= Alpha-squared Omega dynamo threshold
{title2=$\Omega_0^2=4\eta^2\ell^4$}
For constant coefficients and a transverse wavenumber $\ell\ne0$, a planar <mean-field dynamo> retaining both alpha feedback and shear has dispersion $(\sigma+\eta p)^2=\alpha^2p+i\alpha\Omega k$, $p=k^2+\ell^2$. Marginal stability gives $\alpha_m^2=4\eta^4p^4/(4\eta^2p^3+\Omega^2k^2)$. Differentiate with respect to $k^2$: an interior minimum solves $4\eta^2p^3+\Omega^2(3k^2-\ell^2)=0$. Its derivative with respect to $k^2$ is positive, so a unique positive minimizing $k^2$ exists only above the displayed shear threshold. Below it the minimum is at $k=0$. The sign of the marginal oscillation frequency is fixed by $2\eta p\omega=\alpha\Omega k$.
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