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Beta

2015-04-17T20:13:39Z

Mmessmer: added more details to normalized 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>

:<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.
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'' 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>

== 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>

:<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.
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>

== See also ==

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

== References ==
<references />

File:Troyon limit.png

2015-04-17T19:43:34Z

Mmessmer: Empirically determined troyon limit

Empirically determined troyon limit

Beta

2015-04-13T19:13:39Z

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

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>

:<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.
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'' 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>

== Normalized beta ==

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>

:<math>\beta_N = \beta_t \frac{a B_0}{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.
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>

== See also ==

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

== References ==
<references />