The modern ground- and space-based instruments (DST, SST, DKIST, SDO, Hinode, Solar Orbiter) provide solar physicists with ample observations of solar plasma processes, i.e. magnetic bright points, spicules, plasma flows, structure of magnetic fields etc. at different temporal and spatial scales. However, direct measurements of important plasma properties such as e.g. magnetic field strength in the corona, using traditional observational techniques is incredibly difficult. Fortunately, magnetohydrodynamic (MHD) waves which permeate almost all structures observed in the solar atmosphere can be used as a proxy to determine the properties of the plasma, through a tool known as solar magnetoseismology. Therefore, advanced theoretical modelling becomes essential to explain the ever increasing quality of observational results and provide more accurate information about MHD wave propagation and solar atmospheric plasma properties.
MHD waves in a spatially non-uniform plasma
When it comes to an analytical description of wave properties in a solar plasma, the traditional technique of solving the linearised MHD equations for small perturbations is usually adopted. This method ultimately obtains a dispersion relation which relates the frequency of the wave and its wavenumber along with known characteristic properties of the background plasma. Pioneering work  first applied this in a solar context for a uniform magnetic slab model and provided an analytical description of magnetoacoustic kink and sausage modes split into two physical categories namely surface and body waves. In a non uniform magnetic waveguide however (see Figure 1) the governing differential equations develop coefficients which are spatially varying, along the coordinate of inhomogeneity. As a result, the governing equations now have no known closed form analytical solution and consequently no dispersion relation can be obtained, therefore, a numerical approach must be adopted.
We present a numerical approach based on the shooting method and bisection method to obtain the eigenvalues for a magnetic slab with an arbitrary non-uniform background plasma and/or plasma flow . Real frequencies are obtained such that information about trapped modes of the system can be analysed, complex frequencies such as those in the leaky or continuum regimes are left for future work. The initial wave phase space is used as the domain to find eigenvalues that provide exact solutions to satisfy the relevant boundary conditions of the waveguide, namely the continuity of perturbation of radial displacement and total pressure. Additional information from the definitions of the sausage mode and the kink mode are utilised to obtain the relevant (anti-)symmetric eigenfunctions. The governing equations describing these properties are derived and solved numerically as no analytical solution exists without making simplifying assumptions about the model. The values of wave frequency and wavenumber that satisfy both boundary conditions simultaneously will be classified as a solution and used in further analyses of the wave modes.
This is an extremely powerful numerical tool as, provided the initial equilibrium is stable, a wave analysis of any non-uniform or non-linear plasma can be investigated without the need for a dispersion relation. It should also be noted that this numerical approach is not limited to a purely planar geometry, a cylindrical or spherical geometry would only modify the mathematical vector operators used in the initial analytical description – the physics of the numerical tool still remains the same.
Non-uniform plasma density in a coronal slab
We investigate the properties of magnetoacoustic waves in a coronal slab with a non-uniform background plasma flow modelled with the profiles in Figure 2. A sinc(x) profile models the spatial distribution seen in intensity images of magnetic bright points . The width of the Gaussian profiles is determined by a parameter W, where a smaller W indicates a more inhomogeneous profile. The numerical algorithm obtains the eigenvalues plotted on the dispersion diagram (Figure 3) for each case which allows the resulting eigenfunctions for total pressure and horizontal perturbation of velocity to be calculated.
In Figure 4 the eigenfunctions for the different cases of non-uniform equilibria are shown for the slow body sausage and kink modes. It can be seen by comparing the uniform and extreme non-uniform cases that additional nodes and points of inflexion are present for the non-uniform equilibrium case which may cause difficulty when interpreting the wave modes of observed/simulated perturbed velocity fields in non-uniform waveguides.
- A numerical approach has been developed to obtain the eigenvalues for magnetoacoustic waves in an arbitrary symmetrically non-uniform magnetic slab.
- The algorithm is heavily tested against numerous well known analytical results and successfully obtains the correct eigenvalues for both the sausage and kink modes in the long and short wavelength limits.
- Investigations of non-uniform plasma and background flow modelled as a series of Gaussian profiles reveal that slow body modes are more affected by the non-uniform equilibria. For investigation, analysis and discussion into the non-uniform background flow case, refer to .
- Additional nodes and points of inflexion appear in the resulting eigenfunctions which may be of interest to observers when interpreting observational results of MHD waves in the highly structured non-uniform solar atmosphere.
- Future work investigating similar cases to those considered in this work but in the case of a cylindrical model can be found in .