Magnetic field in a finite toroidal domain
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AL, NONLINEAR, AND SOFT MATTER PHYSICS
Magnetic Field in a Finite Toroidal Domain V. I. Ilgisonis and A. A. Skovoroda Russian Research Centre Kurchatov Institute, Moscow, 123182 Russia email: [email protected] Received November 23, 2009
Abstract—The magnetic field structure in a domain surrounded by a closed toroidal magnetic surface is ana lyzed. It is shown that ergodization of magnetic field lines is possible even in a regular field configuration (with nonvanishing toroidal component). A unified approach is used to describe magnetic fields with nested toroidal (possibly asymmetric) flux surfaces, magnetic islands, and ergodic field lines. DOI: 10.1134/S1063776110050201
1. INTRODUCTION Field lines of a solenoidal vector field in a finite domain can form closed loops, wind around surfaces forming topological tori, and ergodically fill subvol umes of a given region. Interest in field line structures of these types is usually associated with general prob lems in regular and stochastic dynamics of Hamilto nian systems (e.g., see [1, 2]). Even though ergodic fields are definitely more general topologically, the toroidal surface structure of phase trajectories of a dynamical system dictated by initial or boundary con ditions is known to survive under relatively weak per turbations according to the KAM theorem. Examples of ergodic fields have been found both analytically and numerically. However, analytical examples are usually either constructed perturbatively, so that the solenoidality and/or ergodicity condition is satisfied only asymptotically, or involve fields whose magnitude vanishes or goes to infinity at some singular points inside the given region (e.g., see [3]). Most numerical examples of ergodic fields have similar properties or occupy unbounded domains [4]. Toroidal magnetic plasma confinement devices (tokamaks, stellarators, etc.) are generally believed to offer possibilities for generating fields with nested tor oidal flux surfaces, ensuring that the equilibrium plasma pressure vanishes at the last closed flux surface. Such a surface can be formed by using a perfectly con ducting wall or external coils. Since the equilibrium equation is nonlinear, the problem of plasma confine ment in the region inside a predefined last closed flux surface generally has more than one solution, and the possible bifurcations may either preserve the nested fluxsurface structure [5] or destroy it, leading to the formation of “magnetic island” structures [6]. Forma tion of magnetic islands by the splitting of rational magnetic surfaces induced in tokamaks by symmetry breaking perturbations is well studied theoretically [7, 8] and experimentally [9]. The fields inside such
islands generally have their own nested helical flux surfaces. The possible chaotic behavior of field lines in the separatrix layer at the divertor has also been inves tigated [10]. However, simple examples illustrating the ergodization of magnetic field lines in the volume bounded by a flux surface have never been presented, and their fea
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