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The role of integration time in determining a steady-state through data assimilation

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Marotzke, J. (1992). The role of integration time in determining a steady-state through data assimilation. Journal of Physical Oceanography, 22(12), 1556-1567. doi:10.1175/1520-0485(1992)022<1556:TROITI>2.0.CO;2.

The length of time an ocean model and its adjoint should be integrated in determining a steady state compatible with observed data is investigated. The starting point is based upon a suggestion that only one time step is required. This method fails to converge to an acceptable solution when applied to a general circulation model (GCM) of the North Atlantic. Using a very coarse resolution GCM in an idealized geometry, the problem is traced to the interplay of convective adjustment and the very short integration time. The general assimilation technique is explored using a very simple model, a linear first-order equation with forcing and damping. The model is unable to provide a dynamical coupling between the forcing and the model response, owing to a mismatch of integration time and adjustment time scale. Coupling can be enforced in the simple linear model through a careful choice of weighting factors, a strategy excluded in the GCM due to the presence of very fast processes like convective adjustment. An integration over a sufficiently long time can avoid the problems encountered. Experiments with the idealized GCM prove successful for longer integrations, and a tentative upper limit of 50 years is given for inversions aiming at the main thermocline structure.