Retention capacity of correlated surfaces

We extend the water retention model [C. L. Knecht et al., Phys. Rev. Lett. 108, 045703 (2012)] to correlated random surfaces. We find that the retention capacity of discrete random landscapes is strongly affected by spatial correlations among the heights. This phenomenon is related to the emergence of power-law scaling in the lake volume distribution. We also solve the uncorrelated case exactly for a small lattice and present bounds on the retention of uncorrelated landscapes.

Figure 1

Snapshot of the water retention model on a square lattice of size $16\times 16$, with number of levels $n=4$. The colors of the cubes indicate the landscape height, which varies from zero (black) to $n-1=3$ [light gray (light orange)]. The retention capacity of this landscape is 72 (lattice units). Dark gray (blue) transparent squares indicate the surfaces of lakes of retained water.

Figure 2

Retention model with continuous heights on a lattice of size $512\times 512$. Surfaces of lakes are indicated in dark gray (blue). The Hurst exponent is $H=0.75$, see, e.g., Refs. [3, 4, 5, 6, 7] for details.

Figure 3

Main plot: Retention $R_n^{\left(L\right)}$ of an $n$-level system of size $L^2$ as function of $L$. Results are averages over at least ${10}^4$ samples. Inset: Data for two and three levels, showing their crossing behavior. Random numbers have been generated with the algorithms of Refs. [34, 35].

Figure 4

Retention per site $R_n^{\left(L\right)}/L^2$ of an $n$-level system of size $L^2$ as function of $L$. The arrows indicate the lower bounds on $R_n^{\left(L\right)}/L^2$, for $n$ levels and large $L$, given by Eq. (5), with $p_c=0.59274602$ [36].

Figure 5

Retention fraction $R_2^{\left(L\right)}/L^2$ as a function of the density of zeros $p$ in the landscape for different $L$. The horizontal line indicates the site percolation threshold $p_c=0.59274602$ [36]. The other straight solid line is the plot of $f\left(p\right)=p$.

Figure 6

$R_2^{\left(L\right)}/\left(L^{2}p\right)$ as a function of the density of zeros $p$ in the landscape.

Figure 7

Snapshots of the water retention model with $n=2$ levels. One half of the sites has height 1, shown as white squares, and the other half has height 0. The sites with height 0 are colored in dark gray (blue) if they retain water, and they are colored in light gray (red) otherwise (draining to the boundary). $H$ increases from (a) to (d) and the lattice size is $L=512$. For $H=0$, it is known that the boundaries of the lakes are fractal with dimension $3/2$ [7, 8, 11], which corresponds to a Gaussian free field or ${\text{SLE}}_4$ [39]. Furthermore, for $0\le H\le 1$, it is believed that the boundary fractal dimension is $(3-H)/2$ [7, 8, 40].

Figure 8

Retention fraction $R_2^{\left(L\right)}/L^2$ for $n=2$ levels as function of the Hurst exponent $H$ for different lattice sizes $L$.

Figure 9

Uncorrelated regime and evolution with $H$. Main plot: Number of lakes of size $s$ per site as function of $s$ with $H=-1$ and for different lattice sizes $L$. Inset: Evolution of the lake size distribution with $H$, for fixed $L=4096$. The sites have height 0 or 1 with equal probability $1/2$. Results are averages over ${10}^4$ samples.

Figure 10

Correlated percolation regime. Number of lakes per site for $H=-0.25$ and different lattice sizes $L$. The solid dark gray (blue) line is a guide to the eye with slope $-2.02$.

Figure 11

Main plot: Number of lakes as function of the lake volume for different nonnegative $H$ and $L=4096$. Straight black lines are guides to the eye with slope $2-H/2$ [7, 8, 33]. Inset: Number of lakes, rescaled by $L^{2-H}=L^{1.25}$ [33], as a function of the lake volume, for $H=0.75$ and different lattice sizes $L$. The solid dark gray (blue) line is a guide to the eye with slope $2-H/2=1.625$ [7, 8, 33].

Figure 12

Illustration of the notation of the analysis on the $L=3$ square lattice. Note that the heights of the four corner sites (filled, gray) do not influence the volume retained by the lattice.

Figure 13

Retention $R_n^{\left(3\right)}$ of an $n$-level system of size $L=3$ as function of $n$. The solid line is the exact result and the squares show the simulation data. The numerical results are averages over ${10}^7$ samples.