Dynamics of impurities in a three-dimensional volume-preserving map

We study the dynamics of inertial particles in three-dimensional incompressible maps, as representations of volume-preserving flows. The impurity dynamics has been modeled, in the Lagrangian framework, by a six-dimensional dissipative bailout embedding map. The fluid-parcel dynamics of the base map is embedded in the particle dynamics governed by the map. The base map considered for the present study is the Arnold-Beltrami-Childress (ABC) map. We consider the behavior of the system both in the aerosol regime, where the density of the particle is larger than that of the base flow, as well as the bubble regime, where the particle density is less than that of the base flow. The phase spaces in both the regimes show rich and complex dynamics with three types of dynamical behaviors—chaotic structures, regular orbits, and hyperchaotic regions. In the one-action case, the aerosol regime is found to have periodic attractors for certain values of the dissipation and inertia parameters. For the aerosol regime of the two-action ABC map, an attractor merging and widening crisis is identified using the bifurcation diagram and the spectrum of Lyapunov exponents. After the crisis an attractor with two parts is seen, and trajectories hop between these parts with period 2. The bubble regime of the embedded map shows strong hyperchaotic regions as well as crisis induced intermittency with characteristic times between bursts that scale as a power law behavior as a function of the dissipation parameter. Furthermore, we observe a riddled basin of attraction and unstable dimension variability in the phase space in the bubble regime. The bubble regime in the one-action case shows similar behavior. This study of a simple model of impurity dynamics may shed light upon the transport properties of passive scalars in three-dimensional flows. We also compare our results with those seen earlier in two-dimensional flows.

Figure 1

The $x\text{−}y\text{−}z$ phase space of the ABC map in (a) the one-action case for the parameter values $(A,B,C)=(1.5,0.08,0.16)$ and (b) the two-action case for the parameter values $(A,B,C)=(2.0,1.5,0.08)$.

Figure 2

The $x\text{−}y\text{−}z$ phase space of the embedded two-action ABC map in (a) the aerosol regime for the parameter values $(A,B,C)=(2.0,1.5,0.08)$ and $(\alpha ,\gamma )=(0.7,3.6)$ and (b) the bubble regime for the parameter values $(A,B,C)=(2.0,1.5,0.08)$ and $(\alpha ,\gamma )=(2.0,2.2).$

Figure 3

The bifurcation diagram for the embedded two-action ABC map in the aerosol regime for the parameter values $(A,B,C)=(2.0,1.5,0.16)$ and $\alpha =0.7$.

Figure 4

The precrisis scenario for the embedded two-action ABC map in the aerosol regime for the parameter values $(A,B,C)=(2.0,1.5,0.16)$ and $(\alpha ,\gamma )=(0.7,2.2)$ for (a) the attractor in $x\text{−}y\text{−}z$ phase space (500 transients discarded) and (b) the time series.

Figure 5

The postcrisis scenario for the embedded ABC map in the aerosol regime for the parameter values $(A,B,C)=(2.0,1.5,0.16)$ and $(\alpha ,\gamma )=(0.7,2.3)$ for (a) the attractor in $x\text{−}y\text{−}z$ phase space (500 transients discarded) and (b) the time series.

Figure 6

The Lyapunov exponents for the embedded two-action ABC map in the aerosol regime for the parameter values $(A,B,C)=(2.0,1.5,0.16)$ and $\alpha =0.7$ for (a) the largest Lyapunov exponent and (b) the full spectrum $\left({\lambda }_1>{\lambda }_2>{\lambda }_3>{\lambda }_4>{\lambda }_5>{\lambda }_6\right)$.

Figure 7

Plot of the fraction of positive time-50 Lyapunov exponents vs time, in the aerosol regime for the parameter values $(A,B,C)=(2.0,1.5,0.16)$ and $(\alpha ,\gamma )=(0.7,2.2)$. UDV is clearly seen.

Figure 8

The hopping in the aerosol regime for the parameter values $(A,B,C)=(2.0,1.3,0.16)$ and $(\alpha ,\gamma )=(0.7,2.89)$.

Figure 9

Statistics of the transients before the hopping in the aerosol regime for the parameter values $(A,B,C)=(2.0,1.3,0.16)$ and $(\alpha ,\gamma )=(0.7,2.89)$ for (a) the probability distribution and (b) the corresponding log-log plot of the reverse cumulative distribution [22].

