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MHD equilibrium: Difference between revisions
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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> | ||
== 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. | 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 grad(''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 implies that ''p'' is constant along any field line (since grad(''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 three dimensions, the existence of flux surfaces (nested or not) is not guaranteed. | In three dimensions, the existence of flux surfaces (nested or not) is not guaranteed. | ||
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Since the field line trajectories are described by Hamiltonian equations, the [[:Wikipedia:Kolmogorov-Arnold-Moser_theorem|KAM theorem]] is relevant; however, it should be noted that the force balance equation does not describe any detail on scales smaller than the gyroradius. | Since the field line trajectories are described by Hamiltonian equations, the [[:Wikipedia:Kolmogorov-Arnold-Moser_theorem|KAM theorem]] is relevant; however, it should be noted that the force balance equation does not describe any detail on scales smaller than the gyroradius. | ||
== 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) | ||