Axisymmetric poloidal fields

If a poloidal field, $(\mbox{\boldmath$\nabla$}\times)^2(\mbox{$\mathcal P$}[\mathbf{u}]\mathbf{r})$; or, equivalently, its defining scalar, $\mbox{$\mathcal P$}[\mathbf{u}]$; is independent of the azimuth, $\phi$, the field-lines are confined to planes passing through the line $\sin\theta=0$ (the $z$-axis) and are identical in each such plane. Then,

$$(\mbox{\boldmath$\nabla$}\times)^2(\mbox{$\mathcal P$}[\math... ...ac{\partial \psi}{\partial r}\;\mbox{\boldmath$\hat{\theta}$},$$ (B.29)

where

$$\psi\equiv r\sin\theta\frac{\partial \mbox{$\mathcal P$}[\mathbf{u}]}{\partial \theta}$$ (B.30)

is Stokes's stream-function which is constant along the field-lines (Lamb 1932, pp. 125-6). This connection between axisymmetric poloidal fields and Stokes's stream-function was pointed out by Moffatt (1978, p. 21), though he misprinted the relation.