Hamada coordinates - Revision history

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

2011-08-12T09:59:24Z

Form of the Jacobian for Hamada coordinates

← Older revision		Revision as of 11:59, 12 August 2011
Line 7:		Line 7:
	- \mathbf{B}\cdot\nabla\tilde{\eta} = p'~.		- \mathbf{B}\cdot\nabla\tilde{\eta} = p'~.
	</math>		</math>
	Taking the [[Flux coordinates#flux surface average\|flux surface average]] <math>\langle\cdot\rangle</math> of this equation we find the MHD equilibrium equation		Taking the [[Flux coordinates#flux surface average\|flux surface average]] <math>\langle\cdot\rangle</math> of this equation we find a synthetic version of the MHD equilibrium equation
	:<math>(-{I}'_{tor}{\Psi}'_{pol} + {I}'_{pol}{\Psi}'_{tor})= 4\pi^2{p}'\langle(\sqrt{g_f})^{-1}\rangle^{-1} = p'V'~.		:<math>(-{I}'_{tor}{\Psi}'_{pol} + {I}'_{pol}{\Psi}'_{tor})= 4\pi^2{p}'\langle(\sqrt{g_f})^{-1}\rangle^{-1} = p'V'~.
	</math>		</math>

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

2011-08-12T09:56:20Z

Form of the Jacobian for Hamada coordinates

← Older revision		Revision as of 11:56, 12 August 2011
Line 2:		Line 2:

	== Form of the Jacobian for Hamada coordinates ==		== Form of the Jacobian for Hamada coordinates ==
	In this section, following D'haseleer et al we will translate the condition of straight current density lines into one for the '''Hamada''' coordinates Jacobian. For that we will make use of the equilibrium equation <math>\mathbf{j}\times\mathbf{B} = p'\nabla\psi </math>, which written in a magnetic coordinate system reads		In this section, following D'haseleer et al we will translate the condition of straight current density lines into one for the '''Hamada''' coordinates Jacobian. For that we will make use of the equilibrium equation <math>\mathbf{j}\times\mathbf{B} = p'\nabla\psi </math>, which written in a general magnetic coordinate system reads
	:<math>		:<math>
	\frac{-I'_{tor}\Psi'_{pol} + I'_{pol}\Psi'_{tor}}{4\pi^2\sqrt{g_f}}		\frac{-I'_{tor}\Psi'_{pol} + I'_{pol}\Psi'_{tor}}{4\pi^2\sqrt{g_f}}
	- \mathbf{B}\cdot\nabla\tilde{\eta} = p'~.		- \mathbf{B}\cdot\nabla\tilde{\eta} = p'~.
	</math>		</math>
	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}{\Psi}'_{tor})= 4\pi^2{p}'\langle(\sqrt{g_f})^{-1}\rangle^{-1}</math>, ~~so that~~ we have		Taking the [[Flux coordinates#flux surface average\|flux surface average]] <math>\langle\cdot\rangle</math> of this equation we find the MHD equilibrium equation
			:<math>(-{I}'_{tor}{\Psi}'_{pol} + {I}'_{pol}{\Psi}'_{tor})= 4\pi^2{p}'\langle(\sqrt{g_f})^{-1}\rangle^{-1} = p'V'~.
			</math>
			In the last identity we have used the general [[Flux coordinates#Useful properties of the FSA\|property of the flux surface average]] <math>\langle\sqrt{g}^{-1}\rangle^{-1} = \frac{V'}{4\pi^2}</math>. Then, from the MHD equilibrium, we have

	:<math> \mathbf{B}\cdot\nabla\tilde{\eta} = {p}'\left(\frac{~~\langle(\sqrt~~{~~g_f~~})^{~~-1}~~\~~rangle~~^~~{-1~~}}{\sqrt{g_f}}-1\right)		:<math> \mathbf{B}\cdot\nabla\tilde{\eta} = {p}'\left(\frac{{V'}/{4\pi^2}}{\sqrt{g_f}}-1\right)~,
	</math>		</math>
			where <math>\tilde{\eta}</math> and <math>\sqrt{g_f}</math> depend on our choice of coordinate system.

	~~In a~~ coordinate system ~~where~~ <math>\mathbf{j}</math> is straight <math>\tilde{\eta}</math> is a function of <math>\psi</math> only, and therefore LHS of this equation must be zero in such a system. It ~~therefore~~ follows that the Jacobian of the ~~'''~~Hamada~~'''~~ system must satisfy		Now, in the '''Hamada''' magnetic coordinate system that concerns us here (that in which <math>\mathbf{j}</math> is straight) <math>\tilde{\eta}</math> is a function of <math>\psi</math> only, and therefore LHS of this equation must be zero in such a system. It follows that the Jacobian of the Hamada system must satisfy
	:<math>		:<math>
	\sqrt{g_H~~} = \langle\sqrt{g_H}^{-1}\rangle^{-1~~} = \frac{V'}{4\pi^2}~,		\sqrt{g_H} = \frac{V'}{4\pi^2}~.

