Beta - Revision history

Admin at 07:45, 17 March 2024

2024-03-17T07:45:06Z

← Older revision		Revision as of 09:45, 17 March 2024
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	== Beta limit ==		== Beta limit ==

	The upper limit of <math>\beta_N</math> has been determined ~~empirically~~ by Troyon to 0.028. Often <math>\beta</math> is expressed in percent, in which case <math>\beta_N = 2.8</math>. This limit results from many different numerical studies determined to find the overall <math>\beta</math> limit out of many different MHD instabilities, such as [[external kink modes]], [[ballooning kink modes]], [[internal modes]], [[localized modes]], etc. <ref name="freidberg"></ref>		The upper limit of <math>\beta_N</math> has been determined numerically by Troyon to 0.028. Often <math>\beta</math> is expressed in percent, in which case <math>\beta_N = 2.8</math>. This limit results from many different numerical studies determined to find the overall <math>\beta</math> limit out of many different MHD instabilities, such as [[external kink modes]], [[ballooning kink modes]], [[internal modes]], [[localized modes]], etc. <ref name="freidberg"></ref>

	Empirical evaluation from the data of different tokamaks raises this value slightly to <math>\beta_N = 3.5</math>, although significantly higher values have been achieved.		Empirical evaluation from the data of different tokamaks raises this value slightly to <math>\beta_N = 3.5</math>, although significantly higher values have been achieved.

Admin at 19:55, 13 March 2024

2024-03-13T19:55:34Z

← Older revision		Revision as of 21:55, 13 March 2024
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	:<math>\frac{1}{\beta} = \frac{1}{\beta_p} + \frac{1}{\beta_t}</math>		:<math>\frac{1}{\beta} = \frac{1}{\beta_p} + \frac{1}{\beta_t}</math>

	== Normalized beta~~, beta limit~~ ==		== Normalized beta ==
	[[File:Troyon limit.png\|~~200px~~\|thumb\|Troyon Limit<ref>ITER Physics Expert Group on Disruptions, Plasma Control, and MHD, ''ITER Physics Basis Chapter 3: MHD stability, operational limits and disruptions'', [[doi:10.1088/0029-5515/39/12/303\|Nucl. Fusion '''39 ''' (1999) 2251-2389]]</ref>]]		[[File:Troyon limit.png\|300px\|thumb\|Troyon Limit<ref>ITER Physics Expert Group on Disruptions, Plasma Control, and MHD, ''ITER Physics Basis Chapter 3: MHD stability, operational limits and disruptions'', [[doi:10.1088/0029-5515/39/12/303\|Nucl. Fusion '''39 ''' (1999) 2251-2389]]</ref>]]
	<math>\beta</math> is often expressed in terms of the normalized beta (or Troyon factor)<ref>F. Troyon, R. Gruber, H. Saurenmann, S. Semenzato and S. Succi, ''MHD-Limits to Plasma Confinement'', [[doi:10.1088/0741-3335/26/1A/319\|Plasma Phys. Control. Fusion '''26''' (1984) 209]]</ref>, an operational parameter indicating how close the plasma is to reaching destabilising major MHD activity. Its definition is (for tokamaks):		<math>\beta</math> is often expressed in terms of the normalized beta (or Troyon factor)<ref>F. Troyon, R. Gruber, H. Saurenmann, S. Semenzato and S. Succi, ''MHD-Limits to Plasma Confinement'', [[doi:10.1088/0741-3335/26/1A/319\|Plasma Phys. Control. Fusion '''26''' (1984) 209]]</ref>, an operational parameter indicating how close the plasma is to reaching destabilising major MHD activity. Its definition is (for tokamaks):
	<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) {{ISBN\|3540242171}}</ref>		<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) {{ISBN\|3540242171}}</ref>
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	where <math>B_T</math> is the toroidal magnetic field in T, <math>a</math> is the minor radius in m, and <math>I_p</math> is the plasma current in MA.		where <math>B_T</math> is the toroidal magnetic field in T, <math>a</math> is the minor radius in m, and <math>I_p</math> is the plasma current in MA.
	The ~~value~~ of <math>\beta_N</math> has been determined ~~numerically~~ by Troyon to 0.028. Often <math>\beta</math> is expressed in percent, in which case <math>\beta_N = 2.8</math>. This limit results from many different numerical studies determined to find the overall <math>\beta</math> limit out of many different MHD instabilities, such as [[external kink modes]], [[ballooning kink modes]], [[internal modes]], [[localized modes]], etc. <ref name="freidberg"></ref> ~~<br>~~
			== Beta limit ==

