The wide range of spatial and temporal scales present in turbulent flow problems forms a computational bottleneck, which must be circumvented by a coarse graining procedure. The effects of the unresolved fluid motions enter the coarse-grained partial differential equations (PDEs) as an unclosed subgrid scale (SGS) term. Despite the coarse-graining procedure, the SGS term still contains a high number of unclosed degrees of freedom. To lower the unclosed degrees of freedom by several orders of magnitude, we propose to close the system at the level of ordinary differential equations (ODE), rather than at the original coarse-grained PDE level. This is possible if we derive new SGS models designed to only capture a small set of spatially-integrated quantities of interest (QoIs). Examples of such QoIs include climate-like statistics of the full model, for instance the global energy of the system. We show that we can track these QoIs with a low-dimensional set of unclosed components. This simplifies subsequent machine learning, now we only need to learn a data-driven model for the unclosed components, striking a balance between physical insight and data-driven modelling. We derive the new SGS model for two-dimensional turbulence in a doubly periodic square domain, and show that it produces the same statistics for our quantities of interest as the full model.