Lagrangian velocity autocorrelation ($R_L(s)=\mathbb{E}[\mathbf{u}(\mathbf{X}(t+s),t+s)\cdot \mathbf{u}(\mathbf{X}(t),t)]$)

For stationary statistics along a Lagrangian trajectory $\mathbf{X}$, the velocity autocorrelation depends on the lag $s$. Integrating velocity twice gives the mean-square displacement $2{\int }_0^t(t-s)R_L(s)ds$. If $R_L$ is integrable, the long-time displacement is diffusive. Its normalized one-sided area is the Lagrangian integral time.