Magnetic curvature - Revision history

Admin: /* Field line curvature */

2011-08-10T18:45:20Z

Field line curvature

← Older revision		Revision as of 20:45, 10 August 2011
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	and its magnitude is equal to the inverse radius of curvature.		and its magnitude is equal to the inverse radius of curvature.

	A plasma is stable against curvature-driven instabilities (e.g., ballooning modes) when		A plasma is stable against curvature-driven [[Plasma instability\|instabilities]] (e.g., ballooning modes) when

	:<math>\vec \kappa \cdot \vec \nabla p < 0</math>		:<math>\vec \kappa \cdot \vec \nabla p < 0</math>

Admin at 09:13, 26 September 2010

2010-09-26T09:13:29Z

← Older revision		Revision as of 11:13, 26 September 2010
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	The tangent plane to any flux surface is spanned up by two tangent vectors: one is the normalized magnetic field vector (discussed above), and the other is		The tangent plane to any flux surface is spanned up by two tangent vectors: one is the normalized magnetic field vector (discussed above), and the other is

	:<math>\vec b_\perp = \frac{\vec \nabla \psi ~~\times \vec B~~ }{\|\vec \nabla \psi \times \vec B\|}</math>		:<math>\vec b_\perp = \frac{\vec \nabla \psi}{\|\vec \nabla \psi\|} \times \frac{\vec B}{\|\vec B\|}</math>

	The corresponding perpendicular curvature is		The corresponding perpendicular curvature (the curvature of the flux surface in the direction perpendicular to the magnetic field) is

	:<math>\vec \kappa_\perp = \vec b_\perp \cdot \vec \nabla \vec b_\perp</math>		:<math>\vec \kappa_\perp = \vec b_\perp \cdot \vec \nabla \vec b_\perp</math>

Admin at 09:09, 26 September 2010

2010-09-26T09:09:56Z

← Older revision		Revision as of 11:09, 26 September 2010
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	The magnetic curvature is defined by		== Field line curvature ==

			The magnetic field line curvature is defined by

	:<math>\vec \kappa = \vec b \cdot \vec \nabla \vec b</math>		:<math>\vec \kappa = \vec b \cdot \vec \nabla \vec b</math>
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	<ref>[http://link.aps.org/doi/10.1103/RevModPhys.76.1071 A.H. Boozer, ''Physics of magnetically confined plasmas'', Rev. Mod. Phys. '''76''' (2004) 1071]</ref>		<ref>[http://link.aps.org/doi/10.1103/RevModPhys.76.1071 A.H. Boozer, ''Physics of magnetically confined plasmas'', Rev. Mod. Phys. '''76''' (2004) 1071]</ref>

	== Normal curvature ==		=== Normal curvature ===

			The component of the curvature perpendicular to the [[Flux surface\|flux surface]] is

	:<math>\kappa_N = \vec \kappa \cdot \frac{\vec \nabla \psi}{\|\vec \nabla \psi\|}</math>		:<math>\kappa_N = \vec \kappa \cdot \frac{\vec \nabla \psi}{\|\vec \nabla \psi\|}</math>
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	Here, ψ is a [[Flux surface\|flux surface]] label (such as the poloidal flux).		Here, ψ is a [[Flux surface\|flux surface]] label (such as the poloidal flux).

	== Geodesic curvature ==		=== Geodesic curvature ===

			The component of the field line curvature parallel to the [[Flux surface\|flux surface]] is

	:<math>\kappa_G = \vec \kappa \cdot \left (\frac{\vec \nabla \psi}{\|\vec \nabla \psi\|} \times \frac{\vec B}{\|\vec B\|}\right )</math>		:<math>\kappa_G = \vec \kappa \cdot \left (\frac{\vec \nabla \psi}{\|\vec \nabla \psi\|} \times \frac{\vec B}{\|\vec B\|}\right )</math>

			== Flux surface curvature ==

			The tangent plane to any flux surface is spanned up by two tangent vectors: one is the normalized magnetic field vector (discussed above), and the other is

			:<math>\vec b_\perp = \frac{\vec \nabla \psi \times \vec B }{\|\vec \nabla \psi \times \vec B\|}</math>

			The corresponding perpendicular curvature is

			:<math>\vec \kappa_\perp = \vec b_\perp \cdot \vec \nabla \vec b_\perp</math>

			and one can again define the corresponding normal and geodesic curvature components in analogy with the above.

