= Rapidly fluctuating alpha effect
{title2=$\overline{\widetilde\alpha\widetilde B}=-C\partial_x\overline B$}
In a <Parker dynamo wave> model, an <alpha effect> of the form $\alpha=\epsilon^{-1}\alpha_0\sin(\omega t/\epsilon)$ has zero time average but can generate a finite averaged coupling. The <method of multiple scales> gives $\widetilde A=-(\alpha_0/\omega)\cos(\omega\tau)\overline B$ and $\widetilde B=-(\Omega\alpha_0/\omega^2)\sin(\omega\tau)\partial_x\overline B$, where $\tau=t/\epsilon$. Consequently $C=\Omega\alpha_0^2/(2\omega^2)$. With diffusion operator $\partial_x^2-\ell^2$, the averaged <Fourier mode> growth exponents are $-k^2-\ell^2\pm|k|\sqrt{\Omega C}$. For freely adjustable real $k$, growth requires $\Omega C>4\ell^2$. The result is an asymptotic averaged model for fixed nonzero $\omega$, not an exact replacement at finite $\epsilon$.
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