Scaling in erosion of landscapes: Renormalization group analysis of a model with turbulent mixing

TL;DR

This paper applies renormalization group analysis to a landscape erosion model with turbulent mixing, revealing complex fixed point structures and potential scaling behavior in the infrared range.

Contribution

It extends the landscape erosion model by incorporating anisotropic turbulent advection and derives the full set of renormalization constants and critical exponents.

Findings

Existence of two fixed point surfaces with possible infrared attraction

Model exhibits nonuniversal critical exponents depending on fixed point coordinates

Derivation of one-loop counterterms and renormalization group functions

Abstract

The model of landscape erosion, introduced in [{\it Phys. Rev. Lett.} {\bf 80}: 4349 (1998); {\it J. Stat. Phys.} {\bf 93}: 477 (1998)] and modified in [{\it Theor. Math. Phys.} - in press; arXiv:1602.00432], is advected by anisotropic velocity field. The field is Gaussian with vanishing correlation time and the pair correlation function of the form $\propto \delta(t-t') / k_{\bot}^{d-1+\xi}$, where $k_{\bot}=|{\bf k}_{\bot}|$ and ${\bf k}_{\bot}$ is the component of the wave vector, perpendicular to a certain preferred direction -- the $d$-dimensional generalization of the ensemble introduced by Avellaneda and Majda [{\it Commun. Math. Phys.} {\bf 131}: 381 (1990)]. Analogous to the case without advection, the model is multiplicatively renormalizable and has infinitely many coupling constants. The one-loop counterterm is derived in a closed form in terms of the certain function $V(h)$, entering the original stochastic equation, and its derivatives with respect to the height field $h(t,{\bf x})$. The full infinite set of the one-loop renormalization constants, $\beta$-functions and anomalous dimensions is obtained from the Taylor expansion of the counter-term. Instead of a two-dimensional surface of fixed points there is two such surfaces; they are likely to contain infrared attractive region(s). If that is the case, the model exhibits scaling behaviour in the infrared range. The corresponding critical exponents are nonuniversal because they depend on the coordinates of the fixed points on the surface; they also satisfy certain universal exact relation.