Parameterization of Turbulent-Induced Correlated Phytoplankton and Nutrient Fluctuations in an Upwelling Sea Surface Mixed Layer and Their Effect on Mean Production

Many fundamental biological dynamical processes in the sea are strongly non-linear. Thus, physical turbulence in the sea not only enhances advective transport processes by eddy diffusivity but also importantly induces turbulent fluctuations in biological state variables which can correlate in the mean and effect mean biological dynamics. Goodman and Robinson (Proc. Roy. Soc A, 2008, 484, 555-572; doi: 10.1098/rspa. 2007.0251) have formulated a theory of turbulent biological-physical interactions in terms of probability density functions (pdfs) and applied it to the simple example of nutrient (N) and seed phytoplankton (P) upwelling into a surface turbulent layer. The mean uptake is to the sum of the product of the means and the often neglected correlation fluctuations . and are positive, but is found to be negative and to significantly reduce . Here we parameterize by with a proportionality constant a function of two non-dimensional parameters: i) the ratio of the biological uptake time scale to the upwelling time, and ii) the ratio of the upwelling time scale to the turbulent diffusive time (Peclet number). An ADR (advective-diffusive-reactive) model is then formulated for the mean fields , and solved analytically for the intense turbulent limit of small Peclet number by perturbation methods. Agreement of , values in the mixed layer with the pdf theory is excellent. The ADR results provide useful insight into coupled turbulent physical-biological dynamical processes.