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

Hyeonggeun Yun, Quanling Deng

TL;DR

The paper develops an Eulerian data assimilation framework (EuDA) based on the Conditional Gaussian Nonlinear System (CGNS) to filter a two-layer quasi-geostrophic model directly in the physical domain. By formulating the state with coupled vorticity and streamfunction fields, and leveraging CGNS for closed-form conditional updates, the method achieves accurate posterior means with low RMSE and high pattern correlation across grid refinements, while remaining computationally efficient. The approach is demonstrated on both pure QG dynamics and a coupled sea-ice–ocean scenario, including an application where Eulerian fields inform Lagrangian floe trajectories through a linear drag coupling. Collectively, the work shows that EuDA-CGNS provides a scalable, interpretable, and physics-consistent framework for multiscale geophysical data assimilation in physical space, with potential extensions to higher-layer models and Lagrangian–Eulerian data fusion.

Abstract

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.

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

TL;DR

The paper develops an Eulerian data assimilation framework (EuDA) based on the Conditional Gaussian Nonlinear System (CGNS) to filter a two-layer quasi-geostrophic model directly in the physical domain. By formulating the state with coupled vorticity and streamfunction fields, and leveraging CGNS for closed-form conditional updates, the method achieves accurate posterior means with low RMSE and high pattern correlation across grid refinements, while remaining computationally efficient. The approach is demonstrated on both pure QG dynamics and a coupled sea-ice–ocean scenario, including an application where Eulerian fields inform Lagrangian floe trajectories through a linear drag coupling. Collectively, the work shows that EuDA-CGNS provides a scalable, interpretable, and physics-consistent framework for multiscale geophysical data assimilation in physical space, with potential extensions to higher-layer models and Lagrangian–Eulerian data fusion.

Abstract

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.

Paper Structure

This paper contains 15 sections, 19 equations, 7 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem statement and numerical schemes
  3. Two-layer quasi-geostrophic (QG) model
  4. Numerical schemes
  5. Temporal discretization
  6. Spatial discretization
  7. Numerical setup
  8. Eulerian data assimilation (EuDA)
  9. Conditional Gaussian nonlinear system (CGNS)
  10. EuDA for recovery of the vorticity
  11. EuDA for recovery of the second layer
  12. Numerical experiments
  13. EuDA for recovery of the vorticity and the second-layer flow
  14. An application to sea ice floe dynamics with QG ocean flow
  15. Conclusion

Figures (7)

  • Figure 1: Flowchart of the proposed method.
  • Figure 2: Simulation of true and recovered $q$ at $t=1.98$ with sinusoidal streamfunction.
  • Figure 3: Simulation of true and recovered $q$ at $t=1.98$ with Gaussian streamfunction.
  • Figure 4: Simulation of true and recovered $\psi_2$ at $t=1.98$ with sinusoidal streamfunction.
  • Figure 5: Simulation of true and recovered $\psi_2$ at $t=1.98$ with Gaussian streamfunction.
  • ...and 2 more figures