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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. | ||
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 === | === Transforming between Magnetic coordinates systems === | ||
There are infinitely many systems of magnetic coordinates. Any transformation of the angles of the from | There are infinitely many systems of magnetic coordinates. Any transformation of the angles of the from | ||