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.

Efficient and effective modeling of complex systems, incorporating cloud physics and precipitation, is essential for accurate climate modeling and forecasting. However, simulating these systems is computationally demanding since microphysics has crucial contributions to the dynamics of moisture and precipitation. In this paper, appropriate stochastic models are developed for the phase-transition dynamics of water, focusing on the precipitating quasi-geostrophic (PQG) model as a prototype. By treating the moisture, phase transitions, and latent heat release as integral components of the system, the PQG model constitutes a set of partial differential equations (PDEs) that involve Heaviside nonlinearities due to phase changes of water. Despite systematically characterizing the precipitation physics, expensive iterative algorithms are needed to find a PDE inversion at each numerical integration time step. As a crucial step toward building an effective stochastic model, a computationally efficient Markov jump process is designed to randomly simulate transitions between saturated and unsaturated states that avoids using the expensive iterative solver. The transition rates, which are deterministic, are derived from the physical fields, guaranteeing physical and statistical consistency with nature. Furthermore, to maintain the consistent spatial pattern of precipitation, the stochastic model incorporates an adaptive parameterization that automatically adjusts the transitions based on spatial information. Numerical tests show the stochastic model retains critical properties of the original PQG system while significantly reducing computational demands. It accurately captures observed precipitation patterns, including the spatial distribution and temporal variability of rainfall, alongside reproducing essential dynamic features such as potential vorticity fields and zonal mean flows.

There are two main strategies for improving the projection-based reduced order model (ROM) accuracy—(i) improving the ROM, that is, adding new terms to the standard ROM; and (ii) improving the ROM basis, that is, constructing ROM bases that yield more accurate ROMs. In this paper, we use the latter. We propose two new Lagrangian inner products that we use together with Eulerian and Lagrangian data to construct two new Lagrangian ROMs, which we denote $\alpha$-ROM and $\lambda$-ROM. We show that both Lagrangian ROMs are more accurate than the standard Eulerian ROMs, that is, ROMs that use standard Eulerian inner product and data to construct the ROM basis. Specifically, for the quasi-geostrophic equations, we show that the new Lagrangian ROMs are more accurate than the standard Eulerian ROMs in approximating not only Lagrangian fields (e.g., the finite time Lyapunov exponent (FTLE)), but also Eulerian fields (e.g., the streamfunction). In particular, the $\alpha$-ROM can be orders of magnitude more accurate than the standard Eulerian ROMs. We emphasize that the new Lagrangian ROMs do not employ any closure modeling to model the effect of discarded modes (which is standard procedure for low-dimensional ROMs of complex nonlinear systems). Thus, the dramatic increase in the new Lagrangian ROMs’ accuracy is entirely due to the novel Lagrangian inner products used to build the Lagrangian ROM basis.