Cartesian magnetic flux function ($\psi$)

For a $y$-independent solenoidal magnetic field, the transverse components can be written $B_x=-{\psi }_z$ and $B_z={\psi }_x$. Contours of $\psi$ are transverse magnetic field lines. The independent axial component $B_y$ can twist the flux tube. Under the MHD induction equation, an additive time-dependent gauge can be chosen so that $D\psi /Dt=0$.

In a $y$-independent magnetostatic equilibrium with no axial pressure or gravitational force, $\nabla \psi \times \nabla B_y=0$. Thus $B_y=b(\psi )$ on a regular connected flux region. The remaining balance is $[{\nabla }^2\psi +b{b}^{\prime }]\nabla \psi /{\mu }_0+\nabla p+\rho \nabla \Phi =0$. Constancy on separate components of a level set does not automatically give one global single-valued $b$.