Saffman decay law ($\mathcal{E}\propto (t-t_{\ast }{)}^{-6/5}$)

With nonzero invariant $L$, a self-preserving integral scale of turbulence gives $L\propto \mathcal{E}{\ell }^3$. Combining this with freely decaying high-Reynolds-number viscous dissipation $ϵ\propto {\mathcal{E}}^{3/2}/\ell$ gives $\mathcal{E}\propto (t-t_{\ast }{)}^{-6/5}$ and $\ell \propto (t-t_{\ast }{)}^{2/5}$. The closure assumptions, not the invariant alone, determine this exponent.