A semi-analytical solver for the Grad-Shafranov equation

TL;DR

This paper introduces a semi-analytical method to solve the linearized Grad-Shafranov equation for plasma equilibrium, enabling efficient modeling of various magnetic configurations in toroidal plasmas.

Contribution

It develops a semi-analytical solver that relates eigenvalues to physical parameters and uses nonlinear fitting to find equilibrium solutions with specified plasma shapes and current profiles.

Findings

Accurately reproduces experimental plasma parameters.

Handles reversed magnetic shear configurations.

Provides a computationally efficient equilibrium modeling approach.

Abstract

In toroidally confined plasmas, the Grad-Shafranov equation, in general a non-linear PDE, describes the hydromagnetic equilibrium of the system. This equation becomes linear when the kinetic pressure is proportional to the poloidal magnetic flux and the squared poloidal current is a quadratic function of it. In this work, the eigenvalue of the associated homogeneous equation is related with the safety factor on the magnetic axis, the plasma beta and the Shafranov shift, then, the adjustable parameters of the particular solution are bounded through physical constrains. The poloidal magnetic flux becomes a linear superposition of independent solutions and its parameters are adjusted with a non-linear fitting algorithm. This method is used to find hydromagnetic equilibria with normal and reversed magnetic shear and defined values of the elongation, triangularity, aspect-ratio, and X-point(s). The resultant toroidal and poloidal beta, the safety factor at the $95\%$ flux surface and the plasma current are in agreement with usual experimental values for high beta discharges and the model can be used locally to describe reversed magnetic shear equilibria.