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POD-Galerkin Reduced Order Modeling of the El Nino-Southern Oscillation (ENSO)

Yusuf Aydogdu, Navaratnam Sri Namachchivaya

TL;DR

This work develops a POD-Galerkin reduced order framework for a five-field, Majda-type ENSO PDE model, reducing the dynamics to a compact set of $n=4$ modes. By projecting the last three active components onto POD bases and enforcing a Galerkin projection, the authors derive a low-dimensional system of ODEs that captures the coupled ocean–atmosphere–SST dynamics with high fidelity (over $95\%$ of the full-order model). The approach leverages a functional-analytic POD foundation to guarantee optimal mode selection and demonstrates the ability to reconstruct key ENSO features, including Kelvin and Rossby wave interactions, using a compact reduced representation. The methodology enables efficient climate simulations and provides a basis for future extensions to stochastic forcing and data assimilation via reduced-order filtering.

Abstract

Reduced order modeling (ROM) aims to mitigate computational complexity by reducing the size of a high-dimensional state space. In this study, we demonstrate the efficiency, accuracy, and stability of proper orthogonal decomposition (POD)-Galerkin ROM when applied to the El Nino Southern Oscillation model, which integrates coupled atmosphere, ocean, and sea surface temperature (SST) mechanisms in the equatorial Pacific. While POD identifies the most energetic modes of a system from simulation data, the Galerkin projection maps the governing equations onto these reduced modes to derive a simplified dynamical system. Leveraging the unique coupling properties of the model, we propose a novel approach to formulate a reduced order model derived from Galerkin projection. Our approach achieves remarkable computational efficiency, requiring only four POD modes. The results provide highly stable and accurate solutions over 95% compared to the high-dimensional full-order model (FOM), highlighting the potential of POD-Galerkin reduction for efficient and accurate climate simulations.

POD-Galerkin Reduced Order Modeling of the El Nino-Southern Oscillation (ENSO)

TL;DR

This work develops a POD-Galerkin reduced order framework for a five-field, Majda-type ENSO PDE model, reducing the dynamics to a compact set of modes. By projecting the last three active components onto POD bases and enforcing a Galerkin projection, the authors derive a low-dimensional system of ODEs that captures the coupled ocean–atmosphere–SST dynamics with high fidelity (over of the full-order model). The approach leverages a functional-analytic POD foundation to guarantee optimal mode selection and demonstrates the ability to reconstruct key ENSO features, including Kelvin and Rossby wave interactions, using a compact reduced representation. The methodology enables efficient climate simulations and provides a basis for future extensions to stochastic forcing and data assimilation via reduced-order filtering.

Abstract

Reduced order modeling (ROM) aims to mitigate computational complexity by reducing the size of a high-dimensional state space. In this study, we demonstrate the efficiency, accuracy, and stability of proper orthogonal decomposition (POD)-Galerkin ROM when applied to the El Nino Southern Oscillation model, which integrates coupled atmosphere, ocean, and sea surface temperature (SST) mechanisms in the equatorial Pacific. While POD identifies the most energetic modes of a system from simulation data, the Galerkin projection maps the governing equations onto these reduced modes to derive a simplified dynamical system. Leveraging the unique coupling properties of the model, we propose a novel approach to formulate a reduced order model derived from Galerkin projection. Our approach achieves remarkable computational efficiency, requiring only four POD modes. The results provide highly stable and accurate solutions over 95% compared to the high-dimensional full-order model (FOM), highlighting the potential of POD-Galerkin reduction for efficient and accurate climate simulations.

Paper Structure

This paper contains 13 sections, 6 theorems, 63 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model Description : El Niño Southern Oscillations
  3. Proper Orthogonal Decomposition
  4. POD-Galerkin Method Applied to the ENSO Model
  5. Results
  6. Conclusion and Future Work
  7. Appendix A
  8. Proof of Theorem \ref{['thm opt']}
  9. Operators $\Gamma$ and $\Gamma^\ast$
  10. Changing the problem
  11. Eigenvalues and eigenfunctions of $\Gamma^\ast \Gamma$
  12. Eigenvalues and eigenvectors of $C$
  13. Appendix B

Key Result

Theorem 1

Given the snapshots $w(t_1),\ldots,w(t_M)$ define the $M \times M$ matrix $C$ by The matrix $C$ is symmetric and non-negative, with eigenvalues $\lambda_1\ge \lambda_2 \ge \cdots \ge 0$ (counted with multiplicities) and corresponding orthonormal eigenvectors $\gamma^1, \gamma^2, \ldots$ in $\mathbb{R}^M$. Then the minimization problem mu has solution $\mu_n = \sum_{\alpha > n} \ and

Figures (4)

  • Figure 1: (a) POD Modes $\tilde{\Psi}_{\alpha}(x)$ and (b) POD time coefficients $q_{\alpha}(t)$
  • Figure 2: (a) Projected initial conditions (b) reduced order equations $a_{\alpha}(t)$ from POD-Galerkin projection and (c) reconstruction of SST time series at the right boundary with different number of modes
  • Figure 3: Full order model (FOM), POD and POD-Galerkin (POD-G) reconstructions of SST component $T(t,x)$ using four modes
  • Figure 4: Reconstruction errors ($L1$) of full order SST component $T(t,x)$ using POD and POD-Galerkin methods with four modes

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Corollary 7.1
  • Remark 7.1
  • Proposition 7.1