Process length scales and longitudinal damping in karst conduits

Simple mathematical models often allow an intuitive grasp of the function of physical systems. We develop a mathematical framework to investigate reactive or dissipative transport processes within karst conduits. Specifically, we note that for processes that occur within a characteristic timescale, advection along the conduit produces a characteristic process length scale. We calculate characteristic length scales for the propagation of thermal and electrical conductivity signals along karst conduits. These process lengths provide a quantitative connection between karst conduit geometry and the signals observed at a karst spring. We show that water input from the porous/fractured matrix is also characterized by a length scale and derive an approximation that accounts for the influence of matrix flow on the transmission of signals through the aquifer. The single conduit model is then extended to account for conduits with changing geometries and conduit flow networks, demonstrating how these concepts can be applied in more realistic conduit geometries. We introduce a recharge density function, φ_R, which determines the capability of an aquifer to damp a given signal, and cast previous explanations of spring variability within this framework. Process lengths are a general feature of karst conduits and surface streams, and we conclude with a discussion of other potential applications of this conceptual and mathematical framework.