2025
Chekroun, Mickaël D.; Liu, Honghu; McWilliams, James C.
Non-Markovian Reduced Models to Unravel Transitions in Non-equilibrium Systems Journal Article
In: Journal of Physics A: Mathematical and Theoretical, vol. 58, no. 4, pp. 045204, 2025.
@article{2734,
title = {Non-Markovian Reduced Models to Unravel Transitions in Non-equilibrium Systems},
author = {Mickaël D. Chekroun and Honghu Liu and James C. McWilliams},
url = {https://doi.org/10.1088/1751-8121/ada7ad},
year = {2025},
date = {2025-01-01},
journal = {Journal of Physics A: Mathematical and Theoretical},
volume = {58},
number = {4},
pages = {045204},
abstract = {This work proposes a general framework for analyzing noise-driven transitions in spatially extended non-equilibrium systems and explains the emergence of coherent patterns beyond the instability onset. The framework relies on stochastic parameterization formulas to reduce the complexity of the original equations while preserving the essential dynamical effects of unresolved scales. The approach is flexible and operates for both Gaussian noise and non-Gaussian noise with jumps.
Our stochastic parameterization formulas offer two key advantages. First, they can approximate stochastic invariant manifolds when these manifolds exist. Second, even when such manifolds break down, our formulas can be adapted through a simple optimization of its constitutive parameters. This allows us to handle scenarios with weak time-scale separation where the system has undergone multiple transitions, resulting in large-amplitude solutions not captured by invariant manifolds or other time-scale separation methods.
The optimized stochastic parameterizations capture then how small-scale noise impacts larger scales through the system’s nonlinear interactions. This effect is achieved by the very fabric of our parameterizations incorporating non-Markovian (memory-dependent) coefficients into the reduced equation. These coefficients account for the noise’s past influence, not just its current value, using a finite memory length that is selected for optimal performance. The specific "memory" function, which determines how this past influence is weighted, depends on both the strength of the noise and how it interacts with the system’s nonlinearities.
Remarkably, training our theory-guided reduced models on a single noise path effectively learns the optimal memory length for out-of-sample predictions. This approach retains indeed good accuracy in predicting noise-induced transitions, including rare events, when tested against a large ensemble of different noise paths. This success stems from our ‘‘hybrid" approach, which combines analytical understanding with data-driven learning. This combination avoids a key limitation of purely data-driven methods: their struggle to generalize to unseen scenarios, also known as the "extrapolation problem."
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This work proposes a general framework for analyzing noise-driven transitions in spatially extended non-equilibrium systems and explains the emergence of coherent patterns beyond the instability onset. The framework relies on stochastic parameterization formulas to reduce the complexity of the original equations while preserving the essential dynamical effects of unresolved scales. The approach is flexible and operates for both Gaussian noise and non-Gaussian noise with jumps.
Our stochastic parameterization formulas offer two key advantages. First, they can approximate stochastic invariant manifolds when these manifolds exist. Second, even when such manifolds break down, our formulas can be adapted through a simple optimization of its constitutive parameters. This allows us to handle scenarios with weak time-scale separation where the system has undergone multiple transitions, resulting in large-amplitude solutions not captured by invariant manifolds or other time-scale separation methods.
The optimized stochastic parameterizations capture then how small-scale noise impacts larger scales through the system’s nonlinear interactions. This effect is achieved by the very fabric of our parameterizations incorporating non-Markovian (memory-dependent) coefficients into the reduced equation. These coefficients account for the noise’s past influence, not just its current value, using a finite memory length that is selected for optimal performance. The specific "memory" function, which determines how this past influence is weighted, depends on both the strength of the noise and how it interacts with the system’s nonlinearities.
Remarkably, training our theory-guided reduced models on a single noise path effectively learns the optimal memory length for out-of-sample predictions. This approach retains indeed good accuracy in predicting noise-induced transitions, including rare events, when tested against a large ensemble of different noise paths. This success stems from our ‘‘hybrid" approach, which combines analytical understanding with data-driven learning. This combination avoids a key limitation of purely data-driven methods: their struggle to generalize to unseen scenarios, also known as the "extrapolation problem."
Our stochastic parameterization formulas offer two key advantages. First, they can approximate stochastic invariant manifolds when these manifolds exist. Second, even when such manifolds break down, our formulas can be adapted through a simple optimization of its constitutive parameters. This allows us to handle scenarios with weak time-scale separation where the system has undergone multiple transitions, resulting in large-amplitude solutions not captured by invariant manifolds or other time-scale separation methods.
The optimized stochastic parameterizations capture then how small-scale noise impacts larger scales through the system’s nonlinear interactions. This effect is achieved by the very fabric of our parameterizations incorporating non-Markovian (memory-dependent) coefficients into the reduced equation. These coefficients account for the noise’s past influence, not just its current value, using a finite memory length that is selected for optimal performance. The specific "memory" function, which determines how this past influence is weighted, depends on both the strength of the noise and how it interacts with the system’s nonlinearities.
Remarkably, training our theory-guided reduced models on a single noise path effectively learns the optimal memory length for out-of-sample predictions. This approach retains indeed good accuracy in predicting noise-induced transitions, including rare events, when tested against a large ensemble of different noise paths. This success stems from our ‘‘hybrid" approach, which combines analytical understanding with data-driven learning. This combination avoids a key limitation of purely data-driven methods: their struggle to generalize to unseen scenarios, also known as the "extrapolation problem."
Chekroun, Mickaël D.; Zagli, Niccolò; Lucarini, Valerio
Kolmogorov modes and linear response of jump-diffusion models Journal Article
In: Reports on Progress in Physics, vol. 88, pp. 127601, 2025.
@article{chekroun2025kolmogorov,
title = {Kolmogorov modes and linear response of jump-diffusion models},
author = {Mickaël D. Chekroun and Niccolò Zagli and Valerio Lucarini},
url = {https://doi.org/10.1088/1361-6633/ae2206},
doi = {10.1088/1361-6633/ae2206},
year = {2025},
date = {2025-01-01},
urldate = {2025-01-01},
journal = {Reports on Progress in Physics},
volume = {88},
pages = {127601},
abstract = {We present a generalization of linear response theory(LRT) for mixed jump-diffusion models—which combine both Gaussian and Lévy noise forcings that interact with the nonlinear dynamics—by deriving a comprehensive set of response formulas that accounts for perturbations to both the drift term and the jumps law. This class of models is particularly relevant for parameterizing the effects of unresolved scales in complex systems. Our formulas help thus quantifying uncertainties in either what needs to be parameterized (e.g. the jumps law), or measuring dynamical changes due to perturbations of the drift term (e.g. parameter variations). By generalizing the concepts of Kolmogorov operators and Green’s functions, we obtain new forms of fluctuation-dissipation relations. The resulting response is decomposed into contributions from the eigenmodes of the Kolmogorov operator, providing a fresh look into the intimate relationship between a system’s natural and forced variability. We demonstrate the theory’s predictive power with two distinct climate-centric applications. First, we apply our framework to a paradigmatic El Nin ̃o-Southern Oscillation model subject to state-dependent jumps and additive white noise, showing how the theory accurately predicts the system’s response to perturbations and how Kolmogorov modes can be used to diagnose its complex time variability. In a second, more challenging application, we use our LRT to perform accurate climate change projections in the Ghil–Sellers energy balance climate model, which is a spatially-extended model forced here by a spatio-temporal α-stable process. This work provides a comprehensive approach to climate modeling and prediction that enriches Hasselmann’s program, with implications for understanding climate sensitivity, detection and attribution of climate change, and assessing the risk of climate tipping points. Our results may find applications beyond the realm of climate, and seem of relevance for epidemiology, biology, finance, and quantitative social sciences, among others.},
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We present a generalization of linear response theory(LRT) for mixed jump-diffusion models—which combine both Gaussian and Lévy noise forcings that interact with the nonlinear dynamics—by deriving a comprehensive set of response formulas that accounts for perturbations to both the drift term and the jumps law. This class of models is particularly relevant for parameterizing the effects of unresolved scales in complex systems. Our formulas help thus quantifying uncertainties in either what needs to be parameterized (e.g. the jumps law), or measuring dynamical changes due to perturbations of the drift term (e.g. parameter variations). By generalizing the concepts of Kolmogorov operators and Green’s functions, we obtain new forms of fluctuation-dissipation relations. The resulting response is decomposed into contributions from the eigenmodes of the Kolmogorov operator, providing a fresh look into the intimate relationship between a system’s natural and forced variability. We demonstrate the theory’s predictive power with two distinct climate-centric applications. First, we apply our framework to a paradigmatic El Nin ̃o-Southern Oscillation model subject to state-dependent jumps and additive white noise, showing how the theory accurately predicts the system’s response to perturbations and how Kolmogorov modes can be used to diagnose its complex time variability. In a second, more challenging application, we use our LRT to perform accurate climate change projections in the Ghil–Sellers energy balance climate model, which is a spatially-extended model forced here by a spatio-temporal α-stable process. This work provides a comprehensive approach to climate modeling and prediction that enriches Hasselmann’s program, with implications for understanding climate sensitivity, detection and attribution of climate change, and assessing the risk of climate tipping points. Our results may find applications beyond the realm of climate, and seem of relevance for epidemiology, biology, finance, and quantitative social sciences, among others.
2024
Koren, Ilan; Dror, Tom; Altaratz, Orit; Chekroun, Mickaël D.
Cloud Versus Void Chord Length Distributions (LvL) as a Measure for Cloud Field Organization Journal Article
In: Geophysical Research Letters, vol. 51, no. 11, pp. e2024GL108435., 2024.
@article{2536,
title = {Cloud Versus Void Chord Length Distributions (LvL) as a Measure for Cloud Field Organization},
author = {Ilan Koren and Tom Dror and Orit Altaratz and Mickaël D. Chekroun},
url = {https://doi.org/10.1029/2024GL108435},
year = {2024},
date = {2024-01-01},
journal = {Geophysical Research Letters},
volume = {51},
number = {11},
pages = {e2024GL108435.},
abstract = {Cloud organization impacts the radiative effects and precipitation patterns of the cloud field. Deviating from randomness, clouds exhibit either clustering or a regular grid structure, characterized by the spacing between clouds and the cloud size distribution. The two measures are coupled but do not fully define each other. Here, we present the deviation from randomness of the cloud- and void-chord length distributions as a measure for both factors. We introduce the LvL representation and an associated 2D score that allow for unambiguously quantifying departure from well-defined baseline randomness in cloud spacing and sizes. This approach demonstrates sensitivity and robustness in classifying cloud field organization types. Its delicate sensitivity unravels the temporal evolution of a single cloud field, providing novel insights into the underlying governing processes.},
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Cloud organization impacts the radiative effects and precipitation patterns of the cloud field. Deviating from randomness, clouds exhibit either clustering or a regular grid structure, characterized by the spacing between clouds and the cloud size distribution. The two measures are coupled but do not fully define each other. Here, we present the deviation from randomness of the cloud- and void-chord length distributions as a measure for both factors. We introduce the LvL representation and an associated 2D score that allow for unambiguously quantifying departure from well-defined baseline randomness in cloud spacing and sizes. This approach demonstrates sensitivity and robustness in classifying cloud field organization types. Its delicate sensitivity unravels the temporal evolution of a single cloud field, providing novel insights into the underlying governing processes.
Gutiérrez, Manuel Santos; Chekroun, Mickaël D.; Koren, Ilan
Gibbs states and Brownian models for haze and cloud droplets Journal Article
In: Science Advances, vol. 10, no. 46, pp. eadq7518, 2024.
@article{2535,
title = {Gibbs states and Brownian models for haze and cloud droplets},
author = {Manuel Santos Gutiérrez and Mickaël D. Chekroun and Ilan Koren},
url = {https://www.science.org/doi/10.1126/sciadv.adq7518},
year = {2024},
date = {2024-01-01},
journal = {Science Advances},
volume = {10},
number = {46},
pages = {eadq7518},
abstract = {Cloud microphysics studies include how tiny cloud droplets grow and become rain. This is crucial for understanding cloud properties like size, life span, and impact on climate through radiative effects. Small weak-updraft clouds near the haze-to-cloud transition are especially difficult to measure and understand. They are abundant but hard to capture by satellites. Köhler’s theory explains initial droplet growth but struggles with large particle groups. Here, we present a stochastic, analytical framework building on Köhler’s theory to account for (monodisperse) aerosols and cloud droplet interaction through competitive growth in a limited water vapor field. These interactions are modeled by sink terms, while fluctuations in supersaturation affecting droplet growth are modeled by nonlinear white noise terms. Our results identify hysteresis mechanisms in the droplet activation and deactivation processes. Our approach allows for multimodal cloud’s droplet size distributions supported by laboratory experiments, offering a different perspective on haze-to-cloud transition and small cloud formation.},
keywords = {},
pubstate = {published},
tppubtype = {article}
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Cloud microphysics studies include how tiny cloud droplets grow and become rain. This is crucial for understanding cloud properties like size, life span, and impact on climate through radiative effects. Small weak-updraft clouds near the haze-to-cloud transition are especially difficult to measure and understand. They are abundant but hard to capture by satellites. Köhler’s theory explains initial droplet growth but struggles with large particle groups. Here, we present a stochastic, analytical framework building on Köhler’s theory to account for (monodisperse) aerosols and cloud droplet interaction through competitive growth in a limited water vapor field. These interactions are modeled by sink terms, while fluctuations in supersaturation affecting droplet growth are modeled by nonlinear white noise terms. Our results identify hysteresis mechanisms in the droplet activation and deactivation processes. Our approach allows for multimodal cloud’s droplet size distributions supported by laboratory experiments, offering a different perspective on haze-to-cloud transition and small cloud formation.
Chekroun, Mickaël D.; Liu, Honghu
Effective reduced models from delay differential equations: Bifurcations, tipping solution paths, and ENSO variability Journal Article
In: Physica D, vol. 460, pp. 134058, 2024.
@article{2516,
title = {Effective reduced models from delay differential equations: Bifurcations, tipping solution paths, and ENSO variability},
author = {Mickaël D. Chekroun and Honghu Liu},
url = {https://doi.org/10.1016/j.physd.2024.134058},
year = {2024},
date = {2024-01-01},
journal = {Physica D},
volume = {460},
pages = {134058},
abstract = {Conceptual delay models have played a key role in the analysis and understanding of El Niño-Southern Oscillation (ENSO) variability. Based on such delay models, we propose in this work a novel scenario for the fabric of ENSO variability resulting from the subtle interplay between stochastic disturbances and nonlinear invariant sets emerging from bifurcations of the unperturbed dynamics.
To identify these invariant sets we adopt an approach combining Galerkin-Koornwinder (GK) approximations of delay differential equations and center-unstable manifold reduction techniques. In that respect, GK approximation formulas are reviewed and synthesized, as well as analytic approximation formulas of center-unstable manifolds. The reduced systems derived thereof enable us to conduct a thorough analysis of the bifurcations arising in a standard delay model of ENSO. We identify thereby a saddle-node bifurcation of periodic orbits co-existing with a subcritical Hopf bifurcation, and a homoclinic bifurcation for this model. We show furthermore that the computation of unstable periodic orbits (UPOs) unfolding through these bifurcations is considerably simplified from the reduced systems.
These dynamical insights enable us in turn to design a stochastic model whose solutions—as the delay parameter drifts slowly through its critical values—produce a wealth of temporal patterns resembling ENSO events and exhibiting also decadal variability. Our analysis dissects the origin of this variability and shows how it is tied to certain transition paths between invariant sets of the unperturbed dynamics (for ENSO’s interannual variability) or simply due to the presence of UPOs close to the homoclinic orbit (for decadal variability). In short, this study points out the role of solution paths evolving through tipping “points” beyond equilibria, as possible mechanisms organizing the variability of certain climate phenomena.},
keywords = {},
pubstate = {published},
tppubtype = {article}
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Conceptual delay models have played a key role in the analysis and understanding of El Niño-Southern Oscillation (ENSO) variability. Based on such delay models, we propose in this work a novel scenario for the fabric of ENSO variability resulting from the subtle interplay between stochastic disturbances and nonlinear invariant sets emerging from bifurcations of the unperturbed dynamics.
To identify these invariant sets we adopt an approach combining Galerkin-Koornwinder (GK) approximations of delay differential equations and center-unstable manifold reduction techniques. In that respect, GK approximation formulas are reviewed and synthesized, as well as analytic approximation formulas of center-unstable manifolds. The reduced systems derived thereof enable us to conduct a thorough analysis of the bifurcations arising in a standard delay model of ENSO. We identify thereby a saddle-node bifurcation of periodic orbits co-existing with a subcritical Hopf bifurcation, and a homoclinic bifurcation for this model. We show furthermore that the computation of unstable periodic orbits (UPOs) unfolding through these bifurcations is considerably simplified from the reduced systems.
These dynamical insights enable us in turn to design a stochastic model whose solutions—as the delay parameter drifts slowly through its critical values—produce a wealth of temporal patterns resembling ENSO events and exhibiting also decadal variability. Our analysis dissects the origin of this variability and shows how it is tied to certain transition paths between invariant sets of the unperturbed dynamics (for ENSO’s interannual variability) or simply due to the presence of UPOs close to the homoclinic orbit (for decadal variability). In short, this study points out the role of solution paths evolving through tipping “points” beyond equilibria, as possible mechanisms organizing the variability of certain climate phenomena.
To identify these invariant sets we adopt an approach combining Galerkin-Koornwinder (GK) approximations of delay differential equations and center-unstable manifold reduction techniques. In that respect, GK approximation formulas are reviewed and synthesized, as well as analytic approximation formulas of center-unstable manifolds. The reduced systems derived thereof enable us to conduct a thorough analysis of the bifurcations arising in a standard delay model of ENSO. We identify thereby a saddle-node bifurcation of periodic orbits co-existing with a subcritical Hopf bifurcation, and a homoclinic bifurcation for this model. We show furthermore that the computation of unstable periodic orbits (UPOs) unfolding through these bifurcations is considerably simplified from the reduced systems.
These dynamical insights enable us in turn to design a stochastic model whose solutions—as the delay parameter drifts slowly through its critical values—produce a wealth of temporal patterns resembling ENSO events and exhibiting also decadal variability. Our analysis dissects the origin of this variability and shows how it is tied to certain transition paths between invariant sets of the unperturbed dynamics (for ENSO’s interannual variability) or simply due to the presence of UPOs close to the homoclinic orbit (for decadal variability). In short, this study points out the role of solution paths evolving through tipping “points” beyond equilibria, as possible mechanisms organizing the variability of certain climate phenomena.
Lucarini, Valerio; Chekroun, Mickaël D.
Detecting and Attributing Change in Climate and Complex Systems: Foundations, Green’s Functions, and Nonlinear Fingerprints Journal Article
In: Physical Review Letters, vol. 133, pp. 244201, 2024.
