# Propagation of Long-Wavelength Nonlinear Slow Sausage Waves in Stratified Magnetic Flux Tubes

## Abstract

The propagation of nonlinear, long-wavelength, slow sausage waves in an expanding magnetic flux tube, embedded in a non-magnetic stratified environment, is discussed. The governing equation for surface waves, which is akin to the Leibovich–Roberts equation, is derived using the method of multiple scales. The solitary wave solution of the equation is obtained numerically. The results obtained are illustrative of a solitary wave whose properties are highly dependent on the degree of stratification.

## Keywords

Magnetohydrodynamics Nonlinear waves## 1 Introduction

The emergence of magnetic flux in the Sun is inhomogeneous, with isolated magnetic flux tubes being a common form of structuring. These tubes form an “elastic medium” and may therefore act as waveguides. The propagation of linear waves along magnetic cylinders has been studied extensively (see, for example, Defouw, 1976 or Roberts, 1981, or the reviews by Andries *et al.*, 2009, De Moortel, 2009, Mathioudakis, Jess, and Erdélyi, 2013 or Wang, 2011). Aspects of the propagation of nonlinear waves have also been studied, with the relevant theory and a number of important results summarised in Ruderman (2003), Ruderman (2006) and Ballai and Ruderman (2011).

The propagation of nonlinear MHD waves (solitons) was first studied in the context of solar physics by Roberts and Mangeney (1982). The propagation of nonlinear wave modes in slabs was discussed more extensively in a series of papers. Merzljakov and Ruderman (1985) derived the Benjamin–Ono equation governing wave propagation in a vertical magnetic slab embedded in a field-free atmosphere. They also estimated the energy transfer of a propagating soliton in a vertically homogeneous atmosphere, and showed that a soliton solution may exist in a stratified magnetic slab. Next, Molotovshchikov and Ruderman (1987) derived an equation governing the propagation of nonlinear slow sausage waves along a magnetic flux tube, again, in a vertically homogeneous atmosphere, but with the added feature of a non-zero external magnetic field \((B_{e} \ne 0)\). Independently, Roberts (1985) obtained the governing equations in both geometries, in the case where the external magnetic field is zero \((B_{e} = 0)\). The findings of Molotovshchikov and Ruderman (1987) were similar to those of Roberts (1985), with the exception that, while their equation possessed a more complicated form of the dispersive term, it was open to numerical analysis. These results, together with those of Molotovshchikov (1989), confirmed the existence of solitary MHD wave solutions to the governing equation whose properties on collision are similar to those of solitons.

Several recent developments of the aforementioned models may be found. A discussion of the use of the thin flux tube approximation, its limitations, and the two-mode approximation may be found in Zhugzhda (2002). Zhugzhda (2004) studied the solutions of slow nonlinear MHD equations in the form of shock waves, while Zhugzhda (2005) derived a new set of equations without making use of the long-wavelength approximation. Numerically, Sakai *et al.* (2000) studied the impact that gravitational stratification has on the upward and downward propagation of nonlinear MHD waves in flux tubes, while Erdélyi and Fedun (2006) performed simulations modelling excitation, time-dependent propagation, and interaction of solitary waves in solar atmospheric plasmas. More recently, Chargeishvili and Japaridze (2016) found that the propagation of a modulated MHD soliton may cause the temperature of a plasma to rise in the peripheral regions of a magnetic flux tube.

From an observational point of view, Zaqarashvili, Kukhianidze, and Khodachenko (2010) suggested that a series of observations by the *Solar Optical Telescope* on board the *Hinode* satellite confirms the existence of slow sausage solitons, propagating in a stratified atmosphere. However, their theoretical analysis of the observations made use of a model of a soliton in a magnetic slab, where a tube would have been better suited.

The present work deals with extending the research of Molotovshchikov and Ruderman (1987) so that it includes the effects of gravitational stratification and radius expansion of the flux tube. We begin by establishing how the undisturbed magnetic field, pressures and densities are related to the varying radius of the tube, and to each other. Next, we employ the thin flux tube approximation and make use of the method of multiple scales to expand the ideal MHD equations. We then reduce these equations to a single nonlinear integro-differential equation. The equation obtained by Molotovshchikov and Ruderman (1987) is then shown to be a simplified case of that obtained here. Lastly, we reduce the governing equation to a form more suitable to analysis, and we numerically obtain a series of solitary wave solutions. The properties of these solutions, including speed–amplitude relations, dependence on stratification, and width–amplitude relations, are then discussed.

## 2 The Basic State

Inside the tube, the equilibrium quantities, namely the kinetic pressure \(p_{0}\), the density \(\rho_{0}\), and the magnetic induction \(\boldsymbol {B}_{0}\) depend on \(r\) and \(z\) only. We suppose that the azimuthal component of the magnetic field is zero, such that \(\boldsymbol {B}_{0} = (B_{r0}, 0, B_{z0})\), and also that \(\nabla \times \boldsymbol {B}_{0} = 0\). Finally, it is assumed that the atmospheric density scale height \(H\) is much greater that the radius of the tube \((r_{0} \ll H)\), such that the effect of gravity is weak, but not negligible.

