# MHD equilibrium

The static, single-fluid, ideal Magneto-Hydrodynamic (MHD) equilibrium of a near-Maxwellian magnetically confined plasma is obtained by solving the force balance equation [1]

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

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. [2]

## Flux surfaces

An important concept in this context is the flux surface, which is a surface such that B is everywhere perpendicular to its normal. The force balance equation implies that p is constant along any field line (since ∇p is perpendicular to B), which is an expression of the underlying assumption that transport along the magnetic field lines is much faster than transport perpendicular to it. The force balance equation also implies that the surface p = constant is a flux surface (assuming flux surfaces exist).

In three dimensions (as opposed to the effectively two-dimensional axisymmetric situation), the existence of flux surfaces (nested or not) is not guaranteed. [3] Assuming an initial situation with nested magnetic surfaces, the rotational transform of the field line on the surface may either be irrational so that the field line covers the surface entirely (ergodically), or rational. In the latter case, the field line does not cover a surface but constitutes a one-dimensional structure. Physically, a rational surface is sensitive to small perturbations and flute-like instabilities may develop that lead to the formation of magnetic islands and stochastic regions (assuming non-zero resistivity). Since the field line trajectories are described by Hamiltonian equations, the KAM theorem is relevant.

It should be noted that the force balance equation does not describe any detail on scales smaller than the gyroradius. In combination with the existence of stochastic field regions this means that the concept of flux surface can only be approximate and not exact. Furthermore, the force balance equation depends on a number of assumptions, such as that of static equilibrium, whereas fusion-grade plasmas are clearly strongly driven systems far from equilibrium. Nevertheless, ideal MHD equilibrium is extremely useful for the description and understanding of magnetically confined plasmas.

## Numerical codes

In two dimensions (assuming axisymmetry), the force balance equation reduces to the Grad-Shafranov equation. A large number of codes is available to evaluate MHD equilibria.

### 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)