MHD equilibrium

The static, ideal Magneto-HydroDynamic (MHD) equilibrium of a near-Maxwellian magnetically confined plasma is obtained by solving the force balance equation

${\vec {\nabla }}p={\vec {j}}\times {\vec {B}}$

where

$\mu _{0}{\vec {j}}={\vec {\nabla }}\times {\vec {B}}$

is the plasma current, subject to appropriate boundary conditions. The word "static" refers to the assumption of zero flow, while the word "ideal" refers to the absence of resistivity. Here, the pressure p is assumed to be isotropic, but a generalization for non-isotropic pressure is possible.

In two dimensions (assuming axisymmetry), the force balance equation reduces to the Grad-Shafranov equation.

In three dimensions, the existence of flux surfaces (nested or not) is not guaranteed.

A large number of codes is available to evaluate MHD equilibria.

2-D codes

• EFIT
• FBT
• HBT

3-D codes

• VMEC (nested flux surfaces)
• NEAR (nested flux surfaces)
• IPEC (nested flux surfaces)
• HINT (islands)
• PIES (islands)
• SIESTA (islands, fixed boundary)
• BETA (finite difference)