= Mean-field shearing-wave transient amplification
{title2=$t_{\rm peak}\simeq|\alpha|T/(\eta k_0)$}
For $k(t)=k_0\sqrt{1+(t/T)^2}$, factor resistive damping out of the shearing-mode equations by $R=e^{-D}G$, $S=e^{-D}H$, $D=\eta\int k^2dt$. They imply $G''=\alpha^2k^2G$, so a growing <WKB approximation> has $G\propto k^{-1/2}e^{|\alpha|\int kdt}$ and $H\simeq i\operatorname{sgn}(\alpha)kG$. Its <magnetic energy> is proportional to $k\exp[2|\alpha|\int kdt-2\eta\int k^2dt]$. At long times the positive quadratic exponent is overtaken by negative cubic resistive damping. A well-separated growing interval exists for sufficiently small positive <magnetic diffusivity>, with $\eta k_0/|\alpha|\ll1$ at fixed nonzero $|\alpha|k_0T$, and the displayed leading peak time. This is transient amplification, not an asymptotically self-sustaining dynamo.
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