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Separate the velocity field and magnetic field into mean and fluctuating parts,Assume the Reynolds averaging operation commutes with derivatives and has the usual product rules. Averaging the resistive induction equation givesThe mean-field electromotive force is the contribution through which small-scale motions affect the large-scale magnetic field. If the mean field varies slowly compared with the fluctuation scales, its response can be expanded in the mean field and its spatial derivatives. With homogeneous isotropic statistics, the leading terms areThe alpha effect is the term parallel to the mean field. Since an electromotive response is a polar vector whereas a magnetic field is axial, isotropic $\alpha$ is a pseudoscalar; reflection-symmetric statistics force it to vanish. Nonzero kinetic helicity supplies the required handedness. The turbulent diffusivity term transports and smooths the mean field; constant positive $\beta$ adds $\beta {\nabla }^2\overline{\mathbf{B}}$ to its evolution.

One controlled calculation is first-order smoothing. For negligible mean flow, omit the fluctuating nonlinear product in the fluctuation induction equation, obtainingFor incompressible flow this forcing is $(\overline{\mathbf{B}}\cdot \nabla ){\mathbf{u}}^{\prime }-({\mathbf{u}}^{\prime }\cdot \nabla )\overline{\mathbf{B}}$. A short-memory approximation of duration $\tau$ replaces the inverse temporal response by multiplication by $\tau$. Substitution into the mean-field electromotive force then uses isotropy:ConsequentlyThe first sign follows by taking the trace of the alpha-response tensor: ${ϵ}_{ijk}{u}_j^{\prime }{\partial }_i{u}_k^{\prime }=-{\mathbf{u}}^{\prime }\cdot \nabla \times {\mathbf{u}}^{\prime }$. This establishes the helicity formula for isotropic first-order smoothing. Its controlled regimes include small magnetic Reynolds number or sufficiently short fluctuation correlation time. Outside such a closure, the coefficients involve response-weighted time correlations; an instantaneous helicity formula is not universal.

Take zero mean flow locally and incompressible fluctuations. Subtracting the averaged induction equation givesThe first-order smoothing approximation omits the second term on the right when solving for $\mathbf{b}$, but retains $⟨\mathbf{u}\times \mathbf{b}⟩$ in the mean equation. For a locally uniform, slowly changing mean magnetic field the forcing becomes $(\overline{\mathbf{B}}\cdot \nabla )\mathbf{u}$.

There are two controlled ways to make the omitted fluctuation term small. At small eddy magnetic Reynolds number $R_m=u_{\mathrm{rms}}\ell /\eta$, diffusion gives $b/B\sim R_m$, so the omitted $ub/\ell$ is smaller than the retained $uB/\ell$ by $R_m$. Alternatively, if $u_{\mathrm{rms}}\tau /\ell \ll 1$ for the fluctuation response time $\tau$, then $b/B\sim u_{\mathrm{rms}}\tau /\ell$ and the short-memory response is perturbative. Mean-field scale separation is additionally required. Merely calling the turbulence weak or assuming that the mean magnetic field is small does not justify the omission.

With homogeneous initial transients removed, the causal linear solution isInsert this into the electromotive correlation. Isotropy gives ${\alpha }_{ij}=\alpha {\delta }_{ij}$, so taking the trace yields the helicity formula for isotropic first-order smoothing:Using the alternating-tensor sign givesThe minus sign follows from ${ϵ}_{imn}=-{ϵ}_{min}$. This derivation specifies the diffusion and correlation weighting, rather than asserting that alpha is proportional to instantaneous helicity for arbitrary turbulence.

If $\eta \tau /{\ell }^2\ll 1$, diffusion over a correlation time is negligible. Define the helicity correlation time byFor an ordinary decaying correlation with nonzero mean kinetic helicity, this gives the familiar estimateIt is the kinetic helicity, not magnetic helicity. Opposite flow handedness reverses the alpha effect. The time-integral formula remains meaningful if the equal-time helicity vanishes or its correlation changes sign, when replacing it by a single positive correlation time may fail. In the same weak-diffusion short-memory approximation the gradient forcing $-(\mathbf{u}\cdot \nabla )\overline{\mathbf{B}}$ gives $\beta =\frac{1}{3}{\int }_0^{\infty }⟨\mathbf{u}(t)\cdot \mathbf{u}(t-\tau )⟩d\tau \simeq {\tau }_U{u}_{\mathrm{rms}}^2/3$; the two correlation times need not be equal.

In the small-$R_m$, slowly varying quasistatic regime, diffusion is instead essential. The response givesFor velocity concentrated at wavenumber $k_0$, this becomes $\alpha =-⟨\mathbf{u}\cdot \mathbf{\varpi }⟩/(3\eta k_0^2)$. Its effective memory is the magnetic diffusion time on the eddy scale. The controlled low-$R_m$ setting is developed in Moffatt's analysis of turbulent dynamo action; the causal-response calculation above also covers short-memory turbulence without assuming that diffusion is always negligible.