Beta limit

Beta limit means there is a limit for the value of beta beyond which the plasma will encounter a serious disruption. Early calculation of the beta limit on JET shows that the maximal $\beta_t$ obtained is proportional to $I_p / (a B_{t 0})$ for $I_p [\ensuremath{\operatorname{MA}}] / a B_{t 0} \leqslant 3$, where $I_p [\ensuremath{\operatorname{MA}}]$ is the plasma current in mega Ampere, $a$ is the plasma minor radius in meter, $B_{t 0}$ is the toroidal magnetic field in Tesla. This scaling relation $\beta_{t \max} \propto I_p / (a B_{t 0})$ is often called Troyon scaling. This scaling relation motivates the definition a normalized beta, $\beta_N$, which is defined by

$$\beta_N = 10^8 \frac{a B_{t 0}}{I_p} \beta_t,$$ (91)

where all quantities are in SI units. Equation (91) can also be written as

$$\beta_N = 10^8 \frac{\langle p \rangle}{I_p / a},$$ (92)

which indicates that $\beta_N$ can be considered as the efficiency of the plasma current in confining the plasma. The normalized beta $\beta_N$ is an operational parameter indicating how close the plasma is to reach destabilizing major MHD activities. Its typical value is of order unit. The maximum value of $\beta_N$ before the onset of deleterious instability is typically 3.5, although significantly higher values, e.g., $\beta_N = 7.2$, have been achieved in the low aspect ratio tokamak NSTX[11]. The ability to increase the value of $\beta_N$ can be considered to be the ability of controlling the major MHD instabilities, and thus can be used to characterize how well a tokamak device is operated. One goal of EAST tokamak during 2015-2016 is to sustain a plasma with $\beta_N \geqslant 2$ for at least 10 seconds.

(check** The tearing mode, specifically the neoclassical tearing mode (NTM) is expected to set the beta limit in a reactor.)

(**check: Tokamak experiments have found that it is easier to achieve high $\beta_N$ in large $I_p$ plasmas than in small $I_p$ plasmas. However, experiments found it is easier to achieve high $\beta_p$ in small $I_p$ plasmas than in large $I_p$ plasmas. Examining the expression of $\beta_N$ and $\beta_p$ given by Eqs. (90) and Eq. (92), respectively, we recognize that pressure limit should have a scaling of $\langle p \rangle \propto I_p^{\alpha}$ with $1 < \alpha < 2$. )

Tokamak experiments have also found that it is easier to achieve higher $\beta_t$ in low $B_t$ plasmas than in higher $B_t$ plasmas, which indicates that the efficiency of the magnetic field in confining plasma is a decreasing function of the magnitude of the magnetic field.