Magnetohydrodynamic waves in the solar corona : a mathematical investigation of the role of resonant absorption and phase mixing in coronal heating

Abstract

Background: The Sun is a massive and highly dynamic ball of plasma, and oscillations in kinetic and magnetic energy are commonplace throughout its atmosphere. Since the plasma conducts electricity, we model the fluid using magnetohydrodynamics (MHD) instead of hydrodynamics which is used for non-ionised fluids. We study two MHD wave phenomena, namely, phase mixing and resonant absorption. These are both phenomena that occur exclusively in MHD fluids and do not occur in hydrodynamic fluids. We study their implications for the coronal heating problem and coronal seismology. The solar surface is significantly denser than the atmosphere, and we model it as a solid wall. In other words, we impose line-tied boundary conditions at the solar surface where the velocity is set equal to zero.

Aims: 1) The first research chapter introduces some of the key properties of footpoint driven Alfvén waves (a type of MHD wave) which are relevant for the rest of this thesis. 2) The third chapter calculates an upper bound for the heat that linear phase-mixed Alfvén waves can produce at observed frequencies and amplitudes to assess its viability as a coronal heating mechanism. 3) The fourth chapter tests if line-tied boundary conditions still apply in a resonant absorption experiment where the transverse length-scales can be very short.

Methods: We take an analytic and theoretical approach to solving each problem and then check the results numerically.

Results: 1) We show that the growth of energy in closed loops for a sinusoidal footpoint driver is highly dependent on the driver frequency. If a resonance is excited, then the energy grows quadratically with time, and for a broadband driver, the energy grows linearly on average. If the loop is partially closed (i.e. only a fraction of the wave amplitude reflects at the boundary), the energy will converge towards a steady-state in which the energy of the loop remains constant with time. 2) We calculate an upper bound for the heat produced by phase-mixed Alfvén waves and find that it is, on average, too small to play a significant role in coronal heating. 3) We show that if the length-scales perpendicular or parallel to the boundary is sufficiently short, imposing line-tied boundary conditions may no longer be valid. However, researchers may wish to continue to use them in their models for their simplicity and ability to significantly reduce computation time if they understand and are aware of their limitations.