Flux coordinates
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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 <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
- <math>
\mathbf{e}^i = \{\nabla\psi, \nabla\theta, \nabla\phi\} </math> and the dual covariant basis defined as
- <math>
\mathbf{e}_i= \frac{\partial\mathbf{x}}{\partial{u^i}} \to \mathbf{e}_i\cdot\mathbf{e}^j = \delta_{i}^{j} \to \mathbf{e}_i = \frac{\mathbf{e}^j\times\mathbf{e}^k}{|\mathbf{e}^i\cdot\mathbf{e}^j\times\mathbf{e}^k|} = \sqrt{g}\;\mathbf{e}^j\times\mathbf{e}^k ~, </math> where <math>(i,j,k)</math> are cyclic permutations of <math>(1,2,3)</math> and we have used the notation <math>(u^1, u^2, u^3) = (\psi,\theta,\phi)</math>. The Jacobian <math>\sqrt{g}</math> is defined below.
Any vector field <math>\mathbf{B}</math> can be represented as
- <math>
\mathbf{B} = (\mathbf{B}\cdot\mathbf{e}^i)\mathbf{e}_i = B^i\mathbf{e}_i </math> or
- <math>
\mathbf{B} = (\mathbf{B}\cdot\mathbf{e}_i)\mathbf{e}^i = B_i\mathbf{e}^i ~. </math> In particular any basis vector <math>\mathbf{e}_i = (\mathbf{e}_i\cdot\mathbf{e}_j)\mathbf{e}^j</math>. The metric tensor is defined as
- <math>
g_{ij} = \mathbf{e}_i\cdot\mathbf{e}_j \; ; \; g^{ij} = \mathbf{e}^i\cdot\mathbf{e}^j \; ; \; g^j_i = \mathbf{e}_i\cdot\mathbf{e}^j = \delta_i^j ~. </math>
The metric tensors can be used to raise or lower indices. Take
- <math>
\mathbf{B} = B_i\mathbf{e}^i = B_i g^{ij}\mathbf{e}_j = B^j\mathbf{e}_j~, </math> so that
- <math>
B^j = g^{ij} B_i~. </math>
Jacobian
The Jacobian of the coordinate transformation <math>\mathbf{x}(\psi, \theta, \phi)</math> is defined as
- <math>
J = \det\left(\frac{\partial(x,y,z)}{\partial(\psi,\theta,\phi)}\right) = \frac{\partial\mathbf{x}}{\partial{\psi}}\cdot\frac{\partial\mathbf{x}}{\partial{\theta}} \times \frac{\partial\mathbf{x}}{\partial{\phi}} </math> and that of the inverse transformation
- <math>
J^{-1} = \det\left(\frac{\partial(\psi,\theta,\phi)}{\partial(x,y,z)}\right) = \nabla{\psi}\cdot\nabla{\theta} \times \nabla{\phi} </math> It can be seen that <ref name='Dhaeseleer'></ref> <math>g \equiv \det(g_{ij}) = J^2 \Rightarrow J = \sqrt{g}</math>
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 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 <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)~, </math> 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
</math> 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 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\phi > 0</math>.
Flux Surface Average
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} </math> where <math>\delta \mathcal{V}</math> is the volume confined between two flux surfaces. It is therefore a volume average over an infinitesimal spatial region rather than a surface average. To avoid confusion, we denote volume elements or domains with the calligraphic <math>\mathcal{V}</math>. Capital <math>V</math> is reserved for the flux label (coordinate) defined as the volume within a flux surface.
