= Axisymmetric magnetic winding
{title2=$\partial_tB_\phi=R\mathbf B_p\cdot\nabla\Omega$}
For prescribed axisymmetric <differential rotation> $\mathbf u=R\Omega\mathbf e_\phi$, the <ideal magnetohydrodynamic induction equation> and <Gauss's law for magnetism> give
$$
\partial_t\mathbf B_p=0,\qquad\partial_tB_\phi=R\mathbf B_p\cdot\nabla\Omega.
$$
The poloidal field is unchanged while rotational shear generates a <toroidal magnetic field>. For time-independent $\Omega$, the toroidal component grows linearly until neglected dynamical feedback becomes relevant.
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