23. Phosphorus Cycles in Lakes and Rivers: Modeling, Analysis, and Simulation

From spring to summer period, a large number of lakes are laced with thick layers of algae implicitly representing a serious problem with respect to the fish stock as well as other important organisms and at the end for the complete biological diversity of species. Consequently, the investigation of the cause-and-effect chain represents an important task concerning the protection of the natural environment. Often such situations are enforced by an oversupply of nutrient. As phosphorus is the limiting nutrient element for most of all algae growth processes an advanced knowledge of the phosphorus cycle is essential. In this context the chapter gives a survey on our recent progress in modeling and numerical simulation of plankton spring bloom situations caused by eutrophication via phosphorus accumulation. Due to the underlying processes we employ the shallow water equations as the fluid dynamic part coupled with additional equations describing biogeochemical processes of interest within both the water layer and the sediment. Depending on the model under consideration one is faced with significant requirements like positivity as well as conservativity in the context of stiff source terms. The numerical method used to simulate the dynamic part and the evolution of the phosphorus and different biomass concentrations is based on a second-order finite volume scheme extended by a specific formulation of the modified Patankar approach to satisfy the natural requirements to be unconditionally positivity preserving as well as conservative due to stiff transition terms. Beside a mathematical analysis, several test cases are shown which confirm both the theoretical results and the applicability of the complete numerical scheme. In particular, the flow field and phosphorus dynamics for the West Lake in Hangzhou, China are computed using the previously stated mass and positivity preserving finite volume scheme.