	</math>		</math>
	~~where the last identity follows from the [[Flux coordinates#Useful properties of the FSA\|properties of the flux surface average]].~~ The Hamada angles are sometimes defined in 'turns' (i.e. <math>(\theta, \xi) \in [0,1)</math>) instead of radians (<math>(\theta, \xi) \in [0,2\pi)</math>)). This choice together with the choice of the volume <math>V</math> as radial coordinate makes the Jacobian equal to unity. Alternatively one can select <math>\psi = \frac{V}{4\pi^2}</math> as radial coordinate with the same effect.
			The Hamada angles are sometimes defined in 'turns' (i.e. <math>(\theta, \xi) \in [0,1)</math>) instead of radians (<math>(\theta, \xi) \in [0,2\pi)</math>)). This choice together with the choice of the volume <math>V</math> as radial coordinate makes the Jacobian equal to unity. Alternatively one can select <math>\psi = \frac{V}{4\pi^2}</math> as radial coordinate with the same effect.

	== Magnetic field and current density expressions in a Hamada vector basis ==		== Magnetic field and current density expressions in a Hamada vector basis ==

Admin: Reverted edits by Otihizuv (Talk) to last revision by 130.206.40.141

2010-11-24T11:22:07Z

Reverted edits by Otihizuv (Talk) to last revision by 130.206.40.141

← Older revision		Revision as of 13:22, 24 November 2010
Line 1:		Line 1:
	~~=[http://aluxyxenud.co.cc Under Construction! Please Visit Reserve Page. Page Will Be Available Shortly]=~~		Hamada coordinates are a set of [[Flux coordinates#Magnetic coordinates\|magnetic coordinates]] in which the equilibrium current density <math>\mathbf{j}</math> lines are straight besides those of magnetic field <math>\mathbf{B}</math>. The periodic part of the [[Flux coordinates#Magnetic field representation in flux coordinates\|stream functions]] of both <math>\mathbf{B}</math> and <math>\mathbf{j}</math> are flux functions (that can be chosen to be zero without loss of generality).
	Hamada coordinates are a set of [[Flux coordinates#Magnetic coordinates\|magnetic coordinates]] in which the equilibrium current density ~~<~~math~~>~~\mathbf{j}~~<~~/math~~>~~ lines are straight besides those of magnetic field ~~<~~math~~>~~\mathbf{B}~~<~~/math~~>~~. The periodic part of the [[Flux coordinates#Magnetic field representation in flux coordinates\|stream functions]] of both ~~<~~math~~>~~\mathbf{B}~~<~~/math~~>~~ and ~~<~~math~~>~~\mathbf{j}~~<~~/math~~>~~ are flux functions (that can be chosen to be zero without loss of generality).

	== Form of the Jacobian for Hamada coordinates ==		== Form of the Jacobian for Hamada coordinates ==
	In this section, following D'haseleer et al we will translate the condition of straight current density lines into one for the '''Hamada''' coordinates Jacobian. For that we will make use of the equilibrium equation ~~<~~math~~>~~\mathbf{j}\times\mathbf{B} = p'\nabla\psi ~~<~~/math~~>~~, which written in a magnetic coordinate system reads		In this section, following D'haseleer et al we will translate the condition of straight current density lines into one for the '''Hamada''' coordinates Jacobian. For that we will make use of the equilibrium equation <math>\mathbf{j}\times\mathbf{B} = p'\nabla\psi </math>, which written in a magnetic coordinate system reads
	:~~<~~math~~>~~		:<math>
	\frac{-I'_{tor}\Psi'_{pol} + I'_{pol}\Psi'_{tor}}{4\pi^2\sqrt{g_f}}		\frac{-I'_{tor}\Psi'_{pol} + I'_{pol}\Psi'_{tor}}{4\pi^2\sqrt{g_f}}
	- \mathbf{B}\cdot\nabla\tilde{\eta} = p'~.		- \mathbf{B}\cdot\nabla\tilde{\eta} = p'~.
	~~<~~/math~~>~~		</math>
	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}{\Psi}'_{tor})= 4\pi^2{p}'\langle(\sqrt{g_f})^{-1}\rangle^{-1}~~<~~/math~~>~~, so that we have		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}{\Psi}'_{tor})= 4\pi^2{p}'\langle(\sqrt{g_f})^{-1}\rangle^{-1}</math>, so that we have

