Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part II: Stochastic Hopf Bifurcation

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. 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. 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 H\"ormander theorem, it is first shown that the geometry of the unperturbed limit cycle, in particular its isochrons, is essential to understand the effect of noise and the phenomenon of phase diffusion. In addition, it is shown that the 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, 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 approach. This approach is not limited to low-dimensional systems and the reduction method presented in part I is applied to a stochastic model relevant to climate dynamics in part III

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

Growth, acidification and proteolysis performance of two co-cultures (Lactobacillus plantarum-Bifidobacterium longum and Streptococcus thermophilus-Bifidobacterium longum)

In this study, the fermentation and proteolysis of two co-cultures were investigated. Two fermented cow skim-milk lactoferrin 1 (LF1) and lactoferrin 2 (LF2) were prepared. LF1 was inoculated with Bifidobacterium longum (Bf I) and Streptococcus thermophillus (St I) while LF2 was inoculated with Lactobacillus plantarum (Lb O) and B. longum (Bf I). Incubation was at 42°C for 8 h. The enumeration revealed bacterial growth in all fermented milk. Maximum growth of (Lb O) and (Bf I) was observed when mixed together after 2 h of fermentation in comparison with Bf I and St I with values of 109 and 3.108 cfu/ml, respectively. The kinetics of acidification (pH and lactic acid production) gave significant values (p < 0.01) for LF2 when compared to LF1 and sterile milk (LS). The proteolytic activity (functions α-NH2 released in M/mg) and total proteins (in μg/mg) gave significant values (p < 0.05) for LF2 when compared to LF1. Two mixed cultures (Lb O- Bf I) and (St I-Bf I) showed proteolysis of ß-lactoglobulin (ß-Lg) and -lactalbumin (α-la)

Time-domain numerical simulations of multiple scattering to extract elastic effective wavenumbers

Elastic wave propagation is studied in a heterogeneous 2-D medium consisting of an elastic matrix containing randomly distributed circular elastic inclusions. The aim of this study is to determine the effective wavenumbers when the incident wavelength is similar to the radius of the inclusions. A purely numerical methodology is presented, with which the limitations usually associated with low scatterer concentrations can be avoided. The elastodynamic equations are integrated by a fourth-order time-domain numerical scheme. An immersed interface method is used to accurately discretize the interfaces on a Cartesian grid. The effective field is extracted from the simulated data, and signal-processing tools are used to obtain the complex effective wavenumbers. The numerical reference solution thus-obtained can be used to check the validity of multiple scattering analytical models. The method is applied to the case of concrete. A parametric study is performed on longitudinal and transverse incident plane waves at various scatterers concentrations. The phase velocities and attenuations determined numerically are compared with predictions obtained with multiple scattering models, such as the Independent Scattering Approximation model, the Waterman-Truell model, and the more recent Conoir-Norris model.Comment: Waves in Random and Complex Media (2012) XX

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

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

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

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

Investigating possible ethnicity and sex bias in clinical examiners: an analysis of data from the MRCP(UK) PACES and nPACES examinations

Bias of clinical examiners against some types of candidate, based on characteristics such as sex or ethnicity, would represent a threat to the validity of an examination, since sex or ethnicity are 'construct-irrelevant' characteristics. In this paper we report a novel method for assessing sex and ethnic bias in over 2000 examiners who had taken part in the PACES and nPACES (new PACES) examinations of the MRCP(UK)