			The upper limit of <math>\beta_N</math> has been determined empirically by Troyon to 0.028. Often <math>\beta</math> is expressed in percent, in which case <math>\beta_N = 2.8</math>. This limit results from many different numerical studies determined to find the overall <math>\beta</math> limit out of many different MHD instabilities, such as [[external kink modes]], [[ballooning kink modes]], [[internal modes]], [[localized modes]], etc. <ref name="freidberg"></ref>

	Empirical evaluation from the data of different tokamaks raises this value slightly to <math>\beta_N = 3.5</math>, although significantly higher values have been achieved.		Empirical evaluation from the data of different tokamaks raises this value slightly to <math>\beta_N = 3.5</math>, although significantly higher values have been achieved.
	<ref>S.A. Sabbagh et al, ''Resistive wall stabilized operation in rotating high beta NSTX plasmas'', [[doi:10.1088/0029-5515/46/5/014\|Nucl. Fusion '''46''' (2006) 635-644]]</ref>		<ref>S.A. Sabbagh et al, ''Resistive wall stabilized operation in rotating high beta NSTX plasmas'', [[doi:10.1088/0029-5515/46/5/014\|Nucl. Fusion '''46''' (2006) 635-644]]</ref>

Andrew.maris: Removed the reference to the Greenwald limit. The operational parameter used to indicate how close the plasma is to the Greenwald limit is the Greenwald fraction (n/n_G). See Martin Greenwald 2002 Plasma Phys. Control. Fusion 44 R27

2023-07-07T15:25:21Z

Removed the reference to the Greenwald limit. The operational parameter used to indicate how close the plasma is to the Greenwald limit is the Greenwald fraction (n/n_G). See Martin Greenwald 2002 Plasma Phys. Control. Fusion 44 R27

← Older revision		Revision as of 17:25, 7 July 2023
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	== Normalized beta, beta limit ==		== Normalized beta, beta limit ==
	[[File:Troyon limit.png\|200px\|thumb\|Troyon Limit<ref>ITER Physics Expert Group on Disruptions, Plasma Control, and MHD, ''ITER Physics Basis Chapter 3: MHD stability, operational limits and disruptions'', [[doi:10.1088/0029-5515/39/12/303\|Nucl. Fusion '''39 ''' (1999) 2251-2389]]</ref>]]		[[File:Troyon limit.png\|200px\|thumb\|Troyon Limit<ref>ITER Physics Expert Group on Disruptions, Plasma Control, and MHD, ''ITER Physics Basis Chapter 3: MHD stability, operational limits and disruptions'', [[doi:10.1088/0029-5515/39/12/303\|Nucl. Fusion '''39 ''' (1999) 2251-2389]]</ref>]]
	<math>\beta</math> is often expressed in terms of the normalized beta (or Troyon factor)<ref>F. Troyon, R. Gruber, H. Saurenmann, S. Semenzato and S. Succi, ''MHD-Limits to Plasma Confinement'', [[doi:10.1088/0741-3335/26/1A/319\|Plasma Phys. Control. Fusion '''26''' (1984) 209]]</ref>, an operational parameter indicating how close the plasma is to reaching ~~the [[Greenwald limit]] or a~~ destabilising major MHD activity. Its definition is (for tokamaks):		<math>\beta</math> is often expressed in terms of the normalized beta (or Troyon factor)<ref>F. Troyon, R. Gruber, H. Saurenmann, S. Semenzato and S. Succi, ''MHD-Limits to Plasma Confinement'', [[doi:10.1088/0741-3335/26/1A/319\|Plasma Phys. Control. Fusion '''26''' (1984) 209]]</ref>, an operational parameter indicating how close the plasma is to reaching destabilising major MHD activity. Its definition is (for tokamaks):
	<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) {{ISBN\|3540242171}}</ref>		<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) {{ISBN\|3540242171}}</ref>

