Nonuniversality in the erosion of tilted landscapes

Abstract

The anisotropic model for landscapes erosion proposed by Pastor-Satorras and Rothman in [R. Pastor-Satorras and D. H. Rothman, Phys. Rev. Lett. 80, 4349 (1998)] is believed to capture the physics of erosion at intermediate length scale ( 3 km), and to account for the large value of the roughness exponent α observed in real data at this scale. Our study of this modelconducted using the nonperturbative renormalization group (NPRG) -concludes on the nonuniversality of this exponent because of the existence of a line of fixed points. Thus the roughness exponent depends (weakly) on the details of the soil and the erosion mechanisms. We conjecture that this feature, while preserving the generic scaling observed in real data, could explain the wide spectrum of values of α measured for natural landscapes.

Static Approach to Renormalization Group Analysis of Stochastic Models with Spatially Quenched Noise

“…This result is not surprising if we remember that the upper critical dimension for the white noise is d * = 2. Indeed, the model would only predict trivial scaling in its upper critical dimension, the fact that was confirmed in [3]. The spatially quenched disorder, however, yields upper critical dimension d * = 4.…”

“…One way to study scaling behavior of a fluctuating surface is to describe the system by stochastic differential equation for the height field h(x) subjected to a random force f (x) where x = {t, x} and x is the spatial coordinates at the moment t. Recently it was shown that the choice of this random force (or noise) can dramatically affect the scaling behavior. Precisely, the choice of the quenched disorder in the Pastor-Satorras-Rothman model of a landscape erosion [1,2] proved to be the decisive factor leading to a type of scaling that explained previously thought inconclusive experimental data [3].…”

“…This "dynamic" approach holds true both for the white noise (1.1) and for the spatially quenched disorder (1.2) (to the models studied in the present paper it was applied, e.g., in [3,46]). For the latter case, though, another (static) approach is possible.…”

“…All of this and especially striking difference between results for different noises reported in [3] drives the importance of considering spatially quenched noise in various stochastic models.…”

“…The second type of noise proposed for the model in [1,2] and studied in [3] is the "columnar" noise [29]…”