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<ref>F. Alladio, F. Chrisanti, ''Analysis of MHD equilibria by toroidal multipolar expansions'', Nucl. Fusion '''26''' (1986) 1143</ref>
<ref>F. Alladio, F. Chrisanti, ''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>
== 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.
=== Function coordinates and basis vector ===
Given the spatial dependence of a coordinate set <math>(\psi(\mathbf{x}),\theta(\mathbf{x}),\phi(\mathbf{x}))</math>
we can calculate the contravariant basis vectors
:<math>
\mathbf{e}^i = \{\nabla\psi, \nabla\theta, \nabla\phi\}
</math>
and the dual covariant basis defined as
:<math>
\mathbf{e}_i= \frac{\partial\mathbf{x}}{\partial{u^i}}
\to
\mathbf{e}_i\cdot\mathbf{e}^j
= \delta_{i}^{j} \to \mathbf{e}_i
= \frac{\mathbf{e}^j\times\mathbf{e}^k}{|\mathbf{e}^i\cdot\mathbf{e}^j\times\mathbf{e}^k|}
= \sqrt{g}\;\mathbf{e}^j\times\mathbf{e}^k ~,
</math>
where <math>(i,j,k)</math> are cyclic permutations of <math>(1,2,3)</math> and we have used the notation <math>(u^1, u^2, u^3) = (\psi,\theta,\phi)</math>. The Jacobian <math>\sqrt{g}</math> is defined below.
Any vector field <math>\mathbf{B}</math> can be represented as
:<math>
\mathbf{B}
= (\mathbf{B}\cdot\mathbf{e}^i)\mathbf{e}_i
= B^i\mathbf{e}_i
</math>
or
:<math>
\mathbf{B}
= (\mathbf{B}\cdot\mathbf{e}_i)\mathbf{e}^i
= B_i\mathbf{e}^i ~.
</math>
In particular any basis vector <math>\mathbf{e}_i = (\mathbf{e}_i\cdot\mathbf{e}_j)\mathbf{e}^j</math>. The metric tensor is defined as
:<math>
g_{ij}
= \mathbf{e}_i\cdot\mathbf{e}_j
\; ; \;
g^{ij}
= \mathbf{e}^i\cdot\mathbf{e}^j
\; ; \;
g^j_i 
= \mathbf{e}_i\cdot\mathbf{e}^j = \delta_i^j ~.
</math>
=== Jacobian ===
The Jacobian of the coordinate transformation <math>\mathbf{x}(\psi, \theta, \phi)</math> is defined as
:<math>
J = \det\left(\frac{\partial(x,y,z)}{\partial(\psi,\theta,\phi)}\right) = \frac{\partial\mathbf{x}}{\partial{\psi}}\cdot\frac{\partial\mathbf{x}}{\partial{\theta}} \times \frac{\partial\mathbf{x}}{\partial{\phi}}
</math>
and that of the inverse transformation
:<math>
J^{-1} = \det\left(\frac{\partial(\psi,\theta,\phi)}{\partial(x,y,z)}\right) = \nabla{\psi}\cdot\nabla{\theta} \times \nabla{\phi}
</math>
It can be seen that <ref name='Dhaeseleer'></ref> <math>g \equiv \det(g_{ij}) = J^2 \Rightarrow J = \sqrt{g}</math>
== Flux coordinates ==
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 <math>(\Psi_{tor})</math> or poloidal <math>(\Psi_{pol})</math> magnetic fluxes or the volume contained within the flux surface <math>V</math>. 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. We will denote a generic flux surface label by <math>\psi</math>.
=== Flux Surface Average ===
The flux surface average of a function <math>\Phi</math> is defined as the limit
:<math>
\langle\Phi\rangle = \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\int_{\delta \mathcal{V}} \Phi\; d\mathcal{V}
</math>
where <math>\delta \mathcal{V}</math> is the volume confined between two flux surfaces. It is therefore a ''volume average'' over an infinitesimal spatial region rather than a surface average. To avoid confusion, we denote volume elements or domains with the calligraphic <math>\mathcal{V}</math>. Capital <math>V</math> is reserved for the flux label (coordinate) defined as the volume within a flux surface.
