2.1. Relevance of the ballistic phenomenology

The present formulation of the ballistic phenomenology addresses the question of the temporal evolution of the mean-square separation of an ensemble of pairs ( $D_{k}^{2}$ in ref (2.1) is the mean-square separation of this ensemble and the growth rates $S_{2}(D_{k})$ and $t_{k}(D_{K})$ have a statistical meaning only). It is important to stress that although this formulation is limited to the second moment of the statistics of pair separation (possible extensions to higher moments are briefly discussed in the last section), it has the benefit of not being just heuristic, but to stand a priori on solid statistical grounds, which will be shown later to give a posteriori an accurate and quantitative description of the time evolution of pairs mean-square separation. First, the short-term ballistic statistical relation for the mean-square separation in (1.1a ), $\langle (\boldsymbol{D}-\boldsymbol{D}_{0})^{2}\rangle =S_{2}(D_{0})t^{2}+O(t^{3})$ (which shows that to the leading order, pairs statistics is related to second-order velocity increments) is purely kinematic and hence unquestionable. Second, as already discussed in the introduction, this leading role of short-term ballistic events in turbulent dispersion statistics and its connection with second-order velocity statistics has been emphasized in several theoretical studies and it was accurately observed in high-resolution Lagrangian tracking experiments and numerical simulations, with robust scale dependent separation rates and characteristic time scale (see, for instance, Bourgoin et al. Reference Bourgoin, Ouellette, Xu, Berg and Bodenschatz2006; Ouellette et al. Reference Ouellette, Xu, Bourgoin and Bodenschatz2006; Bitane et al. Reference Bitane, Homann and Bec2012). This contrasts with the heuristic approach by Thalabard et al. (Reference Thalabard, Krstulovic and Bec2014), who applied a similar ballistic scenario to derive iteratively the temporal evolution of separation for individual pairs (rather than directly for the mean-square separation of an ensemble of pairs), with a growth rate and a ballistic time scale hence defined locally for each individual particle pair (although they still use the average dissipation ${\it\epsilon}$ to define the local ballistic time scale). They then address only in a second stage the statistical implications for an ensemble of such individual iterations (following a continuous-time-random-walk model, similar to the approach by Sokolov et al. Reference Sokolov, Klafter and Blumen2000). This approach has the benefit of allowing them to easily investigate the full statistics of separations built by this process (and not only the mean-square separation); however, it leads to the probably unrealistic result that the separation growth is governed at leading order by third-order velocity increments. Since their approach is heuristic, this has probably only minor implications for most of the qualitative results they obtain, but it may give a misleading picture of the relevant physical ingredients at play and it may also lead to misleading conclusions if it is applied to more subtle effects, as for instance the temporal asymmetry of dispersion, for which the actual role of third-order increments may be crucial (this will be discussed in § 4).

Even if the elementary scale-dependent ballistic process considered here for the means square separation does indeed rely on solid concepts and observations, several approximations still need to be discussed regarding the proposed iterative approach.

(i) To derive the iterative scheme (2.1) we have used the approximation $\langle D^{2}\rangle -\langle D_{0}^{2}\rangle =S_{2}(D_{0})t^{2}$ for the elementary short-term ballistic process in (1.1a ) (which rigorously applies to $\langle (\boldsymbol{D}-\boldsymbol{D}_{0})^{2}\rangle$ ). This is a common approximation, which is equivalent to neglecting the term $\langle \boldsymbol{D}_{0}\boldsymbol{\cdot }{\it\delta}_{\boldsymbol{r}}\boldsymbol{u}\rangle$ (Batchelor Reference Batchelor1950). It is known to hold for statistically homogeneous flows, although it was shown to possibly fail (at very short times) in experiments with non-homogeneous and anisotropic flows (Ouellette et al. Reference Ouellette, Xu, Bourgoin and Bodenschatz2006; Salazar & Collins Reference Salazar and Collins2009).

(ii) A more severe approximation concerns the fact that in the iterative process, the separation rate $S_{2}(D_{k})$ and the ballistic time of flight $t_{k}(D_{k})$ are both estimated assuming a unique value for the separation $D_{k}$ , hence neglecting the statistical fluctuations of pairs square separation, which in the real process explore a whole statistical distribution. In the present implementation of the iterative ballistic model, the separation rate $S_{2}(D_{k})$ and the typical time scale $t_{k}(D_{k})$ will be simply estimated at the scale corresponding to the mean-square separation $D_{k}^{2}$ . As it will be shown, this apparently naïve approximation still gives an accurate quantitative description of the dispersion process when compared to recent numerical simulations of pair dispersion in 3D turbulence. This approximation is also supported by the experimental evidence in Bourgoin et al. (Reference Bourgoin, Ouellette, Xu, Berg and Bodenschatz2006) that the short-term ballistic regime given by (1.1a ) is robustly observed even if the initial separation is coarsely binned across inertial scales, and when the growth rate is simply estimated from the mean separation in each bin (note that bins with width up to 100 % of the mean were considered in those experiments, with still an excellent confirmation of the elementary ballistic process (1.1a )). More quantitatively this approximation can be expected to hold as long as the width, ${\it\delta}D^{2}$ , of the distribution of particles square separation remains smaller than the mean-square separation $\langle D^{2}\rangle$ itself. If ${\it\delta}D^{2}$ is estimated as the standard deviation of particles square separation ( ${\it\delta}D^{2}=\sqrt{\langle (D^{2}-\langle D^{2}\rangle )^{2}\rangle }$ ), this condition would be related to the flatness of the separation distribution. With this respect, it would be interesting, in a further study, to investigate the ratio ${\it\delta}D^{2}/\langle D^{2}\rangle$ and its time evolution based on available experimental and numerical data for the statistical distribution of particles separation, in order to explore more quantitatively the possible range of validity of this approximation.

(iii) Finally, a concrete implementation of the iterative scheme (2.1) requires the expressions for the scale-dependent separation rate $S_{2}(D_{k})$ and the ballistic time of flight $t_{k}(D_{k})$ to be specified. This is discussed in the following subsection.

