Flux coordinates: Difference between revisions

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== General curvilinear coordinates ==
== 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.


=== Coordinates and basis vectors ===  
=== Coordinates and basis vectors ===  
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.  
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 as  
We can calculate the contravariant basis vectors as  
:&lt;math&gt;
:<math>
\mathbf{e}^i = \{\nabla\psi, \nabla\theta, \nabla\phi\}  
\mathbf{e}^i = \{\nabla\psi, \nabla\theta, \nabla\phi\}  
&lt;/math&gt;
</math>
and the dual covariant basis defined as
and the dual covariant basis defined as
:&lt;math&gt;
:<math>
\mathbf{e}_i= \frac{\partial\mathbf{x}}{\partial{u^i}}
\mathbf{e}_i= \frac{\partial\mathbf{x}}{\partial{u^i}}
\to
\to
Line 17: Line 16:
= \frac{\mathbf{e}^j\times\mathbf{e}^k}{|\mathbf{e}^i\cdot\mathbf{e}^j\times\mathbf{e}^k|}
= \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 ~,
= \sqrt{g}\;\mathbf{e}^j\times\mathbf{e}^k ~,
&lt;/math&gt;
</math>
where &lt;math&gt;(i,j,k)&lt;/math&gt; are cyclic permutations of &lt;math&gt;(1,2,3)&lt;/math&gt; and we have used the notation &lt;math&gt;(u^1, u^2, u^3) = (\psi,\theta,\phi)&lt;/math&gt;. The Jacobian &lt;math&gt;\sqrt{g}&lt;/math&gt; 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.


Any vector field &lt;math&gt;\mathbf{B}&lt;/math&gt; can be represented as
Any vector field <math>\mathbf{B}</math> can be represented as
:&lt;math&gt;
:<math>
\mathbf{B}  
\mathbf{B}  
= (\mathbf{B}\cdot\mathbf{e}^i)\mathbf{e}_i
= (\mathbf{B}\cdot\mathbf{e}^i)\mathbf{e}_i
= B^i\mathbf{e}_i
= B^i\mathbf{e}_i
&lt;/math&gt;
</math>
or
or
:&lt;math&gt;
:<math>
\mathbf{B}  
\mathbf{B}  
= (\mathbf{B}\cdot\mathbf{e}_i)\mathbf{e}^i
= (\mathbf{B}\cdot\mathbf{e}_i)\mathbf{e}^i
= B_i\mathbf{e}^i ~.
= B_i\mathbf{e}^i ~.
&lt;/math&gt;
</math>
In particular any basis vector &lt;math&gt;\mathbf{e}_i = (\mathbf{e}_i\cdot\mathbf{e}_j)\mathbf{e}^j&lt;/math&gt;. The metric tensor is defined as
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
:&lt;math&gt;
:<math>
g_{ij}  
g_{ij}  
= \mathbf{e}_i\cdot\mathbf{e}_j
= \mathbf{e}_i\cdot\mathbf{e}_j
Line 42: Line 41:
g^j_i   
g^j_i   
= \mathbf{e}_i\cdot\mathbf{e}^j = \delta_i^j ~.
= \mathbf{e}_i\cdot\mathbf{e}^j = \delta_i^j ~.
&lt;/math&gt;
</math>


The metric tensors can be used to ''raise'' or ''lower'' indices. Take
The metric tensors can be used to ''raise'' or ''lower'' indices. Take
:&lt;math&gt;
:<math>
\mathbf{B}  
\mathbf{B}  
= B_i\mathbf{e}^i = B_i g^{ij}\mathbf{e}_j = B^j\mathbf{e}_j~,
= B_i\mathbf{e}^i = B_i g^{ij}\mathbf{e}_j = B^j\mathbf{e}_j~,
&lt;/math&gt;
</math>
so that
so that
:&lt;math&gt;
:<math>
B^j = g^{ij} B_i~.
B^j = g^{ij} B_i~.
&lt;/math&gt;
</math>


=== Jacobian ===  
=== Jacobian ===  
The Jacobian of the coordinate transformation &lt;math&gt;\mathbf{x}(\psi, \theta, \phi)&lt;/math&gt; is defined as
The Jacobian of the coordinate transformation <math>\mathbf{x}(\psi, \theta, \phi)</math> is defined as
:&lt;math&gt;
:<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}}
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}}
&lt;/math&gt;
</math>
and that of the inverse transformation  
and that of the inverse transformation  
:&lt;math&gt;
:<math>
J^{-1} = \det\left(\frac{\partial(\psi,\theta,\phi)}{\partial(x,y,z)}\right) = \nabla{\psi}\cdot\nabla{\theta} \times \nabla{\phi}
J^{-1} = \det\left(\frac{\partial(\psi,\theta,\phi)}{\partial(x,y,z)}\right) = \nabla{\psi}\cdot\nabla{\theta} \times \nabla{\phi}
&lt;/math&gt;
</math>
It can be seen that &lt;ref name='Dhaeseleer'&gt;&lt;/ref&gt; &lt;math&gt;g \equiv \det(g_{ij}) = J^2 \Rightarrow J = \sqrt{g}&lt;/math&gt;
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 ===
=== Some surface elements ===


Consider a surface defined by a constant value of &lt;math&gt;\phi&lt;/math&gt;. Then, the surface element is
Consider a surface defined by a constant value of <math>\phi</math>. Then, the surface element is
:&lt;math&gt;
:<math>
d{\mathbf S}_\phi = \mathbf{e}_\psi\times\mathbf{e}_\theta d\psi d\theta = \sqrt{g}\,  \nabla\phi d\psi d\theta .
d{\mathbf S}_\phi = \mathbf{e}_\psi\times\mathbf{e}_\theta d\psi d\theta = \sqrt{g}\,  \nabla\phi d\psi d\theta .
&lt;/math&gt;  
</math>  


