Local alpha-Omega dynamo wave dispersion ($(p+{\eta }_T{k}^2{)}^2=i\alpha Sk$)

Take a right-handed local Cartesian frame with $x$ towards the equator, $y$ azimuthal and $z$ radially outwards in the northern hemisphere. A local radial shear $U_y=Sz$ and a poloidal magnetic field represented by $A\hat{\mathbf{y}}$ give the alpha-Omega dynamo equations $A_t=\alpha B+{\eta }_T{A}_{xx}$ and $B_t=S{A}_x+{\eta }_T{B}_{xx}$. Substitution of a Fourier mode proportional to $e^{pt+ikx}$ gives the displayed dispersion relation. The growing branch has $\mathrm{Re}p=-{\eta }_T{k}^2+\sqrt{|\alpha Sk|/2}$ and $\mathrm{Im}p=\mathrm{sgn}(\alpha Sk)\sqrt{|\alpha Sk|/2}$. Its phase velocity is $-\mathrm{Im}p/k$: it propagates equatorwards when $\alpha S<0$, polewards when $\alpha S>0$. The result assumes local constant coefficients and no meridional circulation in a star; advection and spatially separated field regeneration can alter migration.