MHD Stability Theory

Abstract

As in the earlier discussion on waves, disturbances from an initial equilibrium are usually assumed small enough to justify linearisation of the perturbed equations in the mathematical model, as an important first step to define the evolution of a complex system in this chapter. Normal mode analysis involving Fourier forms for the disturbance is therefore often invoked when the initial state is one-dimensional (dependent on only one spatial variable), except that the time exponent is assumed to carry a real part, often with the additional imaginary part (when there is an accompanying oscillation). The sign of the real part then determines whether or not the disturbance grows or decays exponentially—i.e. whether or not the system is linearly unstable or stable, respectively. Identification of the stabilising and destabilising forces is particularly important in MHD stability analysis, in both laboratory and astrophysical applications. In the case of the ideal MHD model, a variational formulation (an energy principle) permits the analysis of more complicated geometries, such as in modern experiments in controlled thermonuclear fusion research. Important stabilising and destabilising forces are identified under this formulation, which is then applied to investigate the stability of cylindrical and toroidal configurations. More direct analysis is usually followed to investigate instability in non-ideal MHD models, but we demonstrate an extension of the variational formulation that includes viscosity. Although the main application we consider is magnetic confinement in thermonuclear research, the final section on Hall instability includes an aspect relevant to laser-driven fusion, and some topics of interest in astrophysics and solar physics are discussed there and elsewhere. The additional bibliography for this chapter once again provides suggested further reading.