As for a surface defined by a constant value of &lt;math&gt;\theta&lt;/math&gt;:
As for a surface defined by a constant value of <math>\theta</math>:
:&lt;math&gt;
:<math>
d{\mathbf S}_\theta = \mathbf{e}_\phi\times\mathbf{e}_\psi d\psi d\phi = \sqrt{g}\,  \nabla\theta d\psi d\phi ,
d{\mathbf S}_\theta = \mathbf{e}_\phi\times\mathbf{e}_\psi d\psi d\phi = \sqrt{g}\,  \nabla\theta d\psi d\phi ,
&lt;/math&gt;
</math>
or a constant &lt;math&gt;\psi&lt;/math&gt; surface:  
or a constant <math>\psi</math> surface:  
:&lt;math&gt;
:<math>
d{\mathbf S}_\psi = \mathbf{e}_\theta\times\mathbf{e}_\phi d\theta d\phi = \sqrt{g}\,  \nabla\psi d\theta d\phi .
d{\mathbf S}_\psi = \mathbf{e}_\theta\times\mathbf{e}_\phi d\theta d\phi = \sqrt{g}\,  \nabla\psi d\theta d\phi .
&lt;/math&gt;
</math>


=== Gradient, Divergence and Curl in curvilinear coordinates ===
=== Gradient, Divergence and Curl in curvilinear coordinates ===
The gradient of a function f is naturally given in the contravariant basis vectors:
The gradient of a function f is naturally given in the contravariant basis vectors:
:&lt;math&gt;
:<math>
  \nabla f = \frac{\partial f}{\partial u^i}\nabla u^i = \frac{\partial f}{\partial u^i}\mathbf{e}^i~.
  \nabla f = \frac{\partial f}{\partial u^i}\nabla u^i = \frac{\partial f}{\partial u^i}\mathbf{e}^i~.
&lt;/math&gt;
</math>
The divergence of a vector &lt;math&gt;\mathbf{A}&lt;/math&gt; 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
:&lt;math&gt;
:<math>
\nabla\cdot\mathbf{A} = \frac{1}{\sqrt{g}}\frac{\partial}{\partial u^i}(\sqrt{g}A^i)~,  
\nabla\cdot\mathbf{A} = \frac{1}{\sqrt{g}}\frac{\partial}{\partial u^i}(\sqrt{g}A^i)~,  
&lt;/math&gt;
</math>
while the curl is  
while the curl is  
:&lt;math&gt;
:<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}{\partial u^i}(\sqrt{g}A_j)\mathbf{e}_k
&lt;/math&gt;
</math>
given in terms of the covariant base vectors, where &lt;math&gt;\varepsilon_{ijk}&lt;/math&gt; 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 ==
== 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, where &lt;math&gt;\psi&lt;/math&gt; is the flux surface label and &lt;math&gt;\theta, \phi&lt;/math&gt; are &lt;math&gt;2\pi&lt;/math&gt;-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 &lt;math&gt;(\Psi_{tor})&lt;/math&gt; or poloidal &lt;math&gt;(\Psi_{pol})&lt;/math&gt; magnetic fluxes or the volume contained within the flux surface &lt;math&gt;V&lt;/math&gt;. By single valued we mean to ensure that any flux label &lt;math&gt;\psi_1 = f(\psi_2)&lt;/math&gt; is a monotonous function of any other flux label  &lt;math&gt;\psi_2&lt;/math&gt;, so that the function  &lt;math&gt;f&lt;/math&gt; is invertible at least in a volume containing the region of interest. We will denote a generic flux surface label by &lt;math&gt;\psi&lt;/math&gt;.
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 &lt;math&gt;d\psi(V)/dV &gt; 0&lt;/math&gt;, the toroidal angle increases in the clockwise direction when seen from above and the poloidal angle increases such that &lt;math&gt; \nabla\psi\cdot\nabla\theta\times\nabla\phi &gt; 0&lt;/math&gt;.
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 ===
=== Flux Surface Average ===
The Flux Surface Average (FSA) of a function &lt;math&gt;\Phi&lt;/math&gt; is defined as the limit
The Flux Surface Average (FSA) of a function <math>\Phi</math> is defined as the limit
:&lt;math&gt;
:<math>
\langle\Phi\rangle = \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\int_{\delta \mathcal{V}} \Phi\; d\mathcal{V}
\langle\Phi\rangle = \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\int_{\delta \mathcal{V}} \Phi\; d\mathcal{V}
&lt;/math&gt;
</math>
where &lt;math&gt;\delta \mathcal{V}&lt;/math&gt; 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 &lt;math&gt;\mathcal{V}&lt;/math&gt;. Capital &lt;math&gt;V&lt;/math&gt; is reserved for the flux label (coordinate) defined as the volume within a flux surface.
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 &lt;math&gt;d\mathcal{V} = \sqrt{g} d\psi d\theta d\phi&lt;/math&gt;
Introducing the differential volume element <math>d\mathcal{V} = \sqrt{g} d\psi d\theta d\phi</math>
:&lt;math&gt;
:<math>
\langle\Phi\rangle  
\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
= \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
= \frac{d\psi}{d V}\int_0^{2\pi}\int_0^{2\pi}\Phi\; \sqrt{g} d\theta d\phi
&lt;/math&gt;
</math>
or, noting that &lt;math&gt;\langle 1\rangle = 1&lt;/math&gt;, we have &lt;math&gt;\frac{dV}{d\psi} = \int_0^{2\pi}\int_0^{2\pi} \sqrt{g} d\theta d\phi&lt;/math&gt; and  
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
we get to a more practical form of the Flux Surface Average
:&lt;math&gt;
:<math>
\langle\Phi\rangle  
\langle\Phi\rangle  
= \frac{\int_0^{2\pi}\int_0^{2\pi}\Phi\; \sqrt{g} d\theta d\phi}
= \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}  
{\int_0^{2\pi}\int_0^{2\pi} \sqrt{g} d\theta d\phi}  
&lt;/math&gt;
</math>


