Transport, diffusion, and energy studies in the Arnold-Beltrami-Childress map

We study the transport and diffusion properties of passive inertial particles described by a six-dimensional dissipative bailout embedding map. The base map chosen for the study is the three-dimensional incompressible Arnold-Beltrami-Childress (ABC) map chosen as a representation of volume preserving flows. There are two distinct cases: the two-action and the one-action cases, depending on whether two or one of the parameters $(A,B,C)$ exceed 1. The embedded map dynamics is governed by two parameters ($\alpha ,\gamma$), which quantify the mass density ratio and dissipation, respectively. There are important differences between the aerosol $(\alpha <1)$ and the bubble $(\alpha >1)$ regimes. We have studied the diffusive behavior of the system and constructed the phase diagram in the parameter space by computing the diffusion exponents $\eta$. Three classes have been broadly classified—subdiffusive transport ($\eta <1$), normal diffusion ($\eta \approx 1$), and superdiffusion ($\eta >1$) with $\eta \approx 2$ referred to as the ballistic regime. Correlating the diffusive phase diagram with the phase diagram for dynamical regimes seen earlier, we find that the hyperchaotic bubble regime is largely correlated with normal and superdiffusive behavior. In contrast, in the aerosol regime, ballistic superdiffusion is seen in regions that largely show periodic dynamical behaviors, whereas subdiffusive behavior is seen in both periodic and chaotic regimes. The probability distributions of the diffusion exponents show power-law scaling for both aerosol and bubbles in the superdiffusive regimes. We further study the Poincáre recurrence times statistics of the system. Here, we find that recurrence time distributions show power law regimes due to the existence of partial barriers to transport in the phase space. Moreover, the plot of average particle kinetic energies versus the mass density ratio for the two-action case exhibits a devil's staircase–like structure for higher dissipation values. We explain these results and discuss their implications for realistic systems.

Figure 1

(a) The configuration space of the one-action ABC map for $(A,B,C)=(1.5,0.08,0.16)$ and (b) the configuration space of the two-action ABC map for $(A,B,C)=(2,1.5,0.08)$. We only indicate regular trajectories. The configuration space of the two-action does not have any invariant sheets leading to the global transport of trajectories. The plots in (c) and (d) show the $Y=0.5$ plane in the one and the two-action cases, respectively. In (c), the elliptic orbits indicate the existence of tubes and invariant surfaces are reflected in the form of lines spanning $\frac{X}{2\pi }=0$ to 1. These surfaces cover the $Y$ plane dividing the phase space in many isolated partitions. The configuration space of the two-action in (d) does not have any invariant sheets leading to global transport of trajectories. We show trajectories for 25 initial conditions randomly chosen from the uniform distribution of angles in the interval $[0,2\pi ]$. The thickness of the $Y$ plane used here is 0.01.

Figure 2

The phase diagram of the embedded one-action ABC map for parameter values $(A,B,C)=(1.5,0.08,0.16)$ [periodic orbits, blue (P); chaotic behavior, red (C); hyperchaotic regions, green (H)]. The $\alpha -\gamma$ space is covered by a $400\times 800$ mesh, each element is of size $0.005\times 0.005$. The phase diagram has been plotted using LE-s calculated for 25 000 iterates after discarding the first 5000 iterates as transients.

Figure 3

(a) The attractor in the $X-Y-Z$ configuration space of the embedded one-action map at $\alpha =1.85$ and $\gamma =3.2$. (b) The frequencies of return times for discrete hopping between the two parts of the attractor. The bars in black correspond to the part $R$ for which $\frac{Y}{2\pi }<0.5$ and those in green to the part $L$ for which $\frac{Y}{2\pi }>0.5$. The return times for the part $R$ are $(1,2,3,4)$, whereas that for the part $L$ are $(2,9,10,11,...,15,24,...,27,39)$. The computations are carried out for 50 000 iterates of which 5000 were discarded as transients.

Figure 4

The plots show the variance of the particle cloud evolving with time. (a) Superdiffusive behavior is observed at $(\alpha ,\gamma )=(0.5,3.7)$. (b) A case of normal diffusion is seen at $(\alpha ,\gamma )=(1.21,1.45)$. Trapping regimes show the plateauing of the variance—(c) with nonstationary states at $(\alpha ,\gamma )=(0.33,1.17)$ and (d) stationary states at $(\alpha ,\gamma )=(1.21,2.75)$. The stationary states are identified when $\eta <0.92$ together with fluctuations in ${\sigma }^2\left(t\right)$ taking values below $1\%$ during the last 2000 iterations.

Figure 5

The location of the attractor in the cover space for (a) nonstationary states at $(\alpha ,\gamma )=(0.33,1.17)$ and (b) stationary states at $(\alpha ,\gamma )=(1.21,2.75)$. The trajectories behave as though they are trapped in the neighborhood of these attractors and their dispersion grows sublinearly with time.

Figure 6

The phase diagrams showing the four diffusion regions in the embedding map for the two-action case in the $(\alpha ,\gamma )$ parameter space: the ballistic regime is marked by the color red with the label “B,” the superdiffusive regime is marked in yellow with the label “S,” normal diffusive regimes are marked in green with the label “N,” subdiffusion with stationary states is marked in blue with the label “T,” and subdiffusion with nonstationary states is shown by black with the label “${\mathrm{T}}^{*}$.” Panels (a) and (b) correspond to the aerosol regime and the bubble regime, respectively. The fraction of the phase space occupied by different diffusive regimes are indicated in (c) for the aerosol and (d) for the bubble regimes.

