# Boozer coordinates

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Boozer coordinates are a set of magnetic coordinates in which the diamagnetic ${\displaystyle \nabla \psi \times \mathbf {B} }$ lines are straight besides those of magnetic field ${\displaystyle \mathbf {B} }$. The periodic part of the stream function of ${\displaystyle \mathbf {B} }$ 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 ${\displaystyle \mathbf {B} \cdot }$ we get

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

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

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

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

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

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

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

## Contravariant representation of the magnetic field in Boozer coordinates

Using this Jacobian in the general form of the magnetic field in magnetic coordinates one gets.

${\displaystyle \mathbf {B} =2\pi {\frac {d\Psi _{pol}}{dV}}{\frac {B^{2}}{\langle B^{2}\rangle }}\mathbf {e} _{\theta }+2\pi {\frac {d\Psi _{tor}}{dV}}{\frac {B^{2}}{\langle B^{2}\rangle }}\mathbf {e} _{\phi }}$

so, in Boozer coordinates,

${\displaystyle B^{\theta }=2\pi {\frac {d\Psi _{pol}}{dV}}{\frac {B^{2}}{\langle B^{2}\rangle }}\quad {\text{and}}\quad B^{\phi }=2\pi {\frac {d\Psi _{tor}}{dV}}{\frac {B^{2}}{\langle B^{2}\rangle }}}$

## Covariant representation of the magnetic field in Boozer coordinates

The covariant representation of the field is also relatively simple when using Boozer coordinates, since the angular covariant ${\displaystyle B}$-field components are flux functions in these coordinates

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

The covariant ${\displaystyle B}$-field components are explicitly

${\displaystyle B_{\psi }=-{\tilde {\eta }}\quad ,\quad B_{\theta }={\frac {I_{tor}}{2\pi }}\quad {\text{and}}\quad B_{\phi }={\frac {I_{pol}^{d}}{2\pi }}~.}$

It then follows that

${\displaystyle \nabla \psi \times \mathbf {B} =\nabla \psi \times \nabla \left({\frac {I_{tor}}{2\pi }}\theta +{\frac {I_{pol}^{d}}{2\pi }}\phi \right)~,}$

and then the 'diamagnetic' lines are straight in Boozer coordinates and given by ${\displaystyle {I_{tor}}\theta +{I_{pol}^{d}}\phi =\mathrm {const.} }$.

It is also useful to know the expression of the following object in Boozer coordinates

${\displaystyle {\frac {\nabla V\times \mathbf {B} }{B^{2}}}=-{\frac {2\pi I_{pol}^{d}}{\langle B^{2}\rangle }}\mathbf {e} _{\theta }+{\frac {2\pi I_{tor}}{\langle B^{2}\rangle }}\mathbf {e} _{\phi }~.}$

The above expressions adopt very simple forms for the 'vacuum' field, i.e. one with ${\displaystyle \nabla \times \mathbf {B} =0}$. In this case ${\displaystyle I_{tor}=0}$ and ${\displaystyle {\tilde {\eta }}=0}$ leaving, e.g.

${\displaystyle \mathbf {B} ={\frac {I_{pol}^{d}}{2\pi }}\nabla \phi ,\quad ({\text{for a vacuum field)}}}$

In a low-${\displaystyle \beta }$ stellarator the equilibrium magnetic field is approximatelly given by the vauum value.