# Corrugation Instability of a Coronal Arcade

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## Abstract

We analyse the behaviour of linear magnetohydrodynamic perturbations of a coronal arcade modelled by a half-cylinder with an azimuthal magnetic field and non-uniform radial profiles of the plasma pressure, temperature, and the field. Attention is paid to the perturbations with short longitudinal (in the direction along the arcade) wavelengths. The radial structure of the perturbations, either oscillatory or evanescent, is prescribed by the radial profiles of the equilibrium quantities. Conditions for the corrugation instability of the arcade are determined. It is established that the instability growth rate increases with decreases in the longitudinal wavelength and the radial wave number. In the unstable mode, the radial perturbations of the magnetic field are stronger than the longitudinal perturbations, creating an almost circularly corrugated rippling of the arcade in the longitudinal direction. For coronal conditions, the growth time of the instability is shorter than one minute, decreasing with an increase in the temperature. Implications of the developed theory for the dynamics of coronal active regions are discussed.

## Keywords

Active regions, models Waves, magnetohydrodynamic Instabilities## 1 Introduction

Arcades of plasma loops are a typical feature of solar coronal active regions. Examples of coronal arcades are sets of postflare loops in two-ribbon flares and closed magnetic configurations overlying eruptive magnetic ropes. Arcades are key ingredients of the standard model of a flare and an inverse-polarity prominence equilibrium. Dynamical processes in arcade structures are important for understanding several key basic processes in the corona, such as energy balance and release, and prominence formation and evolution. Often, these processes are observed to evolve quasi-periodically. For example, light curves of solar and stellar flares often have quasi-periodic pulsations that could be associated with oscillatory processes (*e.g.* Nakariakov and Melnikov, 2009) in the flaring arcades. A transverse oscillatory motion with a period of about 1000 s in an arcade situated above an emerging magnetic rope was detected by Kim, Nakariakov, and Cho (2014). Consecutive ignition of quasi-periodic hard X-ray emission with a period of about a minute, propagating along the magnetic polarity inversion line in a flaring arcade, was observed in a solar flare by Grigis and Benz (2005) and Bogachev *et al.* (2005). A similar effect was detected in the microwave emission of a solar flare (*e.g.* Kim *et al.*, 2013). A recent comprehensive statistical analysis demonstrated that phenomenon is associated with successive triggering of the energy release process in different loops of the flaring arcade (Kuznetsov *et al.*, 2016). Oscillatory processes in the magnetic arcades supporting quiescent prominences might also be responsible for prominence oscillations (Arregui, Oliver, and Ballester, 2012).

A likely mechanism for the oscillatory processes in arcades is connected with magnetohydrodynamic (MHD) oscillations that have been intensively modelled for more than two decades (see, *e.g.* Goossens, Erdélyi, and Ruderman, 2011, for a comprehensive review). Several studies have addressed perturbations in the plane perpendicular to the arcade axis. In particular, the propagation of fast magnetoacoustic waves in coronal potential arcades, excited by photospheric footpoint motions, was studied in Cadez, Oliver, and Ballester (1996). The excitation of resonant shear Alfvén oscillations in coronal arcades driven by toroidal footpoint motions was described by Ruderman *et al.* (1997). Oliver, Hood, and Priest (1996) considered a general two-dimensional arcade equilibrium structure with no longitudinal magnetic field component, and derived governing equations for the coupled slow and fast modes, while the Alfvén mode was decoupled from the magnetoacoustic modes. The propagation of impulsively generated Alfvén waves in an arcade was numerically modelled by Chmielewski *et al.* (2014). Magnetohydrodynamic waves in sheared coronal arcades, including propagation along the axis of the arcade, were studied by *e.g.* Arregui, Oliver, and Ballester (2004b,a) in the zero-\(\beta\) limit, and the properties of coupled fast and Alfvén modes were established. Slow waves propagating along the axis of an arcade, *i.e.* along the neutral line, have been proposed as a candidate process to explain the progression of flaring energy releases along the neutral line that are typical for two-ribbon flares (Nakariakov and Zimovets, 2011). Fast waves guided by a plasma arcade have been considered as a possible mechanism for kink oscillations in coronal active regions (Hindman and Jain, 2015).

