In this work we study a plankton ecosystem model in a turbulent flow. The plankton model we consider contains logistic growth with a spatially varying background carrying capacity and the flow dynamics are generated using the two-dimensional (2D) Navier-Stokes equations. We characterize the system in terms of a dimensionless parameter, $\gamma \equiv T_B/T_F$, which is the ratio of the ecosystem biological time scales $T_B$ and the flow time scales $T_F$. We integrate this system numerically for different values of $\gamma$ until the mean plankton reaches a statistically stationary state and examine how the steady-state mean and variance of plankton depends on $\gamma$. Overall we find that advection in the presence of a nonuniform background carrying capacity can lead to very different plankton distributions depending on the time scale ratio $\gamma$. For small $\gamma$ the plankton distribution is very similar to the background carrying capacity field and has a mean concentration close to the mean carrying capacity. As $\gamma$ increases the plankton concentration is more influenced by the advection processes. In the largest $\gamma$ cases there is a homogenization of the plankton concentration and the mean plankton concentration approaches the harmonic mean, ${⟨1/K⟩}^{-1}$. We derive asymptotic approximations for the cases of small and large $\gamma$. We also look at the dependence of the power spectra exponent, $\beta$, on $\gamma$ where the power spectrum of plankton is $\propto k^{-\beta }$. We find that the power spectra exponent closely obeys $\beta =1+2/\gamma$ as predicted by earlier studies using simple models of chaotic advection.

• Received 16 March 2009

DOI:https://doi.org/10.1103/PhysRevE.79.061902