Flux coordinates - Revision history

Admin at 10:49, 26 January 2023

2023-01-26T10:49:42Z

← Older revision		Revision as of 12:49, 26 January 2023
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	Particular choices of G can be made so as to simplify the description of other fields. The most commonly used magnetic coordinate systems are:		Particular choices of G can be made so as to simplify the description of other fields. The most commonly used magnetic coordinate systems are:
	<ref name='Dhaeseleer'>W.D. D'haeseleer, ''Flux coordinates and magnetic field structure: a guide to a fundamental tool of plasma theory'', Springer series in computational physics, Springer-Verlag (1991) ISBN 3540524193</ref>		<ref name='Dhaeseleer'>W.D. D'haeseleer, ''Flux coordinates and magnetic field structure: a guide to a fundamental tool of plasma theory'', Springer series in computational physics, Springer-Verlag (1991) {{ISBN\|3540524193}}</ref>
	* [[Hamada coordinates]]. <ref>S. Hamada, Nucl. Fusion '''2''' (1962) 23</ref><ref>[~~http~~:~~//dx.doi.org/~~10.1063/1.1706651 J.M. Greene and J.L Johnson, ''Stability Criterion for Arbitrary Hydromagnetic Equilibria'', Phys. Fluids '''5''' (1962) 510]</ref> In these coordinates, both the magnetic field lines and current lines corresponding to the [[MHD equilibrium]] are straight. Referring to the definitions above, both <math>\tilde\nu</math> and <math>\tilde\eta</math> are zero in Hamada coordinates.		* [[Hamada coordinates]]. <ref>S. Hamada, Nucl. Fusion '''2''' (1962) 23</ref><ref>[[doi:10.1063/1.1706651\|J.M. Greene and J.L Johnson, ''Stability Criterion for Arbitrary Hydromagnetic Equilibria'', Phys. Fluids '''5''' (1962) 510]]</ref> In these coordinates, both the magnetic field lines and current lines corresponding to the [[MHD equilibrium]] are straight. Referring to the definitions above, both <math>\tilde\nu</math> and <math>\tilde\eta</math> are zero in Hamada coordinates.
	* [[Boozer coordinates]]. <ref>[~~http~~:~~//dx.doi.org/~~10.1063/1.863297 A.H. Boozer, ''Plasma equilibrium with rational magnetic surfaces'', Phys. Fluids '''24''' (1981) 1999]</ref><ref>[~~http~~:~~//dx.doi.org/~~10.1063/1.863765 A.H. Boozer, ''Establishment of magnetic coordinates for a given magnetic field'', Phys. Fluids '''25''' (1982) 520]</ref> In these coordinates, the magnetic field lines corresponding to the [[MHD equilibrium]] are straight and so are the ''diamagnetic lines '', i.e. the integral lines of <math>\nabla\psi\times\mathbf{B}</math>. Referring to the definitions above, both <math>\tilde\nu</math> and <math>\tilde\chi</math> are zero in Boozer coordinates.		* [[Boozer coordinates]]. <ref>[[doi:10.1063/1.863297\|A.H. Boozer, ''Plasma equilibrium with rational magnetic surfaces'', Phys. Fluids '''24''' (1981) 1999]]</ref><ref>[[doi:10.1063/1.863765\|A.H. Boozer, ''Establishment of magnetic coordinates for a given magnetic field'', Phys. Fluids '''25''' (1982) 520]]</ref> In these coordinates, the magnetic field lines corresponding to the [[MHD equilibrium]] are straight and so are the ''diamagnetic lines '', i.e. the integral lines of <math>\nabla\psi\times\mathbf{B}</math>. Referring to the definitions above, both <math>\tilde\nu</math> and <math>\tilde\chi</math> are zero in Boozer coordinates.

	== References ==		== References ==
	<references />		<references />

Arturo: /* Useful properties of FSA */

2013-11-28T10:04:29Z

Useful properties of FSA

← Older revision		Revision as of 12:04, 28 November 2013
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	The two identities above are the basic simplifying properties of the FSA: The first cancels the contribution of 'conservative forces' like the pressure gradient or electrostatic electric fields. The second reduces the number of spatial variables to only the radial one. Further, it is possible to show that, if <math>\nabla\cdot\Gamma = 0</math> then <math>\langle\Gamma\cdot\nabla V\rangle = 0</math> and not simply constant as the above suggests. This can be seen by simply using Gauss' theorem		The two identities above are the basic simplifying properties of the FSA: The first cancels the contribution of 'conservative forces' like the pressure gradient or electrostatic electric fields. The second reduces the number of spatial variables to only the radial one. Further, it is possible to show that, if <math>\nabla\cdot\Gamma = 0</math> then <math>\langle\Gamma\cdot\nabla V\rangle = 0</math> and not simply constant as the above suggests. This can be seen by simply using Gauss' theorem