@article{2515,
title = {Detecting and Attributing Change in Climate and Complex Systems: Foundations, Green’s Functions, and Nonlinear Fingerprints},
author = {Valerio Lucarini and Mickaël D. Chekroun},
url = {https://doi.org/10.1103/PhysRevLett.133.244201},
year = {2024},
date = {2024-01-01},
journal = {Physical Review Letters},
volume = {133},
pages = {244201},
abstract = {Detection and attribution (DA) studies are cornerstones of climate science, providing crucial evidence for policy decisions. Their goal is to link observed climate change patterns to anthropogenic and natural drivers via the optimal fingerprinting method (OFM). We show that response theory for nonequilibrium systems offers the physical and dynamical basis for OFM, including the concept of causality used for attribution. Our framework clarifies the method’s assumptions, advantages, and potential weaknesses. We use our theory to perform DA for prototypical climate change experiments performed on an energy balance model and on a low-resolution coupled climate model. We also explain the underpinnings of degenerate fingerprinting, which offers early warning indicators for tipping points. Finally, we extend the OFM to the nonlinear response regime. Our analysis shows that OFM has broad applicability across diverse stochastic systems influenced by time-dependent forcings, with potential relevance to ecosystems, quantitative social sciences, and finance, among others.
},
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Detection and attribution (DA) studies are cornerstones of climate science, providing crucial evidence for policy decisions. Their goal is to link observed climate change patterns to anthropogenic and natural drivers via the optimal fingerprinting method (OFM). We show that response theory for nonequilibrium systems offers the physical and dynamical basis for OFM, including the concept of causality used for attribution. Our framework clarifies the method’s assumptions, advantages, and potential weaknesses. We use our theory to perform DA for prototypical climate change experiments performed on an energy balance model and on a low-resolution coupled climate model. We also explain the underpinnings of degenerate fingerprinting, which offers early warning indicators for tipping points. Finally, we extend the OFM to the nonlinear response regime. Our analysis shows that OFM has broad applicability across diverse stochastic systems influenced by time-dependent forcings, with potential relevance to ecosystems, quantitative social sciences, and finance, among others.
Chekroun, Mickaël D.; Liu, H.; Srinivasan, K.; McWilliams, James C.
The High-Frequency and Rare Events Barriers to Neural Closures of Atmospheric Dynamics Journal Article
In: Journal of Physics: Complexity, vol. 5, pp. 025004, 2024.
@article{2403,
title = {The High-Frequency and Rare Events Barriers to Neural Closures of Atmospheric Dynamics},
author = {Mickaël D. Chekroun and H. Liu and K. Srinivasan and James C. McWilliams},
url = {https://iopscience.iop.org/article/10.1088/2632-072X/ad3e59/meta},
year = {2024},
date = {2024-01-01},
journal = {Journal of Physics: Complexity},
volume = {5},
pages = {025004},
abstract = {Recent years have seen a surge in interest for leveraging neural networks to parameterize small-scale or fast processes in climate and turbulence models. In this short paper, we point out two fundamental issues in this endeavor. The first concerns the difficulties neural networks may experience in capturing rare events due to limitations in how data is sampled. The second arises from the inherent multiscale nature of these systems. They combine high-frequency components (like inertia-gravity waves) with slower, evolving processes (geostrophic motion). This multiscale nature creates a significant hurdle for neural network closures. To illustrate these challenges, we focus on the atmospheric 1980 Lorenz model, a simplified version of the Primitive Equations that drive climate models. This model serves as a compelling example because it captures the essence of these difficulties.},
keywords = {},
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Recent years have seen a surge in interest for leveraging neural networks to parameterize small-scale or fast processes in climate and turbulence models. In this short paper, we point out two fundamental issues in this endeavor. The first concerns the difficulties neural networks may experience in capturing rare events due to limitations in how data is sampled. The second arises from the inherent multiscale nature of these systems. They combine high-frequency components (like inertia-gravity waves) with slower, evolving processes (geostrophic motion). This multiscale nature creates a significant hurdle for neural network closures. To illustrate these challenges, we focus on the atmospheric 1980 Lorenz model, a simplified version of the Primitive Equations that drive climate models. This model serves as a compelling example because it captures the essence of these difficulties.
Srinivasan, Kaushik; Chekroun, Mickaël D.; McWilliams, James C.
Turbulence closure with small, local neural networks: Forced two-dimensional and β-plane flows Journal Article
In: Journal of Advances in Modeling Earth Systems, vol. 16, no. 4, pp. e2023MS003795, 2024.
@article{2397,
title = {Turbulence closure with small, local neural networks: Forced two-dimensional and β-plane flows},
author = {Kaushik Srinivasan and Mickaël D. Chekroun and James C. McWilliams},
url = {https://doi.org/10.1029/2023MS003795},
year = {2024},
date = {2024-01-01},
journal = {Journal of Advances in Modeling Earth Systems},
volume = {16},
number = {4},
pages = {e2023MS003795},
abstract = {We parameterize sub-grid scale (SGS) fluxes in sinusoidally forced two-dimensional turbulence on the β-plane at high Reynolds numbers (Re ~25,000) using simple 2-layer convolutional neural networks (CNN) having only O(1000) parameters, two orders of magnitude smaller than recent studies employing deeper CNNs with 8–10 layers; we obtain stable, accurate, and long-term online or a posteriori solutions at 16× downscaling factors. Our methodology significantly improves training efficiency and speed of online large eddy simulations runs, while offering insights into the physics of closure in such turbulent flows. Our approach benefits from extensive hyperparameter searching in learning rate and weight decay coefficient space, as well as the use of cyclical learning rate annealing, which leads to more robust and accurate online solutions compared to fixed learning rates. Our CNNs use either the coarse velocity or the vorticity and strain fields as inputs, and output the two components of the deviatoric stress tensor, Sd. We minimize a loss between the SGS vorticity flux divergence (computed from the high-resolution solver) and that obtained from the CNN-modeled Sd, without requiring energy or enstrophy preserving constraints. The success of shallow CNNs in accurately parameterizing this class of turbulent flows implies that the SGS stresses have a weak non-local dependence on coarse fields; it also aligns with our physical conception that small-scales are locally controlled by larger scales such as vortices and their strained filaments. Furthermore, 2-layer CNN-parameterizations are more likely to be interpretable.
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We parameterize sub-grid scale (SGS) fluxes in sinusoidally forced two-dimensional turbulence on the β-plane at high Reynolds numbers (Re ~25,000) using simple 2-layer convolutional neural networks (CNN) having only O(1000) parameters, two orders of magnitude smaller than recent studies employing deeper CNNs with 8–10 layers; we obtain stable, accurate, and long-term online or a posteriori solutions at 16× downscaling factors. Our methodology significantly improves training efficiency and speed of online large eddy simulations runs, while offering insights into the physics of closure in such turbulent flows. Our approach benefits from extensive hyperparameter searching in learning rate and weight decay coefficient space, as well as the use of cyclical learning rate annealing, which leads to more robust and accurate online solutions compared to fixed learning rates. Our CNNs use either the coarse velocity or the vorticity and strain fields as inputs, and output the two components of the deviatoric stress tensor, Sd. We minimize a loss between the SGS vorticity flux divergence (computed from the high-resolution solver) and that obtained from the CNN-modeled Sd, without requiring energy or enstrophy preserving constraints. The success of shallow CNNs in accurately parameterizing this class of turbulent flows implies that the SGS stresses have a weak non-local dependence on coarse fields; it also aligns with our physical conception that small-scales are locally controlled by larger scales such as vortices and their strained filaments. Furthermore, 2-layer CNN-parameterizations are more likely to be interpretable.
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2023
Chekroun, Mickaël D.; Liu, Honghu; McWilliams, James C.
Optimal parameterizing manifolds for anticipating tipping points and higher-order critical transitions Journal Article
In: Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 33, no. 9, pp. 093126, 2023.
@article{2440,
title = {Optimal parameterizing manifolds for anticipating tipping points and higher-order critical transitions},
author = {Mickaël D. Chekroun and Honghu Liu and James C. McWilliams},
url = {https://doi.org/10.1063/5.0167419},
year = {2023},
date = {2023-01-01},
journal = {Chaos: An Interdisciplinary Journal of Nonlinear Science},
volume = {33},
number = {9},
pages = {093126},
abstract = {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.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
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.
Liu, Huan; Koren, Ilan; Altaratz, Orit; Chekroun, Mickaël D.
Opposing trends of cloud coverage over land and ocean under global warming Journal Article
In: Atmospheric Chemistry and Physics Discussions, vol. 23, pp. 6559–6569, 2023.
@article{2439,
title = {Opposing trends of cloud coverage over land and ocean under global warming},
author = {Huan Liu and Ilan Koren and Orit Altaratz and Mickaël D. Chekroun},
url = {https://doi.org/10.5194/acp-23-6559-2023},
year = {2023},
date = {2023-01-01},
journal = {Atmospheric Chemistry and Physics Discussions},
volume = {23},
pages = {6559–6569},
abstract = {Clouds play a key role in Earth’s energy budget and water cycle. Their response to global warming contributes the largest uncertainty to climate prediction. Here, by performing an empirical orthogonal function analysis on 42 years of reanalysis data of global cloud coverage, we extract an unambiguous trend and El-Niño–Southern-Oscillation-associated modes. The trend mode translates spatially to decreasing trends in cloud coverage over most continents and increasing trends over the tropical and subtropical oceans. A reduction in near-surface relative humidity can explain the decreasing trends in cloud coverage over land. Our results suggest potential stress on the terrestrial water cycle and changes in the energy partition between land and ocean, all associated with global warming.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Clouds play a key role in Earth’s energy budget and water cycle. Their response to global warming contributes the largest uncertainty to climate prediction. Here, by performing an empirical orthogonal function analysis on 42 years of reanalysis data of global cloud coverage, we extract an unambiguous trend and El-Niño–Southern-Oscillation-associated modes. The trend mode translates spatially to decreasing trends in cloud coverage over most continents and increasing trends over the tropical and subtropical oceans. A reduction in near-surface relative humidity can explain the decreasing trends in cloud coverage over land. Our results suggest potential stress on the terrestrial water cycle and changes in the energy partition between land and ocean, all associated with global warming.
Lucarini, Valerio; Chekroun, Mickaël D.
Theoretical tools for understanding the climate crisis from Hasselmann’s programme and beyond Journal Article
In: Nature Review Physics, vol. 5, pp. 744-765, 2023.
@article{2393,
title = {Theoretical tools for understanding the climate crisis from Hasselmann’s programme and beyond},
author = {Valerio Lucarini and Mickaël D. Chekroun},
url = {https://doi.org/10.1038/s42254-023-00650-8},
year = {2023},
date = {2023-01-01},
journal = {Nature Review Physics},
volume = {5},
pages = {744-765},
abstract = {Klaus Hasselmann’s revolutionary intuition in climate science was to use the stochasticity associated with fast weather processes to probe the slow dynamics of the climate system. Doing so led to fundamentally new ways to study the response of climate models to perturbations, and to perform detection and attribution for climate change signals. Hasselmann’s programme has been extremely influential in climate science and beyond. In this Perspective, we first summarize the main aspects of such a programme using modern concepts and tools of statistical physics and applied mathematics. We then provide an overview of some promising scientific perspectives that might clarify the science behind the climate crisis and that stem from Hasselmann’s ideas. We show how to perform rigorous and data-driven model reduction by constructing parameterizations in systems that do not necessarily feature a timescale separation between unresolved and resolved processes. We outline a general theoretical framework for explaining the relationship between climate variability and climate change, and for performing climate change projections. This framework enables us seamlessly to explain some key general aspects of climatic tipping points. Finally, we show that response theory provides a solid framework supporting optimal fingerprinting methods for detection and attribution.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Klaus Hasselmann’s revolutionary intuition in climate science was to use the stochasticity associated with fast weather processes to probe the slow dynamics of the climate system. Doing so led to fundamentally new ways to study the response of climate models to perturbations, and to perform detection and attribution for climate change signals. Hasselmann’s programme has been extremely influential in climate science and beyond. In this Perspective, we first summarize the main aspects of such a programme using modern concepts and tools of statistical physics and applied mathematics. We then provide an overview of some promising scientific perspectives that might clarify the science behind the climate crisis and that stem from Hasselmann’s ideas. We show how to perform rigorous and data-driven model reduction by constructing parameterizations in systems that do not necessarily feature a timescale separation between unresolved and resolved processes. We outline a general theoretical framework for explaining the relationship between climate variability and climate change, and for performing climate change projections. This framework enables us seamlessly to explain some key general aspects of climatic tipping points. Finally, we show that response theory provides a solid framework supporting optimal fingerprinting methods for detection and attribution.
Chekroun, Mickaël D.; Liu, Honghu; McWilliams, James C.; Wang, Shouhong
Transitions in Stochastic Non-equilibrium Systems: Efficient Reduction and Analysis Journal Article
In: Journal of Differential Equations, vol. 346, no. 10, pp. 145-204, 2023.
@article{2332,
title = {Transitions in Stochastic Non-equilibrium Systems: Efficient Reduction and Analysis},
author = {Mickaël D. Chekroun and Honghu Liu and James C. McWilliams and Shouhong Wang},
url = {https://doi.org/10.1016/j.jde.2022.11.025},
year = {2023},
date = {2023-01-01},
journal = {Journal of Differential Equations},
volume = {346},
number = {10},
pages = {145-204},
abstract = {A central challenge in physics is to describe non-equilibrium systems driven by randomness, such as a randomly growing interface, or fluids subject to random fluctuations that account e.g. for local stresses and heat fluxes in the fluid which are not related to the velocity and temperature gradients. For deterministic systems with infinitely many degrees of freedom, normal form and center manifold theory have shown a prodigious efficiency to often completely characterize how the onset of linear instability translates into the emergence of nonlinear patterns, associated with genuine physical regimes. However, in presence of random fluctuations, the underlying reduction principle to the center manifold is seriously challenged due to large excursions caused by the noise, and the approach needs to be revisited.
In this study, we present an alternative framework to cope with these difficulties exploiting the approximation theory of stochastic invariant manifolds, on one hand, and energy estimates measuring the defect of parameterization of the high-modes, on the other. To operate for fluid problems subject to stochastic stirring forces, these error estimates are derived under assumptions regarding dissipation effects brought by the high-modes in order to suitably counterbalance the loss of regularity due to the nonlinear terms. As a result, the approach enables us to analyze, from reduced equations of the stochastic fluid problem, the occurrence in large probability of a stochastic analogue to the pitchfork bifurcation, as long as the noise’s intensity and the eigenvalue’s magnitude of the mildly unstable mode scale accordingly.
In the case of SPDEs forced by a multiplicative noise in the orthogonal subspace of e.g. its mildly unstable mode, our parameterization formulas show that the noise gets transmitted to this mode via non-Markovian coefficients, and that the reduced equation is only stochastically driven by the latter. These coefficients depend explicitly on the noise path’s history, and their memory content is self-consistently determined by the intensity of the random force and its interaction through the SPDE’s nonlinear terms. Applications to a stochastic Rayleigh-Bénard problem are detailed, for which conditions for a stochastic pitchfork bifurcation (in large probability) to occur, are clarified.
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},
keywords = {},
pubstate = {published},
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A central challenge in physics is to describe non-equilibrium systems driven by randomness, such as a randomly growing interface, or fluids subject to random fluctuations that account e.g. for local stresses and heat fluxes in the fluid which are not related to the velocity and temperature gradients. For deterministic systems with infinitely many degrees of freedom, normal form and center manifold theory have shown a prodigious efficiency to often completely characterize how the onset of linear instability translates into the emergence of nonlinear patterns, associated with genuine physical regimes. However, in presence of random fluctuations, the underlying reduction principle to the center manifold is seriously challenged due to large excursions caused by the noise, and the approach needs to be revisited.
In this study, we present an alternative framework to cope with these difficulties exploiting the approximation theory of stochastic invariant manifolds, on one hand, and energy estimates measuring the defect of parameterization of the high-modes, on the other. To operate for fluid problems subject to stochastic stirring forces, these error estimates are derived under assumptions regarding dissipation effects brought by the high-modes in order to suitably counterbalance the loss of regularity due to the nonlinear terms. As a result, the approach enables us to analyze, from reduced equations of the stochastic fluid problem, the occurrence in large probability of a stochastic analogue to the pitchfork bifurcation, as long as the noise’s intensity and the eigenvalue’s magnitude of the mildly unstable mode scale accordingly.
In the case of SPDEs forced by a multiplicative noise in the orthogonal subspace of e.g. its mildly unstable mode, our parameterization formulas show that the noise gets transmitted to this mode via non-Markovian coefficients, and that the reduced equation is only stochastically driven by the latter. These coefficients depend explicitly on the noise path’s history, and their memory content is self-consistently determined by the intensity of the random force and its interaction through the SPDE’s nonlinear terms. Applications to a stochastic Rayleigh-Bénard problem are detailed, for which conditions for a stochastic pitchfork bifurcation (in large probability) to occur, are clarified.
[["fid":"2012","view_mode":"default","fields":"format":"default","type":"media","field_deltas":"1":"format":"default","attributes":"height":"187","width":"583","class":"media-element file-default","data-delta":"1"]]
In this study, we present an alternative framework to cope with these difficulties exploiting the approximation theory of stochastic invariant manifolds, on one hand, and energy estimates measuring the defect of parameterization of the high-modes, on the other. To operate for fluid problems subject to stochastic stirring forces, these error estimates are derived under assumptions regarding dissipation effects brought by the high-modes in order to suitably counterbalance the loss of regularity due to the nonlinear terms. As a result, the approach enables us to analyze, from reduced equations of the stochastic fluid problem, the occurrence in large probability of a stochastic analogue to the pitchfork bifurcation, as long as the noise’s intensity and the eigenvalue’s magnitude of the mildly unstable mode scale accordingly.
In the case of SPDEs forced by a multiplicative noise in the orthogonal subspace of e.g. its mildly unstable mode, our parameterization formulas show that the noise gets transmitted to this mode via non-Markovian coefficients, and that the reduced equation is only stochastically driven by the latter. These coefficients depend explicitly on the noise path’s history, and their memory content is self-consistently determined by the intensity of the random force and its interaction through the SPDE’s nonlinear terms. Applications to a stochastic Rayleigh-Bénard problem are detailed, for which conditions for a stochastic pitchfork bifurcation (in large probability) to occur, are clarified.
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2022
Chekroun, Mickaël D.; Koren, Ilan; Liu, Honghu; Liu, Huan
Generic generation of noise-driven chaos in stochastic time delay systems: Bridging the gap with high-end simulations Journal Article
In: Science Advances, vol. 8, no. 46, pp. eabq7137, 2022.
