Elliptical instability in a rotating spheroid

Abstract

This paper concerns the elliptical instability of a flow in a rotating deformed sphere. The aim of our work is to analyse the characteristics of this instability experimentally and to provide a theoretical model which accounts for the observations. For this purpose, an elastic and transparent hollow sphere has been moulded in a silicone gel block. The flow is visualized using Kalliroscope flakes illuminated with a laser sheet as the sphere is set into rotation and compressed by two rollers. The elliptical instability occurs by the appearance of the so-called 'spin-over' mode whose growth rate and saturation amplitude are measured by video image analysis at different Ekman numbers. Growth rates are predicted well by the linear stability analysis. A nonlinear model is developed and is shown to describe correctly the saturated regimes observed in the experiments. At low Ekman numbers, a secondary instability leading to an intermittent regime is also discovered.

Elliptical instability of the flow in a rotating shell

“…The experiment of Seyed-Mahmoud et al (2004) and our own experimental observations presented in the next section confirm this expectation. Moreover, as in Lacaze et al (2004), the inner shear layers induced by the boundary layer eruptions at the critical latitude (Hollerbach and Kerswell, 1995) are expected not to significantly modify our results and will not be considered in the following.…”

“…The derivation of the inviscid growth rate is classical and has already been done many times in different configurations (Moore and Saffman, 1975;Tsai and Widnall, 1976;Eloy et al, 2003;Lacaze et al, 2004). The method consists in applying a condition of solvability for the corrected flow at order ε.…”

“…Aldridge et al (1997), and Seyed-Mahmoud et al (2000 have performed computations and built as already mentioned, a rotating deformable shell where they observed the presence of the elliptical instability. Using the technique invented by Malkus in 1989(Malkus, 1989, and more recently used and extended to triangular distortions by Eloy et al (2003), we have applied an elliptical constraint to a deformable rotating sphere (Lacaze et al, 2004) and visualized the spin-over mode generated by the elliptical instability. This mode is a solid body rotation around an axis aligned along with the stretching direction.…”

“…Therefore, it is of some importance to study the effect of inner rotating solid bodies in the development of the elliptical instability in spherical geometry. The present article concerns an extension of our previous work on the spin-over mode of the elliptical instability in a full sphere (Lacaze et al, 2004) to the case of a rotating shell. In a first part, we present our theoretical model of the flow contained in the shell.…”

“…In a second part, experimental results are presented. We used the same technique as before (Lacaze et al, 2004) but this time a solid small inner sphere is suspended by a thin wire in the center of a hollow transparent deformable external sphere. The ratio η between both sphere radius has been chosen equal to 1/3 that is close to the value encountered in the Earth core.…”