Nature of the intermediate-time regime in turbulent pair dispersion: non-local Batchelor-Taylor transition versus Richardson $t^3$ scaling

Richardson's 1926 law predicts that the mean-square separation of a pair of passively advected tracers in turbulence grows as $\langle|\mathbf{r}(t)-\mathbf{r}(0)|^2\rangle = g\varepsilon t^3$ at intermediate times, when the separation lies in the inertial range and only eddies of the current separation scale contribute (local dispersion). Elsinga, Ishihara and Hunt (J. Fluid Mech. 932, A17, 2022) reassessed this cornerstone using DNS up to $Re_\lambda\approx1131$: inside the intense 'significant shear layers' that organize high-Reynolds-number turbulence, pair dispersion is strongly non-local and Taylor-like; a literature survey finds no undisputed $t^3$ regime for initial separations $r_0$ inside the inertial range (only near $r_0\approx4\eta$); and they propose instead that the intermediate regime is a transition region connecting the initial Batchelor ($t^2$) regime to the final Taylor ($t^1$) regime, bridged by a power law with empirical endpoints (Batchelor end near $3t_B$, Taylor onset near $8T_L$). The open problem is to establish, theoretically and with data at Reynolds numbers and initial separations beyond those available, the true nature and quantitative description of this intermediate regime: whether a local Richardson regime exists for inertial-range $r_0$ at finite Reynolds number and emerges as $Re_\lambda\to\infty$, or whether the non-local, $r_0$-dependent transition persists; and, if the transition picture holds, to characterize its functional form and derive (or establish the universality of) its parameters. Later work (2022-2026) sharpens but does not resolve the question: a published Physical Review Letters exchange re-analyzes the highest available DNS with conflicting conclusions, and a new experimental apparatus capable of Lagrangian tracking up to $Re_\lambda\approx6000$ has not yet reported pair-dispersion results.

Background

Pair dispersion is the statistical growth of the separation $\mathbf{r}(t)$ between two passive tracer particles advected by a turbulent flow, and it underlies turbulent mixing, pollutant transport, and scalar-variance evolution. For homogeneous isotropic turbulence with mean energy dissipation rate $\varepsilon$, Kolmogorov length and time scales $\eta=(\nu^3/\varepsilon)^{1/4}$ and $\tau_\eta$, rms velocity $U$, Lagrangian integral time $T_L$, and Taylor-scale Reynolds number $Re_\lambda$, three classical regimes describe $\langle|\mathbf{r}(t)-\mathbf{r}(0)|^2\rangle$: (i) the Batchelor regime at short times, $\langle|\mathbf{r}(t)-\mathbf{r}(0)|^2\rangle = S_2(r_0)\,t^2$ with $S_2$ the second-order Eulerian velocity structure function, valid up to a few times the Batchelor time $t_B=(r_0^2/\langle\varepsilon\rangle)^{1/3}$ that marks the memory of the initial separation $r_0$; (ii) Richardson's intermediate regime, $\langle|\mathbf{r}(t)-\mathbf{r}(0)|^2\rangle = g\varepsilon t^3$ with Richardson constant $g\approx0.55$, predicted by Obukhov and Batchelor from Kolmogorov 1941 phenomenology; and (iii) Taylor's final diffusive regime, $\langle|\mathbf{r}(t)-\mathbf{r}(0)|^2\rangle = 12U^2T_L\,t$, once the separation exceeds the large scale $L$. The classical derivations of the Richardson regime assume that $r(t)$ stays within the inertial range ($\eta\ll r\ll L$, at high $Re_\lambda$) and that dispersion is local, i.e. only eddies of scale comparable to the instantaneous separation control the relative velocity. Richardson himself expressed the law as a scale-dependent diffusivity $K\propto r^{4/3}$; later frameworks include diffusion closures, Bourgoin's ballistic cascade phenomenology (J. Fluid Mech. 772, 678-704, 2015), and Lagrangian stochastic models (Borgas and Yeung, J. Fluid Mech. 503, 125-160, 2004; Devenish and Thomson, J. Fluid Mech. 867, 877-905, 2019). A persistent empirical embarrassment is that a clean, decade-spanning $t^3$ regime is essentially only seen for initial separations $r_0\approx4\eta$, which lies in the dissipation range rather than the inertial range, with other $r_0$ approaching $t^3$ from above or below (Sawford, Yeung and Hackl 2008; Bitane, Homann and Bec 2012-2013; Buaria, Sawford and Yeung, Phys. Fluids 27, 105101, 2015, at $Re_\lambda$ up to 1000, the highest Lagrangian DNS available). Elsinga, Ishihara and Hunt (J. Fluid Mech. 932, A17, 2022) reassessed the whole picture. Using $Re_\lambda\approx1131$ DNS they showed that pairs released inside the 'significant shear layers' (thin, intense sheets of elevated dissipation that bound the small-scale motions with large-scale velocity differences of order $U$) disperse in a Taylor-like, highly non-local way with essentially no role for intermediate-scale eddies, and that the layers' contribution to the overall mean-square separation stays significant as $Re_\lambda$ grows (volume fraction $\sim Re_\lambda^{-1}$, dispersion magnitude $\sim Re_\lambda$). Their literature survey concluded there is at present no clear and indisputable evidence for a Richardson $t^3$ regime in the inertial range, and that reported $t^3$ ranges are short (under a decade) and can be artifacts of virtual-origin fits, in which a free time offset $t_0$ in $\langle r^2(t)\rangle \propto (t-t_0)^3$ is adjusted and can make a limited non-cubic range of data appear cubic. As an alternative they modeled the intermediate range as a transition region connecting the Batchelor and Taylor regimes, bridged in a logarithmic plot by a straight power law whose endpoints are fixed empirically: the end of the Batchelor regime at $t\approx3t_B$ and the onset of the Taylor regime at $t\approx cT_L$ with $c\approx8$ (yielding a 'Richardson constant' $g'\approx0.56$ at the single $r_0\approx4\eta$ where the bridge is exactly $t^3$). The model reproduces the observed $r_0$-dependent approach to $t^3$ and returns $t^3$ for all $r_0$ only in the limit $Re_\lambda\to\infty$. The authors are explicit that the bridge shape, the two empirical coefficients, the generality of the shear-layer mechanism, and the model predictions at $Re_\lambda$ far above the available data are all unresolved. The question immediately intersects a live controversy: Tan and Ni's experiments (Phys. Rev. Lett. 128, 114502, 2022) claimed the $t^3$ law is reached only at a critical $r_0\approx3\eta$ independent of $Re_\lambda$, while Buaria's Comment (Phys. Rev. Lett. 130, 029401, 2023) re-analyzed the same highest-Reynolds-number DNS and argued that $t^3$ holds over a widening range $2<r_0/\eta<8$ with a trend toward wider ranges at higher $Re_\lambda$, explicitly deferring the decision to data at Reynolds numbers beyond those currently available.

