Pattern Formation in Force-free Magnetic Fields and Beltrami Flows

Abstract

Abstract Force-free magnetic fields and Beltrami flows, which are selenoidal vector fields and satisfy the condition that the field vector is everywhere parallel to its curl, have complex topological structures and usually show chaotic behavior. The lines of force are determined by the equations of a 3-D (dimensional) dynamical system. In the integrable case, all lines of force lie on some families of tori. If the integrable solution undergoes a small perturbation, most of the original tori still exist but undergo a slight distortion (KAM tori). Near the original heteroclinic cycles emerges a chaotic layer. By superposition of the basic solutions of force-free magnetic fields one can get very complicated pictures: a single line of force could be space filling within some subspace of a 3-D region, which has a fractional dimension and a positive Lyapunov exponent, i.e. one gets a chaotic line of force or a fractal. At the same time there are still ordered regions in the chaotic surroundings. Tubes of force which are tangled or self-knotted embed in the chaotic sea. The KAM tori can also be disrupted through resonances, leading to increased chaotic regions. Thus, the effect of nonlinear dynamics plays an important role in the pattern formation of force-free magnetic fields and Beltrami flows.