Diffusive large-scale pair dispersion ($\mathbb{E}|\mathbf{l}(t){|}^2\sim 2t{\int }_0^{\infty }{R}_{\mathrm{rel}}(s)ds$)

At separations well above the integral scale of turbulence, two fluid velocities can decorrelate spatially. If the relative-velocity autocorrelation $R_{\mathrm{rel}}$ is stationary and integrable, the displacement identity used in Taylor turbulent dispersion gives the displayed long-time linear mean-square growth. With speed $U$ and Lagrangian integral time of order $L/U$, its relative diffusivity is of order $UL$. The characteristic separation grows as $t^{1/2}$, unlike inertial-range Richardson pair dispersion.