Introducing the differential volume element <math>d\mathcal{V} = \sqrt{g} d\psi d\theta d\phi</math>
- <math>
\langle\Phi\rangle = \lim_{\delta \mathcal{V} \to 0} \frac{1}{\delta \mathcal{V}}\int_{\delta \mathcal{V}} \Phi\; \sqrt{g} d\psi d\theta d\phi = \frac{d\psi}{d V}\int_0^{2\pi}\int_0^{2\pi}\Phi\; \sqrt{g} d\theta d\phi </math> or, noting that <math>\langle 1\rangle = 1</math>, we have <math>\frac{dV}{d\psi} = \int_0^{2\pi}\int_0^{2\pi} \sqrt{g} d\theta d\phi</math> and we get to a more practical form of the Flux Surface Average
- <math>
\langle\Phi\rangle = \frac{\int_0^{2\pi}\int_0^{2\pi}\Phi\; \sqrt{g} d\theta d\phi} {\int_0^{2\pi}\int_0^{2\pi} \sqrt{g} d\theta d\phi} </math>
Note that <math>dS = |\nabla\psi|\sqrt{g}d\theta d\phi</math>, so the FSA is a surface integral weighted by <math>|\nabla V|^{-1}</math> :
- <math>
\langle\Phi\rangle = \frac{d\psi}{d V}\int_0^{2\pi}\int_0^{2\pi}\Phi\; \sqrt{g} d\theta d\phi = \frac{d\psi}{d V}\int_{S(\psi)}\frac{\Phi}{|\nabla\psi|}\; dS = \int_{S(\psi)}\frac{\Phi}{|\nabla V|}\; dS </math>
Applying Gauss' theorem to the definition of FSA we get to the identity
- <math>
\langle\nabla\cdot\Gamma\rangle = \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\int_{\delta \mathcal{V}} \nabla\cdot\Gamma\; d\mathcal{V} = \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\int_{S(\delta \mathcal{V})} \Gamma\cdot \frac{\nabla V}{|\nabla V|}dS = \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\left(\langle\Gamma\cdot\nabla V\rangle_{S(V+\delta \mathcal{V})} - \langle\Gamma\cdot\nabla V\rangle_{S(V)} \right) = \frac{d}{dV}\langle\Gamma\cdot\nabla V\rangle~. </math>
Useful properties of FSA
Some useful properties of the FSA are
- <math> \langle\nabla\cdot\Gamma\rangle = \frac{d}{dV}\langle\Gamma\cdot\nabla V\rangle = \frac{1}{V'}\frac{d}{d\psi}V'\langle\Gamma\cdot\nabla \psi\rangle</math>
- <math> \int_{\mathcal{V}}\nabla\cdot\Gamma\; d\mathcal{V} = \langle\Gamma\cdot\nabla V\rangle = V'\langle\Gamma\cdot\nabla \psi\rangle \qquad \mathrm{where~} \mathcal{V} \mathrm{~is~the~volume~enclosed~by~a~flux~surface.}
</math>
- <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 \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{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{for~any~toroidal~ angle~} \phi ~ (\mathrm{Note:}~ \phi(\mathbf{x})~\mathrm{is~not~single~valued})
</math>
- <math> \langle \sqrt{g}^{-1}\rangle = \frac{4\pi^2}{V'}
</math>
In the above <math>V' = \frac{dV}{d\psi}</math>. Some vector identities are useful to derive the above identities.
Magnetic field representation in flux coordinates
Contravariant Form
Any solenoidal vector field <math>\mathbf{B}</math> can be written as <math> \mathbf{B} = \nabla\alpha\times\nabla\nu </math> called its Clebsch representation. For a magnetic field with flux surfaces <math>(\psi = \mathrm{const}\; , \; \nabla\psi\cdot\mathbf{B} = 0)</math> we can choose, say, <math>\alpha</math> to be the flux surface label <math>\psi</math>
- <math>
\mathbf{B} = \nabla\psi\times\nabla\nu </math> Field lines are then given as the intersection of the constant-<math>\psi</math> and constant-<math>\nu</math> surfaces. This form provides a general expression for <math>\mathbf{B}</math> in terms of the covariant basis vectors of a flux coordinate system
- <math>
\mathbf{B} = \frac{\partial\nu}{\partial\theta}\nabla\psi\times\nabla\theta + \frac{\partial\nu}{\partial\phi}\nabla\psi\times\nabla\phi = \frac{1}{\sqrt{g}}\frac{\partial\nu}{\partial\theta}\mathbf{e}_\phi -\frac{1}{\sqrt{g}}\frac{\partial\nu}{\partial\phi}\mathbf{e}_\theta = B^\phi\mathbf{e}_\phi + B^\theta\mathbf{e}_\theta~. </math> in terms of the function <math>\nu</math>, sometimes referred to as the magnetic field's stream function.
It is worthwhile to note that the Clebsch form of <math> \mathbf{B} </math> corresponds to a magnetic vector potential <math> \mathbf{A} = \nu\nabla\psi </math> (or <math> \mathbf{A} = \psi\nabla\nu </math> as they differ only by the Gauge transformation <math> \mathbf{A} \to \mathbf{A} - \nabla (\psi\nu)</math>).
The general form of the stream function is
- <math>
\nu(\psi,\theta,\phi) = \frac{1}{2\pi}(\Psi_{tor}'\theta - \Psi_{pol}'\phi) + \tilde{\nu}(\psi,\theta,\phi) </math> where <math>\tilde{\nu}</math> is a differentiable function periodic in the two angles. This general form can be derived by using the fact that <math> \mathbf{B}</math> is a physical function (hence singe-valued). The specific form for the coefficients in front of the secular terms (i.e. the non-periodic terms) can be obtained from the FSA properties .
