On fluid flows in precessing spheres in the mantle frame of reference

Abstract

We investigate, through both asymptotic and numerical analysis, precessionally driven flows of a homogeneous fluid confined in a spherical container that rotates rapidly with angular velocity Ω and precesses slowly with angular velocity Ωp about an axis that is fixed in space. The precessionally driven flows are primarily characterized by two dimensionless parameters: the Ekman number E providing the measure of relative importance between the viscous force and the Coriolis force, and the Poincaré number Po quantifying the strength of the Poincaré forcing. When E is small but fixed and |Po| is sufficiently small, we derive a time-dependent asymptotic solution for the weakly precessing flow that satisfies the nonslip boundary condition in the mantle frame of reference. No prior assumption about the spatial-temporal structure of the precessing flow is made in the asymptotic analysis. A solvability condition is derived to determine the spatial structure of the precessing flow, via a selection from a complete spectrum of spherical inertial modes in the mantle frame. The weakly precessing flow within the bulk of the fluid is characterized by an inertial wave moving retrogradely. Direct numerical simulation of the same problem in the same frame of reference shows a satisfactory agreement between the time-dependent asymptotic solution and the nonlinear numerical simulation for sufficiently small Poincaré numbers.