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Hamada coordinates: Difference between revisions
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\sqrt{g_H} = \langle\sqrt{g_H}^{-1}\rangle^{-1} = \frac{V'}{4\pi^2}~, | \sqrt{g_H} = \langle\sqrt{g_H}^{-1}\rangle^{-1} = \frac{V'}{4\pi^2}~, | ||
</math> | </math> | ||
where the last idenity follows from the [[Flux coordinates#Useful properties of the FSA|properties of the flux surface average]]. The Hamada angles are sometimes defined in | where the last idenity follows from the [[Flux coordinates#Useful properties of the FSA|properties of the flux surface average]]. The Hamada angles are sometimes defined in 'turns' (i.e. <math>(\theta, \xi) \in [0,1)</math>) instead of radians (<math>(\theta, \xi) \in [0,2\pi)</math>)). This choice together with the choice of the volume <math>V</math> as radial coordinate makes the Jacobian equal to unity. Alternatively one can select <math>\psi = \frac{V}{4\pi^2}</math> as radial coordinate with the same effect. | ||
== Magnetic field and current density expressions in Hamada vector basis == | == Magnetic field and current density expressions in Hamada vector basis == | ||
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\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~. | ||
</math> | </math> | ||
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 current density contravariant looks alike | ||
:<math> | |||
\mu_0\mathbf{j} = 2\pi I_{pol}'(V)\mathbf{e}_\theta + 2\pi I_{tor}'(V)\mathbf{e}_\phi~. | |||
</math> | |||
The [[Flux coordinates#Convariant Form |covariant expression]] is less clean | The [[Flux coordinates#Convariant Form |covariant expression]] of the magnetic field 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 | 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> | ||