An analytical phytoplankton model and its application in the tidal freshwater James River

This study constructs an analytical model to understand the phytoplankton dynamics in a one-dimensional system by integrating physical and biological processes. We present an analytical solution of phytoplankton distribution along the longitudinal direction of a tidal river. The solution has two components: water age and accumulative growth. The water age represents the overall effect of physical transport due to advection, while the accumulative growth represents the accumulation of phytoplankton biomass due to biological processes during the transport. In addition, an alternative solution with nonlinear reaction term is given to account for the feedback mechanism of nonlinear processes that affect phytoplankton growth rate. The nonlinear reaction term can be also used to approximate the effect of physical mixing on phytoplankton distribution. The analytical solutions can be further used to predict the temporal variation and spatial distribution of phytoplankton along the tidal river. They also serve as a mathematical tool to unite different field phytoplankton observations. We applied the model in the Tidal Freshwater Region (TFR) of James River under different dynamic conditions and the model results match well with the observations, which validates the theory on different time scales. Furthermore, an analysis is conducted for the Local Chlorophyll-a Maximum (LCM) in the TFR. It shows that the condition for LCM is biologically controlled, but its location is regulated by river flow and river geomorphology.