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Stochastic geometric models with non-stationary spatial correlations in Lagrangian fluid flows. (English) Zbl 1431.37068

The authors obtain data-driven stochastic models of geophysical fluid dynamics with nonstationary spatial correlations representing the dynamical behaviour of oceanic currents. The paper is inspired by spatio-temporal observations from satellites of the spatial paths of objects drifting near the surface of the ocean in the National Oceanic and Atmospheric Administration’s “Global Drifter Program”. The Lagrangian paths of these freely drifting instruments track the ocean currents – the satellite readings of their positions approximate the motion of a fluid parcel as a curve parametrized by time.
Three models are considered. The first one, in which the spatial correlations are time-independent, is taken from [D. D. Holm, Proc. A, R. Soc. Lond. 471, No. 2176, Article ID 20140963, 19 p. (2015; Zbl 1371.35219)]. Two new models introduce two symmetry breaking mechanisms for which the spatial correlations may be advected by the flow. The models are obtained by the use of symmetry reduction of stochastic variational principles. The authors derive stochastic Hamiltonian systems, whose momentum maps, conservation laws, and Lie-Poisson bracket structures are used in the study of the stochastic models of geophysical fluid dynamics.

MSC:

37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
37H10 Generation, random and stochastic difference and differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
86A05 Hydrology, hydrography, oceanography

Citations:

Zbl 1371.35219

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References:

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