Axisymmetric magnetic winding (${\partial }_t{B}_{\varphi }=R{\mathbf{B}}_p\cdot \nabla \Omega$)

For prescribed axisymmetric differential rotation $\mathbf{u}=R\Omega {\mathbf{e}}_{\varphi }$, the ideal magnetohydrodynamic induction equation and Gauss's law for magnetism giveThe 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.

For Keplerian rotation, initial $B_R\propto R^{-1}$ and thermal pressure $p\propto R^{-2}$, axisymmetric magnetic winding gives $B_{\varphi }=-(3/2)\Omega t{B}_R$. If the initial plasma beta is ${\beta }_p>1$, the formal equality of magnetic pressure and gas pressure occurs atThe magnetic-to-thermal pressure ratio decreases with radius, so magnetic pressure dominates inside this radius. In a disk with a finite radial range, an equality radius exists in the disk only when this formal radius lies within that range.

A stationary axisymmetric magnetic field in ideal purely rotational flow requires constant angular velocity along the magnetic field lines of its poloidal magnetic field. Otherwise axisymmetric magnetic winding generates a time-dependent toroidal component.