@article{2379,
title = {Generic generation of noise-driven chaos in stochastic time delay systems: Bridging the gap with high-end simulations},
author = {Mickaël D. Chekroun and Ilan Koren and Honghu Liu and Huan Liu},
url = {https://www.science.org/doi/10.1126/sciadv.abq7137},
year = {2022},
date = {2022-01-01},
journal = {Science Advances},
volume = {8},
number = {46},
pages = {eabq7137},
abstract = {Nonlinear time delay systems produce inherently delay-induced periodic oscillations, which are, however, too idealistic compared to observations. We exhibit a unified stochastic framework to systematically rectify such oscillations into oscillatory patterns with enriched temporal variabilities through generic, nonlinear responses to stochastic perturbations. Two paradigms of noise-driven chaos in high dimension are identified, fundamentally different from chaos triggered by parameter-space noise. Noteworthy is a low-dimensional stretch-and-fold mechanism, leading to stochastic strange attractors exhibiting horseshoe-like structures mirroring turbulent transport of passive tracers. The other is high-dimensional , with noise acting along the critical eigendirection and transmitted to “deeper” stable modes through nonlinearity, leading to stochastic attractors exhibiting swarm-like behaviors with power-law and scale break properties. The theory is applied to cloud delay models to parameterize missing physics such as intermittent rain and Lagrangian turbulent effects. The stochastically rectified model reproduces with fidelity complex temporal variabilities of open-cell oscillations exhibited by high-end cloud simulations.
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keywords = {},
pubstate = {published},
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Nonlinear time delay systems produce inherently delay-induced periodic oscillations, which are, however, too idealistic compared to observations. We exhibit a unified stochastic framework to systematically rectify such oscillations into oscillatory patterns with enriched temporal variabilities through generic, nonlinear responses to stochastic perturbations. Two paradigms of noise-driven chaos in high dimension are identified, fundamentally different from chaos triggered by parameter-space noise. Noteworthy is a low-dimensional stretch-and-fold mechanism, leading to stochastic strange attractors exhibiting horseshoe-like structures mirroring turbulent transport of passive tracers. The other is high-dimensional , with noise acting along the critical eigendirection and transmitted to “deeper” stable modes through nonlinearity, leading to stochastic attractors exhibiting swarm-like behaviors with power-law and scale break properties. The theory is applied to cloud delay models to parameterize missing physics such as intermittent rain and Lagrangian turbulent effects. The stochastically rectified model reproduces with fidelity complex temporal variabilities of open-cell oscillations exhibited by high-end cloud simulations.
[["fid":"2069","view_mode":"default","fields":"format":"default","type":"media","field_deltas":"1":"format":"default","attributes":"height":"264","width":"617","class":"media-element file-default","data-delta":"1"]]
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Dror, Tom; Silverman, Vered; Altaratz, Orit; Chekroun, Mickaël D.; Koren, Ilan
Uncovering the Large-Scale Meteorology That Drives Continental, Shallow, Green Cumulus Through Supervised Classification Journal Article
In: Geophysical Research Letters, 2022.
@article{2353,
title = {Uncovering the Large-Scale Meteorology That Drives Continental, Shallow, Green Cumulus Through Supervised Classification},
author = {Tom Dror and Vered Silverman and Orit Altaratz and Mickaël D. Chekroun and Ilan Koren},
url = {https://doi.org/10.1029/2021GL096684},
year = {2022},
date = {2022-01-01},
journal = {Geophysical Research Letters},
abstract = {One of the major sources of uncertainty in climate prediction results from the limitations in representing shallow cumulus (Cu) in models. Recently, a class of continental shallow convective Cu was shown to share distinct morphological properties and to emerge globally mostly over forests and vegetated areas, thus named greenCu. Using machine-learning supervised classification, we identify greenCu fields over three regions, from the tropics to mid- and higher-latitudes, and establish a novel satellite-based data set called greenCuDb, consisting of 1° × 1° sized, high-resolution MODIS images. Using greenCuDb in conjunction with ERA5 reanalysis data, we create greenCu composites for different regions and reveal that greenCu are driven by similar large-scale meteorological conditions, regardless of their geographical locations throughout the world’s continents. These conditions include distinct profiles of temperature, humidity and large-scale vertical velocity. The boundary layer is anomalously warm and moderately humid, and is accompanied by a strong large-scale subsidence in the free troposphere.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
One of the major sources of uncertainty in climate prediction results from the limitations in representing shallow cumulus (Cu) in models. Recently, a class of continental shallow convective Cu was shown to share distinct morphological properties and to emerge globally mostly over forests and vegetated areas, thus named greenCu. Using machine-learning supervised classification, we identify greenCu fields over three regions, from the tropics to mid- and higher-latitudes, and establish a novel satellite-based data set called greenCuDb, consisting of 1° × 1° sized, high-resolution MODIS images. Using greenCuDb in conjunction with ERA5 reanalysis data, we create greenCu composites for different regions and reveal that greenCu are driven by similar large-scale meteorological conditions, regardless of their geographical locations throughout the world’s continents. These conditions include distinct profiles of temperature, humidity and large-scale vertical velocity. The boundary layer is anomalously warm and moderately humid, and is accompanied by a strong large-scale subsidence in the free troposphere.
Chekroun, Mickaël D.; Dijkstra, Henk A.; Şengül, Taylan; Wang, Shouhong
Transitions of zonal flows in a two- layer quasi-geostrophic ocean model Journal Article
In: Nonlinear Dynamics, 2022.
@article{2351,
title = {Transitions of zonal flows in a two- layer quasi-geostrophic ocean model},
author = {Mickaël D. Chekroun and Henk A. Dijkstra and Taylan Şengül and Shouhong Wang},
url = {https://link.springer.com/article/10.1007/s11071-022-07529-w},
year = {2022},
date = {2022-01-01},
journal = {Nonlinear Dynamics},
abstract = {We consider a 2-layer quasi-geostrophic ocean model where the upper layer is forced by a steady Kolmogorov wind stress in a periodic channel domain, which allows to mathematically study the nonlinear development of the resulting flow. The model supports a steady parallel shear flow as a response to the wind stress. As the maximal velocity of the shear flow (equivalently the maximal amplitude of the wind forcing) exceeds a critical threshold, the zonal jet destabilizes due to baroclinic instability and we numerically demonstrate that a first transition occurs. We obtain reduced equations of the system using the formalism of dynamic transition theory and establish two scenarios which completely describe this first transition. The generic scenario is that two modes become critical and a Hopf bifurcation occurs as a result. Under an appropriate set of parameters describing midlatitude oceanic flows, we show that this first transition is continuous: a supercritical Hopf bifurcation occurs and a stable time periodic solution bifurcates. We also investigate the case of double Hopf bifurcations which occur when four modes of the linear stability problem simultaneously destabilize the zonal jet. In this case we prove that, in the relevant parameter regime, the flow exhibits a continuous transition accompanied by a bifurcated attractor homeomorphic to S^3. The topological structure of this attractor is analyzed in detail and is shown to depend on the system parameters. In particular, this attractor contains(stable or unstable) time-periodic solutions and a quasi-periodic solution.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We consider a 2-layer quasi-geostrophic ocean model where the upper layer is forced by a steady Kolmogorov wind stress in a periodic channel domain, which allows to mathematically study the nonlinear development of the resulting flow. The model supports a steady parallel shear flow as a response to the wind stress. As the maximal velocity of the shear flow (equivalently the maximal amplitude of the wind forcing) exceeds a critical threshold, the zonal jet destabilizes due to baroclinic instability and we numerically demonstrate that a first transition occurs. We obtain reduced equations of the system using the formalism of dynamic transition theory and establish two scenarios which completely describe this first transition. The generic scenario is that two modes become critical and a Hopf bifurcation occurs as a result. Under an appropriate set of parameters describing midlatitude oceanic flows, we show that this first transition is continuous: a supercritical Hopf bifurcation occurs and a stable time periodic solution bifurcates. We also investigate the case of double Hopf bifurcations which occur when four modes of the linear stability problem simultaneously destabilize the zonal jet. In this case we prove that, in the relevant parameter regime, the flow exhibits a continuous transition accompanied by a bifurcated attractor homeomorphic to S^3. The topological structure of this attractor is analyzed in detail and is shown to depend on the system parameters. In particular, this attractor contains(stable or unstable) time-periodic solutions and a quasi-periodic solution.
2021
Chekroun, Mickaël D.; Liu, Honghu; McWilliams, James C.
Stochastic rectification of fast oscillations on slow manifold closures Journal Article
In: Proc. Natl. Acad. Sci., vol. 118, no. 48, pp. E2113650118, 2021.
@article{2352,
title = {Stochastic rectification of fast oscillations on slow manifold closures},
author = {Mickaël D. Chekroun and Honghu Liu and James C. McWilliams},
url = {https://doi.org/10.1073/pnas.2113650118},
year = {2021},
date = {2021-01-01},
journal = {Proc. Natl. Acad. Sci.},
volume = {118},
number = {48},
pages = {E2113650118},
abstract = {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.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
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.
Dror, Tom; Chekroun, Mickaël D.; Koren, Ilan; Altaratz, Orit
Deciphering organization of GOES-16 green cumulus through the empirical orthogonal function (EOF) lens Journal Article
In: Atmospheric Chemistry and Physics, vol. 21, pp. 12261–12272, 2021.
@article{2321,
title = {Deciphering organization of GOES-16 green cumulus through the empirical orthogonal function (EOF) lens},
author = {Tom Dror and Mickaël D. Chekroun and Ilan Koren and Orit Altaratz},
url = {https://doi.org/10.5194/acp-21-12261-2021},
year = {2021},
date = {2021-01-01},
journal = {Atmospheric Chemistry and Physics},
volume = {21},
pages = {12261–12272},
abstract = {A subset of continental shallow convective cumulus (Cu) cloud fields has been shown to have distinct spatial properties and to form mostly over forests and vegetated areas, thus referred to as “green Cu” (Dror et al., 2020). Green Cu fields are known to form organized mesoscale patterns, yet the underlying mechanisms, as well as the time variability of these patterns, are still lacking understanding. Here, we characterize the organization of green Cu in space and time, by using data-driven organization metrics and by applying an empirical orthogonal function (EOF) analysis to a high-resolution GOES-16 dataset. We extract, quantify, and reveal modes of organization present in a green Cu field, during the course of a day. The EOF decomposition is able to show the field’s key organization features such as cloud streets, and it also delineates the less visible ones, as the propagation of gravity waves (GWs) and the emergence of a highly organized grid on a spatial scale of hundreds of kilometers, over a time period that scales with the field’s lifetime. Using cloud fields that were reconstructed from different subgroups of modes, we quantify the cloud street’s wavelength and aspect ratio, as well as the GW-dominant period.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
A subset of continental shallow convective cumulus (Cu) cloud fields has been shown to have distinct spatial properties and to form mostly over forests and vegetated areas, thus referred to as “green Cu” (Dror et al., 2020). Green Cu fields are known to form organized mesoscale patterns, yet the underlying mechanisms, as well as the time variability of these patterns, are still lacking understanding. Here, we characterize the organization of green Cu in space and time, by using data-driven organization metrics and by applying an empirical orthogonal function (EOF) analysis to a high-resolution GOES-16 dataset. We extract, quantify, and reveal modes of organization present in a green Cu field, during the course of a day. The EOF decomposition is able to show the field’s key organization features such as cloud streets, and it also delineates the less visible ones, as the propagation of gravity waves (GWs) and the emergence of a highly organized grid on a spatial scale of hundreds of kilometers, over a time period that scales with the field’s lifetime. Using cloud fields that were reconstructed from different subgroups of modes, we quantify the cloud street’s wavelength and aspect ratio, as well as the GW-dominant period.
Charó, Gisela D.; Chekroun, Mickaël D.; Sciamarella, Denisse; Ghil, Michael
Noise-driven topological changes in chaotic dynamics Journal Article
In: Chaos, vol. 31, no. 10, pp. 103115, 2021.
@article{2309,
title = {Noise-driven topological changes in chaotic dynamics},
author = {Gisela D. Charó and Mickaël D. Chekroun and Denisse Sciamarella and Michael Ghil},
url = {https://doi.org/10.1063/5.0059461},
year = {2021},
date = {2021-01-01},
journal = {Chaos},
volume = {31},
number = {10},
pages = {103115},
abstract = {Noise modifies the behavior of chaotic systems in both quantitative and qualitative ways. To study these modifications, the present work compares the topological structure of the deterministic Lorenz (1963) attractor with its stochastically perturbed version. The deterministic attractor is well known to be “strange” but it is frozen in time. When driven by multiplicative noise, the Lorenz model’s random attractor (LORA) evolves in time. Algebraic topology sheds light on the most striking effects involved in such an evolution. In order to examine the topological structure of the snapshots that approximate LORA, we use branched manifold analysis through homologies—a technique originally introduced to characterize the topological structure of deterministically chaotic flows—which is being extended herein to nonlinear noise-driven systems. The analysis is performed for a fixed realization of the driving noise at different time instants in time. The results suggest that LORA’s evolution includes sharp transitions that appear as topological tipping points.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Noise modifies the behavior of chaotic systems in both quantitative and qualitative ways. To study these modifications, the present work compares the topological structure of the deterministic Lorenz (1963) attractor with its stochastically perturbed version. The deterministic attractor is well known to be “strange” but it is frozen in time. When driven by multiplicative noise, the Lorenz model’s random attractor (LORA) evolves in time. Algebraic topology sheds light on the most striking effects involved in such an evolution. In order to examine the topological structure of the snapshots that approximate LORA, we use branched manifold analysis through homologies—a technique originally introduced to characterize the topological structure of deterministically chaotic flows—which is being extended herein to nonlinear noise-driven systems. The analysis is performed for a fixed realization of the driving noise at different time instants in time. The results suggest that LORA’s evolution includes sharp transitions that appear as topological tipping points.
Gutiérrez, Manuel Santos; Lucarini, Valerio; Chekroun, Mickaël D.; Ghil, Michael
Reduced-order models for coupled dynamical systems: Data-driven methods and the Koopman operator Journal Article
In: Chaos, vol. 31, pp. 053116, 2021.
@article{2255,
title = {Reduced-order models for coupled dynamical systems: Data-driven methods and the Koopman operator},
author = {Manuel Santos Gutiérrez and Valerio Lucarini and Mickaël D. Chekroun and Michael Ghil},
url = {https://doi.org/10.1063/5.0039496},
year = {2021},
date = {2021-01-01},
journal = {Chaos},
volume = {31},
pages = {053116},
abstract = {Providing efficient and accurate parameterizations for model reduction is a key goal in many areas of science and technology. Here, we present a strong link between data-driven and theoretical approaches to achieving this goal. Formal perturbation expansions of the Koopman operator allow us to derive general stochastic parameterizations of weakly coupled dynamical systems. Such parameterizations yield a set of stochastic integrodifferential equations with explicit noise and memory kernel formulas to describe the effects of unresolved variables. We show that the perturbation expansions involved need not be truncated when the coupling is additive. The unwieldy integrodifferential equations can be recast as a simpler multilevel Markovian model, and we establish an intuitive connection with a generalized Langevin equation. This connection helps setting up a parallelism between the top-down, equation-based methodology herein and the well-established empirical model reduction (EMR) methodology that has been shown to provide efficient dynamical closures to partially observed systems. Hence, our findings, on the one hand, support the physical basis and robustness of the EMR methodology and, on the other hand, illustrate the practical relevance of the perturbative expansion used for deriving the parameterizations.
Parameterizations aim to reduce the complexity of high-dimensional dynamical systems. Here, a theory-based and a data-driven approach for the parameterization of coupled systems are compared, showing that both yield the same stochastic multilevel structure. The results provide very strong support to the use of empirical methods in model reduction and clarify the practical relevance of the proposed theoretical framework.
},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Providing efficient and accurate parameterizations for model reduction is a key goal in many areas of science and technology. Here, we present a strong link between data-driven and theoretical approaches to achieving this goal. Formal perturbation expansions of the Koopman operator allow us to derive general stochastic parameterizations of weakly coupled dynamical systems. Such parameterizations yield a set of stochastic integrodifferential equations with explicit noise and memory kernel formulas to describe the effects of unresolved variables. We show that the perturbation expansions involved need not be truncated when the coupling is additive. The unwieldy integrodifferential equations can be recast as a simpler multilevel Markovian model, and we establish an intuitive connection with a generalized Langevin equation. This connection helps setting up a parallelism between the top-down, equation-based methodology herein and the well-established empirical model reduction (EMR) methodology that has been shown to provide efficient dynamical closures to partially observed systems. Hence, our findings, on the one hand, support the physical basis and robustness of the EMR methodology and, on the other hand, illustrate the practical relevance of the perturbative expansion used for deriving the parameterizations.
Parameterizations aim to reduce the complexity of high-dimensional dynamical systems. Here, a theory-based and a data-driven approach for the parameterization of coupled systems are compared, showing that both yield the same stochastic multilevel structure. The results provide very strong support to the use of empirical methods in model reduction and clarify the practical relevance of the proposed theoretical framework.
Parameterizations aim to reduce the complexity of high-dimensional dynamical systems. Here, a theory-based and a data-driven approach for the parameterization of coupled systems are compared, showing that both yield the same stochastic multilevel structure. The results provide very strong support to the use of empirical methods in model reduction and clarify the practical relevance of the proposed theoretical framework.
2020
Wang, Peng; McWilliams, James C.; Uchiyama, Yusuke; Chekroun, Mickaël D.; Yi, Daling Li
Effects of wave streaming and wave variations on nearshore wave-driven circulation Journal Article
In: J. Phys. Oceanograhy, vol. 50, no. 10, pp. 3025-3041, 2020.
@article{2182,
title = {Effects of wave streaming and wave variations on nearshore wave-driven circulation},
author = {Peng Wang and James C. McWilliams and Yusuke Uchiyama and Mickaël D. Chekroun and Daling Li Yi},
url = {https://journals.ametsoc.org/jpo/article-abstract/50/10/3025/354467/Effects-of-Wave-Streaming-and-Wave-Variations-on?redirectedFrom=fulltext},
year = {2020},
date = {2020-01-01},
journal = {J. Phys. Oceanograhy},
volume = {50},
number = {10},
pages = {3025-3041},
abstract = {Wave streaming is a near-bottom mean current induced by the bottom drag on surface gravity waves. Wave variations include the variations in wave heights, periods, and directions. Here we use numerical simulations to study the effects of wave streaming and wave variations on the circulation that is driven by incident surface waves. Wave streaming induces an inner-shelf Lagrangian overturning circulation, which links the inner shelf with the surf zone. Wave variations cause along shore-variable wave breaking that produces surf eddies; however, such eddies can be suppressed by wave streaming. Moreover, with passive tracers we show that wave streaming and wave variations together enhance the cross- shelf material transport.},
keywords = {},
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tppubtype = {article}
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Wave streaming is a near-bottom mean current induced by the bottom drag on surface gravity waves. Wave variations include the variations in wave heights, periods, and directions. Here we use numerical simulations to study the effects of wave streaming and wave variations on the circulation that is driven by incident surface waves. Wave streaming induces an inner-shelf Lagrangian overturning circulation, which links the inner shelf with the surf zone. Wave variations cause along shore-variable wave breaking that produces surf eddies; however, such eddies can be suppressed by wave streaming. Moreover, with passive tracers we show that wave streaming and wave variations together enhance the cross- shelf material transport.
Chekroun, Mickaël D.; Liu, Honghu
Optimal management of harvested population at the edge of extinction Book Chapter
In: Advances in Nonlinear Biological Systems: Modeling and Optimal Control, J. Kotas (Ed.)., vol. 11, pp. 35-72, AIMS Applied Mathematics Book series. ISBN-10 : 1-60133-025-1, ISBN-13 : 978-1-60133-025-3, 2020.
