Equilibrium of a small toroidal plasma column with an arbitrary current distribution along the cross section

Abstract

Transverse (meridional) cross sections of magnetic surfaces in a toroidal plasma column are circles with displaced centers in the first-approximation expansion in ratio of the column radius ϱ to the torus radius R. In the cross section of every magnetic surface a polar coordinate system of its own can be introduced (with the origin on the center of this cross section), in which the transverse magnetic field on a given magnetic surface has an azimuthal component only that changes according to the law Bω(ϱ, ω) = Bω0(ϱ) [1 + (ϱ/R)Λ(ϱ) cos ω]. The “asymmetry coefficient” Λ(ϱ) of the azimuthal field that is introduced thus for every toroidal magnetic surface is an important characteristic of the toroidal equilibrium configuration. It is shown in this paper that the distance Δ(ϱ) between the centers of the cross sections of two magnetic surfaces, is expressed by a simple formula in Λ(ϱ): Here a1 and ϱ are the radii of the chosen cross sections. In their turn the toroidal corrections to plasma pressure and magnetic field are easily expressed in terms of Δ(ϱ). In a fixed coordinate system ω, ϱ, the origin of which coincides with the center of the cross section of radius a1 of some “initial” magnetic surface, the correction to the transverse magnetic field is expressed by the formulas Bϱ(1) (ϱ, ω) = [Δ(ϱ)/R] Bω0 (ϱ) sin ω, Bω(1) (ϱ, ω) = [Bω(0)(ϱ) ϱΛ (ϱ)/R + Δ(ϱ) d Bω0(ϱ)/dϱ] cos ω, and to the longitudinal field by Bφ(1) (ϱ, ω) = [—Bφ0 (ϱ)ϱ/R + Δ(ϱ) d Bφ0(ϱ)/d φ] cos ω.The correction to the pressure is p(1)(ϱ, ω) = Δ(ϱ) cos ω d p0(ϱ)/dϱ. The asymmetry coefficient Λ(ϱ) is determined by a single integration over the values of zero order. In particular, for a continuous plasma column We have also obtained the expression Λ(ϱ) for a tubular (levitron-type) plasma column. Some examples of the calculation of toroidal corrections are given.