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MHD equilibrium: Difference between revisions
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The static, ideal Magneto- | 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> | |||
:<math>\vec \nabla p = \vec j \times \vec B</math> | :<math>\vec \nabla p = \vec j \times \vec B</math> | ||
| Line 11: | Line 12: | ||
Here, the pressure ''p'' is assumed to be isotropic, but a generalization | Here, the pressure ''p'' is assumed to be isotropic, but a generalization | ||
for non-isotropic pressure is possible. | for non-isotropic pressure is possible. | ||
<ref>R.D. Hazeltine, J.D. Meiss, ''Plasma Confinement'', Courier Dover Publications (2003) ISBN 0486432424</ref> | <ref>R.D. Hazeltine, J.D. Meiss, ''Plasma Confinement'', Courier Dover Publications (2003) {{ISBN|0486432424}}</ref> | ||
An important concept in this context is the | == Flux surfaces == | ||
The force balance equation implies that ''p'' is constant along any field line (since | |||
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). | The force balance equation also implies that the surface ''p'' = constant is a flux surface (assuming flux surfaces exist). | ||
In | In three dimensions (as opposed to the ''effectively'' two-dimensional [[axisymmetry|axisymmetric]] situation), the existence of flux surfaces (nested or not) is not guaranteed. | ||
[[:Wikipedia: | <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. | A large number of codes is available to evaluate MHD equilibria. | ||
== 2-D codes == | === 2-D codes === | ||
* [[EFIT]] | * [[EFIT]] | ||
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* [[HBT]] | * [[HBT]] | ||
== 3-D codes == | === 3-D codes === | ||
* [[VMEC]] (nested flux surfaces) | * [[VMEC]] (nested flux surfaces) | ||
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* [[SIESTA]] (islands, fixed boundary) | * [[SIESTA]] (islands, fixed boundary) | ||
* [[BETA]] (finite difference) | * [[BETA]] (finite difference) | ||
== See also == | |||
* [[Flux coordinates]] | |||
== References == | == References == | ||
<references /> | <references /> | ||