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+Flux coordinates in the context of magnetic confinement fusion (MCF) is a set of coordinate functions adapted to the shape of the [[Flux surface|flux surfaces]] of the confining magnetic trap. They consist of one flux label, often termed <math>\psi</math> and two angle-like variables <math>\theta, \phi</math> whose constant contours on the flux <math>(\psi({\mathbf x}) = {\textrm constant})</math> surfaces close either poloidaly (<math>\phi</math>) or toroidallly (<math>\theta</math>).
+In this coordinates, equilibrium vector fields like the magnetic field <math>{\mathbf B}</math> or current density <math>{\mathbf j}</math> have simplified expressions. A particular kind of flux coordinates, generally called [[Flux_coordinates#Magnetic_coordinates|magnetic coordinates]], simplify the <math>{\mathbf B}</math>-field expression further by making field lines look straight in the <math>(\theta, \phi)</math> plane of that family of coordinates. Some popular choices of magnetic coordinate systems are [[Boozer coordinates]] and [[Hamada coordinates]].
+[[Image:CurrentIntegrationCirtuits.png|400px|thumb|right|alt=Flux surface and a <math>\theta</math>-curve and  <math>\phi</math>-curve.|Sample flux surface of the [[TJ-II]] [[Stellarator|stellarator]] and a <math>\theta</math>-curve (yellow) and <math>\phi</math>-curve (red).]]
 == 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 ===
+=== Coordinates and basis vectors ===
-Given the spatial dependence of a coordinate set <math>(\psi(\mathbf{x}),\theta(\mathbf{x}),\phi(\mathbf{x}))</math>
+Let <math>{\mathbf x}</math> be a set of euclidean coordinates on <math>{\mathbb R}^3</math> and let <math>(\psi(\mathbf{x}),\theta(\mathbf{x}),\phi(\mathbf{x}))</math> define a change of coordinates, arbitrary for the time being.
-we can calculate the contravariant basis vectors
+We can calculate the contravariant basis vectors as
 :<math>
 \mathbf{e}^i = \{\nabla\psi, \nabla\theta, \nabla\phi\}
@@ Line 13: / Line 19: @@
 \to
 \mathbf{e}_i\cdot\mathbf{e}^j
-= \delta_{i}^{j} \to \mathbf{e}_i
+= \delta_{i}^{j}~,
-= \frac{\mathbf{e}^j\times\mathbf{e}^k}{|\mathbf{e}^i\cdot\mathbf{e}^j\times\mathbf{e}^k|}
+</math>
+and therefore relates to the contravariant vectors as
+:<math>
+ \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.
+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. Similarly
+:<math>
+\mathbf{e}^i = \frac{\mathbf{e}_j\times\mathbf{e}_k}{\sqrt{g}} ~.
+</math>
 Any vector field <math>\mathbf{B}</math> can be represented as
@@ Line 63: / Line 76: @@
 </math>
 It can be seen that <ref name='Dhaeseleer'></ref> <math>g \equiv \det(g_{ij}) = J^2 \Rightarrow J = \sqrt{g}</math>
+=== Some surface elements ===
+Consider a surface defined by a constant value of <math>\phi</math>. Then, the surface element is
+:<math>
+d{\mathbf S}_\phi = \mathbf{e}_\psi\times\mathbf{e}_\theta d\psi d\theta = \sqrt{g}\,  \nabla\phi d\psi d\theta .
+</math>
+As for a surface defined by a constant value of <math>\theta</math>:
+:<math>
+d{\mathbf S}_\theta = \mathbf{e}_\phi\times\mathbf{e}_\psi d\psi d\phi = \sqrt{g}\,  \nabla\theta d\psi d\phi ,
+</math>
+or a constant <math>\psi</math> surface:
+:<math>
+d{\mathbf S}_\psi = \mathbf{e}_\theta\times\mathbf{e}_\phi d\theta d\phi = \sqrt{g}\,  \nabla\psi d\theta d\phi .
+</math>
 === Gradient, Divergence and Curl in curvilinear coordinates ===
-The gradient of a funcion f is naturally given in the contravariant base vectors:
+The gradient of a function f is naturally given in the contravariant basis vectors:
 :<math>
   \nabla f = \frac{\partial f}{\partial u^i}\nabla u^i = \frac{\partial f}{\partial u^i}\mathbf{e}^i~.
