= Monochromatic small-Reynolds-number mean electromotive force
{title2=$\mathcal E=R_m\mathcal E^{(1)}+R_m^2\mathcal E^{(2)}+\cdots$}
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,
$$
\mathcal E^{(1)}=\langle\mathbf u\times(\mathbf B\cdot\nabla)\mathbf u\rangle,\qquad
\mathcal E^{(2)}=\langle\mathbf u\times\nabla\times[\mathbf u\times(\mathbf B\cdot\nabla)\mathbf u]\rangle.
$$
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
Back to article page
