Axisymmetric magnetostatic Grad-Shafranov system
= Axisymmetric magnetostatic Grad-Shafranov system
{c}
For an axisymmetric vector potential $\mathbf A=\nabla\alpha\times\nabla\phi+\beta\nabla\phi$, define
$$
L=-r\partial_r(r^{-1}\partial_r)-\partial_z^2.
$$
Magnetostatic azimuthal force balance gives $L\alpha=F(\beta)$. In a barotropic equilibrium the remaining balance integrates to
$$
L\beta=F(\beta)F'(\beta)+r^2\rho G(\beta)
$$
for flux functions $F$ and $G$.