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.

Contributions to the Teaching and Learning of Fluid Mechanics

“…Mou, Wang, Wells, Xie, and Iliescu provide a survey of reduced order models which are computational models "whose dimension is significantly lower than those obtained through classical numerical discretizations" [16]. ROMs, in their various forms, have been found to be valuable in several complex computations involving uncertainty quantification, control, and shape optimization and in the numerical simulation of fluid flows.…”

A Cost-Efficient Approach towards Computational Fluid Dynamics Simulations on Quantum Devices

“…By reducing the state-space dimension of the model (or degrees of freedom), an approximation to the original model is computed, which is commonly referred to as a reduced-order model (ROM) [15][16][17][18]. ROMs are small in complexity and cheap in terms of computational time; thus, they can be effectively applied in the early stages of product development, as in conceptual design, virtual prototyping and optimization (such as in naval shape design [19] and wind-driven ocean flows [20]), where highly accurate results…”

A POD-Galerkin reduced order model for the Navier-Stokes equations in stream function-vorticity formulation

“…The work in this paper represents an intermediate step towards the development of new FOM and ROM approaches for the quasi-geostrophic equations that are usually written in terms of stream function and (potential) vorticity. See [25] for a recent review.…”