 </math>
-The divergence of a vector \mathbf{A} is best expressed in terms of its contravariant components
+The divergence of a vector <math>\mathbf{A}</math> is best expressed in terms of its contravariant components
 :<math>
 \nabla\cdot\mathbf{A} = \frac{1}{\sqrt{g}}\frac{\partial}{\partial u^i}(\sqrt{g}A^i)~,
@@ Line 75: / Line 104: @@
 while the curl is
 :<math>
-  \nabla\times\mathbf{A} = \frac{\varepsilon_{ijk}}{\sqrt{g}}\frac{\partial}{\partial u^i}(\sqrt{g}A_j)\mathbf{e}_k
+  \nabla\times\mathbf{A} = \frac{\varepsilon^{ijk}}{\sqrt{g}}\frac{\partial A_j}{\partial u^i}\mathbf{e}_k \Rightarrow
+\left(\nabla\times\mathbf{A}\right)^k = \frac{\varepsilon^{ijk}}{\sqrt{g}}\frac{\partial A_j}{\partial u^i}
 </math>
-given in temr of the covariant base vectors, where <math>\varepsilon_{ijk}</math> is the [[::Wikipedia:Levi-Civita symbol| Levi-Civita]] symbol.
+given in terms of the covariant base vectors, where <math>\varepsilon^{ijk}</math> is the [[::Wikipedia:Levi-Civita symbol| Levi-Civita]] symbol.
 == 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.
+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, where <math>\psi</math> is 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>.
-To avoid ambiguity in the sign of line and surface integrals we impose <math>d\psi(V)/dV > 0</math>, the toroidal angle increases in the clockwise direction when seen from above and the poloidal angle increases such that <math> \nabla\psi\cdot\nabla\theta\times\nabla\xi > 0</math>.
+To avoid ambiguity in the sign of line and surface integrals we impose <math>d\psi(V)/dV > 0</math>, the toroidal angle increases in the clockwise direction when seen from above and the poloidal angle increases such that <math> \nabla\psi\cdot\nabla\theta\times\nabla\phi > 0</math>.
 === Flux Surface Average ===
-The flux surface average of a function <math>\Phi</math> is defined as the limit
+The Flux Surface Average (FSA) 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}
@@ Line 126: / Line 156: @@
 ==== Useful properties of FSA ====
 Some useful properties of the FSA are
+*<math>	\langle \mathbf{B}\cdot\nabla f \rangle = \langle \nabla\cdot(\mathbf{B} f) \rangle = 0~,\qquad \forall~ \mathrm{single~valued~} f(\mathbf{x}), ~ \mathrm{if}~ \nabla\cdot\mathbf{B} = 0 ~\mathrm{and}~ \nabla \psi\cdot\mathbf{B} = 0 </math>
 *<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 \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 \psi\cdot\mathbf{B} = 0 </math>
+The two identities above are the basic simplifying properties of the FSA: The first cancels the contribution of 'conservative forces' like the pressure gradient or electrostatic electric fields. The second reduces the number of spatial variables to only the radial one. Further, it is possible to show that, if <math>\nabla\cdot\Gamma = 0</math> then <math>\langle\Gamma\cdot\nabla V\rangle = 0</math> and not simply constant as the above suggests. This can be seen by simply using Gauss' theorem
-*<math>	\langle \nabla \psi\cdot\nabla\times \mathbf{A} \rangle = 0~.
+*<math> \int_{\mathcal{V}}\nabla\cdot\Gamma\; d\mathcal{V} =
+\int_{S(\mathcal{V})}\Gamma\cdot\nabla V \frac{dS}{|\nabla V|} =
+\langle\Gamma\cdot\nabla V\rangle \qquad \mathrm{where~} \mathcal{V} \mathrm{~is~the~volume~enclosed~by~a~flux~surface.}
+</math>
+The FSA relates to the conventional volume integral between two surfaces labelled by their volumes <math>V_1</math> and <math>V_2</math> as
+*<math> \int_{\mathcal{V}(V_1<V<V_2)} f\; d\mathcal{V} = \int_{V_1}^{V_2} \langle f \rangle\; dV
+</math>
+whereas the conventional surface integral over a <math>\psi = constant</math> is
+*<math> \int_{S(\psi)} f\; dS =  \langle f |\nabla V| \rangle
+</math>
+Other useful properties are
+*<math>	\langle \nabla \psi\cdot\nabla\times \mathbf{A} \rangle = -\langle \nabla\cdot( \nabla\psi\times\mathbf{A}) \rangle = 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>	\langle \mathbf{B}\cdot\nabla \theta\rangle =2\pi\frac{d\Psi_{pol}}{dV} \qquad \mathrm{for~any~poloidal~ angle~} \theta ~ (\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>	\langle \mathbf{B}\cdot\nabla \phi\rangle =2\pi\frac{d\Psi_{tor}}{dV} \qquad \mathrm{for~any~toroidal~ angle~} \phi ~ (\mathrm{Note:}~ \phi(\mathbf{x})~\mathrm{is~not~single~valued})
 </math>
@@ Line 145: / Line 186: @@
 </math>
-In the above  <math>V' = \frac{dV}{d\psi}</math>.
+In the above  <math>V' = \frac{dV}{d\psi}</math>. Some [[:Wikipedia: Vector calculus identities|vector identities]] are useful to derive the above identities.
