J. Noir, P. Cardin, D. Jault and J.P. Masson $^{a}$
$^{a}$ CNRS and Universit\'e Joseph Fourier, Grenoble, France.
The dynamics of a fluid enclosed in an oblate precessing spheroidal container has been theoretically studied for over one century. The motion has been described as an uniform vorticity flow (the Poincar\'e flow) together with some perturbations. It has the geometry (but not the time dependence) of a nearly diurnal inertial eigenmode, which is known as the 'tilt-over mode'. The axis of the mode is contained in an equatorial plane and slowly drifts in a retrograde direction in the inertial reference frame.In the 60's, Stewartson and Greenspan have pointed out a possible resonance effect between the forcing by a retrograde precession and the tilt-over mode when the adimensionalized precession rate, scaled by the rotation rate of the container, and the ellipticity of the interface are comparable. This resonance is non-linear because its locus depends on the fluid rotation as a solid body, $\mathbf {\omega}_f$ (Pais and Lemouel 2001) We present here analytical and experimental investigations of this resonance phenomena. Let's consider a spheroidal container filled with an incompressible fluid, rotating along it axis $\mathbf {\omega}_c$ and precessing at $\mathbf {\Omega}_p= \omega_c\mathbf {\Omega}$. We restrict our analysis to the family of inviscid, steady flows of uniform vorticity. With no-slip and no-penetration conditions, the rotation vector of the fluid is fully determined by the equilibrium, $\mathbf {\Gamma_i}=\mathbf {\Gamma_p} +\mathbf {\Gamma_v}$ between the inertial torque $\mathbf {\Gamma_i}$, the pressure torque $\mathbf {\Gamma_p}$ and the viscous torque $\mathbf {\Gamma_v}$. In the frame of reference rotating at $\omega_f$, its resolution yields the implicit expression (3.19) of (Busse 1968) for $\mathbf {\omega}_f$: \begin{equation}\label{3.19} \mathbf{\omega}_f=\omega_f^2 \mathbf{\hat e_z} + \frac{X \omega_f^2}{X^2+Y^2} \mathbf{\hat e_z} \times\left ( \mathbf {\Omega}\times\mathbf{\hat e_z}\right )+\frac{Y\omega_f^2}{X^2+Y^2}\mathbf{\hat e_z}\times\mathbf {\Omega}, \end{equation} with, \begin{equation}\label{A B} X=(\frac{E}{\omega_f})^{1/2} \lambda_{so}^i +\eta \omega_f^2 + \Omega_p \cos \alpha, \; Y=-(E \omega_f)^{1/2}\lambda_{so}^r, \ \end{equation} where $E$ is the Ekman number. Depending on the values of $\Omega_p$, $\eta$ and $E^{1/2}$, one or three vectors $\mathbf {\omega}_f$ are solutions of (\ref{3.19}). We have shown in this later case that only two of the three solutions are stable. We then investigated this phenomena experimentally using a spinning spheroidal cavity filled with water, set up on a slowly rotating horizontal turntable. The spheroidal cavity has been machined inside two cylindrical plexiglass blocks, The outer radius of the cylindrical casing is $R_c=150$ mm, the major semi-axis of the cavity is $a=125$ mm, and the ellipticity $\eta$ is 1/25. Assuming that the flow is mainly a solid-body rotation, we have used three independent techniques to determine this rotation: \begin{itemize} \item We introduced light ceramic particles that collapse along the fluid rotation axis. A plan sheet of light rotating along the axis of the cavity is set in the plane $(\mathbf {\omega}_c,\mathbf {\omega}_f)$, the angle it makes with $(\mathbf {\omega}_c,\mathbf {\Omega})$ gives the longitude. Latitude was measured on pictures taken at right angle. \item We have shown analytically that a pressure measurement in time at one point of the cavity wall gives access to a complete determination of $\mathbf {\omega}_f$. Measurement on a large range of parameters were performed with this technique. \item Finally, using ultrasonic Doppler anenometry, we determined the angle between the solid body rotation vector and the cavity axis indirectly, through the secondary flow that the differential rotation at the boundary induces within the fluid. \end{itemize} We always observed a unique axis of rotation with en abrupt change in both direction and amplitude for critical values of ($\omega_f,\eta,E^{1/2}$). We obtained a good agreement between predicted directions and their experimental counterparts ,in particular, the abrupt transition occurs for the analytically predicted values. We show analytically that this effect is due to the nonlinear resonance between the the Poincar\'{e} mode and the precession forcing. Therefore, linear theory ($\omega_f=1$) is not appropriate for strong forcing and consequently it is not always possible to superpose linearly different forcings . \\ In a previous precession experiment (Malkus 1968), Malkus has observed rapid changes in the measured torque of the spin motor, and hysteresis behaviour of the system. Although we never reported any hysteresis in our experiment, the observations of Malkus, that are in a very different range of parameters, may related to this non-linear resonance.\\ In the case of the Earth's Core, resonances between the tilt-over mode and some nutations (annual, semi-annual), that may occured during the past, can be described with a linear approximation.\\ The discovery of the retrograde rotation of the planet Venus has prompted many studies (Goldreich and Peale 1970,Yoder 1995), which take into account resonance between the precession rate and orbital frequencies, atmospheric tides and core-mantle friction. In these works, the core velocity relative to the mantle has been calculated using the linear theory but it is very likely that the conditions for rapid changes of the fluid solid body rotation have been met at some stage during the history of Venus. This stimulates us to study how our results can be extended to the small values of the Ekman number that are relevant for the dynamics of the fluid core of planets. \vspace{1cm} References:\\ R. Poincar\'e, Sur la pr\'ecession des corps d\'eformables, Bull. Astr., 27, 321--356, 1910\\ Greenspan, H. P.,The theory of rotating fluids, Cambridge University Press, 1968.\\ {K. Stewartson and P. H. Roberts, On the motion of a liquid in a spheroidal cavity of a precessing rigid body, J. Fluid Mech., 17, 1--20, 1963.\\ W. V. R. Malkus, Precession of the {Earth} as the Cause of Geomagnetism, Science, 160, 259--264, 1968\\ F. H. Busse, Steady fluid flow in a precessing spheroidal shell, J. Fluid Mech., 33, 739--751, 1968.\\ M. A. Pais and J. L. Le Mou\"el, Precession-induced flows in liquid-filled containers and in the {Earth}'s core, Geophys. J. Int.,539--554,2001.\\