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>
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''); it 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 two dimensions (assuming axisymmetry), the force balance equation reduces to the  
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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.
<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 [[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''.


A large number of codes is available to evaluate MHD equilibria.
A large number of codes is available to evaluate MHD equilibria.

Revision as of 07:50, 19 August 2009

The static, ideal Magneto-HydroDynamic (MHD) equilibrium of a near-Maxwellian magnetically confined plasma is obtained by solving the force balance equation

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. [1]

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); it 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 Grad-Shafranov equation.

In three dimensions, the existence of flux surfaces (nested or not) is not guaranteed. [2] Assuming an initial situation with nested magnetic surfaces, the 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.

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)

References

  1. R.D. Hazeltine, J.D. Meiss, Plasma Confinement, Courier Dover Publications (2003) ISBN 0486432424
  2. H. Grad, Toroidal Containment of a Plasma, Phys. Fluids 10 (1967) 137