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

The Earth is submitted to the gravitational effect of different objects,  resulting  in  small  variations  of  the  orientation  of  its  axis  rotation.   The  precession corresponds to the rotation of the body spin axis around the normal to the elliptic plane. The primary flow forced by precession in a sphere is mainly a tilted solid body rotation, a flow of uniform vorticity. In this study we focused on the pseudo-resonance between the precessional forcing  and  the  spin-over  mode,  detected  as  a  peak  of  amplitude  of  the  norm  of the  vorticity  of  the  fluid.   We  show  the  influences  of  both  the  geometry and the application of an uniform external magnetic field on the external boundary, onto this pseudo-resonance.  The major purpose is to validate a semi-analytical model to allow its interpolation to planetary bodies.  We compared the semi-analytical model [Noir and C ́ebron, 2013] with numerical simulations performed with XSHELLS [Schaeffer, 2013],  which give us the components of the fluid vorticity in a precessing frame. We compared also the spin-over mode coefficients, used to simulate the viscous  effect  on  the  model,  with  two  methods :  an  empirical  equation  and  the numerical solver Tintin [Triana et al., 2019], taking into account the solid inner-core size (η=RI/R).  The differential rotation between the flow and the container, obtained with the model and the XSHELLS simulations, show us a verygood agreement especially for a small Ekman number (E= 10^−5), thus the spin-over mode coefficients for small E and η≤0.5.  An increase of the inner-core size  implies  a  decrease  of  the  resonance  amplitude  caused  by  the  supplementary Ekman layer added at the Inner Core Boundary (ICB); nevertheless thecolatitude (αf) and the longitude (φf) of the fluid don’t change significantly.The  application  of  a  uniform  magnetic  field  at  the  CMB  implies  a  decrease of the resonance amplitude, but also a modification of the mean rotation axis direction.  Indeed, the coupling between the viscous flow and the magnetic field induces a modification of the αfand φf, which follow the main direction angle of the magnetic field axis.  We observe small discrepancies between the simulations (XSHELLS and Tintin) and the model but the behavior following different parameters (Po,α angle,Ro,η,β angle, Λ) is well understood.  As a result, we applied the models at few parameter ”realistic values” of planetary objects like terrestrial planets but also ice’s satellites.

References

[Noir and C ́ebron, 2013]  Noir,  J.  and  C ́ebron,  D.  (2013).    Precession-driven flows in non-axisymmetric ellipsoids.Journal of Fluid Mechanics, 737:412–439.

[Schaeffer, 2013]  Schaeffer, N. (2013).  Efficient spherical harmonic transforms aimed  at pseudospectral numerical  simulations.Geochemistry, Geophysics,Geosystems, 14(3):751–758.

[Triana et al., 2019]  Triana, S. A., Rekier, J., Trinh, A., and Dehant, V. (2019).The coupling between inertial and rotational eigenmodes in planets with liq-uid cores.Geophysical Journal International.