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Boozer coordinates: Difference between revisions
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→Form of the Jacobian for Boozer coordinates
(Created page with 'Boozer coordinates are a set of magnetic coordinates in which the diamagnetic <math>\nabla\psi\times\mathbf{B}</math> lines are straight besides the those of magnetic field <math…') |
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\mathbf{B}\cdot\nabla\tilde\chi = B^2 - \frac{1}{4\pi^2\sqrt{g}}\left(I_{tor}\Psi_{pol}' + I_{pol}^d\Psi_{tor}' \right)~, | \mathbf{B}\cdot\nabla\tilde\chi = B^2 - \frac{1}{4\pi^2\sqrt{g}}\left(I_{tor}\Psi_{pol}' + I_{pol}^d\Psi_{tor}' \right)~, | ||
</math> | </math> | ||
where we note that the term in brackets is a flux function. Taking the [[Flux coordinates#flux surface average|flux surface average]] <math>\langle\cdot\rangle</math> of this equation we find <math>(I_{tor}\Psi_{pol}' + I_{pol}^d\Psi_{tor}') = \langle B^2\rangle/ | where we note that the term in brackets is a flux function. Taking the [[Flux coordinates#flux surface average|flux surface average]] <math>\langle\cdot\rangle</math> of this equation we find <math>(I_{tor}\Psi_{pol}' + I_{pol}^d\Psi_{tor}') = 4\pi^2\langle B^2\rangle/\langle(\sqrt{g})^{-1}\rangle = \langle B^2\rangle V' </math>, so that we have | ||
:<math> | |||
\mathbf{B}\cdot\nabla\tilde\chi = B^2 - \frac{1}{4\pi^2\sqrt{g}}\langle B^2\rangle V' ~, | |||
</math> | |||
In Boozer coordinates, the LHS of this equation is zero and therefore we must have <math>\sqrt{g_B}^{ | In Boozer coordinates, the LHS of this equation is zero and therefore we must have | ||
:<math> | |||
\sqrt{g_B} = \frac{V'}{4\pi^2}\frac{\langle B^2\rangle} {B^2} | |||
</math> | |||