Note that &lt;math&gt;dS = |\nabla\psi|\sqrt{g}d\theta d\phi&lt;/math&gt;, so the FSA is a surface integral ''weighted by'' &lt;math&gt;|\nabla V|^{-1}&lt;/math&gt; :
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> :
:&lt;math&gt;
:<math>
\langle\Phi\rangle  
\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_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
= \frac{d\psi}{d V}\int_{S(\psi)}\frac{\Phi}{|\nabla\psi|}\; dS
= \int_{S(\psi)}\frac{\Phi}{|\nabla V|}\; dS
= \int_{S(\psi)}\frac{\Phi}{|\nabla V|}\; dS
&lt;/math&gt;
</math>


Applying Gauss' theorem to the definition of FSA we get to the identity
Applying Gauss' theorem to the definition of FSA we get to the identity
:&lt;math&gt;
:<math>
\langle\nabla\cdot\Gamma\rangle  
\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_{\delta \mathcal{V}} \nabla\cdot\Gamma\; d\mathcal{V}
Line 139: Line 138:
= \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)
= \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~.
= \frac{d}{dV}\langle\Gamma\cdot\nabla V\rangle~.
&lt;/math&gt;
</math>


==== Useful properties of FSA ====
==== Useful properties of FSA ====
Some useful properties of the FSA are
Some useful properties of the FSA are


*&lt;math&gt; \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&lt;/math&gt;
*<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>


*&lt;math&gt; \int_{\mathcal{V}}\nabla\cdot\Gamma\; d\mathcal{V} =  \langle\Gamma\cdot\nabla V\rangle = V'\langle\Gamma\cdot\nabla \psi\rangle \qquad \mathrm{where~} \mathcal{V} \mathrm{~is~the~volume~enclosed~by~a~flux~surface.}
*<math> \int_{\mathcal{V}}\nabla\cdot\Gamma\; d\mathcal{V} =  \langle\Gamma\cdot\nabla V\rangle = V'\langle\Gamma\cdot\nabla \psi\rangle \qquad \mathrm{where~} \mathcal{V} \mathrm{~is~the~volume~enclosed~by~a~flux~surface.}
&lt;/math&gt;
</math>


*&lt;math&gt; \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 &lt;/math&gt;
*<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>


*&lt;math&gt; \langle \nabla \psi\cdot\nabla\times \mathbf{A} \rangle = -\langle \nabla\cdot( \nabla\psi\times\mathbf{A}) \rangle = 0~.
*<math> \langle \nabla \psi\cdot\nabla\times \mathbf{A} \rangle = -\langle \nabla\cdot( \nabla\psi\times\mathbf{A}) \rangle = 0~.
&lt;/math&gt;
</math>


*&lt;math&gt; \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> \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})
&lt;/math&gt;
</math>


*&lt;math&gt; \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> \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})
&lt;/math&gt;
</math>


*&lt;math&gt; \langle \sqrt{g}^{-1}\rangle = \frac{4\pi^2}{V'}
*<math> \langle \sqrt{g}^{-1}\rangle = \frac{4\pi^2}{V'}
&lt;/math&gt;
</math>


In the above  &lt;math&gt;V' = \frac{dV}{d\psi}&lt;/math&gt;. Some [[:Wikipedia: Vector calculus identities|vector identities]] are useful to derive the above identities.
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 ===
=== Magnetic field representation in flux coordinates ===


==== Contravariant Form ====
==== Contravariant Form ====
Any [[:Wikipedia: solenoidal vector field| solenoidal vector field]] &lt;math&gt;\mathbf{B}&lt;/math&gt; can be written as  
Any [[:Wikipedia: solenoidal vector field| solenoidal vector field]] <math>\mathbf{B}</math> can be written as  
&lt;math&gt; \mathbf{B} = \nabla\alpha\times\nabla\nu &lt;/math&gt;
<math> \mathbf{B} = \nabla\alpha\times\nabla\nu </math>
called its Clebsch representation. For a magnetic field with flux surfaces &lt;math&gt;(\psi = \mathrm{const}\; , \; \nabla\psi\cdot\mathbf{B} = 0)&lt;/math&gt; we can choose, say, &lt;math&gt;\alpha&lt;/math&gt; to be the flux surface label &lt;math&gt;\psi&lt;/math&gt;
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>
:&lt;math&gt;
:<math>
\mathbf{B} = \nabla\psi\times\nabla\nu  
\mathbf{B} = \nabla\psi\times\nabla\nu  
&lt;/math&gt;
</math>
Field lines are then given as the intersection of the constant-&lt;math&gt;\psi&lt;/math&gt; and constant-&lt;math&gt;\nu&lt;/math&gt; surfaces. This form provides a general expression for &lt;math&gt;\mathbf{B}&lt;/math&gt; in terms of the covariant basis vectors of a flux coordinate system
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
:&lt;math&gt;
:<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~.
\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~.
&lt;/math&gt;
</math>
in terms of the function &lt;math&gt;\nu&lt;/math&gt;, sometimes referred to as the magnetic field's ''stream function''.
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 &lt;math&gt; \mathbf{B} &lt;/math&gt; corresponds to a [[:Wikipedia: Magnetic potential|magnetic vector potential]]
It is worthwhile to note that the Clebsch form of <math> \mathbf{B} </math> corresponds to a [[:Wikipedia: Magnetic potential|magnetic vector potential]]
&lt;math&gt; \mathbf{A} = \nu\nabla\psi &lt;/math&gt; (or &lt;math&gt; \mathbf{A} = \psi\nabla\nu &lt;/math&gt; as they differ only by the Gauge transformation &lt;math&gt; \mathbf{A} \to \mathbf{A} - \nabla (\psi\nu)&lt;/math&gt;).
<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
The general form of the stream function is
:&lt;math&gt;
:<math>
\nu(\psi,\theta,\phi)  
\nu(\psi,\theta,\phi)  
= \frac{1}{2\pi}(\Psi_{tor}'\theta
= \frac{1}{2\pi}(\Psi_{tor}'\theta
- \Psi_{pol}'\phi)  
- \Psi_{pol}'\phi)  
+ \tilde{\nu}(\psi,\theta,\phi)
+ \tilde{\nu}(\psi,\theta,\phi)
&lt;/math&gt;
</math>
where &lt;math&gt;\tilde{\nu}&lt;/math&gt; is a differentiable function periodic in the two angles. This general form can be derived by using the fact that  &lt;math&gt; \mathbf{B}&lt;/math&gt; 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 [[Flux coordinates#Useful properties of FSA|FSA properties ]].
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 [[Flux coordinates#Useful properties of FSA|FSA properties ]].


