Exploring the Lyapunov instability properties of high-dimensional atmospheric and climate models

The stability properties of intermediate-order climate models are investigated by computing their Lyapunov exponents (LEs). The two models considered are PUMA (Portable University Model of the Atmosphere), a primitive-equation simple general circulation model, and MAOOAM (Modular Arbitrary-Order Ocean-Atmosphere Model), a quasi-geostrophic coupled ocean–atmosphere model on a β-plane. We wish to investigate the effect of the different levels of filtering on the instabilities and dynamics of the atmospheric flows. Moreover, we assess the impact of the oceanic coupling, the dissipation scheme, and the resolution on the spectra of LEs. The PUMA Lyapunov spectrum is computed for two different values of the meridional temperature gradient defining the Newtonian forcing to the temperature field. The increase in the gradient gives rise to a higher baroclinicity and stronger instabilities, corresponding to a larger dimension of the unstable manifold and a larger first LE. The Kaplan–Yorke dimension of the attractor increases as well. The convergence rate of the rate function for the large deviation law of the finite-time Lyapunov exponents (FTLEs) is fast for all exponents, which can be interpreted as resulting from the absence of a clear-cut atmospheric timescale separation in such a model. The MAOOAM spectra show that the dominant atmospheric instability is correctly represented even at low resolutions. However, the dynamics of the central manifold, which is mostly associated with the ocean dynamics, is not fully resolved because of its associated long timescales, even at intermediate orders. As expected, increasing the mechanical atmosphere–ocean coupling coefficient or introducing a turbulent diffusion parametrisation reduces the Kaplan–Yorke dimension and Kolmogorov–Sinai entropy. In all considered configurations, we are not yet in the regime in which one can robustly define large deviation laws describing the statistics of the FTLEs. This paper highlights the need to investigate the natural variability of the atmosphere–ocean coupled dynamics by associating rate of growth and decay of perturbations with the physical modes described using the formalism of the covariant Lyapunov vectors and considering long integrations in order to disentangle the dynamical processes occurring at all timescales

Multi-level Dynamical Systems: Connecting the Ruelle Response Theory and the Mori-Zwanzig Approach

In this paper we consider the problem of deriving approximate autonomous dynamics for a number of variables of a dynamical system, which are weakly coupled to the remaining variables. In a previous paper we have used the Ruelle response theory on such a weakly coupled system to construct a surrogate dynamics, such that the expectation value of any observable agrees, up to second order in the coupling strength, to its expectation evaluated on the full dynamics. We show here that such surrogate dynamics agree up to second order to an expansion of the Mori-Zwanzig projected dynamics. This implies that the parametrizations of unresolved processes suited for prediction and for the representation of long term statistical properties are closely related, if one takes into account, in addition to the widely adopted stochastic forcing, the often neglected memory effects.Comment: 14 pages, 1 figur

Predicting climate change using response theory: global averages and spatial patterns

The provision of accurate methods for predicting the climate response to anthropogenic and natural forcings is a key contemporary scientific challenge. Using a simplified and efficient open-source general circulation model of the atmosphere featuring O(105105) degrees of freedom, we show how it is possible to approach such a problem using nonequilibrium statistical mechanics. Response theory allows one to practically compute the time-dependent measure supported on the pullback attractor of the climate system, whose dynamics is non-autonomous as a result of time-dependent forcings. We propose a simple yet efficient method for predicting—at any lead time and in an ensemble sense—the change in climate properties resulting from increase in the concentration of CO22 using test perturbation model runs. We assess strengths and limitations of the response theory in predicting the changes in the globally averaged values of surface temperature and of the yearly total precipitation, as well as in their spatial patterns. The quality of the predictions obtained for the surface temperature fields is rather good, while in the case of precipitation a good skill is observed only for the global average. We also show how it is possible to define accurately concepts like the inertia of the climate system or to predict when climate change is detectable given a scenario of forcing. Our analysis can be extended for dealing with more complex portfolios of forcings and can be adapted to treat, in principle, any climate observable. Our conclusion is that climate change is indeed a problem that can be effectively seen through a statistical mechanical lens, and that there is great potential for optimizing the current coordinated modelling exercises run for the preparation of the subsequent reports of the Intergovernmental Panel for Climate Change

Phaseolus vulgaris L. Extract: Alpha-amylase inhibition against metabolic syndrome in mice

