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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.

In mean-field electrodynamics, use Reynolds averaging to write ${\mathbf{U}}_{\mathrm{tot}}=\overline{\mathbf{U}}+\mathbf{u}$ and ${\mathbf{B}}_{\mathrm{tot}}=\overline{\mathbf{B}}+\mathbf{b}$, with zero-mean fluctuations. Assume the averaging operation commutes with the derivatives and that resolved quantities vary little across an averaging region. Averaging the induction equation then givesThis is the mean-field electromotive force, a vector correlation with units of velocity times magnetic field. The closure problem is to express this correlation in terms of the resolved field.

In the kinematic regime the flow statistics are prescribed, so the response is linear in the mean magnetic field, after independently excited fluctuation transients have decayed. If the mean-field scale $L$ is much greater than the turbulent scale $\ell$, and its evolution is slow compared with the response memory, expand that response in the local mean magnetic field and its gradients:Without scale separation one generally has a spatially nonlocal and temporally retarded response rather than constant local coefficients.

For homogeneous turbulence the coefficients are independent of position. Invariance under proper rotations makes the rank-two response proportional to ${\delta }_{ij}$, and the isotropic rank-three gradient response proportional to the alternating tensor. Choosing the diffusivity sign convention givesThe alpha effect can regenerate a mean magnetic field, while $\beta$ supplies turbulent magnetic diffusivity. For constant positive $\beta$, $\nabla \times (-\beta \nabla \times \overline{\mathbf{B}})=\beta {\nabla }^2\overline{\mathbf{B}}$ because the magnetic field is solenoidal. Thus the total diffusivity is $\eta +\beta$.

The electromotive vector is a polar vector, whereas the magnetic field is an axial vector. Therefore $\alpha$ is a pseudoscalar and changes sign under reflection; $\beta$ is an ordinary scalar. A nonzero isotropic alpha effect requires broken reflection symmetry, such as nonzero kinetic helicity. If “isotropic” includes reflection symmetry as well as rotations, $\alpha$ must vanish. The dimensions are $[\alpha ]=L/T$ and $[\beta ]=L^2/T$.

Reynolds averaging separates a resolved mean from zero-mean fluctuations. The idealized averaging operation is linear and idempotent, commutes with the derivatives under consideration and treats resolved mean factors as constant inside its averaging scale. Products then satisfy $\overline{fg}=\overline{f}\overline{g}+\overline{f^{\prime }{g}^{\prime }}$. Actual finite spatial filters need scale separation or corrections when these rules are only approximate.