Monochromatic small-Reynolds-number mean electromotive force ($\mathcal{E}=R_m{\mathcal{E}}^{(1)}+R_m^2{\mathcal{E}}^{(2)}+\cdots$)

Suppose a steady solenoidal velocity satisfies $\Delta \mathbf{u}=-\mathbf{u}$, and spatial averaging permits integration by parts without boundary terms. For a uniform test magnetic field $\mathbf{B}$, the small-magnetic Reynolds number expansion gives ${\mathbf{b}}_1=(\mathbf{B}\cdot \nabla )\mathbf{u}$ and $\Delta {\mathbf{b}}_2=-\nabla \times (\mathbf{u}\times {\mathbf{b}}_1)$. Using the self-adjointness of the Laplacian inside the average,The second equality concerns the averaged mean-field electromotive force; it does not assert that the quadratic forcing itself is monochromatic.

// Target: fluid-mechanics.bigb

For $\mathbf{C}=\mathbf{u}\times (\mathbf{B}\cdot \nabla )\mathbf{u}$, the cross product identity $(\mathbf{u}\times \nabla \times \mathbf{C}{)}_i=u_j{\partial }_i{C}_j-u_j{\partial }_j{C}_i$ and integration by parts give ${\mathcal{E}}_i^{(2)}=-⟨{\partial }_i{u}_j{C}_j⟩$. Its alpha tensor isInterchanging $i,j$ reverses the cross product, showing that the coefficient is an antisymmetric matrix. Such a coefficient represents a turbulent magnetic pumping contribution rather than the symmetric part of the alpha effect.

// Target: fluid-mechanics.bigb

The first coefficient of the alpha tensor is ${\alpha }_{ij}^{(1)}={ϵ}_{ikl}⟨u_k{\partial }_j{u}_l⟩$. Its contraction with the Levi-Civita symbol isby incompressibility and vanishing averages of total derivatives. Every antisymmetric second-rank tensor in three dimensions is detected by this contraction, so the coefficient is a symmetric matrix. The proof needs no Fourier analysis.

// Target: fluid-mechanics.bigb