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Structure-preserving stochastic parameterization of a barotropic coupled ocean-atmosphere model with Ornstein--Uhlenbeck noise

Kamal Kishor Sharma, Peter Korn

Abstract

We present the first application of the stochastic advection by Lie transport (SALT) framework to an idealized coupled ocean-atmosphere system. SALT derives stochastic fluid equations from Hamilton's variational principle under a stochastic Lagrangian kinematic assumption, thereby preserving the geometric structure -- Kelvin circulation theorem, Lie-derivative advection operators, and local conservation laws -- of the underlying deterministic equations. The atmospheric component is rendered stochastic while the ocean remains deterministic, following Hasselmann's paradigm of a fast stochastic atmosphere driving a slow climate. The spatial correlation vectors encoding unresolved subgrid transport are estimated from high-resolution simulations via EOF analysis of Lagrangian trajectory differences. A central contribution is the replacement of the standard white-noise temporal model with Ornstein-Uhlenbeck (OU) processes, motivated by strong autocorrelation (decorrelation times of 50-150 time steps) in the dominant EOF modes; the OU process is the unique stationary Gaussian Markov process capable of capturing this temporal memory with a single parameter. Ensemble forecasts exhibit good spread-error agreement over 10-15 time units. Evaluated via the Continuous Ranked Probability Score -- a strictly proper scoring rule measuring discrepancy between the forecast measure and the true conditional distribution -- the stochastic ensemble consistently outperforms a size-matched deterministic ensemble, despite carrying higher RMSE. Well-posedness of the coupled stochastic system is identified as an important open problem.

Structure-preserving stochastic parameterization of a barotropic coupled ocean-atmosphere model with Ornstein--Uhlenbeck noise

Abstract

We present the first application of the stochastic advection by Lie transport (SALT) framework to an idealized coupled ocean-atmosphere system. SALT derives stochastic fluid equations from Hamilton's variational principle under a stochastic Lagrangian kinematic assumption, thereby preserving the geometric structure -- Kelvin circulation theorem, Lie-derivative advection operators, and local conservation laws -- of the underlying deterministic equations. The atmospheric component is rendered stochastic while the ocean remains deterministic, following Hasselmann's paradigm of a fast stochastic atmosphere driving a slow climate. The spatial correlation vectors encoding unresolved subgrid transport are estimated from high-resolution simulations via EOF analysis of Lagrangian trajectory differences. A central contribution is the replacement of the standard white-noise temporal model with Ornstein-Uhlenbeck (OU) processes, motivated by strong autocorrelation (decorrelation times of 50-150 time steps) in the dominant EOF modes; the OU process is the unique stationary Gaussian Markov process capable of capturing this temporal memory with a single parameter. Ensemble forecasts exhibit good spread-error agreement over 10-15 time units. Evaluated via the Continuous Ranked Probability Score -- a strictly proper scoring rule measuring discrepancy between the forecast measure and the true conditional distribution -- the stochastic ensemble consistently outperforms a size-matched deterministic ensemble, despite carrying higher RMSE. Well-posedness of the coupled stochastic system is identified as an important open problem.

Paper Structure

This paper contains 15 sections, 14 equations, 11 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The climate model
  3. Numerical simulations
  4. High-resolution reference simulation
  5. Model calibration and OU noise
  6. OU versus Gaussian noise.
  7. Stochastic model simulation
  8. Consistency with ensemble size and EOF truncation
  9. Stochastic versus deterministic ensemble
  10. CRPS and approximation of the forecast distribution
  11. Conclusion
  12. Non-dimensionalization
  13. Estimation of correlation vectors
  14. AR(1) model for discrete OU processes
  15. Discretization

Figures (11)

  • Figure 1: Time series of atmospheric kinetic energy. Statistical equilibrium is reached around $t=25$.
  • Figure 2: Atmosphere (top two panels) and ocean (bottom two panels) fields at $t=25$, showing multi-scale eddy structure and the onset of statistical equilibrium.
  • Figure 3: Atmospheric vorticity during the equilibrium window ($t=25$--$45$). Eddy counts and structure remain quasi-stationary, confirming suitability for calibration.
  • Figure 4: Time series $a_i(t)$ for the leading modes ($\boldsymbol{\xi}_{1}$--$\boldsymbol{\xi}_{3}$, left) and high modes ($\boldsymbol{\xi}_{51}$--$\boldsymbol{\xi}_{53}$, centre), compared with a Gaussian process realization (right). Leading modes are strongly autocorrelated.
  • Figure 5: Autocorrelation functions of $a_i(t)$: leading modes (left) show decorrelation times of 50--150 steps; high modes (centre) and Gaussian noise (right) decorrelate within a single step.
  • ...and 6 more figures

Theorems & Definitions (1)

  • Remark 1.1: Well-posedness