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The Grad-Shafranov equation governs the poloidal magnetic flux of an axisymmetric magnetohydrostatic equilibrium. Pressure and toroidal-field contributions depend on the flux function. The Grad-Shafranov equation for a force-free magnetic field is its zero-pressure-gradient reduction.

Put $I=R{B}_{\varphi }$ and define ${\Delta }_{\ast }\psi ={\psi }_{RR}-R^{-1}{\psi }_R+{\psi }_{zz}$. Direct cylindrical differentiation givesThe azimuthal component of the Lorentz force is proportional to $I_z{\psi }_R-I_R{\psi }_z$. A force-free magnetic field therefore has parallel meridional gradients of $I$ and $\psi$, implying $I=f(\psi )$ on connected regular flux surfaces. ConsequentlyWith this dependence the poloidal part of $\nabla \times B$ is $f^{\prime }(\psi )B_p$. Its full cross product with $B$ isVanishing force therefore requires ${\Delta }_{\ast }\psi +f{f}^{\prime }=0$ wherever the flux gradient is nonzero, extending by regularity through ordinary isolated nulls. Finally, evaluating the three-dimensional cylindrical divergence gives $R^2\nabla \cdot (R^{-2}\nabla \psi )={\Delta }_{\ast }\psi$, soThis derives the Grad-Shafranov equation for a force-free magnetic field. The function $f$ is set by boundary/current information; it is not an additional fixed constant. Its single-valued global form presumes the usual connected flux-surface labeling. The equation also makes $\nabla \times B=f^{\prime }(\psi )B$, explicitly showing that current is parallel to the magnetic field.

The toroidal component of $(\nabla \times \mathbf{B})\times \mathbf{B}=0$ givesBoth $\psi$ and $r{B}_{\varphi }$ are therefore constant along each poloidal field line, soThe poloidal components of the same force-free equation then reduce to the Grad-Shafranov equation for a force-free magnetic fieldThe poloidal current density isAmpère's law around an azimuthal circle gives the enclosed poloidal electric current:up to the orientation sign. Thus $F$ is the enclosed-current function in units ${\mu }_0/(2\pi )$.