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The static, single-fluid, [[Ideal Magneto-Hydrodynamics|ideal Magneto-Hydrodynamic]] (MHD) equilibrium of a near-Maxwellian magnetically confined plasma is obtained by solving the force balance equation | |||
<ref>J.P. Freidberg, ''Plasma physics and fusion energy'', Cambridge University Press (2007) {{ISBN|0521851076}}</ref> | |||
== 2-D codes == | :<math>\vec \nabla p = \vec j \times \vec B</math> | ||
where ''B'' is the magnetic field (divergence-free) and | |||
:<math>\mu_0 \vec j = \vec \nabla \times \vec B</math> | |||
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. | |||
<ref>R.D. Hazeltine, J.D. Meiss, ''Plasma Confinement'', Courier Dover Publications (2003) {{ISBN|0486432424}}</ref> | |||
== Flux surfaces == | |||
An important concept in this context is the [[Flux surface|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 [[axisymmetry|axisymmetric]] situation), the existence of flux surfaces (nested or not) is not guaranteed. | |||
<ref>[http://dx.doi.org/10.1063/1.1761965 H. Grad, ''Toroidal Containment of a Plasma'', Phys. Fluids '''10''' (1967) 137]</ref> | |||
Assuming an initial situation with nested magnetic surfaces, the [[Rotational transform|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 [[Plasma instability|instabilities]] may develop that lead to the formation of ''[[Magnetic island|magnetic islands]]'' and ''stochastic regions'' (assuming non-zero resistivity). | |||
Since the field line trajectories are described by Hamiltonian equations, the [[:Wikipedia:Kolmogorov-Arnold-Moser_theorem|KAM theorem]] is relevant. | |||
It should be noted that the force balance equation does not describe any detail on scales smaller than the [[Larmor radius|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 | |||
[[:Wikipedia:Grad-Shafranov equation|Grad-Shafranov equation]]. | |||
A large number of codes is available to evaluate MHD equilibria. | |||
=== 2-D codes === | |||
* [[EFIT]] | * [[EFIT]] | ||
* [[FBT]] | |||
* [[HBT]] | |||
== 3-D codes == | === 3-D codes === | ||
* [[VMEC]] (nested flux surfaces) | * [[VMEC]] (nested flux surfaces) | ||
* [[NEAR]] (nested flux surfaces) | |||
* [[IPEC]] (nested flux surfaces) | * [[IPEC]] (nested flux surfaces) | ||
* [[HINT | * [[HINT]] (islands) | ||
* [[PIES]] | * [[PIES]] (islands) | ||
* [[ | * [[SIESTA]] (islands, fixed boundary) | ||
* [[BETA]] (finite difference) | |||
== See also == | |||
* [[Flux coordinates]] | |||
== References == | == References == | ||
<references /> | <references /> |