==== Covariant Form ====
==== Covariant Form ====


If we consider an equilibrium magnetic field such that &lt;math&gt; \mathbf{j}\times\mathbf{B} \propto \nabla\psi&lt;/math&gt;, where &lt;math&gt; \mathbf{j}&lt;/math&gt; is the current density , then both &lt;math&gt; \mathbf{B}\cdot\nabla\psi = 0&lt;/math&gt; and &lt;math&gt; \nabla\times\mathbf{B}\cdot\nabla\psi = 0&lt;/math&gt; and the magnetic field can be written as
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
:&lt;math&gt;
:<math>
\mathbf{B} = \nabla\chi -\eta\nabla\psi  
\mathbf{B} = \nabla\chi -\eta\nabla\psi  
&lt;/math&gt;
</math>
where &lt;math&gt;\chi&lt;/math&gt; is identified as the magnetic ''scalar'' potential. Its general form is
where <math>\chi</math> is identified as the magnetic ''scalar'' potential. Its general form is
:&lt;math&gt;
:<math>
\chi(\psi, \theta, \phi) = \frac{I_{tor}}{2\pi}\theta + \frac{I_{pol}^d}{2\pi}\phi + \tilde\chi(\psi, \theta, \phi)
\chi(\psi, \theta, \phi) = \frac{I_{tor}}{2\pi}\theta + \frac{I_{pol}^d}{2\pi}\phi + \tilde\chi(\psi, \theta, \phi)
&lt;/math&gt;
</math>


[[Image:CurrentIntegrationCirtuits.png|256px|thumb|right|alt=Sample integration circuits for the definitions of currents.|Sample integration circuits for the current definitions.]]
[[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 contribute to the flux of charge through the surface.]]


Note that &lt;math&gt;I&lt;/math&gt; is not the current but &lt;math&gt;\mu_0&lt;/math&gt; 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
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
:&lt;math&gt;
:<math>
\int_S \mu_0\mathbf{j}\cdot d\mathbf{S}  
\int_S \mu_0\mathbf{j}\cdot d\mathbf{S}  
= \int_{\partial S}\mathbf{B}\cdot d\mathbf{l}  
= \int_{\partial S}\mathbf{B}\cdot d\mathbf{l}  
= \oint(\nabla\chi-\eta\nabla\psi)\cdot d\mathbf{l}
= \oint(\nabla\chi-\eta\nabla\psi)\cdot d\mathbf{l}
= \oint(d\chi-\eta d\psi )
= \oint(d\chi-\eta d\psi )
&lt;/math&gt;
</math>
and choosing an integration circuit contained within a flux surface &lt;math&gt;(d\psi = 0)&lt;/math&gt;. Then we get
and choosing an integration circuit contained within a flux surface <math>(d\psi = 0)</math>. Then we get
:&lt;math&gt;
:<math>
\int_S \mu_0\mathbf{j}\cdot d\mathbf{S}
\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~.
= \Delta \chi = \frac{I_{tor}}{2\pi}\Delta\theta + \frac{I_{pol}^d}{2\pi}\Delta\phi~.
&lt;/math&gt;
</math>


If we now choose a ''toroidal'' circuit &lt;math&gt;(\Delta\theta = 0, \Delta\phi = 2\pi)&lt;/math&gt; we get
If we now choose a ''toroidal'' circuit <math>(\Delta\theta = 0, \Delta\phi = 2\pi)</math> we get
:&lt;math&gt;
:<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)~.
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)~.
&lt;/math&gt;
</math>
here the superscript &lt;math&gt;d&lt;/math&gt; 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 &lt;math&gt;\Psi_{pol}&lt;/math&gt; above these lines.   
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
Similarly
:&lt;math&gt;
:<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)~.
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)~.
&lt;/math&gt;
</math>


===== Contravariant Form of the current density =====
===== Contravariant Form of the current density =====


