Multidimensional incompressible stochastic closure of turbulent pair dispersion and intermittency-controlled convergence to Richardson scaling (open problems from Bitane–Homann–Bec 2012)

Bitane, Homann and Bec (Phys. Rev. E 86, 045302, 2012) reinterpreted the Richardson–Obukhov $t^3$ law for turbulent pair dispersion as arising from a diffusive behavior of the velocity difference between two tracers: acceleration differences decorrelate on the Kolmogorov dissipative time, with a multiplicative noise amplitude set by a local dissipation proxy $\epsilon_{\mathrm{loc}}\sim[\delta u^{\parallel}]^3/\delta x$, so that the separation behaves as the integral of a Brownian motion and converges to Richardson scaling on the energy-dissipation time $t_0=S_2(r_0)/(2\epsilon)$. The authors introduced a one-dimensional phenomenological stochastic model and left open four directions that jointly constitute a predictive closure: (i) the multidimensional generalization enforcing incompressibility and the correct vectorial correlations of velocity and acceleration differences; (ii) the quantitative effect of flow intermittency on the convergence time to Richardson scaling and on extreme or slowly separating pairs (high-order moments, distribution tails, exit times); (iii) derivation and validation of the Reynolds-number independence of the effective noise amplitude in the limit of infinite Reynolds number; (iv) systematic characterization of the subleading correction coefficient as a function of initial separation and Reynolds number, including the status of $r_0\approx4\eta$ as an optimal initial separation.

Background

Turbulent relative dispersion concerns the growth of the separation $\delta\mathbf{x}(t)=\mathbf{X}_1(t)-\mathbf{X}_2(t)$ between two tracer (fluid) particles advected by a turbulent flow. Richardson (1926) discovered that the mean-squared separation grows explosively as $\langle|\delta\mathbf{x}(t)|^2\rangle\propto\epsilon t^3$, where $\epsilon$ is the mean kinetic-energy dissipation rate per unit mass; with Kolmogorov 1941 scaling this is the Richardson–Obukhov law. Richardson explained it by a scale-dependent eddy diffusivity for the separation, an idea that underlies most practical Lagrangian stochastic dispersion models (reviewed by Sawford, 2001). This argument, however, requires the velocity difference $\delta\mathbf{u}$ between the two tracers to decorrelate on times short compared with the observation time, which conflicts with turbulence phenomenology: if the separation grows like $t^{3/2}$, the turnover time of the eddies at the scale of the separation grows like $t$, so the relevant structures stay correlated for about the observation time. For times shorter than the Lagrangian correlation time of the initial velocity difference, the separation instead grows ballistically (Batchelor regime), $\langle|\delta\mathbf{x}(t)-\delta\mathbf{x}(0)|^2\rangle\simeq t^2 S_2(r_0)$, where $S_2(r)=\langle|\delta\mathbf{u}|^2\rangle$ is the Eulerian second-order velocity structure function and $r_0$ the initial separation; $\eta=(\nu^3/\epsilon)^{1/4}$ and $\tau_\eta=(\nu/\epsilon)^{1/2}$ are the Kolmogorov dissipative length and time ($\nu$ is the kinematic viscosity), and $R_\lambda$ the Taylor-scale Reynolds number. Stochastic particle models must also satisfy incompressibility, usually imposed through Thomson's (1987) well-mixed condition: a model for the pair $(\delta\mathbf{x},\delta\mathbf{u})$ must reproduce prescribed Eulerian two-point statistics when averaged over uniformly distributed pairs; existing two-particle closures (e.g. Borgas and Yeung, 2004) therefore require Eulerian input. Bitane, Homann and Bec (2012), using direct numerical simulations (DNS) of homogeneous isotropic turbulence at $R_\lambda=460$ and $730$, proposed a different picture: the acceleration difference $\delta\mathbf{a}$ decorrelates on the short dissipative time $\tau_\eta$, so by the central-limit theorem $\delta\mathbf{u}$ diffuses, $\langle|\delta\mathbf{u}(t)|^2\rangle_{r_0}\simeq S_2(r_0)+D\,\epsilon t$ with $D\approx2.1$, and the separation — the integral of this diffusing velocity — follows $t^{3/2}$ in amplitude, i.e. $t^3$ for the mean square, with Richardson constant $g\approx0.52$. Balancing the diffusive growth of $\langle|\delta\mathbf{u}|^2\rangle$ against its initial value gives the convergence time $t_0=S_2(r_0)/(2\epsilon)$, the time to dissipate the kinetic energy at scale $r_0$ — also the time of departure from Batchelor's ballistic regime. The local dissipation rate along a pair is modeled by the proxy $\epsilon_{\mathrm{loc}}\sim[\delta u^{\parallel}]^3/\delta x$, with $\delta u^{\parallel}$ the longitudinal velocity difference; the simulations show $\langle[\delta u^{\parallel}]^3/\delta x\rangle_{r_0}$ rapidly approaches $\approx6\epsilon$. Dimensional estimates for the local Kolmogorov time $\tau_\eta^{\mathrm{loc}}\sim\nu^{1/2}\epsilon_{\mathrm{loc}}^{-1/2}$ and the acceleration-difference amplitude $A\sim\nu^{-1/4}\epsilon_{\mathrm{loc}}^{3/4}$ then give a noise strength $[\tau_\eta^{\mathrm{loc}}]^{1/2}A\sim\epsilon_{\mathrm{loc}}^{1/2}$, independent of the viscosity $\nu$ and thus expected to remain finite at infinite Reynolds number. This yields a one-dimensional phenomenological Markov model, $d\delta x/dt=\delta u^{\parallel}$, $d\delta u^{\parallel}/dt\simeq(\epsilon+\alpha[\delta u^{\parallel}]^3/\delta x)^{1/2}\xi(t)$ (Stratonovich, $\xi$ white noise, $\alpha>0$), whose solutions show a ballistic short-time regime and Richardson $t^3$ at large times. The mean-squared separation carries a subleading correction $\langle|\delta\mathbf{x}(t)-\delta\mathbf{x}(0)|^2\rangle_{r_0}=g\epsilon t^3[1+C\,t_0/t]+\ldots$ with $C\approx1.6$ in the inertial range, $C<0$ near $r_0\sim\eta$, and $C=0$ at $r_0\approx4\eta$, making that initial separation an 'optimal choice' for observing the $t^3$ law. The authors argue that separation moments should be self-similar at all orders (intermittency of the flow — the rare violent fluctuations and scale-dependent statistics of turbulent velocity differences and local dissipation — should not affect the long-time separation statistics), while conceding that intermittency directly affects the convergence time and may produce corrections in other observables such as exit times (the time for a pair to reach a given separation; Scatamacchia, Biferale and Toschi, 2012, had found extreme-event effects). The paper leaves the model one-dimensional: 'Extension to higher dimensions requires accounting for incompressibility and is the subject of ongoing work.'