2.2. Ballistic separation rates and time scales

The separation rate of the elementary ballistic $k$ th step in (2.1) is simply given by the Eulerian second-order structure function at scale $D_{k}$ . In the inertial range of scales of fully developed turbulent flows, under local homogeneity and isotropy assumptions, the second-order structure function is known to follow K41 scaling (with negligible intermittent corrections), which in 3D turbulence can be written as

(2.2) $$\begin{eqnarray}S_{2}(r)={\textstyle \frac{11}{3}}C_{2}{\it\epsilon}^{2/3}r^{2/3},\end{eqnarray}$$

with $r=\sqrt{|\boldsymbol{r}|^{2}}$ and where $C_{2}$ is a universal constant with a well-known value of approximately 2.1; as $C_{2}$ is analytically related to the Kolmogorov constant $C_{K}\simeq C_{2}/4$ (Sreenivasan Reference Sreenivasan1995), characterizing the $-(5/3)$ spectrum of turbulent kinetic energy ( $E(k)=C_{K}{\it\epsilon}^{2/3}k^{-5/3}$ ), we shall equivalently refer to $C_{2}$ as the Kolmogorov constant itself. Note that $S_{2}$ here is the total structure function trace and not only the structure function for one specific component of velocity.

Concerning the typical time scale $t_{0}$ for which the elementary ballistic process in (1.1a ) is supposed to hold for a given initial separation $D_{0}$ , Batchelor proposed in his original article (Batchelor Reference Batchelor1950) that it should be taken as the eddy turnover time at the considered scale: $t_{0}\propto {\it\epsilon}^{-1/3}D_{0}^{2/3}$ . Experimental measurements of relative dispersion in highly turbulent flows with Reynolds number up to $R_{{\it\lambda}}\simeq 800$ (Bourgoin et al. Reference Bourgoin, Ouellette, Xu, Berg and Bodenschatz2006; Ouellette et al. Reference Ouellette, Xu, Bourgoin and Bodenschatz2006) have shown indeed that the time laps for which the ballistic Batchelor regime holds does scale as ${\it\epsilon}^{-1/3}D_{0}^{2/3}$ . Recent numerical simulations by Bitane et al. (Reference Bitane, Homann and Bec2012) have refined the estimation of the characteristic time scale for the ballistic regime. Their simulations show that the duration of the ballistic regime for an initial separation $D_{0}$ at inertial scales is given by $t_{0}(D_{0})\simeq S_{2}(D_{0})/2{\it\epsilon}=(11/6)C_{2}{\it\epsilon}^{-1/3}D_{0}^{2/3}$ . For times larger than this $t_{0}$ they observe a transition of the mean-square separation towards a cubic Richardson regime, as in (1.1b ).

The time scale $t_{0}(D_{0})\simeq S_{2}(D_{0})/2{\it\epsilon}$ observed in the simulations can be more rigourously discussed by considering the Taylor expansion leading to the short-term ballistic regime in (1.1a ) beyond the second order. The third-order expansion for $\langle (\boldsymbol{D}-\boldsymbol{D}_{0})^{2}\rangle$ (or equivalently for $\langle D^{2}\rangle -D_{0}^{2}$ under the approximation discussed previously) can be kinematically written as (Ouellette et al. Reference Ouellette, Xu, Bourgoin and Bodenschatz2006; Bitane et al. Reference Bitane, Homann and Bec2012)

(2.3) $$\begin{eqnarray}\langle \boldsymbol{D}^{2}\rangle -\langle \boldsymbol{D}_{0}^{2}\rangle =S_{2}(D_{0})t^{2}+\langle {\it\delta}_{\boldsymbol{ D}_{0}}\boldsymbol{a}\boldsymbol{\cdot }{\it\delta}_{\boldsymbol{D}_{0}}\boldsymbol{u}\rangle t^{3}+O(t^{4}),\end{eqnarray}$$

where ${\it\delta}_{\boldsymbol{D}_{0}}\boldsymbol{a}$ is the relative pair acceleration and ${\it\delta}_{\boldsymbol{D}_{0}}\boldsymbol{u}$ the relative pair velocity. The third-order coefficient is therefore given by the crossed velocity–acceleration structure function $\langle {\it\delta}_{\boldsymbol{D}_{0}}\boldsymbol{a}\boldsymbol{\cdot }{\it\delta}_{\boldsymbol{D}_{0}}\boldsymbol{u}\rangle$ which can be analytically derived from the Navier–Stokes equation (Mann, Ott & Andersen Reference Mann, Ott and Andersen1999; Hill Reference Hill2006) and shown to be equal to $-2{\it\epsilon}$ for 3D turbulence, under local stationarity, homogeneity and isotropy conditions (note that $\langle {\it\delta}_{\boldsymbol{D}_{0}}\boldsymbol{a}\boldsymbol{\cdot }{\it\delta}_{\boldsymbol{D}_{0}}\boldsymbol{u}\rangle =-2{\it\epsilon}$ is in particular independent of the probed scale $D_{0}$ ). The time $t_{0}(D_{0})=S_{2}(D_{0})/2{\it\epsilon}$ is therefore exactly the time for which the third-order term in (2.3) equals the second-order ballistic term (in absolute value). The role of the third-order term will be further discussed in § 4, however a brief discussion on the choice of the typical time scale for a purely ballistic (quadratic) dominant regime is still required at this stage. Numerical results by Bitane et al. (Reference Bitane, Homann and Bec2012) show indeed that for a given initial separation $D_{0}$ , the extent of the short-term ballistic regime seems to be accurately given by $S_{2}(D_{0})/2{\it\epsilon}$ , hence suggesting to use $t_{k}(D_{k})=S_{2}(D_{k})/2{\it\epsilon}$ as the typical ‘time of flight’ for each elementary ballistic iterative steps in (2.1). This observation raises however the issue that rigorously speaking, considering the negative sign of the third-order coefficient $\langle {\it\delta}_{\boldsymbol{D}_{k}}\boldsymbol{a}\boldsymbol{\cdot }{\it\delta}_{\boldsymbol{D}_{k}}\boldsymbol{u}\rangle =-2{\it\epsilon}$ , the ballistic and the third-order term in (2.3) cancel exactly for $t=t_{k}(D_{k})$ . For times $t\simeq t_{k}$ , negative third-order contributions annihilate the initial quadratic growth. In other words, a dominant ballistic mechanism, which requires terms of order three and higher to be negligible in the Taylor expansion (2.3) only holds for times $t\ll t_{k}=S_{2}(D_{k})/2{\it\epsilon}$ . For the iterative scheme considered here, this indicates that the actual temporal duration of each elementary ballistic step is necessarily much shorter than $t_{k}(D_{k})=S_{2}(D_{k})/2{\it\epsilon}$ . This points towards the necessity to introduce a ballistic persistence parameter ${\it\alpha}<1$ , defining the actual characteristic ballistic time of flight at each iteration step $k$ as $t_{k}^{\prime }(D_{k})={\it\alpha}t_{k}(D_{k})={\it\alpha}S_{2}(D_{k})/2{\it\epsilon}$ . The condition ${\it\alpha}<1$ (and more realistically ${\it\alpha}\ll 1$ ) ensures that the ballistic term is indeed still dominant at the end of each ballistic iteration of duration $t_{k}^{\prime }$ ; we shall refer to this condition as the ballistic approximation.

