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Hamada coordinates: Difference between revisions
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== Magnetic field and current density expressions in Hamada vector basis == | == Magnetic field and current density expressions in Hamada vector basis == | ||
With the form of the Hamada coordinates' Jacobian we can now write the explicit contravariant form of the magnetic field in terms of the '''Hamada''' basis vectors | With the form of the Hamada coordinates' Jacobian we can now write the explicit [[Flux coordinates#Contravariant Form|contravariant form]] of the magnetic field in terms of the '''Hamada''' basis vectors | ||
:<math> | :<math> | ||
\mathbf{B} = 2\pi\Psi_{pol}'(V)\mathbf{e}_\theta + 2\pi\Psi_{tor}'(V)\mathbf{e}_\phi~. | \mathbf{B} = 2\pi\Psi_{pol}'(V)\mathbf{e}_\theta + 2\pi\Psi_{tor}'(V)\mathbf{e}_\phi~. | ||
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This has the nice property of having flux constant contravariant coefficients (functions of the radial coordinate only). | This has the nice property of having flux constant contravariant coefficients (functions of the radial coordinate only). | ||
The covariant expression is less clean | The [[Flux coordinates#Convariant Form |covariant expression]] is less clean | ||
:<math> | :<math> | ||
\mathbf{B} = \frac{I_{tor}}{2\pi}\nabla\theta + \frac{I_{pol}^d}{2\pi}\nabla\phi + \nabla\tilde\chi~. | \mathbf{B} = \frac{I_{tor}}{2\pi}\nabla\theta + \frac{I_{pol}^d}{2\pi}\nabla\phi + \nabla\tilde\chi~. | ||
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\langle B_\theta\rangle = \langle\mathbf{B}\cdot\mathbf{e}_\theta\rangle = \left\langle \frac{I_{tor}}{2\pi} + \frac{\partial \tilde\chi}{\partial \theta}\right\rangle = \frac{I_{tor}}{2\pi} + (V')^{-1}\int\partial_\theta\tilde\chi \sqrt{g} d\theta d\phi = \frac{I_{tor}}{2\pi} | \langle B_\theta\rangle = \langle\mathbf{B}\cdot\mathbf{e}_\theta\rangle = \left\langle \frac{I_{tor}}{2\pi} + \frac{\partial \tilde\chi}{\partial \theta}\right\rangle = \frac{I_{tor}}{2\pi} + (V')^{-1}\int\partial_\theta\tilde\chi \sqrt{g} d\theta d\phi = \frac{I_{tor}}{2\pi} | ||
</math> | </math> | ||
where the integral over <math>\theta</math> is zero because the jacobian in | where the integral over <math>\theta</math> is zero because the jacobian in Hamada coordinates is not a function of this angle. Similarly | ||
:<math> | :<math> | ||
\langle B_\phi\rangle = \langle\mathbf{B}\cdot\mathbf{e}_\phi\rangle = \frac{I^d_{pol}}{2\pi}~. | \langle B_\phi\rangle = \langle\mathbf{B}\cdot\mathbf{e}_\phi\rangle = \frac{I^d_{pol}}{2\pi}~. | ||
</math> | </math> | ||