Fluid and MHD Instabilities: Non-Trivial and Time-Dependent Basic States

It is widely accepted that the coherent structures that we observe on the solar surface are the result of magnetic field interactions across a wide range of scales. The leading suggestion is that flux tubes rise throughout the convection zone via a mechanism called magnetic buoyancy. The mechanism by which large scale, toroidal magnetic fields are generated is understood through dynamo theory, and most dynamo models agree that poloidal magnetic fields are wound up in a location of strong velocity shear called the Solar Tachocline. The solar dynamo, and therefore modelling the Sun is made difficult by the fact that the magnetic field reacts back in the flow through the Lorentz force. In addition to this, the Solar Tachocline lives at the base of the convection zone, and therefore an understanding of solar convection is significantly important. All of this considered, modelling the Sun, and hence magnetic buoyancy can be a complex nonlinear problem. While there have been nonlinear studies into the relationship between shear, convection, and magnetic buoyancy, nonlinear numerical experiments are hard to conduct and the results can be hard to interpret. In addition to this, the interactions between shear flow and magnetic field within the Tachocline are ultimately a time-dependent process, and so linear studies where the basic state is steady are limited in their scope.

With this motivation, we consider a set of linear stability problems where the basic state is either non-trivial or time-dependent, with the ultimate aim to study magnetic buoyancy instabilities of a time-evolving magnetic field generated by the transverse shearing of a perpendicular uniform magnetic field. We consider the joint linear stability of a steady parallel shear flow and magnetic field, which are linked self-consistently. We find that on one end of parameter space, the resultant instabilities are hydrodynamic and can be described through the Orr-Sommerfeld equation. On the opposite end of the parameter space, we find that the dominant instability is a tearing mode, which can be understood through an equation analogous to the Orr-Sommerfeld equation, in which we consider a parallel resistive magnetic field. In the middle of this parameter space, we find that instabilities are possible where both field and flow are important. We find that a hydrodynamically stable flow can enhance the instability of an unstable basic magnetic field. Taking another step towards the main enterprise, we then consider the same process of generating the field and flow, however the shearing takes place in the presence of a stably-stratified and compressible atmosphere. We find that a multitude of instabilities, i.e. vii undular and interchange modes are possible. In addition to this, we derive a stability criterion for magnetic buoyancy in terms of the imposed shear flow. We find that this instability criterion is reasonable at predicting the onset of interchange and undular mode instabilities. Furthermore, we find that 2D undular modes are possible in such a system. This is surprising, as in the absence of a velocity shear, 2D undular modes are understood to be hard to destabilise. In addition to this, viscosity is thought to have a stabilising effect, however we find that increasing the viscosity can lead to destabilisation of the basic state. The role of viscosity is shown through the basic state analysis which shows that the amplitude of the basic state horizontal magnetic field is linearly proportional to viscosity.

We then wish to incorporate time-dependence into the analysis. Performing a time-dependent linear stability analysis is difficult as one cannot perform a typical modal analysis, in which solutions are separated into exponentially decaying/growing waves. The main hurdle is therefore defining a suitable measure of instability. We study the time-dependent linear stability analysis of a classical problem — Rayleigh Benard convection. We consider two types of time-dependent basic temperature fields which are considered within the literature: i) linear cooling at the top boundary, and ii) period modulation of the temperature at the top and bottom boundary. We reproduce the results within the literature and extend the investigation to unstudied asymptotic regimes. We find that when the amplitude of the periodic modulation at the boundaries is increasingly large, then the system becomes asymptotically unstable, i.e. Rac → 0. In this chapter the main aim is to illustrate the difficulties associated with performing a time-dependent linear stability analysis. We show that the definition of a stability boundary is often problem specific, and either involves characteristics of the problem, or defining a global measure.

With this in mind, we investigate magnetic buoyancy instabilities of a time-varying magnetic field. First, we find that the transient evolution of the basic state configuration depends heavily on the initial conditions. Then, through a frozen-in-time analysis, we find a multitude of instabilities are present throughout the evolution of the basic state. Finally, we define a global measure of kinetic and magnetic energy to characterise the onset of instability. We consider two sets of initial conditions for the perturbed quantities, and show that there is an initial condition which results in optimal growth of the disturbed quantities. We then justify the choice of parameters and discuss extensions to the problem considered.