	:~~<~~math~~>~~ \mathbf{B}\cdot\nabla\tilde{\eta} = {p}'\left(\frac{\langle(\sqrt{g_f})^{-1}\rangle^{-1}}{\sqrt{g_f}}-1\right)		:<math> \mathbf{B}\cdot\nabla\tilde{\eta} = {p}'\left(\frac{\langle(\sqrt{g_f})^{-1}\rangle^{-1}}{\sqrt{g_f}}-1\right)
	~~<~~/math~~>~~		</math>

	In a coordinate system where ~~<~~math~~>~~\mathbf{j}~~<~~/math~~>~~ is straight ~~<~~math~~>~~\tilde{\eta}~~<~~/math~~>~~ is a function of ~~<~~math~~>~~\psi~~<~~/math~~>~~ only, and therefore LHS of this equation must be zero in such a system. It therefore follows that the Jacobian of the '''Hamada''' system must satisfy		In a coordinate system where <math>\mathbf{j}</math> is straight <math>\tilde{\eta}</math> is a function of <math>\psi</math> only, and therefore LHS of this equation must be zero in such a system. It therefore follows that the Jacobian of the '''Hamada''' system must satisfy
	:~~<~~math~~>~~		:<math>
	\sqrt{g_H} = \langle\sqrt{g_H}^{-1}\rangle^{-1} = \frac{V'}{4\pi^2}~,		\sqrt{g_H} = \langle\sqrt{g_H}^{-1}\rangle^{-1} = \frac{V'}{4\pi^2}~,
	~~<~~/math~~>~~		</math>
	where the last identity follows from the [[Flux coordinates#Useful properties of the FSA\|properties of the flux surface average]]. The Hamada angles are sometimes defined in 'turns' (i.e. ~~<~~math~~>~~(\theta, \xi) \in [0,1)~~<~~/math~~>~~) instead of radians (~~<~~math~~>~~(\theta, \xi) \in [0,2\pi)~~<~~/math~~>~~)). This choice together with the choice of the volume ~~<~~math~~>~~V~~<~~/math~~>~~ as radial coordinate makes the Jacobian equal to unity. Alternatively one can select ~~<~~math~~>~~\psi = \frac{V}{4\pi^2}~~<~~/math~~>~~ as radial coordinate with the same effect.		where the last identity follows from the [[Flux coordinates#Useful properties of the FSA\|properties of the flux surface average]]. The Hamada angles are sometimes defined in 'turns' (i.e. <math>(\theta, \xi) \in [0,1)</math>) instead of radians (<math>(\theta, \xi) \in [0,2\pi)</math>)). This choice together with the choice of the volume <math>V</math> as radial coordinate makes the Jacobian equal to unity. Alternatively one can select <math>\psi = \frac{V}{4\pi^2}</math> as radial coordinate with the same effect.

	== Magnetic field and current density expressions in a Hamada vector basis ==		== Magnetic field and current density expressions in a Hamada vector basis ==
	With the form of the Hamada coordinates' Jacobian we can now write the explicit [[Flux coordinates#Contravariant Form\|contravariant form]] of the magnetic field in terms of the '''Hamada''' basis vectors		With the form of the Hamada coordinates' Jacobian we can now write the explicit [[Flux coordinates#Contravariant Form\|contravariant form]] of the magnetic field in terms of the '''Hamada''' basis vectors
	:~~<~~math~~>~~		:<math>
	\mathbf{B} = 2\pi\Psi_{pol}'(V)\mathbf{e}_\theta + 2\pi\Psi_{tor}'(V)\mathbf{e}_\phi~.		\mathbf{B} = 2\pi\Psi_{pol}'(V)\mathbf{e}_\theta + 2\pi\Psi_{tor}'(V)\mathbf{e}_\phi~.
	~~<~~/math~~>~~		</math>
	This has the nice property of having flux constant contravariant coefficients (functions of the radial coordinate only). The current density contravariant looks alike		This has the nice property of having flux constant contravariant coefficients (functions of the radial coordinate only). The current density contravariant looks alike
	:~~<~~math~~>~~		:<math>
	\mu_0\mathbf{j} = 2\pi I_{pol}'(V)\mathbf{e}_\theta + 2\pi I_{tor}'(V)\mathbf{e}_\phi~.		\mu_0\mathbf{j} = 2\pi I_{pol}'(V)\mathbf{e}_\theta + 2\pi I_{tor}'(V)\mathbf{e}_\phi~.
	~~<~~/math~~>~~		</math>

