Investigation of the properties of MHD waves in the presence of non-uniform equilibria and flow using a unique numerical approach

In this thesis we have developed a numerical eigensolver, which is capable of obtaining the magnetoacoustic wave solutions of a given equilibrium in either a Cartesian or cylindrical geometry, for which it is applied to a number of different case studies in the context of waveguides in the solar atmosphere. The eigensolver is tested against known analytical results, with increasing complexity, where the previously obtained solutions are correctly retrieved. The equilibrium is allowed to be symmetrically non-uniform in the internal region of the waveguide, whereas the external region must be uniform. The development of this eigensolver has endless possibilities and greatly advances investigations of previous analytical studies.

Previous analytical studies of magnetoacoustic waves in solar waveguides have been restricted to consider only a limited selection of possible equilibria, such that a mathematical analysis can be conducted. If an equilibrium is too complex, an analytical description breaks down and a relationship describing the wave dispersion cannot be obtained. The work presented in this thesis aims to bridge the gap between realistic modelling of solar waveguides and providing a description of the waves that can propagate. In Chapter 2 we introduce the numerical eigensolver and describe the physics of the algorithm, along with the fundamental properties of MHD waves which it relies on. Additionally, we discuss potential avenues to improve the eigensolver that were not possible within the time frame of this project. Within the context of applying this numerical eigensolver, we have made an original contribution to knowledge in three areas:

- In Chapter 3 ``The effect of non-uniform plasma density and flow on magnetoacoustic wave modes in a magnetic slab geometry'', we test the numerical eigensolver against previously obtained results for magnetoacoustic waves in photopsheric and coronal slabs. Modelling the waveguide in a Cartesian geometry, we then extend this study to consider a non-uniform plasma with a density modelled as a series of Gaussian profiles and also a sinc(x) function. The analysis is conducted under both photospheric and coronal conditions with the resulting eigenfunctions displayed for both scenarios. A magnetic slab in the presence of a non-uniform internal background plasma flow is then presented for which the governing equations are derived, eigenvalues obtained and eigenfunctions displayed. The implications that these results may have for observational interpretations of wave modes in the solar atmosphere is discussed.

- In Chapter 4 ``The effect of non-uniform plasma density and flow on magnetoacoustic wave modes in a magnetic cylinder geometry'', we conduct a similar analysis to that presented in the previous chapter, however with a focus on the wave properties when the waveguide is modelled as a magnetic cylinder. The numerical eigensolver is again tested against previously obtained analytical results of a uniform cylinder before modelling the plasma density and background flow inside the waveguide as spatially non-uniform. The eigenfunctions are again obtained which then allows both a 2D and 3D visualisation of the perturbations to be shown. A discussion about the effect that a non-uniform equilibrium has on the perturbed eigenfunctions is presented, with a focus on the possible onset of instabilities, comparison to previous similar studies and the implications for observational results.

- In Chapter 5 ``Effect of non-linear twist and rotational flow on MHD wave modes of a magnetic cylinder'', we use the numerical eigensolver to retrieve the solutions of a more complicated scenario investigating a twisted magnetic flux tube. The previously obtained analytical results are retrieved and our analysis compliments that of previous studies by considering the modified continua due to the twisted magnetic field. We then investigate the wave modes that can exist in a rotating flux tube under both coronal and photospheric conditions where the rotational flow is modelled using a linear profile. This analysis is further extended to investigate the effect that a nonlinear rotational flow has on the properties of MHD waves and the resulting continuum regions. In both cases the eigenfunctions are calculated and we find that in the presence of a background rotational flow under photospheric conditions, the slow surface kink mode and the fast surface kink mode in the thin flux tube limit appear indistinguishable to observers.

These findings have implications for the theory of MHD waves in non-uniform plasmas and their use in providing accurate estimates of local plasma properties when used as a proxy for atmospheric-seismology.

We summarise our findings and discuss potential further investigations and improvements in Chapter 6.