Mathematical Modeling of Oceanic Phytoplankton Blooms in Chaotic Flows

Phytoplankton blooms are geochemically and ecologically important, but current understanding of the causes of phytoplankton blooms provides little power to predict the timing, extent, and distribution of blooms. In this thesis, I discuss the implications of light limitation for phytoplankton bloom patchiness and community structure by modeling phytoplankton as reacting tracers in a reaction-advection-diffusion system. I use non-dimensional parameters to develop general principles of bloom dynamics with light limitation that can be tested against empirical results. I first develop an exact Lagrangian model of advection and diffusion with no-flux boundaries to represent a vertical water column and compare the properties of tracers in that model to known properties of a tracer in a turbulent mixed layer. I then use both numerical simulations and exact solutions to systematically build towards understanding the individual interactions between reaction, advection, and diffusion in a reactionadvection-diffusion system with non-linear source terms. I find that spatially variable forcing is required to develop patchiness in a reaction-advection-diffusion system. I also analyze the emergence of temporal heterogeneity in bulk phytoplankton concentrations when the ratio of physical and biological timescales is ∼ O(1) and conclude that non-steady equilibria are due to the degree of spatial heterogeneity as well as to advection and diffusion rates. Finally, I demonstrate one possible mechanism for an interaction between light limitation, turbulence, and predator-prey dynamics.