153 research outputs found
Data-adaptive harmonic spectra and multilayer Stuart-Landau models
Harmonic decompositions of multivariate time series are considered for which
we adopt an integral operator approach with periodic semigroup kernels.
Spectral decomposition theorems are derived that cover the important cases of
two-time statistics drawn from a mixing invariant measure.
The corresponding eigenvalues can be grouped per Fourier frequency, and are
actually given, at each frequency, as the singular values of a cross-spectral
matrix depending on the data. These eigenvalues obey furthermore a variational
principle that allows us to define naturally a multidimensional power spectrum.
The eigenmodes, as far as they are concerned, exhibit a data-adaptive character
manifested in their phase which allows us in turn to define a multidimensional
phase spectrum.
The resulting data-adaptive harmonic (DAH) modes allow for reducing the
data-driven modeling effort to elemental models stacked per frequency, only
coupled at different frequencies by the same noise realization. In particular,
the DAH decomposition extracts time-dependent coefficients stacked by Fourier
frequency which can be efficiently modeled---provided the decay of temporal
correlations is sufficiently well-resolved---within a class of multilayer
stochastic models (MSMs) tailored here on stochastic Stuart-Landau oscillators.
Applications to the Lorenz 96 model and to a stochastic heat equation driven
by a space-time white noise, are considered. In both cases, the DAH
decomposition allows for an extraction of spatio-temporal modes revealing key
features of the dynamics in the embedded phase space. The multilayer
Stuart-Landau models (MSLMs) are shown to successfully model the typical
patterns of the corresponding time-evolving fields, as well as their statistics
of occurrence.Comment: 26 pages, double columns; 15 figure
Statistical and dynamical properties of covariant lyapunov vectors in a coupled atmosphere-ocean model—multiscale effects, geometric degeneracy, and error dynamics
We study a simplified coupled atmosphere-ocean model using the formalism of covariant Lyapunov vectors (CLVs), which link physically-based directions of perturbations to growth/decay rates. The model is obtained via a severe truncation of quasi-geostrophic equations for the two fluids, and includes a simple yet physically meaningful representation of their dynamical/thermodynamical coupling. The model has 36 degrees of freedom, and the parameters are chosen so that a chaotic behaviour is observed. There are two positive Lyapunov exponents (LEs), sixteen negative LEs, and eighteen near-zero LEs. The presence of many near-zero LEs results from the vast time-scale separation between the characteristic time scales of the two fluids, and leads to nontrivial error growth properties in the tangent space spanned by the corresponding CLVs, which are geometrically very degenerate. Such CLVs correspond to two different classes of ocean/atmosphere coupled modes. The tangent space spanned by the CLVs corresponding to the positive and negative LEs has, instead, a non-pathological behaviour, and one can construct robust large deviations laws for the finite time LEs, thus providing a universal model for assessing predictability on long to ultra-long scales along such directions. Interestingly, the tangent space of the unstable manifold has substantial projection on both atmospheric and oceanic components. The results show the difficulties in using hyperbolicity as a conceptual framework for multiscale chaotic dynamical systems, whereas the framework of partial hyperbolicity seems better suited, possibly indicating an alternative definition for the chaotic hypothesis. They also suggest the need for an accurate analysis of error dynamics on different time scales and domains and for a careful set-up of assimilation schemes when looking at coupled atmosphere-ocean models
On quantifying the climate of the nonautonomous lorenz-63 model
The Lorenz-63 model has been frequently used to inform our understanding of the Earth's climate and provide insight for numerical weather and climate prediction. Most studies have focused on the autonomous (time invariant) model behaviour in which the model's parameters are constants. Here we investigate the properties of the model under time-varying parameters, providing a closer parallel to the challenges of climate prediction, in which climate forcing varies with time. Initial condition (IC) ensembles are used to construct frequency distributions of model variables and we interpret these distributions as the time-dependent climate of the model. Results are presented that demonstrate the impact of ICs on the transient behaviour of the model climate. The location in state space from which an IC ensemble is initiated is shown to significantly impact the time it takes for ensembles to converge. The implication for climate prediction is that the climate may, in parallel with weather forecasting, have states from which its future behaviour is more, or less, predictable in distribution. Evidence of resonant behaviour and path dependence is found in model distributions under time varying parameters, demonstrating that prediction in nonautonomous nonlinear systems can be sensitive to the details of time-dependent forcing/parameter variations. Single model realisations are shown to be unable to reliably represent the model's climate; a result which has implications for how real-world climatic timeseries from observation are interpreted. The results have significant implications for the design and interpretation of Global Climate Model experiments. Over the past 50 years, insight from research exploring the behaviour of simple nonlinear systems has been fundamental in developing approaches to weather and climate prediction. The analysis herein utilises the much studied Lorenz-63 model to understand the potential behaviour of nonlinear systems, such as the 5 climate, when subject to time-varying external forcing, such as variations in atmospheric greenhouse gases or solar output. Our primary aim is to provide insight which can guide new approaches