Hamiltonian approach to the magnetostatic equilibrium problem

Abstract

The purpose of this paper is to investigate the classical scalar-pressure magnetostatic equilibrium problem for nonsymmetric configurations in the framework of a Hamiltonian approach. Requiring that the equilibrium admits locally a family of nested toroidal magnetic surfaces, the Hamiltonian equations describing the magnetic flux lines in such a subdomain are obtained for a general canonical curvilinear coordinate system. The properties of such a coordinate system are investigated and a representation of the magnetic field is obtained. Its basic feature is that the magnetic field must fulfill suitable periodicity constraints to be imposed on arbitrary rational magnetic surfaces for general nonsymmetric toroidal equilibria, i.e., it is quasisymmetric. Implications for the existence of magnetostatic equilibria are pointed out. In particular, it is proven that a generalized equilibrium equation exists for such quasisymmetric equilibria, which extends the Grad–Shafranov equation to fully three-dimensional configurations. As an application, a representation is obtained for generalized helically symmetric equilibrium, extending the definition given by Nührenberg and Zille [Phys. Lett. A 129, 113 (1988)]. Since the new representation overcomes the inconsistency exhibited by the previous representation near the magnetic axis, pointed out by Garren and Boozer [Phys. Fluids B 3, 2805, 2822 (1991)], it appears potentially useful to interpret the numerical findings of quasihelical equilibria obtained so far.