# Nonlinear Generation of Fluting Perturbations by Kink Mode

## Abstract

We study the excitation of fluting perturbations in a magnetic tube by an initially imposed kink mode. We use the ideal magnetohydrodynamic (MHD) equations in the cold-plasma approximation. We also use the thin-tube approximation and scale the dependent and independent variables accordingly. Then we assume that the dimensionless amplitude of the kink mode is small and use it as an expansion parameter in the regular perturbation method. We obtain the expression for the tube boundary perturbation in the second-order approximation. This perturbation is a superposition of sausage and fluting perturbations. The amplitude of the fluting perturbation takes its maximum at the middle of the tube, and it monotonically decreases with the distance from the middle of the tube.

## Keywords

Sun, corona Magnetic fields Magnetohydrodynamics Waves Oscillations## 1 Introduction

After transverse oscillations of coronal magnetic loops were first observed by the *Transition Region and Coronal Explorer* (TRACE) in 1998 and were reported by Aschwanden *et al.* (1999) and Nakariakov *et al.* (1999), they attracted enhanced attention from theorists. These oscillations were interpreted as fast kink standing waves in magnetic flux tubes. Initially the simplest model of a straight homogeneous magnetic tube (*e.g.* Ryutov and Ryutova, 1976; Edwin and Roberts, 1983) was used for the theoretical studies of coronal-loop transverse oscillations. Later more sophisticated models taking into account such effects as the plasma density variation along and across a tube, the presence of flows, and loop cooling were developed. For a review of the theory of coronal-loop oscillations see, *e.g.*, Ruderman and Erdélyi (2009).

The majority of studies on coronal-loop kink oscillations were carried out in the approximation of linear magnetohydrodynamics (MHD). Studies of nonlinear coronal-loop kink oscillations are sparse. Ruderman (1992) and Ruderman, Goossens, and Andries (2010) analytically studied nonlinear propagating kink waves. Ruderman and Goossens (2014) also analytically investigated the effect of nonlinearity on standing kink waves in magnetic tubes with the density varying along the tube. There were also a few numerical studies of nonlinear kink oscillations of magnetic tubes (*e.g.* Terradas *et al.*, 2008; Magyar, Van Doorselaere, and Marcu, 2015; Magyar and Van Doorselaere, 2016).

This study is motivated by the discussion in a meeting of an international group led by G. Verth and R. Morton at the International Space Science Institute (ISSI). In this meeting (Terradas, Magyar, and Van Doorsselaere 2017) presented the results of the study of nonlinear kink oscillations of a magnetic tube. In particular, they reported the appearance of a fluting perturbation of the tube boundary excited by an initially imposed kink oscillation. The period of the fluting perturbation was equal to the half period of the kink oscillation, and its amplitude took its maximum at the centre of the magnetic tube. Some of the meeting participants insisted that the fluting perturbation must be the first harmonic of the first fluting mode. The amplitude of this harmonic is zero at the tube centre.

Ruderman, Goossens, and Andries (2010) and Ruderman and Goossens (2014) predicted the excitation of the fluting perturbation with the frequency equal to the double frequency of the kink mode. They also predicted that the amplitude of the fluting perturbation is proportional to the amplitude of the kink mode squared. However, their results cannot be directly compared with those reported by Terradas, Magyar, and Van Doorsselaere (2017) because Ruderman, Goossens, and Andries (2010) studied propagating waves, and Ruderman and Goossens (2014) concentrated on the nonlinearity effect on the kink oscillations of a magnetic tube strongly stratified in the longitudinal direction.

This article aims to study analytically the excitation of a fluting perturbation by an initially imposed kink mode and compare the analytical results with the numerical results obtained by Terradas, Magyar, and Van Doorsselaere (2017). The article is organised as follows. In the next section we formulate the problem and write down the governing equations and boundary conditions. In Section 3 we use the regular perturbation method to study the excitation of fluting perturbation. Section 4 contains the summary of the results obtained and our conclusions.

