Stable critical behavior and fast field annihilation in a magnetic field reversal model

We show that the Lu (Phys. Rev. Lett. 74(13) (1995) 2511) model, which is k nown to exhibit some properties of a system in self-organized criticality ( SOC) [Lu, 1995; Klimas et al. (J. Geophys. Res. 105 (2000) (A8),18,765-18,7 80.)], can be obtained through a reduction of the resistive MHD system to a n idealized one-dimensional limit. Resistivity in this model is anomalous a nd localized and is due to the excitation of an idealized current-driven in stability at positions where large spatial gradients appear in the magnetic field distribution, We note that, by reversing the reduction to the ideali zed one-dimensional limit, the Lu model presents an opportunity to construc t a true MHD system that incorporates kinetic phenomena when small spatial scales are generated which may evolve into SOC under some conditions. We st udy the evolution of this model in a driven magnetic field reversal configu ration on a high-resolution spatial grid. It has been shown earlier that th e behavior of several parameters that are global measures of the state of t he field reversal suggests that the reversal can evolve into SOC (Klimas et al., 2000). Here, we study the internal dynamics of the field reversal dur ing the unloading phase of a loading-unloading cycle. Unloading is due to i nternal, localized, dynamic field annihilation; no flux is lost by the syst em through its boundaries. For this continuum model, we define an "avalanch e' as a group of unstable grid points that are contiguous in position and t ime. We demonstrate scale-free power-law size and duration distributions fo r these avalanches during the unloading phase of a loading-unloading cycle. We further demonstrate the stability of these distributions; they do not e volve significantly as the unloading progresses. Box counting statistics on the position-time plane show that the avalanches can be characterized as i ntermittent one-dimensional structures; gaps in these otherwise one-dimensi onal structures lower their dimension to below one. The stable scale-free a valanche size and duration distributions, plus the fractal structure of the avalanches at small scales, provide further evidence that solutions of the continuum Lu model in a field reversal configuration can evolve into SOC. (C) 2001 Elsevier Science Ltd. All rights reserved.