Analytical Solution of Advection-Dispersion Boundary Value Processes in Environmental Flows

The one-dimensional advection-dispersion equation with streamwise boundaries has been used to model a wide range of real-world processes including groundwater contaminant transport, atmospheric plume deposition, and the movement of planktonic organisms and migratory animals through riverine systems. Imposing boundary conditions at upstream and downstream locations complicates the analysis of this equation, and processes that require such boundary conditions are, therefore, typically modeled using approximations that are valid only under certain conditions. Approximations typically involve a simplification of the mass influx into the system, require a large separation between source and boundary, or work only in either strongly advection- or strongly dispersion-dominated systems. Here, we circumvent these limitations by providing an analytical solution to a class of advection-dispersion equations that are broadly applicable to the processes described above as well as many other physical and biological problems. Our solution makes it possible to relax the assumptions required under previous analytic approximations, making it a more flexible approach for modeling advection-dispersion processes under realistic field conditions. We show that the solution is in good agreement with random walk simulations. To illustrate the broad utility of the method, we also apply it to an empirical data set from a population of migratory fish moving through a river system in the California Central Valley and demonstrate that it describes the data more accurately than do existing methods.