Solution to the Grad-Shafranov Boundary Value Problem for a Thermonuclear Plasma contained in a Toroidal Vase With a Conducting Wall under the Assumption of Constant Sources to the Equilibrium in Flux Space

TL;DR

This paper presents a method to solve the Grad-Shafranov equation for plasma equilibrium in a toroidal confinement device, assuming constant sources and boundary conditions, using a combination of particular and multipole solutions.

Contribution

It introduces a novel analytical approach combining particular and multipole solutions to solve the Grad-Shafranov boundary value problem under specific assumptions.

Findings

Derived explicit solutions for plasma equilibrium in a toroidal vessel.

Validated the method for constant source conditions in flux space.

Provided a framework for analytical solutions in magnetically confined plasmas.

Abstract

A method is proposed to solve the Grad-Shafranov partial differential equation for the poloidal flux function associated with the equilibrium of a plasma magnetically confined in an axisymmetric torus under the assumption that the sources to the equilibrium (the gradient of the plasma pressure and the gradient of half the squared toroidal field function in flux space) are constant, and subjected to the condition of constant value of the poloidal flux at the toroidal boundary. The solution to the equation is written as the sum of the particular solution and a combination of N multipole solutions of the lowest orders in the variables of the toroidal-polar coordinate system having the pole at the centre of the torus cross section.