Steady flows in the core of precessing planets : effects of the geometry and an applied magnetic field.
Authors: Laguerre Raphael , Houliez Aymeric , Cébron David , Dehant Véronique 
Published: March 2020

&lt;p&gt;The Earth is submitted to the gravitational effect of different objects, &amp;#160;resulting &amp;#160;in &amp;#160;small &amp;#160;variations &amp;#160;of &amp;#160;the &amp;#160;orientation &amp;#160;of &amp;#160;its &amp;#160;axis &amp;#160;rotation. &amp;#160; The &amp;#160;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 &amp;#160;and &amp;#160;the &amp;#160;spin-over &amp;#160;mode, &amp;#160;detected &amp;#160;as &amp;#160;a &amp;#160;peak &amp;#160;of &amp;#160;amplitude &amp;#160;of &amp;#160;the &amp;#160;norm &amp;#160;of the &amp;#160;vorticity &amp;#160;of &amp;#160;the &amp;#160;fluid. &amp;#160; We &amp;#160;show &amp;#160;the &amp;#160;influences &amp;#160;of &amp;#160;both &amp;#160;the &amp;#160;geometry and the application of an uniform external magnetic field on the external boundary, onto this pseudo-resonance. &amp;#160;The major purpose is to validate a semi-analytical model to allow its interpolation to planetary bodies. &amp;#160;We compared the semi-analytical model [Noir and C &amp;#769;ebron, 2013] with numerical simulations performed with XSHELLS [Schaeffer, 2013], &amp;#160;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 &amp;#160;effect &amp;#160;on &amp;#160;the &amp;#160;model, &amp;#160;with &amp;#160;two &amp;#160;methods : &amp;#160;an &amp;#160;empirical &amp;#160;equation &amp;#160;and &amp;#160;the numerical solver Tintin [Triana et al., 2019], taking into account the solid inner-core size (&amp;#951;=RI/R). &amp;#160;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^&amp;#8722;5), thus the spin-over mode coefficients for small E and &amp;#951;&amp;#8804;0.5. &amp;#160;An increase of the inner-core size &amp;#160;implies &amp;#160;a &amp;#160;decrease &amp;#160;of &amp;#160;the &amp;#160;resonance &amp;#160;amplitude &amp;#160;caused &amp;#160;by &amp;#160;the &amp;#160;supplementary Ekman layer added at the Inner Core Boundary (ICB); nevertheless thecolatitude (&amp;#945;f) and the longitude (&amp;#966;f) of the fluid don&amp;#8217;t change significantly.The &amp;#160;application &amp;#160;of &amp;#160;a &amp;#160;uniform &amp;#160;magnetic &amp;#160;field &amp;#160;at &amp;#160;the &amp;#160;CMB &amp;#160;implies &amp;#160;a &amp;#160;decrease of the resonance amplitude, but also a modification of the mean rotation axis direction. &amp;#160;Indeed, the coupling between the viscous flow and the magnetic field induces a modification of the &amp;#945;fand &amp;#966;f, which follow the main direction angle of the magnetic field axis. &amp;#160;We observe small discrepancies between the simulations (XSHELLS and Tintin) and the model but the behavior following different parameters (Po,&amp;#945; angle,Ro,&amp;#951;,&amp;#946; angle, &amp;#923;) is well understood. &amp;#160;As a result, we applied the models at few parameter &amp;#8221;realistic values&amp;#8221; of planetary objects like terrestrial planets but also ice&amp;#8217;s satellites.&lt;/p&gt;&lt;p&gt;References&lt;/p&gt;&lt;p&gt;[Noir and C &amp;#769;ebron, 2013] &amp;#160;Noir, &amp;#160;J. &amp;#160;and &amp;#160;C &amp;#769;ebron, &amp;#160;D. &amp;#160;(2013). &amp;#160; &amp;#160;Precession-driven flows in non-axisymmetric ellipsoids.Journal of Fluid Mechanics, 737:412&amp;#8211;439.&lt;/p&gt;&lt;p&gt;[Schaeffer, 2013] &amp;#160;Schaeffer, N. (2013). &amp;#160;Efficient spherical harmonic transforms aimed &amp;#160;at pseudospectral numerical &amp;#160;simulations.Geochemistry, Geophysics,Geosystems, 14(3):751&amp;#8211;758.&lt;/p&gt;&lt;p&gt;[Triana et al., 2019] &amp;#160;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.&lt;/p&gt;