Much less attention has been received by MHD instabilities in plasma arcades, which may also be responsible for the progression of flaring energy releases along the neutral line in two-ribbon flares. Another interesting topic that requires understanding regarding MHD instabilities in arcades is the abrupt destabilisation that could trigger a coronal mass ejection. Likewise, the source of the filamentation of an arcade, *i.e.* the appearance of corrugated fine structure that is often detected in post-flare plasma arcades, is of interest. These processes require consideration of MHD perturbations that have a characteristic scale along the arcade axis (*i.e.*, across the field) much shorter than the lengths of the loops that form the arcade. These perturbations, with a spatial scale much shorter than the parallel field-aligned scale when the field is azimuthal, could be referred to as corrugated perturbations, and the associated instability could be called a corrugation instability. On the other hand, as these perturbations have a small scale across the field, they could also be referred to as ‘‘ballooning’’, but this term must be used with caution in the coronal context, as this term is usually applied in solar physics to perturbations with a high azimuthal wave number \(m\).

In solar physics, only a few studies have been dedicated to ballooning modes so far. These works were mainly dedicated to the study of slightly twisted magnetic ropes, where the ballooning perturbations were associated with high values of \(m\). Ruderman (2007) carried out a theoretical analysis of long-wavelength non-axisymmetric modes, including ballooning modes, of a compressible zero-beta weakly twisted magnetic rope surrounded by a straight and homogeneous magnetic field. The main attention was paid to the kink mode. Karami and Bahari (2012) generalised this study and addressed the oscillation period ratios of different harmonics. Costa and González (2006, 2008) developed an energy-principle-based formalism for the estimation of the non-axisymmetric mode parameters, *i.e.* \(m>0\). Cheremnykh *et al.* (2017) showed that in a thin magnetic tube with a curved magnetic field, the stable \(m=1\) mode is dominant when the azimuthal magnetic field outside the tube is absent. However, in the presence of an external magnetic field, this mode becomes unstable (Cheremnykh, Kryshtal, and Tkachenko, 2017). Stepanov, Kopylova, and Tsap (2008) and Tsap *et al.* (2008) demonstrated that the ballooning instability of arch-shaped twisted magnetic ropes might cause the formation of helmet-shaped structures in the lower solar corona, and might also produce a quasi-periodic modulation of gyrosynchrotron emission in flares.

On the other hand, the theory of the ballooning oscillations and instability of highly twisted magnetic ropes, relevant to the study of the corrugation instability of an arcade, has been intensively elaborated in the context of the terrestrial magnetosphere. Gold (1959) suggested very early on that the hot plasma MHD instability (later coined the ballooning instability) can explain various eruptive phenomena in the terrestrial magnetotail. Walker (1987) showed that the MHD modes in the ballooning limit are represented by the coupled Alfvén and slow magnetosonic branches, while the fast magnetosonic mode is evanescent. Burdo, Cheremnykh, and Verkhoglyadova (2000) showed that the coupling of the Alfvén and slow magnetoacoustic modes gives rise to a ballooning instability in the dipole field. These authors also formulated for the first time the instability criterion outside the local approximation. Cheremnykh and Parnowski (2004), Agapitov, Cheremnykh, and Parnowski (2008) and Leonovich and Kozlov (2013b) considered the instability in a plasma with an inhomogeneity along the field lines. Ma, Hirose, and Liu (2014) studied the instability in the Hall-MHD framework. Klimushkin, Mager, and Pilipenko (2012) and Mager and Klimushkin (2017) generalised the theory of ballooning perturbations accounting for kinetic effects. The exploitation of the coronal–magnetospheric analogy opens up interesting opportunities for knowledge transfer (Nakariakov *et al.*, 2016).

The aim of this paper is to further develop the linear theory of corrugated perturbations in a plasma arcade with an arbitrary radial profile of the equilibrium quantities. The shear of the magnetic field is neglected. Mathematically, this work is based on the set of equations for the coupled Alfvén and slow magnetoacoustic modes derived in Klimushkin (1997, 1998) and Cheremnykh and Danilova (2011), and the formalism developed for the study of the spatial structure of the unstable modes in the terrestrial magnetosphere (Klimushkin, 2006; Klimushkin and Chen, 2006; Mazur, Fedorov, and Pilipenko, 2013; Cheremnykh, Klimushkin, and Kostarev, 2014; Cheremnykh, Klimushkin, and Mager, 2016). In Section 2 we review the derivation of the governing equations that describe the radial structure of the perturbations. Section 4 addresses the derivation and study of the dispersion relation for the coupled Alfvén and slow modes with large longitudinal wave numbers perpendicular to the field, which determines the wave vector radial component in the WKB approximation. In Section 5 we study the possibility of mode trapping in the radial direction across the magnetic shells, derive the eigenvalues of the perturbations, consider the conditions under which the modes are unstable, and estimate the growth rates. In Section 6 we summarise the results and discuss their possible observational manifestations.

