Force Equation arc field function: force due to ±q sources stationary background line current generalized hoop force : arc coordinates
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
side view (x = 0) end view (y = 0) initialdisplaced Line-Tied Displacements
Numerical Simulation of an Unstable Flux Rope Titov & Démoulin (1999)Török & Kliem (2005)
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
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?