Stability of Beltrami Fields for Compressible Fluid Flows

This notebook is intended to give a first introduction to hydrodynamic instability. This includes both the physical concepts and several useful mathematical manipulations. The first chapter discusses the concept of linear instability theory and uses a simple wave equation to demonstrate the linearization and calculation of temporal and spatial growth. The second is an introduction to magnetohydrodynamics. In the chapter 3, we give a brief survey of Beltrami fields and ABC flows. In the chapter 4 we study the stability of the magnetostatic equilibrium through a relaxation of a magnetic field B in perfectly conducting compressible and viscous fluid. We establish stability criterion of a large class of Beltrami flows to any admissible displacement about the equilibrium configuration. We show that the field is stable to any displacement with the same periodicity as the basic flow, except the case where perturbations with wavelength much greater than the scale of the basic flow are included.