2.3. Explicit formulation of the iterative ballistic model

We can now explicitly write the complete iterative scheme of the iterative ballistic phenomenology as

(2.4) $$\begin{eqnarray}D_{k+1}^{2}=D_{k}^{2}+S_{2}(D_{k})t_{k}^{\prime 2}(D_{k})\quad \text{with}\left\{\begin{array}{@{}l@{}}S_{2}(D_{k})=C{\it\epsilon}^{2/3}D_{k}^{2/3},\quad \\ t_{k}^{\prime }(D_{k})={\it\alpha}t_{k}={\it\alpha}S_{2}(D_{k})/2{\it\epsilon},\quad \end{array}\right.\end{eqnarray}$$

where the only parameters in the model are $C$ (which is directly related to the Kolmogorov constant, and which we shall refer to as the Kolmogorov constant as well) and the persistence parameter ${\it\alpha}$ .

For 3D turbulence $C=C^{3D}=(11/3)C_{2}^{3D}$ , with $C_{2}^{3D}\simeq 2.1$ the 3D Kolmogorov constant (hence, $C^{3D}\simeq 7.7$ ). The model can also be applied to the inverse cascade regime of scales of 2D turbulence, for which $S_{2}$ also obeys K41 scalings, with $C=C^{2D}=(8/3)C_{2}^{2D}$ , where the 2D Kolmogorov constant is $C_{2}^{2D}\simeq 0.17C_{K}^{2D}\simeq 13.2$ (Lindborg Reference Lindborg1999), with $C_{K}^{2D}\sim 6$ (Boffetta & Ecke Reference Boffetta and Ecke2012) the spectral 2D Kolmogorov constant (hence, $C^{2D}\simeq 35.3$ ).

The persistent parameter is the only adjustable parameter of the model. Based on numerical observations by Bitane et al. (Reference Bitane, Homann and Bec2012), empirically suggesting ${\it\alpha}=1$ , we shall also briefly discuss in the sequel this particular case (where $t_{k}^{\prime }=t_{k}=S_{2}(D_{k})/2{\it\epsilon}$ ) as a situation ‘free of adjustable parameter’, although as pointed previously ${\it\alpha}=1$ is unlikely to be a relevant choice as it is incompatible with the ballistic approximation. Note that in the iterative scheme similarly implemented by Thalabard et al. (Reference Thalabard, Krstulovic and Bec2014), ${\it\alpha}=1$ is assumed (for each individual particle pair).

The next paragraphs show that the present scale-dependent iterative scheme (2.4) builds by itself a long-term super-diffusive $t^{3}$ regime ‘à la Richardson’ and gives a relevant description of the whole dispersion process, both for short-term and long-term regimes, as well as for the transition between one and the other.

2.4. From short-term ballistic steps to long-term Richardson dispersion

Substituting the explicit expressions for $S_{2}(D_{k})$ and $t_{k}^{\prime }(D_{k})$ in (2.4) into the iteration equation for $D_{k}^{2}$ , leads to a simple geometrical progression (and hence to an exponential growth of separation with the iteration number) both for the mean-square separation $D_{k}^{2}$ and the ballistic time scale $t_{k}^{\prime }$ :

(2.5a ) $$\begin{eqnarray}\displaystyle & D_{k}^{2}=\mathscr{A}^{k}D_{0}^{2}, & \displaystyle\end{eqnarray}$$

(2.5b ) $$\begin{eqnarray}\displaystyle & t_{k}^{\prime }=\mathscr{A}^{k/3}t_{0}^{\prime }. & \displaystyle\end{eqnarray}$$

(2.6) $$\begin{eqnarray}\mathscr{A}=1+\frac{{\it\alpha}^{2}C^{3}}{4}\end{eqnarray}$$

and

(2.7) $$\begin{eqnarray}t_{0}^{\prime }=\frac{{\it\alpha}C}{2}{\it\epsilon}^{-1/3}D_{0}^{2/3}.\end{eqnarray}$$

From there, it is trivial algebra to relate the mean-square separation at the iteration $k$ to the total iteration time $T_{k}=\sum _{j=0}^{k-1}t_{j}^{\prime }$ (with $T_{0}=0$ ):

(2.8) $$\begin{eqnarray}\left(\frac{D_{k}^{2}}{D_{0}^{2}}\right)^{1/3}=1+(\mathscr{A}^{1/3}-1)\frac{T_{k}}{t_{0}^{\prime }},\end{eqnarray}$$

which can be equivalently written as

(2.9) $$\begin{eqnarray}D_{k}^{2}=g{\it\epsilon}\left[T_{k}+\left(\frac{D_{0}^{2}}{g{\it\epsilon}}\right)^{1/3}\right]^{3}\end{eqnarray}$$

with

(2.10) $$\begin{eqnarray}g=\left[2\frac{\mathscr{A}^{1/3}-1}{{\it\alpha}C}\right]^{3}=\left[2\frac{\left(\displaystyle 1+\frac{{\it\alpha}^{2}C^{3}}{4}\right)^{1/3}-1}{{\it\alpha}C}\right]^{3}.\end{eqnarray}$$

Before confronting these results to experimental and numerical data, several points are worth being briefly discussed.

