Toroidal coordinates: Difference between revisions

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Toroidal coordinates (view source)

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[[File:Toroidal coordinates.png|400px|thumb|right|A simple toroidal coordinate system]]
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Coordinate systems used in toroidal systems:
 
== Cartesian coordinates ==
 
(''X'', ''Y'', ''Z'')
<ref>[[:Wikipedia:Cartesian coordinate system]]</ref>
 
== Cylindrical coordinates ==
 
(''R'', ''&phi;'', ''Z''), where
<ref>[[:Wikipedia:Cylindrical coordinate system]]</ref>
* ''R<sup>2</sup> = X<sup>2</sup> + Y<sup>2</sup>'', and
* tan ''&phi;'' = ''Y/X''.
 
''&phi;'' is called the ''toroidal angle'' (and not the ''cylindrical'' angle, at least not in the context of magnetic confinement).
 
Cylindrical symmetry (symmetry under rotation over &phi;) is referred to as ''axisymmetry''.
 
== Simple toroidal coordinates ==
 
(''r'', ''&phi;'', ''&theta;''), where
* ''R'' = ''R<sub>0</sub>'' + ''r'' cos ''&theta;'', and
* ''Z'' = ''r'' sin ''&theta;''
''R<sub>0</sub>'' corresponds to the torus axis and is called the ''major radius'', while ''r'' is called the ''minor radius'', and ''&theta;'' the ''poloidal angle''.
 
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>[http://link.aip.org/link/?PHPAEN/5/973/1 R.L. Miller et al, ''Noncircular, finite aspect ratio, local equilibrium model'', Phys. Plasmas '''5''' (1998) 973]</ref>
* letting ''R<sub>0</sub>'' depend on ''r'' (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 (''&epsilon;''), triangularity (''&kappa;''), etc. (to account for non-circular flux surface cross sections)
 
== Toroidal coordinates ==
 
(''&zeta;'', ''&eta;'', ''&phi;''), where
<ref>Morse and Feshbach, ''Methods of theoretical physics'', McGraw-Hill, New York, 1953 ISBN 007043316X</ref>
<ref>[[:Wikipedia:Toroidal coordinates]]</ref>
 
:<math>R = R_p \frac{\sinh \zeta}{\cosh \zeta - \cos \eta}</math>
 
:<math>Z = R_p \frac{\sin \eta}{\cosh \zeta - \cos \eta}</math>
 
where ''R<sub>p</sub>'' is the pole of the coordinate system.
Surfaces of constant ''&zeta;'' are tori with major radii ''R = R<sub>p</sub>''/tanh ''&zeta;'' and minor radii ''r = R<sub>p</sub>''/sinh ''&zeta;''.
At ''R = R<sub>p</sub>'', ''&zeta;'' = &infin;, while at infinity and at ''R = 0, &zeta; = 0''.
The coordinate ''&eta;'' is a poloidal angle and runs from 0 to 2&pi;.
This system is orthogonal.
 
The Laplace equation separates in this system of coordinates, thus allowing an expansion of the vacuum magnetic field in toroidal harmonics.
<ref>F. Alladio, F. Crisanti, ''Analysis of MHD equilibria by toroidal multipolar expansions'', Nucl. Fusion '''26''' (1986) 1143</ref>
<ref>[http://dx.doi.org/10.1016/0010-4655(94)90112-0 B.Ph. van Milligen and A. Lopez Fraguas, ''Expansion of vacuum magnetic fields in toroidal harmonics'', Computer Physics Communications '''81''', Issues 1-2 (1994) 74-90]</ref>
 
== Magnetic coordinates ==
Magnetic coordinates are a particular type of [[flux coordinates]] in which the magnetic field lines are straight lines. 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.
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>
* [[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 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.
<ref>K. Miyamoto, ''Controlled fusion and plasma physics'', Vol. 21 of Series in
Plasma Physics, CRC Press (2007) ISBN 1584887095</ref>
 
== References ==
<references />
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