Magnetic Field Line Diffusion at the Onset of Stochasticity

Abstract

Dedicated to Professor Dieter Pfirsch on his 60th BirthdayThe Ham iltonian equations o f a particle in a random set of waves just above the stochasticity threshold are considered both theoretically and numerically. First we derive the diffusion coefficient and the autocorrelation time perturbatively without using the thermodynamic limit, and we discuss the relevance of the H am iltonian problem for particle acceleration and magnetic field line flow. Then we integrate the equations for an ensemble o f magnetic field lines numerically for a model problem [15] and show the time evolution of moments and correlations. Twice above the threshold we observe diffusive behaviour from the beginning, but the diffusion coefficient includes also the non-resonant modes. Just at threshold we find first a short phase o f free acceleration, later a diffusion which is slower than predicted by the theoretical formula. The best way to analyze the problem is in terms o f cumulants, but a reliable com pari son with any theory requires also a time integration of the corresponding kinetic equations.