel to the Ω-axis. Its dynamic stability depends on the amplitude, Ω ₁, of the sinusoidally varying part of Ω and on the ratio, b/a, of the lengths of the principal axes of S, assumed spheroidal. In (Ω ₁/Ω ₀, b/a) parameter space where Ω ₀ is the average Ω, islands are shown to exist where the constant vorticity states are dynamically unstable. These are surrounded by a sea in which they are stable. When the fluid is slightly viscous, a state in the stable sea retains its uniform vorticity structure except in a viscous boundary layer on S in which the flow acquires a component parallel to the Ω-axis. For (Ω ₁/Ω ₀, b/a) on an island where the uniform vorticity state is unstable, an “alternative flow” exists, which is three-dimensional and is examined here. Assuming that the core is electrically conducting, kinematic dynamos are sought. Uniform vorticity flow appears to be non-regenerative but, when it is stable and viscosity acts to create a sufficiently strong boundary layer flow, dynamo action may occur. It is shown that the alternative flow that exists on an instability island in (Ω ₁/Ω ₀,?b/a) space can be vigorously regenerative.