@inbook{2119,
title = {Optimal management of harvested population at the edge of extinction},
author = {Mickaël D. Chekroun and Honghu Liu},
url = {https://arxiv.org/abs/2007.11785},
year = {2020},
date = {2020-01-01},
booktitle = {Advances in Nonlinear Biological Systems: Modeling and Optimal Control, J. Kotas (Ed.).},
volume = {11},
pages = {35-72},
publisher = {AIMS Applied Mathematics Book series. ISBN-10 : 1-60133-025-1, ISBN-13 : 978-1-60133-025-3},
organization = {AIMS Applied Mathematics Book series. ISBN-10 : 1-60133-025-1, ISBN-13 : 978-1-60133-025-3},
abstract = {Optimal control of harvested population at the edge of extinction in an unprotected area, is considered. The underlying population dynamics is governed by a Kolmogorov-Petrovsky-Piskunov equation with a harvesting term and space-dependent coefficients while the control consists of transporting individuals from a natural reserve. The nonlinear optimal control problem is approximated by means of a Galerkin scheme. Convergence result about the optimal controlled solutions and error estimates between the corresponding optimal controls, are derived. For certain parameter regimes, nearly optimal solutions are calculated from a simple logistic ordinary differential equation (ODE) with a harvesting term, obtained as a Galerkin approximation of the original partial differential equation (PDE) model. A critical allowable fraction of the reserve’s population is inferred from the reduced logistic ODE with a harvesting term. This estimate obtained from the reduced model allows us to distinguish sharply between survival and extinction for the full PDE itself, and thus to declare whether a control strategy leads to success or failure for the corresponding rescue operation while ensuring survival in the reserve’s population. In dynamical terms, this result illustrates that although continuous dependence on the forcing may hold on finite-time intervals, a high sensitivity in the system’s response may occur in the asymptotic time. We believe that this work, by its generality, establishes bridges interesting to explore between optimal control problems of ODEs with a harvesting term and their PDE counterpart.
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Optimal control of harvested population at the edge of extinction in an unprotected area, is considered. The underlying population dynamics is governed by a Kolmogorov-Petrovsky-Piskunov equation with a harvesting term and space-dependent coefficients while the control consists of transporting individuals from a natural reserve. The nonlinear optimal control problem is approximated by means of a Galerkin scheme. Convergence result about the optimal controlled solutions and error estimates between the corresponding optimal controls, are derived. For certain parameter regimes, nearly optimal solutions are calculated from a simple logistic ordinary differential equation (ODE) with a harvesting term, obtained as a Galerkin approximation of the original partial differential equation (PDE) model. A critical allowable fraction of the reserve’s population is inferred from the reduced logistic ODE with a harvesting term. This estimate obtained from the reduced model allows us to distinguish sharply between survival and extinction for the full PDE itself, and thus to declare whether a control strategy leads to success or failure for the corresponding rescue operation while ensuring survival in the reserve’s population. In dynamical terms, this result illustrates that although continuous dependence on the forcing may hold on finite-time intervals, a high sensitivity in the system’s response may occur in the asymptotic time. We believe that this work, by its generality, establishes bridges interesting to explore between optimal control problems of ODEs with a harvesting term and their PDE counterpart.
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Chekroun, Mickaël D.; Koren, Ilan; Liu, Honghu
Efficient reduction for diagnosing Hopf bifurcation in delay differential systems: Applications to cloud-rain models Journal Article
In: Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 30, pp. 053130, 2020.
@article{2050,
title = {Efficient reduction for diagnosing Hopf bifurcation in delay differential systems: Applications to cloud-rain models},
author = {Mickaël D. Chekroun and Ilan Koren and Honghu Liu},
url = {https://doi.org/10.1063/5.0004697},
year = {2020},
date = {2020-01-01},
journal = {Chaos: An Interdisciplinary Journal of Nonlinear Science},
volume = {30},
pages = {053130},
abstract = {By means of Galerkin-Koornwinder (GK) approximations, an efficient reduction approach to the Stuart-Landau (SL) normal form and center manifold is presented for a broad class of nonlinear systems of delay differential equations (DDEs) that covers the cases of discrete as well as distributed delays. The focus is on the Hopf bifurcation as the consequence of the critical equilibrium’s destabilization resulting from an eigenpair crossing the imaginary axis. The nature of the resulting Hopf bifurcation (super- or subcritical) is then characterized by the inspection of a Lyapunov coefficient easy to determine based on the model’s coefficients and delay parameter. We believe that our approach, which does not rely too much on functional analysis considerations but more on analytic calculations, is suitable to concrete situations arising in physics applications.
Thus, using this GK approach to the Lyapunov coefficient and SL normal form, the occurrence of Hopf bifurcations in the cloud-rain delay models of Koren and Feingold (KF) on one hand, and Koren, Tziperman and Feingold (KTF), on the other, are analyzed. Noteworthy is the existence for the KF model of large regions of the parameter space for which subcritical and supercritical Hopf bifurcations coexist. These regions are determined in particular by the intensity of the KF model’s nonlinear effects. ‘‘Islands’’ of supercritical Hopf bifurcations are shown to exist within a subcritical Hopf bifurcation ‘‘sea;’’ these islands being bordered by double-Hopf bifurcations occurring when the linearized dynamics at the critical equilibrium exhibit two pairs of purely imaginary eigenvalues.
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By means of Galerkin-Koornwinder (GK) approximations, an efficient reduction approach to the Stuart-Landau (SL) normal form and center manifold is presented for a broad class of nonlinear systems of delay differential equations (DDEs) that covers the cases of discrete as well as distributed delays. The focus is on the Hopf bifurcation as the consequence of the critical equilibrium’s destabilization resulting from an eigenpair crossing the imaginary axis. The nature of the resulting Hopf bifurcation (super- or subcritical) is then characterized by the inspection of a Lyapunov coefficient easy to determine based on the model’s coefficients and delay parameter. We believe that our approach, which does not rely too much on functional analysis considerations but more on analytic calculations, is suitable to concrete situations arising in physics applications.
Thus, using this GK approach to the Lyapunov coefficient and SL normal form, the occurrence of Hopf bifurcations in the cloud-rain delay models of Koren and Feingold (KF) on one hand, and Koren, Tziperman and Feingold (KTF), on the other, are analyzed. Noteworthy is the existence for the KF model of large regions of the parameter space for which subcritical and supercritical Hopf bifurcations coexist. These regions are determined in particular by the intensity of the KF model’s nonlinear effects. ‘‘Islands’’ of supercritical Hopf bifurcations are shown to exist within a subcritical Hopf bifurcation ‘‘sea;’’ these islands being bordered by double-Hopf bifurcations occurring when the linearized dynamics at the critical equilibrium exhibit two pairs of purely imaginary eigenvalues.
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Thus, using this GK approach to the Lyapunov coefficient and SL normal form, the occurrence of Hopf bifurcations in the cloud-rain delay models of Koren and Feingold (KF) on one hand, and Koren, Tziperman and Feingold (KTF), on the other, are analyzed. Noteworthy is the existence for the KF model of large regions of the parameter space for which subcritical and supercritical Hopf bifurcations coexist. These regions are determined in particular by the intensity of the KF model’s nonlinear effects. ‘‘Islands’’ of supercritical Hopf bifurcations are shown to exist within a subcritical Hopf bifurcation ‘‘sea;’’ these islands being bordered by double-Hopf bifurcations occurring when the linearized dynamics at the critical equilibrium exhibit two pairs of purely imaginary eigenvalues.
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Chekroun, Mickaël D.; Hong, Youngjoon; Temam, Roger
Enriched numerical scheme for singularly perturbed barotropic quasi-geostrophic equations Journal Article
In: Journal of Computational Physics, vol. 416, pp. 109493, 2020.
@article{2048,
title = {Enriched numerical scheme for singularly perturbed barotropic quasi-geostrophic equations},
author = {Mickaël D. Chekroun and Youngjoon Hong and Roger Temam},
url = {https://doi.org/10.1016/j.jcp.2020.109493},
year = {2020},
date = {2020-01-01},
journal = {Journal of Computational Physics},
volume = {416},
pages = {109493},
abstract = {Singularly perturbed barotropic Quasi-Geostrophic (QG) models are considered. A boundary layer analysis is presented and the convergence of solutions is studied. Based on the rigorous analysis of the underlying boundary layer problems, an enriched spectral method (ESM) is proposed to solve the QG models. It consists of adding to the Legendre basis functions, analytically-determined boundary layer elements called “correctors," with the aim of capturing most of the complex behavior occurring near the boundary with such elements. Through detailed numerical experiments, it is shown that high-accuracy is often reached by the ESM scheme with only a relatively low number N of basis functions, when compared to approximations based on spectral elements which typically display non-physical oscillations throughout the physical domain, for such values of N. The key to success relies on our analytically-based boundary layer elements, which, due to their highly nonlinear nature, are able to capture most of the steep gradients occurring in the problem’s solution, near the boundary. Our numerical results include multi-dimensional as well as time-dependent problems.
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Singularly perturbed barotropic Quasi-Geostrophic (QG) models are considered. A boundary layer analysis is presented and the convergence of solutions is studied. Based on the rigorous analysis of the underlying boundary layer problems, an enriched spectral method (ESM) is proposed to solve the QG models. It consists of adding to the Legendre basis functions, analytically-determined boundary layer elements called “correctors," with the aim of capturing most of the complex behavior occurring near the boundary with such elements. Through detailed numerical experiments, it is shown that high-accuracy is often reached by the ESM scheme with only a relatively low number N of basis functions, when compared to approximations based on spectral elements which typically display non-physical oscillations throughout the physical domain, for such values of N. The key to success relies on our analytically-based boundary layer elements, which, due to their highly nonlinear nature, are able to capture most of the steep gradients occurring in the problem’s solution, near the boundary. Our numerical results include multi-dimensional as well as time-dependent problems.
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Tantet, Alexis; Chekroun, Mickaël D.; Dijkstra, Henk A.; Neelin, J. David
Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part II: Stochastic Hopf Bifurcation Journal Article
In: Journal of Statistical Physics, vol. 179, pp. 1403–1448, 2020.
@article{2047,
title = {Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part II: Stochastic Hopf Bifurcation},
author = {Alexis Tantet and Mickaël D. Chekroun and Henk A. Dijkstra and J. David Neelin},
url = {https://link.springer.com/article/10.1007/s10955-020-02526-y},
year = {2020},
date = {2020-01-01},
journal = {Journal of Statistical Physics},
volume = {179},
pages = {1403–1448},
abstract = {The spectrum of the generator (Kolmogorov operator) of a diffusion process, referred to as the Ruelle-Pollicott (RP) spectrum, provides a detailed characterization of correlation functions and power spectra of stochastic systems via decomposition formulas in terms of RP resonances; see Part I of this contribution (Chekroun et al. in Theory J Stat. https://doi.org/10.1007/s10955-020-02535-x, 2020). Stochastic analysis techniques relying on the theory of Markov semigroups for the study of the RP spectrum and a rigorous reduction method is presented in Part I Chekroun et al. (2020). This framework is here applied to study a stochastic Hopf bifurcation in view of characterizing the statistical properties of nonlinear oscillators perturbed by noise, depending on their stability. In light of the Hörmander theorem, it is first shown that the geometry of the unperturbed limit cycle, in particular its isochrons, i.e., the leaves of the stable manifold of the limit cycle generalizing the notion of phase, is essential to understand the effect of the noise and the phenomenon of phase diffusion. In addition, it is shown that the RP spectrum has a spectral gap, even at the bifurcation point, and that correlations decay exponentially fast. Explicit small-noise expansions of the RP eigenvalues and eigenfunctions are then obtained, away from the bifurcation point, based on the knowledge of the linearized deterministic dynamics and the characteristics of the noise. These formulas allow one to understand how the interaction of the noise with the deterministic dynamics affect the decay of correlations. Numerical results complement the study of the RP spectrum at the bifurcation point, revealing useful scaling laws. The analysis of the Markov semigroup for stochastic bifurcations is thus promising in providing a complementary approach to the more geometric random dynamical system (RDS) approach. This approach is not limited to low-dimensional systems and the reduction method presented in Chekroun et al. (2020) is applied to a stochastic model relevant to climate dynamics in the third part of this contribution (Tantet et al. in J Stat Phys. https://doi.org/10.1007/s10955-019-02444-8, 2019).
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The spectrum of the generator (Kolmogorov operator) of a diffusion process, referred to as the Ruelle-Pollicott (RP) spectrum, provides a detailed characterization of correlation functions and power spectra of stochastic systems via decomposition formulas in terms of RP resonances; see Part I of this contribution (Chekroun et al. in Theory J Stat. https://doi.org/10.1007/s10955-020-02535-x, 2020). Stochastic analysis techniques relying on the theory of Markov semigroups for the study of the RP spectrum and a rigorous reduction method is presented in Part I Chekroun et al. (2020). This framework is here applied to study a stochastic Hopf bifurcation in view of characterizing the statistical properties of nonlinear oscillators perturbed by noise, depending on their stability. In light of the Hörmander theorem, it is first shown that the geometry of the unperturbed limit cycle, in particular its isochrons, i.e., the leaves of the stable manifold of the limit cycle generalizing the notion of phase, is essential to understand the effect of the noise and the phenomenon of phase diffusion. In addition, it is shown that the RP spectrum has a spectral gap, even at the bifurcation point, and that correlations decay exponentially fast. Explicit small-noise expansions of the RP eigenvalues and eigenfunctions are then obtained, away from the bifurcation point, based on the knowledge of the linearized deterministic dynamics and the characteristics of the noise. These formulas allow one to understand how the interaction of the noise with the deterministic dynamics affect the decay of correlations. Numerical results complement the study of the RP spectrum at the bifurcation point, revealing useful scaling laws. The analysis of the Markov semigroup for stochastic bifurcations is thus promising in providing a complementary approach to the more geometric random dynamical system (RDS) approach. This approach is not limited to low-dimensional systems and the reduction method presented in Chekroun et al. (2020) is applied to a stochastic model relevant to climate dynamics in the third part of this contribution (Tantet et al. in J Stat Phys. https://doi.org/10.1007/s10955-019-02444-8, 2019).
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Chekroun, Mickaël D.; Tantet, A.; Dijkstra, Henk A.; Neelin, J. David
Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part I: Theory Journal Article
In: Journal of Statistical Physics, vol. 179, pp. 1366–1402, 2020.
@article{2005,
title = {Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part I: Theory},
author = {Mickaël D. Chekroun and A. Tantet and Henk A. Dijkstra and J. David Neelin},
url = {https://link.springer.com/article/10.1007/s10955-020-02535-x},
year = {2020},
date = {2020-01-01},
journal = {Journal of Statistical Physics},
volume = {179},
pages = {1366–1402},
abstract = {A theory of Ruelle–Pollicott (RP) resonances for stochastic differential systems is presented. These resonances are defined as the eigenvalues of the generator (Kolmogorov operator) of a given stochastic system. By relying on the theory of Markov semigroups, decomposition formulas of correlation functions and power spectral densities (PSDs) in terms of RP resonances are then derived. These formulas describe, for a broad class of stochastic differential equations (SDEs), how the RP resonances characterize the decay of correlations as well as the signal’s oscillatory components manifested by peaks in the PSD. It is then shown that a notion reduced RP resonances can be rigorously defined, as soon as the dynamics is partially observed within a reduced state space V. These reduced resonances are obtained from the spectral elements of reduced Markov operators acting on functions of the state space V, and can be estimated from series. They inform us about the spectral elements of some coarse-grained version of the SDE generator. When the time-lag at which the transitions are collected from partial observations in V, is either sufficiently small or large, it is shown that the reduced RP resonances approximate the (weak) RP resonances of the generator of the conditional expectation in V, i.e. the optimal reduced system in V obtained by averaging out the contribution of the unobserved variables. The approach is illustrated on a stochastic slow-fast system for which it is shown that the reduced RP resonances allow for a good reconstruction of the correlation functions and PSDs, even when the time-scale separation is weak. The companions articles, Part II and Part III, deal with further practical aspects of the theory presented in this contribution. One important byproduct consists of the diagnosis usefulness of stochastic dynamics that RP resonances provide. This is illustrated in the case of a stochastic Hopf bifurcation in Part II. There, it is shown that such a bifurcation has a clear manifestation in terms of a geometric organization of the RP resonances along discrete parabolas in the left half plane. Such geometric features formed by (reduced) RP resonances are extractable from time series and allow thus for providing an unambiguous “signature” of nonlinear oscillations embedded within a stochastic background. By relying then on the theory of reduced RP resonances presented in this contribution, Part III addresses the question of detection and characterization of such oscillations in a high-dimensional stochastic system, namely the Cane–Zebiak model of El Niño-Southern Oscillation subject to noise modeling fast atmospheric fluctuations.
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A theory of Ruelle–Pollicott (RP) resonances for stochastic differential systems is presented. These resonances are defined as the eigenvalues of the generator (Kolmogorov operator) of a given stochastic system. By relying on the theory of Markov semigroups, decomposition formulas of correlation functions and power spectral densities (PSDs) in terms of RP resonances are then derived. These formulas describe, for a broad class of stochastic differential equations (SDEs), how the RP resonances characterize the decay of correlations as well as the signal’s oscillatory components manifested by peaks in the PSD. It is then shown that a notion reduced RP resonances can be rigorously defined, as soon as the dynamics is partially observed within a reduced state space V. These reduced resonances are obtained from the spectral elements of reduced Markov operators acting on functions of the state space V, and can be estimated from series. They inform us about the spectral elements of some coarse-grained version of the SDE generator. When the time-lag at which the transitions are collected from partial observations in V, is either sufficiently small or large, it is shown that the reduced RP resonances approximate the (weak) RP resonances of the generator of the conditional expectation in V, i.e. the optimal reduced system in V obtained by averaging out the contribution of the unobserved variables. The approach is illustrated on a stochastic slow-fast system for which it is shown that the reduced RP resonances allow for a good reconstruction of the correlation functions and PSDs, even when the time-scale separation is weak. The companions articles, Part II and Part III, deal with further practical aspects of the theory presented in this contribution. One important byproduct consists of the diagnosis usefulness of stochastic dynamics that RP resonances provide. This is illustrated in the case of a stochastic Hopf bifurcation in Part II. There, it is shown that such a bifurcation has a clear manifestation in terms of a geometric organization of the RP resonances along discrete parabolas in the left half plane. Such geometric features formed by (reduced) RP resonances are extractable from time series and allow thus for providing an unambiguous “signature” of nonlinear oscillations embedded within a stochastic background. By relying then on the theory of reduced RP resonances presented in this contribution, Part III addresses the question of detection and characterization of such oscillations in a high-dimensional stochastic system, namely the Cane–Zebiak model of El Niño-Southern Oscillation subject to noise modeling fast atmospheric fluctuations.
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Chekroun, Mickaël D.; Liu, Honghu; McWilliams, James C.
