  This article studies the intimate relationship between two filtering
algorithms for continuous data assimilation, the synchronization filter and the
nudging filter, in the paradigmatic context of the two-dimensional (2D)
Navier-Stokes equations (NSE) for incompressible fluids. In this setting, the
nudging filter can formally be viewed as an affine perturbation of the 2D NSE.
Thus, in the degenerate limit of zero nudging parameter, the nudging filter
converges to the solution of the 2D NSE. However, when the nudging parameter of
the nudging filter is large, the perturbation becomes singular. It is shown
that in the singular limit of infinite nudging parameter, the nudging filter
converges to the synchronization filter. In establishing this result, the
article fills a notable gap in the literature surrounding these algorithms.
Numerical experiments are then presented that confirm the theoretical results
and probes the issue of selecting a nudging strategy in the presence of
observational noise. In this direction, an adaptive nudging strategy is
proposed that leverages the insight gained from the relationship between the
synchronization filter and the nudging filter that produces measurable
improvement over the constant nudging strategy.
