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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 == | ||
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== Covariant representation of the magnetic field in Boozer coordinates == | == Covariant 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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\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} | ||