Betchov relation ($⟨{\omega }_i{\omega }_j{S}_{ij}⟩=-4⟨abc⟩$)

For differentiable homogeneous incompressible flow, $⟨\mathrm{tr}{A}^3⟩=0$. One divergence representation is $\mathrm{tr}{A}^3={\partial }_i[u_j({\partial }_j{u}_k)({\partial }_k{u}_i)-u_i\mathrm{tr}{A}^2/2]$. Splitting the velocity gradient tensor into strain and rotation gives $⟨{\omega }_i{\omega }_j{S}_{ij}⟩=-4⟨abc⟩$. It connects mean vortex stretching to the cubic strain statistic, without requiring isotropy.