Admin at 10:32, 26 January 2023

2023-01-26T10:32:35Z

← Older revision		Revision as of 12:32, 26 January 2023
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	Plasma performance is often expressed in terms of beta (<math>\beta</math>), defined as:		Plasma performance is often expressed in terms of beta (<math>\beta</math>), defined as:
	<ref name="freidberg">J.P. Freidberg, ''Plasma physics and fusion energy'', Cambridge University Press (2007) ISBN 0521851076</ref>		<ref name="freidberg">J.P. Freidberg, ''Plasma physics and fusion energy'', Cambridge University Press (2007) {{ISBN\|0521851076}}</ref>

	:<math>\beta = \frac{\left \langle p \right \rangle}{B^2/2\mu_0}</math>		:<math>\beta = \frac{\left \langle p \right \rangle}{B^2/2\mu_0}</math>
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	[[File:Troyon limit.png\|200px\|thumb\|Troyon Limit<ref>ITER Physics Expert Group on Disruptions, Plasma Control, and MHD, ''ITER Physics Basis Chapter 3: MHD stability, operational limits and disruptions'', [[doi:10.1088/0029-5515/39/12/303\|Nucl. Fusion '''39 ''' (1999) 2251-2389]]</ref>]]		[[File:Troyon limit.png\|200px\|thumb\|Troyon Limit<ref>ITER Physics Expert Group on Disruptions, Plasma Control, and MHD, ''ITER Physics Basis Chapter 3: MHD stability, operational limits and disruptions'', [[doi:10.1088/0029-5515/39/12/303\|Nucl. Fusion '''39 ''' (1999) 2251-2389]]</ref>]]
	<math>\beta</math> is often expressed in terms of the normalized beta (or Troyon factor)<ref>F. Troyon, R. Gruber, H. Saurenmann, S. Semenzato and S. Succi, ''MHD-Limits to Plasma Confinement'', [[doi:10.1088/0741-3335/26/1A/319\|Plasma Phys. Control. Fusion '''26''' (1984) 209]]</ref>, an operational parameter indicating how close the plasma is to reaching the [[Greenwald limit]] or a destabilising major MHD activity. Its definition is (for tokamaks):		<math>\beta</math> is often expressed in terms of the normalized beta (or Troyon factor)<ref>F. Troyon, R. Gruber, H. Saurenmann, S. Semenzato and S. Succi, ''MHD-Limits to Plasma Confinement'', [[doi:10.1088/0741-3335/26/1A/319\|Plasma Phys. Control. Fusion '''26''' (1984) 209]]</ref>, an operational parameter indicating how close the plasma is to reaching the [[Greenwald limit]] or a destabilising major MHD activity. Its definition is (for tokamaks):
	<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) ISBN 3540242171</ref>		<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) {{ISBN\|3540242171}}</ref>

	:<math>\beta_N = \beta \frac{a B_T}{I_p}</math>		:<math>\beta_N = \beta \frac{a B_T}{I_p}</math>

Admin: Corrected Freidberg

2015-04-20T07:05:50Z

Corrected Freidberg

← Older revision		Revision as of 09:05, 20 April 2015
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	Plasma performance is often expressed in terms of beta (<math>\beta</math>), defined as:		Plasma performance is often expressed in terms of beta (<math>\beta</math>), defined as:
	<ref>J.P. Freidberg, ''Plasma physics and fusion energy'', Cambridge University Press (2007) ISBN 0521851076</ref>		<ref name="freidberg">J.P. Freidberg, ''Plasma physics and fusion energy'', Cambridge University Press (2007) ISBN 0521851076</ref>

