Shear-driven field-line opening and the loss of a force-free magnetostatic equilibrium

This paper discusses a quasi-static evolution of a force-free magnetic field under slow sheared footpoint motions on the plasma's boundary, an important problem with applications to the solar and accretion disk coronae. The main qualitative features of the evolution (such as field-line expansion and opening) are considered, and a comparison is made between two different geometrical settings: the Cartesian case with translational symmetry along a straight line, and the axisymmetric case with axial symmetry around the rotation axis. The main question addressed in the paper is whether a continuous sequence of force-free equilibria describes the evolution at arbitrarily large values of the footpoint displacement, or the sequence ends abruptly and the system exhibits a loss of equilibrium at a finite footpoint displacement. After a formal description of the problem, a review/discussion of the extensive previous work on the subject is given. After that, a series of simple scaling-type arguments, explaining the key essential reason for the main qualitative difference between the two geometry types, is presented. It is found that, in the Cartesian case, force-free equilibria exist at arbitrarily large values of shear and the field approaches the open state only at infinite shear, whereas in the axisymmetric case the field opens up already at a finite shear.