	*<math> \int_{\mathcal{V}}\nabla\cdot\Gamma\; d\mathcal{V} = \langle\Gamma\cdot\nabla V\rangle \qquad \mathrm{where~} \mathcal{V} \mathrm{~is~the~volume~enclosed~by~a~flux~surface.}		*<math> \int_{\mathcal{V}}\nabla\cdot\Gamma\; d\mathcal{V} =
			\int_{S(\mathcal{V})}\Gamma\cdot\nabla V \frac{dS}{\|\nabla V\|} =
			\langle\Gamma\cdot\nabla V\rangle \qquad \mathrm{where~} \mathcal{V} \mathrm{~is~the~volume~enclosed~by~a~flux~surface.}
	</math>		</math>
	The FSA relates to the conventional volume integral between two surfaces labelled by their volumes <math>V_1</math> and <math>V_2</math> as		The FSA relates to the conventional volume integral between two surfaces labelled by their volumes <math>V_1</math> and <math>V_2</math> as

Arturo: /* Gradient, Divergence and Curl in curvilinear coordinates */

2012-10-25T10:42:02Z

Gradient, Divergence and Curl in curvilinear coordinates

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	while the curl is		while the curl is
	:<math>		:<math>
	\nabla\times\mathbf{A} = \frac{\~~varepsilon_~~{ijk}}{\sqrt{g}}\frac{\partial A_j}{\partial u^i}\mathbf{e}_k \Rightarrow		\nabla\times\mathbf{A} = \frac{\varepsilon^{ijk}}{\sqrt{g}}\frac{\partial A_j}{\partial u^i}\mathbf{e}_k \Rightarrow
	\left(\nabla\times\mathbf{A}\right)^k = \frac{\~~varepsilon_~~{ijk}}{\sqrt{g}}\frac{\partial A_j}{\partial u^i}		\left(\nabla\times\mathbf{A}\right)^k = \frac{\varepsilon^{ijk}}{\sqrt{g}}\frac{\partial A_j}{\partial u^i}
	</math>		</math>
	given in terms of the covariant base vectors, where <math>\~~varepsilon_~~{ijk}</math> is the [[::Wikipedia:Levi-Civita symbol\| Levi-Civita]] symbol.		given in terms of the covariant base vectors, where <math>\varepsilon^{ijk}</math> is the [[::Wikipedia:Levi-Civita symbol\| Levi-Civita]] symbol.

	== Flux coordinates ==		== Flux coordinates ==

Arturo: /* Gradient, Divergence and Curl in curvilinear coordinates */

2012-10-25T10:40:28Z

Gradient, Divergence and Curl in curvilinear coordinates

← Older revision		Revision as of 12:40, 25 October 2012
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	while the curl is		while the curl is
	:<math>		:<math>
	\nabla\times\mathbf{A} = \frac{\varepsilon_{ijk}}{\sqrt{g}}\frac{\partial}{\partial u^i}~~(\sqrt{g}A_j)~~\mathbf{e}_k \Rightarrow		\nabla\times\mathbf{A} = \frac{\varepsilon_{ijk}}{\sqrt{g}}\frac{\partial A_j}{\partial u^i}\mathbf{e}_k \Rightarrow
	\left(\nabla\times\mathbf{A}\right)^k = \frac{\varepsilon_{ijk}}{\sqrt{g}}\frac{\partial}{\partial u^i}~~(\sqrt{g}A_j)~~		\left(\nabla\times\mathbf{A}\right)^k = \frac{\varepsilon_{ijk}}{\sqrt{g}}\frac{\partial A_j}{\partial u^i}
	</math>		</math>
	given in terms of the covariant base vectors, where <math>\varepsilon_{ijk}</math> is the [[::Wikipedia:Levi-Civita symbol\| Levi-Civita]] symbol.		given in terms of the covariant base vectors, where <math>\varepsilon_{ijk}</math> is the [[::Wikipedia:Levi-Civita symbol\| Levi-Civita]] symbol.

