 Antisymmetry of the second-order monochromatic alpha tensor

ID: antisymmetry-of-the-second-order-monochromatic-alpha-tensor

For

C = u \times (B \cdot \nabla) u

, the cross product identity

(u \times \nabla \times C)_{i} = u_{j} \partial_{i} C_{j} - u_{j} \partial_{j} C_{i}

E_{i}^{(2)} = - ⟨ \partial_{i} u_{j} C_{j} ⟩

α_{ij}^{(2)} = - ⟨ \partial_{i} u \cdot (u \times \partial_{j} u)⟩ = ⟨ u \cdot (\partial_{i} u \times \partial_{j} u)⟩ .

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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