= Antisymmetry of the second-order monochromatic alpha tensor
{title2=$\alpha^{(2)}_{ij}=-\alpha^{(2)}_{ji}$}
For $\mathbf C=\mathbf u\times(\mathbf B\cdot\nabla)\mathbf u$, the <cross product> identity $(\mathbf u\times\nabla\times\mathbf C)_i=u_j\partial_iC_j-u_j\partial_jC_i$ and <integration by parts> give $\mathcal E^{(2)}_i=-\langle\partial_i u_jC_j\rangle$. Its <alpha tensor> is
$$
\alpha^{(2)}_{ij}=-\langle\partial_i\mathbf u\cdot(\mathbf u\times\partial_j\mathbf u)\rangle
=\langle\mathbf u\cdot(\partial_i\mathbf u\times\partial_j\mathbf u)\rangle.
$$
Interchanging $i,j$ 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
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