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Toroidal coordinates: Difference between revisions
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[[File:Toroidal coordinates.png|400px|thumb|right|A toroidal coordinate system]] | [[File:Toroidal coordinates.png|400px|thumb|right|A simple toroidal coordinate system]] | ||
Coordinate systems used in toroidal systems: | Coordinate systems used in toroidal systems: | ||
== Cartesian == | == Cartesian coordinates == | ||
(''X'', ''Y'', ''Z'') | (''X'', ''Y'', ''Z'') | ||
<ref>[[:Wikipedia:Cartesian coordinate system]]</ref> | <ref>[[:Wikipedia:Cartesian coordinate system]]</ref> | ||
== Cylindrical == | == Cylindrical coordinates == | ||
(''R'', ''φ'', ''Z''), where | (''R'', ''φ'', ''Z''), where | ||
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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 == | == Simple toroidal coordinates == | ||
(''r'', ''φ'', ''θ''), where | (''r'', ''φ'', ''θ''), where | ||
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* 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 == | == Toroidal coordinates == | ||
(''ζ'', ''η'', ''φ''), where | (''ζ'', ''η'', ''φ''), where | ||
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<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> | ||
== General | == General curvilinear coordinates == | ||
Here we briefly review the basic definitions of a general [[:Wikipedia:Curvilinear coordinates | curvilinear coordinate system]] for later convenience when discussing toroidal flux coordinates and magnetic coordinates. | Here we briefly review the basic definitions of a general [[:Wikipedia:Curvilinear coordinates | curvilinear coordinate system]] for later convenience when discussing toroidal flux coordinates and magnetic coordinates. | ||
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It can be seen that <ref name='Dhaeseleer'></ref> <math>g \equiv \det(g_{ij}) = J^2 \Rightarrow J = \sqrt{g}</math> | It can be seen that <ref name='Dhaeseleer'></ref> <math>g \equiv \det(g_{ij}) = J^2 \Rightarrow J = \sqrt{g}</math> | ||
== Flux | == Flux coordinates == | ||
A flux coordinate set is one that includes a flux surface label as a coordinate. A flux surface label is a function that is constant and single valued on each flux surface. In our naming of the general curvilinear coordinates we have already adopted the usual flux coordinate convention for toroidal equilibrium with nested flux surfaces with <math>\psi</math> being the flux surface label and <math>\theta, \phi</math> are <math>2\pi</math>-periodic poloidal and toroidal-like angles. | A flux coordinate set is one that includes a [[Flux surface|flux surface]] label as a coordinate. A flux surface label is a function that is constant and single valued on each flux surface. In our naming of the general curvilinear coordinates we have already adopted the usual flux coordinate convention for toroidal equilibrium with nested flux surfaces with <math>\psi</math> being the flux surface label and <math>\theta, \phi</math> are <math>2\pi</math>-periodic poloidal and toroidal-like angles. | ||
Different flux surface labels can be chosen like toroidal or poloidal magnetic fluxes or the volume contained within the flux surface. By single valued we mean to ensure that any flux label <math>\psi_1 = f(\psi_2)</math> is a monotonous function of any other flux label <math>\psi_2</math>, so that the function <math>f</math> is invertible at least in a volume containing the region of interest. | Different flux surface labels can be chosen like toroidal or poloidal magnetic fluxes or the volume contained within the flux surface. By single valued we mean to ensure that any flux label <math>\psi_1 = f(\psi_2)</math> is a monotonous function of any other flux label <math>\psi_2</math>, so that the function <math>f</math> is invertible at least in a volume containing the region of interest. | ||
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</math> | </math> | ||
== Magnetic == | == Magnetic coordinates == | ||
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. | 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: | ||