Relevance of conservative numerical schemes for an Ensemble Kalman Filter

Abstract

In this article we study the relevance of quantities conserved by a numerical scheme for data assimilation through statistical equilibrium mechanics. We use the Ensemble Kalman Filter with perturbed observations as a data assimilation method. We consider three Arakawa discretizations of the quasi‐geostrophic model that either preserve energy (Arakawa E), or enstrophy (Arakawa Z), or both (Arakawa EZ). We perform a twin experiment, where observations are generated from the Hamiltonian particle‐mesh (HPM) method, which preserves energy and an infinite number of Casimirs though trivially. Due to the chosen initial conditions and conservation laws of the HPM, the true probability density function (PDF) is skewed, while due to the conservation laws of an Arakawa discretization the modelled PDF is normal. Numerical experiments show that, if observations of stream function are assimilated, the choice of a numerical scheme is crucial for a good reconstruction of the time‐averaged fields and PDF estimation. Arakawa E completely fails to reproduce the true nonlinear behaviour, Arakawa Z is sensitive to localization and inflation, and Arakawa EZ provides the best estimate. If observations of potential vorticity are assimilated, a good time‐averaged field reconstruction is independent of a numerical scheme. The PDF estimations are comparable for all Arakawa discretizations. For obtaining non‐zero skewness, localization has to be applied even for a very large ensemble size of 600. Inflation, however, deteriorates skewness estimation for both small and large ensemble sizes.