Fixed-Boundary Equilibrium Codes
Fixed-boundary solvers prescribe the plasma edge shape and compute only the interior flux surfaces, trading realism for speed and precision.
The fixed-boundary problem
A fixed-boundary equilibrium code solves the Grad-Shafranov equation inside a prescribed last closed flux surface. The boundary shape is an input, often parameterized by elongation, triangularity, squareness, and a major and minor radius. Because the domain is fixed, the solver can use a flux-aligned coordinate mesh and reach very high accuracy in the interior profiles of pressure, safety factor, and current density.
This is the natural tool when the plasma shape is already decided and the interest is in the detailed internal structure: the q-profile, the magnetic shear, the bootstrap current fraction, or the metric coefficients needed by downstream transport and stability codes.
Coordinate systems
Many fixed-boundary codes use straight-field-line or Boozer-like coordinates so that field lines appear as straight lines on the flux-surface chart. This representation is convenient for gyrokinetic and MHD stability analysis, which require accurate geometric metrics on each surface.
Inverse and direct formulations
In the inverse formulation the code solves for R(psi,theta) and Z(psi,theta), the flux-surface geometry, given the boundary and the profile functions. In the direct formulation it solves for psi(R,Z) on a rectangular grid. Inverse solvers give smooth surface-averaged quantities directly and are favored as front ends for integrated modeling.
Where fixed-boundary fits
Fixed-boundary equilibria are the standard input for core transport studies because the boundary is held constant while the profiles evolve. For the Hyperion breeder they let designers scan pressure and current profiles at a locked negative-triangularity shape to map the accessible operating space in simulation before hardware exists.
- Boundary shape is an input, not an output
- High interior accuracy on flux-aligned meshes
- Provides metrics for stability and gyrokinetics
- Cannot answer coil-current or position-control questions