Reduced Order Models for the Quasi-Geostrophic Equations: A Brief Survey

Abstract

Reduced order models (ROMs) are computational models whose dimension is significantly lower than those obtained through classical numerical discretizations (e.g., finite element, finite difference, finite volume, or spectral methods). Thus, ROMs have been used to accelerate numerical simulations of many query problems, e.g., uncertainty quantification, control, and shape optimization. Projection-based ROMs have been particularly successful in the numerical simulation of fluid flows. In this brief survey, we summarize some recent ROM developments for the quasi-geostrophic equations (QGE) (also known as the barotropic vorticity equations), which are a simplified model for geophysical flows in which rotation plays a central role, such as wind-driven ocean circulation in mid-latitude ocean basins. Since the QGE represent a practical compromise between efficient numerical simulations of ocean flows and accurate representations of large scale ocean dynamics, these equations have often been used in the testing of new numerical methods for ocean flows. ROMs have also been tested on the QGE for various settings in order to understand their potential in efficient numerical simulations of ocean flows. In this paper, we survey the ROMs developed for the QGE in order to understand their potential in efficient numerical simulations of more complex ocean flows: We explain how classical numerical methods for the QGE are used to generate the ROM basis functions, we outline the main steps in the construction of projection-based ROMs (with a particular focus on the under-resolved regime, when the closure problem needs to be addressed), we illustrate the ROMs in the numerical simulation of the QGE for various settings, and we present several potential future research avenues in the ROM exploration of the QGE and more complex models of geophysical flows.

References in corpus (12)

• Proper Orthogonal Decomposition Closure Models For Turbulent Flows: A Numerical Comparison
• Extreme learning machine for reduced order modeling of turbulent geophysical flows
• A non-intrusive reduced order modeling framework for quasi-geostrophic turbulence
• POD/DEIM Reduced-Order Strategies for Efficient Four Dimensional Variational Data Assimilation
• Data-Driven Variational Multiscale Reduced Order Models
• Approximate deconvolution large eddy simulation of a barotropic ocean circulation model
• A stabilized proper orthogonal decomposition reduced-order model for large scale quasigeostrophic ocean circulation
• A long short-term memory embedding for hybrid uplifted reduced order models
• Approximate deconvolution large eddy simulation of a stratified two-layer quasigeostrophic ocean model
• Continuous Data Assimilation Reduced Order Models of Fluid Flow
• Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: applications to Coanda effect in cardiology
• A Two-Level Finite Element Discretization of the Streamfunction Formulation of the Stationary Quasi-Geostrophic Equations of the Ocean