Mean enstrophy balance ($\dot{Z}=⟨{\omega }_i{\omega }_j{S}_{ij}⟩-\nu ⟨|\nabla \times \mathbf{\omega }{|}^2⟩$)

For unforced homogeneous incompressible flow, the vorticity equation gives $\dot{Z}=⟨{\omega }_i{\omega }_j{S}_{ij}⟩-\nu ⟨|\nabla \times \mathbf{\omega }{|}^2⟩$. Curl and full-gradient squared averages agree because the vorticity is divergence free and boundary divergences average to zero. In established high-Reynolds-number turbulence, slow integral-time evolution makes production and destruction nearly balance.