# Magnetohydrodynamic Waves in an Asymmetric Magnetic Slab

## Abstract

Analytical models of solar atmospheric magnetic structures have been crucial for our understanding of magnetohydrodynamic (MHD) wave behaviour and in the development of the field of solar magneto-seismology. Here, an analytical approach is used to derive the dispersion relation for MHD waves in a magnetic slab of homogeneous plasma enclosed on its two sides by non-magnetic, semi-infinite plasma with different densities and temperatures. This generalises the classic magnetic slab model, which is symmetric about the slab. The dispersion relation, unlike that governing a symmetric slab, cannot be decoupled into the well-known sausage and kink modes, *i.e.* the modes have mixed properties. The eigenmodes of an asymmetric magnetic slab are better labelled as quasi-sausage and quasi-kink modes. Given that the solar atmosphere is highly inhomogeneous, this has implications for MHD mode identification in a range of solar structures. A parametric analysis of how the mode properties (in particular the phase speed, eigenfrequencies, and amplitudes) vary in terms of the introduced asymmetry is conducted. In particular, avoided crossings occur between quasi-sausage and quasi-kink surface modes, allowing modes to adopt different properties for different parameters in the external region.

## Keywords

Coronal seismology Magnetic fields, photosphere Waves, magnetohydrodynamic Waves, modes## 1 Introduction

Dynamic solar events have been widely observed to induce perturbations in the magnetically dominated coronal plasma (Banerjee *et al.*, 2007; McLaughlin, Hood, and de Moortel, 2011; Arregui, Oliver, and Ballester, 2012; Mathioudakis, Jess, and Erdélyi, 2013; Komm *et al.*, 2015). The inhomogeneous structuring of the plasma parameters determines the characteristics and speed of the waves propagating in these structures. The detection and analysis of waves provides an indirect way to gain information about the structure of the magnetic waveguide environment through the techniques of solar magneto-seismology, see *e.g.* reviews by Nakariakov and Verwichte (2005), Andries *et al.* (2009), Ruderman and Erdélyi (2009), De Moortel and Nakariakov (2012). Analytical models of waves in solar magnetic environments are crucial in the accurate employment of this technique.

Linear MHD waves that propagate along magnetic field and density stratifications have been widely researched in Cartesian geometry to gain first insight into the often complex magnetic structures, including *e.g.* wave propagation in sunspots, coronal loops, prominences, or coronal hole boundaries, to name a few applications. One of the simplest approaches is when a single planar magnetic and/or density interface is considered and the properties of the available surface waves that propagate along this interface are deduced (Roberts, 1981a). Adding a second planar interface to this model, in order to form a simple way of mimicking structuring, gives a magnetic slab where the external environment is not magnetised and has a uniform but different density from the internal slab density (Roberts, 1981b). Next, a homogeneous magnetic field may then be added to the external plasma (Edwin and Roberts, 1982), and the eigenvalues, *i.e.* allowed wave modes of a magnetised slab system, can be determined. After this initial surge in analytical linear models of solar atmospheric magnetic structures, Cartesian geometry was replaced in favour of cylindrical geometry, for which significant research has led to the development and applications of solar magneto-seismology (Rosenberg, 1970; Uchida, 1970; Zajtsev and Stepanov, 1975; Roberts, Edwin, and Benz, 1984; Goossens, Andries, and Aschwanden, 2002; see a review by Ruderman and Erdélyi, 2009) and hypothesised mechanisms for solar plasma heating including mechanisms such as phase mixing (Heyvaerts and Priest, 1983) and resonant absorption (see a review by Goossens, Erdélyi, and Ruderman, 2011, with plenty of references), to name a few. This trend is largely a result of ubiquitous observations of solar atmospheric magnetic flux tube oscillations, which lend themselves more naturally to cylindrical geometry.

The purpose of the present work is to generalise the isolated magnetic slab model by investigating the linear wave physics that arises when the density and temperature of one external plasma is different from that of the other, and external densities and temperatures are both different from those inside the slab. The interest in this generalisation comes from the asymmetry of the system, which gives rise to asymmetric quasi-sausage and quasi-kink eigenmodes that demonstrate mixed properties.

