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Hamada coordinates: Difference between revisions
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Hamada coordinates are a set of [[Flux coordinates#Magnetic coordinates|magnetic coordinates]] in which the equilibrium current density <math>\mathbf{j}</math> lines are straight besides | Hamada coordinates are a set of [[Flux coordinates#Magnetic coordinates|magnetic coordinates]] in which the equilibrium current density <math>\mathbf{j}</math> lines are straight besides those of magnetic field <math>\mathbf{B}</math>. The periodic part of the stream functions of both <math>\mathbf{B}</math> and <math>\mathbf{j}</math> are flux functions (that can be chosen to be zero without loss of generality). | ||
== Form of the Jacobian for Hamada coordinates == | == Form of the Jacobian for Hamada coordinates == | ||
In this section, following D'haseleer et al we will translate the condition of straight current density lines into one for the Hamada coordinates Jacobian. For that we will make use of the equilibrium equation <math>\mathbf{j}\times\mathbf{B} = p'\nabla\psi </math>, which written in a magnetic coordinate system reads | In this section, following D'haseleer et al we will translate the condition of straight current density lines into one for the '''Hamada''' coordinates Jacobian. For that we will make use of the equilibrium equation <math>\mathbf{j}\times\mathbf{B} = p'\nabla\psi </math>, which written in a magnetic coordinate system reads | ||
:<math> | :<math> | ||
\frac{-I'_{tor}\Psi'_{pol} + I'_{pol}\Psi'_{tor}}{4\pi^2\sqrt{g_f}} | \frac{-I'_{tor}\Psi'_{pol} + I'_{pol}\Psi'_{tor}}{4\pi^2\sqrt{g_f}} | ||
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</math> | </math> | ||
In a coordinate system where <math>\mathbf{j}</math> is straight <math>\tilde{\eta}</math> is a function of <math>\psi</math> only, and therefore LHS of this equation must be zero in such a system. It therefore follows that the Jacobian of the Hamada system must satisfy | In a coordinate system where <math>\mathbf{j}</math> is straight <math>\tilde{\eta}</math> is a function of <math>\psi</math> only, and therefore LHS of this equation must be zero in such a system. It therefore follows that the Jacobian of the '''Hamada''' system must satisfy | ||
:<math> | :<math> | ||
\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]]. | 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 == | |||
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 | |||
:<math> | |||
\mathbf{B} = 2\pi\Psi_{pol}'(V)\mathbf{e}_\theta + 2\pi\Psi_{tor}'(V)\mathbf{e}_\phi~. | |||
</math> | |||
This has the nice property of having flux constant contravariant coefficients (functions of the radial coordinate only). | |||
The covariant expression is less clean | |||
:<math> | |||
\mathbf{B} = \frac{I_{tor}}{2\pi}\nabla\theta + \frac{I_{pol}^d}{2\pi}\nabla\phi + \nabla\tilde\chi~. | |||
</math> | |||
with contributions from the periodic part of the magnetic scalar potential <math>\tilde\chi</math> to all the covariant components. Nonetheless, the '''flux surface averaged Hamada covariant <math>B</math>-field angular components''' have simple expressions, i.e | |||
:<math> | |||
\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> | |||
where the integral over <math>\theta</math> is zero because the jacobian in Hamda coordinates is not a function of this angle. Similarly | |||
:<math> | |||
\langle B_\phi\rangle = \langle\mathbf{B}\cdot\mathbf{e}_\phi\rangle = \frac{I^d_{pol}}{2\pi}~. | |||
</math> | |||