Saffman integral ($L$)

The Saffman integral is $L=\int ⟨\mathbf{u}(\mathbf{x})\cdot \mathbf{u}(\mathbf{x}+\mathbf{r})⟩d^{3}r$. With integrable trace correlation it equals the large-volume limit of $⟨|{\int }_V\mathbf{u}dV{|}^2⟩/|V|$. It is nonnegative and measures momentum-fluctuation density. Under appropriate unforced far-field flux conditions it is invariant.

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.

A finite nonzero Saffman integral gives $E(k)=L{k}^2/(4{\pi }^2)+o(k^2)$ in the standard isotropic spectrum normalization. This follows by expanding the radial spectral transform at small wavenumber. A further analytic term requires additional correlation moments.