The Grad-Shafranov Equation
The nonlinear elliptic PDE that sets the equilibrium of an axisymmetric magnetically confined plasma.
What it describes
The Grad-Shafranov equation gives the magnetohydrodynamic force balance of an axisymmetric toroidal plasma. It reduces the vector equilibrium condition to a single scalar equation for the poloidal flux function psi(R,Z), where R is major radius and Z is height. Every tokamak equilibrium reconstruction begins here.
In cylindrical coordinates the equation reads R d/dR( (1/R) dpsi/dR ) + d2psi/dZ2 = -mu0 R^2 dp/dpsi - F dF/dpsi, where p(psi) is the pressure profile and F(psi) = R B_toroidal is the poloidal current function. Both source terms are free functions of psi, which is what makes the problem nonlinear.
Why it is elliptic
The left side is the Shafranov operator, a second-order elliptic operator similar to a modified Laplacian. Because the right side depends on psi through p and F, the equation is self-consistent: the flux determines the currents, which in turn determine the flux.
How it is solved numerically
The standard approach is Picard iteration on a fixed rectangular (R,Z) grid. One guesses psi, evaluates the right-hand-side source, solves the linear elliptic problem with a fast Poisson solver (cyclic reduction or multigrid), then updates psi and repeats until the free boundary and profiles converge. The plasma boundary is found as a psi contour, either limiter-defined or by locating the X-point of a divertor.
- Fixed-boundary solvers prescribe the last closed flux surface and solve inside it
- Free-boundary solvers add external coil currents and solve the whole domain
- Codes such as EFIT reconstruct equilibria by fitting to magnetic diagnostics
For the Hyperion breeder, a spherical tokamak with negative triangularity -0.30 and strong shaping, Grad-Shafranov solutions with the Shafranov shift give the starting point for stability and transport analysis. The low aspect ratio makes the R-dependence in the operator especially important.