2. Overview

The nature of the heavy-particle trajectories is graphically illustrated in figure 1 of Biferale et al. (Reference Biferale, Lanotte, Scatamacchia and Toschi2014). In this picture, ensembles of trajectories are shown emanating from a small source of length scale ${\it\eta}=({\it\nu}^{3}/{\it\epsilon})^{1/4}$ , estimated by Kolmogorov (Reference Kolmogorov1941) for the parameters of kinematic viscosity ${\it\nu}$ and local turbulent energy dissipation rate ${\it\epsilon}$ . The trajectories shown in blue for heavy particles form a more coherent cluster than the red tracer trajectories as the flow moves right to left from the source.

Figure 1.

An ensemble of tracer particles with $\mathit{St}=0$ (red) and heavy particles with $\mathit{St}=5$ (blue), simultaneously emitted from a source of size ${\sim}{\it\eta}$ . Trajectories are recorded from the emission time, up to the time $t=75{\it\tau}_{{\it\eta}}$ after the emission.

The Stokes number shown is the ratio of the particle response time ${\it\tau}_{s}$ to the Lagrangian viscous time scale ${\it\tau}_{{\it\eta}}=({\it\nu}/{\it\epsilon})^{1/2}$ : $\mathit{St}={\it\tau}_{s}/{\it\tau}_{{\it\eta}}$ . Here $\mathit{St}=0$ are tracers and $\mathit{St}=5$ for heavy particles.

For fixed scales of the energy containing eddies of the turbulence, say velocity scale ${\it\sigma}$ and length scale $L$ , define a Reynolds number $\mathit{Re}={\it\sigma}L/{\it\nu}$ , implying that ${\it\eta}=\mathit{Re}^{-3/4}L$ , ${\it\tau}_{{\it\eta}}=\mathit{Re}^{-1/2}L/{\it\sigma}$ .

For geophysical flows we anticipate the limit of large Reynolds number $\mathit{Re}\gg 1$ , and we see that the length scale ${\it\eta}$ reduces to zero at a faster rate than the corresponding time scale ${\it\tau}_{{\it\eta}}$ which is the relevant time scale for Lagrangian processes.

The estimates of turbulent scaling, first promoted by Richardson (Reference Richardson1926b ) and Kolmogorov (Reference Kolmogorov1941), is to predict parameters such as velocity differences separated by a length scale $r$ , where viscous or molecular parameters are not important and only ‘inertial’ scales matter: for example

(2.1) $$\begin{eqnarray}\langle (u_{1}(x+r)-u_{1}(x))^{2}\rangle \sim C_{kol}({\it\epsilon}r)^{2/3}r\gg {\it\eta},\end{eqnarray}$$

where $\langle \bullet \rangle$ means an averaging operation of a velocity difference (say over the $x$ direction) and $C_{kol}\approx 2.13$ is a constant (Sreenivasan Reference Sreenivasan1995). Similarly, the analogous Lagrangian result is

(2.2) $$\begin{eqnarray}\langle (u_{1}(t+{\it\tau})-u_{1}(t))^{2}\rangle \sim C_{0}{\it\epsilon}{\it\tau},\quad {\it\tau}\gg {\it\tau}_{{\it\eta}}\end{eqnarray}$$

first examined for viscous scaling corrections by Sawford (Reference Sawford1991) estimating that $C_{0}\approx 6$ .

Viscous scaling means that the resolved dynamic scales of turbulence are smaller in the Lagrangian frame than in the Eulerian frame and for a moderate- $\mathit{Re}$ computational simulation it is difficult to unambiguously observe Lagrangian scaling. The intermittency corrections to scaling for higher moments, say $\langle (u_{1}(x+r)-u_{1}(x))^{q}\rangle$ for $q>2$ , are also difficult to untangle from viscous scaling effects at moderate Reynolds number, and difficult to sample sufficiently in geophysical flows for increasingly rare extreme events, which even for highly non-Gaussian multifractal behaviour are highly infrequent (Sornette Reference Sornette2004).

The work of Biferale et al. (Reference Biferale, Lanotte, Scatamacchia and Toschi2014) seeks to untangle this complex world of multiple scaling by using a lens of heavy particle trajectories, which span the Lagrangian tracer view to the Eulerian view for increasingly more massive particles: a very massive particle remains stationary and samples Eulerian velocities in time. In addition, sophisticated diagnostics are used by Biferale et al. (Reference Biferale, Lanotte, Scatamacchia and Toschi2014) including: high-order moments, probability density functions (emphasising the tails) and exit-time statistics which examine the random times taken for particles of a fixed initial separation to move apart by fixed amounts. In the latter case, heavy particles initially close together can have rare extremely long exit times to separate from ${\it\eta}$ to inertial range scales.

3. Future

The goal of the science of heavy particle motions, at least for particle loadings that do not feedback and influence the underlying turbulence, is to help develop models of heavy particle motions and separations. Studies such as those of Fung & Vassilicos (Reference Fung and Vassilicos1998), Thomson (Reference Thomson1990), Thomson & Devenish (Reference Thomson and Devenish2005) and Borgas & Yeung (Reference Borgas and Yeung2004) show high sensitivity to scaling effects requiring difficult parameterisations. Models for heavy particles are even less well developed and currently cannot incorporate intermittency and complex scaling parameterisations. It is worth noting the famous role of Kolmogorov in the development of stochastic models through the fundamental Chapman–Kolmogorov equation (Gardiner Reference Gardiner2009), which has played a pivotal role in the development of stochastic models for tracer advection in turbulence. The scaling of increments and changes, correlations and dependencies, are all important for model fidelity. Only through ongoing discovery about the scaling behaviour of particle trajectories will we be able to advance the modelling of these trajectories for routine geophysical prediction, say for clouds, pollution or clustering behaviour (Falkovich, Gawȩdzki & Vergassola Reference Falkovich, Gawȩdzki and Vergassola2001; Wang et al. Reference Wang, Franklin, Ayala and Grabowski2006; Thomson & Wilson Reference Thomson, Wilson, Lin, Brunner, Gerbig, Stohl, Luhar and Webley2013). The work of Biferale et al. (Reference Biferale, Lanotte, Scatamacchia and Toschi2014) is an important contribution to this ongoing task.