(i) We can note that relations (4.4a ) and (4.4b ) show that in the present model, particles dispersion eventually proceeds with a similar discrete scheme than in the GV04 model by Goto & Vassilicos (Reference Goto and Vassilicos2004) (although the GV04 model was originally derived only for 2D turbulence, with a phenomenology related to separation bursts by specific hyperbolic points in the flow). It would be interesting in future studies to push further the comparison between both models. For instance, the relation $\mathscr{A}=1+(({\it\alpha}^{2}C^{3})/4)$ in the present model, suggests that the two parameters in the GV04 model can actually be related to each other, via the Kolmogorov constant. On the other hand, the GV04 phenomenology may help giving a physical interpretation, at least for the 2D turbulence case (in terms of the density of stagnation points for instance) for the persistence parameter ${\it\alpha}$ introduced here on mainly kinematic considerations (to warrant the ballistic approximation validity). Similarly, the question of a possible phenomenological connection between both approaches in 3D turbulence may also be addressed.

(ii) Expression (2.9) shows that in the short term, dispersion may still exhibit an apparent Richardson-like cubic regime $D_{k}^{2}=g{\it\epsilon}(T_{k}-T_{origin})^{3}$ , as long as a negative virtual time origin is introduced: $T_{origin}=-(D_{0}^{2}/g{\it\epsilon})^{1/3}=-2t_{0}/Cg^{1/3}$ , which depends on the initial separation $D_{0}$ . This justifies and validates the necessity of introducing such a negative time shift in the experiments by Ott & Mann (Reference Ott and Mann2000), to devise a cubic regime from the available short time measurements. This also suggests that the experimental data by Ott & Mann can be fitted with only one free parameter (namely the Richardson constant $g$ ) as the time shift $T_{origin}$ only depends on $g$ and the known initial separation and energy dissipation rate. It would be interesting to revisit this experimental data with this vision.

(iii) Expression (2.9) has an asymptotic long-term behaviour $D_{k}^{2}=g{\it\epsilon}T_{k}^{3}$ , showing that the elementary scale-dependent ballistic steps, trivially build a cubic regime, independent on initial separation, à la Richardson. In addition, (2.10) relates the Richardson constant $g$ to the two parameters of the ballistic phenomenology (namely the Kolmogorov constant $C$ and the persistence parameter ${\it\alpha}$ ).

3.1. Determination of relevant values of model parameters

We shall focus in the sequel on the case of 3D turbulence, hence taking for the parameter $C$ the value $C^{3D}=(11/3)C_{2}^{3D}\simeq 7.7$ . Later, in § 4, we shall also briefly discuss some aspects of the inverse cascade regime of 2D turbulence, in which case $C^{2D}=(8/3)C_{2}^{2D}\simeq 35.3$ will be used for $C$ .

If we forget for a moment about the persistence parameter (hence, considering the typical ballistic time scale as being simply $t_{k}=S_{2}(D_{k})/2{\it\epsilon}$ , assuming ${\it\alpha}=1$ , as empirically suggested by Bitane et al. simulations and as done in Thalabard et al.), relation (2.10) then directly connects the Richardson constant $g$ to the Kolmogorov constant only. Figure 3(a) shows the predicted evolution of $g$ as a function of $C$ for the case ${\it\alpha}=1$ . Interestingly this figure shows that the predicted value for $g$ when $C=C^{3D}\simeq 7.7$ is then $g^{3D}\simeq 1.0$ . It is appealing that, although this value is slightly larger than the well accepted value $g^{3D}\simeq 0.5{-}0.6$ , it is still a reasonable estimate considering the simplicity of the model and the lack of adjustable parameter. However, figure 3(a) also shows that for ${\it\alpha}=1$ , $g$ tends asymptotically to the limit $g_{\infty }=2$ as the Kolmogorov constant increases. This clearly shows a physical limit of the case ${\it\alpha}=1$ (beyond the previous considerations requesting ${\it\alpha}<1$ for the ballistic approximation to hold), as numerical simulations of pair dispersion in the inverse cascade regime of 2D turbulence report values of $g^{2D}$ larger than two (Boffetta & Sokolov (Reference Boffetta and Sokolov2002b ) report for instance a value $g^{2D}\simeq 3.8$ while Faber & Vassilicos (Reference Faber and Vassilicos2009) report $g^{2D}\simeq 6.9$ ). This observation, raised by one of the anonymous referees of the first version of this article, also supports the necessity of considering the persistence parameter ${\it\alpha}$ .

Figure 3.

(a) Dependency of the Richardson constant $g$ on the Kolmogorov constant $C$ in the ballistic mode when the persistence parameter is fixed to ${\it\alpha}=1$ . (b) Prediction of the Richardson constant $g$ on the persistence parameter ${\it\alpha}$ , when the Kolmogorov constant is fixed to $C=C^{3D}=7.7$ .

Figure 4.

Growth of the mean-square separation between particles predicted as the iterative scale dependent ballistic process (defined by (2.4)) propagates starting from several initial separations $D_{0}$ in the inertial range of turbulence. The mean-square separation $\langle D^{2}\rangle$ is normalized by the square of the dissipative scale ${\it\eta}^{2}$ and time is normalized by the dissipative time scale ${\it\tau}_{{\it\eta}}$ . Turbulent energy dissipation rate and dissipative scales have been chosen to match experiments in Bourgoin et al. (Reference Bourgoin, Ouellette, Xu, Berg and Bodenschatz2006), although as shown in figure 5, the results are independent of this particular choice when the data is properly normalized. It can be seen in this plot that at long times, a dispersive regime ‘à la Richardson’, cubic in time and independent of initial separation, is naturally built by the iterative scale-dependent ballistic process.

To determine the optimal value of the persistence parameter ${\it\alpha}$ for the case of 3D turbulence, we therefore use relation (2.10) with the value for the Kolmogorov constant prescribed to $C=C^{3D}\simeq 7.7$ and we seek for the value of ${\it\alpha}$ for which the well-accepted value for the Richardson constant, $g=g^{3D}=0.55\pm 0.05$ , is retrieved. Figure 3(b) shows the dependency of $g$ predicted by the iterative ballistic phenomenology as a function of the persistence parameter ${\it\alpha}$ for $C=C^{3D}\simeq 7.7$ . We see that $g\simeq 0.55\pm 0.05$ for ${\it\alpha}\simeq 0.118\pm 0.007$ (a second solution exists for ${\it\alpha}\simeq 2.57>1$ which is out of the range of validity of the ballistic approximation, requiring ${\it\alpha}<1$ and which will therefore not be considered). The value ${\it\alpha}=0.12$ will therefore be used in the sequel for the persistence parameter in 3D turbulence. It can also be noted in figure 3(b) that the evolution of $g$ with ${\it\alpha}$ has a mild maximum around ${\it\alpha}\simeq 0.4$ , so that the dependency of $g$ on ${\it\alpha}$ is relatively weak in the range $0.1<{\it\alpha}<1$ , where the Richardson spans the relatively narrow range $0.4<g<1.3$ .

