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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.
Perturbation-theoretic methods have found wide applications in many areas of physics, ranging from classical celestial mechanics to quantum physics. At various occasions, they have proven useful for addressing scientific problems where an explicit analytical treatment is not possible, but the system under study can be considered as a minor modification of another problem where such an analytical solution can be obtained. This work combines basic concepts of perturbation theory with Lagrangian flow networks, a novel tool that allows characterizing structural properties of flows by means of complex network theory. Specifically, by means of an eigenvector decomposition of the associated transition matrix (i.e., a spatial discretization of the Perron-Frobenius operator of the flow), it is studied how minor modifications affect the steady-state distribution of passively advected particles in a given flow pattern. One main potential field of application of the developed framework is atmospheric and ocean physics, where the proposed approach may help in developing more efficient strategies for coping with contaminations of fluid or gaseous media by oil spills, radioactive substances, non-reactive chemicals, or volcanic aerosols (e.g., to determine the most efficient positions for removing the contaminating substances or anticipate their temporary trapping and natural precipitation)
