 Axisymmetric magnetostatic Grad-Shafranov system

For an axisymmetric vector potential $\mathbf{A}=\nabla \alpha \times \nabla \varphi +\beta \nabla \varphi$, defineMagnetostatic azimuthal force balance gives $L\alpha =F(\beta )$. In a barotropic equilibrium the remaining balance integrates tofor flux functions $F$ and $G$.