to climate model experimental design and thereby better address the uncertainties associated with climate change prediction. We use ensembles of simulations to generate distributions which 10 we refer to as the \climate" of the time-variant Lorenz-63 model. In these ensemble experiments a model parameter is varied in a number of ways which can be seen as paralleling both idealised and realistic variations in external forcing of the real climate system. Our results demonstrate that predictability of climate distributions under time varying forcing can be highly sensitive to 15 the specification of initial states in ensemble simulations. This is a result which at a superficial level is similar to the well-known initial condition sensitivity in weather forecasting, but with different origins and different implications for ensemble design. We also demonstrate the existence of resonant behaviour and a dependence on the details of the \forcing" trajectory, thereby highlighting 20 further aspects of nonlinear system behaviour with important implications for climate prediction. Taken together, our results imply that current approaches to climate modeling may be at risk of under-sampling key uncertainties likely to be significant in predicting future climate
Effects of stochastic parametrization on extreme value statistics
Extreme geophysical events are of crucial relevance to our daily life: they threaten human lives and cause property damage. To assess the risk and reduce losses, we need to model and probabilistically predict these events. Parametrizations are computational tools used in the Earth system models, which are aimed at reproducing the impact of unresolved scales on resolved scales. The performance of parametrizations has usually been examined on typical events rather than on extreme events. In this paper, we consider a modified version of the two-level Lorenz’96 model and investigate how two parametrizations of the fast degrees of freedom perform in terms of the representation of extreme events. One parametrization is constructed following Wilks [Q. J. R. Meteorol. Soc. 131, 389–407 (2005)] and is constructed through an empirical fitting procedure; the other parametrization is constructed through the statistical mechanical approach proposed by Wouters and Lucarini [J. Stat. Mech. Theory Exp. 2012, P03003 (2012); J. Stat. Phys. 151, 850–860 (2013)]. The two strategies show different advantages and disadvantages. We discover that the agreement between parametrized models and true model is in general worse when looking at extremes rather than at the bulk of the statistics. The results suggest that stochastic parametrizations should be accurately and specifically tested against their performance on extreme events, as usual optimization procedures might neglect them.
The provision of accurate parametrizations is a task of paramount importance in many scientific areas and specifically in weather and climate modeling. Parametrizations are needed for representing accurately and efficiently the impact of the scales of motions and of the processes that cannot be explicitly represented by the numerical model. Parametrizations are usually constructed in order to optimize the overall performance of the model, thus aiming at an accurate representation of the bulk of the statistics. Nonetheless, numerical models are key to estimating, anticipating, and predicting extreme events. Here, we analyze critically in a simple yet illustrative example the performance of parametrizations in describing extreme events, and we conclude that good performance on typical conditions cannot be in any way extrapolated for rare conditions, which could, nonetheless, be of great practical relevance
Évolution de la teneur glucidique et protéique dans les tubercules de topinambour (Helianthus tuberosus L.)
Cette note concerne l'étude de la production de matière sèche, de glucides totaux et de protéine pendant les différents stades du développement des tubercules de topinambour. (Helianthus tuberosus L.). La matière sèche varie de 17 à 20% matière frafche, les glucides totaux de 8,85 à 14,5% matière frafche dont 75% de fructose (6,71 à 12,3% matière frafche) et les proMines de 9,20 à 12,85% du poids sec. La production moyenne de glucides totaux à l'hectare se situe en année normale à 6,5 Tlha (5,9-7,2); les tubercules perdent une fraction de leur glucides en hiver. Les sous-produits d'extraction des glucides contiennent des résidus sucrés et des acides aminés et constituent un aliment dont la valeur nutritive n'est pas négligeable en période hivernale.Carbohydrates and proteins evolution in the Jerusalem artichoke (HelianthUil tubero.UII L.) tubersThis note concerns a study of production of dry matter, total carbohydrate and protein at aIl stages of development of the tubers of the Jerusalem Artichoke (Helianthus tuberosus L.). Dry matter ranges from 17 to 20% fresh matter, overall carbohydrates from 8,85 to 14,5% fresh matter, of wich 75% is fructose (6,71 à 12,3% matière frafche) and protein from 9,20 to 12,85% dry matter. The average overall carbohydrate production per hectare amounts to 6,5 Tlha (5,9-7,2) in a normal year; the tubers lose a fraction oftheir carbohydrates in winter. The byproduct from extraction of carbohydrates contain residual carbohydrates and amino-acids and orovide a food of significant nutrition al value for the win ter season
Response formulae for n-point correlations in statistical mechanical systems and application to a problem of coarse graining
