Stability and damping in a dynamical model of hillslope hydrology

Abstract

A stability analysis is performed on the two‐state variable, integral‐balance model of hillslope hydrology developed by Duffy [1996]. The motivation for the research is to develop a physically based, low‐dimensional representation of inflow/outflow behavior within a hillslope‐stream setting. Stability criteria are developed for the model equilibria and are evaluated using the results of numerical solutions of Richards' equation for a convex‐concave hillslope geometry. We show that for homogeneous hillslopes of three widely varying soil types the single moisture equilibrium is classified as a stable node for low precipitation rates and a stable spiral for wet conditions. The spiral equilibrium indicates that the hillslope system is lightly damped, and transient oscillations of the state variables are expected for high precipitation rates. The timescale of these oscillations is of the order of several days to weeks for the model hillslopes examined. Furthermore, we demonstrate that the model contains a Hopf bifurcation from a stable static equilibrium to a stable limit cycle. The amplitude, phase, and frequency of the limit cycle are determined analytically using second‐order averaging. However, this behavior is shown to be nonphysical for the particular homogeneous soils and hillslope geometry investigated. Implications of lightly damped behavior in the hillslope system include moisture oscillations in the field under wet conditions and difficulty in numerical solution of Richards' equation.