Persistent patterns and multifractality in fluid mixing

Persistent patterns in periodically driven dynamics have been reported in a wide variety of contexts ranging from table-top and ocean-scale fluid mixing systems to the weak quantum-classical transition in open Hamiltonian systems. We illustrate a common framework for the emergence of these patterns by considering a simple measure of structure maintenance provided by the average radius of the scalar distribution in transform space. Within this framework, scaling laws related to both the formation and persistence of patterns in phase space are presented. Further, preliminary results linking the scaling exponents associated with the persistent patterns to the multifractal nature of the advective phase-space geometry are shown.

Figure 1

The two extremes of near-integrable behavior where $K={10}^{-4}$ (solid line) and near uniform hyperbolicity for $K=100$ (dashed line) are contrasted. $D={10}^{-4}$ in both cases. Analytic, $K=0$ shown in dotted line.

Figure 2

(a) ${\chi }^2\left(t\right)$ for mixed phase-space dynamics. $K=2.1$ (solid), $K=3.5$ (dashed), $K=7$ (dotted), and $K=8$ (dot-dashed). $D={10}^{-4}$ in each case. Note that all the curves go through the stages (a) rapid increase, (b) turnover, (c) decrease, and (d) steady state. Inset: saturation value of ${\chi }^2$ as a function of $D$ for $K=7$. (b) Persistent pattern for $K=2.1$ and $t=30$. (c) Persistent pattern for $K=7$ at $t=30$.

Figure 3

${\chi }^2$ scaled by the $D^{-\gamma }$ and time scaled by $\text{ln}1/D$ plotted for $K=2.1$ and values of $D$ from $5\times {10}^{-6}–{10}^{-4}$. The higher values of scaled ${\chi }^2$ correspond to smaller $D$.

Figure 4

Singularity spectrum with error bars for $K=7$ (circles, solid line), $K=9$ (triangles, dotted line), and $K=12$ (crosses, dashed line). The values of $\gamma$ are 0.8, 0.96, and 0.86, respectively.

Figure 5

Saturation value ${\chi }_{\infty }^2$ over a wider range of diffusivity $D$, exhibiting different scaling over ranges of diffusivity. Parameter values shown are $K=4.5$ (squares), $K=5.5$ (diamonds), $K=6$ (circles), and $K=7$ (asterisks).

Figure 6

Part of the classical phase space for $K=6$ showing the presence of a small stable region. Note that a symmetric partner also exists elsewhere.

Figure 7

(Color online) (a) ${\chi }^2\left(t\right)$ for the advection diffusion equation for the sin-sin flow; $\omega =0.25,\epsilon =0.35$. $D=2.5\times {10}^{-4}$ (solid), $D={10}^{-3}$ (dot-dashed), and $D={10}^{-2}$ (dashed). Time measured in periods of the velocity field. Inset: saturation value of ${\chi }^2$ as a function of $D$; (b) persistent patterns for the three diffusivities (diffusivity decreasing from the top) at three different periods: $T=22$ (left), $T=23$ (middle), and $T=24$ (right).

Figure 8

Scaling exponent $\gamma$ plotted against the corresponding finite-time Hausdorff dimension, $H_D$. The straight line is a fit drawn primarily to guide the eyes.