Problem Statement

The problem, as left open by Bitane, Homann and Bec (2012) and surviving the subsequent literature, is to turn the one-dimensional phenomenological stochastic model of inertial-range pair separation into a validated predictive closure of turbulent relative dispersion. Its four jointly-linked components, all four required for a complete closure, are: (1) Construct a multidimensional stochastic Markov model for the separation vector and velocity difference $(\delta\mathbf{x},\delta\mathbf{u})$ — the multidimensional generalization of $d\delta x/dt=\delta u^{\parallel}$, $d\delta u^{\parallel}/dt\simeq(\epsilon+\alpha[\delta u^{\parallel}]^3/\delta x)^{1/2}\xi(t)$ — that enforces incompressibility of the induced relative-motion dynamics (a requirement the source states without elaborating it; standard formulations include Thomson's well-mixed condition, which requires Eulerian input, whereas the source's construction is deliberately designed to require none, so an admissible answer must state the incompressibility condition it enforces as an explicit, checkable property of the constructed dynamics, justify why that condition is the correct notion of incompressibility for the pair model, and demonstrate that the model satisfies it) and correctly represents the vectorial correlations of $\delta\mathbf{u}$ and $\delta\mathbf{a}$, i.e. the conditional covariance structure $\langle\delta\mathbf{a}\otimes\delta\mathbf{a}\mid\delta\mathbf{x},\delta\mathbf{u}\rangle$; this is the model itself, and it must fix the time scales and state variables on which pair separation is genuinely Markovian. (2) Derive and validate, from the model and/or targeted simulations at higher Reynolds numbers and broader scale separations, the dependence of the effective noise amplitude on the local dissipation proxy $[\delta u^{\parallel}]^3/\delta x$, testing the predicted viscosity-independence and finite infinite-Reynolds-number limit of the amplitude $[\tau_\eta^{\mathrm{loc}}]^{1/2}A\sim\epsilon_{\mathrm{loc}}^{1/2}$; this is the model's calibration. (3) Quantify how intermittency of the flow controls (a) the time of convergence to the Richardson law $\langle|\delta\mathbf{x}(t)-\delta\mathbf{x}(0)|^2\rangle=g\epsilon t^3$, (b) the tails of the separation distribution and slowly separating pairs, and whether the source's predicted self-similarity of high-order separation moments survives intermittency-induced deviations, and (c) exit-time statistics — the source predicts self-similar separation moments at all orders but leaves the intermittency dependence of the convergence time and of non-moment observables undetermined; (3)(b) is resolved by an intermittency-controlled prediction of the separation-distribution tails and of slowly separating pairs, together with the high-order self-similarity verdict reported either way; this is the model's domain of validity. (4) Characterize the subleading correction coefficient $C$ in $\langle|\delta\mathbf{x}(t)-\delta\mathbf{x}(0)|^2\rangle=g\epsilon t^3[1+C\,t_0/t]+\mathrm{h.o.t.}$ as a systematic function of the initial separation $r_0$ and the Reynolds number, including the status of $r_0\approx4\eta$ as an optimal initial separation for observing Richardson scaling; these are the model's finite-time corrections. An answer that resolves only a subset of these components — for example (1) alone, or (1) and (2) — is a partial result that does not resolve the task.

References

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