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

Abstract

We show that the Lu (Phys. Rev. Lett. 74(13) (1995) 2511) model, which is known to exhibit some properties of a system in self-organized criticality (SOC) [Lu, 1995; Klimas et al. (J. Geophys. Res. 105 (2000) (A8), 18,765–18,780.)], can be obtained through a reduction of the resistive MHD system to an idealized one-dimensional limit. Resistivity in this model is anomalous and localized and is due to the excitation of an idealized current-driven instability at positions where large spatial gradients appear in the magnetic field distribution. We note that, by reversing the reduction to the idealized one-dimensional limit, the Lu model presents an opportunity to construct a true MHD system that incorporates kinetic phenomena when small spatial scales are generated which may evolve into SOC under some conditions. We study the evolution of this model in a driven magnetic field reversal configuration on a high-resolution spatial grid. It has been shown earlier that the behavior of several parameters that are global measures of the state of the field reversal suggests that the reversal can evolve into SOC (Klimas et al., 2000). Here, we study the internal dynamics of the field reversal during the unloading phase of a loading–unloading cycle. Unloading is due to internal, localized, dynamic field annihilation; no flux is lost by the system through its boundaries. For this continuum model, we define an “avalanche” as a group of unstable grid points that are contiguous in position and time. We demonstrate scale-free power-law size and duration distributions for these avalanches during the unloading phase of a loading–unloading cycle. We further demonstrate the stability of these distributions; they do not evolve significantly as the unloading progresses. Box counting statistics on the position–time plane show that the avalanches can be characterized as intermittent one-dimensional structures; gaps in these otherwise one-dimensional structures lower their dimension to below one. The stable scale-free avalanche 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.