Landau angular-momentum argument for turbulent decay

For a confined incompressible flow with no velocity flux across the boundary, an integrated divergence identity gives $⟨|\int \mathbf{x}\times \mathbf{u}dV{|}^2⟩=-\iint r^2⟨\mathbf{u}\cdot {\mathbf{u}}^{\prime }⟩dVd{V}^{\prime }$. In a large approximately homogeneous region with short correlations, the latter is approximately $VI$. Conservation of angular momentum in an isolated torque-free cloud then motivates conservation of the Loitsyansky integral. A fixed subvolume of infinite turbulence is not an isolated cloud: advective transport, pressure torques, wall stresses and long-range correlations can invalidate the argument.