Society-induced changes to the environment are altering the effectiveness of existing management strategies for sustaining natural and agricultural ecosystem productivity. At the watershed scale, natural and agro-ecosystems represent complex spatiotemporal stochastic processes. In time, they respond to random rainfall events, evapotranspiration and other losses that are spatially variable because of heterogeneities in soil properties, root distributions, topography, and other factors. To quantify the environmental impact of anthropogenic activities, it is essential that we characterize the evolution of space and time patterns of ecosystem fluxes (e.g., energy, water, and nutrients). Such a characterization then provides a basis for assessing and managing future anthropogenic risks to the sustainability of ecosystem productivity.
To characterize the space and time evolution of watershed scale processes, this dissertation introduces a mean field approach to watershed hydrology. Mean field theory (also known as self-consistent field theory) is commonly used in statistical physics when modeling the space-time behavior of complex systems. The mean field theory approximates a complex multi-component system by considering a lumped (or average) effect of all individual components acting on a single component. Thus, the many body problem is reduced to a one body problem. For watershed hydrology, a mean field theory reduces the numerous point component effects to more tractable watershed averages resulting in a consistent method for linking the average watershed fluxes (evapotranspiration, runoff, etc.) to the local fluxes at each point.
The starting point for this work is a general point description of the soil moisture, rainfall, and runoff system. For this system, we find the joint PDF that describes the temporal variability of the soil water, rainfall, and runoff processes. Since this approach does not account for the spatial variability of runoff, we introduce a probabilistic storage (ProStor) framework for constructing a lumped (unit area) rainfall-runoff response from the spatial distribution of watershed storage. This framework provides a basis for unifying and extending common event-based hydrology models (e.g. Soil Conservation Service curve number (SCS-CN) method) with more modern semi-distributed models (e.g. Variable Infiltration Capacity (VIC) model, the Probability Distributed (PDM) model, and TOPMODEL). In each case, we obtain simple equations for the fractions of the different source areas of runoff, the spatial variability of runoff and soil moisture, and the average runoff value (i.e., the so-called runoff curve). Finally, we link the temporal and spatial descriptions with a mean field approach for watershed hydrology. By applying this mean field approach, we upscale the point description with the spatial distribution of soil moisture and parameterize the numerous local interactions related to lateral fluxes of soil water in terms of its average. With this approach, we then derive PDFs that represent the space and time distribution of soil water and associated watershed fluxes such as evapotranspiration and runoff.
Bartlett, Mark Stephan, Jr. (2016): A Mean Field Approach to Watershed Hydrology. PhD Dissertation, Duke University, Durham, North Carolina.