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Boozer coordinates: Difference between revisions
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→Covariant representation of the magnetic field in Boozer coordinates
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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)~, | ||
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</math> | </math> | ||
== | == Contravariant representation of the magnetic field in Boozer coordinates == | ||
Using this Jacobian in the general form of the magnetic field in [[Flux coordinates # Magnetic coordinates|magnetic coordinates]] one gets. | Using this Jacobian in the general form of the magnetic field in [[Flux coordinates#Magnetic coordinates|magnetic coordinates]] one gets. | ||
:<math> | :<math> | ||
\mathbf{B} = 2\pi\frac{d\Psi_{pol}}{dV}\frac{B^2}{\langle B^2\rangle}\mathbf{e}_\theta + | \mathbf{B} = 2\pi\frac{d\Psi_{pol}}{dV}\frac{B^2}{\langle B^2\rangle}\mathbf{e}_\theta + | ||
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</math> | </math> | ||
== | == Covariant representation of the magnetic field in Boozer coordinates == | ||
The | 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 | The covariant <math>B</math>-field components are explicitly | ||
:<math> | :<math> | ||
B_\psi = -\tilde{\eta} | B_\psi = -\tilde{\eta} | ||
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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. | |||