Richardson pair dispersion ($⟨r^2(t)⟩\sim gϵt^3$)

Richardson pair dispersion concerns relative separation of two marked fluid elements, rather than their centre-of-mass motion. In an inertial range, scale-local relative diffusivity is of order ${ϵ}^{1/3}{r}^{4/3}$. A self-similar closure gives $d⟨r^2⟩/dt=A{ϵ}^{1/3}⟨r^2{⟩}^{2/3}$ and hence a $t^3$ regime after loss of memory of initial separation. Stationarity and inertial-range separation alone do not imply this law at arbitrarily short times.

For positive initial separation below the Kolmogorov length scale, a phenomenological pair-dispersion model has smooth-flow exponential stretching, inertial-range Richardson pair dispersion, and diffusive large-scale pair dispersion beyond the integral scale of turbulence. A positive Lyapunov exponent is needed for the first regime, scale locality and initial-memory loss for the second, and integrable relative-velocity autocorrelation for the third. The inertial-range crossing time is of order $L^{2/3}{ϵ}^{-1/3}$, one outer eddy turnover time.

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.

The dimensionless Richardson constant is the coefficient in the mean-square pair-separation law. Under the simple self-similar diffusivity closure, $g=(A/3{)}^3$. Its universality requires scale locality, adequate scale separation and loss of initial-condition memory; a finite-time fit need not isolate this asymptotic coefficient.