The fundamental partial differential equation describing axisymmetric MHD equilibrium in tokamaks — the mathematical backbone of every toroidal confinement design.
The Grad-Shafranov (GS) equation is a second-order, nonlinear, elliptic partial differential equation for the poloidal magnetic flux function ψ(R, Z) in an axisymmetric toroidal plasma. It was derived independently by Harold Grad and Hyman Rubin in the United States (1958) and by Vitaly Shafranov in the Soviet Union (1957). The equation reads:
Starting from the ideal MHD force balance j × B = ∇p and imposing axisymmetry (∂/∂φ = 0), one shows that both p and F are functions of ψ alone — they are constant on each flux surface. Substituting the expressions for the magnetic field components in terms of ψ into Ampère's law yields the GS equation. The two free functions p(ψ) and F(ψ) must be specified as inputs or determined self-consistently from transport models.[2]
Because the GS equation is nonlinear (the source terms depend on ψ, which is the unknown), it is solved iteratively. Two broad classes of solution exist:
Fixed-boundary solvers prescribe the plasma boundary shape and find the internal flux distribution. These are used for scoping studies and stability analysis. Free-boundary solvers compute ψ throughout the entire domain, including vacuum regions, and self-consistently determine the plasma shape produced by a given set of external coil currents. Free-boundary codes — such as EFIT, CORSICA, and FREEGS — are indispensable for machine design and real-time plasma control.[3]
The GS equation encodes the relationship between plasma shape, pressure, current, and the external magnetic field. Every major design parameter — elongation, triangularity, Shafranov shift, safety factor profile, and bootstrap current fraction — emerges from or feeds into GS solutions. Accurate, fast GS solvers are a prerequisite for real-time shape control in burning-plasma experiments such as ITER.