Relevance of conserved quantities for data assimilation

In this paper we study relevance of quantities conserved by a numerical scheme for data assimilation. We consider three Arakawa discretizations of the quasigeostrophic model that either preserve energy, or enstrophy, or both and an Ensemble Kalman Filter data assimilation method. Numerical experiments show that if non-local observations are assimilated, the choice of a numerical scheme is crucial for a good reconstruction of the mean fields, namely the Arakawa scheme that preserves both energy and enstrophy provides with the best estimate, while the Arakawa scheme that preserves only energy completely fails. If local observations are assimilated, a good mean field reconstruction is independent of a numerical scheme, though it becomes necessary to apply localization and inflation. Furthermore, we show that even though an estimated probability den- sity function is normal due to the Ensemble Kalman Filter assumptions, while the true probability density function is taken to be skewed, the estimated mean fields still exhibit the nonlinear behavior. This indicates the ability of the Ensemble Kalman Filter to reproduce nonlinear large-scale behavior without reproducing non-Gaussian small-scale behavior.