## 2 Problem Formulation and Governing Equations

Below we assume that the tube is thin: \(R/L = \epsilon \ll 1\). In accordance with this assumption we introduce the stretching variable \(Z = \epsilon z\). The characteristic alfvénic time related to the tube radius is \(R/V_{A}\), where \(V_{A} = B(\mu_{0}\rho)^{-1/2}\) is the Alfvén speed. It can be the Alfvén speed either inside or outside the tube because we assume that the density ratio \([\rho_{i}/\rho_{e}]\) is not large, implying that the two Alfvén speeds are of the same order. On the other hand, the oscillation period is of the order of \(L/V_{A} = \epsilon^{-1}R/V_{A}\). This inspires us to introduce the “slow” time \(T = \epsilon t\). Below we assume that the maximum tube axis displacement is of the order of \(aR\), where \(a \ll 1\). The quantity \(a\) can be considered as the dimensionless amplitude of the tube kink oscillation. Later we shall assume that, although \(a\) is small, \(a \gg \epsilon\).

An important property of this system of equations is that it does not contain \(v_{z}\). What is also worth nothing is that the \(z\)-component of the induction equation reduces to Equation 11 which shows that the motion is incompressible in the leading-order approximation with respect to \(\epsilon\).

The system of Equations 7 – 11 with the boundary conditions in Equations 13 and 14 is used in the next section to study the generation of fluting perturbations by a kink mode.

## 3 Generation of Fluting Perturbations

### 3.1 The First Order Approximation

### 3.2 The Second-Order Approximation

## 4 Summary and Conclusions

In this article we studied the excitation of a fluting perturbation of the boundary of a magnetic flux tube by an imposed kink oscillation. We used the MHD equations in the approximation of cold plasmas, *i.e.* we neglected the plasma pressure in comparison with the magnetic pressure. We also used the thin-tube approximation and scaled the dependent and independent variables accordingly. Using this approximation enables us to eliminate the velocity component parallel to the background magnetic field and reduce the axial component of the induction equation to the condition that the plasma motion is incompressible.

To study the excitation of fluting perturbations by a kink oscillation we used the regular perturbation method where the dimensionless amplitude of the kink oscillation is used as a small parameter. The solution of the first-order approximation describes the kink oscillation. To solve the equations of the second-order approximation we imposed the periodicity conditions with respect to time on the dependent variables. The axial component of the induction equation contains the time derivative of the axial component of the magnetic-field perturbation. However, as we have already pointed out, in the thin-tube approximation this equation reduces to the condition that the plasma motion is incompressible and, thus, the time derivative is eliminated. As a result, the order of the system of MHD equations with respect to time reduces from five to four. This implies that we can impose only four periodicity conditions with respect to time. Hence, we imposed the conditions that the radial and azimuthal components of the velocity and magnetic-field perturbation are periodic functions of time with the zero averages, while we did not impose any conditions on the magnetic pressure perturbation.

We solved the equations of the second-order approximation and obtained the expression for the tube boundary perturbation. This perturbation is a superposition of sausage and fluting perturbation. Both perturbations oscillate with the frequency equal to the double frequency of the kink mode. The amplitude of the fluting perturbation takes its maximum at the middle of the tube and monotonically decreases with the distance from the middle point. The first overtone with respect to the axial variables of the first fluting mode is not excited. These results are in a complete agreement with the numerical results obtained by Terradas, Magyar, and Van Doorsselaere (2017).

## Notes

### Acknowledgements

This article was inspired by a discussion at a workshop in the International Space Science Institute (ISSI), Bern, Switzerland, in March 2017. The author gratefully acknowledges the support from ISSI through the team “Towards Dynamic Solar Atmospheric Magneto-Seismology with New Generation Instrumentation” lead by R. Morton and G. Verth. He also acknowledges the financial support from the Science and Technology Facilities Council (STFC).

### Disclosure of Potential Conflicts of Interests

The author declares that he has no conflicts of interest.

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