Arturo: /* Covariant Form */

2012-08-28T06:30:28Z

Covariant Form

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	[[Image:CurrentIntegrationCirtuits.png\|256px\|thumb\|right\|alt=Sample integration circuits for the definitions of currents.\|Sample integration circuits for the current definitions.]]		[[Image:CurrentIntegrationCirtuits.png\|256px\|thumb\|right\|alt=Sample integration circuits for the definitions of currents.\|Sample integration circuits for the current definitions.]]
	[[Image:CurrentIntegrationCirtuitsPoloidalCurrent.png\|256px\|thumb\|right\|alt=Sample surface for the definition of the current though a disc.\|Sample surface for the definition of the current though a disc. Note that only the current of more external surfaces contribute to the flux of charge through the surface.]]		[[Image:CurrentIntegrationCirtuitsPoloidalCurrent.png\|256px\|thumb\|right\|alt=Sample surface for the definition of the current though a disc.\|Sample surface for the definition of the current though a disc. Note that only the current of more external surfaces (those enclosing the one drawn here) contribute to the flux of charge through the surface.]]

	Note that <math>I</math> is not the current but <math>\mu_0</math> times the current. The functional dependence on the angular variables is again motivated by the single-valuedness of the magnetic field. The particular form of the coefficients can be obtained noting that		Note that <math>I</math> is not the current but <math>\mu_0</math> times the current. The functional dependence on the angular variables is again motivated by the single-valuedness of the magnetic field. The particular form of the coefficients can be obtained noting that

Arturo: /* Transforming between Magnetic coordinates systems */

2011-11-24T15:13:01Z

Transforming between Magnetic coordinates systems

← Older revision		Revision as of 17:13, 24 November 2011
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	G(\psi, \theta_f, \phi_f) = \sum_{m,n} G_{mn}(\psi) e^{i(m\theta_f + n\phi_f)}		G(\psi, \theta_f, \phi_f) = \sum_{m,n} G_{mn}(\psi) e^{i(m\theta_f + n\phi_f)}
	</math>		</math>
	and <math>G_{00} = 0 </math>.		and <math>G_{00} = 0 </math> guarantees periodicity is preserved.

Arturo: /* Transforming between Magnetic coordinates systems */

2011-11-03T21:31:49Z

Transforming between Magnetic coordinates systems

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	where		where
	:<math>		:<math>
	G(\psi, \theta_f, \phi_f) = \sum_{m,n} G_{mn}(\psi) e^{i(m\theta_f + ~~nphi_f~~)}		G(\psi, \theta_f, \phi_f) = \sum_{m,n} G_{mn}(\psi) e^{i(m\theta_f + n\phi_f)}
	</math>		</math>
	and <math>G_{00} = 0 </math>.		and <math>G_{00} = 0 </math>.

Arturo: /* Transforming between Magnetic coordinates systems */

2011-11-03T21:31:11Z

Transforming between Magnetic coordinates systems

← Older revision		Revision as of 23:31, 3 November 2011
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	which can be turned into an algebraic equation on the Fourier components of <math>G</math>		which can be turned into an algebraic equation on the Fourier components of <math>G</math>
	:<math>		:<math>
	G_{nm} = \frac{-i}{\Psi_{pol}'n + \Psi_{tor}'m}\left(\frac{\sqrt{g_f}}{\sqrt{g_F}}\right)_{nm}~.		G_{mn} = \frac{-i}{\Psi_{pol}'m + \Psi_{tor}'n}\left(\frac{\sqrt{g_f}}{\sqrt{g_F}}\right)_{mn}~.
	</math>		</math>
	where		where
	:<math>		:<math>
	G(\psi, \theta_f, \phi_f) = \sum_{n,m} G_{nm}(\psi) e^{i(n\theta_f + ~~m\phi_f~~)}		G(\psi, \theta_f, \phi_f) = \sum_{m,n} G_{mn}(\psi) e^{i(m\theta_f + nphi_f)}
	</math>		</math>
	and <math>G_{00} = 0 </math>.		and <math>G_{00} = 0 </math>.

Arturo: /* Useful properties of FSA */

2011-11-03T20:27:59Z

Useful properties of FSA

← Older revision		Revision as of 22:27, 3 November 2011
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	==== Useful properties of FSA ====		==== Useful properties of FSA ====
	Some useful properties of the FSA are		Some useful properties of the FSA are
			*<math> \langle \mathbf{B}\cdot\nabla f \rangle = \langle \nabla\cdot(\mathbf{B} f) \rangle = 0~,\qquad \forall~ \mathrm{single~valued~} f(\mathbf{x}), ~ \mathrm{if}~ \nabla\cdot\mathbf{B} = 0 ~\mathrm{and}~ \nabla \psi\cdot\mathbf{B} = 0 </math>
	*<math> \langle\nabla\cdot\Gamma\rangle = \frac{d}{dV}\langle\Gamma\cdot\nabla V\rangle = \frac{1}{V'}\frac{d}{d\psi}V'\langle\Gamma\cdot\nabla \psi\rangle</math>		*<math> \langle\nabla\cdot\Gamma\rangle = \frac{d}{dV}\langle\Gamma\cdot\nabla V\rangle = \frac{1}{V'}\frac{d}{d\psi}V'\langle\Gamma\cdot\nabla \psi\rangle</math>

