Relation of $g$ with the poloidal electric current

As is discussed in Sec. 2.1, to satisfy the force balance in the toroidal direction, $g \equiv R B_{\phi}$ must be a magnetic surface function, i.e., $g = g (\Psi)$. Using this, equations (37) and (38) are written

$$\mu_0 J_R = - \frac{1}{R} \frac{\partial g}{\partial \Psi} \frac{\partial \Psi}{\partial Z} = \frac{\partial g}{\partial \Psi} B_R,$$ (57)

and

$$\mu_0 J_Z = \frac{1}{R} \frac{\partial g}{\partial \Psi} \frac{\partial \Psi}{\partial R} = \frac{\partial g}{\partial \Psi} B_z,$$ (58)

respectively. The above two equations imply that

$$\frac{J_R}{J_Z} = \frac{B_R}{B_Z},$$ (59)

which implies that the projections of $\mathbf {B}$ lines and $\mathbf{J}$ lines on the poloidal plane are identical to each other. This indicates that the $\mathbf{J}$ surfaces coincide with the magnetic surfaces. Using this and $\nabla \cdot \mathbf{J}= 0$, and following the same steps in Sec. 1.4, we obtain

$$I_{\ensuremath{\operatorname{pol}}} = \frac{1}{\mu_0} 2 \pi [g (\Psi_2) - g (\Psi_1)],$$ (60)

where $I_{\ensuremath{\operatorname{pol}}}$ is the poloidal current enclosed by the two magnetic surfaces, the positive direction of $I_{\ensuremath{\operatorname{pol}}}$ is chosen to be in the clockwise direction when observers look along $\hat{\ensuremath{\boldsymbol{\phi}}}$. Equation (60) indicates that the difference of $g$ between two magnetic surface is proportional to the poloidal current. For this reason, $g$ is usually call the ``poloidal current function''.

In the above, we see that the relation of $g$ with the poloidal electric current is similar to that of $\Psi$ with the poloidal magnetic flux. This similarity is due to the following differential relations:

$$\left\{ \begin{array}{l} \mathbf{B}= \nabla \times \mathbf{A}\\ \Psi = R A_{\phi} \end{array} \right.$$

$$\left\{ \begin{array}{l} \mu_0 \mathbf{J}= \nabla \times \mathbf{B}\\ g = R B_{\phi} \end{array} \right.$$