Toroidal force balance

Next, consider the $\phi$ component of Eq. (43), which is written

$$J_Z B_R - J_R B_Z = \frac{1}{R} \frac{\partial P}{\partial \phi} .$$ (45)

Since $P = P (\Psi)$, which implies $\partial P / \partial \phi = 0$, equation (45) reduces to

$$J_Z B_R - J_R B_Z = 0$$ (46)

Using the expressions of the poloidal current density (37) and (38) in the force balance equation (46) yields

$$\frac{\partial g}{\partial R} B_R + \frac{\partial g}{\partial Z} B_Z = 0,$$ (47)

which can be further written

$$\mathbf{B} \cdot \nabla g = 0.$$ (48)

According to the same reasoning for the pressure, we conclude that Eq. (48) is equivalent to $g = g (\Psi)$. (The function $g$ defined here is usually called the ``poloidal current function'' in tokamak literature. The reason for this name is discussed in Sec. 2.3.)