  We study a nonlinear-nudging modification of the Azouani-Olson-Titi
continuous data assimilation (downscaling) algorithm for the 2D incompressible
Navier-Stokes equations. We give a rigorous proof that the nonlinear-nudging
system is globally well-posed, and moreover that its solutions converge to the
true solution exponentially fast in time. Furthermore, we also prove that, once
the error has decreased below a certain order one threshold, the convergence
becomes double-exponentially fast in time, up until a precision determined by
the sparsity of the observed data. In addition, we demonstrate the
applicability of the analytical and sharpness of the results computationally.
