 Solution

For a barotropic fluid, $\nabla p=\rho \nabla h$. Magnetostatic force balance and $L\alpha =F(\beta )$ giveThe left side is a gradient multiplied by $\rho$, so taking the curl shows that $(L\beta -F{F}^{\prime })/(r^2\rho )$ is constant on each $\beta$ surface. Absorbing the fixed factor ${\mu }_0$ into an arbitrary function $G$ gives the Axisymmetric magnetostatic Grad-Shafranov system