2019/05/27 by A. Renaud, Renaud, Antoine, Jacques Vanneste +2
Biochemistry, Genetics and Molecular Biology · Environmental Science · Mathematics · Physics and Astronomy · #Chemical Physics (physics.chem-ph) #Diffusion and Search Dynamics #Ecosystem dynamics and resilience #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Fractional Differential Equations Solutions #Mathematical Biology Tumor Growth #Theoretical and Computational Physics #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.1905.11086
openalex publication_date 2019/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We examine the dispersion of a passive scalar released in an incompressible\nfluid flow in an unbounded domain. The flow is assumed to be spatially\nperiodic, with zero spatial average, and random in time, in the manner of the\nrandom-phase alternating sine flow which we use as an exemplar. In the\nlong-time limit, the scalar concentration takes the same, predictable form for\nalmost all realisations of the flow, with a Gaussian core characterised by an\neffective diffusivity, and large-deviation tails characterised by a rate\nfunction (which can be evaluated by computing the largest Lyapunov exponent of\na family of random-in-time partial differential equations). We contrast this\nsingle-realisation description with that which applies to the average of the\nconcentration over an ensemble of flow realisations. We show that the\nsingle-realisation and ensemble-average effective diffusivities are identical\nbut that the corresponding rate functions are not, and that the\nensemble-averaged description overestimates the concentration in the tails\ncompared with that obtained for single-flow realisations. This difference has a\nmarked impact for scalars reacting according to the\nFisher--Kolmogorov--Petrovskii--Piskunov (FKPP) model. Such scalars form an\nexpanding front whose shape is approximately independent of the flow\nrealisation and can be deduced from the single-realisation large-deviation rate\nfunction. We test our predictions against numerical simulations of the\nalternating sine flow.\n