MHD equilibrium: Difference between revisions

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The static, ideal Magneto-HydroDynamic (MHD) equilibrium of a near-Maxwellian magnetically confined plasma is obtained by solving the force balance equation
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 surface'', which is a surface such that ''B'' is everywhere perpendicular to its normal.
== Flux surfaces ==
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.
 
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 &nabla;''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 two dimensions (assuming axisymmetry), the force balance equation reduces to the  
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:Grad-Shafranov equation|Grad-Shafranov equation]].
<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.


In three dimensions, the existence of flux surfaces (nested or not) is not guaranteed.
== Numerical codes ==
<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 [[Magnetic shear|rotational transform]] of the field line on the surface may either be irrational and cover the surface entirely, or rational. In the latter case, the surface is sensitive to small perturbations and (assuming non-zero resistivity) may break up to form ''magnetic islands'' and ''stochastic regions''. In this respect, the [[:Wikipedia:Kolmogorov-Arnold-Moser_theorem|KAM theorem]] for Hamiltonian systems is relevant.


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]]
Line 32: Line 43:
* [[HBT]]
* [[HBT]]


== 3-D codes ==
=== 3-D codes ===


* [[VMEC]] (nested flux surfaces)
* [[VMEC]] (nested flux surfaces)
Line 41: Line 52:
* [[SIESTA]] (islands, fixed boundary)
* [[SIESTA]] (islands, fixed boundary)
* [[BETA]] (finite difference)
* [[BETA]] (finite difference)
== See also ==
* [[Flux coordinates]]


== References ==
== References ==
<references />
<references />

Revision as of 11:35, 26 January 2023

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]

where B is the magnetic field (divergence-free) and

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.

2-D codes

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)

See also

References

  1. J.P. Freidberg, Plasma physics and fusion energy, Cambridge University Press (2007) ISBN 0521851076
  2. R.D. Hazeltine, J.D. Meiss, Plasma Confinement, Courier Dover Publications (2003) ISBN 0486432424
  3. H. Grad, Toroidal Containment of a Plasma, Phys. Fluids 10 (1967) 137