Basic Principles of Stochastic Transport

Abstract

Transport in stochastic magnetic fields is reviewed. In the first part, the topic is motivated by commenting on the intricatenesses of known (nonlinear) transport theories. In the second part, non‐integrable magnetic field line systems, their generation and Hamiltonian description are discussed. The symplectic mapping is introduced as the adequate tool for the analysis of the statistics of magnetic field lines. Transport along the unstable and stable manifolds of hyperbolic fixed points is an effective mechanism for heat transfer from the hot core to the plasma boundary. The third part deals with anomalous test particle transport theories starting from stochastic Liouville‐type models. Several theories are based on the V‐Langevin equation in the guiding center limit. In fusion devices, the mean magnetic fields are sufficiently strong to support the small gyro‐radii assumption over a broad area, at least for the electrons. The question remains in what way finite Larmor radii influence the transport, especially in regions where the guiding center assumption fails. Indeed, in tokamaks such areas can be found, e.g. in the vicinity of hyperbolic points. Then, the more general A‐Langevin equation has to be used. Based on the latter description, first for small Kubo numbers, the well known transport coefficients (formulated by Rechester and Rosenbluth, Kadomtsev and Pogutse, and others) are recovered. Second, for large Kubo numbers, new transport regimes are identified.