Boozer coordinates: Difference between revisions

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Boozer coordinates are a set of [[Flux coordinates # Magnetic coordinates|magnetic coordinates]] in which the diamagnetic <math>\nabla\psi\times\mathbf{B}</math> lines are straight besides those of magnetic field <math>\mathbf{B}</math>. The periodic part of the stream function of <math>\mathbf{B}</math> and the scalar magnetic potential are flux functions (that can be chosen to be zero without loss of generality) in this coordinate system.
Boozer coordinates are a set of [[Flux coordinates#Magnetic coordinates|magnetic coordinates]] in which the diamagnetic <math>\nabla\psi\times\mathbf{B}</math> lines are straight besides those of magnetic field <math>\mathbf{B}</math>. The periodic part of the stream function of <math>\mathbf{B}</math> 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 ==
== Form of the Jacobian for Boozer coordinates ==
Multiplying the covariant representation of the magnetic field by <math>\mathbf{B}\cdot</math> we get
Multiplying the [[Flux coordinates#Covariant Form|covariant representation]] of the magnetic field by <math>\mathbf{B}\cdot</math> we get
:<math>
:<math>
  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~.
  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~.
</math>
</math>
Now, using the known form of the contravariant components of the magnetic field for a magnetic coordinate system we get
Now, using the known form of the [[Flux coordinates#Magnetic coordinates|contravariant components]] of the magnetic field for a magnetic coordinate system we get
:<math>
:<math>
  \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)~,
  \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)~,
</math>
</math>
where we note that the term in brackets is a flux function. Taking the [[Flux coordinates#flux surface average|flux surface average]] <math>\langle\cdot\rangle</math> of this equation we find <math>(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' </math>, so that we have
where we note that the term in brackets is a flux function. Taking the [[Flux coordinates#Flux Surface Average|flux surface average]] <math>\langle\cdot\rangle</math> of this equation we find <math>(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' </math>, so that we have
:<math>
:<math>
  \mathbf{B}\cdot\nabla\tilde\chi = B^2  - \frac{1}{4\pi^2\sqrt{g}}\langle B^2\rangle V' ~,
  \mathbf{B}\cdot\nabla\tilde\chi = B^2  - \frac{1}{4\pi^2\sqrt{g}}\langle B^2\rangle V' ~,
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== Contravariant representation of the magnetic field in Boozer coordinates ==
== Contravariant representation of the magnetic field in Boozer coordinates ==
In Boozer coordinates the angular covariant <math>B</math>-field components are flux functions
Using this Jacobian in the general form of the magnetic field in [[Flux coordinates#Magnetic coordinates|magnetic coordinates]] one gets.
:<math>
\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
</math>
so, in Boozer coordinates,
:<math>
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}
</math>
 
== Covariant representation of the magnetic field in Boozer coordinates ==
The [[Flux_coordinates#Covariant_Form|covariant representation]] of the field is also relatively simple when using Boozer coordinates, since the angular covariant <math>B</math>-field components are flux functions in these coordinates
:<math>
:<math>
\mathbf{B} =  -\tilde\eta\nabla\psi + \frac{I_{tor}}{2\pi}\nabla\theta + \frac{I_{pol}^d}{2\pi}\nabla\phi~.
\mathbf{B} =  -\tilde\eta\nabla\psi + \frac{I_{tor}}{2\pi}\nabla\theta + \frac{I_{pol}^d}{2\pi}\nabla\phi~.
</math>
</math>
The covariant <math>B</math>-field components are explicitly
:<math>
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}~.
</math>
It then follows that
It then follows that
:<math>
:<math>
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\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~.
\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~.
</math>
</math>
The above expressions adopt very simple forms for the 'vacuum' field, i.e. one with <math>\nabla\times\mathbf{B} = 0</math>. In this case <math>I_{tor} = 0</math> and <math>\tilde{\eta} = 0</math> leaving, e.g.
:<math>
\mathbf{B} =  \frac{I_{pol}^d}{2\pi}\nabla\phi,\quad (\text{for a vacuum field)}
</math>
In a [[Beta|low-<math>\beta</math>]] stellarator the equilibrium magnetic field is approximatelly given by the vauum value.

Latest revision as of 12:29, 11 October 2012

Boozer coordinates are a set of magnetic coordinates in which the diamagnetic lines are straight besides those of magnetic field . The periodic part of the stream function of 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 we get

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

where we note that the term in brackets is a flux function. Taking the flux surface average of this equation we find , so that we have

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

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.

so, in Boozer coordinates,

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 -field components are flux functions in these coordinates

The covariant -field components are explicitly

It then follows that

and then the 'diamagnetic' lines are straight in Boozer coordinates and given by .

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

The above expressions adopt very simple forms for the 'vacuum' field, i.e. one with . In this case and leaving, e.g.

In a low- stellarator the equilibrium magnetic field is approximatelly given by the vauum value.