Possible Cross-Section for a Coronal Loop of Given Shape?
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We aim to answer the question about the cross-section of a planar coronal loop with a prescribed shape. We restrict the analysis to coronal loops embedded in a planar potential magnetic field. Then we carry out the analysis in the leading-order approximation with respect to the small parameter ϵ equal to the ratio of the characteristic size of the loop cross-section to the loop length. We show that, in this approximation, the loop cross-section can be prescribed arbitrarily at one of its footpoints. Then the loop cross-section at any other point is obtained by stretching or compressing the prescribed loop cross-section in the direction that is perpendicular to the loop axis and in the plane of the loop. The variation of the coefficient of stretching or compression along the loop can be chosen arbitrarily. In particular, it follows from this result that we can consider a planar loop of arbitrary shape and assume that its cross-section is circular everywhere and has a constant radius.
KeywordsSun Plasma Coronal loops Magnetohydrodynamics Magnetic equilibrium
Standing kink waves in coronal magnetic loops were first observed by the Transition Region and Corona Explorer (TRACE) on 14 July 1998. These observations were reported by Aschwanden et al. (1999) and Nakariakov et al. (1999). Later, similar observations were reported by, e.g., Schrijver and Brown (2000), Aschwanden et al. (2002), Schrijver, Aschwanden, and Title (2002), Aschwanden (2006), and Aschwanden and Schrijver (2011).
Since the standing kink waves in coronal loops were first observed, they have attracted great attention from theorists. In the first theoretical works related to these waves, the simplest model of a coronal magnetic loop was used, which is a straight homogeneous magnetic tube. The theory of kink-wave propagation in straight homogeneous magnetic tubes was developed about two decades before they were first observed (e.g. Ryutov and Ryutova, 1976; Edwin and Roberts, 1983). Then more complicated models were developed with properties of coronal magnetic loops such as density variation along and across the loop, loop expansion, curvature, torsion, and magnetic twist. For a recent review of the theory of standing kink waves in coronal magnetic loops see Ruderman and Erdélyi (2009).
A very popular approximation used in the theory of kink waves in coronal loops is the thin-tube (TT) approximation. It is based on the fact that the size of the loop cross-section is much smaller than the loop length, so the ratio of these two quantities can be considered as a small parameter. The use of the TT approximation greatly simplifies the theory of kink waves in coronal loops. It was assumed in most studies that the loop cross-section is circular. It is also typical to neglect the loop curvature. In particular, Dymova and Ruderman (2005) showed that in the TT approximation, kink waves in a straight magnetic tube with a circular cross-section with constant radius and the density varying along the loop are described by a wave equation. Ruderman, Verth, and Erdélyi (2008) and Verth and Erdélyi (2008) generalized this result to include the cross-section radius variation along the loop.
At present, it is not possible to arrive at any conclusion about the shape of the loop cross-section, so it is not clear how relevant the model of a loop with a circular cross-section is. Ruderman (2003) and Morton and Ruderman (2011) studied kink oscillations of a loop with an elliptic cross-section. They found that the main effect of the cross-section ellipticity is that the frequencies of oscillations polarized along the small and large axis of the cross-section are different.
Van Doorsselaere et al. (2004) analytically and Terradas, Oliver, and Ballester (2006) numerically studied the effect of loop curvature on kink oscillations of coronal loops (see also the review by Van Doorsselaere, Verwichte, and Terradas, 2009). They showed that the effect of curvature is very small in the TT approximation and can be neglected. However, these authors assumed that the loop has a half-circle shape and the loop cross-section is a circle of constant radius. Ruderman (2009) studied kink oscillations of a planar curved loop of arbitrary shape embedded in a potential planar magnetic field. He showed that, in general, the loop cross-section is elliptic. The ellipse axis perpendicular to the loop plane is constant, while the second ellipse axis varies along the loop. As a result, the loop kink oscillation polarized in the direction perpendicular to the loop plane has a frequency different from that of the oscillation polarized in the loop plane.
The loop shape is important first of all because it determines the variation of the plasma density along the loop when the dependence of the temperature on the height in the corona is given. The density variation along the loop, in turn, affects the ratio of frequencies of the first overtone and fundamental mode of kink oscillations. This ratio is one of the most important parameters in coronal seismology because it is used to estimate the atmospheric scale height in the corona in the framework of the method developed by Andries, Arregui, and Goossens (2005) (see also the review by Andries et al., 2009).
In the seminal article by Andries, Arregui, and Goossens (2005), the loop was assumed to have a half-circle shape and a circular cross-section of constant radius. Dymova and Ruderman (2006) were the first to study the effect of the loop shape on the frequency ratio. These authors considered loops with the shape of an arc of a circle and a circular cross-section of constant radius, and they studied the dependence of the frequency ratio on the ratio of the loop height to the distance between the loop footpoints. Later, this study was continued by other authors. Morton and Erdélyi (2009) studied kink oscillations of a coronal loop with an elliptic shape and calculated the dependence of the frequency ratio on the ellipticity parameter. Orza and Ballai (2013) considered asymmetric loops and studied the dependence of the frequency ratio on the loop asymmetry. Both Morton and Erdélyi (2009) and Orza and Ballai (2013) assumed that the loop cross-section is a circle of constant radius.
It is very easy to obtain an equilibrium with a loop of half-circle shape and circular cross-section of constant radius embedded in a potential planar magnetic field. The same is true for a loop with an arc-of-circle shape. However, in view of the study by Ruderman (2009), it is not at all clear if it is possible to obtain an equilibrium with a planar loop of, for example, elliptic shape with a circular cross-section of constant radius embedded in a planar potential magnetic field. Moreover, it is not even clear whether a planar loop with an arbitrary shape can be embedded in a planar potential magnetic field. Hence, important questions arise: Can a loop with an arbitrary shape be embedded in a planar potential magnetic field? If the loop shape is given and the loop is embedded in a planar potential magnetic field, then what can be its cross-section? In particular, is it possible to have a loop with the circular cross-section of constant radius and with given shape?
This article aims to answer these questions. It is organized as follows: In the next section we formulate the problem. In Section 3 we obtain the solution describing an equilibrium with a planar magnetic tube embedded in a planar potential magnetic field and study the variation of the loop cross-section along the loop. Section 4 contains the summary and our conclusions.
2 Problem Formulation
3 Loop Cross-Section
4 Summary and Conclusions
We have answered the question about the kind of cross-section a planar coronal loop with a prescribed shape can have. This question arises in relation to works such as those of Morton and Erdélyi (2009) and Orza and Ballai (2013), where the authors considered kink oscillations of curved planar loops with prescribed shapes and assumed that the loop cross-section is circular everywhere and has a constant radius.
In our analysis we assumed that the loop is embedded in a planar potential magnetic field. We also assumed that the loop axis is described by an analytical function. We showed that the loop cross-section can be prescribed arbitrarily at one of the loop footpoints. Then the loop cross-section at any other point is obtained by stretching or compressing the prescribed loop cross-section in the direction that is perpendicular to the loop axis and in the plane of the loop. The variation of the coefficient of stretching/compression along the loop can be chosen arbitrarily. Note that this result is only valid in the leading-order approximation with respect to the small parameter ϵ, which is the ratio of the characteristic size of the loop cross-section to the loop length.
In particular, it follows from the result we obtained that we can consider a planar loop of arbitrary shape and assume that its cross-section is circular everywhere and has a constant radius.
The author acknowledges the support by an STFC grant (R/131168), and by the Russian Fund for Fundamental Research (RFBR) grant (13-02-00656).
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