Network and geometric characterization of three-dimensional fluid transport between two layers

We consider transport in a fluid flow of arbitrary complexity but with a dominant flow direction. One of the situations in which this occurs is when describing by an effective flow the dynamics of sufficiently small particles immersed in a turbulent fluid and vertically sinking because of their weight. We develop a formalism characterizing the dynamics of particles released from one layer of fluid and arriving in a second one after traveling along the dominant direction. The main ingredient in our study is the definition of a two-layer map that describes the Lagrangian transport between both layers. We combine geometric approaches and probabilistic network descriptions to analyze the two-layer map. From the geometric point of view, we express the properties of lines, surfaces, and densities transported by the flow in terms of singular values related to Lyapunov exponents, and define a specific quantifier, the finite depth Lyapunov exponent. Within the network approach, degrees and an entropy are considered to characterize transport. We also provide relationships between both methodologies. The formalism is illustrated with numerical results for a modification of the ABC flow, a model commonly studied to characterize three-dimensional chaotic advection.

Figure 1

Illustration of the dynamics of a rectangular surface element, lying on the upper layer at the release time $t_0$, and with area $d{A}_0$, until leaving a footprint of area $d{A}_{\mathrm{acc}}$ on the lower layer when arriving there. See the discussion and the Appendix for details.

Figure 2

Sketch of the bipartite network construction. Particles travel from the upper layer to the bottom one. Nodes are the boxes $A_i, i=1,...,M_0$ on which the upper layer is partitioned, and $B_j, j=1,...,M_z$, partitioning the lower one. Two nodes are linked if some trajectory joins them.

Figure 3

Histogram of the times to reach the second layer at $z=0$, starting from $z_0=10$.

Figure 4

The travel time, $\omega$, from release layer ($z_0=10$) to collecting layer ($z=0$) displayed at the initial and final position of each particle.

Figure 5

Maximal FDLE ${\overline{\lambda }}_1$ for dynamics under the modified ABC flow, displayed at the initial particle locations in the release layer $z_0=10$, and for the collecting layer at (a) $z=2\pi$, (b) $z=4$, and (c) $z=0$.

Figure 6

(a) The density factor, $F$, computed as $F={\left({\overline{\mathrm{\Lambda }}}_1{\overline{\mathrm{\Lambda }}}_2\right)}^{-1}$ at the collecting layer $z=0$, giving the relative density of collected particles if the density in the release layer $z_0=10$ is uniform. (b) The stretching factor $S$, computed as $S={\left({\mathrm{\Lambda }}_1{\mathrm{\Lambda }}_2\right)}^{-1}$. (c) The projection factor $P$ from Eq. (13). We have checked that $F=SP$ to good accuracy. Note the logarithmic scale in the color maps.

Figure 7

Out-degree and in-degree in the release ($z_0=10$) and the collecting ($z=0$) layers, respectively.

Figure 8

Entropy $H\left(i\right)$ for transport from the release layer at $z_0=10$ to the collecting layer at $z=0$, displayed on the release layer.

Figure 9

(a) The in-strength $S_{\mathrm{IN}}\left(j\right)$ in the lower layer $z=0$, which gives the accumulated density in that layer starting from a unit-density uniform release at $z_0=10$. (b) Density factor averaged on each box of the accumulation layer, i.e., ${\langle F\rangle }_{B_j}={\langle {\left({\overline{\mathrm{\Lambda }}}_1{\overline{\mathrm{\Lambda }}}_2\right)}^{-1}\rangle }_{B_j}$.

Figure 10

(a) Scatter plot of values of $K_{\mathrm{OUT}}\left(i\right)$ vs ${\langle \overline{\mathrm{\Lambda }}\rangle }_{A_i}$. The red diagonal indicates the fulfillment of Eq. (26). (b) Scatter plot of values of $H\left(i\right)$ vs ${\langle log\overline{\mathrm{\Lambda }}\rangle }_{A_i}$. The red diagonal indicates the fulfillment of Eq. (27). Dots are colored according to the value of ${\langle {\overline{\mathrm{\Lambda }}}_2\rangle }_{A_i}$.

Figure 11

Probability density function of the values of ${\overline{\mathrm{\Lambda }}}_2$ on the release layer.

Figure 12

Sketch (in a two-dimensional situation) of the footprint or projection in the direction of motion, $\mathcal{P}\mathbf{q}$, of a vector $\mathbf{q}=(q_x,q_y)$ onto a horizontal collecting layer (in fact a collecting line) when arriving there with velocity $\mathbf{v}=(v_x,v_y)$. We have $\mathcal{P}\mathbf{q}=(q_x-q_z{v}_x/v_z)\widehat{\mathbf{x}}$, where $\widehat{\mathbf{x}}$ is the unit vector in the direction of the collecting line.