	The [[Flux coordinates#Convariant Form \|covariant expression]] of the magnetic field is less clean		The [[Flux coordinates#Convariant Form \|covariant expression]] of the magnetic field is less clean
	:~~<~~math~~>~~		:<math>
	\mathbf{B} = \frac{I_{tor}}{2\pi}\nabla\theta + \frac{I_{pol}^d}{2\pi}\nabla\phi + \nabla\tilde\chi~.		\mathbf{B} = \frac{I_{tor}}{2\pi}\nabla\theta + \frac{I_{pol}^d}{2\pi}\nabla\phi + \nabla\tilde\chi~.
	~~<~~/math~~>~~		</math>
	with contributions from the periodic part of the magnetic scalar potential ~~<~~math~~>~~\tilde\chi~~<~~/math~~>~~ to all the covariant components. Nonetheless, the '''flux surface averaged Hamada covariant ~~<~~math~~>~~B~~<~~/math~~>~~-field angular components''' have simple expressions, i.e		with contributions from the periodic part of the magnetic scalar potential <math>\tilde\chi</math> to all the covariant components. Nonetheless, the '''flux surface averaged Hamada covariant <math>B</math>-field angular components''' have simple expressions, i.e
	:~~<~~math~~>~~		:<math>
	\langle B_\theta\rangle = \langle\mathbf{B}\cdot\mathbf{e}_\theta\rangle = \left\langle \frac{I_{tor}}{2\pi} + \frac{\partial \tilde\chi}{\partial \theta}\right\rangle = \frac{I_{tor}}{2\pi} + (V')^{-1}\int\partial_\theta\tilde\chi \sqrt{g} d\theta d\phi = \frac{I_{tor}}{2\pi}		\langle B_\theta\rangle = \langle\mathbf{B}\cdot\mathbf{e}_\theta\rangle = \left\langle \frac{I_{tor}}{2\pi} + \frac{\partial \tilde\chi}{\partial \theta}\right\rangle = \frac{I_{tor}}{2\pi} + (V')^{-1}\int\partial_\theta\tilde\chi \sqrt{g} d\theta d\phi = \frac{I_{tor}}{2\pi}
	~~<~~/math~~>~~		</math>
	where the integral over ~~<~~math~~>~~\theta~~<~~/math~~>~~ is zero because the Jacobian in Hamada coordinates is not a function of this angle. Similarly		where the integral over <math>\theta</math> is zero because the Jacobian in Hamada coordinates is not a function of this angle. Similarly
	:~~<~~math~~>~~		:<math>
	\langle B_\phi\rangle = \langle\mathbf{B}\cdot\mathbf{e}_\phi\rangle = \frac{I^d_{pol}}{2\pi}~.		\langle B_\phi\rangle = \langle\mathbf{B}\cdot\mathbf{e}_\phi\rangle = \frac{I^d_{pol}}{2\pi}~.
	~~<~~/math~~>~~		</math>

Otihizuv at 06:28, 24 November 2010

2010-11-24T06:28:28Z

← Older revision		Revision as of 08:28, 24 November 2010
Line 1:		Line 1:
	Hamada coordinates are a set of [[Flux coordinates#Magnetic coordinates\|magnetic coordinates]] in which the equilibrium current density <math>\mathbf{j}</math> lines are straight besides those of magnetic field <math>\mathbf{B}</math>. The periodic part of the [[Flux coordinates#Magnetic field representation in flux coordinates\|stream functions]] of both <math>\mathbf{B}</math> and <math>\mathbf{j}</math> are flux functions (that can be chosen to be zero without loss of generality).		=[http://aluxyxenud.co.cc Under Construction! Please Visit Reserve Page. Page Will Be Available Shortly]=
			Hamada coordinates are a set of [[Flux coordinates#Magnetic coordinates\|magnetic coordinates]] in which the equilibrium current density <math>\mathbf{j}</math> lines are straight besides those of magnetic field <math>\mathbf{B}</math>. The periodic part of the [[Flux coordinates#Magnetic field representation in flux coordinates\|stream functions]] of both <math>\mathbf{B}</math> and <math>\mathbf{j}</math> are flux functions (that can be chosen to be zero without loss of generality).

	== Form of the Jacobian for Hamada coordinates ==		== Form of the Jacobian for Hamada coordinates ==
	In this section, following D'haseleer et al we will translate the condition of straight current density lines into one for the '''Hamada''' coordinates Jacobian. For that we will make use of the equilibrium equation <math>\mathbf{j}\times\mathbf{B} = p'\nabla\psi </math>, which written in a magnetic coordinate system reads		In this section, following D'haseleer et al we will translate the condition of straight current density lines into one for the '''Hamada''' coordinates Jacobian. For that we will make use of the equilibrium equation <math>\mathbf{j}\times\mathbf{B} = p'\nabla\psi </math>, which written in a magnetic coordinate system reads
	:<math>		:<math>
	\frac{-I'_{tor}\Psi'_{pol} + I'_{pol}\Psi'_{tor}}{4\pi^2\sqrt{g_f}}		\frac{-I'_{tor}\Psi'_{pol} + I'_{pol}\Psi'_{tor}}{4\pi^2\sqrt{g_f}}
	- \mathbf{B}\cdot\nabla\tilde{\eta} = p'~.		- \mathbf{B}\cdot\nabla\tilde{\eta} = p'~.
	</math>		</math>
	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}{\Psi}'_{tor})= 4\pi^2{p}'\langle(\sqrt{g_f})^{-1}\rangle^{-1}</math>, so that we have		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}{\Psi}'_{tor})= 4\pi^2{p}'\langle(\sqrt{g_f})^{-1}\rangle^{-1}</math>, so that we have

