Advertisement

Solar Physics

, Volume 290, Issue 2, pp 423–435 | Cite as

Possible Cross-Section for a Coronal Loop of Given Shape?

  • M. S. RudermanEmail author
Open Access
Article

Abstract

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.

Keywords

Sun Plasma Coronal loops Magnetohydrodynamics Magnetic equilibrium 

1 Introduction

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

We consider a planar potential magnetic field in the upper part of the xz-plane defined by z≥0. The two components of the magnetic field can be expressed in terms of the magnetic flux function [ψ] as
$$ B_x = \frac{\partial\psi}{\partial z}, \qquad B_z = - \frac{\partial\psi}{\partial x}, $$
(1)
where ψ satisfies the Laplace equation
$$ \frac{\partial^2\psi}{\partial x^2} + \frac{\partial^2\psi}{\partial z^2} = 0. $$
(2)
We assume that the loop axis is defined by the parametric equations
$$ x = x_0(s), \qquad z = z_0(s), $$
(3)
where z 0(0)=z 0(L)=0, L is the loop length, s is the length counted along the loop axis, and z 0(s)≥0 for 0≤sL. We denote this curve as \(\mathcal{C}\). Since s is the distance along the loop, we have the relation
$$ {x'_0}^2 + {z'_0}^2 = 1, $$
(4)
where the prime indicates the derivative with respect to s. We introduce the size of the loop cross-section as the greatest distance between two points at the cross-section boundary. In what follows we assume that the ratio of the cross-section size to L is low and does not exceed ϵ≪1.
For simplicity, we assume below that z 0(s) monotonically increases in the interval (0,s a ) and monotonically decreases in the interval (s a ,L), where s a corresponds to the loop apex. As for x 0(s), we assume that it either monotonically increases in the whole interval (0,L), or monotonically decreases in the interval (0,s 1), and then monotonically increases in the interval (s 1,L), or monotonically decreases in the interval (0,s 1), then monotonically increases in the interval (s 1,s 2), and then again monotonically decreases in the interval (s 2,L), where 0<s 1<s a <s 2<L. We also assume that the loop axis is a concave curve. The three possible shapes of the loop are shown in Figures 1a, b, and c. Other assumptions are that the distance between the loop footpoints is of the order of L, and the smallest radius of the loop curvature is also of the order of L.
Figure 1

Three different loop shapes.

3 Loop Cross-Section

We start by introducing a local curvilinear coordinate system in the vicinity of the loop axis. To do this, we draw straight lines normal to the loop axis at each point (see Figure 2). We denote the length along a straight line measured from the point of its intersection with the loop axis as u. The quantity u is negative when it is measured inside the region bounded by the loop axis and the x-axis, and positive when it is measured outside. This is a local coordinate system because the normal lines, in general, intersect at some distance from the loop axis. However, we only use this coordinate system in the vicinity of the loop axis defined by the condition |u|≤d. We show in Appendix A that it is enough to impose the restriction that d is smaller than half of the minimum curvature radius of the loop axis and also smaller than half of the distance between the loop footpoints. Hence, if we assume that these two quantities are much larger than ϵL, then we can take d much larger than ϵL. The equation of the normal line crossing the loop axis at the point (x 0,z 0) is
$$ x = x_0(s) - uz'_0(s), \qquad z = z_0(s) + ux'_0(s) . $$
(5)
These equations give the relation between the curvilinear coordinates (s,u) and Cartesian coordinates (x,z). Differentiating the identities xx(s(x,z),u(x,z)) and zz(s(x,z),u(x,z)) with respect to x and z yields
$$ \frac{\partial s}{\partial x} = \frac{1}{\Delta}\frac{\partial z}{\partial u},\qquad \frac{\partial s}{\partial z} =-\frac{1}{\Delta}\frac{\partial x}{\partial u},\qquad \frac{\partial u}{\partial x} =-\frac{1}{\Delta}\frac{\partial z}{\partial s},\qquad \frac{\partial u}{\partial z} =\frac{1}{\Delta}\frac{\partial x}{\partial s}, $$
(6)
where
$$ \Delta= \frac{\partial x}{\partial s}\frac{\partial z}{\partial u} - \frac{\partial x}{\partial u} \frac{\partial z}{\partial s} . $$
(7)
Using Equations (4) and (5), we obtain
$$ \Delta= 1 - \kappa u, \quad\kappa= x'_0 z''_0 - x''_0 z'_0 , $$
(8)
where κ is the loop-axis curvature. The curvature [κ] is positive if the tangent vector \((x'_{0},z'_{0})\) to the loop axis rotates counter-clockwise when s increases and negative otherwise. It follows from the assumption that the loop axis is a concave curve that its curvature is everywhere negative.
Figure 2

