Nonequilibrium and current sheet formation in line-tied magnetic fields

Parker's model of coronal heating [E. N. Parker, Astrophys. J. 174, 499 (1972)] is considered within the framework of ideal reduced magnetohydrodynamics. It is shown that there can be at most one smooth magnetostatic equilibrium for a given smooth footpoint mapping between two end plates to which field lines are line-tied. If such a smooth equilibrium is deformed continuously by further footpoint motion so that it becomes unstable, there is no other smooth equilibrium for the plasma to relax to, and the system tends to a nonequilibrium state containing singular currents ("current sheets"). It is shown that this process can occur as the system relaxes asymptotically to a state of minimum energy (possibly in infinite time). Numerical simulations that begin from smooth initial conditions containing current layers are presented. As the current layers become increasingly intense due to footpoint motion and eventually cross a threshold for instability, the magnetic relaxation observed in the simulation shows a tendency to form nonequilibrium states with current sheets. A necessary geometrical criterion that determines the sites of current sheet formation in models without nulls or closed field lines is given. According to this criterion, the rate of velocity amplification, analogous to the Lyapunov exponent in nonlinear dynamics, becomes unbounded at singularities.