An Eulerian Data Assimilation Method for Two-Layer Quasi-Geostrophic Model in Physical Domain

Data assimilation (DA) integrates observational data with numerical models to improve the prediction of complex physical systems. However, traditional DA methods often struggle with nonlinear dynamics and multi-scale variability, particularly when implemented directly in the physical domain. To address these challenges, this work develops an Eulerian Data Assimilation (EuDA) method with the Conditional Gaussian Nonlinear System (CGNS). The proposed approach enables the treatment of systems with non-periodic boundaries and provides a more intuitive representation of localized and time-dependent phenomena. The work considers a simplified physical domain inspired by sea-ice floe trajectories and ocean eddy recovery in the Arctic regions, where the dynamics are modeled by a two-layer quasi-geostrophic (QG) system. The QG equations are numerically solved using forward-Euler time stepping and centered finite-difference schemes. CGNS provides a nonlinear filter as it offers an analytical and continuous formulation for filtering a nonlinear system. Model performance is assessed using normalized root mean square error (RMSE) and pattern correlation (Corr) of the posterior mean. The results show that both metrics improve monotonically with refining timesteps, while RMSE converges to approximately 0.1, which is the noise strength, and Corr increases from 0.64 to 0.92 as the grid resolution becomes finer. Lastly, a coupled scenario with sea-ice particles advected by the two-layer QG flow under a linear drag force is examined, demonstrating the flexibility of the EuDA-CGNS framework in capturing coupled ice-ocean interactions. These findings demonstrate the effectiveness of exploiting the two-layer QG model in the physical domain to capture multiscale flow features.