## 2 Governing Equations

*z*-pinch, with the magnetic surfaces represented by nested coaxial cylinders (Figure 1). In the solar coronal context, a similar model with a purely poloidal magnetic field has recently been used by Kaneko

*et al.*(2015). The azimuthal angle \(\varphi\) shows the direction along the field lines. At \(\varphi=0, \pi/2\), which corresponds to the photosphere, the line-tying boundary conditions are applied. The longitudinal \(y\) coordinate is directed along the cylinder axis,

*i.e.*the magnetic neutral line. All parameters of the equilibrium are taken to depend only on the radial coordinate \(r\), that is, the field line curvature radius. The coordinates \(\varphi\), \(y\), and \(r\) are locally orthogonal to each other. In the following, we focus on ideal MHD processes.

*e.g.*Klimushkin, 1997), that is, it can be represented as a two-dimensional gradient of some scalar (“a potential”):

*i.e.*in the \(y\) and \(r\) directions, are small in comparison with the spatial scales of the equilibrium quantities. A necessary condition for this is a high \(k_{y}\) value: \(k_{y} \gg\kappa\).

Considering that the perturbations with spatial scale in the radial direction are much smaller than the local radius of the field curvature, we can use the WKB approximation with respect to the radial coordinate \(r\). That is, the \(r\) dependence of the perturbations can be represented in the form \(\exp{i \int k_{r}(r, \omega) \mathrm{d}r}\), where \(k_{r}\) is the radial component of the wave vector. The perpendicular component of the wave vector \(k_{\perp}\) is \(k_{\perp}^{2}=(k_{r}^{2}+k_{y}^{2})^{1/2}\).

*e.g.*Walker, 1987). The equations must be supplemented with boundary conditions. On the azimuthal coordinate, the line-tying boundary conditions must be applied (Hood, 1986), which means that the plasma displacement \({\boldsymbol {\xi}}\) must vanish on the photosphere, where \(\varphi=0\) and \(\pi\):

*e.g.*Klimushkin, 1998). Then, the equality \({\boldsymbol {\xi}}_{\parallel}=0\) imposes the boundary condition for the \(\Theta\) function:

Equations 6 and 7 supplemented with boundary conditions 9 and 10 are the governing equations used for the analysis presented in the paper.

## 3 Alfvén and Slow-Mode Eigenfunctions

Equations 6 and 7 describe the coupling of Alfvén and slow modes. We consider two auxiliary problems connected with uncoupled modes.

Note that the cases \(\omega=\Omega_{\mathrm{A}N}\) and \(\omega =\Omega_{\mathrm{C}N}\) correspond to well-known Alfvén and slow-mode resonances, respectively (*e.g.* Cheng *et al.*, 1993; Klimushkin, 1997, 1998; Leonovich and Kozlov, 2013a).

## 4 Dispersion Relations for the Coupled MHD Modes

The symmetric and antisymmetric (with respect to the arcade apex (\(\varphi=\pi/2\))) functions \(\Phi\) must be considered separately. In this paper, the convolution product \(\langle\Theta_{N} \Phi_{N'} \rangle\) is considered as non-negligible only for the lowest relevant \(N\) and \(N'\) numbers.

### 4.1 Alfvén Mode Eigenfunctions Symmetric with Respect to \(\varphi=\pi/2\)

*e.g.*Cheremnykh, Klimushkin, and Kostarev, 2014). The cut-off surfaces are determined by the equation

*i.e.*for the small \(\beta\) typical for coronal arcades.

### 4.2 Alfvén Mode Eigenfunctions Antisymmetric with Respect to \(\varphi=\pi/2\)

### 4.3 General Properties of the Solutions for Symmetric and Antisymmetric Alfvén Modes

In both symmetric and antisymmetric cases, the discriminant \(D\) is a positive value, thus the squares of the cut-off frequencies have real values. This agrees with the well-known MHD theorem that is a consequence of the energetic principle, stating that the wave frequency squared of linear perturbations in ideal MHD is always real (Frieman and Rotenberg, 1960; Kadomtsev, 1966). Hence, no over-stabilities can occur in the considered case.