Magnetic structures in the corona where solar magneto-seismology has previously been employed (Nakariakov and Ofman, 2001; Nakariakov and Verwichte, 2005) are often better modelled by cylindrical than Cartesian geometry. Instead, structures closer to the photosphere, such as the magnetic canopy, provide an asymmetric slab structure for application of this work. The magnetic canopy is a region of dominant magnetic field parallel to the surface of the Sun between the much less magnetised photosphere and the chromosphere that connects the magnetic field lines between active regions. Another application may be to oscillations in magnetic bright points (MBPs). MBPs are often elongated vertical magnetic structures between granular cells of different densities and temperatures. Further information on lower atmospheric solar magneto-seismology can be found in *e.g.* de Pontieu and Erdélyi (2006).

First, a derivation of the dispersion relation is presented in Section 2. The eigenmode solutions of the dispersion relation are compared to those of a symmetric slab in Section 2.1. The properties of the waves in an incompressible and low-beta plasma are discussed in Sections 3.2 and 3.3. Wide and thin slab approximations are made in Sections 3.4 and 3.5. A numerical procedure is employed in Section 4 to investigate the effect that varying the external densities has on wave dispersion.

## 2 Derivation of the Dispersion Relation

### 2.1 Comparison with a Symmetric Slab

There is an intrinsic difference between perturbations along symmetric and asymmetric magnetic slabs. The dispersion relation governing an asymmetric slab is a single equation, whereas the dispersion relation governing a symmetric slab (Roberts, 1981a) consists of two independent equations, corresponding to the sausage and kink eigenmodes.

### 2.2 Asymmetric Eigenmodes

There is a rich spectrum of MHD waves supported by a magnetic slab. In a symmetric slab, the dispersion relation, Equation (23), consists of two decoupled equations that correspond to the two types of fundamental wave supported by the slab: the “sausage” and “kink” MHD waves. Sausage and kink modes have been observed, for example, in chromospheric fibrils, where the ubiquity of these waves has been linked to coronal heating (Morton *et al.*, 2012).

*i.e.*they can have nodes inside the slab), that is, they exist when \(m_{0}^{2}<0\), which occurs when

Body modes are also affected by the asymmetric external environment (for visualisation see Figures 3c and 3d). Local maxima and minima in wave power are shifted towards the external plasma of higher density for a quasi-kink body mode and towards the external plasma of lower density for a quasi-sausage mode. However, body modes depend only weakly on the external plasma parameters and are therefore less affected than surface modes. This is shown analytically in Sections 3.4 and 3.5.

## 3 Analytical Solutions

In this section, simplifications are made to the dispersion relation, Equation (20), and approximate dispersion relation, Equation (22). Thin slab, wide slab, low-beta, and incompressible approximations are made.

### 3.1 Spurious Solutions

The solution to the exact and approximate dispersion relations, Equations (20) and (22), given by \(\omega =kv_{\mathrm{A}}\), is spurious and does not correspond to an eigenmode. To see this, observe that \(m_{0}=0\) for this solution, which leads to a linear (rather than oscillatory or exponential) solution to the governing differential equation, Equation (2.20). The same can be said for the solutions \(\omega=kc_{0}\) and \(\omega=kc_{\mathrm{T}}\). This rules out the possibility of the existence of pure sound waves and pure Alfvén waves.

### 3.2 Incompressible Approximation

### 3.3 Low-Beta Approximation

The following section concerns the case when the magnetic pressure strongly dominates the gas pressure within the slab, *i.e.* \(\beta:=2\mu_{0}{p_{0}}/B_{0}^{2}\ll1\). This is known as the low-beta approximation and corresponds to the Alfvén speed dominating the sound speed in the slab; this provides a good approximation of the solar coronal environment. However, for application here an external magnetic field needs to be considered; this is the subject of a follow-up article.

*e.g.*Roberts 1981b), the dispersion relation reduces to a quadratic expression in \(\omega^{2}\) whose solutions are the fast sausage and kink surface modes given by

Unfortunately, for the more general case of an asymmetric slab of low-beta plasma, the dispersion relation does not reduce to an analytically solvable equation. However, we find numerically that there are two fast surface modes. The quasi-sausage surface mode is not present for small \(kx_{0}\), but becomes a solution at an intermediate value of \(kx_{0}\) with phase speed \(\omega^{2}/k^{2}=\min {(c_{1}^{2},c_{2}^{2})}\). The quasi-kink surface mode is present for all values of \(kx_{0}\). Qualitatively, the solutions for a low-beta plasma are analogous to the fast quasi-sausage and quasi-kink mode solutions, discussed later in Sections 3.4 and 3.5.

