Magnetic field induced by elliptical instability in a rotating spheroid

Abstract

The tidal or the elliptical instability of the rotating fluid flows is generated by the resonant interaction of the inertial waves. In a slightly elliptically deformed rotating sphere, the most unstable linear mode is called the spin-over mode, and is a solid body rotation versus an axis aligned with the maximum strain direction. In the non-viscous case, this instability corresponds to the median moment of the inertial instability of the solid rotating bodies. This analogy is furthermore illustrated by an elliptical top experiment, which shows the expected inviscid heteroclinic behaviour. In geophysics, the elliptical instability may appear in the molten liquid cores of the rotating planets, which are slightly deformed by the tidal gravitational effects of the close bodies. It may then participate in the general outer core dynamics and possibly the geodynamo process. In this context, Kerswell and Malkus (Kerswell, R.R. and Malkus, W.V.R., Tidal instability as the source for Io's magnetic signature. Geophys. Res. Lett., 1998, 25, 603-606) showed that the puzzling magnetic field of the Jovian satellite Io may indeed be induced by the elliptically unstable motions of its liquid core that deflect the Jupiter's magnetic field. Our magnetohydrodynamics (MHD) experiment is a toy-experiment of this geophysical situation and demonstrates for the first time the possibility of an induction of a magnetic field by the flow motions due to the elliptical instability. A full analytical calculation of the magnetic dipole induced by the spin-over is presented. Finally, exponential growths of this induced magnetic field in a slightly deformed rotating sphere filled with galinstan liquid metal are measured for different rotating rates. Their growth rates compare well with the theoretical predictions in the limit of a vanishing Lorentz force.

On the effects of an imposed magnetic field on the elliptical instability in rotating spheroids

“…We derive a nonlinear and viscous model of its development under an imposed magnetic field valid for low values of the magnetic Reynolds number, based on the hydrodynamical model of Lacaze et al [5] and including the magnetic damping term determined by Thess & Zikanov [17]. These results are validated experimentally using an extended version of the set-up of Lacaze et al [16], with stronger imposed magnetic fields. These results are then extended to the large magnetic Reynolds number, large Reynolds number limit relevant to planetary applications, using a short-wavelength Lagrangian theory [18].…”

“…This article, which completes and extends the previous works of Lacaze et al [16] and Thess & Zikanov [17], is organized as follow. We first focus on the so-called spin-over mode, which corresponds to the simplest mode of the elliptical instability in spheroids, excited at the smallest values of the Reynolds number above threshold.…”

“…In the limit Λ ∼ , Rm → 0, this theory can be extended to include the magnetic field effect perturbatively. We will not go as far, as the combination of the results of [5], [16] and [17], allows us to describe the linear and nonlinear dynamics of the dominant spin-over mode, which is the only mode accessible to purely hydrodynamical experiments using the present device in a spherical geometry with a fixed strain field [19]. The spinover mode is mainly a solid body rotation around an inclined axis, whose horizontal projection Ω H is aligned with the axis stretched by the strain field, at polar angle in the vicinity of −45 o in the (x, y) plane (see figure 1).…”

“…This also implies that the viscous terms in the nonlinear system of [5] may be used here. Finally, there will be no significant contributions to the external and internal magnetic fields due to the boundary layer, which would make the field deviate from the field induced by the non-viscous spin-over mode, calculated in [16].…”

“…Even if the hydrodynamic of the elliptical instability is today well known, its planetary consequences are still controversial and necessitate a full understanding of the magnetohydrodynamic (MHD) of the elliptical instability, which remains a mostly open question (e.g. [16]). Understanding the MHD of the elliptical instability is also important in metallurgic applications, especially regarding its role in the transition from two to three-dimensional MHD-turbulence [17].…”