Journal of Conference Abstracts

Anisotropic Correlations and the Coupling of Length Scales in Eroding Landscapes

Kelvin Chan (kechan@segovia.mit.edu), Romualdo Pastor-Satorras (romu@segovia.mit.edu) & Daniel H. Rothman (dan@segovia.mit.edu)

Department of Earth, Atmospheric, and Planetary Sciences, MIT, Cambridge, MA, 02139, U.S.A.

The observation that topography may be fractal implies that the underlying equations of erosion may be nonlinear. This, in turn, implies that topographic evolution at different scales is coupled. Here we address one way in which this coupling may be measured. Importantly, the results apply to topography that need not explicitly exhibit scale invariance.

We begin with the observation that the erosion of a slope with fixed inclination is intrinsically anisotropic due to the presence of a preferred direction for material transport. We formulate a stochastic equation to model this process. At zeroth order in the anisotropy, our theory is linear and it predicts that the height-height correlation function is anisotropic at the level of direction-dependent prefactors. On the other hand, a nonlinear theory constructed from the first higher-order contribution of the anisotropy predicts that the correlations decay algebraically with characteristic exponents that depend on direction. Measurements made from a digital map of a submarine canyon are in good agreement with the nonlinear theory. We have also investigated several desert environments. In these cases, the predicted scaling is not conclusively observed, but the anisotropy of the height-height correlation functions is ubiquitous.

The inclination of real landscapes, however, is rarely fixed as in our theory; slopes vary not only with space but also with the length scale at which they are measured. Using digital elevation maps of real landscapes, we find that the anisotropy of the correlations at small scales is strongly coupled to the topographic structure at much larger scales. These results suggest that the coupling of local anisotropy to large-scale structure is a fundamental physical property of eroding landscapes. They also indicate that the original, time-of-deposition, large-scale structure of subsequently deformed sedimentary basins may be reconstructed from small-scale topographic fluctuations.