Hamilton flow generated by field lines near a toroidal magnetic surface
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TICAL, NONLINEAR, AND SOFT MATTER PHYSICS
Hamilton Flow Generated by Field Lines near a Toroidal Magnetic Surface A. A. Skovoroda National Research Center Kurchatov Institute, Moscow, 123182 Russia email: [email protected] Received September 7, 2012
Abstract—A method is described for obtaining the Hamiltonian of a vacuum magnetic field in a given 3D toroidal magnetic surface (superconducting shell). This method is used to derive the expression for the inte grable surface Hamiltonian in the form of the expansion of a rotational transform of field lines on embedded nearboundary magnetic surfaces into a Taylor series in the distance from the boundary. This expansion con tains the value of the rotational transform and its shear at the boundary surface. It is shown that these quan tities are related to the components of the first and second quadratic forms of the boundary surface. DOI: 10.1134/S1063776113060228
1. INTRODUCTION It is well known that a magnetic field generates the Hamilton flow of field lines, which may contain mag netic surfaces (invariant Kolmogorov–Arnold–Moser (KAM) tori), which are the most important elements of fusion plasma traps. Under irrational rotational transformation of field lines on a smooth magnetic surface (nonresonant case on the master torus), a pos itive measure of other magnetic closetomaster (slave) surfaces exists near this surface in accordance with the KAM theory [1], which generally substanti ates the prospects of longterm fusion plasma confine ment in magnetic traps. Moreover, the relative mea sure coated with slave surfaces in a small but macro scopic neighborhood of the master torus is close to unity and rapidly tends to unity upon a decrease in the neighborhood (stickiness phenomenon observed for tori) [2, 3]. In this study, we consider the inner problem with fixation of the 3D master toroidal magnetic surface (e.g., in the form of a superconducting shell). When a smooth closed toroidal surface S in the form r = rS(θ, ζ) is specified in vacuum under the requirement that it is magnetic (B ⋅ nS = 0), we arrive at the inner Neumann boundaryvalue problem for the Laplace equation: Δϕ = 0, ∇ϕ ⋅ n S = 0, (1) which has a unique solution. Here, B = ∇ϕ is the potential magnetic field with a multivalued potential ϕ, nS is the unit vector of the normal to the surface, and θ and ζ are arbitrary poloidal and toroidal coordinates with a period of 2π. It should be noted that we consider the magnetic geometry typical of toroidal traps for fusion plasmas, in which the magnetic field modulus does not vanish anywhere and all field lines lie on the torus, circum
venting its major axis, B ⋅ ∇ζ > 0. The general repre sentation of such a potential field has the form ˜ ), (2) 2πB = F∇ ( ζ + ϕ ˜ is the where F is a positive dimensional constant and ϕ dimensionless periodic part of potential ϕ with a zero mean value. Thus, the geometry of surface S deter mines the geometry of magnetic field B in the entire volume bounded by the surface. Knowledge of B obtained from the numerical
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