= Local alpha-Omega dynamo wave dispersion
{title2=$(p+\eta_T k^2)^2=i\alpha Sk$}
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 $A\widehat{\mathbf y}$ 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 $\operatorname{Re}p=-\eta_Tk^2+\sqrt{|\alpha Sk|/2}$ and $\operatorname{Im}p=\operatorname{sgn}(\alpha Sk)\sqrt{|\alpha Sk|/2}$. Its phase velocity is $-\operatorname{Im}p/k$: 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.
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