9. Van Tend & Kuperus (1978)

10. Three-Dimensional Configuration of Titov & Démoulin (1999) 3. line-current 1. flux rope 2. magnetic charges 3 field sources

11. Initial Configuration (Titov & Démoulin) Normal Field at Surface flux-rope footprint I 0 = I I : flux rope current I 0 : sub-surface line current

14. Force Equation arc field function: force due to ±q sources stationary background line current generalized hoop force : arc coordinates

15. Other Equations Current, I, from flux conservation: Requires numerical integration Radial variation: Assume Lundquist solution, a = a 0 (I/I 0 ), l i = 1 Ignore tapering (end effect occuring over a distance of a – a 0 ) force arc length apex surface

16. side view (x = 0) end view (y = 0) initialdisplaced Line-Tied Displacements

20. Kliem & Török (2004) current density

21. Numerical Simulation of an Unstable Flux Rope Titov & Démoulin (1999)Török & Kliem (2005)

22. Simulation of Torus* Instability *nonhelical kink (see Bateman 1973) 1. no subsurface line current 2. subcritical twist for helical kink 3. torus center near surface

23. What is the efficiency of energy conversion as a function of reconnection rate? How does reconnection work in a current sheet whose length grows at a rapid rate? Can we determine the equilibria and stability properties of more realistic line-tied field configurations?