Fractional advection-dispersion equations for modeling transport at the Earth surface

Characterizing the collective behavior of particle transport on the Earth surface is a key ingredient in describing landscape evolution. We seek equations that capture essential features of transport of an ensemble of particles on hillslopes, valleys, river channels, or river networks, such as mass conservation, superdiffusive spreading in flow fields with large velocity variation, or retardation due to particle trapping. Development of stochastic partial differential equations such as the advection-dispersion equation (ADE) begins with assumptions about the random behavior of a single particle: possible velocities it may experience in a flow field and the length of time it may be immobilized. When assumptions underlying the ADE are relaxed, a fractional ADE (fADE) can arise, with a non-integer-order derivative on time or space terms. Fractional ADEs are nonlocal; they describe transport affected by hydraulic conditions at a distance. Space fractional ADEs arise when velocity variations are heavy tailed and describe particle motion that accounts for variation in the flow field over the entire system. Time fractional ADEs arise as a result of power law particle residence time distributions and describe particle motion with memory in time. Here we present a phenomenological discussion of how particle transport behavior may be parsimoniously described by a fADE, consistent with evidence of superdiffusive and subdiffusive behavior in natural and experimental systems.