Hamada coordinates

Hamada coordinates are a set of magnetic coordinates in which the equilibrium current density $\mathbf {j}$ lines are straight besides the 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 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 $(-{\dot {I}}_{tor}{\Psi }'_{pol}+{I}'_{pol}{\Psi }'_{tor})=4\pi ^{2}{p}'\langle ({\sqrt {g_{f}}})^{-1}\rangle ^{-1}$, so that we have

$\mathbf {B} \cdot \nabla {\tilde {\eta }}={p}'\left({\frac {\langle ({\sqrt {g_{f}}})^{-1}\rangle ^{-1}}{\sqrt {g_{f}}}}-1\right)$

In a coordinate system where $\mathbf {j}$ is straight ${\tilde {\eta }}$ is a function of $\psi$ only, and therefore LHS of this equation must be zero in such a system. It therefore follows that the Jacobian of the Hamada system must satisfy

${\sqrt {g_{H}}}=\langle {\sqrt {g_{H}}}^{-1}\rangle ^{-1}={\frac {V'}{4\pi ^{2}}}~,$

where the last idenity follows from the properties of the flux surface average.