Figure 7

The distribution of diffusion exponents for the two-action case in (a) the aerosol and (b) the bubble regimes. Power-law scaling is observed in the the long tail: (a) In the aerosol regime in the window $1.05<\eta <1.98$, associated with larger peak at $\eta \approx 2$, has power-law scaling $1.429\pm 0.079$ for the cumulative distribution. (b) In the bubble regime, the long-tail associated with larger peak at $\eta =1$, within $1.02<\eta <1.98$, has a power-law scaling with exponent $1.630\pm 0.055$ for the reverse cumulative distribution.

Figure 8

The phase diagrams showing the four diffusion regions in the embedding map for the one-action case in the $(\alpha ,\gamma )$ parameter space: the ballistic regime is marked by the color red with the label “B,” the superdiffusive regime is marked in yellow with the label “S,” the normal diffusive regimes are marked in green with the label “N,” subdiffusion with stationary states is marked by the color blue with the label “T,” and subdiffusion with nonstationary states is shown in black with the label “${\mathrm{T}}^{*}$.” Panel (a) shows the aerosol regime, and panel (b) shows the bubble regime. The fraction of the phase space occupied by different diffusive regimes are indicated in (c) for the aerosol and (d) for the bubble regimes.

Figure 9

The distribution of diffusion exponents for the one-action case in (a) the aerosol and (b) the bubble regimes. Power-law scaling is observed in the long tail in diffusion exponent distribution. (a) In the aerosol regime in the window $1.01<\eta <1.92$, associated with larger peak at $\eta \approx 2$, has power-law scaling $3.381\pm 0.120$ for the cumulative distribution. (b) In the bubble regime, the long-tail associated with larger peak at $\eta =1$, within $1.02<\eta <1.82$, has a power-law scaling with exponent $2.032\pm 0.062$ for the reverse cumulative distribution.

Figure 10

The schematic illustration of the recurrence phenomena in an invariant set $\mathrm{\Gamma }$. A trajectory starting in a small subset $\xi$ is revisiting the subset in finite time. The diagram shows only the first three recurrences corresponding to recurrence times ${\tau }_1, {\tau }_2$, and ${\tau }_3$.

Figure 11

Recurrence time distributions in (a) the hyperchaotic regime $(\alpha ,\gamma )=(1.6,3.7)$ for the two-action case (b) the chaotic regime $(\alpha ,\gamma )=(0.9,1)$ for the two-action case, (c) the hyperchaotic regime $(\alpha ,\gamma )=(1.85,3.2)$ for the one-action case, and (d) the chaotic regime $(\alpha ,\gamma )=(0.85,2.0)$ for the one-action case. The plateaus indicate very long recurrences due to the trapping of particles in the tubular regions.

Figure 12

This figure demonstrates the stickiness of the trajectories around the tubes in the embedded two-action case in the hyperchaotic regime at $(\alpha ,\gamma )=(1.6,3.7)$. The plot shows the $X-Z$ space at the $Y=0.5$ plane. The darker regions correspond to the neighborhoods of the invariant tubes of the base ABC map. Trajectories wander inside the tubes briefly, resulting in plateaus in the corresponding recurrence time distribution. Regular structures and islands in red indicate the elliptical invariant regions in the base ABC map (two-action) [see Fig. 1].

Figure 13

The phase diagram of the embedded two-action ABC map for parameter values $(A,B,C)=(2.0,1.3,0.16)$ [periodic orbits, blue (P); chaotic behavior, red (C); hyperchaotic regions, green (H)]. The $\alpha -\gamma$ space is covered by a $400\times 800$ mesh, each of size $0.005\times 0.005$. A total of 25 000 iterates have been calculated in each case. The phase diagram has been plotted for 25 000 iterates discarding first 5000 iterates as transients.

Figure 14

(a) Self-similar staircase in the variation of $\langle E\left(\alpha \right)\rangle$ for the aerosol regime in the two-action case at $\gamma =3.5$. These jumps in $\langle E\left(\alpha \right)\rangle$ indicate that the particles move on trajectories confined inside different tubes with increasing $\alpha$. (b) The corresponding bubble regime shows entirely different behavior. For $\gamma =3.5$, the energies are small due to the fact that the trajectories get concentrated around the invariant tubes, whereas for $\gamma =0.5$, the trajectories transport over the entire phase space. The one-action case shows similar behavior. The energy values have been computed for $10000$ iterations and averaged over 200 particles.

Figure 15

All the plots show the $Y=0.5$ plane. The left panel: (a) regular regions in the ABC map; (c), (e), and (g) show the phase space of the embedded ABC map in the aerosol regime for $\alpha =\{0.25,0.60,0.75\}$, respectively, at $\gamma =3.5$. The trajectories go inside the regular regions. The right panel: (b) chaotic regime in the ABC map; (d), (f), and (h) show the phase space of the embedded ABC map in the bubble regime for $\alpha =\{1.25,1.50,1.80\}$, respectively at $\gamma =3.5$. The trajectories are expelled out of the regular regions.