Reduced-Order Modeling and Data Assimilation of the El Niño–Southern Oscillation

Abstract

Simulations of complex fluid dynamics problems or climate models take weeks to complete even when run parallel in state-of-the-art supercomputers. Given computational resource constraints and the need for adaptable simulation settings, cost-efficient and accurate algorithms are essential. In this thesis, we explore stable, efficient, and accurate methodologies when applied to the El Niño–Southern Oscillation (ENSO), which integrates coupled atmosphere, ocean, and sea surface temperature (SST) mechanisms in the equatorial Pacific. ENSO is one of the most influential and complex climate phenomena, affecting weather patterns across the globe.

ENSO consists of irregular oscillations between warm (El Niño) and cold (La Niña) phases in the Pacific Ocean, significantly impacting global weather patterns. Due to ENSO's inherent complexity and uncertainties, it is particularly suited for stochastic modeling. By modeling these uncertainties, stochastic simulations offer a more accurate representation of ENSO's variability, including its irregular periods and amplitudes. We first study the effects of stochastic perturbations on ENSO dynamics and introduce novel modeling and numerical schemes based on the Wiener Chaos Expansion (WCE). The key idea behind WCE is the explicit discretization of white noise through Fourier expansion. We also compare these methods with Monte Carlo (MC) simulations. Our findings demonstrate that the simulation of the linear stochastic ENSO model driven by the Ornstein-Uhlenbeck process using WCE requires far less computational resources and gives more accurate results compared to MC ensembles. This part of the thesis provides an alternative efficient approach for simulations of stochastic climate models and quantification of statistical moments,i.e, mean and variance.

In the next stage of this research, we explore a reduced-order modeling (ROM) framework based on the POD-Galerkin method when applied to a nonlinear ENSO model. POD-Galerkin reduced order modeling aims to reduce the computational complexity and present high-dimensional problems~(usually PDEs) with reduced-order equations (ODEs). POD modes are optimal in capturing the system’s dominant features, making it particularly effective for reducing the dimensionality of systems governed by PDEs. By capturing the full-order ENSO (PDE) model with only four modes and four reduced-order equations, we achieve a substantial reduction in computational complexity without significant loss of accuracy. Due to the special properties of the model, we introduce a novel approach using different POD bases, but the same time coefficients for all model components. Moreover, we employ machine learning methods to explore different ROM and model discovery techniques in this part.

The final part of this thesis focuses on the data assimilation of the nonlinear stochastic ENSO model, which forms the core results of this research. We first demonstrate the validity of POD-Galerkin reduced order modeling for the stochastic ENSO driven by the Ornstein-Uhlenbeck process. We project SPDEs onto POD modes derived from the deterministic model and introduce the reduced-order stochastic equations (SDEs). After setting up the filtering framework, we combine these equations and artificial observations in the Pacific ocean, based on realistic experiments, to estimate the ENSO-related SST anomalies. We employ particle filters and test the efficiency using different number of particles and ensembles. From novel ENSO modeling to uncertainty quantification, from reduced order modeling to nonlinear filtering, this thesis provides a promising approach for accurate and efficient predictions of ENSO-related climate variables.