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.

翻译：数据同化（DA）通过将观测数据与数值模型相结合，以提升复杂物理系统的预测能力。然而，传统DA方法在处理非线性动力学和多尺度变异性时常常面临困难，尤其是在物理域中直接实施时。为应对这些挑战，本研究开发了一种结合条件高斯非线性系统（CGNS）的欧拉数据同化（EuDA）方法。所提出的方法能够处理具有非周期边界的系统，并为局部化和时间相关现象提供更直观的表征。该工作考虑了一个受北极地区海冰浮冰轨迹和海洋涡旋恢复启发的简化物理域，其中动力学由双层准地转（QG）系统建模。QG方程采用前向欧拉时间步进和中心有限差分格式进行数值求解。CGNS提供了一个非线性滤波器，因为它为非线性系统的滤波提供了解析且连续的公式。模型性能通过后验均值的归一化均方根误差（RMSE）和模式相关性（Corr）进行评估。结果表明，随着时间步长的细化，这两个指标均单调改善：RMSE收敛至约0.1（即噪声强度），而Corr则从0.64增至0.92，同时网格分辨率变得更高。最后，研究考察了在线性拖曳力作用下由双层QG流平流输送的海冰粒子耦合情景，展示了EuDA-CGNS框架在捕捉冰-海洋耦合相互作用方面的灵活性。这些发现证明了在物理域中利用双层QG模型捕捉多尺度流动特征的有效性。