# 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 $ \mathbf{B} $. The periodic part of the stream functions of both $ \mathbf{B} $ and $ \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 $ \mathbf{j}\times\mathbf{B} = p'\nabla\psi $, which written in a general 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 $ \langle\cdot\rangle $ of this equation we find a synthetic version of the MHD equilibrium equation

- $ (-{I}'_{tor}{\Psi}'_{pol} + {I}'_{pol}{\Psi}'_{tor})= 4\pi^2{p}'\langle(\sqrt{g_f})^{-1}\rangle^{-1} = p'V'~. $

In the last identity we have used the general property of the flux surface average $ \langle\sqrt{g}^{-1}\rangle^{-1} = \frac{V'}{4\pi^2} $. Then, from the MHD equilibrium, we have

- $ \mathbf{B}\cdot\nabla\tilde{\eta} = {p}'\left(\frac{{V'}/{4\pi^2}}{\sqrt{g_f}}-1\right)~, $

where $ \tilde{\eta} $ and $ \sqrt{g_f} $ depend on our choice of coordinate system.

Now, in the **Hamada** magnetic coordinate system that concerns us here (that in which $ \mathbf{j} $ is straight) $ \tilde{\eta} $ is a function of $ \psi $ only, and therefore LHS of this equation must be zero in such a system. It follows that the Jacobian of the Hamada system must satisfy

- $ \sqrt{g_H} = \frac{V'}{4\pi^2}~. $

The Hamada angles are sometimes defined in 'turns' (i.e. $ (\theta, \xi) \in [0,1) $) instead of radians ($ (\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 $ \psi = \frac{V}{4\pi^2} $ as radial coordinate with the same effect.

## Magnetic field and current density expressions in a 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

- $ \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 current density contravariant looks alike

- $ \mu_0\mathbf{j} = 2\pi I_{pol}'(V)\mathbf{e}_\theta + 2\pi I_{tor}'(V)\mathbf{e}_\phi~. $

The covariant expression of the magnetic field 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 Hamada coordinates is not a function of this angle. Similarly

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