Beta

Plasma performance is often expressed in terms of beta ($ \beta $), defined as:

$ \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

$ \beta $ is often expressed in terms of the normalized beta (or Troyon factor), 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):

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

See also

• Internal inductance
• Wikipedia:Beta_(plasma_physics)