A subordinated kinematic wave equation for heavy‐tailed flow responses from heterogeneous hillslopes

Abstract

Analytical expressions of hillslope‐scale subsurface stormflow discharge are currently restricted to hillslopes with homogeneous or mildly heterogeneous conductivity fields. In steep, straight hillslopes with uniform recharge these exhibit a classical piston flow response, which arises from an assemblage of impulses all moving at a constant velocity but with different starting locations. Heterogeneity within a hillslope soil creates variations in the downslope velocity of these impulses, which may lead to nonpiston flow responses with either exponential or heavy (power law) tails. The presence of heavy tails suggests that heterogeneity imparts a temporal memory on the motion of the impulses. Using this assumption, a subordinated kinematic wave equation is proposed for moderately to highly heterogeneous hillslopes. This equation convolves the piston response from a homogenous hillslope with a stable subordinator. The stable subordinator randomizes the time that impulses spend in motion and produces nonpiston solutions with heavy tails. Through comparisons of synthetic data generated from numerical hillslope simulations with physically realistic parameters, this equation faithfully reproduces both early and late time characteristics of heavy‐tailed flow responses from moderate to highly heterogenous hillslopes. A systematic evaluation of hillslope responses under different degrees of heterogeneity revealed a quantitative link between the statistical properties of the heterogeneous random fields and the parameters of the subordination framework. This suggests that the subordinator can be parameterized with the physical measurement of hillslope properties.