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Flux coordinates: Difference between revisions
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while the curl is | while the curl is | ||
:<math> | :<math> | ||
\nabla\times\mathbf{A} = \frac{\ | \nabla\times\mathbf{A} = \frac{\varepsilon^{ijk}}{\sqrt{g}}\frac{\partial A_j}{\partial u^i}\mathbf{e}_k \Rightarrow | ||
\left(\nabla\times\mathbf{A}\right)^k = \frac{\ | \left(\nabla\times\mathbf{A}\right)^k = \frac{\varepsilon^{ijk}}{\sqrt{g}}\frac{\partial A_j}{\partial u^i} | ||
</math> | </math> | ||
given in terms of the covariant base vectors, where <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 == | == Flux coordinates == | ||
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==== Useful properties of FSA ==== | ==== Useful properties of FSA ==== | ||
Some useful properties of the FSA are | Some useful properties of the FSA are | ||
*<math> \langle \mathbf{B}\cdot\nabla f \rangle = \langle \nabla\cdot(\mathbf{B} f) \rangle = 0~,\qquad \forall~ \mathrm{single~valued~} f(\mathbf{x}), ~ \mathrm{if}~ \nabla\cdot\mathbf{B} = 0 ~\mathrm{and}~ \nabla \psi\cdot\mathbf{B} = 0 </math> | |||
*<math> \langle\nabla\cdot\Gamma\rangle = \frac{d}{dV}\langle\Gamma\cdot\nabla V\rangle = \frac{1}{V'}\frac{d}{d\psi}V'\langle\Gamma\cdot\nabla \psi\rangle</math> | *<math> \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> | ||
The two identities above are the basic simplifying properties of the FSA: The first cancels the contribution of 'conservative forces' like the pressure gradient or electrostatic electric fields. The second reduces the number of spatial variables to only the radial one. Further, it is possible to show that, if <math>\nabla\cdot\Gamma = 0</math> then <math>\langle\Gamma\cdot\nabla V\rangle = 0</math> and not simply constant as the above | The two identities above are the basic simplifying properties of the FSA: The first cancels the contribution of 'conservative forces' like the pressure gradient or electrostatic electric fields. The second reduces the number of spatial variables to only the radial one. Further, it is possible to show that, if <math>\nabla\cdot\Gamma = 0</math> then <math>\langle\Gamma\cdot\nabla V\rangle = 0</math> and not simply constant as the above suggests. This can be seen by simply using Gauss' theorem | ||
*<math> \int_{\mathcal{V}}\nabla\cdot\Gamma\; d\mathcal{V} = \langle\Gamma\cdot\nabla V\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} = | ||
\int_{S(\mathcal{V})}\Gamma\cdot\nabla V \frac{dS}{|\nabla V|} = | |||
\langle\Gamma\cdot\nabla V\rangle \qquad \mathrm{where~} \mathcal{V} \mathrm{~is~the~volume~enclosed~by~a~flux~surface.} | |||
</math> | </math> | ||
The FSA relates to the conventional volume integral as | The FSA relates to the conventional volume integral between two surfaces labelled by their volumes <math>V_1</math> and <math>V_2</math> as | ||
*<math> \int_{\mathcal{V}(V_1<V<V_2)} f\; d\mathcal{V} = \int_{V_1}^{V_2} \langle f \rangle\; dV | *<math> \int_{\mathcal{V}(V_1<V<V_2)} f\; d\mathcal{V} = \int_{V_1}^{V_2} \langle f \rangle\; dV | ||
</math> | </math> | ||
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[[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 (those enclosing the one drawn here) contribute to the flux of charge through the surface.]] | ||
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 | 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 | ||
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which can be turned into an algebraic equation on the Fourier components of <math>G</math> | which can be turned into an algebraic equation on the Fourier components of <math>G</math> | ||
:<math> | :<math> | ||
G_{ | G_{mn} = \frac{-i}{\Psi_{pol}'m + \Psi_{tor}'n}\left(\frac{\sqrt{g_f}}{\sqrt{g_F}}\right)_{mn}~. | ||
</math> | </math> | ||
where | where | ||
:<math> | :<math> | ||
G(\psi, \theta_f, \phi_f) = \sum_{n | G(\psi, \theta_f, \phi_f) = \sum_{m,n} G_{mn}(\psi) e^{i(m\theta_f + n\phi_f)} | ||
</math> | </math> | ||
and <math>G_{00} = 0 </math>. | and <math>G_{00} = 0 </math> guarantees periodicity is preserved. | ||
Particular choices of G can be made so as to simplify the description of other fields. The most commonly used magnetic coordinate systems are: | Particular choices of G can be made so as to simplify the description of other fields. The most commonly used magnetic coordinate systems are: | ||
<ref name='Dhaeseleer'>W.D. D'haeseleer, ''Flux coordinates and magnetic field structure: a guide to a fundamental tool of plasma theory'', Springer series in computational physics, Springer-Verlag (1991) ISBN 3540524193</ref> | <ref name='Dhaeseleer'>W.D. D'haeseleer, ''Flux coordinates and magnetic field structure: a guide to a fundamental tool of plasma theory'', Springer series in computational physics, Springer-Verlag (1991) {{ISBN|3540524193}}</ref> | ||
* [[Hamada coordinates]]. <ref>S. Hamada, Nucl. Fusion '''2''' (1962) 23</ref><ref>[ | * [[Hamada coordinates]]. <ref>S. Hamada, Nucl. Fusion '''2''' (1962) 23</ref><ref>[[doi:10.1063/1.1706651|J.M. Greene and J.L Johnson, ''Stability Criterion for Arbitrary Hydromagnetic Equilibria'', Phys. Fluids '''5''' (1962) 510]]</ref> In these coordinates, both the magnetic field lines and current lines corresponding to the [[MHD equilibrium]] are straight. Referring to the definitions above, both <math>\tilde\nu</math> and <math>\tilde\eta</math> are zero in Hamada coordinates. | ||
* [[Boozer coordinates]]. <ref>[ | * [[Boozer coordinates]]. <ref>[[doi:10.1063/1.863297|A.H. Boozer, ''Plasma equilibrium with rational magnetic surfaces'', Phys. Fluids '''24''' (1981) 1999]]</ref><ref>[[doi:10.1063/1.863765|A.H. Boozer, ''Establishment of magnetic coordinates for a given magnetic field'', Phys. Fluids '''25''' (1982) 520]]</ref> In these coordinates, the magnetic field lines corresponding to the [[MHD equilibrium]] are straight and so are the ''diamagnetic lines '', i.e. the integral lines of <math>\nabla\psi\times\mathbf{B}</math>. Referring to the definitions above, both <math>\tilde\nu</math> and <math>\tilde\chi</math> are zero in Boozer coordinates. | ||
== References == | == References == | ||
<references /> | <references /> | ||