Glassy dynamics of landscape evolution

Abstract

Soil creeps imperceptibly downhill, but also fails catastrophically to create landslides. Despite the importance of these processes as hazards and in sculpting landscapes, there is no agreed-upon model that captures the full range of behavior. Here we examine the granular origins of hillslope soil transport by discrete element method simulations and reanalysis of measurements in natural landscapes. We find creep for slopes below a critical gradient, where average particle velocity (sediment flux) increases exponentially with friction coefficient (gradient). At critical gradient there is a continuous transition to a dense-granular flow rheology. Slow earthflows and landslides thus exhibit glassy dynamics characteristic of a wide range of disordered materials; they are described by a two-phase flux equation that emerges from grain-scale friction alone. This glassy model reproduces topographic profiles of natural hillslopes, showing its promise for predicting hillslope evolution over geologic timescales.

Keywords: creep; dynamical phase transition; granular flow; landscape evolution.

Fig. 1.

Landslide and creep phenomenology. (A) Rapid landslide in San Salvador, El Salvador. Image courtesy of Wikimedia Commons/USGS. (B) Slow earthflow in Serranía del Interior, Venezuela. Image courtesy of Wikimedia Commons/Fev. (C) Ranges of surface velocities observed for various types of slow and rapid landslides. The datapoints in red, brown, magenta, and green correspond to the observations reported or documented by Cruden and Varnes (4), Hungr et al. (5), Hilley et al. (6), and Saunders and Young (7), respectively. (D) Schematic cross-section of a soil-mantled hillslope.

Fig. 2.

Hillslope DEM simulations. (A–C) Snapshots correspond to time $t=140$ (s) after inclining the granular hillslope; particle colors represent their downslope ($x$-dir) instantaneous velocities, while black lines show time-averaged downslope velocities. The bulk (macroscopic) angle of repose for this set of simulations is $\theta \approx 24.6^{○}$. (A) $\theta =24.5^{○}$ corresponds to a hillslope just below onset of dense flow at the surface, where the pack is almost fully creeping. (B) $\theta =24.6^{○}$ is right at the transition point. (C) $\theta =28^{○}$ shows a fully developed dense granular flow in more than half of the model depth. D and E show time-averaged downslope ($x$-dir) velocity ($u_x\left(z\right)$) and local friction coefficient ($\mu ={\tau }_{xz}/{\sigma }_n$) profiles, respectively, for the granular hillslope at $\theta =28^{○}$. The critical depth $z_c$ and critical downslope velocity $u_x\left(z_c\right)$ at the transition to creep are indicated. The value $z_c$ is further used in E to determine the critical friction coefficient ${\mu }_c$.

Fig. 3.

General flow behavior in simulations. (A) DEM results showing local downslope velocity ($u_x\left(z\right)$) as a function of local friction coefficient ($\mu$) for four different inclinations, below and above the bulk angle of repose. (B) DEM results showing normalized local downslope velocity ($\frac{u_x\left(z\right)}{u_x\left(z_c\right)}$) as a function of normalized local friction coefficient ($\frac{\mu }{{\mu }_c}$) for the four different inclinations shown in A. Dashed line illustrates an exponential scaling for creep regime with critical gradient ${\mu }_c=0.3$ and a power-law scaling for the range of large flux with a power-law exponent $\beta =5/2$. Relations are not fitted to the data; they are shown only for illustrative purposes.

Fig. 4.

Field data showing measured sediment flux $q_s$ vs. hillslope gradient $S$ for natural hillslopes reported previously in the literature. (A) Gabet (57). (B) Group A basins in Martin (59) and Martin and Church (58). (C) Group B basins in Martin (59). (D) Yoo et al. (56). (E) Rescaled data combined from A–D. Note that normalized flux $q_s/q_{sc}$ is equivalent to normalized velocity while normalized gradient ($S/S_c$) is equivalent to normalized friction, allowing comparison with numerical results (Fig. 3B). The dashed line in E shows the same bipartite flux relation as in Fig. 3B for illustration purposes: i.e., an exponential flux relation for gradients below critical gradient and a power-law relation for larger gradients.

Fig. 5.

Hillslope topography of the OCR derived from publicly available airborne LiDAR data (67). (A) Regional perspective view, showing locations of two example hillslopes. (B and C) The elevation–distance (B) and gradient–distance (C) relationships for representative profiles of hillslopes 1 (black dots) and 2 (red dots) in A. Blue dashed line is the prediction of the “glassy” flux model with $S_c=0.5$ and $\beta =5/2$. See SI Appendix, Figs. S6–S8 for more examples.