= Toroidal magnetic-field winding equation
{title2=$D(B_\phi/(\rho R))/Dt=(\mathbf B_p/\rho)\cdot\nabla\Omega$}
In axisymmetric <ideal magnetohydrodynamics>, $\Omega=u_\phi/R$ is the material angular velocity. Cylindrical basis-vector derivatives and <mass conservation> give the displayed winding equation. Thus $B_\phi/(\rho R)$ 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.
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