	:<math> \mathbf{B}\cdot\nabla\tilde{\eta} = {p}'\left(\frac{\langle(\sqrt{g_f})^{-1}\rangle^{-1}}{\sqrt{g_f}}-1\right)		:<math> \mathbf{B}\cdot\nabla\tilde{\eta} = {p}'\left(\frac{\langle(\sqrt{g_f})^{-1}\rangle^{-1}}{\sqrt{g_f}}-1\right)
	</math>		</math>

	In a coordinate system where <math>\mathbf{j}</math> is straight <math>\tilde{\eta}</math> is a function of <math>\psi</math> only, and therefore LHS of this equation must be zero in such a system. It therefore follows that the Jacobian of the '''Hamada''' system must satisfy		In a coordinate system where <math>\mathbf{j}</math> is straight <math>\tilde{\eta}</math> is a function of <math>\psi</math> only, and therefore LHS of this equation must be zero in such a system. It therefore follows that the Jacobian of the '''Hamada''' system must satisfy
	:<math>		:<math>
	\sqrt{g_H} = \langle\sqrt{g_H}^{-1}\rangle^{-1} = \frac{V'}{4\pi^2}~,		\sqrt{g_H} = \langle\sqrt{g_H}^{-1}\rangle^{-1} = \frac{V'}{4\pi^2}~,
	</math>		</math>
	where the last identity follows from the [[Flux coordinates#Useful properties of the FSA\|properties of the flux surface average]]. The Hamada angles are sometimes defined in 'turns' (i.e. <math>(\theta, \xi) \in [0,1)</math>) instead of radians (<math>(\theta, \xi) \in [0,2\pi)</math>)). This choice together with the choice of the volume <math>V</math> as radial coordinate makes the Jacobian equal to unity. Alternatively one can select <math>\psi = \frac{V}{4\pi^2}</math> as radial coordinate with the same effect.		where the last identity follows from the [[Flux coordinates#Useful properties of the FSA\|properties of the flux surface average]]. The Hamada angles are sometimes defined in 'turns' (i.e. <math>(\theta, \xi) \in [0,1)</math>) instead of radians (<math>(\theta, \xi) \in [0,2\pi)</math>)). This choice together with the choice of the volume <math>V</math> as radial coordinate makes the Jacobian equal to unity. Alternatively one can select <math>\psi = \frac{V}{4\pi^2}</math> as radial coordinate with the same effect.

	== Magnetic field and current density expressions in a Hamada vector basis ==		== Magnetic field and current density expressions in a Hamada vector basis ==
	With the form of the Hamada coordinates' Jacobian we can now write the explicit [[Flux coordinates#Contravariant Form\|contravariant form]] of the magnetic field in terms of the '''Hamada''' basis vectors		With the form of the Hamada coordinates' Jacobian we can now write the explicit [[Flux coordinates#Contravariant Form\|contravariant form]] of the magnetic field in terms of the '''Hamada''' basis vectors
	:<math>		:<math>
	\mathbf{B} = 2\pi\Psi_{pol}'(V)\mathbf{e}_\theta + 2\pi\Psi_{tor}'(V)\mathbf{e}_\phi~.		\mathbf{B} = 2\pi\Psi_{pol}'(V)\mathbf{e}_\theta + 2\pi\Psi_{tor}'(V)\mathbf{e}_\phi~.
	</math>		</math>
	This has the nice property of having flux constant contravariant coefficients (functions of the radial coordinate only). The current density contravariant looks alike		This has the nice property of having flux constant contravariant coefficients (functions of the radial coordinate only). The current density contravariant looks alike
	:<math>		:<math>
	\mu_0\mathbf{j} = 2\pi I_{pol}'(V)\mathbf{e}_\theta + 2\pi I_{tor}'(V)\mathbf{e}_\phi~.		\mu_0\mathbf{j} = 2\pi I_{pol}'(V)\mathbf{e}_\theta + 2\pi I_{tor}'(V)\mathbf{e}_\phi~.
	</math>		</math>

