Impact of discontinuous deformation upon the rate of chaotic mixing

Mixing in smoothly deforming systems is achieved by repeated stretching and folding of material, and has been widely studied. However, for the classes of materials that also admit discontinuous deformation, the theory of mixing based on the assumption of smooth deformation does not apply. Discontinuous deformation provides additional topological freedom for material transport and results in different Lagrangian coherent structures forbidden in smoothly deforming systems. We uncover the impact of discontinuous deformation on mixing rates, showing that mixing can be either enhanced or impeded depending on the local stability of the underlying smooth map.

Figure 1

Parameters $({\gamma }_1,{\gamma }_2)$ of the map ${\mathrm{\Lambda }}_1$ such that the period 1 point at the origin is elliptic (gray) and hyperbolic (white). The curves ${\gamma }_1=0,{\gamma }_2=0,4+{\gamma }_1{\gamma }_2=0$ separate regions of different stabilities.

Figure 2

Orbits of sets of differently colored particles under (a, c) the linear map ${\mathrm{\Lambda }}_1$ and (b, d) the discontinuous map ${\mathrm{\Lambda }}_2$. In all cases ${\gamma }_1=-a=0.04$. (a, b) ${\gamma }_2=-1$. (c, d) ${\gamma }_2=1$. (e) The web of preimages of the Lagrangian discontinuity for the case in panel (b).

Figure 3

The initial scalar field $c_0(x,y)=cos(\pi y/2)$ in the square domain ${[-2,2)}^2$ that is used to demonstrate the impact of cutting on mixing using the maps ${\mathrm{\Lambda }}_1,{\mathrm{\Lambda }}_2$.

Figure 4

Iterates of the scalar field $c_0$ (Fig. 3) under the maps ${\mathrm{\Lambda }}_1,{\mathrm{\Lambda }}_2$. Videos of each sequence can be found in the Supplemental Material [37]. (a) $({\gamma }_1,{\gamma }_2,a)=(1,-1,-1)$ corresponding to $\theta =\pi /3$. (b) $({\gamma }_1,{\gamma }_2,a)=(2-\sqrt{2},-1,\sqrt{2}-2)$ corresponding to $\theta =\pi /4$. (c) $({\gamma }_1,{\gamma }_2,a)=(0.04,-1,-0.04)$ corresponding to $\theta /\pi \approx 0.0638$.

Figure 5

Log-linear plot of the evolution of the mix-norm of the initial scalar field $c_0$ (Fig. 3) under the maps ${\mathrm{\Lambda }}_1$ (open symbols connected by dashed lines) and ${\mathrm{\Lambda }}_2$ (closed symbols connected by solid lines). Blue squares: $({\gamma }_1,{\gamma }_2,a)=(1,-1,-1)$ corresponding to Fig. 4. Red triangles: $({\gamma }_1,{\gamma }_2,a)=(2-\sqrt{2},-1,\sqrt{2}-2)$ corresponding to Fig. 4. Black circles: $({\gamma }_1,{\gamma }_2,a)=(0.04,-1,-0.04)$ corresponding to Fig. 4.

Figure 6

Particle orbits (colored arrows or points) and web of preimages of the Lagrangian discontinuities (thick black) for (a) $\theta =\pi /3$, (b) $\theta =\pi /4$, and (c) $\theta /\pi$ irrational corresponding to $({\gamma }_1,{\gamma }_2)=(0.04,-1)$. (a) The blue, green, and brown orbits are period 6, and the red orbit is period 12. (b) The blue, green, and brown orbits are period 8, and the red orbit is period 1944, though only the first nine iterates are shown. (c) The blue, brown, and red orbits all have infinite period and fill an ellipse (blue, brown) or a pair of ellipses (red).

Figure 7

Iterates of the scalar field $c_0$ (Fig. 3) under the maps ${\mathrm{\Lambda }}_1$ and ${\mathrm{\Lambda }}_2$. Videos of each sequence can be found in the Supplemental Material [37]. (a) $({\gamma }_1,{\gamma }_2,a)=(0.04,1,-0.04)$. (b) $({\gamma }_1,{\gamma }_2,a)=(0.16,1,-0.16)$.

Figure 8

Log-linear plot of the evolution of the mix-norm of the initial scalar field $c_0$ (Fig. 3) under the maps ${\mathrm{\Lambda }}_1$ (open symbols) and ${\mathrm{\Lambda }}_2$ (closed symbols). Blue squares: $({\gamma }_1,{\gamma }_2,a)=(0.04,1,-0.04)$. Red triangles: $({\gamma }_1,{\gamma }_2,a)=(0.16,1,-0.16)$. In each case the decay rate predicted by Eq. (10) is shown dashed.