Nonlocal Transport Theories in Geomorphology: Mathematical Modeling of Broad Scales of Motion

Most geomorphic transport laws proposed to date are local in character, that is, they express material flux at a point (e.g., sediment flux, tracer concentration, and so on) as a function of geomorphic quantities at that point only, such as, elevation gradient, bed shear stress, local entrainment rate, and so on. We present here recent research efforts that argue that nonlocal constitutive laws, in which the flux at a point depends on the conditions in some larger neighborhood around this point in space and/or in time, present a physically motivated alternative to linear and nonlinear diffusion due to their ability to naturally incorporate the presence of heterogeneities known to exist in geomorphic systems. Moreover, this class of models has the potential to eliminate the scale dependence of local nonlinear constitutive laws, which typically require appropriate closure terms. A particularly attractive subclass of these nonlocal constitutive laws involves fractional (noninteger) derivatives in space and/or in time and provides a rich class of models extensively studied in other fields of science. In this chapter, we present examples of nonlocal transport models in a variety of geomorphologic applications, including tracer dispersal in rivers, hillslope sediment transport, and landscape evolution modeling. Nonlocal transport theories is a new and rapidly evolving field of study in the earth sciences and is anticipated to bring new insight into the interpretation of observations and lead to the development of a broader class of models that can explain the stochastic and complex behavior of geomorphic systems over a broad range of space-time scales.