Boozer coordinates - Revision history

Arturo: /* Covariant representation of the magnetic field in Boozer coordinates */

2012-10-11T10:29:09Z

Covariant representation of the magnetic field in Boozer coordinates

← Older revision		Revision as of 12:29, 11 October 2012
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	\mathbf{B} = \frac{I_{pol}^d}{2\pi}\nabla\phi,\quad (\text{for a vacuum field)}		\mathbf{B} = \frac{I_{pol}^d}{2\pi}\nabla\phi,\quad (\text{for a vacuum field)}
	</math>		</math>
	In a low-<math>\beta</math> stellarator the equilibrium magnetic field is approximatelly given by the vauum value.		In a [[Beta\|low-<math>\beta</math>]] stellarator the equilibrium magnetic field is approximatelly given by the vauum value.

Arturo: /* Covariant representation of the magnetic field in Boozer coordinates */

2012-10-11T10:26:25Z

Covariant representation of the magnetic field in Boozer coordinates

← Older revision		Revision as of 12:26, 11 October 2012
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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 low-<math>\beta</math> stellarator the equilibrium magnetic field is approximatelly given by the vauum value.

Arturo: /* Form of the Jacobian for Boozer coordinates */

2012-08-21T15:33:26Z

Form of the Jacobian for Boozer coordinates

← Older revision		Revision as of 17:33, 21 August 2012
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	== 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)~,

Arturo: /* Contravariant representation of the magnetic field in Boozer coordinates */

2012-03-14T10:17:21Z

Contravariant representation of the magnetic field in Boozer coordinates

← Older revision		Revision as of 12:17, 14 March 2012
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	</math>		</math>

	== ~~Contravariant~~ representation of the magnetic field in Boozer coordinates ==		== 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		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>

Arturo: /* Covariant representation of the magnetic field in Boozer coordinates */

2012-03-14T10:17:07Z

Covariant representation of the magnetic field in Boozer coordinates

← Older revision		Revision as of 12:17, 14 March 2012
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	</math>		</math>

	== ~~Covariant~~ representation of the magnetic field in Boozer coordinates ==		== 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>

Arturo: /* Contravariant representation of the magnetic field in Boozer coordinates */

2012-03-14T10:16:04Z

Contravariant representation of the magnetic field in Boozer coordinates

← Older revision		Revision as of 12:16, 14 March 2012
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	== Contravariant representation of the magnetic field in Boozer coordinates ==		== Contravariant representation of the magnetic field in Boozer coordinates ==
	The ~~contravariant~~ 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		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~.

Arturo at 10:12, 14 March 2012

2012-03-14T10:12:29Z

← Older revision		Revision as of 12:12, 14 March 2012
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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 +
Line 40:		Line 40:
	\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 ~~then~~		The covariant <math>B</math>-field components are explicitly
	:<math>		:<math>
	B_\psi = -\tilde{\eta}		B_\psi = -\tilde{\eta}

Arturo: /* Form of the Jacobian for Boozer coordinates */

2012-03-14T10:10:08Z

Form of the Jacobian for Boozer coordinates

← Older revision		Revision as of 12:10, 14 March 2012
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	\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' ~,

Arturo: /* Contravariant representation of the magnetic field in Boozer coordinates */

2012-03-14T10:07:45Z

Contravariant representation of the magnetic field in Boozer coordinates

@@ Line 40: / Line 40: @@
 \mathbf{B} =  -\tilde\eta\nabla\psi + \frac{I_{tor}}{2\pi}\nabla\theta + \frac{I_{pol}^d}{2\pi}\nabla\phi~.
 </math>
 It then follows that
 :<math>

Arturo: /* Covariant representation of the magnetic field in Boozer coordinates */

2012-03-14T10:04:15Z

Covariant representation of the magnetic field in Boozer coordinates

← Older revision		Revision as of 12:04, 14 March 2012
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	\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 +
	2\pi\frac{d\Psi_{tor}}{dV}\frac{B^2}{\langle B^2\rangle}\mathbf{e}_\phi		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>		</math>