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Toroidal coordinates: Difference between revisions
→Magnetic coordinates
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== Magnetic coordinates == | == Magnetic coordinates == | ||
Magnetic coordinates are a particular type of flux coordinates in which the magnetic field lines are straight lines. In mathematical terms this implies that the periodic part of the magnetic field's stream function is zero in these coordinates so the magnetic field reads | Magnetic coordinates are a particular type of flux coordinates in which the magnetic field lines are straight lines. In mathematical terms this implies that the periodic part of the magnetic field's stream function is zero in these coordinates so the magnetic field reads | ||
: | :<math> | ||
\mathbf{B} = \nabla\psi\times \left( \frac{\Psi_{tor}'}{2\pi}\theta_f | |||
- \frac{\Psi_{pol}'}{2\pi}\phi_f \right) | |||
</math> | |||
Magnetic coordinates adapt to the magnetic field, and therefore to the [[MHD equilibrium]] (also see [[Flux surface]]). | Magnetic coordinates adapt to the magnetic field, and therefore to the [[MHD equilibrium]] (also see [[Flux surface]]). | ||
Magnetic coordinates simplify the description of the magnetic field. | Magnetic coordinates simplify the description of the magnetic field. | ||
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<ref name='Dhaeseleer'>W.D. D'haeseleer, ''Flux coordinates and magnetic field structure: a guide to a fundamental tool of plasma theory'', Springer series in computational physics, Springer-Verlag (1991) ISBN 3540524193</ref> | <ref name='Dhaeseleer'>W.D. D'haeseleer, ''Flux coordinates and magnetic field structure: a guide to a fundamental tool of plasma theory'', Springer series in computational physics, Springer-Verlag (1991) ISBN 3540524193</ref> | ||
* Hamada coordinates. <ref>S. Hamada, Nucl. Fusion '''2''' (1962) 23</ref><ref>[http://dx.doi.org/10.1063/1.1706651 J.M. Greene and J.L Johnson, ''Stability Criterion for Arbitrary Hydromagnetic Equilibria'', Phys. Fluids '''5''' (1962) 510]</ref> In these coordinates, both the field lines and current lines corresponding to the [[MHD equilibrium]] are straight. | * Hamada coordinates. <ref>S. Hamada, Nucl. Fusion '''2''' (1962) 23</ref><ref>[http://dx.doi.org/10.1063/1.1706651 J.M. Greene and J.L Johnson, ''Stability Criterion for Arbitrary Hydromagnetic Equilibria'', Phys. Fluids '''5''' (1962) 510]</ref> In these coordinates, both the field lines and current lines corresponding to the [[MHD equilibrium]] are straight. | ||
* Boozer coordinates. <ref>[http://dx.doi.org/10.1063/1.863297 A.H. Boozer, ''Plasma equilibrium with rational magnetic surfaces'', Phys. Fluids '''24''' (1981) 1999]</ref><ref>[http://dx.doi.org/10.1063/1.863765 A.H. Boozer, ''Establishment of magnetic coordinates for a given magnetic field'', Phys. Fluids '''25''' (1982) 520]</ref> In these coordinates, the field lines corresponding to the [[MHD equilibrium]] are straight. | * Boozer coordinates. <ref>[http://dx.doi.org/10.1063/1.863297 A.H. Boozer, ''Plasma equilibrium with rational magnetic surfaces'', Phys. Fluids '''24''' (1981) 1999]</ref><ref>[http://dx.doi.org/10.1063/1.863765 A.H. Boozer, ''Establishment of magnetic coordinates for a given magnetic field'', Phys. Fluids '''25''' (1982) 520]</ref> In these coordinates, the field lines corresponding to the [[MHD equilibrium]] are straight and so are the ''diamagnetic lines '', i.e. the integral lines of <math>\nabla\psi\times\mathbf{B}</math>. | ||
These two coordinate systems are related. | These two coordinate systems are related. | ||
<ref>K. Miyamoto, ''Controlled fusion and plasma physics'', Vol. 21 of Series in | <ref>K. Miyamoto, ''Controlled fusion and plasma physics'', Vol. 21 of Series in | ||