Separatrix crossing and large scale diffusion in low-frequency three-wave systems

Abstract

The $\mathbf{E}\ifmmode\times\else\texttimes\fi{}\mathbf{B}$ guiding center diffusion in three low-frequency two-dimensional electrostatic waves is considered. It is shown that the stochastic guiding center (GC) diffusion can be explained and predicted with the help of the rules of the adiabatic theory of Hamiltonian systems, i.e., (i) conservation of the canonical action except at separatrix crossing times and (ii) time evolution of the canonical action determined by the surfaces enclosed by the separatrices of the potential. The probability distributions are calculated. Our demonstration applies at least for isotropically distributed wave vectors, very high Kubo number $K,$ and closed equipotentials. A statistical analysis of the dynamics shows that the GC motion is a spaced constrained random walk governed by a ``complete trapping'' scaling law for diffusion: $\overline{D}{(K)=K}^{0}.$ This result is demonstrated both semianalytically and numerically.