Self-Organized Criticality in Riverbank Systems

Abstract

Where and when do natural rivers become unstable? To answer this question, we visually estimated bank-failure extent in 100-m increments along 180 km of riverbanks in three watersheds of the northern Yellowstone ecosystem. The riverbank data reveal precise power-law relationships between the number of bank failures of a given size throughout each watershed and the magnitude of those bank failures. The slopes of log-log graphs (i.e., the exponent τ) of bank-failure magnitude versus failure frequency in alluvial reaches vary from 1.07 to 1.44, while τ for all reaches combined (alluvial, colluvial, and bedrock) varies from 1.18 to 1.53, suggesting that lower-gradient, alluvial streams are more susceptible to large bank failures. Cellular automata simulations of riverbanks show similar power-law failure relationships, as do bank-erosion data from a long-term independent dataset from another location. These power-law structures can be interpreted as the spatial signal of a self-organized critical (SOC) system, in which local instabilities function to generate broader-scale order. SOC systems are considered to be at the “edge of chaos,” where local processes interact to make prediction of specific failure events impossible, although probability distribution prediction of the magnitude and spatial frequency of those events is possible. A critical structure of this sort is to be expected in bank failures along a stream given a nonlinear diffusive system such as a drainage basin. If riverbanks are, in fact, part of a critical system, then long-term local or watershed-wide stability is an unlikely or even impossible engineering or restoration goal. The existence of criticality in natural stream settings suggests that local human alterations designed to increase channel stability, while changing the local frequency of small failures, will only encourage an increase in the magnitude of system-wide, low-frequency large failures. A restoration or stabilization effort will not eliminate the bank instability. Instead, it will transfer that instability to neighboring riverbank areas.

Key Words:

• bank stability
• fluvial geomorphology
• self-organized criticality
• stream restoration
• width adjustment

Acknowledgments

This work was supported by the Wild Waters Initiative of Yellowstone Ecosystem Studies and by the U.S. Environmental Protection Agency, through a research grant to Duncan Patten, Rick Lawrence, Andrew Marcus, and Wayne Minshall. Thanks are due to Jim Rasmussen, Carl Legleiter, Wendy Bigler, Chad Heidtke, Justin Balhiser, Brent Nickola, and Keith Van Etten for their outstanding field assistance in Yellowstone National Park. Karen Wynn Fonstad designed and produced the study area map. Robert Crabtree and the Northern Yellowstone Field Station staff provided critical logistical support. We also wish to thank Jonathan Phillips, Stephen Darby, and an anonymous reviewer for their insightful comments on the manuscript.

Notes

^aPercent of each 100-meter left bank reach that is actively failing.

^bPercent of each 100-meter right bank reach that is actively failing.

^cBankfull width.

^dBankfull depth.

^eLocal channel slope.

^fEstimated two-year flood cross-sectional stream power.

^gEstimated two-year flood mean stream power.

^aLeft (L) or right (R) bank facing downstream.

^bNumber of 100-m reaches surveyed.

^cSlope of the fit power-law equation.

^dSlope-intercept.

^eCoefficient of determination for the fit power-law.

^fLevel of significance for each regression.