Plane-layer alpha-Omega dynamo threshold ($|D{|}_c=32/(3\sqrt{3}),|k_c|=\pi /(\sqrt{3}d)$)

For a mean-field dynamo with uniform alpha, shear $\pi V/d$, diffusivity $\eta >0$ and fixed vertical wavenumber $\pi /d$, a horizontal Fourier mode obeys $(s+\eta (k^2+{\pi }^2/d^2){)}^2=i\alpha \pi Vk/d$. With $D=\alpha V{d}^2/({\pi }^2{\eta }^2)$ and $h=|k|d/\pi$, the growing branch has positive real part exactly when $|D|>2(1+h^2{)}^2/h$. Differentiation gives $3h^4+2h^2-1=0$, hence $h=1/\sqrt{3}$. Equality is a neutral oscillation, not exponential growth. The sign of $D$ reverses wave propagation without changing the growth threshold.