= Bounded-modulation alpha-Omega growth estimate
{title2=$g_{\max}=\Theta(|\alpha_0\Omega|^{2/3}\eta^{-1/3})$}
In the two-component <alpha-Omega dynamo> with $|f(t)|\leq1$, 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.
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