With the decrease in \(\beta\), the lower cut-off frequency \(\Omega_{-}\) in both cases tends to the slow-mode resonance frequency (\(\Omega _{\mathrm{C2}}\) or \(\Omega_{\mathrm{C1}}\)), while the higher cut-off frequency \(\Omega_{+}\) tends to the Alfvénic resonant frequency (\(\Omega_{\mathrm{A1}}\) or \(\Omega_{\mathrm{A2}}\)). Thus, we can conclude that the frequencies \(\Omega_{-}\) and \(\Omega_{\mathrm{C}}\) characterise the slow-mode oscillation branch, and \(\Omega_{+}\) and \(\Omega_{\mathrm{A}}\) characterise the Alfvénic oscillation branch. The instability can develop only on the slow-mode branch. This is in accordance with the results of earlier papers (Mazur, Fedorov, and Pilipenko, 2012; Kozlov *et al.*, 2014; Cheremnykh, Klimushkin, and Kostarev, 2014). However, the instability threshold depends on the symmetry properties of the Alfvén mode. Clearly, for low and moderate \(\beta\) values, the instability threshold is lower for the symmetric Alfvén mode than for the antisymmetric mode.

*cf.*the body modes, see Edwin and Roberts (1983), or the leaky modes (

*e.g.*Verwichte, Foullon, and Nakariakov, 2006). Outside these regions, the modes have exponential radial dependences. The cut-off frequencies \(\Omega_{\pm}\) prescribe the cylindrical surfaces where the behaviour of the perturbation changes from oscillatory to evanescent in the radial direction. Equation 45 shows that there are mode localisation regions of two kinds. One of them is bounded by the Alfvénic resonance surface where \(\omega =\Omega_{\mathrm{A}}(r)\) and the higher cut-off shell where \(\omega =\Omega_{+}(r)\). In the following, this region is referred to as the Alfvénic mode localisation region. The other region is bounded by the surface of the cusp resonance where \(\omega=\Omega_{\mathrm{C}}(r)\) and the lower cut-off shell where \(\omega=\Omega_{-}(r)\). This is the slow-mode localisation region. Because of the inequality \(\Omega _{-}<\Omega_{\mathrm{A}}<\Omega_{+}\), these regions do not overlap with each other.

The spatial structure of the perturbation in the mode localisation regions in the considered geometry was studied by Klimushkin (1997) and Cheremnykh, Klimushkin, and Mager (2016). In both Alfvénic and slow-mode localisation regions, the wave was found to have an oscillatory structure across the magnetic shells in the radial direction in the annulus between the cut-off shell and the resonant shell. In such an annulus, the local radial component of the wave vector changes from zero to infinity. On the resonant magnetic surface, the mode is absorbed through mode coupling and subsequent dissipation. Outside these regions, the mode is evanescent. However, this structure corresponds to a stable plasma configuration when all the cut-off and resonant frequencies have real values. However, the lower cut-off frequency \(\Omega_{-}\) can take imaginary values. This case corresponds to the corrugation instability.

## 5 Radial Structure of the Unstable Modes

The instability takes place when \(\omega^{2}<0\) for a given \(k_{r}^{2}\) value. At the resonances, \(k_{r}^{2} \to\infty\), and the modes of the considered plasma configuration are always stable since the resonant frequencies squared are always positive. At the cut-off frequencies, the situation is different. Equations 28 and 39 show that the Alfvénic (the higher) cut-off frequency squared \(\Omega_{+}^{2}\) is always positive, while the slow-mode (the lower) cut-off frequency squared \(\Omega_{-}^{2}\) can be negative, provided that either condition 32 or 42 is fulfilled. This is the instability condition. Only the slow-mode oscillation branch can be unstable, in agreement with Mazur, Fedorov, and Pilipenko (2012).

*i.e.*the wave-trapping region) formed by the radial non-uniformity of the equilibrium parameters, the radial component of the wave vector must satisfy the Bohr–Sommerfeld quantisation condition

First, Equation 50 shows that in the unstable case \(\gamma _{n}^{2}>0\), the inequality \(\Delta< l\) holds. As follows from Equation 47, this regime occurs when the cut-off shells are situated in the region where the slow-mode cut-off frequency squared is negative, \(\Omega_{-}^{2}<0\). Vice versa, the stable regime, with \(\gamma _{n}^{2}<0\), occurs when the cut-off shells are situated outside the unstable region with \(\Omega_{-}^{2}<0\). That is, in the radial direction, only the central part of the resonator is situated inside the unstable region: \(\Delta> l\).

*i.e.*vertical direction. The unstable region has a small size in the direction of the arcade axis, while it includes the whole magnetic flux tube between the footpoints, see Figure 5. A perturbation of this kind could be called a ‘‘tongue’’.

