  This paper investigates the recently introduced data-driven correction
reduced order model (DDC-ROM) in the numerical simulation of the
quasi-geostrophic equations. The DDC-ROM uses available data to model the
correction term that is generally used to represent the missing information in
low-dimensional ROMs. Physical constraints are added to the DDC-ROM to create
the constrained data-driven correction reduced order model (CDDC-ROM) in order
to further improve its accuracy and stability. Finally, the DDC-ROM is tested
on time intervals that are longer than the time interval over which it was
trained. The numerical investigation shows that, for low-dimensional ROMs, both
the DDC-ROM and CDDC-ROM perform better than the standard Galerkin ROM (G-ROM)
and the CDDC-ROM provides the best results.
