Flux coordinates: Difference between revisions
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</math> | </math> | ||
=== Jacobian === | === Length element === | ||
The squared length of a differential change in the position vector is | |||
:<math> | |||
ds^2 = d\mathbf{x}\cdot d\mathbf{x} = \left(\frac{\partial\mathbf{x}}{\partial{u^i}}du^i\right)\cdot \left(\frac{\partial\mathbf{x}}{\partial{u^j}}du^j\right) = g_{ij}du^idu^j | |||
</math> | |||
=== Jacobian and volume element=== | |||
The Jacobian of the coordinate transformation <math>\mathbf{x}(\psi, \theta, \phi)</math> is defined as | The Jacobian of the coordinate transformation <math>\mathbf{x}(\psi, \theta, \phi)</math> is defined as | ||
:<math> | :<math> | ||
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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} | ||
</math> | </math> | ||
It can be seen that <ref name='Dhaeseleer'></ref> <math>g \equiv \det(g_{ij}) = J^2 \Rightarrow J = \sqrt{g}</math> | It can be seen that <ref name='Dhaeseleer'></ref> <math>g \equiv \det(g_{ij}) = J^2 \Rightarrow J = \sqrt{g}</math>. | ||
The differential volume element in the curvilinear coordinates is <math>d\mathcal{V} = \sqrt{g} d\psi d\theta d\phi</math> | |||
=== Some surface elements === | === Some surface elements === | ||
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:<math> | :<math> | ||
\nu(\psi,\theta,\phi) | \nu(\psi,\theta,\phi) | ||
= \frac{1}{2\pi}(\ | = \frac{1}{2\pi}(\Psi'_{tor}\theta | ||
- \ | - \Psi'_{pol}\phi) | ||
+ \tilde{\nu}(\psi,\theta,\phi) | + \tilde{\nu}(\psi,\theta,\phi) | ||
</math> | </math> | ||
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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 /> | ||