Data-Driven Variational Multiscale Reduced Order Models for the Quasi-Geostrophic Equations

This paper investigates the recently introduced data-driven variational multiscale reduced order model (DD-VMS-ROM) in the numerical simulation of the quasi-geostrophic equations (QGE). The new DD-VMS-ROM framework centers around the hierarchical structure of the variational multiscale (VMS) methodology and utilizes data to increase the ROM accuracy at a modest computational cost. Thus, instead of phenomenological models used in VMS for standard numerical discretizations (e.g., eddy viscosity models), we utilize available data to construct new structural VMSROM closure models. Specifically, we build ROM operators (vectors, matrices, and tensors) that are closest to the true ROM closure terms evaluated with the available data. Physical constraints are also added to the DD-VMS-ROM to further improve its accuracy and stability. We test the new DD-VMSROM in the numerical simulation of the QGE at a Reynolds number Re=450 and a Rossby number Ro = 0.0036.