Taking the curl of the covariant form of &lt;math&gt;\mathbf{B}&lt;/math&gt; the equilibrium current density &lt;math&gt;\mathbf{j}&lt;/math&gt; can be written as
Taking the curl of the covariant form of <math>\mathbf{B}</math> the equilibrium current density <math>\mathbf{j}</math> can be written as
: &lt;math&gt;
: <math>
\mu_0\mathbf{j} = \nabla\psi\times\nabla\eta~.
\mu_0\mathbf{j} = \nabla\psi\times\nabla\eta~.
&lt;/math&gt;
</math>
By very similar arguments as those used for &lt;math&gt;\mathbf{B}&lt;/math&gt; (note that both &lt;math&gt;\mathbf{B}&lt;/math&gt; and &lt;math&gt;\mathbf{j}&lt;/math&gt; are solenoidal fields tangent to the flux surfaces) it can be shown that the general expression for &lt;math&gt;\eta&lt;/math&gt; is
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
:&lt;math&gt;
:<math>
\eta(\psi,\theta,\phi) = \frac{1}{2\pi}({I}_{tor}'\theta
\eta(\psi,\theta,\phi) = \frac{1}{2\pi}({I}_{tor}'\theta
- {I}_{pol}'\phi)  
- {I}_{pol}'\phi)  
+ \tilde{\eta}(\psi,\theta,\phi)~.
+ \tilde{\eta}(\psi,\theta,\phi)~.
&lt;/math&gt;
</math>
Note that the poloidal current is now defined through a ribbon and not a disc. The two currents are related as &lt;math&gt;\nabla\cdot\mathbf{j} = 0&lt;/math&gt; implies
Note that the poloidal current is now defined through a ribbon and not a disc. The two currents are related as <math>\nabla\cdot\mathbf{j} = 0</math> implies
:&lt;math&gt;
:<math>
I_{pol} + I_{pol}^d = \oint_{\psi=0}\mathbf{B}\cdot d\mathbf{l} \Rightarrow I_{pol}' + (I_{pol}^d)' = 0 ~,  
I_{pol} + I_{pol}^d = \oint_{\psi=0}\mathbf{B}\cdot d\mathbf{l} \Rightarrow I_{pol}' + (I_{pol}^d)' = 0 ~,  
&lt;/math&gt;
</math>
where the integral is performed along the magnetic axis and therefore does not depend on &lt;math&gt;\psi&lt;/math&gt;. This can be used to show that a expanded version of &lt;math&gt;\mathbf{B}&lt;/math&gt; is given as
where the integral is performed along the magnetic axis and therefore does not depend on <math>\psi</math>. This can be used to show that a expanded version of <math>\mathbf{B}</math> is given as
:&lt;math&gt;
:<math>
\mathbf{B} =  -\tilde\eta\nabla\psi + \frac{I_{tor}}{2\pi}\nabla\theta + \frac{I_{pol}^d}{2\pi}\nabla\phi + \nabla\tilde\chi~.
\mathbf{B} =  -\tilde\eta\nabla\psi + \frac{I_{tor}}{2\pi}\nabla\theta + \frac{I_{pol}^d}{2\pi}\nabla\phi + \nabla\tilde\chi~.
&lt;/math&gt;
</math>


== 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. 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
:&lt;math&gt;
:<math>
\mathbf{B} =  \nabla\psi\times \nabla\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}\phi_f \right)
           = \frac{\Psi_{pol}'}{2\pi\sqrt{g}}\mathbf{e}_\theta + \frac{\Psi_{tor}'}{2\pi\sqrt{g}}\mathbf{e}_\phi~.
           = \frac{\Psi_{pol}'}{2\pi\sqrt{g}}\mathbf{e}_\theta + \frac{\Psi_{tor}'}{2\pi\sqrt{g}}\mathbf{e}_\phi~.
&lt;/math&gt;
</math>
Now a field line is given by &lt;math&gt;\psi = \psi_0&lt;/math&gt; and &lt;math&gt;\Psi_{tor}'\theta_f - \Psi_{pol}'\phi_f =  2\pi\nu_0&lt;/math&gt;.
Now a field line is given by <math>\psi = \psi_0</math> and <math>\Psi_{tor}'\theta_f - \Psi_{pol}'\phi_f =  2\pi\nu_0</math>.


Note that, in general, the contravariant components of the magnetic field in a magnetic coordinate system
Note that, in general, the contravariant components of the magnetic field in a magnetic coordinate system
:&lt;math&gt;
:<math>
B^{\theta_f} = \frac{\Psi_{pol}'}{2\pi\sqrt{g}}\; ;\quad B^{\phi_f} = \frac{\Psi_{tor}'}{2\pi\sqrt{g}}
B^{\theta_f} = \frac{\Psi_{pol}'}{2\pi\sqrt{g}}\; ;\quad B^{\phi_f} = \frac{\Psi_{tor}'}{2\pi\sqrt{g}}
&lt;/math&gt;
</math>
are not flux functions, but their quotient is  
are not flux functions, but their quotient is  
:&lt;math&gt;
:<math>
\frac{B^{\theta_f}}{B^{\phi_f}} = \frac{\Psi_{pol}'}{\Psi_{tor}'} \equiv \frac{\iota}{2\pi}~,
\frac{B^{\theta_f}}{B^{\phi_f}} = \frac{\Psi_{pol}'}{\Psi_{tor}'} \equiv \frac{\iota}{2\pi}~,
&lt;/math&gt;
</math>
&lt;math&gt;\iota&lt;/math&gt; being the [[rotational transform]]. In a magnetic coordinate system the ''poloidal'' &lt;math&gt; \mathbf{B}_P = B^\theta\mathbf{e}_\theta &lt;/math&gt; and ''toroidal'' &lt;math&gt; \mathbf{B}_T = B^\phi\mathbf{e}_\phi&lt;/math&gt; components of the magnetic field are individually divergence-less.
<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  &lt;math&gt; \mathbf{B} &lt;/math&gt; in magnetic coordinates it is easy to obtain the following identities valid for any magnetic coordinate system
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
:&lt;math&gt;
:<math>
\mathbf{e}_\theta\times\mathbf{B} =\frac{1}{2\pi}\nabla\Psi_{tor}~,
\mathbf{e}_\theta\times\mathbf{B} =\frac{1}{2\pi}\nabla\Psi_{tor}~,
&lt;/math&gt;
</math>
:&lt;math&gt;
:<math>
\mathbf{e}_\phi\times\mathbf{B} = -\frac{1}{2\pi}\nabla\Psi_{pol} ~.
\mathbf{e}_\phi\times\mathbf{B} = -\frac{1}{2\pi}\nabla\Psi_{pol} ~.
&lt;/math&gt;
</math>
=== Transforming between Magnetic coordinates systems ===
=== Transforming between Magnetic coordinates systems ===
There are infinitely many systems of magnetic coordinates. Any transformation of the angles of the from
There are infinitely many systems of magnetic coordinates. Any transformation of the angles of the from
:&lt;math&gt;
:<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)
\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)
&lt;/math&gt;
</math>


