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Flux coordinates: Difference between revisions
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(Created page with '== General curvilinear coordinates == Here we briefly review the basic definitions of a general curvilinear coordinate system for later c…') |
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<math>\iota</math> being the [[rotational transform]]. | <math>\iota</math> being the [[rotational transform]]. | ||
It can be easily checked that any transformation of the angles of the from | |||
:<math> | :<math> | ||
\theta_F = \theta_f +\Psi_{pol}' G(\psi, \theta_f, \phi_f)\; ;\quad \phi_F = \phi_f +\Psi_{tor}' G(\psi, \theta_f, \phi_f) | \theta_F = \theta_f +\Psi_{pol}' G(\psi, \theta_f, \phi_f)\; ;\quad \phi_F = \phi_f +\Psi_{tor}' G(\psi, \theta_f, \phi_f) | ||
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where <math>G</math> is periodic in the angles, preserves the straightness of the field lines. The spatial function <math>G(\psi, \theta_f, \phi_f)</math>, is called the ''generating function''. | where <math>G</math> is periodic in the angles, preserves the straightness of the field lines. The spatial function <math>G(\psi, \theta_f, \phi_f)</math>, is called the ''generating function''. | ||
Particular choices of G can be that simplify the description of other fields. The most commonly used magnetic coordinate systems are: | |||
<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. Referring to the definitions above, both <math>\tilde\nu</math> and <math>\tilde\eta</math> are zero in Hamada coordinates. | * 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. Referring to the definitions above, both <math>\tilde\nu</math> and <math>\tilde\eta</math> are zero in Hamada coordinates. | ||
* 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>. Referring to the definitions above, both <math>\tilde\nu</math> and <math>\tilde\chi</math> are zero in Boozer coordinates. | * 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>. Referring to the definitions above, both <math>\tilde\nu</math> and <math>\tilde\chi</math> are zero in Boozer coordinates. | ||
== References == | == References == | ||
<references /> | <references /> | ||