Relation of $\Psi$ with the poloidal magnetic flux

Note that $\Psi$ is defined by $\Psi = R A_{\phi}$, which is actually a component of the vector potential $\mathbf{A}$, thereby not having an obvious physical meaning. Next, we try to find the physical meaning of $\Psi$, i.e., try to find some simple algebraic relation of $\Psi$ with some quantity that can be measured in experiments.

Figure 2: The poloidal magnetic flux $\Psi _p$ between the two magnetic surfaces $\Psi _1$ and $\Psi _2$ is given by $\Psi _p = 2 \pi (\Psi _2 - \Psi _1)$.

In Fig. 2, there are two magnetic surfaces labeled, respectively, by $\Psi = \Psi_1$ and $\Psi = \Psi_2$. Using Gauss's theorem in the toroidal volume between the two magnetic surface, we know that the poloidal magnetic flux through any toroidal ribbons between the two magnetic surfaces is equal to each other. Next, we calculate this poloidal magnetic flux. To make the calculation easy, we select a plane perpendicular to the $Z$ axis, as is shown by the dash line in Fig. 1. In this case, only $B_Z$ contribute to the poloidal magnetic flux, which is written (the positive direction of the plane is chosen in the direction of $\hat{\mathbf{Z}}$)

$$\Psi_p = \int_{R_1}^{R_2} B_z (R, Z) 2 \pi R d R$$

$$= \int_{R_1}^{R_2} \frac{1}{R} \frac{\partial \Psi}{\partial R} 2 \pi R d R$$

$$= 2 \pi \int^{R_2}_{R_1} \frac{\partial \Psi}{\partial R} d R$$

$$= 2 \pi [\Psi_2 - \Psi_1] .$$ (18)

Equation (18) provides a simple physical meaning for $\Psi$, i.e., the difference of $2 \pi \Psi$ between two magnetic surfaces is equal to the poloidal magnetic flux enclosed by the two magnetic surfaces. Noting that we are considering the axisymmetric case, the physical meaning of $\Psi$ can also be stated as: the difference of $\Psi$ between two magnetic surfaces is equal to the poloidal magnetic flux per radian. Due to this relation of $\Psi$ with the poloidal flux $\Psi _p$, $\Psi$ is usually called the ``poloidal magnetic flux function'' in tokamak literature. Note that it is the difference of $\Psi$ between two locations that determines the the physical quantities $\Psi _p$, which is consistent with the fact that gauge transformation (12) dose not change the values of physical quantities.