The Magnetoelliptic Instability in the presence of Inertial Forces

Abstract

The elliptical instability, namely the problem of stability of a two-dimensional flow with elliptical streamlines with respect to three-dimensional perturbations is expected to play an important role as a secondary instability in the transition to turbulence [2]. We investigate the influence of uniform magnetic field, perpendicular to the plane of the flow, and of the inertial forces in a rotating frame (Coriolis force and angular acceleration) on the stability of the flow. Though the magnetic field and the inertial forces in this context have already been studied in isolation [5], [4], there has been no comprehensive investigation of their joint influence. Our study is motivated by possible geophysical and astrophysical applications since the results may be used to comment on the influence of tidal and rotational distortion of the Earth’s (or another planet’s or satelite’s) mantle on the outer core’s stability with possible dynamo implications (together with the stability analysis of the boundary layer at the core-mantle boundary [3], [6]) and the magnetoelliptic instability may prove to be an important factor in turbulent transition and therefore in understanding the energy balance in the Solar corona. The system considered consists of an infinite two-dimensional vortex with elliptical streamlines in which the velocity field is defined as u = γ[−Ey,Ex, 0] with γ > 0 and E ≤ 1 in a frame of reference rotating with angular velocity Ω = Ωez in the presence of a uniform magnetic field perpendicular to the plane of the flow B0 = B0ez. The fluid is inviscid and incompressible. Such configuration permits resonances between inertial waves which can grow exponentially [1]. Perturbations in the form of inertial waves are therefore introduced into the Navier-Stoke’s and induction equations and by application of the Floquet theory we obtain ordinary matrix equations which then are solved numerically to obtain the growth rate dependence on the parameters characterizing the system such as the angle θ between the wave vector and the axis of the vortex (the z -axis),the eccentricity E, the