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

Admin at 10:55, 31 July 2010

2010-07-31T10:55:53Z

← Older revision		Revision as of 12:55, 31 July 2010
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	(good curvature) and unstable otherwise (bad curvature). Here, ''p'' is the pressure.		(good curvature) and unstable otherwise (bad curvature). Here, ''p'' is the pressure.
	<ref>[http://link.aps.org/doi/10.1103/RevModPhys.76.1071 A.H. Boozer, ''Physics of ~~magnetized~~ plasmas'', Rev. Mod. Phys. '''76''' (2004) 1071]</ref>		<ref>[http://link.aps.org/doi/10.1103/RevModPhys.76.1071 A.H. Boozer, ''Physics of magnetically confined plasmas'', Rev. Mod. Phys. '''76''' (2004) 1071]</ref>

	== Normal curvature ==		== Normal curvature ==

Admin at 20:17, 16 February 2010

2010-02-16T20:17:59Z

← Older revision		Revision as of 22:17, 16 February 2010
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	== Normal curvature ==		== Normal curvature ==

	:<math>\kappa_N = \kappa \cdot \frac{\vec \nabla \psi}{\|\vec \nabla \psi\|}</math>		:<math>\kappa_N = \vec \kappa \cdot \frac{\vec \nabla \psi}{\|\vec \nabla \psi\|}</math>

	Here, ψ is a [[Flux surface\|flux surface]] label (such as the poloidal flux).		Here, ψ is a [[Flux surface\|flux surface]] label (such as the poloidal flux).
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	== Geodesic curvature ==		== Geodesic curvature ==

	:<math>\kappa_G = \kappa \cdot \left (\frac{\vec \nabla \psi}{\|\vec \nabla \psi\|} \times \frac{\vec B}{\|\vec B\|}\right )</math>		:<math>\kappa_G = \vec \kappa \cdot \left (\frac{\vec \nabla \psi}{\|\vec \nabla \psi\|} \times \frac{\vec B}{\|\vec B\|}\right )</math>

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

Admin at 20:15, 11 October 2009

2009-10-11T20:15:11Z

← Older revision		Revision as of 22:15, 11 October 2009
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	(good curvature) and unstable otherwise (bad curvature). Here, ''p'' is the pressure.		(good curvature) and unstable otherwise (bad curvature). Here, ''p'' is the pressure.
	<ref>[http://link.aps.org/doi/10.1103/RevModPhys.76.1071 A.H. Boozer, ''Physics of magnetized plasmas'', Rev. Mod. Phys. '''76''' (2004) 1071]</ref>		<ref>[http://link.aps.org/doi/10.1103/RevModPhys.76.1071 A.H. Boozer, ''Physics of magnetized plasmas'', Rev. Mod. Phys. '''76''' (2004) 1071]</ref>

			== Normal curvature ==

			:<math>\kappa_N = \kappa \cdot \frac{\vec \nabla \psi}{\|\vec \nabla \psi\|}</math>

			Here, ψ is a [[Flux surface\|flux surface]] label (such as the poloidal flux).

			== Geodesic curvature ==

			:<math>\kappa_G = \kappa \cdot \left (\frac{\vec \nabla \psi}{\|\vec \nabla \psi\|} \times \frac{\vec B}{\|\vec B\|}\right )</math>

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

Admin at 16:21, 18 August 2009

2009-08-18T16:21:39Z

← Older revision		Revision as of 18:21, 18 August 2009
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	is a unit vector along the magnetic field.		is a unit vector along the magnetic field.
	~~When ''B'' is locally a circle section, then~~ ''κ'' points towards the centre of the ~~circle~~.		''κ'' points towards the local centre of curvature of ''B'',
			and its magnitude is equal to the inverse radius of curvature.

	A plasma is stable against curvature-driven instabilities (e.g., ballooning modes) when		A plasma is stable against curvature-driven instabilities (e.g., ballooning modes) when

Admin at 08:10, 10 August 2009

2009-08-10T08:10:41Z

← Older revision		Revision as of 10:10, 10 August 2009
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	is a unit vector along the magnetic field.		is a unit vector along the magnetic field.
	When ''B'' is locally a circle section, then ''&\kappa;'' points towards the centre of the circle.		When ''B'' is locally a circle section, then ''κ'' points towards the centre of the circle.

	A plasma is stable against curvature-driven instabilities (e.g., ballooning modes) when		A plasma is stable against curvature-driven instabilities (e.g., ballooning modes) when

Admin at 08:10, 10 August 2009

2009-08-10T08:10:14Z

← Older revision		Revision as of 10:10, 10 August 2009
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	is a unit vector along the magnetic field.		is a unit vector along the magnetic field.
			When ''B'' is locally a circle section, then ''&\kappa;'' points towards the centre of the circle.

	A plasma is stable against curvature-driven instabilities (e.g., ballooning modes) when		A plasma is stable against curvature-driven instabilities (e.g., ballooning modes) when

Admin at 07:50, 10 August 2009

2009-08-10T07:50:50Z

← Older revision		Revision as of 09:50, 10 August 2009
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	(good curvature) and unstable otherwise (bad curvature). Here, ''p'' is the pressure.		(good curvature) and unstable otherwise (bad curvature). Here, ''p'' is the pressure.
	<ref>[http://link.aps.org/doi/10.1103/RevModPhys.76.1071 A.H. Boozer, ''Physics of magnetized plasmas'', Rev. Mod. Phys. '''76'' (2004) 1071]</ref>		<ref>[http://link.aps.org/doi/10.1103/RevModPhys.76.1071 A.H. Boozer, ''Physics of magnetized plasmas'', Rev. Mod. Phys. '''76''' (2004) 1071]</ref>

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