 Toroidal magnetic-field winding equation

In axisymmetric ideal magnetohydrodynamics, $\Omega =u_{\varphi }/R$ is the material angular velocity. Cylindrical basis-vector derivatives and mass conservation give the displayed winding equation. Thus $B_{\varphi }/(\rho R)$ is an advected scalar only when ${\mathbf{B}}_p\cdot \nabla \Omega =0$. Differential rotation along a field line otherwise produces toroidal field. This does not violate magnetic flux freezing, which concerns flux through a material surface rather than a cylindrical vector component.