= Isotropic alpha effect of three helical traveling waves
{title2=$\alpha=-\frac{\eta k^2}{3[(\eta k^2)^2+\Omega^2]}\langle\mathbf u\cdot\nabla\times\mathbf u\rangle$}
Three equal-amplitude transverse waves along orthogonal axes, with the same <circular polarization>, each satisfy $\nabla\times\mathbf u_j=\sigma k\mathbf u_j$, $\sigma=\pm1$. Distinct spatial phases have zero cross averages. The periodic <first-order smoothing> response to a uniform test field satisfies $(\partial_t-\eta\Delta)\mathbf b_j=(\overline{\mathbf B}\cdot\nabla)\mathbf u_j$. Writing $a=\eta k^2$, the response is $\mathbf b_j=k\overline B_j[a\partial_\phi\mathbf u_j-\Omega\mathbf u_j]/(a^2+\Omega^2)$. Since $\mathbf u_j\times\partial_\phi\mathbf u_j=-\sigma U^2\mathbf e_j$, summing yields the displayed isotropic <alpha effect>. It includes the resistive and temporal weighting absent from an unqualified instantaneous helicity rule.
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