Steady flows in the core of precessing planets : effects of the geometry and an uniform magnetic field.

Abstract:

The Earth is subjected to the gravitational influence of the sun, the moon and surrounding planets, resulting in small variations of the orientation of its rotation axis. Precession is a long period motion of the rotation axis around the normal to the elliptic plane. This variation acts as a mechanical forcing at the planet’s Core Mantle Boundary (CMB). For a spherical fluid core, the primary flow induced by precession corresponds to a tilted solid-body rotation,i.e a flow of uniform vorticity (Poincaré,1910). The direction and amplitude of rotation result from a balance between the gyroscopic effect of precession and the viscous torque at the CMB (Busse, 1968; Noir et al., 2003). This solid-body rotation resembles the Spin-Over Mode (SOM), the simplest inertial modes in a rotating fluid (Greenspan et al., 1968). In this study we focus on the pseudo-resonance between the precessional forcing and the SOM, detected as a peak in the amplitude of the fluid’s vorticity. We study the impact of the geometry (i.e.the presence of a solid-inner core) and the impact of the uniform external magnetic field. We compare the semi-analytical model of Noir and Cébron (2013) to our own numerical resolution of the nonlinear Navier-Stokes equation based on the XSHELLS solver (Schaeffer, 2013). We find that the semi-analytical solution agrees well with the result of the numerical simulations so long as the inner core is small (aspect ratio η ≤ 0.5). This is especially true in the regime of low viscosity (Ekman number E ≤10⁻⁵). The amplitude of the resonance decreases when η > 0.5, a fact which we attribute to the supplementary Ekman (shear) layers originating from the surface of the Inner-Core Boundary (ICB). In this case, the orientation of the fluid’s mean rotation axis does not change significantly. This changes in the presence of an external uniform magnetic field, in which case the rotation axis tends to align with the direction of the magnetic field. We also estimate the corrections on the SOM decay rate and eigenfrequency from viscosity and the inner core by numerically solving the linearised Navier-Stokes equation (Triana et al., 2019) and comparing it to the semi-analytical solution. We find that both method are in very good agreement.