# Beta

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

${\displaystyle \beta ={\frac {\left\langle p\right\rangle }{B^{2}/2\mu _{0}}}}$

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

${\displaystyle {\frac {1}{\beta }}={\frac {1}{\beta _{p}}}+{\frac {1}{\beta _{t}}}}$

## Normalized beta

Troyon Limit[2]

${\displaystyle \beta }$ is often expressed in terms of the normalized beta (or Troyon factor)[3], an operational parameter indicating how close the plasma is to reaching destabilising major MHD activity. Its definition is (for tokamaks): [4]

${\displaystyle \beta _{N}=\beta {\frac {aB_{T}}{I_{p}}}}$

where ${\displaystyle B_{T}}$ is the toroidal magnetic field in T, ${\displaystyle a}$ is the minor radius in m, and ${\displaystyle I_{p}}$ is the plasma current in MA.

## Beta limit

The upper limit of ${\displaystyle \beta _{N}}$ has been determined numerically by Troyon to 0.028. Often ${\displaystyle \beta }$ is expressed in percent, in which case ${\displaystyle \beta _{N}=2.8}$. This limit results from many different numerical studies determined to find the overall ${\displaystyle \beta }$ 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 ${\displaystyle \beta _{N}=3.5}$, although significantly higher values have been achieved. [5]

## References

1. J.P. Freidberg, Plasma physics and fusion energy, Cambridge University Press (2007) ISBN 0521851076
2. 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
3. F. Troyon, R. Gruber, H. Saurenmann, S. Semenzato and S. Succi, MHD-Limits to Plasma Confinement, Plasma Phys. Control. Fusion 26 (1984) 209
4. K. Miyamoto, Plasma Physics and Controlled Nuclear Fusion, Springer-Verlag (2005) ISBN 3540242171
5. S.A. Sabbagh et al, Resistive wall stabilized operation in rotating high beta NSTX plasmas, Nucl. Fusion 46 (2006) 635-644