Free-Boundary Equilibrium Solvers
Free-boundary codes solve the plasma equilibrium together with the external coil fields, letting the plasma shape and boundary emerge from the solution.
Fixed vs free boundary
A fixed-boundary solver assumes the plasma boundary shape is prescribed and solves the Grad-Shafranov equation only inside it. A free-boundary solver instead includes the external poloidal-field coils and conducting structures, so the boundary is determined self-consistently by the balance between plasma current and coil fields.
Coupling to coils
The vacuum field from each coil is computed with Green's functions for the axisymmetric elliptic operator. The plasma contributes its own field through the toroidal current distribution. The solver finds psi everywhere such that the interior satisfies Grad-Shafranov and the exterior matches the superposed coil fields, with the last closed flux surface defined by a limiter contact point or an X-point.
Uses
- Designing coil sets that produce a target plasma shape
- Predicting equilibria across a discharge as coil currents ramp
- Providing consistent flux maps for control-system and disruption studies
- Evolving the equilibrium in time when coupled to circuit equations
Time-dependent operation
Coupled to the coil-and-vessel circuit equations, a free-boundary solver becomes an evolution code: it steps the equilibrium forward as currents in the coils and induced currents in passive structures change. This is essential for simulating startup, shape control, and the response to transients.
Kronos context
The compact, high-elongation, negative-triangularity shape of the Hyperion breeder places demanding requirements on the poloidal-field coil set. Free-boundary solvers are the tool for checking that a candidate coil arrangement can hold the intended shape with realistic currents.
Because these solvers combine plasma and circuit physics, they sit at the boundary between equilibrium codes and control simulation, and are frequently embedded in larger integrated workflows.