where &lt;math&gt;G&lt;/math&gt; is periodic in the angles, preserves the straightness of the field lines (as can be easily checked by direct substitution). The spatial function &lt;math&gt;G(\psi, \theta_f, \phi_f)&lt;/math&gt;, is called the ''generating function''. It can be obtained from a [[magnetic differential equation]] if we know the Jacobians of the two magnetic coordinate systems &lt;math&gt; \sqrt{g_f}&lt;/math&gt; and &lt;math&gt; \sqrt{g_F}&lt;/math&gt;. In fact taking &lt;math&gt;\mathbf{B}\cdot\nabla&lt;/math&gt; on any of the transformation of the angles and using the known expressions for the contravariant components of &lt;math&gt;\mathbf{B}&lt;/math&gt; in magnetic coordinates we get
where <math>G</math> is periodic in the angles, preserves the straightness of the field lines (as can be easily checked by direct substitution). The spatial function <math>G(\psi, \theta_f, \phi_f)</math>, is called the ''generating function''. It can be obtained from a [[magnetic differential equation]] if we know the Jacobians of the two magnetic coordinate systems <math> \sqrt{g_f}</math> and <math> \sqrt{g_F}</math>. In fact taking <math>\mathbf{B}\cdot\nabla</math> on any of the transformation of the angles and using the known expressions for the contravariant components of <math>\mathbf{B}</math> in magnetic coordinates we get
:&lt;math&gt;
:<math>
  2\pi\mathbf{B}\cdot\nabla G = \frac{1}{\sqrt{g_F}} - \frac{1}{\sqrt{g_f}}~.
  2\pi\mathbf{B}\cdot\nabla G = \frac{1}{\sqrt{g_F}} - \frac{1}{\sqrt{g_f}}~.
&lt;/math&gt;
</math>
The LHS of this equation has a particularly simple form when one uses a magnetic coordinate system. For instance, if we write &lt;math&gt;\mathbf{B}&lt;/math&gt; in terms of the original magnetic coordinate system we get
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
:&lt;math&gt;
:<math>
  (\Psi_{pol}'\partial_{\theta_f} + \Psi_{tor}'\partial_{\phi_f}) G = \frac{\sqrt{g_f}}{\sqrt{g_F}} - 1~.
  (\Psi_{pol}'\partial_{\theta_f} + \Psi_{tor}'\partial_{\phi_f}) G = \frac{\sqrt{g_f}}{\sqrt{g_F}} - 1~.
&lt;/math&gt;
</math>
which can be turned into an algebraic equation on the Fourier components of &lt;math&gt;G&lt;/math&gt;
which can be turned into an algebraic equation on the Fourier components of <math>G</math>
:&lt;math&gt;
:<math>
   G_{nm} = \frac{-i}{\Psi_{pol}'n + \Psi_{tor}'m}\left(\frac{\sqrt{g_f}}{\sqrt{g_F}}\right)_{nm}~.
   G_{nm} = \frac{-i}{\Psi_{pol}'n + \Psi_{tor}'m}\left(\frac{\sqrt{g_f}}{\sqrt{g_F}}\right)_{nm}~.
&lt;/math&gt;
</math>
where  
where  
:&lt;math&gt;
:<math>
  G(\psi, \theta_f, \phi_f) = \sum_{n,m} G_{nm}(\psi) e^{i(n\theta_f + m\phi_f)}
  G(\psi, \theta_f, \phi_f) = \sum_{n,m} G_{nm}(\psi) e^{i(n\theta_f + m\phi_f)}
&lt;/math&gt;
</math>
and &lt;math&gt;G_{00} = 0 &lt;/math&gt;.
and <math>G_{00} = 0 </math>.
   
   


Particular choices of G can be made so as to simplify the description of other fields. The most commonly used magnetic coordinate systems are:
Particular choices of G can be made so as to simplify the description of other fields. The most commonly used magnetic coordinate systems are:
&lt;ref name='Dhaeseleer'&gt;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&lt;/ref&gt;
<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]]. &lt;ref&gt;S. Hamada, Nucl. Fusion '''2''' (1962) 23&lt;/ref&gt;&lt;ref&gt;[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]&lt;/ref&gt; In these coordinates, both the magnetic field lines and current lines corresponding to the [[MHD equilibrium]] are straight. Referring to the definitions above, both &lt;math&gt;\tilde\nu&lt;/math&gt; and &lt;math&gt;\tilde\eta&lt;/math&gt; are zero in Hamada coordinates.
* [[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 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]]. &lt;ref&gt;[http://dx.doi.org/10.1063/1.863297 A.H. Boozer, ''Plasma equilibrium with rational magnetic surfaces'', Phys. Fluids '''24''' (1981) 1999]&lt;/ref&gt;&lt;ref&gt;[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]&lt;/ref&gt; 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 &lt;math&gt;\nabla\psi\times\mathbf{B}&lt;/math&gt;. Referring to the definitions above, both &lt;math&gt;\tilde\nu&lt;/math&gt; and &lt;math&gt;\tilde\chi&lt;/math&gt; are zero in Boozer 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 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 ==
&lt;references /&gt;
<references />

Revision as of 13:21, 24 November 2010

General curvilinear coordinates

Here we briefly review the basic definitions of a general curvilinear coordinate system for later convenience when discussing toroidal flux coordinates and magnetic coordinates.

Coordinates and basis vectors

Let be a set of euclidean coordinates on and let define a change of coordinates, arbitrary for the time being. We can calculate the contravariant basis vectors as

and the dual covariant basis defined as

where are cyclic permutations of and we have used the notation . The Jacobian is defined below.

