Flux-surface preservation during magnetic-tube expansion

Suppose the Cartesian magnetic flux function is advected, the initial axial velocity is zero, $B_y=f(\psi )$ initially, and $\nabla \cdot \mathbf{u}=g(\psi ,t)$. Then $B_y=f(\psi )\exp [-{\int }_0^{t}g(\psi ,s)ds]$ remains a function of the flux label and time. The axial Lorentz force vanishes because its transverse derivative along a magnetic field line vanishes. Smooth ideal evolution therefore preserves zero axial motion.