Problem Statement

For pairs of passive fluid tracers in homogeneous isotropic turbulence, determine the nature and quantitative description of the intermediate-time regime of relative dispersion that connects the initial Batchelor regime ($\langle|\mathbf{r}(t)-\mathbf{r}(0)|^2\rangle = S_2(r_0)\,t^2$) to the final Taylor regime ($\langle|\mathbf{r}(t)-\mathbf{r}(0)|^2\rangle = 12U^2T_L\,t$). The overarching objective is a validated answer to: does a local, inertial-range Richardson regime ($\langle|\mathbf{r}(t)-\mathbf{r}(0)|^2\rangle = g\varepsilon t^3$ with $g$ universal) exist for initial separations $r_0$ inside the inertial range at finite Reynolds number and emerge as $Re_\lambda\to\infty$, or is the intermediate regime instead a non-local, $r_0$-dependent transition region whose observed approach to $t^3$ near $r_0\approx4\eta$ is a finite-Reynolds-number crossover artifact, as proposed by Elsinga, Ishihara and Hunt? The scientific goal is a resolution of this dichotomy, whether by first-principles theory, by new pair-dispersion evidence, or by decisive re-analysis of existing data; a validated resolution of the dichotomy by any one of these three routes constitutes a complete answer to the task. Open aspects identified in the source and its aftermath, which a resolution addresses to the extent the corresponding branch holds, include: pair-dispersion evidence at Reynolds numbers and initial separations beyond those currently available ($r_0$ spanning the dissipation range to the inertial range, from a few $\eta$ to at least $60\eta$) capable of discriminating between the two pictures; if the transition picture holds, the functional form of the intermediate-time evolution beyond the coarse power-law bridge (its smoothness, curvature, and dependence on $r_0$ and $Re_\lambda$), together with the origin or universality of its empirical parameters (the end of the Batchelor regime, of order $3t_B$, and the Taylor-onset constant $c\approx8$), or a predictive replacement for them; the magnitude and high-Reynolds-number persistence of the non-local contribution of significant shear layers to the overall mean-square separation (including statistics of layer volume fraction, lifetime, and orientation, and the behavior of pairs not wholly inside layers); and the unexplained per-Reynolds-number parameter adjustment of Lagrangian stochastic and mechanistic dispersion models. These open aspects are optional scope, each relevant only to the extent its branch holds, not a set of jointly mandatory deliverables; in particular, an aspect that calls for pair-dispersion evidence at Reynolds numbers beyond those currently available is addressed by first-principles theory or by re-analysis of existing data rather than by new measurement.

References

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