Toroidal coordinates - Revision history

Admin: /* Simple toroidal coordinates */

2023-12-12T10:13:31Z

Simple toroidal coordinates

← Older revision		Revision as of 12:13, 12 December 2023
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	:<math>Z(r,\theta) = Z_0(r) + \kappa(r) r \sin(\theta + \zeta \sin 2 \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 \theta and \phi. 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>		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 ==

Eldond: /* Simple toroidal coordinates */ add COCOS warning: the directions of theta and phi vary in different conventions

2023-03-27T16:53:33Z

Simple toroidal coordinates: add COCOS warning: the directions of theta and phi vary in different conventions

← Older revision		Revision as of 18:53, 27 March 2023
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	:<math>R(r,\theta) = R_0(r) + r \cos(\theta + \arcsin \delta \sin \theta)</math>		:<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>		:<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 \theta and \phi. 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 ==

Admin: /* Simple toroidal coordinates */

2023-01-31T08:38:02Z

Simple toroidal coordinates

← Older revision		Revision as of 10:38, 31 January 2023
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	* 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>

	:<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)</math>
	Z(r,\theta) = Z_0(r) + \kappa(r) r \sin(\theta + \zeta \sin 2 \theta) </math>		:<math>Z(r,\theta) = Z_0(r) + \kappa(r) r \sin(\theta + \zeta \sin 2 \theta) </math>

	== Toroidal coordinates ==		== Toroidal coordinates ==

Admin at 10:34, 26 January 2023

2023-01-26T10:34:17Z

Admin: Updated doi references and mathematical expressions

2015-04-17T08:23:41Z

Updated doi references and mathematical expressions

← Older revision		Revision as of 10:23, 17 April 2015
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	== Cylindrical coordinates ==		== Cylindrical coordinates ==

	(''R'', ~~''&~~phi~~;''~~, ''Z''), where		<math>(R, \phi, Z)</math>, where
	<ref>[[:Wikipedia:Cylindrical coordinate system]]</ref>		<ref>[[:Wikipedia:Cylindrical coordinate system]]</ref>
	* ~~''R~~<~~sup~~>2~~</sup>~~ = X~~<sup>~~2~~</sup>~~ + Y~~<sup>~~2</~~sup~~>'', and		* <math>R^2 = X^2 + Y^2</math>, and
	* tan ~~''&~~phi~~;''~~ = ''Y/X''.		* <math>\tan \phi = Y/X</math>.

	~~''&~~phi~~;''~~ is called the ''toroidal angle'' (and not the ''cylindrical'' angle, at least not in the context of magnetic confinement).		<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 φ) is referred to as ''[[axisymmetry]]''.		Cylindrical symmetry (symmetry under rotation over <math>\phi</math>) is referred to as ''[[axisymmetry]]''.

	== Simple toroidal coordinates ==		== Simple toroidal coordinates ==

	(''r'', ~~''&~~phi~~;''~~, ~~''&~~theta~~;''~~), where		<math>(r, \phi, \theta)</math>, where
	* ''R'' = ~~''R<sub>0</sub>''~~ + ''r'' cos ~~''&~~theta~~;''~~, and		* <math>R = R_0 + r \cos \theta</math>, and
	* ''Z'' = ''r'' sin ~~''&~~theta~~;''~~		* <math>Z = r \sin \theta</math>
	~~''R~~<~~sub~~>0</~~sub~~>'' corresponds to the torus axis and is called the ''major radius'', while ''0 ≤ r ≤ a'' is called the ''minor radius'', and ''θ'' the ''poloidal angle''.		<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 ~~''R~~<~~sub~~>0</~~sub~~>~~/a''~~ is called the ''aspect ratio'' of the torus.		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>~~[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>		<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 ~~''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>		* 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]] (~~''&~~kappa~~;''~~), [[triangularity]] (~~''&~~delta~~;''~~), and squareness (~~''&~~zeta~~;''~~) 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>

	:<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 ==

	(~~''&~~zeta~~;''~~, ~~''&~~eta~~;''~~, ~~''&~~phi~~;''~~), 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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	:<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 ~~''R~~<~~sub~~>p</~~sub~~>'' is the pole of the coordinate system.		where <math>R_p</math> is the pole of the coordinate system.
	Surfaces of constant ~~''&~~zeta~~;''~~ are tori with major radii ''R = ~~R<sub>p~~</~~sub~~>~~''/tanh ''ζ''~~ and minor radii ''r = ~~R<sub>p~~</~~sub~~>~~''/sinh ''ζ''~~.		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 ''R = R<~~sub~~>p<~~/sub~~>~~'', ''&~~zeta~~;''~~ = ~~∞~~, while at infinity and at ''R = 0, ζ = 0''.		At <math>R = R_p</math>, <math>\zeta = \infty</math>, while at infinity and at <math>R = 0, \zeta = 0</math>.
	The coordinate ~~''&~~eta~~;''~~ is a poloidal angle and runs from 0 to 2π.		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>~~[http://dx.doi.org/10.1088/0029-5515/26/9/002~~ 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>~~[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>		<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>[~~http~~:~~//dx.doi.org/~~10.1088/0029-5515/2/1-2/005 ~~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.		* [[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>~~[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>.		* [[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.

Admin: /* Simple toroidal coordinates */

2014-06-11T13:06:46Z

Simple toroidal coordinates

← Older revision		Revision as of 15:06, 11 June 2014
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	<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>		<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>		* 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)		* adding [[ellipticity]] (''κ''), [[triangularity]] (''δ''), and squareness (''ζ'') 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)\\
			Z(r,\theta) = Z_0(r) + \kappa(r) r \sin(\theta + \zeta \sin 2 \theta) </math>

	== Toroidal coordinates ==		== Toroidal coordinates ==

Admin at 08:15, 11 June 2014

2014-06-11T08:15:52Z

← Older revision		Revision as of 10:15, 11 June 2014
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	<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>		<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>		* 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 (''ε''), triangularity (''κ''), etc. (to account for non-circular flux surface cross sections)		* adding [[ellipticity]] (''ε''), [[triangularity]] (''κ''), etc. (to account for non-circular flux surface cross sections)

	== Toroidal coordinates ==		== Toroidal coordinates ==

Admin at 07:54, 5 September 2013

2013-09-05T07:54:35Z

← Older revision		Revision as of 09:54, 5 September 2013
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	''φ'' is called the ''toroidal angle'' (and not the ''cylindrical'' angle, at least not in the context of magnetic confinement).		''φ'' 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 φ) is referred to as ''axisymmetry''.		Cylindrical symmetry (symmetry under rotation over φ) is referred to as ''[[axisymmetry]]''.

	== Simple toroidal coordinates ==		== Simple toroidal coordinates ==

Admin: /* Toroidal coordinates */

2011-06-30T19:55:30Z

Toroidal coordinates

← Older revision		Revision as of 21:55, 30 June 2011
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	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>[http://dx.doi.org/10.1088/0029-5515/26/9/002 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>		<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>

Admin: /* Magnetic coordinates */

2011-06-30T19:53:36Z

Magnetic coordinates

← Older revision		Revision as of 21:53, 30 June 2011
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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>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.		* [[Hamada coordinates]]. <ref>[http://dx.doi.org/10.1088/0029-5515/2/1-2/005 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>.		* [[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>.

← Older revision		Revision as of 12:34, 26 January 2023
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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>.
Line 65:		Line 65:
	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 />