Toroidal equilibria in spherical coordinates

TL;DR

This paper introduces a novel formulation of the Grad-Shafranov equation in spherical coordinates to describe toroidal plasma equilibria, providing analytical solutions for different plasma beta regimes.

Contribution

It presents an alternative spherical coordinate approach to the Grad-Shafranov equation, enabling analytical solutions for toroidal plasma equilibria.

Findings

Derived elementary function solutions for low and high beta plasmas

Demonstrated the feasibility of spherical coordinate formulation

Provided benchmarks for plasma equilibrium modeling

Abstract

The standard Grad-Shafranov equation for axisymmetric toroidal plasma equilibrium is customary expressed in cylindrical coordinates with toroidal contours, and through which benchmark equilibria are solved. An alternative approach to cast the Grad-Shafranov equation in spherical coordinates is presented. This equation, in spherical coordinates, is examined for toroidal solutions to describe low $\beta$ Solovev and high $\beta$ plasma equilibria in terms of elementary functions.