 Isotropic alpha effect of three helical traveling waves

Three equal-amplitude transverse waves along orthogonal axes, with the same circular polarization, each satisfy $\nabla \times {\mathbf{u}}_j=\sigma k{\mathbf{u}}_j$, $\sigma =\pm 1$. 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 ${\mathbf{b}}_j=k{\overline{B}}_j[a{\partial }_{\varphi }{\mathbf{u}}_j-\Omega {\mathbf{u}}_j]/(a^2+{\Omega }^2)$. Since ${\mathbf{u}}_j\times {\partial }_{\varphi }{\mathbf{u}}_j=-\sigma U^2{\mathbf{e}}_j$, summing yields the displayed isotropic alpha effect. It includes the resistive and temporal weighting absent from an unqualified instantaneous helicity rule.