	:<math>\beta = \frac{\left \langle p \right \rangle}{B^2/2\mu_0}</math>		:<math>\beta = \frac{\left \langle p \right \rangle}{B^2/2\mu_0}</math>
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	where <math>B_T</math> is the toroidal magnetic field in T, <math>a</math> is the minor radius in m, and <math>I_p</math> is the plasma current in MA.		where <math>B_T</math> is the toroidal magnetic field in T, <math>a</math> is the minor radius in m, and <math>I_p</math> is the plasma current in MA.
	The value of <math>\beta_N</math> has been determined numerically by Troyon to 0.028. Often <math>\beta</math> is expressed in percent, in which case <math>\beta_N = 2.8</math>. This limit results from many different numerical studies determined to find the overall <math>\beta</math> limit out of many different MHD instabilities, such as [[external kink modes]], [[ballooning kink modes]], [[internal modes]], [[localized modes]], etc. <ref~~>J. Freidberg,~~ "~~Plasma Physics and Fusion Energy~~"~~,Cambridge University Press (2007) ISBN 139780511273759~~</ref> <br>		The value of <math>\beta_N</math> has been determined numerically by Troyon to 0.028. Often <math>\beta</math> is expressed in percent, in which case <math>\beta_N = 2.8</math>. This limit results from many different numerical studies determined to find the overall <math>\beta</math> limit out of many different MHD instabilities, such as [[external kink modes]], [[ballooning kink modes]], [[internal modes]], [[localized modes]], etc. <ref name="freidberg"></ref> <br>
	Empirical evaluation from the data of different tokamaks raises this value slightly to <math>\beta_N = 3.5</math>, although significantly higher values have been achieved.		Empirical evaluation from the data of different tokamaks raises this value slightly to <math>\beta_N = 3.5</math>, although significantly higher values have been achieved.
	<ref>S.A. Sabbagh et al, ''Resistive wall stabilized operation in rotating high beta NSTX plasmas'', [[doi:10.1088/0029-5515/46/5/014\|Nucl. Fusion '''46''' (2006) 635-644]]</ref>		<ref>S.A. Sabbagh et al, ''Resistive wall stabilized operation in rotating high beta NSTX plasmas'', [[doi:10.1088/0029-5515/46/5/014\|Nucl. Fusion '''46''' (2006) 635-644]]</ref>

Admin: Updated references

2015-04-20T07:02:54Z

Updated references

Mmessmer: added more details to normalized beta

2015-04-17T20:13:39Z

added more details to normalized beta

← Older revision		Revision as of 22:13, 17 April 2015
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	:<math>\frac{1}{\beta} = \frac{1}{\beta_p} + \frac{1}{\beta_t}</math>		:<math>\frac{1}{\beta} = \frac{1}{\beta_p} + \frac{1}{\beta_t}</math>

	== Normalized beta ==		== Normalized beta, beta limit ==
			[[File:Troyon limit.png\|200px\|thumb\|Troyon Limit<ref>ITER PHYSICS BASIS, Chapter 3</ref>]]
	~~The~~ normalized beta (or Troyon factor)<ref>[[doi:10.1088/0741-3335/26/1A/319\|F. Troyon, R. Gruber, H. Saurenmann, S. Semenzato and S. Succi, ''MHD-Limits to Plasma Confinement'', Plasma Phys. Control. Fusion '''26''' (1984) 209]]</ref> is an operational parameter indicating how close the plasma is to reaching the [[Greenwald limit]] or ~~destabilizing~~ major MHD activity. Its definition is (for tokamaks):		<math>\beta</math> is often expressed in terms of the normalized beta (or Troyon factor)<ref>[[doi:10.1088/0741-3335/26/1A/319\|F. Troyon, R. Gruber, H. Saurenmann, S. Semenzato and S. Succi, ''MHD-Limits to Plasma Confinement'', Plasma Phys. Control. Fusion '''26''' (1984) 209]]</ref>, an operational parameter indicating how close the plasma is to reaching the [[Greenwald limit]] or a destabilising major MHD activity. Its definition is (for tokamaks):
	<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) ISBN 3540242171</ref>		<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) ISBN 3540242171</ref>

