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 B is the magnetic field (divergence-free) and

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

is the plasma current, subject to appropriate boundary conditions. The word "static" refers to the assumption of zero flow, while "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)