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) framework with the Conditional Gaussian Nonlinear System (CGNS). The proposed approach enables the treatment of non-periodic systems and provides a more intuitive representation of localized and time-dependent phenomena. The work considers a physical domain inspired by sea-ice floe trajectories and ocean eddy recovery in the Arctic regions, where the model 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 increasing timesteps, while RMSE converges to approximately 0.1 across all grid sizes and Corr increases from 0.64 to 0.92 as 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，以改进复杂物理系统的预测。然而，传统的DA方法在处理非线性动力学和多尺度变率时常常面临困难，特别是在物理域中直接实施时。为应对这些挑战，本研究结合条件高斯非线性系统（CGNS）发展了一种欧拉数据同化（EuDA）框架。所提出的方法能够处理非周期系统，并为局部化和时间相关现象提供更直观的表征。本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物理域，其中模型动力学由两层准地转（QG）系统描述。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）和模式相关系数（Corr）进行评估。结果表明，随着时间步长的增加，两个指标均单调改善，其中RMSE在所有网格尺寸下收敛至约0.1，而Corr则随着网格分辨率提高从0.64增至0.92。最后，研究考察了海冰粒子在线性拖曳力作用下由两层QG流平流的耦合情景，展示了EuDA-CGNS框架在捕捉海冰-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两层QG模型捕捉多尺度流动特征的有效性。