  In this paper, a dynamic closure modeling approach has been derived to
stabilize the projection-based reduced order models in the long-term evolution
of forced-dissipative dynamical systems. To simplify our derivation without
losing generalizability, the proposed reduced order modeling (ROM) framework is
first constructed by Galerkin projection of the single-layer quasi-geostrophic
equation, a standard prototype of large-scale general circulation models, onto
a set of dominant proper orthogonal decomposition (POD) modes. We then propose
an eddy viscosity closure approach to stabilize the resulting surrogate model
considering the analogy between large eddy simulation (LES) and truncated modal
projection. Our efforts, in particular, include the translation of the dynamic
subgrid-scale model into our ROM setting by defining a test truncation similar
to the test filtering in LES. The a posteriori analysis shows that our approach
is remarkably accurate, allowing us to integrate simulations over long time
intervals at a nominally small computational overhead.
