Grad-Shafranov Equation for Control
The force-balance equation that underlies equilibrium reconstruction, and why its structure shapes real-time control.
Force balance in a torus
An axisymmetric plasma in equilibrium satisfies the Grad-Shafranov equation, a nonlinear elliptic partial differential equation for the poloidal flux function psi. It expresses that the pressure gradient is balanced by the magnetic force: the flux is set by two free functions of psi - the pressure p(psi) and the toroidal field function F(psi) - together with boundary conditions from the external coils.
The equation in words
A second-order elliptic operator acting on psi equals a source term built from the derivatives of p(psi) and F(psi), scaled by major radius. Given the two profile functions and the coil currents, the equation has a unique flux solution. Reconstruction runs this backward: it adjusts the free functions until the resulting flux matches measurements.
Why the structure helps control
Because the equation is linear in the flux for fixed profile functions, much of the coil-to-flux mapping can be precomputed as response matrices. Real-time control exploits this: the influence of each coil on each control point is a fixed matrix, and only the plasma's contribution must be re-estimated each cycle. This is what makes millisecond magnetic control tractable.
The under-determination
External magnetic measurements constrain the boundary and total current well, but constrain the internal profiles weakly. Two different internal current profiles can produce nearly the same external field. Control that only needs the boundary (shape, position) is well-served; control that needs internal profiles (the q profile) needs internal diagnostics to break the ambiguity.
Practical use
Real-time solvers use fixed grids, precomputed Green's functions for the coils, and a small number of profile parameters. The result is a fast, repeatable equilibrium that is good enough for control even if it would not satisfy a physicist doing detailed transport analysis. Control asks for consistency and speed more than ultimate fidelity.