# 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

Troyon Limit[2]

$\beta$ 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]

## 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