Variational approach to closure of nonlinear dynamical systems: Autonomous case Journal Article
In: Journal of Statistical Physics, vol. 179, pp. 1073–1160, 2020.
@article{2004,
title = {Variational approach to closure of nonlinear dynamical systems: Autonomous case},
author = {Mickaël D. Chekroun and Honghu Liu and James C. McWilliams},
url = {https://link.springer.com/article/10.1007%2Fs10955-019-02458-2},
year = {2020},
date = {2020-01-01},
journal = {Journal of Statistical Physics},
volume = {179},
pages = {1073–1160},
abstract = {A general, variational approach to derive low-order reduced systems for nonlinear systems subject to an autonomous forcing, is introduced. The approach is based on the concept of optimal parameterizing manifold (PM) that substitutes the more classical notion of slow manifold or invariant manifold when breakdown of slaving occurs. An optimal PM provides the manifold that describes the average motion of the neglected scales as a function of the resolved scales and allows, in principle, for determining the best vector field of the reduced state space that describes e.g. the dynamics’ slow motion. The underlying optimal parameterizations are approximated by dynamically-based formulas derived analytically from the original equations. These formulas are contingent upon the determination of only a few (scalar) parameters obtained from minimization of cost functionals, depending on training dataset collected from direct numerical simulation. In practice, a training period of length comparable to a characteristic recurrence or decorrelation time of the dynamics, is sufficient for the efficient derivation of optimized parameterizations. Applications to the closure of low-order models of Atmospheric Primitive Equations and Rayleigh-Bénard convection are then discussed. The approach is finally illustrated — in the context of the Kuramoto-Sivashinsky turbulence — as providing efficient closures without slaving for a cutoff scale kc placed within the inertial range and the reduced state space is just spanned by the unstable modes, without inclusion of any stable modes whatsoever. The underlying optimal PMs obtained by our variational approach are far from slaving and allow for remedying the excessive backscatter transfer of energy to the low modes encountered by classical invariant manifold approximations in their standard forms when the latter are used at this cutoff wavelength.
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A general, variational approach to derive low-order reduced systems for nonlinear systems subject to an autonomous forcing, is introduced. The approach is based on the concept of optimal parameterizing manifold (PM) that substitutes the more classical notion of slow manifold or invariant manifold when breakdown of slaving occurs. An optimal PM provides the manifold that describes the average motion of the neglected scales as a function of the resolved scales and allows, in principle, for determining the best vector field of the reduced state space that describes e.g. the dynamics’ slow motion. The underlying optimal parameterizations are approximated by dynamically-based formulas derived analytically from the original equations. These formulas are contingent upon the determination of only a few (scalar) parameters obtained from minimization of cost functionals, depending on training dataset collected from direct numerical simulation. In practice, a training period of length comparable to a characteristic recurrence or decorrelation time of the dynamics, is sufficient for the efficient derivation of optimized parameterizations. Applications to the closure of low-order models of Atmospheric Primitive Equations and Rayleigh-Bénard convection are then discussed. The approach is finally illustrated — in the context of the Kuramoto-Sivashinsky turbulence — as providing efficient closures without slaving for a cutoff scale kc placed within the inertial range and the reduced state space is just spanned by the unstable modes, without inclusion of any stable modes whatsoever. The underlying optimal PMs obtained by our variational approach are far from slaving and allow for remedying the excessive backscatter transfer of energy to the low modes encountered by classical invariant manifold approximations in their standard forms when the latter are used at this cutoff wavelength.
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Tantet, Alexis; Chekroun, Mickaël D.; Neelin, J. David; Dijkstra, Henk A.
In: Journal of Statistical Physics, vol. 179, pp. 1449–1474, 2020.
@article{730,
title = {Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part III: Application to the Cane-Zebiak Model of the El Niño-Southern Oscillation},
author = {Alexis Tantet and Mickaël D. Chekroun and J. David Neelin and Henk A. Dijkstra},
url = {https://link.springer.com/article/10.1007/s10955-019-02444-8},
year = {2020},
date = {2020-01-01},
journal = {Journal of Statistical Physics},
volume = {179},
pages = {1449–1474},
abstract = {The response of a low-frequency mode of climate variability, El Niño–Southern Oscillation, to stochastic forcing is studied in a high-dimensional model of intermediate complexity, the fully-coupled Cane–Zebiak model (Zebiak and Cane 1987), from the spectral analysis of Markov operators governing the decay of correlations and resonances in the power spectrum. Noise-induced oscillations excited before a supercritical Hopf bifurcation are examined by means of complex resonances, the reduced Ruelle–Pollicott (RP) resonances, via a numerical application of the reduction approach of the first part of this contribution (Chekroun et al. 2019) to model simulations. The oscillations manifest themselves as peaks in the power spectrum which are associated with RP resonances organized along parabolas, as the bifurcation is neared. These resonances and the associated eigenvectors are furthermore well described by the small-noise expansion formulas obtained by Gaspard (2002) and made explicit in the second part of this contribution (Tantet et al. 2019). Beyond the bifurcation, the spectral gap between the imaginary axis and the real part of the leading resonances quantifies the diffusion of phase of the noise-induced oscillations and can be computed from the linearization of the model and from the diffusion matrix of the noise. In this model, the phase diffusion coefficient thus gives a measure of the predictability of oscillatory events representing ENSO. ENSO events being known to be locked to the seasonal cycle, these results should be extended to the non-autonomous case. More generally, the reduction approach theorized in Chekroun et al. (2019), complemented by our understanding of the spectral properties of reference systems such as the stochastic Hopf bifurcation, provides a promising methodology for the analysis of low-frequency variability in high-dimensional stochastic systems.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
The response of a low-frequency mode of climate variability, El Niño–Southern Oscillation, to stochastic forcing is studied in a high-dimensional model of intermediate complexity, the fully-coupled Cane–Zebiak model (Zebiak and Cane 1987), from the spectral analysis of Markov operators governing the decay of correlations and resonances in the power spectrum. Noise-induced oscillations excited before a supercritical Hopf bifurcation are examined by means of complex resonances, the reduced Ruelle–Pollicott (RP) resonances, via a numerical application of the reduction approach of the first part of this contribution (Chekroun et al. 2019) to model simulations. The oscillations manifest themselves as peaks in the power spectrum which are associated with RP resonances organized along parabolas, as the bifurcation is neared. These resonances and the associated eigenvectors are furthermore well described by the small-noise expansion formulas obtained by Gaspard (2002) and made explicit in the second part of this contribution (Tantet et al. 2019). Beyond the bifurcation, the spectral gap between the imaginary axis and the real part of the leading resonances quantifies the diffusion of phase of the noise-induced oscillations and can be computed from the linearization of the model and from the diffusion matrix of the noise. In this model, the phase diffusion coefficient thus gives a measure of the predictability of oscillatory events representing ENSO. ENSO events being known to be locked to the seasonal cycle, these results should be extended to the non-autonomous case. More generally, the reduction approach theorized in Chekroun et al. (2019), complemented by our understanding of the spectral properties of reference systems such as the stochastic Hopf bifurcation, provides a promising methodology for the analysis of low-frequency variability in high-dimensional stochastic systems.
2019
Cao, Yining; Chekroun, Mickaël D.; Temam, Roger; Huang, Aimin
Mathematical analysis of the Jin-Neelin model of El Nino-Southern-Oscillation Journal Article
In: Chinese Annals of Mathematics, Series B, vol. 40, no. 1, pp. 1–38, 2019.
@article{1390,
title = {Mathematical analysis of the Jin-Neelin model of El Nino-Southern-Oscillation},
author = {Yining Cao and Mickaël D. Chekroun and Roger Temam and Aimin Huang},
url = {https://link.springer.com/article/10.1007/s11401-018-0115-3},
year = {2019},
date = {2019-01-01},
journal = {Chinese Annals of Mathematics, Series B},
volume = {40},
number = {1},
pages = {1–38},
abstract = {
The Jin-Neelin model for the El Niño–Southern Oscillation (ENSO for short) is considered for which the authors establish existence and uniqueness of global solutions in time over an unbounded channel domain. The result is proved for initial data and forcing that are sufficiently small. The smallness conditions involve in particular key physical parameters of the model such as those that control the travel time of the equatorial waves and the strength of feedback due to vertical-shear currents and upwelling; central mechanisms in ENSO dynamics.
From the mathematical view point, the system appears as the coupling of a linear shallow water system and a nonlinear heat equation. Because of the very different nature of the two components of the system, the authors find it convenient to prove the existence of solution by semi-discretization in time and utilization of a fractional step scheme. The main idea consists of handling the coupling between the oceanic and temperature components by dividing the time interval into small sub-intervals of length k and on each sub-interval to solve successively the oceanic component, using the temperature T calculated on the previous sub-interval, to then solve the sea-surface temperature (SST for short) equation on the current sub-interval. The passage to the limit as k tends to zero is ensured via a priori estimates derived under the aforementioned smallness conditions.
},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
The Jin-Neelin model for the El Niño–Southern Oscillation (ENSO for short) is considered for which the authors establish existence and uniqueness of global solutions in time over an unbounded channel domain. The result is proved for initial data and forcing that are sufficiently small. The smallness conditions involve in particular key physical parameters of the model such as those that control the travel time of the equatorial waves and the strength of feedback due to vertical-shear currents and upwelling; central mechanisms in ENSO dynamics.
From the mathematical view point, the system appears as the coupling of a linear shallow water system and a nonlinear heat equation. Because of the very different nature of the two components of the system, the authors find it convenient to prove the existence of solution by semi-discretization in time and utilization of a fractional step scheme. The main idea consists of handling the coupling between the oceanic and temperature components by dividing the time interval into small sub-intervals of length k and on each sub-interval to solve successively the oceanic component, using the temperature T calculated on the previous sub-interval, to then solve the sea-surface temperature (SST for short) equation on the current sub-interval. The passage to the limit as k tends to zero is ensured via a priori estimates derived under the aforementioned smallness conditions.
The Jin-Neelin model for the El Niño–Southern Oscillation (ENSO for short) is considered for which the authors establish existence and uniqueness of global solutions in time over an unbounded channel domain. The result is proved for initial data and forcing that are sufficiently small. The smallness conditions involve in particular key physical parameters of the model such as those that control the travel time of the equatorial waves and the strength of feedback due to vertical-shear currents and upwelling; central mechanisms in ENSO dynamics.
From the mathematical view point, the system appears as the coupling of a linear shallow water system and a nonlinear heat equation. Because of the very different nature of the two components of the system, the authors find it convenient to prove the existence of solution by semi-discretization in time and utilization of a fractional step scheme. The main idea consists of handling the coupling between the oceanic and temperature components by dividing the time interval into small sub-intervals of length k and on each sub-interval to solve successively the oceanic component, using the temperature T calculated on the previous sub-interval, to then solve the sea-surface temperature (SST for short) equation on the current sub-interval. The passage to the limit as k tends to zero is ensured via a priori estimates derived under the aforementioned smallness conditions.
2018
Kondrashov, Dmitri; Chekroun, Mickaël D.; Ghil, Michael
Data-adaptive harmonic decomposition and prediction of Arctic sea ice extent Journal Article
In: Dynamics and Statistics of the Climate System, vol. 3, no. 1, pp. 1, 2018.
@article{1345,
title = {Data-adaptive harmonic decomposition and prediction of Arctic sea ice extent},
author = {Dmitri Kondrashov and Mickaël D. Chekroun and Michael Ghil},
url = {https://academic.oup.com/climatesystem/article/3/1/dzy001/4925706},
year = {2018},
date = {2018-03-01},
journal = {Dynamics and Statistics of the Climate System},
volume = {3},
number = {1},
pages = {1},
abstract = {Decline in the Arctic sea ice extent (SIE) is an area of active scientific research with profound socio-economic implications. Of particular interest are reliable methods for SIE forecasting on subseasonal time scales, in particular from early summer into fall, when sea ice coverage in the Arctic reaches its minimum. Here, we apply the recent data-adaptive harmonic (DAH) technique of Chekroun and Kondrashov, (2017), Chaos, 27 for the description, modeling and prediction of the Multisensor Analyzed Sea Ice Extent (MASIE, 2006–2016) data set. The DAH decomposition of MASIE identifies narrowband, spatio-temporal data-adaptive modes over four key Arctic regions. The time evolution of the DAH coefficients of these modes can be modelled and predicted by using a set of coupled Stuart–Landau stochastic differential equations that capture the modes’ frequencies and amplitude modulation in time. Retrospective forecasts show that our resulting multilayer Stuart–Landau model (MSLM) is quite skilful in predicting September SIE compared to year-to-year persistence; moreover, the DAH–MSLM approach provided accurate real-time prediction that was highly competitive for the 2016–2017 Sea Ice Outlook.
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Decline in the Arctic sea ice extent (SIE) is an area of active scientific research with profound socio-economic implications. Of particular interest are reliable methods for SIE forecasting on subseasonal time scales, in particular from early summer into fall, when sea ice coverage in the Arctic reaches its minimum. Here, we apply the recent data-adaptive harmonic (DAH) technique of Chekroun and Kondrashov, (2017), Chaos, 27 for the description, modeling and prediction of the Multisensor Analyzed Sea Ice Extent (MASIE, 2006–2016) data set. The DAH decomposition of MASIE identifies narrowband, spatio-temporal data-adaptive modes over four key Arctic regions. The time evolution of the DAH coefficients of these modes can be modelled and predicted by using a set of coupled Stuart–Landau stochastic differential equations that capture the modes’ frequencies and amplitude modulation in time. Retrospective forecasts show that our resulting multilayer Stuart–Landau model (MSLM) is quite skilful in predicting September SIE compared to year-to-year persistence; moreover, the DAH–MSLM approach provided accurate real-time prediction that was highly competitive for the 2016–2017 Sea Ice Outlook.
[["fid":1725,"view_mode":"default","fields":"format":"default","type":"media","field_deltas":"1":"format":"default","attributes":"class":"media-element file-default","data-delta":"1"]]
[["fid":1725,"view_mode":"default","fields":"format":"default","type":"media","field_deltas":"1":"format":"default","attributes":"class":"media-element file-default","data-delta":"1"]]
Kondrashov, Dmitri; Chekroun, Mickaël D.; Berloff, Pavel
Multiscale Stuart-Landau Emulators: Application to Wind-Driven Ocean Gyres Journal Article
In: Fluids, vol. 3, no. 1, pp. 21, 2018.
@article{1313,
title = {Multiscale Stuart-Landau Emulators: Application to Wind-Driven Ocean Gyres},
author = {Dmitri Kondrashov and Mickaël D. Chekroun and Pavel Berloff},
url = {http://www.mdpi.com/2311-5521/3/1/21},
year = {2018},
date = {2018-03-01},
journal = {Fluids},
volume = {3},
number = {1},
pages = {21},
abstract = {The multiscale variability of the ocean circulation due to its nonlinear dynamics remains a big challenge for theoretical understanding and practical ocean modeling. This paper demonstrates how the data-adaptive harmonic (DAH) decomposition and inverse stochastic modeling techniques introduced in (Chekroun and Kondrashov, (2017), Chaos, 27), allow for reproducing with high fidelity the main statistical properties of multiscale variability in a coarse-grained eddy-resolving ocean flow. This fully-data-driven approach relies on extraction of frequency-ranked time-dependent coefficients describing the evolution of spatio-temporal DAH modes (DAHMs) in the oceanic flow data. In turn, the time series of these coefficients are efficiently modeled by a family of low-order stochastic differential equations (SDEs) stacked per frequency, involving a fixed set of predictor functions and a small number of model coefficients. These SDEs take the form of stochastic oscillators, identified as multilayer Stuart–Landau models (MSLMs), and their use is justified by relying on the theory of Ruelle–Pollicott resonances. The good modeling skills shown by the resulting DAH-MSLM emulators demonstrates the feasibility of using a network of stochastic oscillators for the modeling of geophysical turbulence. In a certain sense, the original quasiperiodic Landau view of turbulence, with the amendment of the inclusion of stochasticity, may be well suited to describe turbulence.
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},
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The multiscale variability of the ocean circulation due to its nonlinear dynamics remains a big challenge for theoretical understanding and practical ocean modeling. This paper demonstrates how the data-adaptive harmonic (DAH) decomposition and inverse stochastic modeling techniques introduced in (Chekroun and Kondrashov, (2017), Chaos, 27), allow for reproducing with high fidelity the main statistical properties of multiscale variability in a coarse-grained eddy-resolving ocean flow. This fully-data-driven approach relies on extraction of frequency-ranked time-dependent coefficients describing the evolution of spatio-temporal DAH modes (DAHMs) in the oceanic flow data. In turn, the time series of these coefficients are efficiently modeled by a family of low-order stochastic differential equations (SDEs) stacked per frequency, involving a fixed set of predictor functions and a small number of model coefficients. These SDEs take the form of stochastic oscillators, identified as multilayer Stuart–Landau models (MSLMs), and their use is justified by relying on the theory of Ruelle–Pollicott resonances. The good modeling skills shown by the resulting DAH-MSLM emulators demonstrates the feasibility of using a network of stochastic oscillators for the modeling of geophysical turbulence. In a certain sense, the original quasiperiodic Landau view of turbulence, with the amendment of the inclusion of stochasticity, may be well suited to describe turbulence.
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Pierini, Stefano; Chekroun, Mickaël D.; Ghil, Michael
The onset of chaos in nonautonomous dissipative dynamical systems: a low-order ocean-model case study Journal Article
In: Nonlinear Processes in Geophysics, vol. 25, pp. 671-692, 2018.
@article{1662,
title = {The onset of chaos in nonautonomous dissipative dynamical systems: a low-order ocean-model case study},
author = {Stefano Pierini and Mickaël D. Chekroun and Michael Ghil},
url = {https://doi.org/10.5194/npg-25-671-2018},
year = {2018},
date = {2018-01-01},
journal = {Nonlinear Processes in Geophysics},
volume = {25},
pages = {671-692},
abstract = {A four-dimensional nonlinear spectral ocean model is used to study the transition to chaos induced by periodic forcing in systems that are nonchaotic in the autonomous limit. The analysis relies on the construction of the system’s pullback attractors (PBAs) through ensemble simulations, based on a large number of initial states in the remote past. A preliminary analysis of the autonomous system is carried out by investigating its bifurcation diagram, as well as by calculating a metric that measures the mean distance between two initially nearby trajectories, along with the system’s entropy. We find that nonchaotic attractors can still exhibit sensitive dependence on initial data over some time interval; this apparent paradox is resolved by noting that the dependence only concerns the phase of the periodic trajectories, and that it disappears once the latter have converged onto the attractor. The periodically forced system, analyzed by the same methods, yields periodic or chaotic PBAs depending on the periodic forcing’s amplitude ε. A new diagnostic method – based on the cross-correlation between two initially nearby trajectories – is proposed to characterize the transition between the two types of behavior. Transition to chaos is found to occur abruptly at a critical value εc and begins with the intermittent emergence of periodic oscillations with distinct phases. The same diagnostic method is finally shown to be a useful tool for autonomous and aperiodically forced systems as well.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
A four-dimensional nonlinear spectral ocean model is used to study the transition to chaos induced by periodic forcing in systems that are nonchaotic in the autonomous limit. The analysis relies on the construction of the system’s pullback attractors (PBAs) through ensemble simulations, based on a large number of initial states in the remote past. A preliminary analysis of the autonomous system is carried out by investigating its bifurcation diagram, as well as by calculating a metric that measures the mean distance between two initially nearby trajectories, along with the system’s entropy. We find that nonchaotic attractors can still exhibit sensitive dependence on initial data over some time interval; this apparent paradox is resolved by noting that the dependence only concerns the phase of the periodic trajectories, and that it disappears once the latter have converged onto the attractor. The periodically forced system, analyzed by the same methods, yields periodic or chaotic PBAs depending on the periodic forcing’s amplitude ε. A new diagnostic method – based on the cross-correlation between two initially nearby trajectories – is proposed to characterize the transition between the two types of behavior. Transition to chaos is found to occur abruptly at a critical value εc and begins with the intermittent emergence of periodic oscillations with distinct phases. The same diagnostic method is finally shown to be a useful tool for autonomous and aperiodically forced systems as well.