In the following section, thin and wide slab approximations are made to the approximate dispersion relation, Equation (22), rather than the exact dispersion relation, Equation (20). Here, the aim is to retrieve the variety of wave modes with future applications in mind.

### 3.4 Thin Slab Approximation

Consider the case where the wavelength, \(\lambda\), of the waves propagating in the system is much greater than the width of the slab, \(2x_{0}\), *i.e* \(x_{0}/\lambda=kx_{0}/2\pi\ll1\).

*i.e.*as a solution to Equation (36)) when \(c_{0}< v_{\mathrm{A}}\), \(c_{1}< v_{\mathrm{A}}\), and \(c_{2}< v_{\mathrm{A}}\). For example, in Figure 5a, the minimum of \(c_{1}\) and \(c_{2}\) becomes a new cut-off, causing the fast quasi-sausage surface mode to degenerate for small \(kx_{0}\).

*i.e.*only for body modes. To find these solutions, set \(\omega^{2}=k^{2}c_{\mathrm{T}}^{2}(1+\nu (kx_{0})^{2})\) for some \(\nu>0\) that is to be determined. To see why this form has been chosen, a substitution into the definition of \(m_{0}^{2}\) demonstrates that \(|m_{0}^{2}|\to\infty\) and \(m_{0}x_{0}\) remains bounded as \(kx_{0}\to0\), as required. Using this ansatz, Equation (22) has a countably infinite set of quasi-sausage body solutions, which in the thin slab limit, behave like

### 3.5 Wide Slab Approximation

We now turn our attention to the behaviour of solutions to the dispersion relation in the wide slab limit, \(kx_{0}\gg{1}\). As the width of the slab increases with respect to the wavelength, the slab boundaries have diminishing effect on the modes and each other.

## 4 Numerical Solutions

In this section, the asymmetric dispersion relation is solved numerically, with particular interest placed on the effect of changing the ratio of the external densities.

### 4.1 Density Ratio Variation

More generally, consider an asymmetric slab whose equilibrium conditions are given by Figure 1. In Section 2.1 it was shown that the dispersion relation does not decouple into separate sausage and kink mode equations. However, their characteristics remain by and large, therefore the labels of “quasi-sausage” and “quasi-kink” may be used by referring to the anti-phase and in-phase oscillatory behaviour of the slab boundaries. However, we note that the quasi-kink mode now does not retain the width of the perturbed slab like the symmetric kink mode does.

For a wide slab width, \(kx_{0}\gg1\), Figure 7e illustrates that the eigencurves of the slow surface modes demonstrate a wave phenomenon known as “avoided crossing”. Avoided crossings occur when the phase speeds of two wave modes avoid crossing when a parameter of the system is varied due to constraints preventing them from being equal; it demonstrates a transferral of properties between the two modes and can be used to give insight into the modal structure. There is rich literature regarding avoided crossings for the eigensolutions of a wide range of physical processes including energy level repulsion in quantum physics (Naqvi and Brown, 1972) and coupled spring oscillations in classical mechanics (Novotny, 2010); in MHD the subject has been covered only briefly, for example between fast and slow magneto-acoustic gravity waves in a magnetically stratified plasma by Abdelatif (1990) and Mather and Erdélyi (2016).

*i.e.*half of the magnetic slab model. Each interface of an asymmetric magnetic slab has a distinct “local” phase-speed, and according to Roberts (1981a), these phase speeds are inversely proportional to the density in the non-magnetic region.

When they exist, the fast quasi-sausage and quasi-kink surface modes demonstrate an identical behaviour (not shown). As demonstrated analytically in Equations (34) – (35) and (37) – (40), the body modes are not dependent on internal or external densities to leading order in \(kx_{0}\). This means that body modes demonstrate only a weak dependence on the external densities, and an avoided crossing does not occur between these modes as the external densities are varied.

## 5 Discussion

For the first time, a mathematical model of an isolated magnetic slab in an asymmetric environment has been presented, generalising the classic symmetric magnetic slab model (Roberts, 1981b). Key analytical results demonstrate the fundamental differences between the behaviour of linear waves propagating along asymmetric and symmetric slabs.

