Research Interests

From Chekroun et al. (2011), Physica D, 240 (21): 1685-1700:

 

Vimeo movie: https://vimeo.com/240039610

 

Recent Publications

Chekroun, Mickaël D., Jeroen S.W. Lamb, Christian J. Pangerl, and Martin Rasmussen. Submitted. “A Girsanov approach to slow parameterizing manifolds in the presence of noise”. arXiv's link Abstract
We consider a three-dimensional slow-fast system with quadratic nonlinearity and additive noise. The associated deterministic system of this stochastic differential equation (SDE) exhibits a periodic orbit and a slow manifold. The deterministic slow manifold can be viewed as an approximate parameterization of the fast variable of the SDE in terms of the slow variables. In other words the fast variable of the slow-fast system is approximately "slaved" to the slow variables via the slow manifold. We exploit this fact to obtain a two dimensional reduced model for the original stochastic system, which results in the Hopf-normal form with additive noise. Both, the original as well as the reduced system admit ergodic invariant measures describing their respective long-time behaviour. We will show that for a suitable metric on a subset of the space of all probability measures on phase space, the discrepancy between the marginals along the radial component of both invariant measures can be upper bounded by a constant and a quantity describing the quality of the parameterization. An important technical tool we use to arrive at this result is Girsanov's theorem, which allows us to modify the SDEs in question in a way that preserves transition probabilities. This approach is then also applied to reduced systems obtained through stochastic parameterizing manifolds, which can be viewed as generalized notions of deterministic slow manifolds.
 
Tantet, Alexis, Mickaël D. Chekroun, Henk A. Dijkstra, and J. David Neelin. Submitted. “Mixing Spectrum in Reduced Phase Spaces of Stochastic Differential Equations. Part II: Stochastic Hopf Bifurcation”. ArXiv's Version Abstract

The spectrum of the Markov semigroup of a diffusion process, referred to as the mixing spectrum, provides a detailed characterization of the dynamics of statistics such as the correlation function and the power spectrum. Stochastic analysis techniques for the study of the mixing spectrum and a rigorous reduction method have been presented in the first part (Chekroun et al. 2017) of this contribution. This framework is now applied to the study of a stochastic Hopf bifurcation, to characterize the statistical properties of nonlinear oscillators perturbed by noise, depending on their stability. In light of the H\"ormander 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 mixing spectrum has a spectral gap, even at the bifurcation point, and that correlations decay exponentially fast. Small-noise expansions of the 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 to understand how the interaction of the noise with the deterministic dynamics affect the decay of correlations. Numerical results complement the study of the mixing spectrum at the bifurcation point, revealing interesting 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 systems. This approach is not limited to low-dimensional systems and the reduction method presented in part I will be applied to stochastic models relevant to climate dynamics in Part III.

Cao, Yining, Mickaël D. Chekroun, Roger Temam, and Aimin Huang. 2019. “Mathematical analysis of the Jin-Neelin model of El Nino-Southern-Oscillation.” Chinese Annals of Mathematics, Series B 40 (1): 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.

 

Pierini, Stefano, Mickaël D. Chekroun, and Michael Ghil. 2018. “The onset of chaos in nonautonomous dissipative dynamical systems: a low-order ocean-model case study.” Nonlinear Processes in Geophysics 25: 671-692. Publisher's Version 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.

Kondrashov, Dmitri, Mickaël D. Chekroun, and Michael Ghil. 2018. “Data-adaptive harmonic decomposition and prediction of Arctic sea ice extent.” Dynamics and Statistics of the Climate System 3 (1): 1. Publisher's Version 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.

Kondrashov, Dmitri, Mickaël D. Chekroun, and Pavel Berloff. 2018. “Multiscale Stuart-Landau Emulators: Application to Wind-Driven Ocean Gyres.” Fluids 3 (1): 21. Publisher's Version 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.

Decadal DAH mode

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