To examine the effects of the alpha-amylase inhibitor isoform 1 called phaseolamin, a standardized extract from white kidney beans (Phaseolus vulgaris L) was tested against the hallmarks of metabolic syndrome. The efficacy of a per os repeated treatment with P. vulgaris extract (500 mg/kg) was compared with metformin (100 mg/kg) and atorvastatin (10 mg/kg) in a model of metabolic syndrome evoked by prolonged high fat diet (HFD; week 1 to week 19) in C57BL/6 mice. Bean extract and compounds administration started after metabolic syndrome establishment (week 11). P. vulgaris extract reduced the body weight overtime, as well as effectively lowered glycaemia, triglycerides, and cholesterol. On week 19, bean extract normalized the HFD-evoked tolerance to glucose and insulin. According to the phytochemical characterization, it inhibited the alpha-amylase activity. Animals treated with the extract were rescued from motor impairments and nociceptive threshold alterations induced by HFD. Specific organs analysis revealed that P. vulgaris extract decreased hepatic steatosis and lipid peroxidation in liver. It protected the heart from HFD oxidative alterations increasing the expression of the detoxifying enzymes catalase and glutathione reductase, and normalizing NADH dehydrogenase level. The histological analysis of aorta showed a protection about the development of fatty streaks in the muscular layers. In conclusion, a prolonged treatment with the standardized extract of P. vulgaris significantly reduced several pathological features related to a metabolic syndrome-like condition; a multifactorial approach that candidates this vegetal product as a possible therapeutic option against metabolic syndrome

Ground effects induced by the 2012 seismic sequence in Emilia: implications for seismic hazard assessment in the Po Plain

Since May 16, 2012, a seismic sequence has affected a wide portion of the Emilia Region (northern Italy), chiefly for the Modena and Ferrara Provinces. The first mainshock (Ml 5.9; focal depth, ca. 6 km) occurred on May 20, 2012, with the epicenter located a few kilometers north of Finale Emilia. A second main shock (Ml 5.8; focal depth, ca. 10 km) occurred on May 29, 2012, about 12 km west of the first earthquake, with the epicenter near Medolla. The seismic sequence has been characterized by five other Ml 655 events, and more than 2,300 aftershocks of lower magnitude, until the end July 2012. The distribution of the aftershocks identifies a WNW-ESE-trending zone ca. 40 km long that is characterized by NNE-SSW nearly pure compression, as indicated by the focal mechanisms. This report focuses on the many ground effects that were induced by this seismic sequence, as mainly cracks, liquefaction-type phenomena, and hydrological anomalies. The aim is to provide a complete representation of such effects, to: illustrate their type, size and areal distribution; identify the zones in the affected area that were most prone to the occurrence of ground effects (i.e., more susceptible to local geological instability in the case of earthquake occurrence); carry out an independent assessment of the intensities of the earthquakes through the ESI 2007 intensity scale, which is based only on coseismic effects on the natural environment

How to be causal: time, spacetime, and spectra

I explain a simple definition of causality in widespread use, and indicate how it links to the Kramers Kronig relations. The specification of causality in terms of temporal differential eqations then shows us the way to write down dynamical models so that their causal nature /in the sense used here/ should be obvious to all. To extend existing treatments of causality that work only in the frequency domain, I derive a reformulation of the long-standing Kramers Kronig relations applicable not only to just temporal causality, but also to spacetime "light-cone" causality based on signals carried by waves. I also apply this causal reasoning to Maxwell's equations, which is an instructive example since their casual properties are sometimes debated.Comment: v4 - add Appdx A, "discrete" picture (not in EJP); v5 - add Appdx B, cause classification/frames (not in EJP); v7 - unusual model case; v8 add reference

Intercomparison of the northern hemisphere winter mid-latitude atmospheric variability of the IPCC models

We compare, for the overlapping time frame 1962-2000, the estimate of the northern hemisphere (NH) mid-latitude winter atmospheric variability within the XX century simulations of 17 global climate models (GCMs) included in the IPCC-4AR with the NCEP and ECMWF reanalyses. We compute the Hayashi spectra of the 500hPa geopotential height fields and introduce an integral measure of the variability observed in the NH on different spectral sub-domains. Only two high-resolution GCMs have a good agreement with reanalyses. Large biases, in most cases larger than 20%, are found between the wave climatologies of most GCMs and the reanalyses, with a relative span of around 50%. The travelling baroclinic waves are usually overestimated, while the planetary waves are usually underestimated, in agreement with previous studies performed on global weather forecasting models. When comparing the results of various versions of similar GCMs, it is clear that in some cases the vertical resolution of the atmosphere and, somewhat unexpectedly, of the adopted ocean model seem to be critical in determining the agreement with the reanalyses. The GCMs ensemble is biased with respect to the reanalyses but is comparable to the best 5 GCMs. This study suggests serious caveats with respect to the ability of most of the presently available GCMs in representing the statistics of the global scale atmospheric dynamics of the present climate and, a fortiori, in the perspective of modelling climate change.Comment: 39 pages, 8 figures, 2 table

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

Predictive use of the Maximum Entropy Production principle for Past and Present Climates

In this paper, we show how the MEP hypothesis may be used to build simple climate models without representing explicitly the energy transport by the atmosphere. The purpose is twofold. First, we assess the performance of the MEP hypothesis by comparing a simple model with minimal input data to a complex, state-of-the-art General Circulation Model. Next, we show how to improve the realism of MEP climate models by including climate feedbacks, focusing on the case of the water-vapour feedback. We also discuss the dependence of the entropy production rate and predicted surface temperature on the resolution of the model

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