Parker dynamo wave

A Parker dynamo wave is a traveling or oscillatory mode of a mean-field dynamo coupling poloidal and toroidal magnetic-field components through an alpha effect and shear. A reduced complex-amplitude model can represent fluctuating alpha by a time-dependent coupling. Its periodically switched growth must be calculated from the monodromy matrix of a periodic linear system, rather than by averaging noncommuting generators.

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.

For $\dot{A}=D^{2}f(t)B$ and $\dot{B}=iA$, with $f$ alternating between $+1$ and $-1$ for intervals of length $T$, the monodromy matrix of a periodic linear system has determinant one and trace $\cosh (\sqrt{2}DT)+\cos (\sqrt{2}DT)>2$. Its reciprocal Floquet multipliers give positive dominant Floquet growth rate for $D,T>0$, despite the zero mean of $f$. An exceptional initial state in the stable eigenspace decays.