Abstract
Inspired by spatiotemporal observations from satellites of the trajectories of objects drifting near the surface of the ocean in the National Oceanic and Atmospheric Administration’s “Global Drifter Program”, this paper develops data-driven stochastic models of geophysical fluid dynamics (GFD) with non-stationary spatial correlations representing the dynamical behaviour of oceanic currents. Three models are considered. Model 1 from Holm (Proc R Soc A 471:20140963, 2015) is reviewed, in which the spatial correlations are time independent. Two new models, called Model 2 and Model 3, introduce two different symmetry breaking mechanisms by which the spatial correlations may be advected by the flow. These models are derived using reduction by symmetry of stochastic variational principles, leading to stochastic Hamiltonian systems, whose momentum maps, conservation laws and Lie–Poisson bracket structures are used in developing the new stochastic Hamiltonian models of GFD.
Highlights
This paper develops data-driven stochastic models of fluid dynamics, inspired by spatiotemporal 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”
Given the stochasticity in the flow map, how does one derive the corresponding Eulerian equations of continuum motion? The paper lays out a geometric framework for deriving stochastic Eulerian motion equations for geophysical fluid dynamics (GFD) using the method of symmetry reduction for a modified Hamilton’s principle for fluids with advected quantities
The methodology of this approach has two primary features: (1) stochastic variational principles; and (2) stochastic Hamiltonian formulations. These two primary features will allow us to introduce two new stochastic extensions of Geophysical Fluid Dynamics (GFD), one with advection by the drift velocity of the eigenvectors ξt discussed in Sect. 3, and the other with eigenvectors ξt depending on advected fluid quantities discussed in Sect
Summary
This paper develops data-driven stochastic models of fluid dynamics, inspired by spatiotemporal 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”. Erratic trajectories will be represented stochastically in the paper, and the larger-scale spatial correlations they follow will be modelled as spatiotemporal modulations of the stochasticity, which follow the resolved drift currents. Along this path one sees the effects of the interactions of the drifter with a variety of space and time scales of evolving fluid motion, strongly suggesting a need for modelling the non-stationary statistics of Lagrangian paths (Sykulski et al 2016)
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