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Flux coordinates: Difference between revisions
→Magnetic coordinates
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<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. | <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. | ||
=== Transforming between Magnetic coordinates systems === | |||
There are infinitely many systems of magnetic coordinates. Any transformation of the angles of the from | |||
:<math> | :<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) | ||
</math> | </math> | ||
where <math>G</math> is periodic in the angles, preserves the straightness of the field lines. 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 | 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> | :<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}}~. | ||