Class: equilibrium

The equilibrium class is responsible for computing magnetohydrodynamic (MHD) equilibrium solutions based on the Grad–Shafranov equation, and for extracting critical physics profiles and geometric structures from the result.

Purpose

This class defines and solves the axisymmetric equilibrium of a tokamak, including:

  • Construction of the poloidal magnetic flux (ψ)
  • Extraction of the magnetic field and current components
  • Identification of O- and X-points
  • Post-processing of the Last Closed Flux Surface (LCFS)
  • Evaluation of pressure, magnetic (F²) and kinetic profiles

Properties

Imported Dependencies

  • geo: Geometry class instance with grid and wall info.
  • config: Configuration parameters, including solver and plasma settings.
  • separatrix: Defines the target separatrix geometry.
  • utils: Utility class for differential operators.
  • toroidal_curr: Class with toroidal current calculation methods.
  • const: Structure containing physical constants.
  • MHD_prof: Class for evaluating MHD pressure and F² from ψ.
  • kin_prof: Class for computing kinetic quantities like ne, Te, etc.

Computed Fields

  • psi: Poloidal magnetic flux per unit toroidal angle (Wb / rad).
  • Psi: Total poloidal magnetic flux (Wb).
  • psi_n: Normalised poloidal flux.

Current Densities

  • Jt: Toroidal current density.
  • Jr: Radial current density.
  • Jz: Vertical current density.

Magnetic Fields

  • Bt: Toroidal magnetic field.
  • Br: Radial magnetic field.
  • Bz: Vertical magnetic field.

Pressure and Magnetic Function

  • p: Plasma pressure.
  • F2: Squared poloidal current function (F² = R²·Bt²).

Kinetic Profiles

  • ne, ni: Electron and ion densities.
  • Te, Ti: Electron and ion temperatures.
  • pe, pi: Electron and ion pressures.

Topology

  • Opoint: O-point (magnetic axis) coordinates.
  • Xpoint: X-point coordinates.
  • LCFS: Last Closed Flux Surface (structure with shape and mask).

Methods

import_configuration(geo, config)

Sets up the geometry and configuration for the equilibrium solver, and initializes default solver parameters.

import_classes()

Imports all required helper classes used in solving the equilibrium.

solve_equilibrium(psi)

Solves the Grad–Shafranov equation either from a first guess or a given initial ψ.

  • Iteratively updates the solution using the toroidal current and Grad–Shafranov operator.
  • Applies separatrix and boundary conditions.
  • Tracks convergence and visualizes iteration progress.

compute_profiles()

Computes the full set of physics quantities after equilibrium solution:

  • Evaluates p, F², Bt, Br, Bz, Jr, Jz
  • Computes kinetic quantities: ne, ni, Te, Ti, pe, pi

CriticalPoints(Ip, R, inside_wall, Psi)

Detects the O- and X-points by locating extrema in the ψ gradient within the domain bounded by the wall.

Returns:

  • Opoint: Index of magnetic axis
  • Xpoint: Index of the X-point

equi_pp()

Post-processes the final solution:

  • Interpolates to high-resolution grid
  • Recomputes O/X-points
  • Re-normalizes ψ (ψₙ)
  • Identifies the Last Closed Flux Surface (LCFS)

⚠️ This documentation is still a work in progress. There may be errors or inaccuracies. Please feel free to contact us if you notice any issues.