Predicting the response of a system to perturbations is a key challenge in mathematical and natural sciences. Under suitable conditions on the nature of the system, of the perturbation, and of the observables of interest, response theories allow to construct operators describing the smooth change of the invariant measure of the system of interest as a function of the small parameter controlling the intensity of the perturbation. In particular, response theories can be developed both for stochastic and chaotic deterministic dynamical systems, where in the latter case stricter conditions imposing some degree of structural stability are required. In this paper we extend previous findings and derive general response formulae describing how n-point correlations are affected by perturbations to the vector flow. We also show how to compute the response of the spectral properties of the system to perturbations. We then apply our results to the seemingly unrelated problem of coarse graining in multiscale systems: we find explicit formulae describing the change in the terms describing parameterisation of the neglected degrees of freedom resulting from applying perturbations to the full system. All the terms envisioned by the Mori-Zwanzig theory - the deterministic, stochastic, and non-Markovian terms - are affected at 1st order in the perturbation. The obtained results provide a more comprehesive understanding of the response of statistical mechanical systems to perturbations and contribute to the goal of constructing accurate and robust parameterisations and are of potential relevance for fields like molecular dynamics, condensed matter, and geophysical fluid dynamics. We envision possible applications of our general results to the study of the response of climate variability to anthropogenic and natural forcing and to the study of the equivalence of thermostatted statistical mechanical systems
Estimating model evidence using data assimilation
We review the field of data assimilation (DA) from a Bayesian perspective and show that, in addition to its by now common application to state estimation, DA may be used for model selection. An important special case of the latter is the discrimination between a factual model–which corresponds, to the best of the modeller's knowledge, to the situation in the actual world in which a sequence of events has occurred–and a counterfactual model, in which a particular forcing or process might be absent or just quantitatively different from the actual world. Three different ensemble‐DA methods are reviewed for this purpose: the ensemble Kalman filter (EnKF), the ensemble four‐dimensional variational smoother (En‐4D‐Var), and the iterative ensemble Kalman smoother (IEnKS). An original contextual formulation of model evidence (CME) is introduced. It is shown how to apply these three methods to compute CME, using the approximated time‐dependent probability distribution functions (pdfs) each of them provide in the process of state estimation. The theoretical formulae so derived are applied to two simplified nonlinear and chaotic models: (i) the Lorenz three‐variable convection model (L63), and (ii) the Lorenz 40‐variable midlatitude atmospheric dynamics model (L95). The numerical results of these three DA‐based methods and those of an integration based on importance sampling are compared. It is found that better CME estimates are obtained by using DA, and the IEnKS method appears to be best among the DA methods. Differences among the performance of the three DA‐based methods are discussed as a function of model properties. Finally, the methodology is implemented for parameter estimation and for event attribution
DADA: data assimilation for the detection and attribution of weather and climate-related events
A new nudging method for data assimilation, delay‐coordinate nudging, is presented. Delay‐coordinate nudging makes explicit use of present and past observations in the formulation of the forcing driving the model evolution at each time step. Numerical experiments with a low‐order chaotic system show that the new method systematically outperforms standard nudging in different model and observational scenarios, also when using an unoptimized formulation of the delay‐nudging coefficients. A connection between the optimal delay and the dominant Lyapunov exponent of the dynamics is found based on heuristic arguments and is confirmed by the numerical results, providing a guideline for the practical implementation of the algorithm. Delay‐coordinate nudging preserves the easiness of implementation, the intuitive functioning and the reduced computational cost of the standard nudging, making it a potential alternative especially in the field of seasonal‐to‐decadal predictions with large Earth system models that limit the use of more sophisticated data assimilation procedures
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On the importance of the convergence to climate attractors
Ensemble approaches are becoming widely used in climate research. In contrast to weather forecast, however, in the climatic context one is interested in long-time properties, those arising on the scale of several decades. The well-known strong internal variability of the climate system implies the existence of a related dynamical attractor with chaotic properties. Under the condition of climate change this should be a snapshot attractor, naturally arising in an ensemble-based framework. Although ensemble averages can be evaluated at any instant of time, results obtained during the process of convergence of the ensemble towards the attractor are not relevant from the point of view of climate. In simulations, therefore, attention should be paid to whether the convergence to the attractor has taken place. We point out that this convergence is of exponential character, therefore, in a finite amount of time after initialization relevant results can be obtained. The role of the time scale separation due to the presence of the deep ocean is discussed from the point of view of ensemble simulations
Dimension reduction for systems with slow relaxation
We develop reduced, stochastic models for high dimensional, dissipative
dynamical systems that relax very slowly to equilibrium and can encode long
term memory. We present a variety of empirical and first principles approaches
for model reduction, and build a mathematical framework for analyzing the
reduced models. We introduce the notions of universal and asymptotic filters to
characterize `optimal' model reductions for sloppy linear models. We illustrate
our methods by applying them to the practically important problem of modeling
evaporation in oil spills.Comment: 48 Pages, 13 figures. Paper dedicated to the memory of Leo Kadanof
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