2. Compressible renewing flow

We consider the following velocity field $\boldsymbol{u}=(u_{x},u_{y})$ on a periodic square box ${\it\Omega}=[-L/2,L/2]^{2}$ :

(2.1) $$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}u_{x}=U\sqrt{2(1-C)}\cos (ky+{\it\phi}_{y})+U\sqrt{2C}\cos (kx+{\it\phi}_{x}),\quad \\ u_{y}=0,\quad \end{array}\right.\ 2nT\leqslant t<(2n+1)T\end{eqnarray}$$

and

(2.2) $$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}u_{x}=0,\quad \\ u_{y}=U\sqrt{2(1-C)}\cos (kx+{\it\phi}_{x})+U\sqrt{2C}\cos (ky+{\it\phi}_{y}),\quad \end{array}\right.\ (2n+1)T\leqslant t<2(n+1)T.\end{eqnarray}$$

Here $n\in \mathbb{N}$ , $k=2{\rm\pi}/L$ , $U=\sqrt{\langle u^{2}\rangle }$ is the root-mean-square (r.m.s.) velocity and $C=\langle (\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u})^{2}\rangle /\langle \Vert \boldsymbol{{\rm\nabla}}\boldsymbol{u}\Vert _{F}^{2}\rangle$ is the degree of compressibility of the flow. Here, $\Vert \cdot \Vert _{F}$ is the Frobenius norm and $\langle \cdot \rangle$ denotes a spatial average over the domain ${\it\Omega}$ . Note that $0\leqslant C\leqslant 1$ ; $C=0$ corresponds to an incompressible flow, and $C=1$ to a gradient flow. The angles ${\it\phi}_{x}$ and ${\it\phi}_{y}$ are independent random numbers uniformly distributed over $[0,2{\rm\pi}]$ and change randomly at each time period  $T$ .

The velocity field defined in (2.1) and (2.2) is a sequence of randomly translated sinusoidal profiles, each of which persists for a time $T$ ; the velocity is alternately oriented in the $x$ and $y$ directions. The renewing flow is in principle non-stationary, because the values of the velocity at two different times are either correlated or independent depending on whether or not the two times belong to the same frozen phase. However, it can be regarded as stationary for times much longer than $T$ (Zel’dovich et al. Reference Zel’dovich, Ruzmaikin, Molchanov and Sokoloff1984). The temporal statistics of the flow is characterised by the correlation function:

(2.3) $$\begin{eqnarray}F_{\mathscr{E}}(t)\equiv {\displaystyle \frac{\langle \boldsymbol{u}(\boldsymbol{x},s+t)\boldsymbol{\cdot }\boldsymbol{u}(\boldsymbol{x},s)\rangle _{\mathscr{E}}}{U^{2}}}=\left\{\begin{array}{@{}ll@{}}1-{\displaystyle \frac{t}{T}},\quad & t\leqslant T,\\ 0,\quad & t>T.\end{array}\right.\end{eqnarray}$$

Here $\langle \cdot \rangle _{\mathscr{E}}$ denotes a space–time Eulerian average: $\langle \cdot \rangle _{\mathscr{E}}\equiv T^{-1}L^{-2}\!\int _{0}^{T}\!\int _{{\it\Omega}}\cdot \,\text{d}s\,\text{d}\boldsymbol{x}$ . The correlation time of the flow is: $T_{\mathscr{E}}\equiv \int _{0}^{T}\!F_{\mathscr{E}}(t)\,\text{d}t=T/2$ . A dimensionless measure of the correlation time is given by the Kubo number $\mathit{Ku}\equiv TUk$ ; $\mathit{Ku}$ is proportional to the ratio of $T_{\mathscr{E}}$ and the eddy turnover time of the flow.

Figure 2.

Velocity profile of the renewing flow for (a)  $C=0$ , (b)  $C=1/4$ and (c)  $C=1/2$ . The thick solid grey lines (red online) are the portions of the stagnation lines where $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}<0$ .

The position of a tracer evolves according to the following equation:

(2.4) $$\begin{eqnarray}\dot{\boldsymbol{X}}(t)=\boldsymbol{u}(\boldsymbol{X}(t),t).\end{eqnarray}$$

Figure 1 shows the distribution of tracers in the weakly compressible regime. The transition from the regime of weak compressibility to that of strong compressibility occurs when the maximum Lyapunov exponent of the flow becomes negative. In this case, the flow is no longer chaotic and tracers are attracted to a point-like set. To study this transition, it is useful to consider the set of stagnation points of the flow, in which $\boldsymbol{u}=(0,0)$ . Tracers indeed tend to accumulate in the neighbourhood of these points. From (2.1) and (2.2), we deduce that the set of stagnation points of the renewing flow consists of the two lines