Any vector field can be represented as

or

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B} = (\mathbf{B}\cdot\mathbf{e}_i)\mathbf{e}^i = B_i\mathbf{e}^i ~. }

In particular any basis vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}_i = (\mathbf{e}_i\cdot\mathbf{e}_j)\mathbf{e}^j} . The metric tensor is defined as

The metric tensors can be used to raise or lower indices. Take

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B} = B_i\mathbf{e}^i = B_i g^{ij}\mathbf{e}_j = B^j\mathbf{e}_j~, }

so that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B^j = g^{ij} B_i~. }

Jacobian

The Jacobian of the coordinate transformation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{x}(\psi, \theta, \phi)} is defined as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}} }

and that of the inverse transformation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J^{-1} = \det\left(\frac{\partial(\psi,\theta,\phi)}{\partial(x,y,z)}\right) = \nabla{\psi}\cdot\nabla{\theta} \times \nabla{\phi} }

It can be seen that [1] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \equiv \det(g_{ij}) = J^2 \Rightarrow J = \sqrt{g}}

Some surface elements

Consider a surface defined by a constant value of . Then, the surface element is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d{\mathbf S}_\phi = \mathbf{e}_\psi\times\mathbf{e}_\theta d\psi d\theta = \sqrt{g}\, \nabla\phi d\psi d\theta . }

As for a surface defined by a constant value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d{\mathbf S}_\theta = \mathbf{e}_\phi\times\mathbf{e}_\psi d\psi d\phi = \sqrt{g}\, \nabla\theta d\psi d\phi , }

or a constant surface:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d{\mathbf S}_\psi = \mathbf{e}_\theta\times\mathbf{e}_\phi d\theta d\phi = \sqrt{g}\, \nabla\psi d\theta d\phi . }

Gradient, Divergence and Curl in curvilinear coordinates

The gradient of a function f is naturally given in the contravariant basis vectors:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla f = \frac{\partial f}{\partial u^i}\nabla u^i = \frac{\partial f}{\partial u^i}\mathbf{e}^i~. }

The divergence of a vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{A}} is best expressed in terms of its contravariant components

while the curl is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla\times\mathbf{A} = \frac{\varepsilon_{ijk}}{\sqrt{g}}\frac{\partial}{\partial u^i}(\sqrt{g}A_j)\mathbf{e}_k }

given in terms of the covariant base vectors, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon_{ijk}} is the [[::Wikipedia:Levi-Civita symbol| Levi-Civita]] symbol.

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, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi} is the flux surface label and are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi} -periodic poloidal and toroidal-like angles.

Different flux surface labels can be chosen like toroidal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Psi_{tor})} or poloidal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Psi_{pol})} magnetic fluxes or the volume contained within the flux surface . By single valued we mean to ensure that any flux label Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_1 = f(\psi_2)} is a monotonous function of any other flux label Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_2} , so that the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is invertible at least in a volume containing the region of interest. We will denote a generic flux surface label by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi} .

To avoid ambiguity in the sign of line and surface integrals we impose , the toroidal angle increases in the clockwise direction when seen from above and the poloidal angle increases such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla\psi\cdot\nabla\theta\times\nabla\phi > 0} .

Flux Surface Average

The Flux Surface Average (FSA) of a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi} is defined as the limit

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\Phi\rangle = \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\int_{\delta \mathcal{V}} \Phi\; d\mathcal{V} }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta \mathcal{V}} 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 . Capital Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is reserved for the flux label (coordinate) defined as the volume within a flux surface.

Introducing the differential volume element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\mathcal{V} = \sqrt{g} d\psi d\theta d\phi}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 }

or, noting that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle 1\rangle = 1} , we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dV}{d\psi} = \int_0^{2\pi}\int_0^{2\pi} \sqrt{g} d\theta d\phi} and we get to a more practical form of the Flux Surface Average

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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} }

Note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dS = |\nabla\psi|\sqrt{g}d\theta d\phi} , so the FSA is a surface integral weighted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\nabla V|^{-1}}  :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 }

Applying Gauss' theorem to the definition of FSA we get to the identity

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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~. }

Useful properties of FSA

Some useful properties of the FSA are

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathcal{V}}\nabla\cdot\Gamma\; d\mathcal{V} = \langle\Gamma\cdot\nabla V\rangle = V'\langle\Gamma\cdot\nabla \psi\rangle \qquad \mathrm{where~} \mathcal{V} \mathrm{~is~the~volume~enclosed~by~a~flux~surface.} }
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}) }
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}) }
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \sqrt{g}^{-1}\rangle = \frac{4\pi^2}{V'} }

In the above Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V' = \frac{dV}{d\psi}} . Some vector identities are useful to derive the above identities.

Magnetic field representation in flux coordinates

Contravariant Form

Any solenoidal vector field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}} can be written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B} = \nabla\alpha\times\nabla\nu } called its Clebsch representation. For a magnetic field with flux surfaces Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\psi = \mathrm{const}\; , \; \nabla\psi\cdot\mathbf{B} = 0)} we can choose, say, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} to be the flux surface label Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B} = \nabla\psi\times\nabla\nu }

Field lines are then given as the intersection of the constant-Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi} and constant-Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu} surfaces. This form provides a general expression for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}} in terms of the covariant basis vectors of a flux coordinate system

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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~. }

in terms of the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu} , sometimes referred to as the magnetic field's stream function.

It is worthwhile to note that the Clebsch form of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B} } corresponds to a magnetic vector potential Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{A} = \nu\nabla\psi } (or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{A} = \psi\nabla\nu } as they differ only by the Gauge transformation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{A} \to \mathbf{A} - \nabla (\psi\nu)} ).

The general form of the stream function is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu(\psi,\theta,\phi) = \frac{1}{2\pi}(\Psi_{tor}'\theta - \Psi_{pol}'\phi) + \tilde{\nu}(\psi,\theta,\phi) }

where is a differentiable function periodic in the two angles. This general form can be derived by using the fact that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}} 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 FSA properties .

Covariant Form

If we consider an equilibrium magnetic field such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{j}\times\mathbf{B} \propto \nabla\psi} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{j}} is the current density , then both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}\cdot\nabla\psi = 0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla\times\mathbf{B}\cdot\nabla\psi = 0} and the magnetic field can be written as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B} = \nabla\chi -\eta\nabla\psi }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} is identified as the magnetic scalar potential. Its general form is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi(\psi, \theta, \phi) = \frac{I_{tor}}{2\pi}\theta + \frac{I_{pol}^d}{2\pi}\phi + \tilde\chi(\psi, \theta, \phi) }
Sample integration circuits for the definitions of currents.
Sample integration circuits for the current definitions.
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.