3.2. Practical implementation of the model

Let me now first illustrate the proposed ballistic cascade mechanism, in 3D turbulence, using the values $C^{3D}=7.7$ and ${\it\alpha}=0.12$ for parameters of the model, and with realistic numbers for the energy dissipation rate and initial separations compared to existing experiments and simulations. Concerning the energy dissipation, we use the value corresponding to the experiment by Bourgoin et al. (Reference Bourgoin, Ouellette, Xu, Berg and Bodenschatz2006) at $R_{{\it\lambda}}=815$ : ${\it\epsilon}\simeq 6.25~\text{m}^{2}~\text{s}^{-3}$ (giving a dissipation scale ${\it\eta}\simeq 23~{\rm\mu}\text{m}$ for a flow of water and a dissipation time scale ${\it\tau}_{{\it\eta}}\simeq 0.4~\text{ms}$ ). Several initial separations $D_{0}$ , spanning the range $[{\it\eta};100{\it\eta}]$ will be considered.

Figure 4 shows the mean-square separation obtained then from the iterative ballistic scheme given by (2.4). It can be seen that in the long term the iterative ballistic process eventually leads to a cubic super-diffusive regime ‘à la Richardson’, independent on the initial separation. The inset shows the same data but with the initial mean-square separation $D_{0}^{2}$ subtracted in order to better emphasize the initial ballistic regime at short times and the transition toward the Richardson regime. Note that time $t$ in these plots corresponds to the total discrete iteration time $T_{k}$ defined previously.

In figure 4 time is non-dimensionalized using the dissipative time scale ${\it\tau}_{{\it\eta}}$ while mean-square separations are non-dimensionalized using the square of dissipative scale  ${\it\eta}^{2}$ . However, the natural scales in the iterative ballistic process proposed here are $t_{0}$ (or $t_{0}^{\prime }={\it\alpha}t_{0}$ ) for the time scale and $D_{0}$ for the spatial scale. Relation (2.8) shows indeed that with such non-dimensionalized separations and time, the separation process only depends on the parameters $C$ and ${\it\alpha}$ of the model, via the constant $\mathscr{A}$ (2.6). For the sake of comparison with numerical result we use the same non-dimensionalization as Bitane et al. (Reference Bitane, Homann and Bec2012), based on $t_{0}^{2}S_{2}(D_{0})=(C^{3}D_{0}^{2})/2$ (rather than just $D_{0}^{2}$ , without loss of generality) for the mean-square separation and on $t_{0}$ for the time. With such a choice, relation (2.9) for the evolution of the mean-square separation simply becomes

(3.1) $$\begin{eqnarray}X_{k}^{2}=\frac{g}{2}\left[{\it\tau}_{k}-\frac{2}{Cg^{1/3}}\right]^{3}\end{eqnarray}$$

with $X_{k}^{2}=D_{k}^{2}/t_{0}^{2}S_{2}(D_{0})$ and ${\it\tau}_{k}=T_{k}/t_{0}$ .

The solid black line in figure 5 represents the same data as in figure 4, but with this new non-dimensionalization, which now collapses the mean-square separations for all initial separations into a single curve. I have also reported on the same figure the mean-square separations obtained for various initial separations in the simulations by Bitane et al. (Reference Bitane, Homann and Bec2012). The agreement of the iterative ballistic phenomenology compared to the numerics is almost perfect. The first interesting observation concerns the collapse of the numerical data with this normalization (as already noted by Bitane et al. Reference Bitane, Homann and Bec2012) which is perfectly reproduced by the present phenomenology. Besides, not only the global trend of the mean-square separation evolution is very well described by the model (both for short and long term regimes), but some subtler details are also well captured. For instance, as in the simulation, the transition between the early Batchelor regime and the Richardson regime is robustly found to occur around $t=t_{0}$ , even if the duration of the initial ballistic iteration is $t_{0}^{\prime }={\it\alpha}t_{0}$ (hence significantly shorter than $t_{0}$ in the present case). A mild slowing down of the ballistic separation for $t\lesssim t_{0}$ (prior to the transition towards the cubic regime), present in the simulations is also captured by the iterative ballistic model.

Figure 5.

Growth of the mean-square separation (with the initial separation subtracted) normalized by $t_{0}S_{2}(D_{0})$ as a function of time normalized by $t_{0}$ . The solid black line correspond to the exact same data as in figure 4 from the ballistic model (with $C=7.7$ and ${\it\alpha}=0.12$ ), which with the present normalization collapses onto a single curve, according to relation (3.1). The mid-grey line corresponds to the prediction of the ballistic model with $C=7.7$ and ${\it\alpha}=1$ . Coloured circles correspond to experimental results for the mean-square pair separation in Bourgoin et al. (Reference Bourgoin, Ouellette, Xu, Berg and Bodenschatz2006) for different initial separations. Other symbols correspond to the mean-square separation for different initial separations from the DNS by Bitane et al. (Reference Bitane, Homann and Bec2012). Finally the light grey line shows the prediction from Grossmann and Procaccia’s model (Grossmann & Procaccia Reference Grossmann and Procaccia1984; Grossmann Reference Grossmann1990). Two important aspects to be noted are: (i) the limited temporal range of experimental data, which does not allow to probe the transition between the initial ballistic regime and the late Richardson cubic regime; and (ii) the remarkable agreement between the ballistic model and the numerical simulations.

