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Measure the cloud radius about its moving centre, so uniform sweeping does not count as spreading. In the inertial range, an eddy of size $R$ has velocity difference $u_R\sim (ϵR{)}^{1/3}$ and turnover time $t_R\sim R/u_R$. A scale-local relative diffusivity is thereforegiving the estimateIt requires separation from the molecular and viscous cutoffs and from the integral scale of turbulence, high Reynolds number, approximately steady energy supply, and scale-local relative motion. Mean shear, walls, particle inertia or coherent large-scale strain can invalidate this simple diffusivity model. At very short times a pair retains its initial relative velocity, so a diffusion closure is not yet appropriate.

Let $s(t)=⟨|\delta \mathbf{x}(t){|}^2⟩$. A self-similar pair-separation distribution, together with the relative-diffusivity estimate, givesSelf-similarity is needed to replace a separation-dependent moment such as $⟨r^{4/3}⟩$ by a fixed multiple of $s^{2/3}$; stationarity alone does not do that. Integrating this model givesAfter loss of initial-separation memory, it reduces to the Richardson pair dispersion lawThe required time window is $t\gg t_0\sim (r_0^2/ϵ{)}^{1/3}$ while the separation remains far below the integral scale. The stated stationarity and inertial-range separation conditions alone cannot imply the displayed $t^3$ law at every time: a nonzero release separation already gives $s(0)=r_0^2$, whereas $gϵt^3$ vanishes at zero. At sufficiently short times, $\delta \mathbf{x}(t)-\delta {\mathbf{x}}_0\simeq \delta {\mathbf{u}}_{0}t$, so its mean-square change is ballistic and remembers $r_0$.

The Richardson constant might be universal if the inertial-range relative dynamics, including the normalized separation distribution, become independent of release details and forcing at sufficiently high Reynolds number. This is an additional universality hypothesis. As evidence available before 2008, Boffetta and Sokolov's numerical study estimated a coefficient near $0.55$ and reported agreement with experimental estimates, alongside small deviations from ideal Richardson scaling. Bourgoin and collaborators' 2006 experiment showed substantial dependence on initial separation in the accessible laboratory regime. Values of order $0.5$ are supported in suitable regimes, but finite-time fits do not establish one universally observed $t^3$ coefficient for all releases.