Variational problems for the canonical profiles
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Variational Problems for the Canonical Profiles Yu. N. Dnestrovskij, A. Yu. Dnestrovskij, A. V. Danilov, S. E. Lysenko, and S. V. Cherkasov Nuclear Fusion Institute, Russian Research Centre Kurchatov Institute, pl. Kurchatova 1, Moscow, 123182 Russia Received November 23, 2006; in final form, May 22, 2007
Abstract—A noncontradictory formulation of the variational problem for a canonical profile is proposed that refines the problem posed by B.B. Kadomtsev for a circular plasma cylinder. The results are generalized to a toroidal plasma with an arbitrary cross section. For the problem in toroidal geometry, boundary conditions are proposed with which to single out the Kadomtsev-like solution (the canonical profile) from the solutions to the Euler equation. Canonical profiles for the L- and H-modes are constructed. For a number of interesting examples, it is numerically shown that the second variation of the magnetic energy functional is positive. The canonical profile transport model is outlined, and the relationship between the canonical, numerical, and experimental profiles in tokamaks is briefly discussed. PACS numbers: 52.55.Fa, 52.25.Fi, 52.55.Dy DOI: 10.1134/S1063780X08010017
1. INTRODUCTION
the variational problem. For the H-mode in a circular plasma cylinder, the function i (µ) can be written explicitly: i (µ) = µ2. But for the L-mode in the cylinder and for the L- and H-modes in a toroidal plasma, this function cannot be expressed in explicit form. In what follows, we show that the function i (µ) can be constructed in terms of the solutions to the Euler equation. As for the Euler equation itself, it can be easily generalized from cylindrical to toroidal geometry by averaging the local current density and local poloidal magnetic field over a magnetic surface. This generalization does not require information about the form of the function i (µ). The specific character of the problems for the L- and Hmodes manifests itself in the formulation of the boundary conditions. We propose boundary conditions for canonical profiles with a pedestal in the current and pressure at the plasma edge for the H-mode and without a pedestal for the L-mode. Knowing the solution to the Euler equation with the boundary conditions formulated, we can find the function i (µ) and the form of the integral in conservation condition (1). We can thereby reconstruct the variational problem equivalent to the boundary-value problem formulated. The canonical profiles for the L- and H-modes can be used for comparison with the experimental profiles of the plasma parameters (pressure, temperature, and density) [2]. The canonical profiles for the L- and H-modes correspond to incompletely and completely relaxed plasma states [1]. With this interpretation, the canonical profile for the H-mode can be used to determine the critical plasma temperature and pressure gradients in transport equations [3–5]. Hence, the new results of our work are noncontradictory formulations of the variational problems for a circular plasma cylinder and a toroidal plasm
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