How jets and vortices influence the speed of propagation of plankton blooms: an idealised view

Alexandra Tzella
University of Birmingham

We investigate the influence of flows on the propagation of chemical pulsating fronts evolving inside an infinite channel domain. We focus on the sharp front obtained in Fisher--Kolmogorov--Petrovskii--Piskunov (FKPP) type models in the limit of small molecular diffusivity and fast reaction (large P\'eclet and Damk\""ohler numbers) and on its heuristic approximation by the G equation. This problem arises naturally in oceanographic applications when studying interacting chemical or biological species, such as plankton in the ocean. We introduce a variational formulation that expresses the front speed in terms of periodic trajectories minimising the time of travel across a characteristic length scale of the flow subject to a constraint that differs between the FKPP and G equations. This formulation makes it plain that the FKPP front speed is greater than or equal to the G equation front speed. We study the two front speeds for a class of cellular vortex flows. Using a numerical implementation of the variational formulation, we show that the differences between the two front speeds are modest for a broad range of parameters. However, large differences appear when a strong mean flow opposes front propagation; in particular, we identify a range of parameters for which FKPP fronts can propagate against the flow while G fronts cannot.

This work is in collaboration with Jacques Vanneste, University of Edinburgh.

Presentation (PDF File)

Back to Transport and Mixing in Complex and Turbulent Flows