Toroidal plasma equilibrium with arbitrary current distribution

Abstract

A new System of co-ordinates is found and a method developed to determine the toroidal equilibrium of plasmas with arbitrary current distribution and plasma cross-section. The method depends on knowledge of the equilibrium of a straight plasma column of similar cross-section and similar current distribution. A large aspect ratio is assumed. By successive approximations, better solutions can be obtained. An explicit formula is presented for the poloidal flux of a nearly circular plasma. This can be written in terms of a function related to the asymmetry of the poloidal field due to toroidality. The method works provided that there is only one magnetic axis.

Analytical solutions for Tokamak equilibria with reversed toroidal current

“…This property can be described by a scalar function, a surface quantity w, such as B 0 • r p ¼ 0, where B 0 is the plasma equilibrium magnetic field and the magnetic surfaces are characterized by w ¼ constant. 13 The tokamak equilibrium magnetic field B 0 is obtained from an analytical solution of the Grad-Shafranov equation in these coordinates, 13 such as w ¼ wðr t ; h t Þ. Besides, the intersections of the flux surfaces w p ¼ constant with a toroidal plane are not concentric circles but rather present a Shafranov shift toward the exterior equatorial region.…”

“…Besides, the intersections of the flux surfaces w p ¼ constant with a toroidal plane are not concentric circles but rather present a Shafranov shift toward the exterior equatorial region. 13 For large aspect-ratio and almost circular sections, the solution for the Grad-Shafranov equation can be written in terms of the toroidal coordinates as 13,14 …”

“…To obtain these solutions, in terms of nonorthogonal toroidal polar coordinates, 13 we use a successive approximation method. These toroidal polar coordinates were introduced in previous works to evidence toroidal effects in the equilibrium field geometry.…”

“…These toroidal polar coordinates were introduced in previous works to evidence toroidal effects in the equilibrium field geometry. 13,14 For the considered reversed current profile, our zero order solution of the poloidal magnetic flux function depends only on the radial toroidal polar coordinate and describes nested toroidal magnetic surfaces with a singular surface where the flux gradient is null. The flux surfaces in a poloidal plane are not concentric circles but rather circles shifted toward the exterior equatorial region, as observed in every tokamak (the so called Shafranov shift 1 ).…”

Non-twist field line mappings for tokamaks with reversed magnetic shear

“…We will use a suitable coordinate systems to describe magnetic field line geometry in a tokamak: (r t , θ t , ϕ t ), given by [15] …”