Derivation of Solar Flare Cellular Automata Models from a Subset of the Magnetohydrodynamic Equations

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Abstract
Cellular automata (CA) models account for the power-law distributions found for solar flare hard X-ray observations, but their physics has been unclear. We examine four of these models and show that their criteria and magnetic field distribution rules can be derived by discretizing the MHD diffusion equation as obtained from a simplified Ohm's law. Identifying the discrete MHD with the CA models leads to an expression for the resistivity as a function of the current on the flux tube boundary, as may be expected from current-driven instabilities. Anisotropic CA models correspond to a nonlinear resistivity η(J), while isotropic ones are associated with hyperresistivity η(2J). The discrete equations satisfy the necessary conditions for self-organized criticality (Lu): there is local conservation of a field (magnetic flux), while the nonlinear resistivity provides a rapid dissipation and relaxation mechanism. The approach justifies many features of the CA models that were originally based on intuition.

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