The local coordinate system. The thick line is the loop axis. The length s along the loop is measured from the left footpoint. The coordinate u is measured from the loop axis along a normal line to the loop axis.

With the aid of Equations (4), (5), and (8), we reduce Equation (6) to
$$ \begin{array}{lcl@{\qquad }lcl} \frac{\partial s}{\partial x} &=& \frac{x'_0}{1 - \kappa u} , & \displaystyle\frac{\partial s}{\partial z} &=& \frac{z'_0}{1 - \kappa u} , \\ \frac{\partial u}{\partial x} &=& -\frac {z'_0+ux''_0}{1-\kappa u}, & \frac{\partial u}{\partial z} &=& \frac {x'_0-uz''_0}{1-\kappa u} . \end{array} $$
(9)
Using Equation (8), we obtain
$$ \kappa^2 = \bigl(x'_0 z''_0 - x''_0 z'_0 \bigr)^2 = {x'_0}^2{z''_0}^2 - 2x'_0 z'_0 x''_0 z''_0 + {x''_0}^2{z'_0}^2 . $$
(10)
Now we use Equation (4) to transform this equation to
$$ \kappa^2 = {x''_0}^2 + {z''_0}^2 - {x'_0}^2{x''_0}^2 - 2x'_0 z'_0 x''_0 z''_0 - {z'_0}^2{z''_0}^2 = {x''_0}^2 + {z''_0}^2 - \bigl(x'_0 x''_0 + z'_0 z''_0 \bigr)^2. $$
(11)
In accordance with Equation (4), the second term on the right-hand side of this equation is zero, so eventually we arrive at
$$ \kappa^2 = {x''_0}^2 + {z''_0}^2 . $$
(12)
Using Equations (4), (8), (9), and (12), we rewrite Equation (2) in terms of s and u as
$$ \frac{\partial}{\partial s} \biggl(\frac{1}{1-\kappa u} \frac{\partial\psi}{\partial s} \biggr) + \frac{\partial}{\partial u} \biggl((1-\kappa u) \frac{\partial\psi}{\partial u} \biggr) = 0. $$
(13)
We look for the solution to this equation in the form of power series:
$$ \psi= \sum_{n=0}^\infty u^n \psi_n(s) . $$
(14)
In what follows we assume that the loop axis is a magnetic-field line. This implies that ψ=const. when u=0. Without loss of generality, we can take ψ=0 when u=0. Then ψ 0(s)=0. Now, substituting Equation (14) in Equation (13) and collecting terms with the same power of u, we obtain the infinite set of equations for the coefficients ψ n (s). It is convenient to write the first two equations of this set separately. As a result, we have
$$\begin{aligned} 2\psi_2 =& \kappa\psi_1, \end{aligned}$$
(15)
$$\begin{aligned} 6\psi_3 =& 8\kappa\psi_2 - \psi''_1 - 2\kappa^2\psi_1, \end{aligned}$$
(16)
$$\begin{aligned} (n+1) (n + 2)\psi_{n+2} =& (n+1) (3n+1)\kappa\psi_{n+1} - \psi''_n - n(3n-1)\kappa^2 \psi_n \\ &{}+ \kappa\psi''_{n-1} - \kappa'\psi'_{n-1} - (n-1)^2 \kappa^3\psi_{n-1} , \quad n = 2,3,\dots \end{aligned}$$
(17)
We can see that we can take ψ 1(s) arbitrarily and then calculate ψ n (s), n=2,3, etc., using Equations (15) – (17). It is proved in Appendix B that the series (14) is convergent if x 0(s), y 0(s), and ψ 1(s) are sufficiently “good” functions, i.e. they satisfy all restrictions imposed in Appendix B, and d is sufficiently small, but still much larger than ϵL.
In what follows we use only the first non-zero term in the series in Equation (14) and assume that |u|≲ϵL, so
$$ \psi= u\psi_1(s)\bigl[1 + \mathcal{O}(\epsilon)\bigr] . $$