For comparison, figure 5 also shows the prediction of the ballistic model, with unity persistence parameter ${\it\alpha}=1$ , hence taking $t_{k}^{\prime }=t_{k}=S_{2}(D_{k})/2{\it\epsilon}$ as the characteristic time of the fundamental ballistic process (as originally proposed by Bitane et al. (Reference Bitane, Homann and Bec2012) and in the line of the implementation of the ballistic phenomenology by Thalabard et al.). It is interesting to note that although the long-term separation is then slightly over-estimated (as a result of the Richardson constant being $g=1.0$ in the model with ${\it\alpha}=1$ as discussed previously, instead of $g\simeq 0.55$ ), the global picture for the mean-square separation is still relatively well captured, with no other additional adjustable parameter in this case than the Kolmogorov constant. Overall, the global picture of the temporal growth of the mean-square separation in the present ballistic phenomenology is not very sensitive to the specific value of the persistence parameter (as long as we remain in the range $0.1<{\it\alpha}<1.0$ ): when the persistence parameter is varied in the range $0.1<{\it\alpha}<1.0$ the Batchelor to Richardson transition is always observed to occur for times around $t_{0}$ , with a Richardson constant restricted to the range $0.4<g<1.3$ (see figure 3 b), reasonable compared with the well-accepted value of 0.55. The choice ${\it\alpha}\simeq 0.12$ , remains however optimal in terms of quantitative comparison with available numerical data for the mean-square separation.

Figure 5 also reports the prediction for the mean-square separation of particle pairs from an alternative model of turbulent relative dispersion, proposed by Grossmann & Procaccia in 1984 (Grossmann & Procaccia Reference Grossmann and Procaccia1984; Grossmann Reference Grossmann1990). This model relies on a mean field approach of Navier–Stokes equation and a phenomenological closure assumption to predict the evolution of the mean-square separation of pairs of particles (with initial separation within inertial scales of turbulence) as follows:

(3.2) $$\begin{eqnarray}\langle (\boldsymbol{D}-\boldsymbol{D}_{0})^{2}\rangle =(D_{0}^{4/3}+{\textstyle \frac{2}{3}}C{\it\epsilon}^{2/3}t^{2})^{3/2}-D_{0}^{2}\end{eqnarray}$$

whose long time approximation asymptotically reaches a cubic Richardson regime $\langle (\boldsymbol{D}-\boldsymbol{D}_{0})^{2}\rangle =g^{\prime }{\it\epsilon}t^{3}$ with $g^{\prime }=((2/3)C)^{3/2}$ . Interestingly this model also proposes a direct connection between the Richardson and the Kolmogorov constant $C$ . However, although the prediction given by (3.2) and shown in figure 5 captures correctly the initial ballistic separation, it significantly anticipates the transition towards the Richardson regime which occurs much earlier than in the DNS or the iterative ballistic phenomenology proposed, what results in a significant over-estimation of the actual mean-square separation at long times. This sooner transition is related to the over-estimation of the Richardson constant in Grossmann & Procaccia’s model: $g^{\prime }=((2/3)C)^{3/2}\simeq 12$ (with $C=7.7$ ), what is much larger than the well-accepted value $g\simeq 0.55$ .

Finally, we have also superimposed in figure 5 the mean-square separation of pairs measured in high-resolution particle tracking experiments by Bourgoin et al. (Reference Bourgoin, Ouellette, Xu, Berg and Bodenschatz2006). In those experiments, only the Batchelor ballistic regime was reported, while no hint of Richardson regime was detected. Figure 5 emphasizes a possible reason for the failure in experiments to observe the Richardson regime: the longest experimental tracks did not exceed a few tenths of $t_{0}$ while the separation needs to be tracked for at least a few $t_{0}$ to reasonably detect the transition toward the cubic regime. A simple possible strategy to improve the chances to observe the cubic regime in experiments would simply consist in better controlling the injection of particle pairs in order to achieve sufficiently small initial separations, hence reducing the time $t_{0}$ required for the transition to occur within experimentally accessible tracking time.

4.1. A ballistic cascade phenomenology

Time irreversibility can be simply introduced by pushing up to third order the Taylor expansion leading to the elementary ballistic process (1.1a ), as already presented in (2.3). When pushed to third order, the iterative scheme (2.4), giving the growth of pair separation at iteration between the $k$ th and the $(k+1)$ th iteration then becomes

(4.1) $$\begin{eqnarray}D_{k+1}^{2}=D_{k}^{2}+S_{2}(D_{k})t_{k}^{\prime 2}+S_{au}(D_{k})t_{k}^{\prime 3},\end{eqnarray}$$

where $S_{au}(\boldsymbol{r})=\langle {\it\delta}_{\boldsymbol{r}}\boldsymbol{a}\boldsymbol{\cdot }{\it\delta}_{\boldsymbol{r}}\boldsymbol{u}\rangle$ is the crossed velocity–acceleration structure function.

The relevance of considering the cubic term in the iterative process is also supported by the fact that in the practical implementation of the iterative ballistic phenomenology discussed in the previous section, the optimal value for the persistence parameter ${\it\alpha}$ was found to be of the order of 0.1, while as discussed in § 2.2 neglecting completely the third-order term would require ${\it\alpha}\ll 1$ . With ${\it\alpha}\simeq 0.1$ , the cubic term can still be expected to contribute of the order of 10 % to the quadratic separation at each iteration step. It is therefore reasonable to consider a possible corrective contribution of third-order effects in the ballistic phenomenology.

From a physical point of view, the third-order term has a clear energetic interpretation. As discussed in § 2.2, the crossed velocity–acceleration structure function $\langle {\it\delta}_{\boldsymbol{r}}\boldsymbol{a}\boldsymbol{\cdot }{\it\delta}_{\boldsymbol{r}}\boldsymbol{u}\rangle$ in 3D turbulence is predicted to be $-2{\it\epsilon}$ (at all inertial scales), and hence solely related to the energy dissipation rate ${\it\epsilon}$ . If we were discussing single-particle issues (instead of particle pairs) this would be simply understood as a dissipative correction of the average ballistic motion of the particle, as $\langle \boldsymbol{a}\boldsymbol{\cdot }\boldsymbol{u}\rangle =\langle \text{d}u^{2}/\text{d}t\rangle /2$ is indeed the average dissipation of particle kinetic energy along its Lagrangian path. When the relative motion of particles in a pair is considered, the physical meaning of the crossed velocity–acceleration structure function $S_{au}=\langle {\it\delta}_{\boldsymbol{r}}\boldsymbol{a}\boldsymbol{\cdot }{\it\delta}_{\boldsymbol{r}}\boldsymbol{u}\rangle$ is however somehow subtler. Here $S_{au}$ can indeed be analytically related to the third-order velocity structure function (and, hence, to the energy cascade across scales) directly from Navier–Stokes equation, such that under local stationarity and homogeneity assumptions (see for instance Mann et al. Reference Mann, Ott and Andersen1999; Hill Reference Hill2006):

