# Beta

Plasma performance is often expressed in terms of beta ($ \beta $), defined as:
^{[1]}

- $ \beta = \frac{\left \langle p \right \rangle}{B^2/2\mu_0} $

i.e., the ratio of the plasma pressure to the magnetic pressure.
Here, $ \left \langle p \right \rangle $ is the mean plasma pressure, and $ B $ the mean total field strength.
It is customary to introduce also the *poloidal* beta $ \beta_p $ and the *toroidal* beta $ \beta_t $, in which $ B $ is replaced by the poloidal and toroidal magnetic field component, respectively. One has:

- $ \frac{1}{\beta} = \frac{1}{\beta_p} + \frac{1}{\beta_t} $

## Normalized beta, beta limit

$ \beta $ is often expressed in terms of the normalized beta (or Troyon factor)^{[3]}, 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):
^{[4]}

- $ \beta_N = \beta \frac{a B_T}{I_p} $

where $ B_T $ is the toroidal magnetic field in T, $ a $ is the minor radius in m, and $ I_p $ is the plasma current in MA.
The value of $ \beta_N $ has been determined numerically by Troyon to 0.028. Often $ \beta $ is expressed in percent, in which case $ \beta_N = 2.8 $. This limit results from many different numerical studies determined to find the overall $ \beta $ limit out of many different MHD instabilities, such as external kink modes, ballooning kink modes, internal modes, localized modes, etc. ^{[1]}

Empirical evaluation from the data of different tokamaks raises this value slightly to $ \beta_N = 3.5 $, although significantly higher values have been achieved.
^{[5]}

## See also

## References

- ↑
^{1.0}^{1.1}J.P. Freidberg,*Plasma physics and fusion energy*, Cambridge University Press (2007) ISBN 0521851076 - ↑ ITER Physics Expert Group on Disruptions, Plasma Control, and MHD,
*ITER Physics Basis Chapter 3: MHD stability, operational limits and disruptions*, Nucl. Fusion**39**(1999) 2251-2389 - ↑ F. Troyon, R. Gruber, H. Saurenmann, S. Semenzato and S. Succi,
*MHD-Limits to Plasma Confinement*, Plasma Phys. Control. Fusion**26**(1984) 209 - ↑ K. Miyamoto,
*Plasma Physics and Controlled Nuclear Fusion*, Springer-Verlag (2005) ISBN 3540242171 - ↑ S.A. Sabbagh et al,
*Resistive wall stabilized operation in rotating high beta NSTX plasmas*, Nucl. Fusion**46**(2006) 635-644