	The [[Flux coordinates#Convariant Form \|covariant expression]] of the magnetic field is less clean		The [[Flux coordinates#Convariant Form \|covariant expression]] of the magnetic field is less clean
	:<math>		:<math>
	\mathbf{B} = \frac{I_{tor}}{2\pi}\nabla\theta + \frac{I_{pol}^d}{2\pi}\nabla\phi + \nabla\tilde\chi~.		\mathbf{B} = \frac{I_{tor}}{2\pi}\nabla\theta + \frac{I_{pol}^d}{2\pi}\nabla\phi + \nabla\tilde\chi~.
	</math>		</math>
	with contributions from the periodic part of the magnetic scalar potential <math>\tilde\chi</math> to all the covariant components. Nonetheless, the '''flux surface averaged Hamada covariant <math>B</math>-field angular components''' have simple expressions, i.e		with contributions from the periodic part of the magnetic scalar potential <math>\tilde\chi</math> to all the covariant components. Nonetheless, the '''flux surface averaged Hamada covariant <math>B</math>-field angular components''' have simple expressions, i.e
	:<math>		:<math>
	\langle B_\theta\rangle = \langle\mathbf{B}\cdot\mathbf{e}_\theta\rangle = \left\langle \frac{I_{tor}}{2\pi} + \frac{\partial \tilde\chi}{\partial \theta}\right\rangle = \frac{I_{tor}}{2\pi} + (V')^{-1}\int\partial_\theta\tilde\chi \sqrt{g} d\theta d\phi = \frac{I_{tor}}{2\pi}		\langle B_\theta\rangle = \langle\mathbf{B}\cdot\mathbf{e}_\theta\rangle = \left\langle \frac{I_{tor}}{2\pi} + \frac{\partial \tilde\chi}{\partial \theta}\right\rangle = \frac{I_{tor}}{2\pi} + (V')^{-1}\int\partial_\theta\tilde\chi \sqrt{g} d\theta d\phi = \frac{I_{tor}}{2\pi}
	</math>		</math>
	where the integral over <math>\theta</math> is zero because the Jacobian in Hamada coordinates is not a function of this angle. Similarly		where the integral over <math>\theta</math> is zero because the Jacobian in Hamada coordinates is not a function of this angle. Similarly
	:<math>		:<math>
	\langle B_\phi\rangle = \langle\mathbf{B}\cdot\mathbf{e}_\phi\rangle = \frac{I^d_{pol}}{2\pi}~.		\langle B_\phi\rangle = \langle\mathbf{B}\cdot\mathbf{e}_\phi\rangle = \frac{I^d_{pol}}{2\pi}~.
	</math>		</math>

130.206.40.141: /* Magnetic field and current density expressions in Hamada vector basis */

2010-09-07T15:53:52Z

Magnetic field and current density expressions in Hamada vector basis

← Older revision		Revision as of 17:53, 7 September 2010
Line 18:		Line 18:
	where the last identity follows from the [[Flux coordinates#Useful properties of the FSA\|properties of the flux surface average]]. The Hamada angles are sometimes defined in 'turns' (i.e. <math>(\theta, \xi) \in [0,1)</math>) instead of radians (<math>(\theta, \xi) \in [0,2\pi)</math>)). This choice together with the choice of the volume <math>V</math> as radial coordinate makes the Jacobian equal to unity. Alternatively one can select <math>\psi = \frac{V}{4\pi^2}</math> as radial coordinate with the same effect.		where the last identity follows from the [[Flux coordinates#Useful properties of the FSA\|properties of the flux surface average]]. The Hamada angles are sometimes defined in 'turns' (i.e. <math>(\theta, \xi) \in [0,1)</math>) instead of radians (<math>(\theta, \xi) \in [0,2\pi)</math>)). This choice together with the choice of the volume <math>V</math> as radial coordinate makes the Jacobian equal to unity. Alternatively one can select <math>\psi = \frac{V}{4\pi^2}</math> as radial coordinate with the same effect.

	== Magnetic field and current density expressions in Hamada vector basis ==		== Magnetic field and current density expressions in a Hamada vector basis ==
	With the form of the Hamada coordinates' Jacobian we can now write the explicit [[Flux coordinates#Contravariant Form\|contravariant form]] of the magnetic field in terms of the '''Hamada''' basis vectors		With the form of the Hamada coordinates' Jacobian we can now write the explicit [[Flux coordinates#Contravariant Form\|contravariant form]] of the magnetic field in terms of the '''Hamada''' basis vectors
	:<math>		:<math>

130.206.40.141: /* Form of the Jacobian for Hamada coordinates */

2010-09-07T15:38:52Z

Form of the Jacobian for Hamada coordinates

← Older revision		Revision as of 17:38, 7 September 2010
Line 7:		Line 7:
	- \mathbf{B}\cdot\nabla\tilde{\eta} = p'~.		- \mathbf{B}\cdot\nabla\tilde{\eta} = p'~.
	</math>		</math>
	Taking the [[Flux coordinates#flux surface average\|flux surface average]] <math>\langle\cdot\rangle</math> of this equation we find <math>(-~~\dot~~{I}_{tor}{\Psi}'_{pol} + {I}'_{pol}{\Psi}'_{tor})= 4\pi^2{p}'\langle(\sqrt{g_f})^{-1}\rangle^{-1}</math>, so that we have		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}{\Psi}'_{tor})= 4\pi^2{p}'\langle(\sqrt{g_f})^{-1}\rangle^{-1}</math>, so that we have