Introducing the differential volume element <math>d\mathcal{V} = \sqrt{g} d\psi d\theta d\phi</math>
:<math>
\langle\Phi\rangle
= \lim_{\delta \mathcal{V} \to 0} \frac{1}{\delta \mathcal{V}}\int_{\delta \mathcal{V}} \Phi\; \sqrt{g} d\psi d\theta d\phi
= \frac{d\psi}{d V}\int_0^{2\pi}\int_0^{2\pi}\Phi\; \sqrt{g} d\theta d\phi
</math>
or, noting that <math>\langle 1\rangle = 1</math>, we have <math>\frac{dV}{d\psi} = \int_0^{2\pi}\int_0^{2\pi} \sqrt{g} d\theta d\phi</math> and
we get to a more practical form of the Flux Surface Average
:<math>
\langle\Phi\rangle
= \frac{\int_0^{2\pi}\int_0^{2\pi}\Phi\; \sqrt{g} d\theta d\phi}
{\int_0^{2\pi}\int_0^{2\pi} \sqrt{g} d\theta d\phi}
</math>
Note that <math>dS = |\nabla\psi|\sqrt{g}d\theta d\phi</math>, so the FSA is a surface integral ''weighted by'' <math>|\nabla V|^{-1}</math> :
:<math>
\langle\Phi\rangle
= \frac{d\psi}{d V}\int_0^{2\pi}\int_0^{2\pi}\Phi\; \sqrt{g} d\theta d\phi
= \frac{d\psi}{d V}\int_{S(\psi)}\frac{\Phi}{|\nabla\psi|}\; dS
= \int_{S(\psi)}\frac{\Phi}{|\nabla V|}\; dS
</math>
Applying Gauss' theorem to the definition of FSA we get to the identity
:<math>
\langle\nabla\cdot\Gamma\rangle
= \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\int_{\delta \mathcal{V}} \nabla\cdot\Gamma\; d\mathcal{V}
= \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\int_{S(\delta \mathcal{V})} \Gamma\cdot \frac{\nabla V}{|\nabla V|}dS
= \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\left(\langle\Gamma\cdot\nabla V\rangle_{S(V+\delta \mathcal{V})} - \langle\Gamma\cdot\nabla V\rangle_{S(V)} \right)
= \frac{d}{dV}\langle\Gamma\cdot\nabla V\rangle~.
</math>
==== Useful properties of FSA ====
Some useful properties of the FSA are
*<math> \langle\nabla\cdot\Gamma\rangle = \frac{d}{dV}\langle\Gamma\cdot\nabla V\rangle  = \frac{1}{V'}\frac{d}{d\psi}V'\langle\Gamma\cdot\nabla \psi\rangle</math>
*<math> \int_{\mathcal{V}}\nabla\cdot\Gamma\; d\mathcal{V} =  \langle\Gamma\cdot\nabla V\rangle = V'\langle\Gamma\cdot\nabla \psi\rangle</math>
*<math> \langle \sqrt{g}^{-1}\rangle = \frac{4\pi^2}{V'}
</math>
*<math> \langle \mathbf{B}\cdot\nabla f \rangle = 0~,\qquad \forall~ \mathrm{single~valued~} f(\mathbf{x}), ~ \mathrm{if}~ \nabla\cdot\mathbf{B} = 0 ~\mathrm{and}~ \nabla V\cdot\mathbf{B} = 0 </math>
*<math> \langle \mathbf{B}\cdot\nabla \theta\rangle =2\pi\frac{d\Psi_{pol}}{dV} \qquad (\mathrm{Note:}~ \theta(\mathbf{x})~\mathrm{is~not~single~valued})
</math>
*<math> \langle \mathbf{B}\cdot\nabla \phi\rangle =2\pi\frac{d\Psi_{tor}}{dV} \qquad (\mathrm{Note:}~ \phi(\mathbf{x})~\mathrm{is~not~single~valued})
</math>
where <math>V' = \frac{dV}{d\psi}</math>.
