On mapping models of field lines in a stochastic magnetic field

Mapping models of Hamiltonian systems are discussed using the example of magnetic field lines in magnetically confined fusion plasmas. They are usually constructed in a certain symplectic form by imposing several constraints that make them compatible with a toroidal geometry. The possible symplectic forms of model mappings for Hamiltonian systems are derived using the recently developed method for the construction of symplectic mappings (Abdullaev S.S. 2002 J. Phys. A 35 2811). It is shown that the symplectic mappings may have symmetric and nonsymmetric forms. The generating function of the symmetric mapping depends only on the perturbation field while the one for the nonsymmetric map depends also on the safety factor (or winding number). Mapping models, particularly, a global mapping model in a toroidal system, the tokamap (Balescu R. et al 1998 Phys. Rev. E 58 951), usually used in plasma physics are reviewed and their relations with continuous Hamiltonian systems are discussed. The symmetric form of the tokamap is proposed and compared with the conventional tokamap. The symmetric and nonsymmetric mapping models for field lines in ergodic divertor tokamaks are also considered. It is shown that symmetric mappings closely describe original Hamiltonian systems in comparison with nonsymmetric mappings.