Sausage and kink modes are traditionally thought of as the fundamental modes of symmetric magnetic slab and tube geometries. Unlike that of a symmetric slab, the dispersion relation governing linear waves along a magnetic slab in a non-magnetic asymmetric external environment does not decouple into two equations, which signifies that the eigenmodes of an asymmetric slab are not the pure sausage and kink modes that we are familiar with; instead, they are adjusted by the asymmetry in the external region and demonstrate mixed properties. For example, the quasi-kink mode does not have a spatially constant perturbed slab width like the symmetric kink mode.

*vice versa*for the in-anti-phase mode. A higher spring constant in the left or right springs in this analogy corresponds to a lower density outside the magnetic slab because a higher spring constant in an uncoupled spring gives a higher characteristic frequency. This gives us some motivation as to why the surface modes of the asymmetric magnetic slab have higher amplitudes on different sides for quasi-sausage and quasi-kink modes.

Mixed properties and mode coupling of asymmetric MHD waves are very interesting for solar physics because they signify interaction between different modes, and interaction can mean energy transfer. This indicates a potential plasma heating mechanism: waves with negligible dissipation are excited in the solar interior and propagate through the photosphere and chromosphere until they interact and transfer energy to modes with rapid dissipation in the upper solar atmosphere, *i.e.* corona, where the kinetic and magnetic energy of the wave is converted into thermal energy (Priest, 2014).

There is potential use of the ideas presented here as a diagnostic tool in the emerging field of solar magneto-seismology. Without loss of generality, consider a magnetic slab of plasma in an asymmetric environment with a higher density in the left external plasma than on the right (*e.g.* Figures 2a and 2b). The ratio between the amplitudes of oscillation on the left and right boundaries will be a function of the equilibrium and wave parameters of the slab. This ratio and several other parameters such as the slab width, the wave frequency, and the wavelength are directly observable, and parameters such as the density and temperature in each region can be determined through emission spectra, leaving only the magnetic field parameters unknown. Thus, in the context of solar observations, the cross-slab amplitude ratio could be used to calculate difficult-to-measure solar parameters such as the Alfvén speed and magnetic field strength when observing MHD waves in slab-like structures.

A similar solar magneto-seismological tool that shows its potential is the novel anti-node shift method, which uses the shift due to density structuring of the anti-nodes of standing modes to deduce the parameters of solar environments (Erdélyi and Verth, 2007; Verth *et al.*, 2007; Erdélyi, Hague, and Nelson, 2014). The success of this method indicates the need in the solar physics community for novel solar magneto-seismology methods.

There are clear future generalisations to the asymmetric slab model presented here. The consideration of a constant, uniform shear flow in one or both external environments would generalise the symmetric slab with shear flow studied by Li, Habbal, and Chen (2013), and introduce the additional physics of the Kelvin Helmholtz instability and negative energy waves. This would allow for better application of the asymmetric slab model to dynamic solar environments such as the solar wind, where magnetic structures with flows have been modelled using tube and symmetric slab geometries (Parker, 1965; Nakariakov, Roberts, and Mann, 1996; Taroyan and Erdélyi, 2002; Ruderman, 2010; Terradas *et al.*, 2011; Yu *et al.*, 2016). The generalisation of including a uniform external magnetic field and gravity, either in the direction parallel or perpendicular to the slab, would improve the applicability of the model, at the cost of analytic difficulty. With the completion of the next generation of solar telescopes, including the eagerly awaited Daniel K. Inouye Solar Telescope (DKIST), observations of fine-scale solar magnetic structures look forward to significant improvements in both spatial and temporal resolution; the future looks bright for solar magneto-seismology.

## Notes

### Acknowledgements

The authors would like to thank M.S. Ruderman for a number of fruitful discussions and the anonymous referee for their constructive comments. M. Allcock would like to thank the University Prize Scholarship and the SURE Scheme at the University of Sheffield. R. Erdélyi acknowledges the support from the UK Science and Technology Facilities Council (STFC), the Royal Society, and is also grateful to the Chinese Academy of Sciences Presidents International Fellowship Initiative, Grant No. 2016VMA045 for support received.

## Declaration of Potential Conflicts of Interest

The authors declare that they have no conflicts of interest.

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