Force-free magnetic field

A force-free magnetic field obeys $(\nabla \times \mathbf{B})\times \mathbf{B}=0$, so its electric current is locally parallel to the magnetic field.

Writing $\nabla \times \mathbf{B}=(4\pi /c)\alpha \mathbf{B}$ defines the force-free parameter $\alpha$. Since the divergence of a curl vanishes and $\nabla \cdot \mathbf{B}=0$, it satisfies $\mathbf{B}\cdot \nabla \alpha =0$.

A cylindrical force-free magnetic field depends only on cylindrical radius and has current everywhere parallel to its magnetic field. Regularity on the axis often forces its radial component to vanish.

For an axisymmetric force-free field with ${\mathbf{B}}_p=\nabla \psi \times \nabla \varphi$ and $r{B}_{\varphi }=F(\psi )$, the flux function obeys