Induction magnetic stability with a two-component velocity field

A generalization of the stability results of the induction magnetic field in a simple one space dimensional model for a protoplanetary disc, given in Rüdiger and Shalybkov (2004) and Straughan (2013), is proposed. The model studied in Rüdiger and Shalybkov (2004) and Straughan (2013) arises from the induction equation for the magnetic field in the presence of a sheared one-component velocity flow with Hall and ion-slip effects. Because of applications in some geophysical problems, here we consider a more general two-component velocity field that generalizes the case studied in Straughan (2013) and also includes elliptic (in particular uniform circular motions a = − b) and hyperbolic orbits. We study linear instability and nonlinear stability of the induction magnetic field by means of the classical spectral and energy methods.