Mean field theory (MFT) is commonly used in statistical physics when modeling the space-time behavior of complex systems. The main premise of MFT is to replace multi-component interactions with an effective interaction to an average (i.e. lumped) field value. Thus, a many body problem is reduced to a one body problem. In watershed hydrology, the numerous interactions between watershed points are reduced to points interacting with more tractable watershed (unit area) averages. Through MFT, we consistently link point scale behavior to lumped (unit area) watershed behavior.
We show that MFT links the local rainfall-runoff behavior to the runoff thresholds observed at both the watershed and hillslope scales of experiment catchments. The watershed scale water balance, which includes the lumped local effects, may be coupled to a probabilistic description of seasonal rainfall. Based on this seasonal description, we find an analytical expression for the distribution of the average (unit area) soil water storage. In turn, this seasonal distribution provides analytical expressions for the seasonal distributions of watershed scale evapotranspiration and runoff fluxes. Through MFT, we may disaggregate the average (unit area lumped) fluxes into specific local values explicitly mapped to the watershed area. We map the spatial variation of these fluxes under different seasonal conditions. In comparison to fully-distributed models, this approach is a simpler analytical alternative for testing and refining point scale theories in relation to climatic changes and experimental measurements at the hillslope and watershed scales.
Bartlett, M.S., I. Rodriguez-Iturbe, and A.M. Porporato (2016): A mean field approach to the watershed response under stochastic seasonal forcing. American Geophysical Union 2016 Fall Meeting, San Francisco, CA.
This Paper/Book acknowledges NSF CZO grant support.