Theory of Multipole Solutions to the Sourceless Grad-Shafranov Equation in Plasma Physics

TL;DR

This paper develops a systematic method to construct polynomial solutions to the sourceless Grad-Shafranov equation in plasma physics, expressing solutions in terms of Chebyshev polynomials and providing explicit tables for low-degree solutions.

Contribution

It establishes rules for constructing polynomial solutions to the sourceless Grad-Shafranov equation using Chebyshev polynomials and provides explicit solutions in different coordinate systems.

Findings

Polynomial solutions involve Chebyshev polynomials in the angular variable.

Solutions are expressed as products of radial powers and polynomials with angular-dependent coefficients.

Tables of lowest-degree solutions are provided in toroidal-polar and cylindrical coordinates.

Abstract

The rules to write out any one of the linearly independent functions belonging to the infinite set of those in polynomial form that satisfy the sourceless Grad-Shafranov equation as stated in the toroidal-polar coordinate system are established. It is found that a polynomial solution even in the poloidal angle is given by the product of an integral power of the radial coordinate variable by a complete polynomial of equal degree in this same variable with angular-dependent coefficient functions that are linear combinations of a finite number of Chebyshev polynomials in the cosine of the poloidal angle, the numerical coefficients of these being expressed in terms of the binomial numbers of Pascal's arithmetic triangle. Tables of the ten polynomial solutions of the lowest degrees are provided in variables of the toroidal-polar and of the cylindrical coordinate systems.