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

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