MHD Equilibrium and Force Balance

The static balance grad p = J x B that every confined plasma must satisfy at each instant.

The equilibrium condition

On the slow confinement timescale a plasma is in mechanical equilibrium: inertia is negligible, so the pressure-gradient force is balanced everywhere by the magnetic force:

grad p = J x B

This deceptively simple equation is the foundation of magnetic confinement. It says magnetic forces hold the plasma pressure. Taking B dotted with it shows B . grad p = 0, so pressure is constant along field lines; taking J dotted with it shows current also lies in surfaces of constant pressure.

Flux surfaces emerge

Because both B and J lie in surfaces of constant pressure, those surfaces are the nested magnetic flux surfaces. Equilibrium therefore automatically organizes the plasma into the nested-surface structure that makes toroidal confinement work. In axisymmetry this equation becomes the Grad-Shafranov equation.

The Shafranov shift

• Finite pressure pushes the flux surfaces outward in major radius, the Shafranov shift
• Higher beta gives a larger shift, compressing surfaces on the outboard side
• The shift changes the field-line geometry and hence stability and transport

How it is solved numerically

In toroidal axisymmetry the equilibrium is solved via Grad-Shafranov (Picard iteration on a flux grid). In three dimensions (stellarators, or tokamaks with fields that break symmetry) variational energy-minimization codes find equilibria assuming nested surfaces, or extended methods allow islands and stochastic regions.

Its role

Every stability and transport calculation starts from an equilibrium. Force balance defines the operating point, the beta, and the geometry within which instabilities and turbulence are then evaluated. For the Hyperion breeder, the equilibrium with its negative-triangularity shaping and Shafranov shift is the basis for all subsequent physics analysis.