## 3 The Governing Equation for Nonlinear Surface MHD Waves

After having defined the undisturbed state using Equations (13), we may now begin deriving the equation that governs the propagation of nonlinear small-amplitude slow sausage MHD waves in the long-wavelength approximation. Since the tube is assumed to be axisymmetric, the variables we deal with depend on \(r\) and \(z\) only, and the azimuthal component of the velocity and the magnetic induction are zero.

Let \(\epsilon \ll 1\) be the non-dimensional amplitude of the waves. By the thin-tube approximation, we assume that \(r_{0}/L \sim \epsilon\), where \(L\) is the wavelength. We now introduce a new variable \(\tau = \epsilon ( t - \int \mathrm{d}z/c_{T} )\). Hence, the nonlinearity and dispersion significantly influence the wave when it progresses a distance of the order of \(\epsilon^{-2}r_{0}\). For the effect of the stratification to be of the same order as the effects of nonlinearity and dispersion, we assume that \(\mu = \epsilon^{2}\) in our description of the undisturbed state.

Inside the tube, the horizontal scale is equal to \(r_{0}\). Outside it, however, the horizontal and vertical scales are equal to the wavelength. It is therefore advantageous to introduce the new variable \(r_{e} = \epsilon r\) to be used outside the tube, instead of \(r\).

When deriving Equations (24), we make use of the relation \(r_{e} = \epsilon r\) and expand all the outside variables in series around \(r_{e} = 0\). This means that, for example, \(u_{e}(\epsilon r_{0}) = u_{e}^{(0)} + \epsilon r_{0} (\partial u_{e}/ \partial r_{e})^{(0)} + \cdots\), where the upper zero indices indicate that the variable is taken at \(r_{e} = 0\).

## 4 Numerical Investigation

The initial value problem for Equation (41) was first solved numerically by Leibovich and Randall (1972). When doing so, they used a simplified version of the equation which we will now address briefly.

The results obtained by Molotovshchikov and Ruderman (1987) are similar to our own computations. This is due to the fact that the equation in Molotovshchikov and Ruderman (1987) may be reduced to Equation (44) when the parameter \(\psi\) is equal to one.

## 5 Conclusion

The propagation of long-wavelength small-amplitude slow MHD sausage waves in expanding magnetic flux tubes in a field-free environment, with the effects due to gravity taken into account, is modelled by Equation (41).

It should be noted that in this case the tube is not of constant width, being stratified, and expanding due to the effects of gravity. The vertical gravitational stratification is represented in Equation (41) by the final term. Waves propagating along the tube will therefore be under the influence of three competing effects: nonlinearity, dispersion, and stratification. We have therefore proven that if these three effects are of the same order, a soliton may form and propagate stably.

All of these properties are relevant to the study of coronal heating due to the innate properties of solitons, such as energy conservation during propagation. This makes solitons prime candidates as a possible energy transfer mechanism from the photosphere into the lower atmosphere.

The thin flux tube approximation imposes the condition that \(r_{0} \ll H\), which suggests that, for a photospheric scale height of \(H \approx 200~\mbox{km}\), we may expect a tube radius of \(r_{0} \lesssim 50~\mbox{km}\) at the base.

To illustrate an application of the theory, we choose the following values typical to the photosphere: \(v_{A} = 10~\mbox{km}\,\mbox{s}^{-1}\), \(c_{e} = c_{0} = 8~\mbox{km}\,\mbox{s}^{-1}\), \(\gamma = \frac{5}{3}\), \(\rho_{0} = 5 \times 10^{-3}~\mbox{kg}\,\mbox{m}^{-3}\), and a density ratio of \(\rho_{e0}/\rho_{0} = 0.9\), which yield a wave amplitude of \(\eta \approx 3~\mbox{km}\). In contrast, choosing values typical to the base of the corona, \(v_{A} = 3000~\mbox{km}\,\mbox{s}^{-1}\), \(c_{e} = c_{0} = 200~\mbox{km}\,\mbox{s}^{-1}\), \(\gamma = \frac{5}{3}\), \(\rho_{0} = 1 \times 10^{-10}~\mbox{kg}\,\mbox{m}^{-3}\), and a density ratio of \(\rho_{e0}/\rho_{0} = 0.9\), the amplitude of the wave will increase to \(\eta \approx 24~\mbox{km}\).

Further work may include the numerical calculation of the typical energy of such solitary waves in the photosphere, the distance a disturbance would need to travel before the various forces balance out and a soliton forms, or the interaction of several solitary waves in order to confirm that their properties conform to those of solitons.

## Notes

### Acknowledgements

The authors would like to thank M.S. Ruderman for his many useful suggestions towards obtaining the results in this paper. The authors are also grateful to the Science and Technology Facilities Council (STFC) UK and the support received from the Royal Society for part of this work. M. Barbulescu would also like to thank the members of the Debrecen Heliophysical Observatory, Hungarian Academy of Sciences, for their hospitality during his visit, where part of this work was conducted.

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