Boozer coordinates

Boozer coordinates are a set of magnetic coordinates in which the diamagnetic $\nabla \psi \times \mathbf {B}$ lines are straight besides the those of magnetic field $\mathbf {B}$. The periodic part of the stream function of $\mathbf {B}$ and the scalar magnetic potential are flux functions (that can be chosen to be zero without loss of generality) in this coordinate system.

Form of the Jacobian for Boozer coordinates

Multiplying the covariant representation of the magnetic field by $\mathbf {B} \cdot$ we get

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

Now, using the known form of the contravariant components of the magnetic field for a magnetic coordinate system we get

$\mathbf {B} \cdot \nabla {\tilde {\chi }}=B^{2}-{\frac {1}{4\pi ^{2}{\sqrt {g}}}}\left(I_{tor}\Psi _{pol}'+I_{pol}^{d}\Psi _{tor}'\right)~,$

where we note that the term in brackets is a flux function. Taking the flux surface average $\langle \cdot \rangle$ of this equation we find $(I_{tor}\Psi _{pol}'+I_{pol}^{d}\Psi _{tor}')=4\pi ^{2}\langle B^{2}\rangle /\langle ({\sqrt {g}})^{-1}\rangle =\langle B^{2}\rangle V'$, so that we have

$\mathbf {B} \cdot \nabla {\tilde {\chi }}=B^{2}-{\frac {1}{4\pi ^{2}{\sqrt {g}}}}\langle B^{2}\rangle V'~,$

In Boozer coordinates, the LHS of this equation is zero and therefore we must have

${\sqrt {g_{B}}}={\frac {V'}{4\pi ^{2}}}{\frac {\langle B^{2}\rangle }{B^{2}}}$