(18)
Consider the plane [Π] orthogonal to the loop axis at its left footpoint. Since the vector \(\boldsymbol{e}_{u} = (-z'_{0}(0),x'_{0}(0))\) is orthogonal to the loop axis at this footpoint [e u ∈Π]. We denote the unit vector in the y-direction, which is the direction orthogonal to the plane of the loop, as e y . Since e u e y , these two vectors can be considered as the unit vector of Cartesian coordinates in the plane [Π]. The corresponding coordinates are u and y, and the coordinate origin coincides with the intersection of Π and the loop axis.
Now we consider a closed contour Γ∈Π. It is defined by the equations
$$ u = u_0(t), \qquad y = y_0(t), $$
(19)
where u 0(T)=u 0(0) and y 0(T)=y 0(0). We assume that Γ is a simple contour, i.e. it does not have self-intersections, and the coordinate origin is inside Γ. In addition, we assume that Γ is convex, i.e. any straight line can intersect it at no more than two points. The numbers u, y, and s define orthogonal curvilinear coordinates in three-dimensional space. In these coordinates the equation of the loop axis is u=y=0. The magnetic line crossing Γ at a point with coordinates (u 0(t),y 0(t)) is defined by the equations y=y 0(t) and ψ(u,s)=ψ(u 0(t),0). Varying s from 0 to L, we obtain a part of this line bounded by the two planes orthogonal to the loop axis at the footpoints. Now we also vary t from 0 to T. Then we obtain a magnetic surface consisting of magnetic lines crossing the contour Γ. We can consider this surface as the surface of the magnetic tube. Then, using Equation (18), we write the equation of the surface of the magnetic tube as
$$ y = y_0(t), \qquad u\psi_1(s) = u_0(t) \psi_1(0)\bigl[1+\mathcal{O}(\epsilon)\bigr] , \quad 0 \leq s \leq L, \ 0 \leq t \leq T . $$
(20)
The boundary of the tube cross-section by the plane perpendicular to the tube axis at \(s = \bar{s}\) is defined by the equation
$$ y = y_0(t), \qquad u = u_0(t)\frac{\psi_1(0)}{\psi_1(\bar{s})}\bigl[1 + \mathcal{O}(\epsilon)\bigr] , \quad 0 \leq t \leq T . $$
(21)
We see that, in the leading-order approximation with respect to ϵ, the tube cross-section at \(s = \bar{s}\) is obtained from the tube cross-section at s=0 by stretching or compressing this latter cross-section in the u-direction with the coefficient \(\psi _{1}(0)/\psi_{1}(\bar{s})\). In particular, if we take Γ to be a circle of radius a centred at the loop axis, and we also take ψ 1(s)≡1, then we obtain that in the leading-order approximation with respect to ϵ, the tube cross-section is the circle of radius a centred at the loop axis everywhere. Hence, we can consider a planar loop of arbitrary shape and assume that it has a circular cross-section with constant radius everywhere.

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.

Notes

Acknowledgements

The author acknowledges the support by an STFC grant (R/131168), and by the Russian Fund for Fundamental Research (RFBR) grant (13-02-00656).