(2.5) $$\begin{eqnarray}y=\pm {\displaystyle \frac{1}{k}}\arccos \bigg(\sqrt{{\displaystyle \frac{C}{1-C}}}\cos (kx+{\it\phi}_{x})\bigg)-{\displaystyle \frac{{\it\phi}_{y}}{k}}+2{\rm\pi}m,\end{eqnarray}$$

if the velocity is given by (2.1), or

(2.6) $$\begin{eqnarray}y=\pm {\displaystyle \frac{1}{k}}\arccos \bigg(\sqrt{{\displaystyle \frac{1-C}{C}}}\cos (kx+{\it\phi}_{x})\bigg)-{\displaystyle \frac{{\it\phi}_{y}}{k}}+2{\rm\pi}m,\end{eqnarray}$$

if the velocity is given by (2.2). In (2.5) and (2.6), $m\in \mathbb{Z}$ is such that $(x,y)\in {\it\Omega}$ . The stagnation lines are shown in figure 2 for some representative values of the degree of compressibility. If $C=0$ , the flow consists of parallel periodic ‘channels’ of width $L/2$ . If $0<C<1/2$ , the stagnation lines form barriers that block the motion of tracers over a portion of the domain whose size increases as $C$ approaches $1/2$ ; the width of the periodic channels shrinks accordingly. Finally, if $1/2\leqslant C\leqslant 1$ , the stagnation lines divide the domain into regions that are not linked by any streamline. For these values of $C$ , there are no periodic trajectories, and if $\mathit{Ku}$ is sufficiently large, all tracers collapse onto the stagnation lines. We conclude that the transition from the regime of weak clustering to that of strong clustering must occur for $C\leqslant 1/2$ . However, the critical value of the degree of compressibility depends on $\mathit{Ku}$ .

3. Lagrangian versus Eulerian statistics

The properties of the flow introduced in § 2, namely the r.m.s. velocity, the degree of compressibility and the correlation function, are of an Eulerian nature. If the flow is compressible and has a non-zero correlation time, the Lagrangian counterparts of the aforementioned quantities may be different. Indeed, tracers are attracted towards the stagnation points and hence do not sample the phase space uniformly.

Figure 3.

(a) Ratio of the r.m.s. Lagrangian and Eulerian velocities as a function of the Kubo number $\mathit{Ku}$ and of the degree of compressibility $C$ of the flow. (b) Ratio of the Lagrangian and Eulerian correlation times of the velocity as a function of $C$ and $\mathit{Ku}$ .

We define the r.m.s. Lagrangian velocity as $u_{\mathscr{L}}\equiv \sqrt{\langle u^{2}(\boldsymbol{X }(s),s)\rangle _{\mathscr{L}}}$ , where $u=|\boldsymbol{u}|$ and $\langle \cdot \rangle _{\mathscr{L}}$ is a Lagrangian average over both the random trajectory $\boldsymbol{X}(s)$ and time. Figure 3(a) compares $u_{\mathscr{L}}^{2}$ and its Eulerian counterpart $U^{2}$ for different values of $\mathit{Ku}$ and $C$ . The plot includes 51 values of $C$ between 0 and $1/2$ , and 31 values of $\mathit{Ku}$ ranging from $10^{-1}$ to $10^{2}$ . For each pair $(\mathit{Ku},C)$ , we have computed $u_{\mathscr{L}}^{2}$ by solving (2.4) for $10^{2}$ tracers and for an integration time $t=2\times 10^{3}\,T$ (to integrate (2.4), we have used a fourth-order Runge–Kutta method). At $C=0$ , $u_{\mathscr{L}}$ is the same as $U$ for all values of $\mathit{Ku}$ , because tracers explore the phase space uniformly. Likewise, at $\mathit{Ku}=0$ the Eulerian and the Lagrangian statistics coincide independently of the value of $C$ . By contrast, for $C\neq 0$ and $\mathit{Ku}\neq 0$ , $u_{\mathscr{L}}<U$ because tracers are attracted towards the stagnation lines, where $u=0$ . The ability of the stagnation points to trap tracers strengthens with increasing $C$ and $\mathit{Ku}$ ; hence $u_{\mathscr{L}}$ eventually approaches zero.

Figure 4.

(a) Ratio of the Lagrangian and Eulerian correlation times of the velocity as a function of $\mathit{Ku}$ for $C=1/4$ and $C=1/2$ . The inset shows the Lagrangian correlation time rescaled by $(Uk)^{-1}$ as a function of $\mathit{Ku}$ for $C=1/2$ . (b) Lagrangian correlation function of the velocity for $\mathit{Ku}=0.1$ , $C=1/4$ , $\mathit{Ku}=10$ , $C=1/4$ and $\mathit{Ku}=10$ , $C=1/2$ .

