Instability by Weak Precession of the Flow in a Rotating Sphere

A linear stability analysis is presented of the steady flow in a rapidly rotating sphere undergoing weak precession. It is well-known that without precession all disturbances damp with decay rate proportional to Re−1/2 where Re = a2Ωs/ν is the Reynolds num- ber defined by a the sphere radius, Ωs the the spin angular velocity, and ν the kinematic viscosity of fluid. With precession, two kinds of instability modes exist; one is global and inviscid in nature, and the other is local and viscous. Here we investigate the former by an asymptotic analysis for large Re and small Γ (= Ωp/Ωs), the ratio of the precession and spin angular velocities. It is shown that a weak precession with Γ of order Re−1/2 destabilises disturbances through coupling between an axisymmetric mode (with respect to the spin axis) and the (2,1,1) mode. We find that the neutral curve for the instability behaves asymptotically as Γ = 7.9Re−1/2. For the local modes on the other hand, the neutral curve behaves as Γ ∝ Re−4/5. These results are compared with observations (Goto et al., 2011) for a precessing sphere and spheroid.