Fixed-Boundary Equilibrium Codes
When the plasma shape is prescribed, fixed-boundary solvers compute the interior equilibrium accurately and efficiently for stability and transport studies.
The prescribed-boundary approach
A fixed-boundary code takes the last closed flux surface as a given curve and solves the Grad-Shafranov equation only inside it. Removing the coupling to external coils makes the problem cleaner and faster, and lets analysts specify exactly the shape they want to study.
Flux coordinates
Many fixed-boundary codes work in flux coordinates rather than a Cartesian grid. They map the plasma interior onto surfaces of constant psi, producing a straight-field-line coordinate system in which subsequent stability and transport calculations are far more tractable. Codes in this family solve for the inverse map R(psi,theta), Z(psi,theta).
Inputs and outputs
- Inputs: boundary shape, pressure profile, and a current-related profile (safety factor q or the current function)
- Outputs: high-accuracy flux surfaces, metric coefficients, q profile, magnetic well, and quantities needed by stability codes
Why accuracy matters here
Ideal-MHD stability calculations are sensitive to fine features of the equilibrium, such as the exact shear and the local magnetic curvature. A fixed-boundary solver in flux coordinates delivers the smooth, high-resolution equilibrium that downstream stability codes require, avoiding numerical noise that a coarse Cartesian grid would introduce.
Typical workflow
A common chain is: reconstruct or design an equilibrium, refine it in a fixed-boundary flux-coordinate solver, then hand the metric coefficients to a stability or gyrokinetic code. Each stage demands a self-consistent equilibrium, so the fixed-boundary step is a routine building block of integrated analysis.
For the Hyperion breeder, fixed-boundary solves at the design shape feed both stability screening and turbulent-transport estimates during design iteration.