Joint Instabilities of Sheared Flows and Magnetic Fields

Shear flows and magnetic fields are ubiquitous in astrophysical bodies such as stars and accretion discs. Furthermore,
the interaction between flows and magnetic field plays a key role in the dynamics of plasma fusion devices. Typically,
the flows and magnetic field are both sheared, and it is therefore a problem of fundamental importance to understand
the instabilities that may occur in such a system.

In the absence of magnetic field, the linear stability of a viscous sheared flow is governed by the Orr-Sommerfeld
equation; this is one of the classic problems of hydrodynamics. At the other limit, there are somewhat analogous
instabilities of a fluid of finite electrical conductivity containing a static sheared magnetic field. These are related to
the classical tearing modes that have received considerable attention in both the astrophysical and plasma physics
literature.

In general though, the fluid flow and the magnetic field will both be important players. Previous studies have investigated
configurations which have served as models for systems such as the magnetotail and solar surges. While these
investigations have been fruitful, the prescription of the basic field and flow, while physically motivated, have been
chosen somewhat arbitrarily. It is therefore of interest to consider the instability problem within this more general
framework.

Motivated astrophysically, such as by the dynamics in the solar tachocline, here we consider a self-consistent problem
in which both instabilities can occur. In particular, we consider the stability of equilibrium states arising from the
shearing of a uniform magnetic field by a forced transverse flow. The problem is governed by three non-dimensional
parameters: the Chandrasekhar number, and the flow and magnetic Reynolds numbers. In opposite limits of parameter
space, we recover the predictions of the aforementioned classical problems. As we move through this three-dimensional
parameter space, a range of interactions are possible: We demonstrate the stabilisation of a purely hydrodynamic
instability through the magnetic field, show the existence of a joint instability outlining the physical mechanisms at
play, and demonstrate that under certain conditions, hydrodynamically-stable parallel shear flows lead to instability
growth rates that exceed those of static tearing modes. To conclude, we elucidate the consequences of considering
the linear stability of an evolving background state and show that a quasi-static approach may not be meaningful. In
these circumstances, it therefore becomes essential to perform a stability analysis of a time-varying basic state.