	:<math>\beta_N = \~~beta_t~~ \frac{a ~~B_0~~}{I_p}</math>		:<math>\beta_N = \beta \frac{a B_T}{I_p}</math>

	where <math>~~B_0~~</math> is the ~~external~~ magnetic field in T, <math>a</math> is the minor radius in m, and <math>I_p</math> is the plasma current in MA.		where <math>B_T</math> is the toroidal magnetic field in T, <math>a</math> is the minor radius in m, and <math>I_p</math> is the plasma current in MA.
	~~Typically, the maximum~~ value of <math>\beta_N</math> ~~before~~ the ~~onset~~ of ~~deleterious instability is~~ 3.5, although significantly higher values have been achieved.		The value of <math>\beta_N</math> has been determined numerically by Troyon to 0.028. Often <math>\beta</math> is expressed in percent, in which case <math>\beta_N = 2.8</math>. This limit results from many different numerical studies determined to find the overall <math>\beta</math> limit out of many different MHD instabilities, such as [[external kink modes]], [[ballooning kink modes]], [[internal modes]], [[localized modes]], etc. <ref>J. Freidberg, "Plasma Physics and Fusion Energy",Cambridge University Press (2007) ISBN 139780511273759</ref> <br>
			Empirical evaluation from the data of different tokamaks raises this value slightly to <math>\beta_N = 3.5</math>, although significantly higher values have been achieved.
	<ref>[http://dx.doi.org/10.1088/0029-5515/46/5/014 S.A. Sabbagh et al, ''Resistive wall stabilized operation in rotating high beta NSTX plasmas'', Nucl. Fusion '''46''' (2006) 635-644]</ref>		<ref>[http://dx.doi.org/10.1088/0029-5515/46/5/014 S.A. Sabbagh et al, ''Resistive wall stabilized operation in rotating high beta NSTX plasmas'', Nucl. Fusion '''46''' (2006) 635-644]</ref>

Mmessmer: converted all html to math tags and added some minor information

2015-04-13T19:13:39Z

converted all html to math tags and added some minor information

← Older revision		Revision as of 21:13, 13 April 2015
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	Plasma performance is often expressed in terms of beta (β), defined as:		Plasma performance is often expressed in terms of beta (<math>\beta</math>), defined as:
	<ref>J.P. Freidberg, ''Plasma physics and fusion energy'', Cambridge University Press (2007) ISBN 0521851076</ref>		<ref>J.P. Freidberg, ''Plasma physics and fusion energy'', Cambridge University Press (2007) ISBN 0521851076</ref>

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	i.e., the ratio of the plasma pressure to the magnetic pressure.		i.e., the ratio of the plasma pressure to the magnetic pressure.
	Here, <''p''> is the mean plasma pressure, and ''B'' the mean total field strength.		Here, <math>\left \langle p \right \rangle </math> is the mean plasma pressure, and <math>B</math> the mean total field strength.
	It is customary to introduce also the ''poloidal'' ~~β (&~~beta;<~~sub~~>p</~~sub~~>) and the ''toroidal'' &beta~~; (β~~<~~sub~~>t</~~sub~~>), in which ''B'' is replaced by the poloidal and toroidal magnetic field component, respectively. One has:		It is customary to introduce also the ''poloidal'' beta <math>\beta_p</math> and the ''toroidal'' beta <math>\beta_t</math>, in which <math>B</math> is replaced by the poloidal and toroidal magnetic field component, respectively. One has:

	:<math>\frac{1}{\beta} = \frac{1}{\beta_p} + \frac{1}{\beta_t}</math>		:<math>\frac{1}{\beta} = \frac{1}{\beta_p} + \frac{1}{\beta_t}</math>
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	<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) ISBN 3540242171</ref>		<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) ISBN 3540242171</ref>