Kondrashov, Dmitri; Chekroun, Mickaël D.; Yuan, Xiaojun; Ghil, Michael
Data-adaptive harmonic decomposition and stochastic modeling of Arctic sea ice Book Chapter
In: Advances in Nonlinear Geosciences, pp. 179-205, Springer, A. Tsonis, 2018.
@inbook{1177,
title = {Data-adaptive harmonic decomposition and stochastic modeling of Arctic sea ice},
author = {Dmitri Kondrashov and Mickaël D. Chekroun and Xiaojun Yuan and Michael Ghil},
url = {https://link.springer.com/chapter/10.1007/978-3-319-58895-7_10},
year = {2018},
date = {2018-01-01},
booktitle = {Advances in Nonlinear Geosciences},
pages = {179-205},
publisher = {Springer},
edition = {A. Tsonis},
organization = {Springer},
abstract = {We present and apply a novel method of describing and modeling complex multivariate datasets in the geosciences and elsewhere. Data-adaptive harmonic (DAH) decomposition identifies narrow-banded, spatio-temporal modes (DAHMs) whose frequencies are not necessarily integer multiples of each other. The evolution in time of the DAH coefficients (DAHCs) of these modes can be modeled using a set of coupled Stuart-Landau stochastic differential equations that capture the modes’ frequencies and amplitude modulation in time and space. This methodology is applied first to a challenging synthetic dataset and then to Arctic sea ice concentration (SIC) data from the US National Snow and Ice Data Center (NSIDC). The 36-year (1979–2014) dataset is parsimoniously and accurately described by our DAHMs. Preliminary results indicate that simulations using our multilayer Stuart-Landau model (MSLM) of SICs are stable for much longer time intervals, beyond the end of the twenty-first century, and exhibit interdecadal variability consistent with past historical records. Preliminary results indicate that this MSLM is quite skillful in predicting September sea ice extent.},
keywords = {},
pubstate = {published},
tppubtype = {inbook}
}
We present and apply a novel method of describing and modeling complex multivariate datasets in the geosciences and elsewhere. Data-adaptive harmonic (DAH) decomposition identifies narrow-banded, spatio-temporal modes (DAHMs) whose frequencies are not necessarily integer multiples of each other. The evolution in time of the DAH coefficients (DAHCs) of these modes can be modeled using a set of coupled Stuart-Landau stochastic differential equations that capture the modes’ frequencies and amplitude modulation in time and space. This methodology is applied first to a challenging synthetic dataset and then to Arctic sea ice concentration (SIC) data from the US National Snow and Ice Data Center (NSIDC). The 36-year (1979–2014) dataset is parsimoniously and accurately described by our DAHMs. Preliminary results indicate that simulations using our multilayer Stuart-Landau model (MSLM) of SICs are stable for much longer time intervals, beyond the end of the twenty-first century, and exhibit interdecadal variability consistent with past historical records. Preliminary results indicate that this MSLM is quite skillful in predicting September sea ice extent.
2017
Boers, N.; Chekroun, M. D.; Liu, H.; Kondrashov, D.; Rousseau, D. -D.; Svensson, A.; Bigler, M.; Ghil, M.
Inverse stochastic-dynamic models for high-resolution Greenland ice-core records Journal Article
In: Earth System Dynamics, vol. 8, pp. 1171–1190, 2017.
@article{esd-2017-8,
title = {Inverse stochastic-dynamic models for high-resolution Greenland ice-core records},
author = {N. Boers and M. D. Chekroun and H. Liu and D. Kondrashov and D. -D. Rousseau and A. Svensson and M. Bigler and M. Ghil},
url = {https://www.earth-syst-dynam.net/8/1171/2017/},
doi = {10.5194/esd-2017-8},
year = {2017},
date = {2017-01-01},
journal = {Earth System Dynamics},
volume = {8},
pages = {1171–1190},
abstract = {Proxy records from Greenland ice cores have been studied for several decades, yet many open questions remain regarding the climate variability encoded therein. Here, we use a Bayesian framework for inferring inverse, stochastic-dynamic models from δ18O and dust records of unprecedented, subdecadal temporal resolution. The records stem from the North Greenland Ice Core Project (NGRIP) and we focus on the time interval 59 ka–22 ka b2k. Our model reproduces the dynamical characteristics of both the δ18O and dust proxy records, including the millennial-scale Dansgaard–Oeschger variability, as well as statistical properties such as probability density functions, waiting times and power spectra, with no need for any external forcing. The crucial ingredients for capturing these properties are (i) high-resolution training data; (ii) cubic drift terms; (iii) nonlinear coupling terms between the δ18O and dust time series; and (iv) non-Markovian contributions that represent short-term memory effects.
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Proxy records from Greenland ice cores have been studied for several decades, yet many open questions remain regarding the climate variability encoded therein. Here, we use a Bayesian framework for inferring inverse, stochastic-dynamic models from δ18O and dust records of unprecedented, subdecadal temporal resolution. The records stem from the North Greenland Ice Core Project (NGRIP) and we focus on the time interval 59 ka–22 ka b2k. Our model reproduces the dynamical characteristics of both the δ18O and dust proxy records, including the millennial-scale Dansgaard–Oeschger variability, as well as statistical properties such as probability density functions, waiting times and power spectra, with no need for any external forcing. The crucial ingredients for capturing these properties are (i) high-resolution training data; (ii) cubic drift terms; (iii) nonlinear coupling terms between the δ18O and dust time series; and (iv) non-Markovian contributions that represent short-term memory effects.
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[["fid":"1721","view_mode":"default","fields":"format":"default","type":"media","field_deltas":"1":"format":"default","attributes":"class":"media-element file-default","data-delta":"1"]]
Chekroun, Mickaël D.; Liu, Honghu; McWilliams, James C.
The emergence of fast oscillations in a reduced primitive equation model and its implications for closure theories Journal Article
In: Computers & Fluids, vol. 151, pp. 3-22, 2017, ISSN: 0045-7930.
@article{Chekroun2016,
title = {The emergence of fast oscillations in a reduced primitive equation model and its implications for closure theories},
author = {Mickaël D. Chekroun and Honghu Liu and James C. McWilliams},
url = {http://www.sciencedirect.com/science/article/pii/S004579301630216X},
doi = {http://dx.doi.org/10.1016/j.compfluid.2016.07.005},
issn = {0045-7930},
year = {2017},
date = {2017-01-01},
journal = {Computers & Fluids},
volume = {151},
pages = {3-22},
abstract = {The problem of emergence of fast gravity-wave oscillations in rotating, stratified flow is reconsidered. Fast inertia-gravity oscillations have long been considered an impediment to initialization of weather forecasts, and the concept of a “slow manifold” evolution, with no fast oscillations, has been hypothesized. It is shown on a reduced Primitive Equation model introduced by Lorenz in 1980 that fast oscillations are absent over a finite interval in Rossby number but they can develop brutally once a critical Rossby number is crossed, in contradistinction with fast oscillations emerging according to an exponential smallness scenario such as reported in previous studies, including some others by Lorenz. The consequences of this dynamical transition on the closure problem based on slow variables is also discussed. In that respect, a novel variational perspective on the closure problem exploiting manifolds is introduced. This framework allows for a unification of previous concepts such as the slow manifold or other concepts of “fuzzy” manifold. It allows furthermore for a rigorous identification of an optimal limiting object for the averaging of fast oscillations, namely the optimal parameterizing manifold (PM). It is shown through detailed numerical computations and rigorous error estimates that the manifold underlying the nonlinear Balance Equations provides a very good approximation of this optimal PM even somewhat beyond the emergence of fast and energetic oscillations.
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The problem of emergence of fast gravity-wave oscillations in rotating, stratified flow is reconsidered. Fast inertia-gravity oscillations have long been considered an impediment to initialization of weather forecasts, and the concept of a “slow manifold” evolution, with no fast oscillations, has been hypothesized. It is shown on a reduced Primitive Equation model introduced by Lorenz in 1980 that fast oscillations are absent over a finite interval in Rossby number but they can develop brutally once a critical Rossby number is crossed, in contradistinction with fast oscillations emerging according to an exponential smallness scenario such as reported in previous studies, including some others by Lorenz. The consequences of this dynamical transition on the closure problem based on slow variables is also discussed. In that respect, a novel variational perspective on the closure problem exploiting manifolds is introduced. This framework allows for a unification of previous concepts such as the slow manifold or other concepts of “fuzzy” manifold. It allows furthermore for a rigorous identification of an optimal limiting object for the averaging of fast oscillations, namely the optimal parameterizing manifold (PM). It is shown through detailed numerical computations and rigorous error estimates that the manifold underlying the nonlinear Balance Equations provides a very good approximation of this optimal PM even somewhat beyond the emergence of fast and energetic oscillations.
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2016
Kondrashov, Dmitri; Chekroun, Mickaël D.; Ghil, Michael
Comment on ‘‘Nonparametric forecasting of low-dimensional dynamical systems&$#$39;&$#$39; Journal Article
In: Phys. Rev. E, vol. 93, pp. 036201, 2016.
@article{PhysRevE.93.036201,
title = {Comment on ‘‘Nonparametric forecasting of low-dimensional dynamical systems&$#$39;&$#$39;},
author = {Dmitri Kondrashov and Mickaël D. Chekroun and Michael Ghil},
url = {http://link.aps.org/doi/10.1103/PhysRevE.93.036201},
doi = {10.1103/PhysRevE.93.036201},
year = {2016},
date = {2016-03-01},
journal = {Phys. Rev. E},
volume = {93},
pages = {036201},
publisher = {American Physical Society},
abstract = {[["fid":"291","view_mode":"default","type":"media","field_deltas":,"attributes":"height":"215","width":"275","style":"float: right;","class":"media-element file-default","data-delta":"1","fields":]]The comparison performed in Berry et al. [Phys. Rev. E 91, 032915 (2015)] between the skill in predicting the El Niño-Southern Oscillation climate phenomenon by the prediction method of Berry et al. and the “past-noise” forecasting method of Chekroun et al. [Proc. Natl. Acad. Sci. USA 108, 11766 (2011)] is flawed. Three specific misunderstandings in Berry et al. are pointed out and corrected.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
[["fid":"291","view_mode":"default","type":"media","field_deltas":,"attributes":"height":"215","width":"275","style":"float: right;","class":"media-element file-default","data-delta":"1","fields":]]The comparison performed in Berry et al. [Phys. Rev. E 91, 032915 (2015)] between the skill in predicting the El Niño-Southern Oscillation climate phenomenon by the prediction method of Berry et al. and the “past-noise” forecasting method of Chekroun et al. [Proc. Natl. Acad. Sci. USA 108, 11766 (2011)] is flawed. Three specific misunderstandings in Berry et al. are pointed out and corrected.
Dijkstra, Henk A; Tantet, Alexis; Viebahn, Jan; Mulder, Erik; Hebbink, Mariët; Castellana, Daniele; Pol, Henri; Frank, Jason; Baars, Sven; Wubs, Fred; Chekroun, Mickaël; Kuehn, Christian
A numerical framework to understand transitions in high-dimensional stochastic dynamical systems Journal Article
In: Dynamics and Statistics of the Climate System, vol. 1, no. 1, pp. 1-27, 2016.
@article{doi:10.1093/climsys/dzw003,
title = {A numerical framework to understand transitions in high-dimensional stochastic dynamical systems},
author = {Henk A Dijkstra and Alexis Tantet and Jan Viebahn and Erik Mulder and Mariët Hebbink and Daniele Castellana and Henri Pol and Jason Frank and Sven Baars and Fred Wubs and Mickaël Chekroun and Christian Kuehn},
url = {https://doi.org/10.1093/climsys/dzw003},
doi = {10.1093/climsys/dzw003},
year = {2016},
date = {2016-01-01},
journal = {Dynamics and Statistics of the Climate System},
volume = {1},
number = {1},
pages = {1-27},
abstract = {Dynamical systems methodology is a mature complementary approach to forward simulation which can be used to investigate many aspects of climate dynamics. With this paper, a review is given on the methods to analyze deterministic and stochastic climate models and show that these are not restricted to low-dimensional toy models, but that they can be applied to models formulated by stochastic partial differential equations. We sketch the numerical implementation of these methods and illustrate these by showing results for two canonical problems in climate dynamics.},
keywords = {},
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Dynamical systems methodology is a mature complementary approach to forward simulation which can be used to investigate many aspects of climate dynamics. With this paper, a review is given on the methods to analyze deterministic and stochastic climate models and show that these are not restricted to low-dimensional toy models, but that they can be applied to models formulated by stochastic partial differential equations. We sketch the numerical implementation of these methods and illustrate these by showing results for two canonical problems in climate dynamics.
Chekroun, Mickaël D.; Park, Eunhee; Temam, Roger
The Stampacchia maximum principle for stochastic partial differential equations and applications Journal Article
In: Journal of Differential Equations, vol. 260, no. 3, pp. 2926 - 2972, 2016, ISSN: 0022-0396.
@article{Chekroun20162926,
title = {The Stampacchia maximum principle for stochastic partial differential equations and applications},
author = {Mickaël D. Chekroun and Eunhee Park and Roger Temam},
url = {http://www.sciencedirect.com/science/article/pii/S0022039615005653},
doi = {http://dx.doi.org/10.1016/j.jde.2015.10.022},
issn = {0022-0396},
year = {2016},
date = {2016-01-01},
journal = {Journal of Differential Equations},
volume = {260},
number = {3},
pages = {2926 - 2972},
abstract = {Abstract Stochastic partial differential equations (SPDEs) are considered, linear and nonlinear, for which we establish comparison theorems for the solutions, or positivity results a.e., and a.s., for suitable data. Comparison theorems for §PDEs are available in the literature. The originality of our approach is that it is based on the use of truncations, following the Stampacchia approach to maximum principle. We believe that our method, which does not rely too much on probability considerations, is simpler than the existing approaches and to a certain extent, more directly applicable to concrete situations. Among the applications, boundedness results and positivity results are respectively proved for the solutions of a stochastic Boussinesq temperature equation, and of reaction–diffusion equations perturbed by a non-Lipschitz nonlinear noise. Stabilization results to a Chafee–Infante equation perturbed by a nonlinear noise are also derived.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Abstract Stochastic partial differential equations (SPDEs) are considered, linear and nonlinear, for which we establish comparison theorems for the solutions, or positivity results a.e., and a.s., for suitable data. Comparison theorems for §PDEs are available in the literature. The originality of our approach is that it is based on the use of truncations, following the Stampacchia approach to maximum principle. We believe that our method, which does not rely too much on probability considerations, is simpler than the existing approaches and to a certain extent, more directly applicable to concrete situations. Among the applications, boundedness results and positivity results are respectively proved for the solutions of a stochastic Boussinesq temperature equation, and of reaction–diffusion equations perturbed by a non-Lipschitz nonlinear noise. Stabilization results to a Chafee–Infante equation perturbed by a nonlinear noise are also derived.
Chen, C.; Cane, M.; Henderson, N.; Lee, D.; Chapman, D.; Kondrashov, D.; Chekroun, M. D.
Diversity, nonlinearity, seasonality and memory effect in ENSO simulation and prediction using empirical model reduction Journal Article
In: Journal of Climate, vol. 29, no. 5, pp. 1809–1830, 2016.
@article{377,
title = {Diversity, nonlinearity, seasonality and memory effect in ENSO simulation and prediction using empirical model reduction},
author = {C. Chen and M. Cane and N. Henderson and D. Lee and D. Chapman and D. Kondrashov and M. D. Chekroun},
url = {http://journals.ametsoc.org/doi/abs/10.1175/JCLI-D-15-0372.1},
year = {2016},
date = {2016-01-01},
journal = {Journal of Climate},
volume = {29},
number = {5},
pages = {1809–1830},
abstract = {A suite of empirical model experiments under the empirical model reduction framework are conducted to advance the understanding of [["fid":"314","view_mode":"default","type":"media","field_deltas":"1":,"fields":,"attributes":"height":"266","width":"359","style":"float: right;","class":"media-element file-default","data-delta":"1"]]ENSO diversity, nonlinearity, seasonality, and the memory effect in the simulation and prediction of tropical Pacific sea surface temperature (SST) anomalies. The model training and evaluation are carried out using 4000-yr preindustrial control simulation data from the coupled model GFDL CM2.1. The results show that multivariate models with tropical Pacific subsurface information and multilevel models with SST history information both improve the prediction skill dramatically. These two types of models represent the ENSO memory effect based on either the recharge oscillator or the time-delayed oscillator viewpoint. Multilevel SST models are a bit more efficient, requiring fewer model coefficients. Nonlinearity is found necessary to reproduce the ENSO diversity feature for extreme events. The nonlinear models reconstruct the skewed probability density function of SST anomalies and improve the prediction of the skewed amplitude, though the role of nonlinearity may be slightly overestimated given the strong nonlinear ENSO in GFDL CM2.1. The models with periodic terms reproduce the SST seasonal phase locking but do not improve the prediction appreciably. The models with multiple ingredients capture several ENSO characteristics simultaneously and exhibit overall better prediction skill for more diverse target patterns. In particular, they alleviate the spring/autumn prediction barrier and reduce the tendency for predicted values to lag the target month value.},
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A suite of empirical model experiments under the empirical model reduction framework are conducted to advance the understanding of [["fid":"314","view_mode":"default","type":"media","field_deltas":"1":,"fields":,"attributes":"height":"266","width":"359","style":"float: right;","class":"media-element file-default","data-delta":"1"]]ENSO diversity, nonlinearity, seasonality, and the memory effect in the simulation and prediction of tropical Pacific sea surface temperature (SST) anomalies. The model training and evaluation are carried out using 4000-yr preindustrial control simulation data from the coupled model GFDL CM2.1. The results show that multivariate models with tropical Pacific subsurface information and multilevel models with SST history information both improve the prediction skill dramatically. These two types of models represent the ENSO memory effect based on either the recharge oscillator or the time-delayed oscillator viewpoint. Multilevel SST models are a bit more efficient, requiring fewer model coefficients. Nonlinearity is found necessary to reproduce the ENSO diversity feature for extreme events. The nonlinear models reconstruct the skewed probability density function of SST anomalies and improve the prediction of the skewed amplitude, though the role of nonlinearity may be slightly overestimated given the strong nonlinear ENSO in GFDL CM2.1. The models with periodic terms reproduce the SST seasonal phase locking but do not improve the prediction appreciably. The models with multiple ingredients capture several ENSO characteristics simultaneously and exhibit overall better prediction skill for more diverse target patterns. In particular, they alleviate the spring/autumn prediction barrier and reduce the tendency for predicted values to lag the target month value.
