Channeling of Branched Flow in Weakly Scattering Anisotropic Media

When waves propagate through weakly scattering but correlated, disordered environments they are randomly focused into pronounced branchlike structures, a phenomenon referred to as branched flow, which has been studied in a wide range of isotropic random media. In many natural environments, however, the fluctuations of the random medium typically show pronounced anisotropies. A prominent example is the focusing of tsunami waves by the anisotropic structure of the ocean floor topography. We study the influence of anisotropy on such natural focusing events and find a strong and nonintuitive dependence on the propagation angle which we explain by semiclassical theory.

Figure 1

Branched flow in an anisotropic random medium with an aspect ratio of the correlation lengths of $1/3$ (correlation lengths indicated by scale bars). Color code indicates the logarithmic relative density of Newtonian trajectories emitted from a point source at the center.

Figure 2

Branched flow (lower panels) with plane wave initial conditions. Correlation functions are shown in the insets of the random potentials on the top row. (left) Isotropic potential. Red dashed line indicates onset of branching in the isotropic case. For comparison, the same line is plotted in the two anisotropic cases. (center) Anisotropy $\mathcal{A}=1/3$: The potential has the same correlations in the $y$ direction as the isotropic case but a shorter correlation in the $x$ direction. Surprisingly, although the mean correlation length is smaller than in the isotropic case, the typical focusing distance is larger. (right) Anisotropy $\mathcal{A}=3$: Inverse case to the center row, with much shorter focusing length, even shorter than in the isotropic medium.

Figure 3

(a) Mean distance to the first caustic $d_{\mathrm{fc}}$ for plane wave initial conditions as a function of the anisotropy $\mathcal{A}=b/a$ of the random potential for different angles $\alpha$. Lines: analytical predictions from Eq. (8). Circles: numerical simulations. The inset shows $d_{\mathrm{fc}}$ for $\mathcal{A}=1/2$ for trajectories originating from a point source, as a function of the angle. (The two different symbols correspond to using the angle of the initial velocity or the angle of the position vector of the caustic, respectively). (b) Scintillation index $\mathrm{\Sigma }(\overrightarrow{x})$ measuring the strength of intensity fluctuations in simulated tsunami waves emitted from a point source propagating over isotropic random ocean floor topographies (bathymetries). The peak position of the scintillation index, shown averaged over all angles in (c), scales like the branching length and is indicated by an orange circle. (d) Scintillation index for anisotropic bathymetries ($\mathcal{A}=1/2$). The orange line shows the prediction of Eq. (8), in excellent agreement with the simulation results.

Figure 4

(a) Cuts through the mean ray intensity (averaged over 200 realizations of the random medium) at a constant distance $45{\ell }_x$ from the source for different ${\ell }_y$, i.e., varying $\mathcal{A}$. The cut for infinite $\mathcal{A}$ corresponds to ${\ell }_x\to \infty$ and $45{\ell }_y$, where ${\ell }_y$ is the same as in the isotropic case. SWW (shallow water waves) denotes simulations of tsunami waves propagating in random bathymetries of anisotropy $\mathcal{A}=3$. Left inset: Mean intensity for $\mathcal{A}=5$ in a logarithmic color scale. Red circle indicates the position of the cut. Right inset: Mean intensity for infinite $\mathcal{A}$. Black dashed lines in the inset and the red bars in the main panel are the predictions of Eq. (11). (b) Staying probability $p(\varphi =0,t)$ and standard deviation of the velocity angle $\varphi$ for trajectories started in the $x$ direction ($\alpha =0$) and perpendicular ($\alpha =\pi /2$) for potentials with $\mathcal{A}={\ell }_x/{\ell }_y=5$. Dashed lines are the corresponding diffusion solutions (see text), dashed dotted lines guide the eyes. Curves for different ${\ell }_x$ overlay perfectly, showing the consistency of the simulations (${\ell }_{\max }$ corresponds to the largest ${\ell }_x$). Curves belonging to the right axis show the density perpendicular and opposite to the initial velocity direction (see text for discussion). (c) Standard deviation of the velocity angle for various anisotropies for trajectories starting along the axis with longer correlations ($\alpha =0$), scaled by the corresponding diffusion solution.