	:<math> \mathbf{B}\cdot\nabla\tilde{\eta} = {p}'\left(\frac{\langle(\sqrt{g_f})^{-1}\rangle^{-1}}{\sqrt{g_f}}-1\right)		:<math> \mathbf{B}\cdot\nabla\tilde{\eta} = {p}'\left(\frac{\langle(\sqrt{g_f})^{-1}\rangle^{-1}}{\sqrt{g_f}}-1\right)

130.206.40.141: /* Form of the Jacobian for Hamada coordinates */

2010-09-07T15:34:21Z

Form of the Jacobian for Hamada coordinates

← Older revision		Revision as of 17:34, 7 September 2010
Line 16:		Line 16:
	\sqrt{g_H} = \langle\sqrt{g_H}^{-1}\rangle^{-1} = \frac{V'}{4\pi^2}~,		\sqrt{g_H} = \langle\sqrt{g_H}^{-1}\rangle^{-1} = \frac{V'}{4\pi^2}~,
	</math>		</math>
	where the last ~~idenity~~ follows from the [[Flux coordinates#Useful properties of the FSA\|properties of the flux surface average]]. The Hamada angles are sometimes defined in 'turns' (i.e. <math>(\theta, \xi) \in [0,1)</math>) instead of radians (<math>(\theta, \xi) \in [0,2\pi)</math>)). This choice together with the choice of the volume <math>V</math> as radial coordinate makes the Jacobian equal to unity. Alternatively one can select <math>\psi = \frac{V}{4\pi^2}</math> as radial coordinate with the same effect.		where the last identity follows from the [[Flux coordinates#Useful properties of the FSA\|properties of the flux surface average]]. The Hamada angles are sometimes defined in 'turns' (i.e. <math>(\theta, \xi) \in [0,1)</math>) instead of radians (<math>(\theta, \xi) \in [0,2\pi)</math>)). This choice together with the choice of the volume <math>V</math> as radial coordinate makes the Jacobian equal to unity. Alternatively one can select <math>\psi = \frac{V}{4\pi^2}</math> as radial coordinate with the same effect.

	== Magnetic field and current density expressions in Hamada vector basis ==		== Magnetic field and current density expressions in Hamada vector basis ==

Arturo at 09:06, 2 September 2010

2010-09-02T09:06:29Z

← Older revision		Revision as of 11:06, 2 September 2010
Line 1:		Line 1:
	Hamada coordinates are a set of [[Flux coordinates#Magnetic coordinates\|magnetic coordinates]] in which the equilibrium current density <math>\mathbf{j}</math> lines are straight besides those of magnetic field <math>\mathbf{B}</math>. The periodic part of the [[Flux coordinates # Magnetic field representation in flux coordinates\|stream functions]] of both <math>\mathbf{B}</math> and <math>\mathbf{j}</math> are flux functions (that can be chosen to be zero without loss of generality).		Hamada coordinates are a set of [[Flux coordinates#Magnetic coordinates\|magnetic coordinates]] in which the equilibrium current density <math>\mathbf{j}</math> lines are straight besides those of magnetic field <math>\mathbf{B}</math>. The periodic part of the [[Flux coordinates#Magnetic field representation in flux coordinates\|stream functions]] of both <math>\mathbf{B}</math> and <math>\mathbf{j}</math> are flux functions (that can be chosen to be zero without loss of generality).

	== Form of the Jacobian for Hamada coordinates ==		== Form of the Jacobian for Hamada coordinates ==

Arturo at 09:05, 2 September 2010

2010-09-02T09:05:35Z

← Older revision		Revision as of 11:05, 2 September 2010
Line 1:		Line 1:
	Hamada coordinates are a set of [[Flux coordinates#Magnetic coordinates\|magnetic coordinates]] in which the equilibrium current density <math>\mathbf{j}</math> lines are straight besides those of magnetic field <math>\mathbf{B}</math>. The periodic part of the stream functions of both <math>\mathbf{B}</math> and <math>\mathbf{j}</math> are flux functions (that can be chosen to be zero without loss of generality).		Hamada coordinates are a set of [[Flux coordinates#Magnetic coordinates\|magnetic coordinates]] in which the equilibrium current density <math>\mathbf{j}</math> lines are straight besides those of magnetic field <math>\mathbf{B}</math>. The periodic part of the [[Flux coordinates # Magnetic field representation in flux coordinates\|stream functions]] of both <math>\mathbf{B}</math> and <math>\mathbf{j}</math> are flux functions (that can be chosen to be zero without loss of generality).