(4.2) $$\begin{eqnarray}2\langle {\it\delta}_{\boldsymbol{r}}\boldsymbol{a}\boldsymbol{\cdot }{\it\delta}_{\boldsymbol{r}}\boldsymbol{u}\rangle =\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\langle {\it\delta}_{\boldsymbol{r}}\boldsymbol{u}{\it\delta}_{\boldsymbol{r}}\boldsymbol{u}\boldsymbol{\cdot }{\it\delta}_{\boldsymbol{r}}\boldsymbol{u}\rangle .\end{eqnarray}$$

In 3D turbulence, $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\langle {\it\delta}_{\boldsymbol{r}}\boldsymbol{u}{\it\delta}_{\boldsymbol{r}}\boldsymbol{u}\boldsymbol{\cdot }{\it\delta}_{\boldsymbol{r}}\boldsymbol{u}\rangle =-4{\it\epsilon}$ (what is an exact relation and an alternative version of the Karman–Howarth–Monin relation under local homogeneity and isotropy assumptions (Frisch Reference Frisch1995)), what retrieves the relation $S_{au}^{3D}=-2{\it\epsilon}$ mentioned previously for the case of 3D turbulence. The negative sign in these relations reflects the fact that in 3D turbulence the energy cascade at inertial scales is a direct cascade (energy flows from large to small scales).

Conversely, in the inverse cascade of 2D turbulence $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\langle {\it\delta}_{\boldsymbol{r}}\boldsymbol{u}{\it\delta}_{\boldsymbol{r}}\boldsymbol{u}\boldsymbol{\cdot }{\it\delta}_{\boldsymbol{r}}\boldsymbol{u}\rangle =+4{\it\epsilon}$ (Lindborg Reference Lindborg1999), what leads then to $S_{au}^{2D}=+2{\it\epsilon}$ . The positive sign referring now to the inverse nature of the energy cascade.

The crossed velocity–acceleration structure function therefore carries the signature of the energy flux across scales. In the present context of pair dispersion, the cubic term in (4.1) should therefore be seen as the Lagrangian signature of the energy flux due to the relative motion (relative velocity and relative acceleration) of particles separating from a given scale to larger scales. The negative sign in the relation $S_{au}^{3D}=-2{\it\epsilon}$ in 3D can be interpreted as the fact that in the forward dispersion process, as particles separate (from small to large scales), they climb the energy cascade ‘upstream’, against the energy flux (which flows from large to small scales in the direct 3D cascade), while in the backward case separation pair separation climbs the cascade downstream, with the energy flux. The scenario is reversed in the inverse cascade of 2D turbulence.

The time asymmetry introduced by the third-order correction to the ballistic process in the extended iterative phenomenology therefore accounts, in the Lagrangian framework of pair dispersion, for the scale asymmetry of energy flux in the turbulent cascade. We shall therefore refer to this extension as the ballistic cascade phenomenology.

We can note that when the relation $S_{au}^{3D}=-2{\it\epsilon}$ (for 3D turbulence) is reported in the elementary short-term separation process given by (2.3) (or by (4.1)), we find that for short times, the short-term separation is naturally dominated by the ballistic (quadratic) contribution, with a third-order temporal asymmetry such that the difference between short-term forward and backward separation is $\langle D^{2}\rangle (-t)-\langle D^{2}\rangle (t)=4{\it\epsilon}t^{3}$ (in 3D turbulence). This corresponds to the short-term cubic in time asymmetry of relative dispersion recently investigated by Jucha et al. (Reference Jucha, Xu, Pumir and Bodenschatz2014). We will show here (in the next subsection) that the iterative propagation of this short-term asymmetry also builds the long-term asymmetry, in quantitative agreement with previous studies for the Richardson regime asymmetry. Such a connection between the short-term and long-term asymmetry still remained to be established, as pointed out by Jucha et al. (Reference Jucha, Xu, Pumir and Bodenschatz2014).

4.2. Explicit formulation of the ballistic cascade phenomenology

The extended iterative phenomenology can be explicitly implemented as follows

(4.3) $$\begin{eqnarray}D_{k+1}^{2}=D_{k}^{2}+S_{2}(D_{k})t_{k}^{\prime 2}+S_{au}t_{k}^{\prime 3}\quad \text{with}\left\{\begin{array}{@{}l@{}}S_{2}(D_{k})=C{\it\epsilon}^{2/3}D_{k}^{2/3},\quad \\ S_{au}=C_{au}2{\it\epsilon},\quad \\ t_{k}^{\prime }(D_{k})={\it\alpha}t_{k}={\it\alpha}S_{2}(D_{k})/2{\it\epsilon},\quad \end{array}\right.\end{eqnarray}$$

where the asymmetry coefficient $C_{au}=\pm 1$ depending on the inverse or direct nature of the energy cascade ( $C_{au}^{3D}=-1$ for the 3D cascade in 3D turbulence and $C_{au}^{2D}=+1$ for the inverse cascade in 2D turbulence). We also recall the values for the Kolmogorov constant (discussed previously in § 3.2): $C^{3D}=7.7$ and $C^{2D}=35.3$ .

Time asymmetry between forward and backward dispersion can then be simply considered by changing $t_{k}^{\prime }\rightarrow -t_{k}^{\prime }$ (what only affects the third-order term in (4.3)) or equivalently by reversing the sign of $C_{au}$ when propagating the iterative scheme. Table 1 specifies the values to be used for the asymmetry parameter $C_{au}$ for the different cases.

Table 1.

Values to be used for the asymmetry parameter $C_{au}$ for the different cases.