Pierini, S.; Ghil, M.; Chekroun, M.
Exploring the pullback attractors of a low-order quasigeostrophic ocean model: the deterministic case Journal Article
In: Journal of Climate, vol. 29, no. 11, pp. 4185-4202, 2016.
@article{376,
title = {Exploring the pullback attractors of a low-order quasigeostrophic ocean model: the deterministic case},
author = {S. Pierini and M. Ghil and M. Chekroun},
url = {http://journals.ametsoc.org/doi/abs/10.1175/JCLI-D-15-0848.1?af=R},
year = {2016},
date = {2016-01-01},
journal = {Journal of Climate},
volume = {29},
number = {11},
pages = {4185-4202},
abstract = {A low-order quasigeostrophic double-gyre ocean model is subjected to an aperiodic forcing that mimics time dependence dominated by interdecadal variability. This model is used as a prototype of an unstable and nonlinear dynamical system with time-dependent forcing to explore basic features of climate change in the presence of natural variability. The study relies on the theoretical framework of nonautonomous dynamical systems and of their pullback attractors (PBAs), that is, of the time-dependent invariant sets attracting all trajectories initialized in the remote past. The existence of a global PBA is rigorously demonstrated for this weakly dissipative nonlinear model. Ensemble simulations are carried out and the convergence to PBAs is assessed by computing the probability density function (PDF) of localization of the trajectories. A sensitivity analysis with respect to forcing amplitude shows that the PBAs experience large modifications if the underlying autonomous system is dominated by small-amplitude limit cycles, while less dramatic changes occur in a regime characterized by large-amplitude relaxation oscillations. The dependence of the attracting sets on the choice of the ensemble of initial states is then analyzed. Two types of basins of attraction coexist for certain parameter ranges; they contain chaotic and nonchaotic trajectories, respectively. The statistics of the former does not depend on the initial states whereas the trajectories in the latter converge to small portions of the global PBA. This complex scenario requires separate PDFs for chaotic and nonchaotic trajectories. General implications for climate predictability are finally discussed.},
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A low-order quasigeostrophic double-gyre ocean model is subjected to an aperiodic forcing that mimics time dependence dominated by interdecadal variability. This model is used as a prototype of an unstable and nonlinear dynamical system with time-dependent forcing to explore basic features of climate change in the presence of natural variability. The study relies on the theoretical framework of nonautonomous dynamical systems and of their pullback attractors (PBAs), that is, of the time-dependent invariant sets attracting all trajectories initialized in the remote past. The existence of a global PBA is rigorously demonstrated for this weakly dissipative nonlinear model. Ensemble simulations are carried out and the convergence to PBAs is assessed by computing the probability density function (PDF) of localization of the trajectories. A sensitivity analysis with respect to forcing amplitude shows that the PBAs experience large modifications if the underlying autonomous system is dominated by small-amplitude limit cycles, while less dramatic changes occur in a regime characterized by large-amplitude relaxation oscillations. The dependence of the attracting sets on the choice of the ensemble of initial states is then analyzed. Two types of basins of attraction coexist for certain parameter ranges; they contain chaotic and nonchaotic trajectories, respectively. The statistics of the former does not depend on the initial states whereas the trajectories in the latter converge to small portions of the global PBA. This complex scenario requires separate PDFs for chaotic and nonchaotic trajectories. General implications for climate predictability are finally discussed.
Chekroun, Mickaël D.; Ghil, Michael; Liu, Honghu; Wang, Shouhong
Low-dimensional Galerkin approximations of nonlinear delay differential equations Journal Article
In: Discrete and Continuous Dynamical Systems, vol. 36, no. 8, pp. 4133-4177, 2016, ISSN: 1078-0947.
@article{Chekroun20164133,
title = {Low-dimensional Galerkin approximations of nonlinear delay differential equations},
author = {Mickaël D. Chekroun and Michael Ghil and Honghu Liu and Shouhong Wang},
url = {http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=12348},
doi = {10.3934/dcds.2016.36.4133},
issn = {1078-0947},
year = {2016},
date = {2016-01-01},
journal = {Discrete and Continuous Dynamical Systems},
volume = {36},
number = {8},
pages = {4133-4177},
abstract = {This article revisits the approximation problem of systems of nonlinear delay differential equations (DDEs) by a set of ordinary differential equations (ODEs). We work in Hilbert spaces endowed with a natural inner product including a point mass, and introduce polynomials orthogonal with respect to such an inner product that live in the domain of the linear operator associated with the underlying DDE. These polynomials are then used to design a general Galerkin scheme for which we derive rigorous convergence results and show that it can be numerically implemented via simple analytic formulas. The scheme so obtained is applied to three nonlinear DDEs, two autonomous and one forced: (i) a simple DDE with distributed delays whose solutions recall Brownian motion; (ii) a DDE with a discrete delay that exhibits bimodal and chaotic dynamics; and (iii) a periodically forced DDE with two discrete delays arising in climate dynamics. In all three cases, the Galerkin scheme introduced in this article provides a good approximation by low-dimensional ODE systems of the DDE’s strange attractor, as well as of the statistical features that characterize its nonlinear dynamics.
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This article revisits the approximation problem of systems of nonlinear delay differential equations (DDEs) by a set of ordinary differential equations (ODEs). We work in Hilbert spaces endowed with a natural inner product including a point mass, and introduce polynomials orthogonal with respect to such an inner product that live in the domain of the linear operator associated with the underlying DDE. These polynomials are then used to design a general Galerkin scheme for which we derive rigorous convergence results and show that it can be numerically implemented via simple analytic formulas. The scheme so obtained is applied to three nonlinear DDEs, two autonomous and one forced: (i) a simple DDE with distributed delays whose solutions recall Brownian motion; (ii) a DDE with a discrete delay that exhibits bimodal and chaotic dynamics; and (iii) a periodically forced DDE with two discrete delays arising in climate dynamics. In all three cases, the Galerkin scheme introduced in this article provides a good approximation by low-dimensional ODE systems of the DDE’s strange attractor, as well as of the statistical features that characterize its nonlinear dynamics.
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[["fid":"293","view_mode":"default","type":"media","field_deltas":"1":,"fields":,"attributes":"height":"323","width":"606","style":"display: block; margin-left: auto; margin-right: auto;","class":"media-element file-default","data-delta":"1"]]
Chekroun, M. D.; Liu, H.
2016 IEEE 55th Conference on Decision and Control (CDC), IEEE IEEE, Las Vegas, USA, 2016.
@conference{443,
title = {Post-processing finite-horizon parameterizing manifolds for optimal control of nonlinear parabolic PDEs},
author = {M. D. Chekroun and H. Liu},
url = {http://ieeexplore.ieee.org/document/7798464/},
year = {2016},
date = {2016-01-01},
booktitle = {2016 IEEE 55th Conference on Decision and Control (CDC)},
pages = {1411-1416},
publisher = {IEEE},
address = {Las Vegas, USA},
organization = {IEEE},
abstract = {[["fid":"310","view_mode":"default","type":"media","field_deltas":"1":,"fields":,"attributes":"height":"258","width":"401","style":"float: right;","class":"media-element file-default","data-delta":"1"]]The goal of this article is to propose an efficient way of empirically improving suboptimal solutions designed from the recent method of finite-horizon parameterizing manifolds (PMs) introduced by Chekroun and Liu (Acta Appl. Math., 2015) and concerned with the (sub)optimal control of nonlinear parabolic partial differential equations (PDEs). Given a finite horizon [0, T ] and a reduced low-mode phase space, a finite-horizon PM provides an approximate parameterization of the high modes by the low ones so that the unexplained high-mode energy is reduced — in an L 2-sense — when this parameterization is applied. In Chekroun and Liu (Acta Appl. Math., 2015), various PMs were constructed analytically from the uncontrolled version of the underlying PDE that allow for the design of reduced systems from which low-dimensional suboptimal controllers can be efficiently synthesized. In this article, the analytic approach from Chekroun and Liu (Acta Appl. Math., 2015) is recalled and a post-processing procedure is introduced to improve the PM-based suboptimal controllers. It consists of seeking for a high-mode parametrization aiming to reduce the energy contained in the high modes of the PDE solution, when the latter is driven by a PM-based suboptimal controller. This is achieved by solving simple regression problems. The skills of the resulting empirically post-processed suboptimal controllers are numerically assessed for an optimal control problem associated with a Burgers-type equation.},
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tppubtype = {conference}
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[["fid":"310","view_mode":"default","type":"media","field_deltas":"1":,"fields":,"attributes":"height":"258","width":"401","style":"float: right;","class":"media-element file-default","data-delta":"1"]]The goal of this article is to propose an efficient way of empirically improving suboptimal solutions designed from the recent method of finite-horizon parameterizing manifolds (PMs) introduced by Chekroun and Liu (Acta Appl. Math., 2015) and concerned with the (sub)optimal control of nonlinear parabolic partial differential equations (PDEs). Given a finite horizon [0, T ] and a reduced low-mode phase space, a finite-horizon PM provides an approximate parameterization of the high modes by the low ones so that the unexplained high-mode energy is reduced — in an L 2-sense — when this parameterization is applied. In Chekroun and Liu (Acta Appl. Math., 2015), various PMs were constructed analytically from the uncontrolled version of the underlying PDE that allow for the design of reduced systems from which low-dimensional suboptimal controllers can be efficiently synthesized. In this article, the analytic approach from Chekroun and Liu (Acta Appl. Math., 2015) is recalled and a post-processing procedure is introduced to improve the PM-based suboptimal controllers. It consists of seeking for a high-mode parametrization aiming to reduce the energy contained in the high modes of the PDE solution, when the latter is driven by a PM-based suboptimal controller. This is achieved by solving simple regression problems. The skills of the resulting empirically post-processed suboptimal controllers are numerically assessed for an optimal control problem associated with a Burgers-type equation.
2015
Chekroun, Mickaël D.; Liu, Honghu
Finite-Horizon Parameterizing Manifolds, and Applications to Suboptimal Control of Nonlinear Parabolic PDEs Journal Article
In: Acta Applicandae Mathematicae, vol. 135, no. 1, pp. 81–144, 2015, ISSN: 1572-9036.
@article{Chekroun2015,
title = {Finite-Horizon Parameterizing Manifolds, and Applications to Suboptimal Control of Nonlinear Parabolic PDEs},
author = {Mickaël D. Chekroun and Honghu Liu},
url = {http://dx.doi.org/10.1007/s10440-014-9949-1},
doi = {10.1007/s10440-014-9949-1},
issn = {1572-9036},
year = {2015},
date = {2015-01-01},
journal = {Acta Applicandae Mathematicae},
volume = {135},
number = {1},
pages = {81–144},
abstract = {This article proposes a new approach based on finite-horizon parameterizing manifolds (PMs) for the design of low-dimensional suboptimal controllers to optimal control problems of nonlinear partial differential equations (PDEs) of parabolic type. Given a finite horizon [0,T] and a low-mode truncation of the PDE, a PM provides an approximate parameterization of the uncontrolled high modes by the controlled low ones so that the unexplained high-mode energy is reduced, in an L2-sense, when this parameterization is applied. Analytic formulas of such PMs are derived by application of the method of pullback approximation of the high-modes. These formulas allow for an effective derivation of reduced ODE systems, aimed to model the evolution of the low-mode truncation of the controlled state variable, where the high-mode part is approximated by the PM function applied to the low modes. A priori error estimates between the resulting PM-based low-dimensional suboptimal controller u_R* and the optimal controller u* are derived. These estimates demonstrate that the closeness of u_R* to u*? is mainly conditioned on two factors: (i) the parameterization defect of a given PM, associated respectively with u_R* and u*; and (ii) the energy kept in the high modes of the PDE solution either driven by u_R* or u* itself. The practical performances of such PM-based suboptimal controllers are numerically assessed for various optimal control problems associated with a Burgers-type equation. The numerical results show that a PM-based reduced system allows for the design of suboptimal controllers with good performances provided that the associated parameterization defects and energy kept in the high modes are small enough, in agreement with the rigorous results. The practical performances of such PM-based suboptimal controllers are numerically assessed for optimal control problems associated with a Burgers-type equation; the locally as well as globally distributed cases being both considered. The numerical results show that a PM-based reduced system allows for the design of suboptimal controllers with good performances provided that the associated parameterization defects and energy kept in the high modes are small enough, in agreement with the rigorous results.
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This article proposes a new approach based on finite-horizon parameterizing manifolds (PMs) for the design of low-dimensional suboptimal controllers to optimal control problems of nonlinear partial differential equations (PDEs) of parabolic type. Given a finite horizon [0,T] and a low-mode truncation of the PDE, a PM provides an approximate parameterization of the uncontrolled high modes by the controlled low ones so that the unexplained high-mode energy is reduced, in an L2-sense, when this parameterization is applied. Analytic formulas of such PMs are derived by application of the method of pullback approximation of the high-modes. These formulas allow for an effective derivation of reduced ODE systems, aimed to model the evolution of the low-mode truncation of the controlled state variable, where the high-mode part is approximated by the PM function applied to the low modes. A priori error estimates between the resulting PM-based low-dimensional suboptimal controller u_R* and the optimal controller u* are derived. These estimates demonstrate that the closeness of u_R* to u*? is mainly conditioned on two factors: (i) the parameterization defect of a given PM, associated respectively with u_R* and u*; and (ii) the energy kept in the high modes of the PDE solution either driven by u_R* or u* itself. The practical performances of such PM-based suboptimal controllers are numerically assessed for various optimal control problems associated with a Burgers-type equation. The numerical results show that a PM-based reduced system allows for the design of suboptimal controllers with good performances provided that the associated parameterization defects and energy kept in the high modes are small enough, in agreement with the rigorous results. The practical performances of such PM-based suboptimal controllers are numerically assessed for optimal control problems associated with a Burgers-type equation; the locally as well as globally distributed cases being both considered. The numerical results show that a PM-based reduced system allows for the design of suboptimal controllers with good performances provided that the associated parameterization defects and energy kept in the high modes are small enough, in agreement with the rigorous results.
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[["fid":"1722","view_mode":"default","fields":"format":"default","type":"media","field_deltas":"1":"format":"default","attributes":"class":"media-element file-default","data-delta":"1"]]
Kondrashov, Dmitri; Chekroun, Mickaël D.; Ghil, Michael
Data-driven non-Markovian closure models Journal Article
In: Physica D: Nonlinear Phenomena, vol. 297, pp. 33 - 55, 2015, ISSN: 0167-2789.
@article{Kondrashov201533,
title = {Data-driven non-Markovian closure models},
author = {Dmitri Kondrashov and Mickaël D. Chekroun and Michael Ghil},
url = {http://www.sciencedirect.com/science/article/pii/S0167278914002413},
doi = {http://dx.doi.org/10.1016/j.physd.2014.12.005},
issn = {0167-2789},
year = {2015},
date = {2015-01-01},
journal = {Physica D: Nonlinear Phenomena},
volume = {297},
pages = {33 - 55},
abstract = {Abstract This paper has two interrelated foci: (i) obtaining stable and efficient data-driven closure models by using a multivariate time series of partial observations from a large-dimensional system; and (ii) comparing these closure models with the optimal closures predicted by the Mori–Zwanzig (MZ) formalism of statistical physics. Multilayer stochastic models (MSMs) are introduced as both a generalization and a time-continuous limit of existing multilevel, regression-based approaches to closure in a data-driven setting; these approaches include empirical model reduction (EMR), as well as more recent multi-layer modeling. It is shown that the multilayer structure of MSMs can provide a natural Markov approximation to the generalized Langevin equation (GLE) of the MZ formalism. A simple correlation-based stopping criterion for an EMR–MSM model is derived to assess how well it approximates the GLE solution. Sufficient conditions are derived on the structure of the nonlinear cross-interactions between the constitutive layers of a given MSM to guarantee the existence of a global random attractor. This existence ensures that no blow-up can occur for a broad class of MSM applications, a class that includes non-polynomial predictors and nonlinearities that do not necessarily preserve quadratic energy invariants. The EMR–MSM methodology is first applied to a conceptual, nonlinear, stochastic climate model of coupled slow and fast variables, in which only slow variables are observed. It is shown that the resulting closure model with energy-conserving nonlinearities efficiently captures the main statistical features of the slow variables, even when there is no formal scale separation and the fast variables are quite energetic. Second, an MSM is shown to successfully reproduce the statistics of a partially observed, generalized Lotka–Volterra model of population dynamics in its chaotic regime. The challenges here include the rarity of strange attractors in the model’s parameter space and the existence of multiple attractor basins with fractal boundaries. The positivity constraint on the solutions’ components replaces here the quadratic-energy–preserving constraint of fluid-flow problems and it successfully prevents blow-up.},
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Abstract This paper has two interrelated foci: (i) obtaining stable and efficient data-driven closure models by using a multivariate time series of partial observations from a large-dimensional system; and (ii) comparing these closure models with the optimal closures predicted by the Mori–Zwanzig (MZ) formalism of statistical physics. Multilayer stochastic models (MSMs) are introduced as both a generalization and a time-continuous limit of existing multilevel, regression-based approaches to closure in a data-driven setting; these approaches include empirical model reduction (EMR), as well as more recent multi-layer modeling. It is shown that the multilayer structure of MSMs can provide a natural Markov approximation to the generalized Langevin equation (GLE) of the MZ formalism. A simple correlation-based stopping criterion for an EMR–MSM model is derived to assess how well it approximates the GLE solution. Sufficient conditions are derived on the structure of the nonlinear cross-interactions between the constitutive layers of a given MSM to guarantee the existence of a global random attractor. This existence ensures that no blow-up can occur for a broad class of MSM applications, a class that includes non-polynomial predictors and nonlinearities that do not necessarily preserve quadratic energy invariants. The EMR–MSM methodology is first applied to a conceptual, nonlinear, stochastic climate model of coupled slow and fast variables, in which only slow variables are observed. It is shown that the resulting closure model with energy-conserving nonlinearities efficiently captures the main statistical features of the slow variables, even when there is no formal scale separation and the fast variables are quite energetic. Second, an MSM is shown to successfully reproduce the statistics of a partially observed, generalized Lotka–Volterra model of population dynamics in its chaotic regime. The challenges here include the rarity of strange attractors in the model’s parameter space and the existence of multiple attractor basins with fractal boundaries. The positivity constraint on the solutions’ components replaces here the quadratic-energy–preserving constraint of fluid-flow problems and it successfully prevents blow-up.
