Viscous and rotational splitting of the translational oscillations of Earth's solid inner core

Abstract

The small oscillations of the inner core about its central position present an interesting problem in fluid dynamics with roots in the classical literature. It is a generalization of the problem Stokes [Stokes, G.G., 1851. On the effect of the internal friction of fluids on the motion of pendulums, Trans. Cambridge Philos. Soc. 9, 8.] solved of a sphere oscillating in an unbounded, non-rotating, viscous fluid to the case of a sphere oscillating in the contained, rotating fluid outer core. In the inviscid case, the influence of the Coriolis acceleration splits the oscillation into three distinct motions with differing periods, one along the axis of rotation, one prograde in the equatorial plane, and one retrograde in the equatorial plane. In this paper, we consider the effect of the presumed high viscosity at the inner core boundary on the rotational splitting. We obtain novel exterior analytic solutions of the Poincaré equation for the flow outside the boundary and solve the Ekman boundary-layer problem to match the no-slip condition at the inner core boundary. Continuity of normal displacement at the inner core boundary, and in far-field approximation at the core–mantle boundary, determine the free constants in the two independent Poincaré solutions. Conservation of linear momentum between the inner core, outer core and shell establishes a centre of mass reference frame. Analytic expressions for the pressure and viscous drag are obtained for both the axial and equatorial modes of oscillation. These are then used in the equation of motion for the inner core to deduce viscous and rotational splitting laws. Viscosity is found to reduce the effect of rotation on the periods of all three modes; most noticeably, it reduces the large rotational splitting of the two equatorial modes. We expect the geophysically interesting value of the viscosity just outside the inner core might be recovered from the observed splitting, particularly, that of the two equatorial modes.