Perturbative approach to the nonlinear growth and saturation of the tearing mode for any current gradient, with an application to the forced reconnection problem

The full text follows. Within the traditional framework of low-β reduced MHD, a new systematic perturbation expansion provides the equation ruling the nonlinear growth and the saturation of a single helicity thin magnetic island, in slab or cylindrical geometry, in the Δ' regime characteristic of tearing modes. The relevant small expansion parameter is the width of the nonlinear boundary layer, which is proportional to the magnetic island width w. When the technique is applied to the generic case, characterized by an arbitrary current gradient on the rational surface, the final equation includes all the terms of order w ln(l/w) and of order w, which have the same magnitude for practical purposes. Moreover, the evolution of the mean magnetic field is also given consistently with the island evolution and provides an accurate estimate of the change of magnetic flux occurring during the reconnection process. The technique is also applied to the case where the island is subject to a static forcing provided by an external magnetic perturbation. The ensuing system of coupled evolution equations, for the island width and for the relative phase between the island and the perturbation, exhibits one or three fixed points, depending on the value of Δ' and on the strength of the external perturbation. Of these, only the fixed point having an island width larger than one obtained in the absence of forcing is stable. The island width of the stable fixed point exhibits three scaling regimes in terms of Δ' and of the forcing strength. For large negative Δ' the Hahmn-Kulsrud scaling is recovered. We have also recovered the above results with an alternative, more direct, approach, in which the magnetic flux function is employed as an independent variable, which provides an independent check. Excellent agreement with available numerical data in the appropriate limit was also obtained. Besides giving a systematic solution to long-standing problems, the new techniques appear amenable to application to more general problems involving rotating islands, as in the case of a plasma with sheared mass flow and/or diamagnetic effects, since they do not rely on the condition that certain fields, and in particular the current density, be flux functions. (authors)