On mapping models of field lines in a stochastic magnetic field

Abstract

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.

Mappings of stochastic field lines in poloidal divertor tokamaks

“…The general form of the mapping (25) for the Hamiltonian system (19)- (21) is derived in [26] (see also [27]). In the first order of perturbation amplitude it has the following fluxpreserving form:…”

Stochastization as a possible cause for fast reconnection during MHD mode activity in the ASDEX Upgrade tokamak

“…A properly chosen mapping procedure always conserves the main flux preserving property of the magnetic field, which is important for a correct reproduction of the long-term behaviour of field lines in stochastic regions. In this formalism the equations for magnetic field lines take the Hamiltonian form , , For our purposes we have chosen the symmetric symplectic mapping derived in [9] on the basis of the Hamilton-Jacobi method. In the first order approximation this mapping can be written as follows:…”

Traces of stable and unstable manifolds in heat flux patterns

“…The main goal of mapping models is to replace the original continuous dynamical system, the magnetic field lines, by a discrete iterative map, which runs much faster then the small-step numerical integration. 15,[18][19][20][21] Mappings should be symplectic ͑or flux preserving͒. They should have the same periodic points as the Poincaré map of the original system, and they should show the same regular and chaotic regions as the continuous magnetic field line evaluation.…”