 Helicity formula for isotropic first-order smoothing

For a locally uniform mean magnetic field, first-order smoothing solves ${\partial }_t\mathbf{b}-\eta \Delta \mathbf{b}=(\overline{\mathbf{B}}\cdot \nabla )\mathbf{u}$. Inserting its causal diffusion propagator into the mean-field electromotive force and taking the trace of the isotropic response tensor gives the displayed relation to kinetic helicity. When diffusion during a correlation time is negligible, it reduces to $\alpha =-{\tau }_H⟨\mathbf{u}\cdot \nabla \times \mathbf{u}⟩/3$. At small magnetic Reynolds number, a quasistatic monochromatic velocity instead gives $\alpha =-⟨\mathbf{u}\cdot \nabla \times \mathbf{u}⟩/(3\eta k_0^2)$. The effective correlation time and diffusion weighting should not be silently discarded.