Steady-state solutions to the advection-diffusion equation and ghost coordinates for a chaotic flow

Steady-state solutions to the advection-diffusion equation for a passive scalar, with a chaotic divergence-free flow, are determined using a discrete-time, finite-difference model. The physical system studied is a density of particles diffusing across a chaotic layer. The impact of the advective structures on the solutions is illustrated, with special attention given to the cantori. It is argued that cantori play an important role in restricting transport and that coordinates adapted to cantori, called ghost coordinates, provide a natural framework about which the dynamics may be organized; for example, the averaged density profile becomes a smoothed devil’s staircase in ghost coordinates.

Figure 1

Finite numerical resolution error $e$, scaling for various $D$, plotted against grid size $h$, for $k=0.9716$. The dashed line satisfies $e\sim h^2$.

Figure 2

Poincaré plot showing the stochastic sea (gray dots), a segment of the unstable manifolds (thick line) emanating from the (0, 1), (1, 2), (1, 3), and (1, 4) periodic orbits $(\times )$, and the density contours (black lines) for $D={10}^{-3}$ (left), $D={10}^{-5}$ (middle), and $D={10}^{-7}$ (right). The horizontal $x$ axis is [0.0, 0.5], the vertical $y$ axis is [0.0, 0.7], and the perturbation parameter $k=0.9716$.

Figure 3

Density along the (0,1) unstable manifold, plotted against length from the (0,1) unstable periodic orbit, for the perturbation parameter $k=0.9716$, and various values of diffusion $D$.

Figure 4

(Color online) Density contours (black lines), showing the density inflection, and Poincaré plot (gray dots), across the (0,1) island, for $D={10}^{-3}$ and $k=0.191787$.

Figure 5

Critical island width $W$ against diffusion coefficient $D$. The dashed line satisfies $W=\frac{3}{2}{D}^{1∕3}$

Figure 6

Particle flux $Q$ as a function of perturbation parameter $k$, for various values of diffusion $D$, where $D=\left|Q\left(0\right)\right|$.

Figure 7

Poincaré plot showing the stochastic sea (gray dots), the coordinate boundary (dashed curve), the selected cantori (black dots), and the cantori density contours (black lines) for $D={10}^{-3}$ (left), $D={10}^{-5}$ (middle), and $D={10}^{-7}$ (right). The horizontal $x$ axis range is [0.0, 0.5], the vertical $y$ range is [0.1, 0.7], and the perturbation parameter $k=0.9716$.

Figure 8

Variation $\sigma$ in density $\rho$ along $\left(p,q\right)$ periodic orbits, for the first seven levels of the Farey tree, for the perturbation parameter $k=0.9716$, plotted against frequency $\omega =p∕q$ for $D={10}^{-4}$ (left), $D={10}^{-5}$ (middle), and $D={10}^{-6}$ (right). The upper (lower) row is for the elliptic (hyperbolic) periodic orbits.

Figure 9

Poincaré plot (gray dots), selected ghost circles (black curves), and coordinate boundary (dashed curves) in both $\left(x,y\right)$ coordinates (above) and $\left(\alpha ,s\right)$ ghost coordinates (below).

Figure 10

Density contours (gray lines), selected cantori (black dots), and the average cantori density contours (black lines) plotted in ghost coordinates for $D={10}^{-3}$ (left), $D={10}^{-5}$ (middle), and $D={10}^{-7}$ (right). The horizontal $\alpha$ range is [0.0, 0.5], the vertical $y$ range is [0.0, 0.61803], and the perturbation parameter $k=0.9716$. The thick black line is the (0,1) unstable manifold.

Figure 11

Radial density profiles, for $D={10}^{-5}$ (left) and $D={10}^{-8}$ (middle), in ghost coordinates for the perturbation parameter $k=0.9716$, compared to the devil’s staircase function $C$ (right).