	:<math>\beta_N = \~~beta~~ \frac{a B}{I_p}</math>		:<math>\beta_N = \beta_t \frac{a B_0}{I_p}</math>

	where ~~''B''~~ is in T, ''a'' in m, and ~~''I~~<~~sub~~>p</~~sub~~>'' in MA.		where <math>B_0</math> is the external magnetic field in T, <math>a</math> is the minor radius in m, and <math>I_p</math> is the plasma current in MA.
	Typically, the maximum value of ~~β~~<~~sub~~>N</~~sub~~> before the onset of deleterious instability is 3.5, although significantly higher values have been achieved.		Typically, the maximum value of <math>\beta_N</math> before the onset of deleterious instability is 3.5, although significantly higher values have been achieved.
	<ref>[http://dx.doi.org/10.1088/0029-5515/46/5/014 S.A. Sabbagh et al, ''Resistive wall stabilized operation in rotating high beta NSTX plasmas'', Nucl. Fusion '''46''' (2006) 635-644]</ref>		<ref>[http://dx.doi.org/10.1088/0029-5515/46/5/014 S.A. Sabbagh et al, ''Resistive wall stabilized operation in rotating high beta NSTX plasmas'', Nucl. Fusion '''46''' (2006) 635-644]</ref>

Admin at 18:59, 13 August 2012

2012-08-13T18:59:09Z

← Older revision		Revision as of 20:59, 13 August 2012
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	== Normalized beta ==		== Normalized beta ==

	The normalized beta (or Troyon factor) is an operational parameter indicating how close the plasma is to reaching the [[Greenwald limit]] or destabilizing major MHD activity. Its definition is (for tokamaks):		The normalized beta (or Troyon factor)<ref>[[doi:10.1088/0741-3335/26/1A/319\|F. Troyon, R. Gruber, H. Saurenmann, S. Semenzato and S. Succi, ''MHD-Limits to Plasma Confinement'', Plasma Phys. Control. Fusion '''26''' (1984) 209]]</ref> is an operational parameter indicating how close the plasma is to reaching the [[Greenwald limit]] or destabilizing major MHD activity. Its definition is (for tokamaks):
	<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) ISBN 3540242171</ref>		<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) ISBN 3540242171</ref>

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	* [[Internal inductance]]		* [[Internal inductance]]
			* [[:Wikipedia:Beta_(plasma_physics)]]

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

Admin at 09:40, 11 August 2012

2012-08-11T09:40:08Z

← Older revision		Revision as of 11:40, 11 August 2012
Line 20:		Line 20:
	Typically, the maximum value of β<sub>N</sub> before the onset of deleterious instability is 3.5, although significantly higher values have been achieved.		Typically, the maximum value of β<sub>N</sub> before the onset of deleterious instability is 3.5, although significantly higher values have been achieved.
	<ref>[http://dx.doi.org/10.1088/0029-5515/46/5/014 S.A. Sabbagh et al, ''Resistive wall stabilized operation in rotating high beta NSTX plasmas'', Nucl. Fusion '''46''' (2006) 635-644]</ref>		<ref>[http://dx.doi.org/10.1088/0029-5515/46/5/014 S.A. Sabbagh et al, ''Resistive wall stabilized operation in rotating high beta NSTX plasmas'', Nucl. Fusion '''46''' (2006) 635-644]</ref>

			== See also ==

			* [[Internal inductance]]

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

← Older revision		Revision as of 09:02, 20 April 2015
Line 1:		Line 1:
	Plasma performance is often expressed in terms of beta (<math>\beta</math>), defined as:		Plasma performance is often expressed in terms of beta (<math>\beta</math>), defined as:
	<ref>J.P. Freidberg, ''Plasma physics and fusion energy'', Cambridge University Press (2007) ISBN 0521851076</ref>		<ref>J.P. Freidberg, ''Plasma physics and fusion energy'', Cambridge University Press (2007) ISBN 0521851076</ref>

