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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. 

During the last deglaciation, collapse of the saddle between the North American Cordilleran and Laurentide ice sheets led to rapid ice-sheet mass loss and separation, with meltwater discharge contributing to deglacial sea level rise. We directly date ice-sheet separation at the end of the saddle collapse using 64 10Be exposure ages along an ~1200-km transect of the ice-sheet suture zone. Collapse began in the south by 15.4 ± 0.4 ka and ended by 13.8 ± 0.1 ka at ~56◦N. Ice-sheet model simulations consistent with the 10Be ages find that the saddle collapse contributed 6.2–7.2 m to global mean sea-level rise from ~15.5 ka to ~14.0 ka, or approximately one third of global mean sea-level rise over this period. We determine 3.1–3.6 m of the saddle collapse meltwater was released during Meltwater Pulse 1A ~14.6-14.3 ka, constituting 20–40% of this meltwater pulse’s volume. Because the separation of the Cordilleran and Laurentide ice sheets occurred over 1–2 millennia, the associated release of meltwater during the saddle collapse supplied a smaller contribution to the magnitude of Meltwater Pulse 1A than has been recently proposed. 

Motivated by recent progress in data assimilation, we develop an algorithm to dynamically learn the parameters of a chaotic system from partial observations. Under reasonable assumptions, we supply a rigorous analytical proof that guarantees the convergence of this algorithm to the true parameter values when the system in question is the classic three-dimensional Lorenz system. Such a result appears to be the first of its kind for dynamical parameter estimation of nonlinear systems. Computationally, we demonstrate the efficacy of this algorithm on the Lorenz system by recovering any proper subset of the three non-dimensional parameters of the system, so long as a corresponding subset of the state is observable. We moreover probe the limitations of the algorithm by identifying dynamical regimes under which certain parameters cannot be effectively inferred having only observed certain state variables. In such cases, modifications to the algorithm are proposed that ultimately result in recovery of the parameter. Lastly, computational evidence is provided that supports the efficacy of the algorithm well beyond the hypotheses specified by the theorem, including in the presence of noisy observations, stochastic forcing, and the case where the observations are discrete and sparse in time.