The MHD force balance that determines the shape, position, and stability of a magnetically confined plasma — the starting point for every tokamak and stellarator design.
Plasma equilibrium refers to the steady-state condition in which the forces acting on a magnetically confined plasma are in balance. In the framework of ideal magnetohydrodynamics (MHD), equilibrium is reached when the outward pressure gradient of the plasma is exactly balanced by the inward Lorentz force produced by the interaction of plasma currents with the confining magnetic field. The governing equation is:
Every magnetic confinement fusion device must establish and sustain a well-defined equilibrium before high-performance operation is possible. The equilibrium determines the shape of the plasma cross-section (elongation, triangularity), the position of the magnetic axis (Shafranov shift), and the distribution of pressure and current density across the plasma. Departures from equilibrium can trigger disruptions — sudden losses of confinement that deposit enormous energy on plasma-facing components.[2]
In an axisymmetric equilibrium, the magnetic field lines trace out closed, nested toroidal surfaces called flux surfaces. Pressure is constant on each flux surface, and the equilibrium is fully described by specifying two free functions — the pressure profile p(ψ) and the poloidal current function F(ψ) — as functions of the poloidal magnetic flux ψ. This reduction leads directly to the Grad-Shafranov equation.[1]
In experiments, the equilibrium is not measured directly but is reconstructed from external magnetic diagnostics using codes such as EFIT (Equilibrium FITting). These codes solve the Grad-Shafranov equation iteratively, adjusting the free-function profiles until the computed external fields match the measurements. Accurate equilibrium reconstruction is essential for stability analysis, transport modeling, and performance optimization.[3]
For future reactors, free-boundary equilibrium codes (such as CORSICA, CREATE-NL, or FREEGS) compute the coil currents needed to produce a target plasma shape. The interplay between the poloidal field coil set and the desired equilibrium determines the engineering constraints on superconducting magnets, vacuum vessel geometry, and divertor placement.