4.3. From short-term asymmetric ballistic dispersion to long-term asymmetric Richardson dispersion

As for the case of the purely ballistic model discussed in § 2.4, the new iterative scheme can be explicitly solved by substituting the expressions for $S_{2}$ , $S_{au}$ and $t_{k}^{\prime }$ into the iterative relation in (4.1). Since $S_{au}$ does not depend on the separation $D_{k}$ , the solution is straightforward, and leads to a geometric progression for $D_{k}^{2}$ and $t_{k}^{\prime }$ , analogue to the previous relations (4.4a ) and (4.4b ):

(4.4a ) $$\begin{eqnarray}\displaystyle & D_{k}^{2}=\mathscr{A}^{\prime k}D_{0}^{2}, & \displaystyle\end{eqnarray}$$

(4.4b ) $$\begin{eqnarray}\displaystyle & t_{k}^{\prime }=\mathscr{A}^{\prime k/3}t_{0}^{\prime }. & \displaystyle\end{eqnarray}$$

(4.5) $$\begin{eqnarray}\mathscr{A}^{\prime }=1+\frac{{\it\alpha}^{2}C^{3}}{4}(1+C_{au}{\it\alpha}).\end{eqnarray}$$

Note that setting $C_{au}=0$ (hence, neglecting the third-order term) naturally retrieves $\mathscr{A}^{\prime }=\mathscr{A}=1+({\it\alpha}^{2}C^{3})/4$ (as given by (2.6) for the purely ballistic model).

The temporal evolution of the mean-square separation can therefore be written with an analogous formulation as (2.9) and (2.10)

(4.6) $$\begin{eqnarray}D_{k}^{2}=g{\it\epsilon}\left[T_{k}+\left(\frac{D_{0}^{2}}{g{\it\epsilon}}\right)^{1/3}\right]^{3}\end{eqnarray}$$

with

(4.7) $$\begin{eqnarray}g=\left[2\frac{\mathscr{A}^{\prime 1/3}-1}{{\it\alpha}C}\right]^{3}=\left[2\frac{\left(\displaystyle 1+\frac{{\it\alpha}^{2}C^{3}}{4}\left(1+C_{au}{\it\alpha}\right)\right)^{1/3}-1}{{\it\alpha}C}\right]^{3}.\end{eqnarray}$$

As a result, the third-order corrected model behaves exactly as the purely ballistic model, except for the relation between the Richardson constant and the model parameters which now includes an ${\it\alpha}^{3}$ corrective term in the expression of the constant $\mathscr{A}^{\prime }$ (4.5), associated with the asymmetry coefficient $C_{au}$ .

4.4. Forward versus backward dispersion

We consider here the practical implementation of the ballistic cascade phenomenology, exploring the consequences in terms of temporal asymmetry of the dispersion process in the direct cascade of 3D turbulence and the inverse cascade regime of 2D turbulence.

4.4.1. The case of direct cascade in 3D turbulence

Figure 6 represents the dependency on the persistence parameter ${\it\alpha}$ of the Richardson constant as predicted by the ballistic cascade phenomenology given by relation (4.7) for the case of 3D turbulence ( $C=7.7$ ) in the forward ( $C_{au}=-1$ ) and backward ( $C_{au}=+1$ ) situations ( $g_{fwd}^{3D}$ and $g_{bwd}^{3D}$ are respectively the 3D forward and 3D backward Richardson constants). The figure also shows the ratio $g_{bwd}^{3D}/g_{fwd}^{3D}$ as a function of ${\it\alpha}$ . This ratio is found to be always larger than unity, what shows that for a given value of the persistence parameter, the backward dispersion in 3D turbulence always operates faster than forward dispersion, in qualitative agreement with experiments and simulations by Berg et al. (Reference Berg, Lüthi, Mann and Ott2006). In the context of the present phenomenology, the faster separation in the backward case compared to the forward case is directly related to the positiveness and negativeness of $C_{au}$ in each respective situation. The positive value of $C_{au}$ in the backward case, results in an enhancement of the separation due to the positive third-order corrective term in (4.3) which accelerates in fine the long-term cubic separation compared with the case of the purely ballistic iterative phenomenology. In contrast, the negative value of $C_{au}$ for the forward case results in the reduction of the separation due to the negative third-order corrective term, which decelerates the long-term cubic separation compared with the purely ballistic iterative phenomenology.

Figure 6.

Dependency of the Richardson constant predicted by the ballistic cascade model (with third-order term correction (4.3)) for the forward and backward dispersion problem in direct cascade 3D turbulence, as a function of the persistence parameter. The plot shows the forward Richardson constant $g_{fwd}^{3D}$ (full line, blue online), the backward Richardson constant $g_{bwd}^{3D}$ (dot–dashed line, red online) and the ratio $g_{bwd}^{3D}/g_{fwd}^{3D}$ (dashed line, yellow online). The vertical dot line emphasizes the optimal value for the persistence parameter, determined as the smallest value of this parameter for which the forward Richardson constant predicted by the model matches the values in the literature ( $g_{fwd}^{3D}\simeq 0.55$ ). The horizontal lines emphasize the corresponding values for the forward and backward Richardson constants, as well as for the ratio $g_{bwd}^{3D}/g_{fwd}^{3D}$ (these values are reported on the right-hand side of the plots).

The comparison with experiments and simulations by Berg et al. (Reference Berg, Lüthi, Mann and Ott2006) can be pushed to a quantitative level by considering the optimal value of the persistence parameter, for which $g_{fwd}^{3D}=0.55$ (the well-accepted value for the forward dispersion problem). It can be seen in figure 6 that two values of ${\it\alpha}$ satisfy this condition, one around 0.15, the other around 0.42. There is no a priori obvious choice to select one or the other of these values, except that the validity of the short term Taylor expansion for the mean-square separation (which is the starting point of the present phenomenology) is expected to be more robust for shortest times, what tends to prefer the smallest compatible value for the persistence parameter. We will therefore take ${\it\alpha}\simeq 0.15$ as the optimal value for the persistence parameter, compatible with the well-accepted value $g_{fwd}^{3D}=0.55$ . Figure 6 then shows that the corresponding value for the backward Richardson constant is $g_{bwd}^{3D}\simeq 1.04$ , leading to a ratio $g_{bwd}^{3D}/g_{fwd}^{3D}\simeq 1.9$ , in good quantitative agreement with the experiments and simulations by Berg et al. (Reference Berg, Lüthi, Mann and Ott2006) and with the more recent simulations by Bragg et al. (Reference Bragg, Ireland and Collins2014), who all find a ratio $g_{bwd}^{3D}/g_{fwd}^{3D}\simeq 2$ .

Figure 7.

Same as figure 6, but for the case of 2D inverse cascade. The optimal value for the persistence parameter, is here determined as the smallest value for which the forward Richardson constant predicted by the model matches the value in Faber & Vassilicos (Reference Faber and Vassilicos2009) ( $g_{fwd}^{2D}\simeq 6.9$ ).