Slow body magnetohydrodynamic waves in solar photospheric flux tubes with inhomogeneous equilibria

Intense magnetic waveguides in the solar photosphere, such as pores and sunspots are ideal environments for the propagation of guided waves. However, modelling these photospheric waveguides with varying background quantities such as plasma density and magnetic field has thus far been very limited. Such modelling is required to correctly interpret MHD waves observed in pores and sunspots with resolved inhomogeneities such as light bridges and umbral dots. Theoretical descriptions of waves are very sensitive to the way density is distributed and configuration of waveguides. Current theoretical models assume a homogeneous distribution of plasma parameters and magnetic field. High resolution observations of the last decade show that these assumptions are very crude and alterations from this ideal setup are expected to have a major effect on the property of waves. One major impediment in extending the existing theoretical modelling to more realistic situations was the complexity of the mathematical framework in which waves are investigated.

The aim of my research presented in this Thesis is to address this shortcoming and propose analytical and numerical techniques for wave identification in the presence of inhomogeneous magnetic waveguides. Here, we provide two various types of models that can be used to investigate slow MHD modes in solar photospheric flux tubes in the presence of local equilibrium density, pressure and magnetic field inhomogeneity. In all studied cases, the equilibrium profile inhomogeneity is represented by a local circular enhancement or depletion whose strength, size and position can change.

First, we investigate the propagation characteristics and the spatial structure of slow body eigenmodes in a magnetic flux tube with circular cross section. For analytical progress we assume that the model has constant plasma-$\beta$, assuming that only the plasma equilibrium density has a spatial dependence.

Later, the constant plasma-$\beta$ model (which is a rather restrictive approximation) is relaxed and results of the modification of the properties and morphology of slow body modes are investigated considering a case where not only the equilibrium density as function of coordinates, but also equilibrium pressure and magnetic field, in line with observations and numerical modelling. Analytical progress was made by considering that the plasma pressure and density vary following the same dependency on coordinates, meaning that we are dealing with a constant sound speed, i.e. isothermal equilibrium. Given the complexity of the problem, the task was addressed numerically via the Fourier-Chebyshev Spectral method (FCS), as well as Galerkin Finite Element method (FEM), respectively. The radial and azimuthal variation of eigenfunctions is obtained by solving a Helmholtz-type partial differential equation with Dirichlet boundary conditions for slow body waves.

The inhomogeneous transverse equilibrium density profile results in modified eigenvalues and eigenvectors. In particular, a modification in the equilibrium density distribution leads to a decrease in the eigenvalues and the spatial structure of modes ceases to be global, as the modes migrate towards regions of lower density in the case of the constant plasma-$\beta$ model. Comparing the homogeneous case and the cases corresponding to depleted density enhancement, the dimensionless phase speed undergoes a significant drop in its value (at least 40\%). In contrast to the density enhancement, the slow body modes investigated here preserve their morphology.

Our investigations can be considered as a very first step in studying the properties of waves in sunspots and pores in the presence of local inhomogeneities in the form of umbral dots (UDs) and light bridges (LBs). Multistructure density distributions (as observations show) can often be replaced by a resulting structure that adequately captures the effects of multiple UDs present in the sunspot umbra. This means that the equilibrium density inhomogeneity profile is represented by a single local density inhomogeneity, which simulates the UDs observed in the sunspot umbra region, assuming that they are placed close to each other. Furthermore, the LB observed in sunspots can be modeled by a single local density inhomogeneity that separates the sunspot umbra region. The more complex the shape of the density inhomogeneity, the spatial structure of the higher-order slow-body modes will be modified depending on the considered density shape and the cross-sectional configuration of the waveguide. As a result, the pattern of possible waves loses the high-degree of symmetry one can meet in homogeneous cases, especially for higher-order modes, which means that the pattern is no longer global and, therefore, cannot be easily identified. Local wave observations in sunspots may be a way to identify the location and size of density inhomogeneities in the umbra region.

These intriguing findings have significant ramifications for optimising modal structures that may be viewed, compared, and correlated in observational data as well as for constructing and organising modes in circular structures.