= Plane-layer alpha-Omega dynamo threshold
{title2=$|D|_c=32/(3\sqrt3),\quad |k_c|=\pi/(\sqrt3d)$}
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 $D=\alpha Vd^2/(\pi^2\eta^2)$ 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 $h=1/\sqrt3$. Equality is a neutral oscillation, not exponential growth. The sign of $D$ reverses wave propagation without changing the growth threshold.
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