Free-Boundary Equilibrium Codes
Free-boundary solvers compute the plasma shape and position self-consistently with external coil currents rather than prescribing the boundary.
What free-boundary means
A tokamak equilibrium satisfies the Grad-Shafranov equation, a nonlinear elliptic partial differential equation for the poloidal flux function psi. In a free-boundary calculation the plasma-vacuum interface is not given in advance. Instead the code solves for psi over the whole machine cross-section, including the region occupied by poloidal field coils, and the last closed flux surface emerges where the plasma pressure vanishes and the field is set by the coil currents and plasma current together.
This contrasts with a fixed-boundary code, which prescribes the boundary shape and solves only the interior. Free-boundary is essential when the question is about coil currents, vertical stability, X-point formation, strike-point location, or how a control system must act to hold a shape.
The governing equation
In cylindrical coordinates (R,Z) the Grad-Shafranov operator is applied to psi, with a right-hand side built from p'(psi) and FF'(psi), the pressure and poloidal-current profiles. The source term is nonlinear because it depends on psi through the profile functions, so the solver iterates: guess psi, evaluate the source, solve the linear elliptic problem, update, repeat until convergence.
Coils, currents, and constraints
Coil currents contribute Green's-function terms to psi. A free-boundary code either takes coil currents as inputs (forward problem) or solves for the currents that produce a desired shape (inverse problem). The inverse mode underlies scenario design: specify the plasma boundary, elongation, triangularity, and X-point, and the code returns the coil currents that realize them.
Why it matters for design
For the Hyperion breeder, a strongly shaped spherical tokamak with negative triangularity of -0.30, free-boundary modeling checks that a physical set of poloidal field coils can hold that shape against the outward hoop force and vertical instability. The same tools feed vertical stability growth-rate estimates and the design of the control system, all of which are simulation studies for a machine whose construction begins Q2 2027.
- Solves for plasma shape and position given coils
- Handles X-points, divertor legs, strike points
- Supports forward and inverse (shape-target) modes
- Feeds vertical stability and control-system design