Chekroun, Mickaël D.; Liu, Honghu; Wang, Shouhong
Approximation of Stochastic Invariant Manifolds : Stochastic Manifolds for Nonlinear SPDEs I Book
Springer Briefs in Mathematics, Springer, New York, 2015.
@book{395,
title = {Approximation of Stochastic Invariant Manifolds : Stochastic Manifolds for Nonlinear SPDEs I},
author = {Mickaël D. Chekroun and Honghu Liu and Shouhong Wang},
url = {http://www.springer.com/us/book/9783319124957},
year = {2015},
date = {2015-01-01},
pages = {pp. 127},
publisher = {Springer Briefs in Mathematics, Springer},
address = {New York},
organization = {Springer Briefs in Mathematics, Springer},
abstract = {[["fid":"285","view_mode":"default","type":"media","field_deltas":"1":,"fields":,"attributes":"height":"232","width":"153","style":"float: right;","class":"media-element file-default","data-delta":"1"]]This first volume is concerned with the analytic derivation of explicit formulas for the leading-order Taylor approximations of (local) stochastic invariant manifolds associated with a broad class of nonlinear stochastic partial differential equations. These approximations take the form of Lyapunov-Perron integrals, which are further characterized in Volume II as pullback limits associated with some partially coupled backward-forward systems. This pullback characterization provides a useful interpretation of the corresponding approximating manifolds and leads to a simple framework that unifies some other approximation approaches in the literature. A self-contained survey is also included on the existence and attraction of one-parameter families of stochastic invariant manifolds, from the point of view of the theory of random dynamical systems.
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[["fid":"285","view_mode":"default","type":"media","field_deltas":"1":,"fields":,"attributes":"height":"232","width":"153","style":"float: right;","class":"media-element file-default","data-delta":"1"]]This first volume is concerned with the analytic derivation of explicit formulas for the leading-order Taylor approximations of (local) stochastic invariant manifolds associated with a broad class of nonlinear stochastic partial differential equations. These approximations take the form of Lyapunov-Perron integrals, which are further characterized in Volume II as pullback limits associated with some partially coupled backward-forward systems. This pullback characterization provides a useful interpretation of the corresponding approximating manifolds and leads to a simple framework that unifies some other approximation approaches in the literature. A self-contained survey is also included on the existence and attraction of one-parameter families of stochastic invariant manifolds, from the point of view of the theory of random dynamical systems.
[["fid":"1718","view_mode":"default","fields":"format":"default","type":"media","field_deltas":"2":"format":"default","attributes":"class":"media-element file-default","data-delta":"2"]]
[["fid":"1718","view_mode":"default","fields":"format":"default","type":"media","field_deltas":"2":"format":"default","attributes":"class":"media-element file-default","data-delta":"2"]]
A.Bousquet,; Chekroun, M. D.; Hong, Y.; Temam, R.; Tribbia, J.
Numerical simulations of the humid atmosphere above a mountain Journal Article
In: Mathematics of Climate and Weather Forecasting, vol. 1, no. 1, pp. 96-126, 2015.
@article{394,
title = {Numerical simulations of the humid atmosphere above a mountain},
author = {A.Bousquet and M. D. Chekroun and Y. Hong and R. Temam and J. Tribbia},
url = {http://www.degruyter.com/view/j/mcwf.2015.1.issue-1/mcwf-2015-0005/mcwf-2015-0005.xml},
year = {2015},
date = {2015-01-01},
journal = {Mathematics of Climate and Weather Forecasting},
volume = {1},
number = {1},
pages = {96-126},
abstract = {New avenues are explored for the numerical study of the two dimensional inviscid hydrostatic primitive equations of the atmosphere with humidity and saturation, in presence of topography and subject to physically plausible boundary conditions for the system of equations. Flows above a mountain are classically treated by the so-called method of terrain following coordinate system. We avoid this discretization method which induces errors in the discretization of tangential derivatives near the topography. Instead we implement a first order finite volume method for the spatial discretization using the initial coordinates x and p. A compatibility condition similar to that related to the condition of incompressibility for the Navier- Stokes equations, is introduced. In that respect, a version of the projection method is considered to enforce the compatibility condition on the horizontal velocity field, which comes from the boundary conditions. For the spatial discretization, a modified Godunov type method that exploits the discrete finite-volume derivatives by using the so-called Taylor Series Expansion Scheme (TSES), is then designed to solve the equations. We report on numerical experiments using realistic parameters. Finally, the effects of a random small-scale forcing on the velocity equation is numerically investigated.},
keywords = {},
pubstate = {published},
tppubtype = {article}
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New avenues are explored for the numerical study of the two dimensional inviscid hydrostatic primitive equations of the atmosphere with humidity and saturation, in presence of topography and subject to physically plausible boundary conditions for the system of equations. Flows above a mountain are classically treated by the so-called method of terrain following coordinate system. We avoid this discretization method which induces errors in the discretization of tangential derivatives near the topography. Instead we implement a first order finite volume method for the spatial discretization using the initial coordinates x and p. A compatibility condition similar to that related to the condition of incompressibility for the Navier- Stokes equations, is introduced. In that respect, a version of the projection method is considered to enforce the compatibility condition on the horizontal velocity field, which comes from the boundary conditions. For the spatial discretization, a modified Godunov type method that exploits the discrete finite-volume derivatives by using the so-called Taylor Series Expansion Scheme (TSES), is then designed to solve the equations. We report on numerical experiments using realistic parameters. Finally, the effects of a random small-scale forcing on the velocity equation is numerically investigated.
Chekroun, Mickaël D.; Liu, Honghu; Wang, Shouhong
Springer Briefs in Mathematics, Springer, New York, 2015.
@book{396,
title = {Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations : Stochastic Manifolds for Nonlinear SPDEs II},
author = {Mickaël D. Chekroun and Honghu Liu and Shouhong Wang},
url = {http://www.springer.com/us/book/9783319125190},
year = {2015},
date = {2015-01-01},
pages = {pp. 129},
publisher = {Springer Briefs in Mathematics, Springer},
address = {New York},
organization = {Springer Briefs in Mathematics, Springer},
abstract = {[["fid":282,"view_mode":"default","type":"media","field_deltas":"1":,"fields":,"attributes":"height":"232","width":"153","style":"float: right;","class":"media-element file-default","data-delta":"1"]]In this second volume, a general approach is developed to provide approximate parameterizations of the "small" scales by the "large" ones for a broad class of stochastic partial differential equations (SPDEs). This is accomplished via the concept of parameterizing manifolds (PMs), which are stochastic manifolds that improve, for a given realization of the noise, in mean square error the partial knowledge of the full SPDE solution when compared to its projection onto some resolved modes. Backward-forward systems are designed to give access to such PMs in practice. The key idea consists of representing the modes with high wave numbers as a pullback limit depending on the time-history of the modes with low wave numbers. Non-Markovian stochastic reduced systems are then derived based on such a PM approach. The reduced systems take the form of stochastic differential equations involving random coefficients that convey memory effects. The theory is illustrated on a stochastic Burgers-type equation.
[["fid":"1717","view_mode":"default","fields":"format":"default","type":"media","field_deltas":"2":"format":"default","attributes":"class":"media-element file-default","data-delta":"2"]]},
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[["fid":282,"view_mode":"default","type":"media","field_deltas":"1":,"fields":,"attributes":"height":"232","width":"153","style":"float: right;","class":"media-element file-default","data-delta":"1"]]In this second volume, a general approach is developed to provide approximate parameterizations of the "small" scales by the "large" ones for a broad class of stochastic partial differential equations (SPDEs). This is accomplished via the concept of parameterizing manifolds (PMs), which are stochastic manifolds that improve, for a given realization of the noise, in mean square error the partial knowledge of the full SPDE solution when compared to its projection onto some resolved modes. Backward-forward systems are designed to give access to such PMs in practice. The key idea consists of representing the modes with high wave numbers as a pullback limit depending on the time-history of the modes with low wave numbers. Non-Markovian stochastic reduced systems are then derived based on such a PM approach. The reduced systems take the form of stochastic differential equations involving random coefficients that convey memory effects. The theory is illustrated on a stochastic Burgers-type equation.
[["fid":"1717","view_mode":"default","fields":"format":"default","type":"media","field_deltas":"2":"format":"default","attributes":"class":"media-element file-default","data-delta":"2"]]
[["fid":"1717","view_mode":"default","fields":"format":"default","type":"media","field_deltas":"2":"format":"default","attributes":"class":"media-element file-default","data-delta":"2"]]
2014
Roques, Lionel; Chekroun, Mickaël D.; Cristofol, Michel; Soubeyrand, Samuel; Ghil, Michael
Parameter estimation for energy balance models with memory Journal Article
In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 470, no. 2169, 2014, ISSN: 1364-5021.
@article{Roques20140349,
title = {Parameter estimation for energy balance models with memory},
author = {Lionel Roques and Mickaël D. Chekroun and Michel Cristofol and Samuel Soubeyrand and Michael Ghil},
url = {http://rspa.royalsocietypublishing.org/content/470/2169/20140349},
doi = {10.1098/rspa.2014.0349},
issn = {1364-5021},
year = {2014},
date = {2014-01-01},
journal = {Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences},
volume = {470},
number = {2169},
publisher = {The Royal Society},
abstract = {We study parameter estimation for one-dimensional energy balance models with memory (EBMMs) given localized and noisy temperature measurements. Our results apply to a wide range of nonlinear, parabolic partial differential equations with integral memory terms. First, we show that a space-dependent parameter can be determined uniquely everywhere in the PDE’s domain of definition D, using only temperature information in a small subdomain E⊂D. This result is valid only when the data correspond to exact measurements of the temperature. We propose a method for estimating a model parameter of the EBMM using more realistic, error-contaminated temperature data derived, for example, from ice cores or marine-sediment cores. Our approach is based on a so-called mechanistic-statistical model that combines a deterministic EBMM with a statistical model of the observation process. Estimating a parameter in this setting is especially challenging, because the observation process induces a strong loss of information. Aside from the noise contained in past temperature measurements, an additional error is induced by the age-dating method, whose accuracy tends to decrease with a sample’s remoteness in time. Using a Bayesian approach, we show that obtaining an accurate parameter estimate is still possible in certain cases.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We study parameter estimation for one-dimensional energy balance models with memory (EBMMs) given localized and noisy temperature measurements. Our results apply to a wide range of nonlinear, parabolic partial differential equations with integral memory terms. First, we show that a space-dependent parameter can be determined uniquely everywhere in the PDE’s domain of definition D, using only temperature information in a small subdomain E⊂D. This result is valid only when the data correspond to exact measurements of the temperature. We propose a method for estimating a model parameter of the EBMM using more realistic, error-contaminated temperature data derived, for example, from ice cores or marine-sediment cores. Our approach is based on a so-called mechanistic-statistical model that combines a deterministic EBMM with a statistical model of the observation process. Estimating a parameter in this setting is especially challenging, because the observation process induces a strong loss of information. Aside from the noise contained in past temperature measurements, an additional error is induced by the age-dating method, whose accuracy tends to decrease with a sample’s remoteness in time. Using a Bayesian approach, we show that obtaining an accurate parameter estimate is still possible in certain cases.
Chekroun, M. D.; Neelin, J. D.; Kondrashov, D.; McWilliams, J. C.; Ghil, M.
Rough parameter dependence in climate models and the role of Ruelle-Pollicott resonance Journal Article
In: Proceeding of the National Academy of Sciences, vol. 111, no. 5, pp. 1684—1690, 2014.
@article{391,
title = {Rough parameter dependence in climate models and the role of Ruelle-Pollicott resonance},
author = {M. D. Chekroun and J. D. Neelin and D. Kondrashov and J. C. McWilliams and M. Ghil},
url = {http://www.pnas.org/content/111/5/1684.full?sid=caae9302-0489-4096-86d1-61846dd5bbaa},
year = {2014},
date = {2014-01-01},
journal = {Proceeding of the National Academy of Sciences},
volume = {111},
number = {5},
pages = {1684—1690},
abstract = {[["fid":"308","view_mode":"default","type":"media","field_deltas":"1":,"fields":,"attributes":"height":"164","width":"329","style":"float: right;","class":"media-element file-default","data-delta":"1"]]Despite the importance of uncertainties encountered in climate model simulations, the fundamental mechanisms at the origin of sensitive behavior of long-term model statistics remain unclear. Variability of turbulent flows in the atmosphere and oceans exhibits recurrent large-scale patterns. These patterns, while evolving irregularly in time, manifest characteristic frequencies across a large range of time scales, from intraseasonal through interdecadal. Based on modern spectral theory of chaotic and dissipative dynamical systems, the associated low-frequency variability may be formulated in terms of Ruelle-Pollicott (RP) resonances. RP resonances encode information on the nonlinear dynamics of the system, and an approach for estimating them—as filtered through an observable of the system—is proposed. This approach relies on an appropriate Markov representation of the dynamics associated with a given observable. It is shown that, within this representation, the spectral gap—defined as the distance between the subdominant RP resonance and the unit circle—plays a major role in the roughness of parameter dependences. The model statistics are the most sensitive for the smallest spectral gaps; such small gaps turn out to correspond to regimes where the low-frequency variability is more pronounced, whereas autocorrelations decay more slowly. The present approach is applied to analyze the rough parameter dependence encountered in key statistics of an El-Niño–Southern Oscillation model of intermediate complexity. Theoretical arguments, however, strongly suggest that such links between model sensitivity and the decay of correlation properties are not limited to this particular model and could hold much more generally.
},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
[["fid":"308","view_mode":"default","type":"media","field_deltas":"1":,"fields":,"attributes":"height":"164","width":"329","style":"float: right;","class":"media-element file-default","data-delta":"1"]]Despite the importance of uncertainties encountered in climate model simulations, the fundamental mechanisms at the origin of sensitive behavior of long-term model statistics remain unclear. Variability of turbulent flows in the atmosphere and oceans exhibits recurrent large-scale patterns. These patterns, while evolving irregularly in time, manifest characteristic frequencies across a large range of time scales, from intraseasonal through interdecadal. Based on modern spectral theory of chaotic and dissipative dynamical systems, the associated low-frequency variability may be formulated in terms of Ruelle-Pollicott (RP) resonances. RP resonances encode information on the nonlinear dynamics of the system, and an approach for estimating them—as filtered through an observable of the system—is proposed. This approach relies on an appropriate Markov representation of the dynamics associated with a given observable. It is shown that, within this representation, the spectral gap—defined as the distance between the subdominant RP resonance and the unit circle—plays a major role in the roughness of parameter dependences. The model statistics are the most sensitive for the smallest spectral gaps; such small gaps turn out to correspond to regimes where the low-frequency variability is more pronounced, whereas autocorrelations decay more slowly. The present approach is applied to analyze the rough parameter dependence encountered in key statistics of an El-Niño–Southern Oscillation model of intermediate complexity. Theoretical arguments, however, strongly suggest that such links between model sensitivity and the decay of correlation properties are not limited to this particular model and could hold much more generally.
2013
Chekroun, M. D.; Roux, J.
Homeomorphisms group of normed vector spaces : The conjugacy problem and the Koopman operator Journal Article
In: Discrete and Continuous Dynamical Systems (DCDS-A), vol. 33, no. 9, pp. 3957—3950, 2013.
@article{397,
title = {Homeomorphisms group of normed vector spaces : The conjugacy problem and the Koopman operator},
author = {M. D. Chekroun and J. Roux},
url = {http://www.aimsciences.org/article/doi/10.3934/dcds.2013.33.3957},
year = {2013},
date = {2013-01-01},
journal = {Discrete and Continuous Dynamical Systems (DCDS-A)},
volume = {33},
number = {9},
pages = {3957—3950},
abstract = {This article is concerned with conjugacy problems arising in the homeomorphisms group, Hom(F), of unbounded subsets F of normed vector spaces E. Given two homeomorphisms f and g in Hom(F), it is shown how the existence of a conjugacy may be related to the existence of a common generalized eigenfunction of the associated Koopman operators. This common eigenfunction serves to build a topology on Hom(F), where the conjugacy is obtained as limit of a sequence generated by the conjugacy operator, when this limit exists. The main conjugacy theorem is presented in a class of generalized Lipeomorphisms.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
This article is concerned with conjugacy problems arising in the homeomorphisms group, Hom(F), of unbounded subsets F of normed vector spaces E. Given two homeomorphisms f and g in Hom(F), it is shown how the existence of a conjugacy may be related to the existence of a common generalized eigenfunction of the associated Koopman operators. This common eigenfunction serves to build a topology on Hom(F), where the conjugacy is obtained as limit of a sequence generated by the conjugacy operator, when this limit exists. The main conjugacy theorem is presented in a class of generalized Lipeomorphisms.
Kondrashov, K.; Chekroun, M. D.; Robertson, A. W.; Ghil, M.
Low-order stochastic model and “past-noise forecasting” of the Madden-Julian oscillation Journal Article
In: Geophysical Research Letters, vol. 40, no. 19, pp. 5303—5310, 2013.
@article{392,
title = {Low-order stochastic model and “past-noise forecasting” of the Madden-Julian oscillation},
author = {K. Kondrashov and M. D. Chekroun and A. W. Robertson and M. Ghil},
url = {http://onlinelibrary.wiley.com/doi/10.1002/grl.50991/abstract},
year = {2013},
date = {2013-01-01},
journal = {Geophysical Research Letters},
volume = {40},
number = {19},
pages = {5303—5310},
abstract = {This paper presents a predictability study of the Madden-Julian Oscillation (MJO) that relies on combining empirical model reduction (EMR) with the “past-noise forecasting” (PNF) method. EMR is a data-driven methodology for constructing stochastic low-dimensional models that account for nonlinearity, seasonality and serial correlation in the estimated noise, while PNF constructs an ensemble of forecasts that accounts for interactions between (i) high-frequency variability (noise), estimated here by EMR, and (ii) the low-frequency mode of MJO, as captured by singular spectrum analysis (SSA). A key result is that—compared to an EMR ensemble driven by generic white noise—PNF is able to considerably improve prediction of MJO phase. When forecasts are initiated from weak MJO conditions, the useful skill is of up to 30 days. PNF also significantly improves MJO prediction skill for forecasts that start over the Indian Ocean.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
This paper presents a predictability study of the Madden-Julian Oscillation (MJO) that relies on combining empirical model reduction (EMR) with the “past-noise forecasting” (PNF) method. EMR is a data-driven methodology for constructing stochastic low-dimensional models that account for nonlinearity, seasonality and serial correlation in the estimated noise, while PNF constructs an ensemble of forecasts that accounts for interactions between (i) high-frequency variability (noise), estimated here by EMR, and (ii) the low-frequency mode of MJO, as captured by singular spectrum analysis (SSA). A key result is that—compared to an EMR ensemble driven by generic white noise—PNF is able to considerably improve prediction of MJO phase. When forecasts are initiated from weak MJO conditions, the useful skill is of up to 30 days. PNF also significantly improves MJO prediction skill for forecasts that start over the Indian Ocean.