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Flux coordinates: Difference between revisions
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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 | 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 between two surfaces labelled by their volumes <math>V_1</math> and <math>V_2</math> 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 | ||
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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 /> | ||