Pond fractals in a tidal flat

Studies over the past decade have reported power-law distributions for the areas of terrestrial lakes and Arctic melt ponds, as well as fractal relationships between their areas and coastlines. Here we report similar fractal structure of ponds in a tidal flat, thereby extending the spatial and temporal scales on which such phenomena have been observed in geophysical systems. Images taken during low tide of a tidal flat in Damariscotta, Maine, reveal a well-resolved power-law distribution of pond sizes over three orders of magnitude with a consistent fractal area-perimeter relationship. The data are consistent with the predictions of percolation theory for unscreened perimeters and scale-free cluster size distributions and are robust to alterations of the image processing procedure. The small spatial and temporal scales of these data suggest this easily observable system may serve as a useful model for investigating the evolution of pond geometries, while emphasizing the generality of fractal behavior in geophysical surfaces.

Figure 1

(a) Field site at Days Cove during high tide, (b) low tide later that day, with ponds appearing, (c) unprocessed overhead image, $3.3\times 2.5 {\mathrm{m}}^2$, with reference disk, and (d) processed black (mud) and white (pond) image detecting pond coverage.

Figure 2

Rank-frequency function $\left(f\right)$ for pond areas. $f$ will have slope 1 greater than probability density function (PDF) due to being an integral; $f$ power-law fit has slope of $\alpha +1\approx -0.67$ (dotted yellow line), thus PDF has slope of $\alpha \approx -1.67$. $f$ fit with correlation length scale $\xi \approx 0.80$ m (solid red line), which results in an exponential tailing for large ponds. $r^2$ for both are $>0.99$ over their fitting domain.

Figure 3

Area-perimeter scatterplot for pond fractal dimension. The thick line is the predicted slope of 4/3 from percolation theory for unscreened hulls, with an $r^2>0.85$. The thinner line is representative of a possible transition towards connected ponds with fractal dimension $d_f\approx 2$ in the upper limit of pond sizes, which may not show up distinctly in this data set due to masking by the correlation length scale.