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.
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