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MHD equilibrium: Difference between revisions
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The static, single-fluid, 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> | <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> | ||
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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> | ||
== Flux surfaces == | == Flux surfaces == | ||
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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 (as opposed to the ''effectively'' two-dimensional axisymmetric situation), the existence of flux surfaces (nested or not) is not guaranteed. | 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> | <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 [[ | 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. | 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). | 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. | 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 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. | 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. | 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. | Nevertheless, ideal MHD equilibrium is extremely useful for the description and understanding of magnetically confined plasmas. | ||
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== Numerical codes == | == Numerical codes == | ||
In two dimensions (assuming [[ | In two dimensions (assuming [[axisymmetry]]), the force balance equation reduces to the | ||
[[:Wikipedia:Grad-Shafranov equation|Grad-Shafranov equation]]. | [[: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. | ||
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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 /> | ||