The impact of temporal variability in light-climate on time-averaged primary production and a phytoplankton bloom in a well-mixed estuary

Phytoplankton primary production (PP) in turbid estuaries is often limited by light-availability. Two important factors altering light-climate are solar irradiance at the water surface and exponential light-extinction coefficient within the water column. Additionally, the depth of the water body changes the light-climate and corresponding PP by altering the ratio of the euphotic and mixing depth in a well-mixed estuary. These three parameters are highly variable yet are often assumed to be constant by both experimental scientists and modelers because of a lack of data or to reduce complexity. Because assuming constant parameters introduces an error, we utilize an idealized model of depth-integrated primary production to analyze the (individual) impact of temporal variability in these three parameters. We only consider the main tidal and solar constituents in temporal variability of the forcings and apply a second-order moment approximation to analyze the bias introduced to time-averaged PP estimates by neglecting temporal fluctuations. We demonstrate that the sign and magnitude of this bias are system-specific and depend on two non-dimensional parameters that characterize the system. The first is equivalent to the ratio of mixing and photic depth. The second accounts for typical incident irradiance and the photosynthetic parameters of the phytoplankton population present. To demonstrate the applicability of our approach, we apply the model to two cases in the Scheldt estuary (Belgium) in the brackish and freshwater part. In the first application, we study the impact of fluctuations on phytoplankton in dynamic equilibrium, where biomass is assumed to be constant. We show that variability in solar irradiance has the largest impact on time-averaged PP in dynamic equilibrium, resulting in a 30 percent decrease compared to time-invariant forcing. By comparing with a numerical integrator, we show that a second-order moment approximation correctly predicts the order of magnitude of the impact of temporal variability of the individual parameters. In the second application, we study the impact of fluctuations on unbounded exponential phytoplankton growth. Also here, fluctuations in solar irradiance have the largest impact and lead to a significant decrease in exponential growth. In this case study, we show that temporal fluctuations delay the onset of the biomass by two weeks and decrease the biomass by a factor 14 after two weeks compared to time invariant forcing. Additionally, we show that the temporal fluctuations induce low-frequency variability in phytoplankton biomass with similar periodicity as the spring-neap cycle, making it difficult to observe these phenomena in real-world time series.