Figure 10

The bifurcation diagram for the embedded ABC map in the bubble regime for the parameter values $(A,B,C)=(2.0,1.3,0.05)$ and $\alpha =1.1$.

Figure 11

The precrisis scenario for the embedded two-action ABC map in the bubble regime at the parameter values $(A,B,C)=(2.0,1.3,0.05)$ and $(\alpha ,\gamma )=(1.1,.405)$ for (a) the attractor in $x\text{−}y\text{−}z$ phase space (500 transients discarded) and (b) the time series.

Figure 12

The postcrisis scenario for the embedded two-action ABC map in the bubble regime for the parameter values $(A,B,C)=(2.0,1.3,0.05)$ and $(\alpha ,\gamma )=(1.1,.425)$ for (a) the attractor in $x\text{−}y\text{−}z$ phase space (500 transients discarded) and (b) the time series.

Figure 13

Plots of phenomena observed in the bubble regime at the parameter values $(A,B,C)=(2.0,1.3,0.05)$ and $\alpha =1.1$ for (a) the power law for crisis induced intermittency (the number of initial conditions equals 500, and $\gamma$ varies from 0.26 to ${\gamma }_c=0.415$) and (b) the Lyapunov spectrum $\left({\lambda }_1>{\lambda }_2>{\lambda }_3>{\lambda }_4>{\lambda }_5>{\lambda }_6\right)$.

Figure 14

Plot of the number of positive time-50 Lyapunov exponents as a fraction of the total number, vs time, in the bubble regime for the parameter values $(A,B,C)=(2.0,1.3,0.05)$ and $(\alpha ,\gamma )=(1,2.82)$. The presence of unstable dimension variability is very clear.

Figure 15

The riddling of the basin of attraction in the bubble regime for the parameter values $(A,B,C)=(2.0,1.3,0.05)$ and $(\alpha ,\gamma )=(1.1,2.82)$ for (a) the attractor with patches [the black dot indicates the initial condition (0.1724, 0.9116, 0.3947)], (b) the corresponding time series points, (c) the attractor with stripes [the black dot indicates the initial condition (0.6003, 0.6661, 0.9543)], and (d) the corresponding time series points.

Figure 16

The riddled basin of attraction in the bubble regime for the parameter values $(A,B,C)=(2.0,1.3,0.05)$ and $(\alpha ,\gamma )=1,2.82)$. Initial conditions marked with plus signs $(+)$ evolve to the attractor with stripes, and those marked with asterisks $(*)$ evolve to the attractor with patches.

Figure 17

The phase diagram of the embedded two-action ABC map for parameter values $(A,B,C)=(2.0,1.3,0.05)$ (periodic orbits are marked with the label P, chaotic behaviors are marked with the label C, and hyperchaotic regions are marked with the label H). The $\alpha \text{−}\gamma$ 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 15 000 iterates discarding the first 10 000 iterates as transients.

Figure 18

Phase space plot for the chaotic and hyperchaotic regimes for the two-action case for the parameter values $(A,B,C)=(2.0,1.3,0.05)$ and $\gamma =1.5$ for (a) the aerosol regime $\left(\alpha =0.7\right)$ and (b) the bubble regime $\left(\alpha =1.1\right)$. The last 1000 points of the 10 000 iterates are plotted. See Fig. 17 for the complete phase diagram of the two-action case.

Figure 19

The $x\text{−}y\text{−}z$ phase space of the embedded one-action ABC map for parameter values $(A,B,C)=(1.5,0.08,0.16)$ in (a) the aerosol regime for $(\alpha ,\gamma )=(0.2,2.9)$ and (b) the bubble regime for $(\alpha ,\gamma )=(1.2,2.8).$

Figure 20

One-action map case $(A,B,C)=(1.5,0.08,0.16)$ and $\alpha =0.2$ for (a) the bifurcation diagram and (b) the largest Lyapunov exponent variation.

Figure 21

One-action map case $(A,B,C)=(1.5,0.08,0.16)$ and $\alpha =1.25$ for (a) the bifurcation diagram and (b) the variation of the two largest Lyapunov exponents.