Note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} is not the current but Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_0} 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_S \mu_0\mathbf{j}\cdot d\mathbf{S} = \int_{\partial S}\mathbf{B}\cdot d\mathbf{l} = \oint(\nabla\chi-\eta\nabla\psi)\cdot d\mathbf{l} = \oint(d\chi-\eta d\psi ) }

and choosing an integration circuit contained within a flux surface Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (d\psi = 0)} . Then we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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~. }

If we now choose a toroidal circuit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Delta\theta = 0, \Delta\phi = 2\pi)} we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)~. }

here the superscript Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi_{pol}} above these lines. Similarly

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)~. }
Contravariant Form of the current density

Taking the curl of the covariant form of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}} the equilibrium current density Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{j}} can be written as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_0\mathbf{j} = \nabla\psi\times\nabla\eta~. }

By very similar arguments as those used for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}} (note that both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{j}} are solenoidal fields tangent to the flux surfaces) it can be shown that the general expression for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta} is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta(\psi,\theta,\phi) = \frac{1}{2\pi}({I}_{tor}'\theta - {I}_{pol}'\phi) + \tilde{\eta}(\psi,\theta,\phi)~. }

Note that the poloidal current is now defined through a ribbon and not a disc. The two currents are related as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla\cdot\mathbf{j} = 0} implies

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{pol} + I_{pol}^d = \oint_{\psi=0}\mathbf{B}\cdot d\mathbf{l} \Rightarrow I_{pol}' + (I_{pol}^d)' = 0 ~, }

where the integral is performed along the magnetic axis and therefore does not depend on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi} . This can be used to show that a expanded version of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}} is given as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B} = -\tilde\eta\nabla\psi + \frac{I_{tor}}{2\pi}\nabla\theta + \frac{I_{pol}^d}{2\pi}\nabla\phi + \nabla\tilde\chi~. }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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~. }

Now a field line is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi = \psi_0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi_{tor}'\theta_f - \Psi_{pol}'\phi_f = 2\pi\nu_0} .

Note that, in general, the contravariant components of the magnetic field in a magnetic coordinate system

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B^{\theta_f} = \frac{\Psi_{pol}'}{2\pi\sqrt{g}}\; ;\quad B^{\phi_f} = \frac{\Psi_{tor}'}{2\pi\sqrt{g}} }

are not flux functions, but their quotient is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{B^{\theta_f}}{B^{\phi_f}} = \frac{\Psi_{pol}'}{\Psi_{tor}'} \equiv \frac{\iota}{2\pi}~, }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iota} being the rotational transform. In a magnetic coordinate system the poloidal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}_P = B^\theta\mathbf{e}_\theta } and toroidal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}_T = B^\phi\mathbf{e}_\phi} components of the magnetic field are individually divergence-less.

From the above general form of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B} } in magnetic coordinates it is easy to obtain the following identities valid for any magnetic coordinate system

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}_\theta\times\mathbf{B} =\frac{1}{2\pi}\nabla\Psi_{tor}~, }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}_\phi\times\mathbf{B} = -\frac{1}{2\pi}\nabla\Psi_{pol} ~. }

Transforming between Magnetic coordinates systems

There are infinitely many systems of magnetic coordinates. Any transformation of the angles of the from

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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) }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is periodic in the angles, preserves the straightness of the field lines (as can be easily checked by direct substitution). The spatial function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(\psi, \theta_f, \phi_f)} , is called the generating function. It can be obtained from a magnetic differential equation if we know the Jacobians of the two magnetic coordinate systems and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{g_F}} . In fact taking Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}\cdot\nabla} on any of the transformation of the angles and using the known expressions for the contravariant components of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}} in magnetic coordinates we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi\mathbf{B}\cdot\nabla G = \frac{1}{\sqrt{g_F}} - \frac{1}{\sqrt{g_f}}~. }

The LHS of this equation has a particularly simple form when one uses a magnetic coordinate system. For instance, if we write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}} in terms of the original magnetic coordinate system we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Psi_{pol}'\partial_{\theta_f} + \Psi_{tor}'\partial_{\phi_f}) G = \frac{\sqrt{g_f}}{\sqrt{g_F}} - 1~. }

which can be turned into an algebraic equation on the Fourier components of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_{nm} = \frac{-i}{\Psi_{pol}'n + \Psi_{tor}'m}\left(\frac{\sqrt{g_f}}{\sqrt{g_F}}\right)_{nm}~. }

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(\psi, \theta_f, \phi_f) = \sum_{n,m} G_{nm}(\psi) e^{i(n\theta_f + m\phi_f)} }

and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_{00} = 0 } .


Particular choices of G can be made so as to simplify the description of other fields. The most commonly used magnetic coordinate systems are: [1]

  • Hamada coordinates. [2][3] In these coordinates, both the magnetic field lines and current lines corresponding to the MHD equilibrium are straight. Referring to the definitions above, both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde\nu} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde\eta} are zero in Hamada coordinates.
  • Boozer coordinates. [4][5] 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla\psi\times\mathbf{B}} . Referring to the definitions above, both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde\nu} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde\chi} are zero in Boozer coordinates.

References

  1. ↑ 1.0 1.1 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
  2. ↑ S. Hamada, Nucl. Fusion 2 (1962) 23
  3. ↑ J.M. Greene and J.L Johnson, Stability Criterion for Arbitrary Hydromagnetic Equilibria, Phys. Fluids 5 (1962) 510
  4. ↑ A.H. Boozer, Plasma equilibrium with rational magnetic surfaces, Phys. Fluids 24 (1981) 1999
  5. ↑ A.H. Boozer, Establishment of magnetic coordinates for a given magnetic field, Phys. Fluids 25 (1982) 520
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