The problems of identifying the slow component (e.g., for weather forecast initialization) and of characterizing slow–fast interactions are central to geophysical fluid dynamics. In this
study, the related rectification problem of slow manifold closures is addressed when breakdown of slow-to-fast scales deterministic parameterizations occurs due to explosive emergence of fast oscillations on the slow, geostrophic motion. For such regimes, it is shown on the Lorenz 80 model that if 1) the underlying manifold provides a good approximation of the optimal nonlinear parameterization that averages out the fast variables and 2) the residual dynamics off this manifold is mainly orthogonal to it, then no memory terms are required in the Mori–Zwanzig full closure. Instead, the noise term is key to resolve, and is shown to be, in this case, well modeled by a state-independent noise, obtained by means of networks of stochastic nonlinear oscillators. This stochastic parameterization allows, in turn, for rectifying the momentum-balanced slow manifold, and for accurate recovery of the multiscale dynamics. The approach is promising to be further applied to the closure of other more complex slow–fast systems, in strongly coupled regimes.
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Optimal parameterizing manifolds for anticipating tipping points and higher-order critical transitions
A general, variational approach to derive low-order reduced models from possibly non-autonomous systems is presented. The approach is based on the concept of optimal parameterizing manifold (OPM) that substitutes more classical notions of invariant or slow manifolds when the breakdown of “slaving” occurs, i.e., when the unresolved variables cannot be expressed as an exact functional of the resolved ones anymore. The OPM provides, within a given class of parameterizations of the unresolved variables, the manifold that averages out optimally these variables as conditioned on the resolved ones. The class of parameterizations retained here is that of continuous deformations of parameterizations rigorously valid near the onset of instability. These deformations are produced through the integration of auxiliary backward–forward systems built from the model’s equations and lead to analytic formulas for parameterizations. In this modus operandi, the backward integration time is the key parameter to select per scale/variable to parameterize in order to derive the relevant parameterizations which are doomed to be no longer exact away from instability onset due to the breakdown of slaving typically encountered, e.g., for chaotic regimes. The selection criterion is then made through data-informed minimization of a least-square parameterization defect. It is thus shown through optimization of the backward integration time per scale/variable to parameterize, that skilled OPM reduced systems can be derived for predicting with accuracy higher-order critical transitions or catastrophic tipping phenomena, while training our parameterization formulas for regimes prior to these transitions takes place.
more » « less- Award ID(s):
- 2108856
- NSF-PAR ID:
- 10509767
- Publisher / Repository:
- AIP Publishing
- Date Published:
- Journal Name:
- Chaos: An Interdisciplinary Journal of Nonlinear Science
- Volume:
- 33
- ISSN:
- 1054-1500
- Page Range / eLocation ID:
- 093126
- Subject(s) / Keyword(s):
- Tipping points variational parameterizations model reduction The Stommel–Cessi model
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Physical parameterizations (or closures) are used as representations of unresolved subgrid processes within weather and global climate models or coarse-scale turbulent models, whose resolutions are too coarse to resolve small-scale processes. These parameterizations are typically grounded on physically based, yet empirical, representations of the underlying small-scale processes. Machine learning-based parameterizations have recently been proposed as an alternative solution and have shown great promise to reduce uncertainties associated with the parameterization of small-scale processes. Yet, those approaches still show some important mismatches that are often attributed to the stochasticity of the considered process. This stochasticity can be due to coarse temporal resolution, unresolved variables, or simply to the inherent chaotic nature of the process. To address these issues, we propose a new type of parameterization (closure), which is built using memory-based neural networks, to account for the non-instantaneous response of the closure and to enhance its stability and prediction accuracy. We apply the proposed memory-based parameterization, with differentiable solver, to the Lorenz ’96 model in the presence of a coarse temporal resolution and show its capacity to predict skillful forecasts over a long time horizon of the resolved variables compared to instantaneous parameterizations. This approach paves the way for the use of memory-based parameterizations for closure problems.
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Abstract This study utilizes Deep Neural Networks (DNN) to improve the K‐Profile Parameterization (KPP) for the vertical mixing effects in the ocean's surface boundary layer turbulence. The deep neural networks were trained using 11‐year turbulence‐resolving solutions, obtained by running a large eddy simulation model for Ocean Station Papa, to predict the turbulence velocity scale coefficient and unresolved shear coefficient in the KPP. The DNN‐augmented KPP schemes (KPP_DNN) have been implemented in the General Ocean Turbulence Model (GOTM). The KPP_DNN is stable for long‐term integration and more efficient than existing variants of KPP schemes with wave effects. Three different KPP_DNN schemes, each differing in their input and output variables, have been developed and trained. The performance of models utilizing the KPP_DNN schemes is compared to those employing traditional deterministic first‐order and second‐moment closure turbulent mixing parameterizations. Solution comparisons indicate that the simulated mixed layer becomes cooler and deeper when wave effects are included in parameterizations, aligning closer with observations. In the KPP framework, the velocity scale of unresolved shear, which is used to calculate ocean surface boundary layer depth, has a greater impact on the simulated mixed layer than the magnitude of diffusivity does. In the KPP_DNN, unresolved shear depends not only on wave forcing, but also on the mixed layer depth and buoyancy forcing.
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1063/5.0167419
