A GENERAL THEORY OF RIVERBANK INSTABILITY

Where and when do natural rivers become unstable? We answered this question by mapping percent riverbank failures in one-hundred meter reaches along 180 kilometers of northern Yellowstone Park streams. These data reveal precise power-law magnitude/frequency relationships in the spatial failure patterns. Such power-law structures can be the spatial signal of a self-organized critical (SOC) system, where local instabilities function to generate broader-scale order. Such a critical structure is to be expected given a nonlinear diffusive system such as a drainage basin. Cellular automata simulations of SOC riverbank systems predict similar power-law magnitude/frequency relationships through both time and space.

In log-log magnitude/frequency graphs of the observed mountain river systems, the measured slope of the power-law function (the tau-exponent, t) of spatial failure frequencies in alluvial reaches of these systems (from 1.07 to 1.44) is less than that for all reaches (alluvial, colluvial, and bedrock) combined, suggesting that alluvial reaches are more susceptible to large riverbank instabilities. Changes in the t-exponent are related to differences in average channel slope and in near-channel bank material. If in fact riverbank systems are at a critical state rather than an equilibrium state, then long-term local stability is an unlikely or even impossible engineering or restoration goal. The probable existence of criticality in natural stream settings suggests that local human alterations designed to increase channel stability, while lowering the local frequency of small failures, will only encourage an increase in the magnitude of system-wide, low-frequency large failures. A local stabilization or restoration effort will not eliminate the bank instability, but will instead transfer that instability to adjacent riverbank areas.