Experimental evidence of non-linear resonance effects between retrograde precession and the tilt-over mode within a spheroid

SUMMARY The Poincare flow (also known as the tilt-over mode) in a precessing cavity filled with water is investigated experimentally. Assuming that the flow is mainly a solid-body rotation, we have used three independent techniques to determine this rotation: introduction of light ceramic particles to materialize the fluid rotation axis, pressure measurements at the cavity wall, and ultrasonic Doppler anenometry. With the latter technique, we determine the angle between the solid body rotation vector and the cavity axis indirectly, through the secondary flow - oblique shear layers which are stationary in a frame rotating at the precession rate - that the differential rotation at the boundary induces within the fluid. Rapid changes in the direction of the axis of rotation of the fluid for critical values of the rate of precession and of the rate of rotation are demonstrated. In some cases, the inclination of the fluid rotation vector with respect to the cavity axis much exceeds the inclination of the prescribed precession vector. A torque approach, which can be generalized, shows that this effect is due to a non-linear resonance between the frequencies of the Poincare mode and of precession. As a result, we can determine the validity domain of current models of precession and nutations of planets enclosing a fluid core. 1 I NTR O DUCTION 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 Poincare flow) together with some perturbation. It has the geometry (but not the time dependence) of a nearly diurnal inertial eigenmode, which is known as the 'tilt-over mode' (Toomre 1974). The axis of the mode is contained in an equatorial plane and slowly drifts in a retro- grade direction in the inertial reference frame. Stewartson & Roberts (1963) and Greenspan (1968, p.75) have pointed out a possible res- onance effect between the forcing by a retrograde precession and the tilt-over mode when the non-dimensionalized precession rate, scaled by the rotation rate of the container, and the ellipticity of the interface are comparable. The rotation vector of the fluid ωf consists of an axial component along the rotation vector of the container ωc and of an equatorial component which becomes important when res- onance is approached. Since the eigenperiod of the tilt-over mode depends on the amplitude ωf instead of ωc, the resonance effect is non-linear (Pais & Mouel 2001). This has been investigated by Busse (1968), who has calculated ωf taking into account viscous effects in the boundary layer and the finite angle θ between the two rotation vectors ωc and ωf. From his expression also, we can antici- pate large changes in the direction of the vorticity vector in the case of retrograde precession. The non-linearity makes the position of the rotation axis derived from the analytical asymptotic approach of Busse (1968), in the resonance region, not intuitive. We shall show that there are values of the non-dimensionalized precession rate � = � p/ωc, the ellipticity η, the Ekman number E and the angle α between the axis of the container and the direction of preces- sion, for which three values of ωf are solutions of the implicit equation obtained by Busse. In the geophysical context, excur- sions of ωf may have occurred because of resonance with nu- tations (Toomre 1974; Greff-Lefftz & Legros 1999a,b). The pre- cession example demonstrates that the finite value of θ may have to be taken into account in the analyses of resonance and warns against linear superposition of nutation and precession effects. Fi- nally, we do not know to which extent the theoretical analysis of Busse is valid when ωf strongly diverges from ωc. Numerical and experimental models are thus important to ascertain the theoretical predictions.