Critical Conditions for Phytoplankton

We motivate and analyze a reaction-advection-diiusion model for the dynamics of a phytoplankton species. The reproductive rate of the phytoplankton is determined by the local light intensity. The light intensity decreases with depth due to absorption by water and phytoplankton. Phytoplankton is transported by turbulent diiusion in a water column of given depth. Furthermore, it might be sinking or buoyant depending on its speciic density. Dimensional analysis allows the reduction of the full problem to a problem with four dimensionless parameters that is fully explored. We prove, that the critical parameter regime for which a stationary phytoplankton bloom ceases to exist, can be analyzed by a reduced linearized equation with particular boundary conditions. This problem is mapped exactly to a Bessel function problem, which is evaluated both numerically and by asymptotic expansions. A nal transformation from dimensionless parameters back to laboratory parameters results in a complete set of predictions for the conditions that allow phytoplankton bloom development. Our results show that the conditions for phytoplankton bloom development can be captured by a critical depth, a compensation depth, and zero, one or two critical values of the vertical turbulent diiusion coeecient. These experimentally testable predictions take the form of similarity laws: every plankton-water-light-system characterized by the same dimensionless parameters will show the same dynamics.