A perturbation-theoretic approach to Lagrangian flow networks

Naoya  Fujiwara; Kathrin  Kirchen; Jonathan F.  Donges; Reik V.  Donner

doi:10.1063/1.4978549

ABSTRACT

Complex network approaches have been successfully applied for studying transport processes in complex systems ranging from road, railway, or airline infrastructures over industrial manufacturing to fluid dynamics. Here, we utilize a generic framework for describing the dynamics of geophysical flows such as ocean currents or atmospheric wind fields in terms of Lagrangian flow networks. In this approach, information on the passive advection of particles is transformed into a Markov chain based on transition probabilities of particles between the volume elements of a given partition of space for a fixed time step. We employ perturbation-theoretic methods to investigate the effects of modifications of transport processes in the underlying flow for three different problem classes: efficient absorption (corresponding to particle trapping or leaking), constant input of particles (with additional source terms modeling, e.g., localized contamination), and shifts of the steady state under probability mass conservation (as arising if the background flow is perturbed itself). Our results demonstrate that in all three cases, changes to the steady state solution can be analytically expressed in terms of the eigensystem of the unperturbed flow and the perturbation itself. These results are potentially relevant for developing more efficient strategies for coping with contaminations of fluid or gaseous media such as ocean and atmosphere by oil spills, radioactive substances, non-reactive chemicals, or volcanic aerosols.

ACKNOWLEDGMENTS

N.F. is grateful to the financial supports by JSPS KAKENHI Grant No. 15K16061, and by CREST, Japan Science and Technology Agency. J.F.D. acknowledges the financial support by the BMBF project GLUES, the Stordalen Foundation (via the Planetary Boundary Research Network PB.net) and the Earth League's EarthDoc program. R.V.D. received funding by the German Federal Ministry for Education and Research (BMBF) via the BMBF Young Investigator's Group CoSy-CC²2. A. Barrat, M. Barthelemy, and A. Vespignani, Dynamical Processes on Complex Networks ( Cambridge University Press, 2008). (“Complex Systems Approaches to Understanding Causes and Consequences of Past, Present, and Future Climate Change, Grant No. 01LN1306A”) and the IRTG 1740 “Dynamical Phenomena in Complex Networks” jointly funded by DFG and FAPESP. The authors acknowledge the valuable input from discussions with Kazuyuki Aihara, Jürgen Kurths, Alexis Tantet, and Emilio Hernandéz-Garcia as well as two anonymous reviewers of this manuscript.

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A perturbation-theoretic approach to Lagrangian flow networks

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