Aspects of Nonlinearity and Dissipation in Magnetohydrodynamics

Verwichte, Erwin André Omer (2000). Aspects of Nonlinearity and Dissipation in Magnetohydrodynamics. PhD thesis The Open University.

DOI: https://doi.org/10.21954/ou.ro.0000f963

Abstract

We investigate two projects: (i) the weakly nonlinear evolution of two oppositely travelling waves and (ii) the dissipative instability of a tangential discontinuity.

We show that the ponderomotive force is a basic nonlinear force, which is, to leading-order, proportional to the product of a wave quantity and a gradient of a wave quantity. The ponderomotive force of Alfvén waves corresponds to a magnetic wave pressure force.

The motion of beads on a string and the fluid motions in an oscillating tube are shown to be good mechanical analogues for the weakly nonlinear evolution of bounded Alfvén waves, especially in plasmas with a low plasma β.

We examine, analytically and numerically, the weakly nonlinear evolution of bounded fast magneto-acoustic waves in the cold plasma limit and show that the ponderomotive force moves plasma along the equilibrium magnetic field. The maximum density enhancement is proportional to α²/β, with α the wave amplitude and β of the order of the plasma β, We obtain the wave amplitude and frequency modulation and discuss the problems with the cold-plasma assumption.

The weakly nonlinear interaction of Alfvén pulses is investigated in the cold-plasma limit and for finite-β plasmas. We find excellent agreement between analytical and numerical results. We describe a density enhancement, maximally of order α²β^1/2 at the position of Alfvén pulse excitation, which splits into two slow pulses, We describe the shock-formation of the Alfvén and slow pulses.

The dissipative instability of the tangential discontinuity is examined and applied to coronal hole boundaries. We derive a dispersion relation, which includes weak viscosity and thermal conduction, and is solved for a specific model, using perturbation theory. The effect of viscosity and thermal conduction on stability are discussed, It is shown that dissipation lowers the threshold flow speed for instability.