Rapidly fluctuating alpha effect ($\overline{\tilde{\alpha }\tilde{B}}=-C{\partial }_x\overline{B}$)

In a Parker dynamo wave model, an alpha effect of the form $\alpha ={ϵ}^{-1}{\alpha }_0\sin (\omega t/ϵ)$ has zero time average but can generate a finite averaged coupling. The method of multiple scales gives $\tilde{A}=-({\alpha }_0/\omega )\cos (\omega \tau )\overline{B}$ and $\tilde{B}=-(\Omega {\alpha }_0/{\omega }^2)\sin (\omega \tau ){\partial }_x\overline{B}$, where $\tau =t/ϵ$. 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 $ϵ$.