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Toroidal coordinates: Difference between revisions
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Updated doi references and mathematical expressions
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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 | 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>R(r,\theta) = R_0(r) + r \cos(\theta + \arcsin \delta \sin \theta)\\ | ||
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== 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> | <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 == | ||
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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>[ | * [[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. | ||