	*<math> \langle \mathbf{B}\cdot\nabla f \rangle = \langle \nabla\cdot(\mathbf{B} f) \rangle = 0~,\qquad \forall~ \mathrm{single~valued~} f(\mathbf{x}), ~ \mathrm{if}~ \nabla\cdot\mathbf{B} = 0 ~\mathrm{and}~ \nabla \psi\cdot\mathbf{B} = 0 </math>

	The two identities above are the basic simplifying properties of the FSA: The first cancels the contribution of 'conservative forces' like the pressure gradient or electrostatic electric fields. The second reduces the number of spatial variables to only the radial one. Further, it is possible to show that, if <math>\nabla\cdot\Gamma = 0</math> then <math>\langle\Gamma\cdot\nabla V\rangle = 0</math> and not simply constant as the above ~~suggest~~. This can be seen by simply using Gauss' theorem		The two identities above are the basic simplifying properties of the FSA: The first cancels the contribution of 'conservative forces' like the pressure gradient or electrostatic electric fields. The second reduces the number of spatial variables to only the radial one. Further, it is possible to show that, if <math>\nabla\cdot\Gamma = 0</math> then <math>\langle\Gamma\cdot\nabla V\rangle = 0</math> and not simply constant as the above suggests. This can be seen by simply using Gauss' theorem

	*<math> \int_{\mathcal{V}}\nabla\cdot\Gamma\; d\mathcal{V} = \langle\Gamma\cdot\nabla V\rangle \qquad \mathrm{where~} \mathcal{V} \mathrm{~is~the~volume~enclosed~by~a~flux~surface.}		*<math> \int_{\mathcal{V}}\nabla\cdot\Gamma\; d\mathcal{V} = \langle\Gamma\cdot\nabla V\rangle \qquad \mathrm{where~} \mathcal{V} \mathrm{~is~the~volume~enclosed~by~a~flux~surface.}
	</math>		</math>
	The FSA relates to the conventional volume integral as		The FSA relates to the conventional volume integral between two surfaces labelled by their volumes <math>V_1</math> and <math>V_2</math> as
	*<math> \int_{\mathcal{V}(V_1<V<V_2)} f\; d\mathcal{V} = \int_{V_1}^{V_2} \langle f \rangle\; dV		*<math> \int_{\mathcal{V}(V_1<V<V_2)} f\; d\mathcal{V} = \int_{V_1}^{V_2} \langle f \rangle\; dV
	</math>		</math>

Arturo: /* Useful properties of FSA */

2011-10-14T11:24:29Z

Useful properties of FSA

← Older revision		Revision as of 13:24, 14 October 2011
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	The two identities above are the basic simplifying properties of the FSA: The first cancels the contribution of 'conservative forces' like the pressure gradient or electrostatic electric fields. The second reduces the number of spatial variables to only the radial one. Further, it is possible to show that, if <math>\nabla\cdot\Gamma = 0</math> then <math>\langle\Gamma\cdot\nabla V\rangle = 0</math> and not simply constant as the above suggest. This can be seen by simply using Gauss' theorem		The two identities above are the basic simplifying properties of the FSA: The first cancels the contribution of 'conservative forces' like the pressure gradient or electrostatic electric fields. The second reduces the number of spatial variables to only the radial one. Further, it is possible to show that, if <math>\nabla\cdot\Gamma = 0</math> then <math>\langle\Gamma\cdot\nabla V\rangle = 0</math> and not simply constant as the above suggest. This can be seen by simply using Gauss' theorem

	*<math> \int_{\mathcal{V}}\nabla\cdot\Gamma\; d\mathcal{V} = \langle\Gamma\cdot\nabla V~~\rangle = V'\langle\Gamma\cdot\nabla \psi~~\rangle \qquad \mathrm{where~} \mathcal{V} \mathrm{~is~the~volume~enclosed~by~a~flux~surface.}		*<math> \int_{\mathcal{V}}\nabla\cdot\Gamma\; d\mathcal{V} = \langle\Gamma\cdot\nabla V\rangle \qquad \mathrm{where~} \mathcal{V} \mathrm{~is~the~volume~enclosed~by~a~flux~surface.}
	</math>		</math>
	The FSA relates to the conventional volume integral as		The FSA relates to the conventional volume integral as