Resistive Diffusion of Force-Free Magnetic Fields in a Passive Medium. 11. a Nonlinear Analysis of the One-Dimensional Case

We investigate the force-free magnetic field B = [cos (z, t), sin (z, t), 0] to ascertain when it may develop infinite field gradients while undergoing resistive diffusion in a passive medium (Low). A complete analysis of the nonlinear evolution of B is given, subject to the conditions (i) 4L(z, 0) is an odd and increasing function of the Cartesian coordinate z and (ii) ( I 00, t) = I o for all time, where o is finite and independent of time. if the total rotation of B along the z-axis exceeds half a revolution, i.e., > , we find that B always develops infinite field gradients after a finite period of time, irrespective of the initial distribution (z, 0). The modest criterion > supports our suggestion that this type of instability triggers the eruption of flares in the solar chromosphere. if < , we find that B always evolves toward a uniform field, irrespective of the initial distribution (z, 0). The marginal case = admits an unstable steady state. We use the properties of this steady state to show how a quantitative theory of homologous flares might be constructed. Subject headings: flares, solar - hydromagnetics