# Stochastic geometric models with non-stationary spatial correlations in Lagrangian fluid flows

## TL;DR

This paper develops data-driven stochastic models for oceanic surface flows that incorporate non-stationary spatial correlations, inspired by satellite observations, and introduces two new models with symmetry-breaking mechanisms.

## Contribution

It introduces two novel stochastic models with non-stationary spatial correlations for geophysical fluid dynamics, extending previous models by incorporating flow-advection of correlations.

## Key findings

- Models capture non-stationary spatial correlations in ocean flows
- New models derived using stochastic variational principles and Hamiltonian structures
- Enhanced understanding of oceanic current dynamics through advanced stochastic modeling

## 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 [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.