=== Magnetic field representation in flux coordinates ===
==== Contravariant From ====
Any [[:Wikipedia: solenoidal vector field| solenoidal vector field]] <math>\mathbf{B}</math>  can be written as
<math> \mathbf{B} = \nabla\alpha\times\nabla\nu </math>
called its Clebsch representation. For a magnetic field with flux surfaces <math>(\psi = \mathrm{const}\; , \; \nabla\psi\cdot\mathbf{B} = 0)</math> we can choose, say, <math>\alpha</math> to be the flux surface label <math>\psi</math>
:<math>
\mathbf{B} = \nabla\psi\times\nabla\nu
</math>
Field lines are then given as the intersection of the constant-<math>\psi</math> and constant-<math>\nu</math> surfaces. This form provides a general expression for <math>\mathbf{B}</math> in terms of the covariant basis vectors of a flux coordinate system
:<math>
\mathbf{B} = \frac{\partial\nu}{\partial\theta}\nabla\psi\times\nabla\theta + \frac{\partial\nu}{\partial\phi}\nabla\psi\times\nabla\phi =  \frac{1}{\sqrt{g}}\frac{\partial\nu}{\partial\theta}\mathbf{e}_\phi -\frac{1}{\sqrt{g}}\frac{\partial\nu}{\partial\phi}\mathbf{e}_\theta = B^\phi\mathbf{e}_\phi + B^\theta\mathbf{e}_\theta~.
</math>
in terms of the function <math>\nu</math>, sometimes referred to as the magnetic field's ''stream function''.
It is worthwhile to note that the Clebsch form of <math> \mathbf{B} </math> corresponds to a [[:Wikipedia: Magnetic potential|magnetic vector potential]]
<math> \mathbf{A} = \nu\nabla\psi </math> (or <math> \mathbf{A} = \psi\nabla\nu </math> as they differ only by the Gauge transformation <math> \mathbf{A} \to \mathbf{A} - \nabla (\psi\nu)</math>).
The general form of the stream function is
:<math>
\nu(\psi,\theta,\phi)
= \frac{1}{2\pi}(\Psi_{tor}'\theta
- \Psi_{pol}'\phi)
+ \tilde{\nu}(\psi,\theta,\phi)
</math>
where <math>\tilde{\nu}</math> is a differentiable function periodic in the two angles. This general form can be derived by using the fact that  <math> \mathbf{B}</math> is a physical function (hence singe-valued). The specific form for the coefficients in front of the secular terms (i.e. the non-periodic terms) can be obtained from the [[Toroidal coordinates#Useful properties of FSA|FSA properties ]].
==== Covariant Form ====
If we consider an equilibrium magnetic field such that <math> \mathbf{j}\times\mathbf{B} \propto \nabla\psi</math>, where <math> \mathbf{j}</math> is the current density , then both <math> \mathbf{B}\cdot\nabla\psi = 0</math> and <math> \nabla\times\mathbf{B}\cdot\nabla\psi = 0</math> and the magnetic field can be written as
:<math>
\mathbf{B} = \beta\nabla\psi + \nabla\chi
</math>
where <math>\chi</math> is identified as the magnetic ''scalar'' potential. Its general form is
:<math>
\chi(\psi, \theta, \phi) = \frac{I_{tor}}{2\pi}\theta + \frac{I_{pol}^d}{2\pi}\phi + \tilde\chi(\psi, \theta, \phi)
</math>
[[Image:CurrentIntegrationCirtuits.png|thumb|right|alt=Sample integration circuits for the definitions of currents.|Sample integration circuits for the current definitions.]]
[[Image:CurrentIntegrationCirtuitsPoloidalCurrent.png|thumb|right|alt=Sample surface for the definition of the current though a disc.|Sample surface for the definition of the current though a disc. Note that only the current of more external surfaces contribute to the flux of charge through the surface.]]
