Bounded-modulation alpha-Omega growth estimate ($g_{\max }=\Theta (|{\alpha }_0\Omega {|}^{2/3}{\eta }^{-1/3})$)

In the two-component alpha-Omega dynamo with $|f(t)|\le 1$, apply the weighted energy estimate for two coupled modes with $a=|{\alpha }_0|$, $c=|k\Omega |$, and $d=\eta k^2$. The field-amplitude growth rate is at most $\sqrt{|{\alpha }_0\Omega k|}-\eta k^2$. Optimizing over $k$ gives the upper constant $3/2^{8/3}$ multiplying $({\alpha }_0^2{\Omega }^2/\eta {)}^{1/3}$. The admissible choice $f=1$ gives a lower constant $3/2^{10/3}$, so the maximum possible growth has the displayed scaling. An arbitrary chosen modulation need not grow. The statement concerns the model with freely adjustable wavenumber; physical scale-separation restrictions can constrain that optimization.