Covariant Form
If we consider an equilibrium magnetic field such that <math> \mathbf{j}\times\mathbf{B} \propto \nabla\psi</math>, where <math> \mathbf{j}</math> is the current density , then both <math> \mathbf{B}\cdot\nabla\psi = 0</math> and <math> \nabla\times\mathbf{B}\cdot\nabla\psi = 0</math> and the magnetic field can be written as
- <math>
\mathbf{B} = \nabla\chi -\eta\nabla\psi </math> where <math>\chi</math> is identified as the magnetic scalar potential. Its general form is
- <math>
\chi(\psi, \theta, \phi) = \frac{I_{tor}}{2\pi}\theta + \frac{I_{pol}^d}{2\pi}\phi + \tilde\chi(\psi, \theta, \phi) </math>
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} = \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 ) </math> and choosing an integration circuit contained within a flux surface <math>(d\psi = 0)</math>. Then we get
- <math>
\int_S \mu_0\mathbf{j}\cdot d\mathbf{S} = \Delta \chi = \frac{I_{tor}}{2\pi}\Delta\theta + \frac{I_{pol}^d}{2\pi}\Delta\phi~. </math>
If we now 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)~. </math> here the superscript <math>d</math> is meant to indicate the flux is computed through a disc limited by the integration line, as opposed to the ribbon limited by the integration line on one side and the magnetic axis on the other that was used for the definition of poloidal magnetic flux <math>\Psi_{pol}</math> above these lines. Similarly
- <math>
I_{tor} = \int_S \mu_0\mathbf{j}\cdot d\mathbf{S}\; ; ~\mathrm{with}~ \partial S ~\mathrm{such~that}~ (\Delta\theta = 2\pi, \Delta\phi = 0)~. </math>
Contravariant Form of the current density
Taking the curl of the covariant form of <math>\mathbf{B}</math> the equilibrium current density <math>\mathbf{j}</math> can be written as
- <math>
\mu_0\mathbf{j} = \nabla\psi\times\nabla\eta~. </math> By very similar arguments as those used for <math>\mathbf{B}</math> (note that both <math>\mathbf{B}</math> and <math>\mathbf{j}</math> are solenoidal fields tangent to the flux surfaces) it can be shown that the general expression for <math>\eta</math> is
- <math>
\eta(\psi,\theta,\phi) = \frac{1}{2\pi}({I}_{tor}'\theta - {I}_{pol}'\phi) + \tilde{\eta}(\psi,\theta,\phi)~. </math> Note that the poloidal current is now defined through a ribbon and not a disc. The two currents are related as <math>\nabla\cdot\mathbf{j} = 0</math> implies
- <math>
I_{pol} + I_{pol}^d = \oint_{\psi=0}\mathbf{B}\cdot d\mathbf{l} \Rightarrow I_{pol}' + (I_{pol}^d)' = 0 ~, </math> 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
- <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~. </math>
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
- <math>
\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~.
</math> 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
- <math>
B^{\theta_f} = \frac{\Psi_{pol}'}{2\pi\sqrt{g}}\; ;\quad B^{\phi_f} = \frac{\Psi_{tor}'}{2\pi\sqrt{g}} </math> are not flux functions, but their quotient is
- <math>
\frac{B^{\theta_f}}{B^{\phi_f}} = \frac{\Psi_{pol}'}{\Psi_{tor}'} \equiv \frac{\iota}{2\pi}~, </math> <math>\iota</math> being the rotational transform. 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
- <math>
\theta_F = \theta_f +\Psi_{pol}' G(\psi, \theta_f, \phi_f)\; ;\quad \phi_F = \phi_f +\Psi_{tor}' G(\psi, \theta_f, \phi_f) </math>
where <math>G</math> is periodic in the angles, preserves the straightness of the field lines (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
- <math>
2\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_{nm} = \frac{-i}{\Psi_{pol}'n + \Psi_{tor}'m}\left(\frac{\sqrt{g_f}}{\sqrt{g_F}}\right)_{nm}~.
</math> where
- <math>
G(\psi, \theta_f, \phi_f) = \sum_{n,m} G_{nm}(\psi) e^{i(n\theta_f + m\phi_f)}
</math> 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:
<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>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>A.H. Boozer, Plasma equilibrium with rational magnetic surfaces, Phys. Fluids 24 (1981) 1999</ref><ref>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
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