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Grad–Shafranov Equation

\[\Delta^* \psi + \mu_0 R^2 \frac{dp}{d\psi} + F \frac{dF}{d\psi} = 0\]

Source lecture(s): pc368_lec14_mhd_equilibrium

Physical Meaning

The Grad–Shafranov equation is the fundamental PDE of axisymmetric MHD equilibrium. It relates the poloidal flux \(\psi\) to the pressure profile \(p(\psi)\) and the toroidal field function \(F(\psi) = R B_\phi\).

Variable Definitions

Symbol Definition SI Units
\(\psi(R,Z)\) Poloidal magnetic flux Wb
\(p(\psi)\) Plasma pressure Pa
\(F(\psi) = R B_\phi\) Toroidal field stream function T·m
\(R\) Major radius m
\(\mu_0\) Vacuum permeability N A\(^{-2}\)

Assumptions

  • Axisymmetry: \(\partial/\partial \phi = 0\).
  • Static or steady-state equilibrium.
  • Toroidal current flows in flux surfaces.

Derivation

  1. Begin with \(\mathbf{J}\times\mathbf{B} = \nabla p\) and \(\mathbf{B} = \nabla\times\mathbf{A}\).
  2. In axisymmetry, \(\mathbf{A} = \psi(R,Z) \nabla\phi\).
  3. Express \(\mathbf{B}\) and \(\mathbf{J}\) in terms of \(\psi\) and \(F\).
  4. Substitute into force balance and separate poloidal/radial components.
  5. The resulting equation is the Grad–Shafranov equation above.

Applications

  • Tokamak design: Equilibrium solvers (e.g., EFIT) use this equation.
  • Stellarators: Flux-surface optimization.
  • Z-pinch and spheromak: Axisymmetric equilibria.

Connections to Other Equations