## 6 Discussion and Conclusions

Waves guided by various plasma non-uniformities are often considered as a superposition of eigenmodes of certain spatially localised resonators where the perturbations have an oscillatory structure between some reflecting or refractive walls and are evanescent outside (*e.g.* Nakariakov *et al.*, 2016; Cheremnykh, Klimushkin, and Mager, 2016). In this paper, this approach was applied to the study of unstable small-scale corrugation (or ballooning) perturbations of a plasma arcade filled in with a radially non-uniform plasma of finite temperature. This allowed us to understand the influence of the radial wave structure on the instability of an arcade.

We considered an arcade as a half-cylinder with line-tying boundary conditions at the bottom of the corona. The effects of the magnetic shear were neglected, while the dependence of the perturbation along the axis of the arcade was taken into account. We focused on perturbations with wavelengths along the arcade much smaller than the radius of the arcade. We found that this plasma configuration may be unstable to the corrugated perturbations of the slow magnetoacoustic nature, which is mainly characterised by the radial displacement of the plasma. The arcade is most unstable to the perturbations with low radial harmonic numbers, \(n \approx1\), and very short longitudinal (along the cylinder) wavelengths, in comparison with the radius of the rope. In the azimuthal direction, along the magnetic field lines, the wavelength is of the order of the field line length. Thus, the unstable perturbations cause displacements of the plasma in the radial direction, and have an almost circularly corrugated structure in the direction along the neutral line. The discussed instability is most effective in hot arcades with non-negligible values of the plasma parameter \(\beta\).

The characteristic time of the instability was determined approximately by the ratio of the arcade radius and the sound speed (see Equation 48), *i.e.* the acoustic travel time across the arcade. For example, in an arcade of radius 50 Mm and filled with the plasma of the temperature of 1 MK, the characteristic time of the instability is estimated as about one minute. For hotter arcades, this time would be shorter, *e.g.* about twenty seconds, if the temperature were 10 MK. We should stress that the term ‘‘corrugated’’ highlights the ring-like localised (in the longitudinal direction) perturbations of the arcade. In an arcade, these perturbations are of an interchange nature, and hence could be considered to be similar to the fluting, or ballooning, perturbations of a plasma cylinder with the magnetic field along its axis (*e.g.* considered in Edwin and Roberts, 1983).

With the increase in the radial harmonic number, the ratio of the radial and longitudinal (along the neutral line) components of the magnetic field decreases, as does the instability growth rate. When this ratio approaches unity, the instability ceases. The full set of the instability conditions is determined by inequalities 55 and 57, which prescribe the field-aligned (azimuthal), longitudinal, and radial structure of the unstable corrugated mode.

As in the unstable modes the magnetic field and plasma are displaced mainly in the radial direction, the instability may lead to the formation of tongues that are highly localised in the direction along the arcade, displacing practically a whole line-tied magnetic flux tube in the radial direction. The magnetic field frozen into a plasma tongue pushed by the instability in the radial direction could reconnect with the overlying magnetic field, leading to the heating of the unstable flux tube. The instability can also act as a trigger to a flare or eruption if the host active region is in a subcritical equilibrium state. If the radial profiles of the equilibrium plasma parameters vary slowly along the arcade, the conditions for the corrugation instability could be fulfilled in only certain segments of the arcade, leading to the formation of a non-regular corrugated pattern. This resembles the pronounced irregular variation of the brightness of individual loops, *i.e.* the circularly corrugated rippling of the arcade in the longitudinal direction that is typical of post-flare arcades.

A sequence of the tongue emergences occurring at different locations along the arcade could induce a sequence of secondary flares in the host active region. In particular, this scenario could lead to the chaotic displacements of hard X-ray sources observed by Kuznetsov *et al.* (2016). We would like to stress that as the theory developed in this work considers perturbations with the small scales along the arcade, it is applicable to the short (in the longitudinal direction) arcades as well. Thus, the corrugation instability of plasma arcades may be responsible for several dynamical phenomena that are observed in the solar corona, which justifies the need for a better understanding of this effect.

## Notes

### Acknowledgements

The study of DYK was carried out as part of the Program of the fundamental research of the state academies of Russia for 2013-2020, project II.16.1.3. VMN acknowledges the support of the Russian Science Foundation under grant 16-12-10448. The study of PNM was supported by the Grant of the Russian Foundation for Basic Research No. 16-05-00254. The work of OKC was carried out as part of the Integrated Scientific Program of the National Academy of Sciences of Ukraine on Space Research and the Program of the NASU on plasma physics.

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