Continuum analysis of an avalanche model for solar flares.

ABSTRACT

We investigate the continuum limit of a class of self-organized critical lattice models for solar flares. Such models differ from the classical numerical sandpile model in their formulation of stability criteria in terms of the curvature of the nodal field, and are known to belong to a different universality class. A fourth-order nonlinear hyperdiffusion equation is reverse engineered from the discrete model's redistribution rule. A dynamical renormalization-group analysis of the equation yields scaling exponents that compare favorably with those measured in the discrete lattice model within the relevant spectral range dictated by the sizes of the domain and the lattice grid. We argue that the fourth-order nonlinear diffusion equation that models the behavior of the discrete model in the continuum limit is, in fact, compatible with magnetohydrodynamics (MHD) of the flaring phenomenon in the regime of strong magnetic field and the effective magnetic diffusivity characteristic of strong MHD turbulence.