Stabilities of magnetic flux ropes anchored in the solar surface

Eruptions of a magnetic flux rope in the solar corona can be triggered by certain magnetohydrodynamic (MHD) instabilities, such as the helical

kink instability or the torus instability. The instability threshold depends on the geometry of the flux rope, and it is not a-priori clear which instability will occur or dominate in a given configuration. Furthermore, a magnetic flux rope is anchored in the photosphere, and this line-tying condition imposes an essential constraint on the stability analysis. The modified Titov-Demoulin model (TDm, Titov et al. 2014) provides a three-parameter family of approximate equilibria of a toroidal-arc magnetic flux rope anchored in the photosphere that is smoothly embedded in an ambient bipolar potential field in a spherical geometry. This study examines the stability of a force-free TDm magnetic flux rope in dependence of its geometric parameters. The global MHD MAS code is used to evolve the flux ropes under the line-tying condition. For a given perturbation of the flux rope, a configuration is not stable if it never returns to an equilibrium or if it evolves into a new equilibrium that deviates strongly from the initial equilibrium solution. Our preliminary study suggests that configurations with an X-line below the flux rope are usually not stable, and also confirms that a high-lying flux rope can be kink-stable even for large twist (see, e.g., Torok et al. 2004).