Current sheets and reconnection driven by footpoint motion in two-dimensional coronal loops with x-type neutral lines

The dynamical evolution of a two-dimensional coronal loop from a smooth initial state containing an X-type neutral line is considered. An exact solution of the linearized ideal magnetohydrodynamic equations shows that the amplitude of the current sheet at the separatrix grows exponentially with time while its width shrinks at the same rate. Thus, although there is a strong tendency for a current sheet to form, a true singularity is not realized in finite time. Resistivity intervenes, and the ideal phase is followed by a linear resistive phase in which the dynamics is still exponential in time with a growth rate proportional to S^-1/3, where S is the Lundquist number of the coronal plasma. The linear resistive phase is followed by a helicity-conserving nonlinear phase in which the growth rate is algebraic in time, and the reconnection rate is proportional to S^-1/2, as in the Sweet-Parker model. It is demonstrated that the heating caused by these current sheets can be large enough to account for the energy balance in quiet as well as some active coronal loops.