= Helicity formula for isotropic first-order smoothing
{title2=$\alpha=-\frac13\int_0^\infty\langle\mathbf u(t)\cdot\nabla\times e^{\eta\tau\Delta}\mathbf u(t-\tau)\rangle\,d\tau$}
= Isotropic alpha-helicity relation
{synonym}
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 $\alpha=-\tau_H\langle\mathbf u\cdot\nabla\times\mathbf u\rangle/3$. At small <magnetic Reynolds number>, a quasistatic monochromatic velocity instead gives $\alpha=-\langle\mathbf u\cdot\nabla\times\mathbf u\rangle/(3\eta k_0^2)$. The effective <correlation time> and diffusion weighting should not be silently discarded.
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