Hamada coordinates

Hamada coordinates are a set of magnetic coordinates in which the equilibrium current density $\mathbf {j}$ lines are straight besides those of magnetic field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}} . The periodic part of the stream functions of both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{j}} are flux functions (that can be chosen to be zero without loss of generality).

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{j}\times\mathbf{B} = p'\nabla\psi } , which written in a magnetic coordinate system reads

${\frac {-I'_{tor}\Psi '_{pol}+I'_{pol}\Psi '_{tor}}{4\pi ^{2}{\sqrt {g_{f}}}}}-\mathbf {B} \cdot \nabla {\tilde {\eta }}=p'~.$

Taking the flux surface average Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\cdot\rangle} of this equation we find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-\dot{I}_{tor}{\Psi}'_{pol} + {I}'_{pol}{\Psi}'_{tor})= 4\pi^2{p}'\langle(\sqrt{g_f})^{-1}\rangle^{-1}} , so that we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}\cdot\nabla\tilde{\eta} = {p}'\left(\frac{\langle(\sqrt{g_f})^{-1}\rangle^{-1}}{\sqrt{g_f}}-1\right) }

In a coordinate system where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{j}} is straight ${\tilde {\eta }}$ is a function of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi} 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{g_H} = \langle\sqrt{g_H}^{-1}\rangle^{-1} = \frac{V'}{4\pi^2}~, }

where the last idenity follows from the properties of the flux surface average. The Hamada angles are sometimes defined in `turns' (i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\theta, \xi) \in [0,1)} ) instead of radians (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\theta, \xi) \in [0,2\pi)} )). This choice together with the choice of the volume $V$ as radial coordinate makes the Jacobian equal to unity. Alternatively one can select Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi = \frac{V}{4\pi^2}} 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B} = 2\pi\Psi_{pol}'(V)\mathbf{e}_\theta + 2\pi\Psi_{tor}'(V)\mathbf{e}_\phi~. }

This has the nice property of having flux constant contravariant coefficients (functions of the radial coordinate only).

The covariant expression is less clean

$\mathbf {B} ={\frac {I_{tor}}{2\pi }}\nabla \theta +{\frac {I_{pol}^{d}}{2\pi }}\nabla \phi +\nabla {\tilde {\chi }}~.$

with contributions from the periodic part of the magnetic scalar potential ${\tilde {\chi }}$ to all the covariant components. Nonetheless, the flux surface averaged Hamada covariant $B$-field angular components have simple expressions, i.e

$\langle B_{\theta }\rangle =\langle \mathbf {B} \cdot \mathbf {e} _{\theta }\rangle =\left\langle {\frac {I_{tor}}{2\pi }}+{\frac {\partial {\tilde {\chi }}}{\partial \theta }}\right\rangle ={\frac {I_{tor}}{2\pi }}+(V')^{-1}\int \partial _{\theta }{\tilde {\chi }}{\sqrt {g}}d\theta d\phi ={\frac {I_{tor}}{2\pi }}$

where the integral over $\theta$ is zero because the jacobian in Hamda coordinates is not a function of this angle. Similarly

$\langle B_{\phi }\rangle =\langle \mathbf {B} \cdot \mathbf {e} _{\phi }\rangle ={\frac {I_{pol}^{d}}{2\pi }}~.$