References

  1. Andries, J., Arregui, I., Goossens, M.: 2005, Astrophys. J. Lett. 624, L57.  DOI. ADSCrossRefGoogle Scholar
  2. Andries, J., Van Doorsselaere, T., Roberts, B., Verth, G., Erdélyi, R.: 2009, Space Sci. Rev. 149, 3.  DOI. ADSCrossRefGoogle Scholar
  3. Aschwanden, M.J.: 2006, Phil. Trans. Roy. Soc. A 364, 417.  DOI. ADSCrossRefGoogle Scholar
  4. Aschwanden, M.J., Schrijver, C.J.: 2011, Astrophys. J. 736, 102.  DOI. ADSCrossRefGoogle Scholar
  5. Aschwanden, M.J., Fletcher, L., Schrijver, C.J., Alexander, D.: 1999, Astrophys. J. 520, 880.  DOI. ADSCrossRefGoogle Scholar
  6. Aschwanden, M.J., De Pontieu, B., Schrijver, C.J., Title, A.M.: 2002, Solar Phys. 206, 99.  DOI. ADSCrossRefGoogle Scholar
  7. Dymova, M., Ruderman, M.S.: 2005, Solar Phys. 229, 79. ADS.  DOI. ADSCrossRefGoogle Scholar
  8. Dymova, M., Ruderman, M.S.: 2006, Astron. Astrophys. 459, 241.  DOI. ADSCrossRefGoogle Scholar
  9. Edwin, P.M., Roberts, B.: 1983, Solar Phys. 88, 179.  DOI. ADSCrossRefGoogle Scholar
  10. Morton, R.J., Erdélyi, R.: 2009, Astron. Astrophys. 502, 315.  DOI. ADSCrossRefzbMATHGoogle Scholar
  11. Morton, R.J., Ruderman, M.S.: 2011, Astron. Astrophys. 527, A53.  DOI. ADSCrossRefGoogle Scholar
  12. Nakariakov, V., Ofman, L., DeLuca, E.E., Roberts, B., Davila, J.M.: 1999, Science 285, 862.  DOI. ADSCrossRefGoogle Scholar
  13. Orza, B., Ballai, I.: 2013, Astron. Nachr. 334, 948.  DOI. ADSCrossRefGoogle Scholar
  14. Ruderman, M.S.: 2003, Astron. Astrophys. 409, 287.  DOI. ADSCrossRefGoogle Scholar
  15. Ruderman, M.S.: 2009, Astron. Astrophys. 506, 885.  DOI. ADSCrossRefzbMATHGoogle Scholar
  16. Ruderman, M.S., Erdélyi, R.: 2009, Space Sci. Rev. 149, 199.  DOI. ADSCrossRefGoogle Scholar
  17. Ruderman, M.S., Verth, G., Erdélyi, R.: 2008, Astrophys. J. 686, 694.  DOI. ADSCrossRefGoogle Scholar
  18. Ryutov, D.D., Ryutova, M.P.: 1976, Sov. Phys. JETP 43, 491. ADS. ADSGoogle Scholar
  19. Schrijver, C.J., Brown, D.S.: 2000, Astrophys. J. Lett. 537, L69.  DOI. ADSCrossRefGoogle Scholar
  20. Schrijver, C.J., Aschwanden, M.J., Title, A.M.: 2002, Solar Phys. 206, 69.  DOI. ADSCrossRefGoogle Scholar
  21. Terradas, J., Oliver, R., Ballester, J.L.: 2006, Astrophys. J. Lett. 650, L91.  DOI. ADSCrossRefGoogle Scholar
  22. Van Doorsselaere, T., Debosscher, A., Andries, J., Poedts, S.: 2004, Astron. Astrophys. 424, 1065.  DOI. ADSCrossRefGoogle Scholar
  23. Van Doorsselaere, T., Verwichte, E., Terradas, J.: 2009, Space Sci. Rev. 149, 299.  DOI. ADSCrossRefGoogle Scholar
  24. Verth, G., Erdélyi, R.: 2008, Astron. Astrophys. 486, 1015.  DOI. ADSCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2014

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  1. 1.Solar Physics and Space Plasma Research Centre (SP2RC), School of Mathematics and StatisticsUniversity of SheffieldSheffieldUK
  2. 2.Space Research Institute (IKI) Russian Academy of SciencesMoscowRussia

Personalised recommendations