Efficiency of tokamak magnetic field in confining plasma: Plasma beta

To characterize the efficiency of the magnetic field of tokamaks in confining plasmas, introduce the plasma $\beta$, which is defined as the ratio of the thermal pressure to the magnetic pressure, i.e.,

$$\beta = \frac{p}{B^2 / 2 \mu_0} .$$ (87)

Since the pressure in tokamak plasmas is inhomogeneous, the volume averaged pressure is usually used to define the beta. In tokamak plasmas, the toroidal beta $\beta_t$ and the poloidal beta $\beta_p$ are defined, respectively, by

$$\beta_t = \frac{\langle p \rangle}{B^2_{t 0} / 2 \mu_0},$$ (88)

$$\beta_p = \frac{\langle p \rangle}{B^2_{p a} / 2 \mu_0},$$ (89)

where $\langle \ldots \rangle$ is the volume averaging, $B_{t 0}$ is the vacuum toroidal magnetic field at the magnetic axis (or geometrical center of the plasma), $B_{p a}$ is the averaged poloidal magnetic field on the plasma surface. In tokamaks, the toroidal magnetic field is dominant and thus the the toroidal beta $\beta_t$ (not $\beta_p$) is the usual way to characterize the the efficiency of the magnetic field in confining plasmas. Why do we need $\beta_p$? The short answer is that $\beta_p$ characterizes the efficiency of the plasma current in confining the plasma. This can be seen by using Ampere's law to approximately write the poloidal magnetic field $B_{p a}$ as $B_{p a} \approx \mu_0 I_p / (2 \pi a)$. Then $\beta_p$ is written

$$\beta_p \approx \frac{\langle p \rangle}{I_p^2 / (8 \pi^2 a^2)},$$ (90)

which is the ratio of the pressure to the plasma current, and thus characterizes the efficiency of the plasma current in confining the plasma. Furthermore $\beta_p$ is closely related to the normalized beta $\beta_N$ introduced in the next subsection (refer to Eq. (92)). In addition, $\beta_p$ is proportional to an important current, the so-called bootstrap current, in tokamak plasmas. I will discuss this later.