Self-organized criticality in a two-dimensional cellular automaton model of a magnetic flux tube with background flow

Abstract

We investigate the transition to Self Organized Criticality in a two-dimensional model of a flux tube with a background flow. The magnetic induction equation, represented by a partial differential equation with a stochastic source term, is discretized and implemented on a two dimensional cellular automaton. The energy released by the automaton during one relaxation event is the magnetic energy. As a result of the simulations we obtain the time evolution of the energy release, of the system control parameter, of the event lifetime distribution and of the event size distribution, respectively, and we establish that a Self Organized Critical state is indeed reached by the system. Moreover, energetic initial impulses in the magnetohydrodynamic flow can lead to one dimensional signatures in the magnetic two dimensional system, once the Self Organized Critical regime is established. The applications of the model for the study of Gamma Ray Bursts is briefly considered, and it is shown that some astrophysical parameters of the bursts, like the light curves, the maximum released energy, and the number of peaks in the light curve can be reproduced and explained, at least on a qualitative level, by working in a framework in which the systems settles in a Self Organized Critical state via magnetic reconnection processes in the magnetized Gamma Ray Burst fireball.