The functional dependence on the angular variables is again motivated by the single-valuedness of the magnetic field. The particular form of the coefficients can be obtained noting that
:<math>
\int_S \mu_0\mathbf{j}\cdot d\mathbf{S}
= \int_{\partial S}\mathbf{B}\cdot d\mathbf{l}
= \oint(\beta\nabla\psi + \nabla\chi)\cdot d\mathbf{l}
= \oint(\beta d\psi + d\chi)
</math>
and choosing an integration circuit contained within a flux surface <math>(d\psi = 0)</math>. Then we get
:<math>
\int_S \mu_0\mathbf{j}\cdot d\mathbf{S}
= \Delta \chi = \frac{I_{tor}}{2\pi}\Delta\theta + \frac{I_{pol}^d}{2\pi}\Delta\phi~.
</math>
If we now chose a ''toroidal'' circuit <math>(\Delta\theta = 0, \Delta\phi = 2\pi)</math> we get
:<math>
I_{pol}^d = \int_S \mu_0\mathbf{j}\cdot d\mathbf{S}\; ; ~\mathrm{with}~ \partial S ~\mathrm{such~that}~ (\Delta\theta = 0, \Delta\phi = 2\pi)~.
</math>
here the superscript <math>d</math> is meant to indicate the flux is computed through a disc limited by the integration line, as opposed to the ribbon limited by the integration line on one side and the magnetic axis on the other that was used for the definition of poloidal magnetic flux <math>\Psi_{pol}</math> above these lines. 
Similarly
:<math>
I_{tor} = \int_S \mu_0\mathbf{j}\cdot d\mathbf{S}\; ; ~\mathrm{with}~ \partial S ~\mathrm{such~that}~ (\Delta\theta = 2\pi, \Delta\phi = 0)~.
</math>
===== Contravariant Form of the current density =====
Taking the curl of the covariant form of <math>\mathbf{B}</math> the equilibrium current density <math>\mathbf{j}</math> can be written as
: <math>
\mathbf{j} = \nabla\psi\times\nabla\eta~.
</math>
By very similar arguments as those used for <math>\mathbf{B}</math> (note that both <math>\mathbf{B}</math> and <math>\mathbf{j}</math> are solenoidal fields tangent to the flux surfaces) it can be shown that the general expression for <math>\eta</math> is
:<math>
\eta(\psi,\theta,\phi) = \frac{1}{2\pi}({I}_{tor}'\theta
- {I}_{pol}'\phi)
+ \tilde{\eta}(\psi,\theta,\phi)~.
</math>
Note that the poloidal current is now defined through a ribbon and not a disc.


== Magnetic coordinates ==
== Magnetic coordinates ==
Magnetic coordinates are a particular type of flux coordinates in which the magnetic field lines are straight lines. In mathematical terms this implies that the periodic part of the magnetic field's stream function is zero in these coordinates so the magnetic field reads
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]]).  
:<math>
\mathbf{B} =  \nabla\psi\times \left( \frac{\Psi_{tor}'}{2\pi}\theta_f
- \frac{\Psi_{pol}'}{2\pi}\phi_f \right)~.
</math>
Note that, in general, the contravariant components of the magnetic field in a magnetic coordinate system
:<math>
B^{\theta_f} = \frac{\Psi_{pol}'}{2\pi\sqrt{g}}\; ;\quad B^{\phi_f} = \frac{\Psi_{tor}'}{2\pi\sqrt{g}}
</math>
are not flux functions, but their quotient is
:<math>
\frac{B^{\theta_f}}{B^{\phi_f}} = \frac{\Psi_{pol}'}{\Psi_{tor}'} \equiv \frac{\iota}{2\pi}~,
</math>
<math>\iota</math> being the [[rotational transform]].
 
Any transformation of the angles of the from
:<math>
\theta_F = \theta_f +\Psi_{pol}' G(\psi, \theta_f, \phi_f)\; ;\quad \phi_F = \phi_f +\Psi_{tor}' G(\psi, \theta_f, \phi_f)
</math>
 
where <math>G</math> is periodic in the angles, preserves the straightness of the field lines. The spatial function <math>G(\psi, \theta_f, \phi_f)</math>, is called the ''generating function''.
 
 
 
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:
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