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Beta: Difference between revisions
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added more details to normalized beta
m (converted all html to math tags and added some minor information) |
(added more details to normalized beta) |
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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>]] | |||
<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 = \ | :<math>\beta_N = \beta \frac{a B_T}{I_p}</math> | ||
where <math> | 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> | |||
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> | ||