Joint Instabilities of Sheared Flows and Magnetic Fields

Abstract

<p>Shear flows and magnetic fields are ubiquitous in astrophysical bodies such as stars and accretion discs. Furthermore,<br>the interaction between flows and magnetic field plays a key role in the dynamics of plasma fusion devices. Typically,<br>the flows and magnetic field are both sheared, and it is therefore a problem of fundamental importance to understand<br>the instabilities that may occur in such a system.</p><p>In the absence of magnetic field, the linear stability of a viscous sheared flow is governed by the Orr-Sommerfeld<br>equation; this is one of the classic problems of hydrodynamics. At the other limit, there are somewhat analogous<br>instabilities of a fluid of finite electrical conductivity containing a static sheared magnetic field. These are related to<br>the classical tearing modes that have received considerable attention in both the astrophysical and plasma physics<br>literature.</p><p>In general though, the fluid flow and the magnetic field will both be important players. Previous studies have investigated<br>configurations which have served as models for systems such as the magnetotail and solar surges. While these<br>investigations have been fruitful, the prescription of the basic field and flow, while physically motivated, have been<br>chosen somewhat arbitrarily. It is therefore of interest to consider the instability problem within this more general<br>framework.</p><p>Motivated astrophysically, such as by the dynamics in the solar tachocline, here we consider a self-consistent problem<br>in which both instabilities can occur. In particular, we consider the stability of equilibrium states arising from the<br>shearing of a uniform magnetic field by a forced transverse flow. The problem is governed by three non-dimensional<br>parameters: the Chandrasekhar number, and the flow and magnetic Reynolds numbers. In opposite limits of parameter<br>space, we recover the predictions of the aforementioned classical problems. As we move through this three-dimensional<br>parameter space, a range of interactions are possible: We demonstrate the stabilisation of a purely hydrodynamic<br>instability through the magnetic field, show the existence of a joint instability outlining the physical mechanisms at<br>play, and demonstrate that under certain conditions, hydrodynamically-stable parallel shear flows lead to instability<br>growth rates that exceed those of static tearing modes. To conclude, we elucidate the consequences of considering<br>the linear stability of an evolving background state and show that a quasi-static approach may not be meaningful. In<br>these circumstances, it therefore becomes essential to perform a stability analysis of a time-varying basic state.</p><p> </p><p> </p>