Dynamic Evolution of Current Sheets, Ideal Tearing, Plasmoid Formation and Generalized Fractal Reconnection Scaling Relations

Magnetic reconnection may be the fundamental process allowing energy stored in magnetic fields to be released abruptly, with solar flares and coronal mass ejection being archetypal natural plasma examples. Magnetic reconnection is much too slow of a process to be efficient on the large scales, but accelerates once small enough scales are formed in the system. For this reason, the fractal reconnection scenario was introduced to explain explosive events in the solar atmosphere; it was based on the recursive triggering and collapse via tearing instability of a current sheet originally thinned during the rise of a filament in the solar corona. Here we compare the different fractal reconnection scenarios that have been proposed, and derive generalized scaling relations for the recursive triggering of fast, "ideal"-i.e., Lundquist number independent-tearing in collapsing current sheet configurations with arbitrary current profile shapes. An important result is that the Sweet-Parker scaling with Lundquist number, if interpreted as the aspect ratio of the singular layer in an ideally unstable sheet, is universal and does not depend on the details of the current profile in the sheet. Such a scaling, however, must not be interpreted in terms of stationary reconnection, rather it defines a step in the accelerating sequence of events of the ideal tearing mediated fractal cascade. We calculate scalings for the expected number of plasmoids for such generic profiles and realistic Lundquist numbers, showing that in ideal tearing scenarios a smaller number of plasmoids, by orders of magnitude, is generated compared to the original fractal model.