Exact solutions of the Grad-Shafranov equation via similarity reduction and applications to magnetically confined plasmas

TL;DR

This paper derives exact analytical solutions for the linear and nonlinear Grad-Shafranov equation, including flow effects, using similarity reduction, aiding the design of magnetically confined plasma equilibria.

Contribution

It introduces a method to obtain exact solutions of both linear and nonlinear GS equations with flow, expanding analytical tools for plasma equilibrium modeling.

Findings

Exact solutions for linear GS equation with flow

Analytical solutions for nonlinear GS equations

Force-free solutions in linear and nonlinear regimes

Abstract

We derive exact solutions of a linear form of the Grad-Shafranov (GS) equation, including incompressible equilibrium flow, using ansatz-based similarity reduction methods. The linearity of the equilibrium equation allows linear combinations of solutions in order to obtain axisymmetric MHD equilibria with closed and nested magnetic surfaces which are favorable for the effective confinement of laboratory plasmas. In addition, employing the same reduction methods we obtain analytical solutions for several non-linear forms of the GS equation. In this context analytic force-free solutions in both linear and nonlinear regimes are also derived.