The Lagrangian correlation function of the velocity is defined as

(3.1) $$\begin{eqnarray}F_{\mathscr{L}}(t)\equiv {\displaystyle \frac{\langle \boldsymbol{u}(\boldsymbol{X}(s+t),s+t)\boldsymbol{\cdot }\boldsymbol{u}(\boldsymbol{X}(s),s)\rangle _{\mathscr{L}}}{u_{\mathscr{L}}^{2}}}.\end{eqnarray}$$

The associated Lagrangian correlation time is $T_{\mathscr{L}}\equiv \int _{0}^{T}\!F_{\mathscr{L}}(t)\,\text{d}t$ . If the flow is incompressible ( $C=0$ ) or if it is decorrelated in time ( $\mathit{Ku}=0$ ), then $T_{\mathscr{L}}=T_{\mathscr{E}}$ (figure 3 b). If both $C$ and $\mathit{Ku}$ are non-zero, $T_{\mathscr{L}}$ is smaller than $T_{\mathscr{E}}$ , and the ratio $T_{\mathscr{L}}/T_{\mathscr{E}}$ decreases as $\mathit{Ku}$ or $C$ increase (figure 3 b). Again, this behaviour can be explained by considering that a tracer is attracted towards the stagnation points, where $u=0$ , and hence its velocity decorrelates from the velocity it had at the beginning of the period. As $\mathit{Ku}$ and $C$ increase, a larger fraction of tracers get close to the stagnation points, and therefore the decorrelation is faster. The inset of figure 4(a) also shows that, for $C$ close to $1/2$ , $T_{\mathscr{L}}$ is proportional to $T$ for small values of $\mathit{Ku}$ , whereas it saturates to a value proportional to the eddy turnover time for large values of $\mathit{Ku}$ . Indeed, after that time most of the tracers have reached a stagnation point, and their velocities have completely decorrelated.

Figure 4(b) shows that, for small values of $\mathit{Ku}$ , $F_{\mathscr{L}}(t)$ is the same as $F_{\mathscr{E}}(t)$ irrespective of the value of $C$ . However, at large $\mathit{Ku}$ , not only the Lagrangian correlation time of the velocity but also the functional form of $F_{\mathscr{L}}(t)$ varies with $C$ .

The degree of compressibility experienced by tracers also depends on $\mathit{Ku}$ and $C$ and differs from its Eulerian value if $\mathit{Ku}\neq 0$ and $C\neq 0$ . Let us define the Lagrangian degree of compressibility as $C_{\mathscr{L}}\equiv \langle (\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u})^{2}\rangle _{\mathscr{L}}/\langle \Vert \boldsymbol{{\rm\nabla}}\boldsymbol{u}\Vert _{F}^{2}\rangle _{\mathscr{L}}$ . Then, $C_{\mathscr{L}}>C$ for all non-zero values of $\mathit{Ku}$ and $C$ , because tracers spend more time in high-compressibility regions. For fixed $\mathit{Ku}$ , the increase in compressibility is an increasing function of $C$ , whereas for fixed $C$ it is maximum when $\mathit{Ku}$ is near to unity and vanishes in both the small- and the large- $\mathit{Ku}$ limits (figure 5 a). The non-monotonic behaviour of the Lagrangian compressibility as a function of $\mathit{Ku}$ is due to a peculiar feature of the model flow considered here. The stagnation lines (2.5) and (2.6) towards which the tracers are attracted do not coincide with the regions where the local compressibility of the flow is maximum, i.e. the lines $ky+{\it\phi}_{y}=m{\rm\pi}$ for $2nT\leqslant t<(2n+1)T$ and $kx+{\it\phi}_{x}=m{\rm\pi}$ for $(2n+1)T\leqslant t<2(n+1)T$ . Therefore, in the long-correlated limit the preferential sampling of the regions of strong compressibility is reduced.

4. Fractal clustering

The spatial distribution of tracers within a fluid can be characterised in terms of the Lyapunov dimension (e.g. Sommerer & Ott Reference Sommerer and Ott1993),