	:<math>\beta = \frac{\left \langle p \right \rangle}{B^2/2\mu_0}</math>		:<math>\beta = \frac{\left \langle p \right \rangle}{B^2/2\mu_0}</math>
Line 11:		Line 11:

	== Normalized beta, beta limit ==		== Normalized beta, beta limit ==
	[[File:Troyon limit.png\|200px\|thumb\|Troyon Limit<ref>ITER ~~PHYSICS BASIS~~, Chapter 3</ref>]]		[[File:Troyon limit.png\|200px\|thumb\|Troyon Limit<ref>ITER Physics Expert Group on Disruptions, Plasma Control, and MHD, ''ITER Physics Basis Chapter 3: MHD stability, operational limits and disruptions'', [[doi:10.1088/0029-5515/39/12/303\|Nucl. Fusion '''39 ''' (1999) 2251-2389]]</ref>]]
	<math>\beta</math> is often expressed in terms of the normalized beta (or Troyon factor)<ref>~~[[doi:10.1088/0741-3335/26/1A/319\|~~F. Troyon, R. Gruber, H. Saurenmann, S. Semenzato and S. Succi, ''MHD-Limits to Plasma Confinement'', Plasma Phys. Control. Fusion '''26''' (1984) 209]]</ref>, an operational parameter indicating how close the plasma is to reaching the [[Greenwald limit]] or a destabilising major MHD activity. Its definition is (for tokamaks):		<math>\beta</math> is often expressed in terms of the normalized beta (or Troyon factor)<ref>F. Troyon, R. Gruber, H. Saurenmann, S. Semenzato and S. Succi, ''MHD-Limits to Plasma Confinement'', [[doi:10.1088/0741-3335/26/1A/319\|Plasma Phys. Control. Fusion '''26''' (1984) 209]]</ref>, an operational parameter indicating how close the plasma is to reaching the [[Greenwald limit]] or a destabilising major MHD activity. Its definition is (for tokamaks):
	<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) ISBN 3540242171</ref>		<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) ISBN 3540242171</ref>

Line 20:		Line 20:
	The value of <math>\beta_N</math> has been determined numerically by Troyon to 0.028. Often <math>\beta</math> is expressed in percent, in which case <math>\beta_N = 2.8</math>. This limit results from many different numerical studies determined to find the overall <math>\beta</math> limit out of many different MHD instabilities, such as [[external kink modes]], [[ballooning kink modes]], [[internal modes]], [[localized modes]], etc. <ref>J. Freidberg, "Plasma Physics and Fusion Energy",Cambridge University Press (2007) ISBN 139780511273759</ref> <br>		The value of <math>\beta_N</math> has been determined numerically by Troyon to 0.028. Often <math>\beta</math> is expressed in percent, in which case <math>\beta_N = 2.8</math>. This limit results from many different numerical studies determined to find the overall <math>\beta</math> limit out of many different MHD instabilities, such as [[external kink modes]], [[ballooning kink modes]], [[internal modes]], [[localized modes]], etc. <ref>J. Freidberg, "Plasma Physics and Fusion Energy",Cambridge University Press (2007) ISBN 139780511273759</ref> <br>
	Empirical evaluation from the data of different tokamaks raises this value slightly to <math>\beta_N = 3.5</math>, although significantly higher values have been achieved.		Empirical evaluation from the data of different tokamaks raises this value slightly to <math>\beta_N = 3.5</math>, although significantly higher values have been achieved.
	<ref>~~[http://dx.doi.org/10.1088/0029-5515/46/5/014~~ S.A. Sabbagh et al, ''Resistive wall stabilized operation in rotating high beta NSTX plasmas'', Nucl. Fusion '''46''' (2006) 635-644]</ref>		<ref>S.A. Sabbagh et al, ''Resistive wall stabilized operation in rotating high beta NSTX plasmas'', [[doi:10.1088/0029-5515/46/5/014\|Nucl. Fusion '''46''' (2006) 635-644]]</ref>

	== See also ==		== See also ==