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Plasma performance is often expressed in terms of beta ( | 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> | ||
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, < | 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'' | 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> | ||
== Normalized beta == | == Normalized beta == | ||
[[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): | |||
<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> | |||
< | |||
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. | |||
== Beta limit == | |||
<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. | |||
<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 == | ||
* [[Internal inductance]] | * [[Internal inductance]] | ||
* [[:Wikipedia:Beta_(plasma_physics)]] | |||
== References == | == References == | ||
<references /> | <references /> | ||