(4.1) $$\begin{eqnarray}D_{L}=N+\frac{\ \displaystyle \mathop{\sum }_{i=1}^{N}{\it\lambda}_{i}\ }{|{\it\lambda}_{N+1}|},\end{eqnarray}$$

where ${\it\lambda}_{1}\geqslant {\it\lambda}_{2}\geqslant \cdots \geqslant {\it\lambda}_{d}$ are the Lyapunov exponents of the flow and $N$ is the maximum integer such that $\sum _{i=1}^{N}{\it\lambda}_{i}\geqslant 0$ ( $d$ denotes the dimension of the flow). Three different regimes can be identified. If the flow is incompressible ( $\sum _{i=1}^{d}{\it\lambda}_{i}=0$ ), tracers spread out evenly within the fluid ( $D_{L}=d$ ). In the weakly compressible regime ( $\sum _{i=1}^{d}{\it\lambda}_{i}<0$ and ${\it\lambda}_{1}>0$ ), tracers cluster over a fractal set ( $1<D_{L}<d$ ). In the strongly compressible regime ( $\sum _{i=1}^{d}{\it\lambda}_{i}<0$ and ${\it\lambda}_{1}<0$ ), tracers are attracted to a point-like set ( $D_{L}=0$ ). The transition from the regime of weak compressibility to that of strong compressibility occurs when ${\it\lambda}_{1}$ changes sign.

Figure 5.

(a) Increase in compressibility multiplied by 100, $(C_{\mathscr{L}}-C)\times 100$ , as a function of $\mathit{Ku}$ and $C$ . (b) Maximum Lyapunov exponent of the compressible renewing flow as a function of $\mathit{Ku}$ and $C$ .

For the smooth $d$ -dimensional Kraichnan (Reference Kraichnan1968) flow, the Lyapunov exponents can be calculated exactly (we remind the reader that in the Kraichnan model the velocity field is Gaussian, delta-correlated in time, and statistically homogenous and isotropic). The Lyapunov exponents are ${\it\lambda}_{i}=\mathscr{D}\{d(d-2i+1)-2C[d+(d-2)i]\}$ , where $i=1,\dots ,d$ , $C$ is defined as in § 2, and $\mathscr{D}>0$ determines the amplitude of the fluctuations of the velocity gradient (Le Jan Reference Le Jan1984, Reference Le Jan1985). Thus, for $d=2$ , the Lyapunov dimension of the smooth Kraichnan model is $D_{L}^{0}=2/(1+2C)$ and the weak–strong clustering transition occurs at $C=1/2$ . In time-correlated flows, the prediction of the Kraichnan model is recovered in the small- $\mathit{Ku}$ limit (Boffetta et al. Reference Boffetta, Davoudi, Eckhardt and Schumacher2004; Gustavsson & Mehlig Reference Gustavsson and Mehlig2013b ).

In this section, we study how the weak–strong clustering transition depends on $\mathit{Ku}$ in the compressible renewing flow. To compute the Lyapunov exponents, we have used the method proposed by Benettin et al. (Reference Benettin, Galgani, Giorgilli and Strelcyn1980). We have set the integration time to $t=10^{6}$ for all values of $\mathit{Ku}$ and $C$ in order to ensure the convergence of the stretching rates to their asymptotic values.

The maximum Lyapunov exponent decreases with increasing $C$ (figure 5 b). Its behaviour as a function of $\mathit{Ku}$ is different in the weakly compressible and strongly compressible regimes. For small values of $C$ , ${\it\lambda}_{1}$ is maximum for $\mathit{Ku}$ near to unity, which signals an increase in chaoticity when the correlation time of the flow is comparable to the eddy turnover time. By contrast, for values of $C$ near to $1/2$ , ${\it\lambda}_{1}$ is minimum at $\mathit{Ku}\approx 1$ , in accordance with the fact that the Lagrangian compressibility is maximum for these values of the parameters (figure 5). We also note that if $\mathit{Ku}$ is near to unity, ${\it\lambda}_{1}$ becomes negative for $C<1/2$ ; hence the weak–strong clustering transition occurs at a lower degree of compressibility compared to the short-correlated case.

Figure 6.

(a) Lyapunov dimension $D_{L}$ as a function of $\mathit{Ku}$ and $C$ . The region in which $D_{L}=0$ is coloured in white in the plot. (b) Lyapunov dimension $D_{L}$ rescaled by its value at $\mathit{Ku}=0$ , $D_{L}^{0}$ , as a function of $\mathit{Ku}$ and $C$ .

Analogous conclusions can be reached by studying the behaviour of $D_{L}=1-{\it\lambda}_{1}/{\it\lambda}_{2}$ (figure 6). For fixed $\mathit{Ku}$ , an increase in the Eulerian compressibility yields an increased level of clustering. The behaviour as a function of $\mathit{Ku}$ is not monotonic and depends on the value of $C$ . When $\mathit{Ku}$ is near to unity, the level of clustering is minimum if  $C$ is small and is maximum if $C$ is near to $1/2$ . Furthermore, $D_{L}$ vanishes for values of $C$ smaller than the critical value of the short-correlated case. The most important deviations of $D_{L}$ from the $\mathit{Ku}=0$ prediction are observed for values of $\mathit{Ku}$ greater than unity (figure 6 b).