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Toroidal coordinates: Difference between revisions
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== Cylindrical coordinates == | == Cylindrical coordinates == | ||
( | <math>(R, \phi, Z)</math>, where | ||
<ref>[[:Wikipedia:Cylindrical coordinate system]]</ref> | <ref>[[:Wikipedia:Cylindrical coordinate system]]</ref> | ||
* | * <math>R^2 = X^2 + Y^2</math>, and | ||
* tan | * <math>\tan \phi = Y/X</math>. | ||
<math>\phi</math> 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 | Cylindrical symmetry (symmetry under rotation over <math>\phi</math>) is referred to as ''[[axisymmetry]]''. | ||
== Simple toroidal coordinates == | == Simple toroidal coordinates == | ||
( | <math>(r, \phi, \theta)</math>, where | ||
* | * <math>R = R_0 + r \cos \theta</math>, and | ||
* | * <math>Z = r \sin \theta</math> | ||
<math>R_0</math> corresponds to the torus axis and is called the ''major radius'', while <math>0 \le r \le a</math> is called the ''minor radius'', and ''θ'' the ''poloidal angle''. | |||
The ratio <math>R_0/a</math> is called the ''aspect ratio'' of the torus. | |||
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> | <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 | * 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 ( | * 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> | ||
:<math>R(r,\theta) = R_0(r) + r \cos(\theta + \arcsin \delta \sin \theta)</math> | |||
:<math>Z(r,\theta) = Z_0(r) + \kappa(r) r \sin(\theta + \zeta \sin 2 \theta) </math> | |||
Warning: there are varying conventions for the directions of <math>\theta</math> and <math>\phi</math>. Which convention is used can depend on the local facility, the software being used, or other context. To help reduce confusion, the different tokamak coordinate conventions have been described and codified in the COCOS system.<ref>O. Sauter and S.Yu. Medvedev, ''Tokamak coordinate conventions: COCOS'', [[doi:10.1016/j.cpc.2012.09.010|Computer Physics Communications '''184''', (2013) 293-302]]</ref> | |||
== Toroidal coordinates == | == Toroidal coordinates == | ||
( | <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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:<math>Z = R_p \frac{\sin \eta}{\cosh \zeta - \cos \eta}</math> | :<math>Z = R_p \frac{\sin \eta}{\cosh \zeta - \cos \eta}</math> | ||
where | where <math>R_p</math> is the pole of the coordinate system. | ||
Surfaces of constant | Surfaces of constant <math>\zeta</math> are tori with major radii <math>R = R_p/\tanh \zeta</math> and minor radii <math>r = R_p/\sinh \zeta</math>. | ||
At | At <math>R = R_p</math>, <math>\zeta = \infty</math>, while at infinity and at <math>R = 0, \zeta = 0</math>. | ||
The coordinate | The coordinate <math>\eta</math> is a poloidal angle and runs from 0 to <math>2\pi</math>. | ||
This system is orthogonal. | 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. | 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>F. Alladio, F. Crisanti, ''Analysis of MHD equilibria by toroidal multipolar expansions'', [[doi:10.1088/0029-5515/26/9/002|Nucl. Fusion '''26''' (1986) 1143]]</ref> | ||
<ref> | <ref>B.Ph. van Milligen and A. Lopez Fraguas, ''Expansion of vacuum magnetic fields in toroidal harmonics'', [[doi:10.1016/0010-4655(94)90112-0|Computer Physics Communications '''81''', Issues 1-2 (1994) 74-90]]</ref> | ||
== Magnetic coordinates == | == Magnetic coordinates == | ||
| Line 55: | Line 61: | ||
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, Nucl. Fusion '''2''' (1962) 23</ref><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. | ||
* [[Boozer coordinates]]. <ref> | * [[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>. | ||
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 /> | ||