= Solution
Use the <method of multiple scales>, treating $t$ and $\tau=t/\epsilon$ as independent variables, so differentiation becomes $\partial_t+\epsilon^{-1}\partial_\tau$. Write $L=\partial_x^2-\ell^2$ and average over a fast period $2\pi/|\omega|$, with $\omega\ne0$. The fluctuations can depend slowly on $x,t$ as well as periodically on $\tau$. The $\epsilon^{-1}$ balance in the <poloidal magnetic field> potential equation is
$$
\partial_\tau\widetilde A=\alpha_0\sin(\omega\tau)\overline B.
$$
Integrating and imposing zero fast average determines
$$
\widetilde A=-\frac{\alpha_0}{\omega}\cos(\omega\tau)\overline B.
$$
At order one, the <toroidal magnetic field> equation is $\partial_t\overline B+\partial_\tau\widetilde B=\Omega\partial_x(\overline A+\widetilde A)+L\overline B$. Its average and fluctuating parts give, respectively,
$$
\partial_t\overline B=\Omega\partial_x\overline A+L\overline B,\qquad
\partial_\tau\widetilde B=\Omega\partial_x\widetilde A.
$$
Consequently the zero-average fluctuation is
$$
\widetilde B=-\frac{\Omega\alpha_0}{\omega^2}\sin(\omega\tau)\partial_x\overline B.
$$
The order-one averaged <poloidal magnetic field> potential equation retains the correlation between the fluctuating <alpha effect> and the fluctuating <toroidal magnetic field>. Although alpha has zero average, its product with the magnetic fluctuation does not:
$$
\overline{\widetilde\alpha\widetilde B}
=-\frac{\Omega\alpha_0^2}{\omega^2}\overline{\sin^2(\omega\tau)}\partial_x\overline B
=-C\partial_x\overline B,\qquad
\boxed{C=\frac{\Omega\alpha_0^2}{2\omega^2}}.
$$
Thus the <rapidly fluctuating alpha effect> produces the leading averaged equations
$$
\boxed{\partial_t\overline A=-C\partial_x\overline B+L\overline A,\qquad
\partial_t\overline B=\Omega\partial_x\overline A+L\overline B}.
$$
A higher-order zero-average correction to $A$ accommodates the remaining fast order-one terms; it does not alter this leading averaged balance. These equations describe the small-$\epsilon$ limit, rather than an exact finite-$\epsilon$ identity.
For a <Fourier mode>, let $(\overline A,\overline B)=(a,b)e^{ikx+\sigma t}$ and put $d=k^2+\ell^2$. Then $(\sigma+d)a=-ikCb$ and $(\sigma+d)b=ik\Omega a$. Their determinant gives the <dispersion relation>
$$
(\sigma+d)^2=\Omega Ck^2,\qquad
\boxed{\sigma_\pm=-k^2-\ell^2\pm|k|\sqrt{\Omega C}}.
$$
Here $\Omega C=\Omega^2\alpha_0^2/(2\omega^2)\ge0$, so both <growth rates> are real. For a prescribed nonzero <wavenumber>, growth requires $\Omega C>(k^2+\ell^2)^2/k^2$. If real <wavenumbers> can be chosen freely, the largest <growth rate> occurs at $|k|=\sqrt{\Omega C}/2$ and is $\Omega C/4-\ell^2$. Hence \b[growing averaged modes exist precisely when]
$$
\boxed{\Omega C>4\ell^2}.
$$
A bounded or periodic spatial domain instead requires an admissible <wavenumber> satisfying the preceding mode-by-mode inequality.
The physical feedback differs from an ordinary <Parker dynamo wave>. A constant <alpha effect> couples $B$ directly into $A$, while the <Omega effect> differentiates $A$ once, producing the factor $ik\alpha\Omega$ responsible for an oscillatory phase. Here the fast <alpha effect> first creates a potential fluctuation in temporal quadrature; shear creates a magnetic fluctuation correlated with alpha. The averaged feedback is consequently proportional to $-\partial_x\overline B$. Combining its spatial derivative with the derivative in the <Omega effect> gives the real positive factor $\Omega Ck^2$. In fact $(\partial_t-L)^2\overline B=-\Omega C\partial_x^2\overline B$. The resulting growing fields have fixed spatial phase, so \b[the leading averaged pattern grows without travelling]. Fast within-period oscillations of the original fields remain present.
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