A Numerical Method for Advection-Reaction-Diffusion Equations

with applications to plankton modeling

A mathematical model of plankton dynamics involves three ingredients:

  • Population dynamics: a set of ODEs that describe the food web of the many species forming a spatially homogeneous planktonic community; this is in general awfully complicated. Usually only predator-prey dynamics between zooplankton and phytoplankton is modeled, often with the addition of a nutrient variable.
  • Large-scale transport: plankton is carried along by mesoscale and basin-scale ocean currents. This produces highly complicated lagrangian trajectories.
  • Small-scale mixing: at small scales (less than 100m) plankton is mixed by turbulence of the ocean's mixed layer and by the swimming of single individuals. In general, this leads to non-linear diffusion, but here, for simplicity, we just use Laplacian diffusion.
  • Putting all three together one gets an advection-reaction-diffusion equation for each species considered in the population dynamics model.

    If there were no diffusion, the solution of the advection-reaction problem is straightforward using the method of characteristics.

    In the presence of diffusion, we propose a numerical method that handles the diffusion operator on an auxiliary eulerian grid, while using a particle-based approach for the advection-reaction part. In the limit of no diffusion, one recovers the method of characteristics.

    For a description of the method, please refer to this paper.

    An open issue is the lack of cycles in the satellite observations of chlorophyll (phytoplankton) concentration. Limit cycles (or chaotic non-linear oscillations with a well-defined time scale) are ubiquitous in the solution of the population dynamics ODEs.

    Stirring by mesoscale currents may disguise the oscillations: water patches oscillating with largely different phases may be brought together by mesoscale stirring: a coarse graining observation, such one from a satellite, would average over the phases, canceling any fluctuations.

    Without diffusion this explanation appears to be robust, even without invoking other factors such as patchy nutrient sources: here is a simulation with no diffusion, notice the "out-of-tuning TV" effect at the end. Stand back a few steps from the screen: you won't see any oscillations.


    With a relatively large diffusion coefficient (D=10-5 for both species) we have a rather quick spatial synchronization. You will see the screen flashing as the zooplankton and the phytoplankton follow their own limit cycle at unison in the entire domain.


    With a small diffusion coefficient (D=5*10-7, in the long run, we still observe spatial synchronization. However, the out-of-tune TV effect is still there for quite a while...


    Last update: 04/12/2007