General Grad-Shafranov equation
Phys. Rev. D 113, 104077 – Published 29 May, 2026
DOI: https://doi.org/10.1103/pk8r-rfnv
Abstract
To effectively describe the plasma with strong magnetic field, the force-free electrodynamics was introduced, within which the Grad-Shafranov equation plays the key role. The Grad-Shafranov equation governs the global structure of an electromagnetic field in equilibrium with symmetries. It is widely applicable in a variety of scenarios, such as the tokamak, the solar corona, the magnetosphere of Earth, neutron stars, and black holes, etc. However, in different situations, the Grad-Shafranov equation is expressed differently, and the derivations might be complicated. In this work, via the language of differential form, we provide a general expression of the Grad-Shafranov equation, from which the expression in any specific situation can be quickly obtained. Additionally, we present a Lagrangian density for a scalar field whose on-shell condition is precisely the Grad-Shafranov equation.