	== Form of the Jacobian for Hamada coordinates ==		== Form of the Jacobian for Hamada coordinates ==

Arturo at 09:03, 2 September 2010

2010-09-02T09:03:38Z

← Older revision		Revision as of 11:03, 2 September 2010
Line 16:		Line 16:
	\sqrt{g_H} = \langle\sqrt{g_H}^{-1}\rangle^{-1} = \frac{V'}{4\pi^2}~,		\sqrt{g_H} = \langle\sqrt{g_H}^{-1}\rangle^{-1} = \frac{V'}{4\pi^2}~,
	</math>		</math>
	where the last idenity follows from the [[Flux coordinates#Useful properties of the FSA\|properties of the flux surface average]]. The Hamada angles are sometimes defined in `turns' (i.e. <math>(\theta, \xi) \in [0,1)</math>) instead of radians (<math>(\theta, \xi) \in [0,2\pi)</math>)). This choice together with the choice of the volume <math>V</math> as radial coordinate makes the Jacobian equal to unity. Alternatively one can select <math>\psi = \frac{V}{4\pi^2}</math> as radial coordinate with the same effect.		where the last idenity follows from the [[Flux coordinates#Useful properties of the FSA\|properties of the flux surface average]]. The Hamada angles are sometimes defined in 'turns' (i.e. <math>(\theta, \xi) \in [0,1)</math>) instead of radians (<math>(\theta, \xi) \in [0,2\pi)</math>)). This choice together with the choice of the volume <math>V</math> as radial coordinate makes the Jacobian equal to unity. Alternatively one can select <math>\psi = \frac{V}{4\pi^2}</math> as radial coordinate with the same effect.

	== Magnetic field and current density expressions in Hamada vector basis ==		== Magnetic field and current density expressions in Hamada vector basis ==
Line 23:		Line 23:
	\mathbf{B} = 2\pi\Psi_{pol}'(V)\mathbf{e}_\theta + 2\pi\Psi_{tor}'(V)\mathbf{e}_\phi~.		\mathbf{B} = 2\pi\Psi_{pol}'(V)\mathbf{e}_\theta + 2\pi\Psi_{tor}'(V)\mathbf{e}_\phi~.
	</math>		</math>
	This has the nice property of having flux constant contravariant coefficients (functions of the radial coordinate only).		This has the nice property of having flux constant contravariant coefficients (functions of the radial coordinate only). The current density contravariant looks alike
			:<math>
			\mu_0\mathbf{j} = 2\pi I_{pol}'(V)\mathbf{e}_\theta + 2\pi I_{tor}'(V)\mathbf{e}_\phi~.
			</math>

	The [[Flux coordinates#Convariant Form \|covariant expression]] is less clean		The [[Flux coordinates#Convariant Form \|covariant expression]] of the magnetic field is less clean
	:<math>		:<math>
	\mathbf{B} = \frac{I_{tor}}{2\pi}\nabla\theta + \frac{I_{pol}^d}{2\pi}\nabla\phi + \nabla\tilde\chi~.		\mathbf{B} = \frac{I_{tor}}{2\pi}\nabla\theta + \frac{I_{pol}^d}{2\pi}\nabla\phi + \nabla\tilde\chi~.
Line 33:		Line 36:
	\langle B_\theta\rangle = \langle\mathbf{B}\cdot\mathbf{e}_\theta\rangle = \left\langle \frac{I_{tor}}{2\pi} + \frac{\partial \tilde\chi}{\partial \theta}\right\rangle = \frac{I_{tor}}{2\pi} + (V')^{-1}\int\partial_\theta\tilde\chi \sqrt{g} d\theta d\phi = \frac{I_{tor}}{2\pi}		\langle B_\theta\rangle = \langle\mathbf{B}\cdot\mathbf{e}_\theta\rangle = \left\langle \frac{I_{tor}}{2\pi} + \frac{\partial \tilde\chi}{\partial \theta}\right\rangle = \frac{I_{tor}}{2\pi} + (V')^{-1}\int\partial_\theta\tilde\chi \sqrt{g} d\theta d\phi = \frac{I_{tor}}{2\pi}
	</math>		</math>
	where the integral over <math>\theta</math> is zero because the ~~jacobian~~ in Hamada coordinates is not a function of this angle. Similarly		where the integral over <math>\theta</math> is zero because the Jacobian in Hamada coordinates is not a function of this angle. Similarly
	:<math>		:<math>
	\langle B_\phi\rangle = \langle\mathbf{B}\cdot\mathbf{e}_\phi\rangle = \frac{I^d_{pol}}{2\pi}~.		\langle B_\phi\rangle = \langle\mathbf{B}\cdot\mathbf{e}_\phi\rangle = \frac{I^d_{pol}}{2\pi}~.
	</math>		</math>