Data-Driven Filtering Techniques for Turbulent Flow Models (A Lagrangian Data Assimilation Approach)

Abstract

This work contributes to the understanding of high-dimensional fluid flows, focusing on quasi-geostrophic systems (QG) by addressing the challenge of reconstructing turbulent flow fields from a partial, noisy, and time-sequential set of tracer measurements. More precisely, we aim for an accurate recovery of the Eulerian energy spectra of the geostrophic system of equations using Lagrangian tracers or drifters, which are essential in atmospheric and oceanic measurements as dynamic data collectors that stream real-time data of the velocity field, forwarding that to specialized servers that can convert the data to information of the tracked dynamics through methods in this thesis. We solve the QG equation using numerical simulations to produce the ground truth of the high-dimensional dynamics of the system. A novel hybrid filter, merging Ensemble Kalman Filter (EnKF) and Particle Filter (PF) procedures are developed based on this; the proposed hybrid filter is specifically designed for Lagrangian data assimilation and aims to accurately predict the system’s states. We simulate the complex dynamics of physical systems through a Lagrangian data assimilation design. addressing the challenges of predicting turbulent behaviour in systems where deterministic equations are generally not available or too complicated to be solved analytically. Our approach bridges the estimation-prediction gap by making use of partial, turbulent, and stochastic observations that update reduced dynamics of tracer advection by fluid flows. In particular, the governing fluid model is the QG barotropic flow model. Additionally, we progress from ordinary differential equations (ODEs) to stochastic partial differential equations (PDEs), and their rotational versions, including stochastic Lorenz 63, stochastic L96 with forcing, Shallow Water Equations (on the plane and sphere) and Quasi-Geostrophic Equations. Broadly, this thesis contributes to the advancement of filtering techniques for high-dimensional dynamical systems and provides a practical framework for reconstructing turbulent QG flow fields from partial Lagrangian measurements and offers new tools for atmospheric and oceanic large-scale flows using data assimilation.