 === Magnetic field representation in flux coordinates ===
-==== Contravariant From ====
+==== Contravariant Form ====
 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>
@@ Line 186: / Line 227: @@
 [[Image:CurrentIntegrationCirtuits.png|256px|thumb|right|alt=Sample integration circuits for the definitions of currents.|Sample integration circuits for the current definitions.]]
-[[Image:CurrentIntegrationCirtuitsPoloidalCurrent.png|256px|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.]]
+[[Image:CurrentIntegrationCirtuitsPoloidalCurrent.png|256px|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 (those enclosing the one drawn here) 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
+Note that <math>I</math> is not the current but <math>\mu_0</math> times the current. 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}
@@ Line 201: / Line 242: @@
 </math>
-If we now chose a ''toroidal'' circuit <math>(\Delta\theta = 0, \Delta\phi = 2\pi)</math> we get
+If we now choose 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)~.
@@ Line 235: / Line 276: @@
 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
 :<math>
-\mathbf{B} =  \nabla\psi\times \left( \frac{\Psi_{tor}'}{2\pi}\theta_f
+\mathbf{B} =  \nabla\psi\times \nabla\left( \frac{\Psi_{tor}'}{2\pi}\theta_f
 		- \frac{\Psi_{pol}'}{2\pi}\phi_f \right)
             = \frac{\Psi_{pol}'}{2\pi\sqrt{g}}\mathbf{e}_\theta + \frac{\Psi_{tor}'}{2\pi\sqrt{g}}\mathbf{e}_\phi~.
@@ Line 251: / Line 292: @@
 <math>\iota</math> being the [[rotational transform]]. In a magnetic coordinate system the ''poloidal'' <math> \mathbf{B}_P = B^\theta\mathbf{e}_\theta </math> and ''toroidal'' <math> \mathbf{B}_T = B^\phi\mathbf{e}_\phi</math> components of the magnetic field are individually divergence-less.
+From the above general form of  <math> \mathbf{B} </math> in magnetic coordinates it is easy to obtain the following identities valid for any magnetic coordinate system
+:<math>
+\mathbf{e}_\theta\times\mathbf{B} =\frac{1}{2\pi}\nabla\Psi_{tor}~,
+</math>
+:<math>
+\mathbf{e}_\phi\times\mathbf{B} = -\frac{1}{2\pi}\nabla\Psi_{pol} ~.
+</math>
 === Transforming between Magnetic coordinates systems ===
 There are infinitely many systems of magnetic coordinates. Any transformation of the angles of the from
@@ Line 260: / Line 308: @@
 :<math>
 \pi\mathbf{B}\cdot\nabla G = \frac{1}{\sqrt{g_F}} - \frac{1}{\sqrt{g_f}}~.
 </math>
+The LHS of this equation has a particularly simple form when one uses a magnetic coordinate system. For instance, if we write <math>\mathbf{B}</math> in terms of the original magnetic coordinate system we get
+:<math>
+ (\Psi_{pol}'\partial_{\theta_f} + \Psi_{tor}'\partial_{\phi_f}) G = \frac{\sqrt{g_f}}{\sqrt{g_F}} - 1~.
+</math>
+which can be turned into an algebraic equation on the Fourier components of <math>G</math>
+:<math>
+  G_{mn} = \frac{-i}{\Psi_{pol}'m + \Psi_{tor}'n}\left(\frac{\sqrt{g_f}}{\sqrt{g_F}}\right)_{mn}~.
+</math>
+where
+:<math>
+ G(\psi, \theta_f, \phi_f) = \sum_{m,n} G_{mn}(\psi) e^{i(m\theta_f + n\phi_f)}
+</math>
+and <math>G_{00} = 0 </math> guarantees periodicity is preserved.
 Particular choices of G can be made so as to simplify the description of other fields. The most commonly used magnetic 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. Referring to the definitions above, both <math>\tilde\nu</math> and <math>\tilde\eta</math> are zero in Hamada coordinates.
+* [[Hamada coordinates]]. <ref>S. Hamada, Nucl. Fusion '''2''' (1962) 23</ref><ref>[[doi: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 magnetic field lines and current lines corresponding to the [[MHD equilibrium]] are straight. Referring to the definitions above, both <math>\tilde\nu</math> and <math>\tilde\eta</math> are zero in Hamada coordinates.
-* [[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>. Referring to the definitions above, both <math>\tilde\nu</math> and <math>\tilde\chi</math> are zero in Boozer coordinates.
+* [[Boozer coordinates]]. <ref>[[doi:10.1063/1.863297|A.H. Boozer, ''Plasma equilibrium with rational magnetic surfaces'', Phys. Fluids '''24''' (1981) 1999]]</ref><ref>[[doi: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 magnetic 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>. Referring to the definitions above, both <math>\tilde\nu</math> and <math>\tilde\chi</math> are zero in Boozer coordinates.
 == References ==
 <references />