Steady toroidal induction by spherical differential rotation ($B_{\varphi }=-\frac{B_0\sin \theta \cos \theta }{\eta r^3}{\int }_0^r{x}^4\Omega (x)dx$)

An axial uniform magnetic field and localized radial differential rotation generate a steady toroidal magnetic field. The angular factor $\sin \theta \cos \theta$ is an eigenfunction of the toroidal angular diffusion operator with eigenvalue $-6$. Its radial amplitude satisfies $\eta (f^{\prime \prime }+2f^{\prime }/r-6f/r^2)=-B_{0}r{\Omega }^{\prime }$. Differentiating the integral in the title proves the formula. It is regular at the origin and decays at infinity when $\Omega$ is finite at zero and decays sufficiently rapidly.

With a fixed axial seed field and prescribed localized differential rotation, the winding source is time independent. Early induction gives $B_{\varphi }\sim B_0\Omega t$, while the steady balance has $B_{\varphi }\sim B_0\Omega {\ell }^2/\eta$. Their comparison gives the magnetic diffusion time, not merely the rotation period. It diverges as conductivity becomes infinite. In an unbounded domain, this is a local equilibration estimate on scale $\ell$, not finite-time exact convergence everywhere. Lorentz force feedback may invalidate the prescribed-flow approximation sooner.