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Toroidal coordinates: Difference between revisions
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In order to adapt this simple system better to the [[Flux surface|magnetic surfaces]] of an axisymmetric [[MHD equilibrium]], it may be enhanced by | In order to adapt this simple system better to the [[Flux surface|magnetic surfaces]] of an axisymmetric [[MHD equilibrium]], it may be enhanced by | ||
<ref>R.L. Miller et al, ''Noncircular, finite aspect ratio, local equilibrium model'', [[doi:10.1063/1.872666|Phys. Plasmas '''5''' (1998) 973]]</ref> | <ref>R.L. Miller et al, ''Noncircular, finite aspect ratio, local equilibrium model'', [[doi:10.1063/1.872666|Phys. Plasmas '''5''' (1998) 973]]</ref> | ||
* letting <math>R_0/a</math> depend on <math>r</math> (to account for the [[Shafranov shift]] of flux surfaces) <ref>R.D. Hazeltine, J.D. Meiss, ''Plasma confinement'', Courier Dover Publications (2003) ISBN 0486432424</ref> | * letting <math>R_0/a</math> depend on <math>r</math> (to account for the [[Shafranov shift]] of flux surfaces) <ref>R.D. Hazeltine, J.D. Meiss, ''Plasma confinement'', Courier Dover Publications (2003) {{ISBN|0486432424}}</ref> | ||
* adding [[ellipticity]] (<math>\kappa</math>), [[triangularity]] (<math>\delta</math>), and squareness (<math>\zeta</math>) to account for non-circular flux surface cross sections. A popular simple expression for shaped flux surfaces is: <ref> R.L. Miller, M.S. Chu, J.M. Greene, Y.R. Lin-Liu and R.E. Waltz, ''Noncircular, finite aspect ratio, local equilibrium model'', [[doi:10.1063/1.872666|Phys. Plasmas '''5''' (1998) 973]]</ref> | * adding [[ellipticity]] (<math>\kappa</math>), [[triangularity]] (<math>\delta</math>), and squareness (<math>\zeta</math>) to account for non-circular flux surface cross sections. A popular simple expression for shaped flux surfaces is: <ref> R.L. Miller, M.S. Chu, J.M. Greene, Y.R. Lin-Liu and R.E. Waltz, ''Noncircular, finite aspect ratio, local equilibrium model'', [[doi:10.1063/1.872666|Phys. Plasmas '''5''' (1998) 973]]</ref> | ||
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<math>(\zeta, \eta, \phi)</math>, where | <math>(\zeta, \eta, \phi)</math>, where | ||
<ref>Morse and Feshbach, ''Methods of theoretical physics'', McGraw-Hill, New York, 1953 ISBN 007043316X</ref> | <ref>Morse and Feshbach, ''Methods of theoretical physics'', McGraw-Hill, New York, 1953 {{ISBN|007043316X}}</ref> | ||
<ref>[[:Wikipedia:Toroidal coordinates]]</ref> | <ref>[[:Wikipedia:Toroidal coordinates]]</ref> | ||
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Magnetic coordinates simplify the description of the magnetic field. | Magnetic coordinates simplify the description of the magnetic field. | ||
In 3 dimensions (not assuming axisymmetry), the most commonly used coordinate systems are: | In 3 dimensions (not assuming axisymmetry), the most commonly used 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, [[doi:10.1088/0029-5515/2/1-2/005|Nucl. Fusion '''2''' (1962) 23]]</ref><ref>J.M. Greene and J.L Johnson, ''Stability Criterion for Arbitrary Hydromagnetic Equilibria'', [[doi:10.1063/1.1706651|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, [[doi:10.1088/0029-5515/2/1-2/005|Nucl. Fusion '''2''' (1962) 23]]</ref><ref>J.M. Greene and J.L Johnson, ''Stability Criterion for Arbitrary Hydromagnetic Equilibria'', [[doi:10.1063/1.1706651|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>A.H. Boozer, ''Plasma equilibrium with rational magnetic surfaces'', [[doi:10.1063/1.863297|Phys. Fluids '''24''' (1981) 1999]]</ref><ref>A.H. Boozer, ''Establishment of magnetic coordinates for a given magnetic field'', [[doi:10.1063/1.863765|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>. | * [[Boozer coordinates]]. <ref>A.H. Boozer, ''Plasma equilibrium with rational magnetic surfaces'', [[doi:10.1063/1.863297|Phys. Fluids '''24''' (1981) 1999]]</ref><ref>A.H. Boozer, ''Establishment of magnetic coordinates for a given magnetic field'', [[doi:10.1063/1.863765|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>. | ||
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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 | ||
Plasma Physics, CRC Press (2007) ISBN 1584887095</ref> | Plasma Physics, CRC Press (2007) {{ISBN|1584887095}}</ref> | ||
== References == | == References == | ||
<references /> | <references /> | ||