Convergence of data assimilation by periodic updating in simple Hamiltonian and dissipative systems

In this paper, we study the influence of the interval between data insertion events on the convergence of sequential data assimilation problems. An example of a conservative Hamiltonian system is presented (that of Hénon and Heiles (1964)) where sequential assimilation with periodic data insertion every Δt achieves a more rapid convergence if data is not inserted at the smallest possible update interval, Δt. It is shown analytically that this is true for all Hamiltonian systems when the updated variables produce convergence of the assimilation, because the resolvent matrix then varies as O(Δt2) to highest order. The theory successfully predicts the turnover point for the He´non and Heiles system when a larger Δt leads to slower convergence and also the assimilation interval at which convergence may cease altogether. The application to a simplified low order shallow water model describing coupled Rossby and gravity waves and with a forced-dissipative perturbation extends the previous result to systems which are a more realistic model for the atmosphere and the ocean. Formally, the same behaviour still holds when a realistic dissipation scheme is applied with increasing amplitudes or when strongly dissipative systems, which are not forced-dissipative perturbations of Hamiltonians, are used.