The formation of magnetic singularities by time-dependent collapse of an X -type magnetic field

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Abstract
A self-consistent solution is presented for nonlinear time-dependent collapse of a two-dimensional X-type magnetic field to form a current sheet. A so-called ‘strong magnetic field approximation’ is adopted for highly sub-Alfvenic flow of an ideal low-beta plasma. To lowest order in the Alfven Mach number, the magnetic field evolves through a series of topologically accessible piece-wise potential states with the constraint that the acceleration be perpendicular to the magnetic field. A wide class of solutions is obtained using complex variable theory by assuming the magnetic potential is frozen to the plasma. The current sheet in the basic solution stretches along the x-axis from —/ t to +/ t , and regions of reversed current are found near the ends of the sheet. A current conservation theorem is proved, which states that the total current in the sheet is zero if it forms by collapse of an initially current-free X-point under the strong magnetic field approximation and with the magnetic potential frozen to the plasma. The basic solution is generalized to include other initial states and initial flows. A general numerical method for the evolution of magnetic fields under the strong magnetic field approximation is set up when the magnetic potential is not necessarily frozen to the plasma. This method is applied to an example of the formation of a current sheet with Y-type neutral points at its ends.