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May 6, 2018 - As a result, the flux rope may levitate stably in the corona after catastrophe, ... behavior of coronal magnetic flux ropes in order to explain the.
The Astrophysical Journal, 626:1096–1101, 2005 June 20 # 2005. The American Astronomical Society. All rights reserved. Printed in U.S.A.

DOUBLE CATASTROPHE OF CORONAL FLUX ROPE IN QUADRUPOLAR MAGNETIC FIELD Y. Z. Zhang,1 Y. Q. Hu,2 and J. X. Wang1 Received 2005 February 7; accepted 2005 March 5

ABSTRACT Using a relaxation method based on time-dependent ideal magnetohydrodynamic simulations, we find 2.5dimensional force-free field solutions in spherical geometry, which are associated with an isolated flux rope embedded in a quadrupolar background magnetic field. The background field is of Antiochos type, consisting of a dipolar and an octopolar component with a neutral point somewhere in the equatorial plane. The flux rope is characterized by its magnetic fluxes, including the annular flux p and the axial magnetic flux ’ , and its geometric features described by the height of the rope axis and the length of the vertical current sheet below the rope. It is found that for a given p , the force-free field exhibits a complex catastrophic behavior with respect to increasing ’ . There exist two catastrophic points, and the catastrophic amplitude, measured by the jump in the height of the rope axis, is finite for both catastrophes. As a result, the flux rope may levitate stably in the corona after catastrophe, with a transverse current sheet above and a vertical current sheet below. The magnetic energy threshold for the two successive catastrophes are found to be larger than the corresponding partly open field energy. We argue that it is the transverse current sheet formed above the flux rope that provides a downward Lorentz force on the flux rope and thus keeps the rope levitating stably in the corona. Subject headingg s: Sun: corona — Sun: coronal mass ejections (CMEs) — Sun: flares — Sun: magnetic fields

1. INTRODUCTION

In this paper we take a more complex background field in spherical geometry, a quadrupolar field of the Antiochos type (Antiochos et al. 1999), and discuss the equilibrium properties of the flux rope system with emphasis on the catastrophic behavior of the system. A relaxation method based on time-dependent magnetohydrodynamic (MHD) simulations is used to find equilibrium solutions, and special numerical techniques are taken to keep the magnetic field force-free throughout the computational domain. Quite interestingly, the system has two successive catastrophic points, and the flux rope may stay in the corona in equilibrium after catastrophe with a transverse current sheet above and a vertical current sheet below. The solution procedures for forcefree fields with an isolated flux rope are described in x 2. We discuss the catastrophic behavior of the flux rope system in x 3 and conclude our work in x 4.

Magnetic flux ropes are typical structures in the solar corona, and their eruptions are closely related to solar flares and coronal mass ejections (Forbes 2000; Low 2001). Both analytical and numerical studies have been made to explore the catastrophic behavior of coronal magnetic flux ropes in order to explain the solar explosive phenomena (Forbes & Isenberg 1991; Isenberg et al. 1993; Forbes & Priest 1995; Lin et al. 1998, 2001; Lin & Forbes 2000; Hu & Liu 2000; Hu 2001; Hu et al. 2001, 2003; Hu & Jiang 2001; Li & Hu 2001, 2003; Lin & Ballegooijen 2002). A common conclusion of these studies is that catastrophe exists under certain conditions. In Cartesian geometry, if the background field is partly opened, there exists a catastrophe for the rope system (Hu 2001; Hu & Jiang 2001). On the other hand, if the background field is completely closed, catastrophe exists only for thin ropes with a radius of the cross section less than a certain critical value (Forbes & Priest 1995), but not for flux ropes of large cross section (Hu & Liu 2000; Wang & Hu 2003). In spherical geometry, however, catastrophe exists for the flux rope system with a bipolar background field that may be either partly open or completely closed (Hu et al. 2003; Li & Hu 2003), and the catastrophic amplitude is infinite. In other words, the flux rope escapes to infinity after catastrophe. Also, a catastrophic energy threshold was identified by Li & Hu (2003) for a flux rope embedded in a bipolar background field that is either closed or partly opened. The flux rope remains attached to the solar surface in equilibrium when the energy of the system is below the threshold and erupts to infinity otherwise. No equilibrium has been found with the flux rope levitating in the corona in terms of spherical geometry. The threshold was found to be larger than that of the corresponding fully open field by about 8%.

2. SOLUTION PROCEDURES Following Hu (2004), we adopt a relaxation method based on time-dependent ideal MHD simulations to find force-free field solutions associated with an isolated flux rope. For 2.5dimensional (2.5-D) MHD problems in spherical coordinates (r, , ’), one may introduce a magnetic flux function (t; r; ) related to the magnetic field by   ˆ ˆ þ B’ ; B’ ¼ B’ j; ð1Þ j B ¼ :< r sin  where B’ is the azimuthal component of the magnetic field. Then the 2.5-D ideal MHD equations are cast in the following form: @ þ : = (v) ¼ 0; @t   @v 1 1  þ v = :v þ :p þ L : þ B’ < : < B’ @t     1 GM ˆ þ 2 rˆ ¼ 0; : = : < B’ j þ r sin  r

1 National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China. 2 School of Earth and Space Sciences, University of Science and Technology of China, Hefei 230026, China.

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CORONAL FLUX ROPE IN QUADRUPOLAR MAGNETIC FIELD @ þ v = : ¼ 0; @t   h  v i @B’ B’ v ’ þ r sin  : = ¼ 0; þ :