arXiv subjects

Valerio Lucarini

Publications and source records attributed to Valerio Lucarini.

At least 55 records · Page 3Linked to original sources

arXiv2023

Heterogeneity of the Attractor of the Lorenz '96 Model: Lyapunov Analysis, Unstable Periodic Orbits, and Shadowing Properties

The predictability of weather and climate is strongly state-dependent: special and extremely relevant atmospheric states like blockings are associated with anomalous instability. Indeed, typically, the instability of a chaotic dynamical system can vary considerably across its attractor. Such an attractor is in general densely populated by unstable periodic orbits that can be used to approximate any forward trajectory through the so-called shadowing. Dynamical heterogeneity can lead to the presence of unstable periodic orbits with different number of unstable dimensions. This phenomenon - unstable dimensions variability - implies a serious breakdown of hyperbolicity and has considerable implications in terms of the structural stability of the system and of the possibility to describe accurately its behaviour through numerical models. As a step in the direction of better understanding the properties of high-dimensional chaotic systems, we provide here an extensive numerical study of the dynamical heterogeneity of the Lorenz '96 model in a parametric configuration leading to chaotic dynamics. We show that the detected variability in the number of unstable dimensions is associated with the presence of many finite-time Lyapunov exponents that fluctuate about zero also when very long averaging times are considered. The transition between regions of the attractor with different degrees of instability comes with a significant drop of the quality of the shadowing. By performing a coarse graining based on the shadowing unstable periodic orbits, we can characterize the slow fluctuations of the system between regions featuring, on the average, anomalously high and anomalously low instability. In turn, such regions are associated, respectively, with states of anomalously high and low energy, thus providing a clear link between the microscopic and thermodynamical properties of the system.

nlin.CD↗

arXiv2023

Analysis of a two-layer energy balance model: long time behaviour and greenhouse effect

We study a two-layer energy balance model, that allows for vertical exchanges between a surface layer and the atmosphere. The evolution equations of the surface temperature and the atmospheric temperature are coupled by the emission of infrared radiation by one level, that emission being captured by the other layer, and the effect of all non radiative vertical exchanges of energy. Therefore, an essential parameter is the absorptivity of the atmosphere, denoted $ε_a$. The value of $ε_a$ depends critically on greenhouse gases: increasing concentrations of $CO_2$ and $CH_4$ lead to a more opaque atmosphere with higher values of $ε_a$. First we prove that global existence of solutions of the system holds if and only if $ε_a \in (0, 2)$, and blow up in finite time occurs if $ε_a > 2$. (Note that the physical range of values for $ε_a$ is $(0, 1]$.) Next, we explain the long time dynamics for $ε_a \in (0, 2)$, and we prove that all solutions converge to some equilibrium point. Finally, motivated by the physical context, we study the dependence of the equilibrium points with respect to the involved parameters, and we prove in particular that the surface temperature increases with respect to $ε_a$. This is the key mathematical manifestation of the greenhouse effect.

math.AP↗

arXiv2023

A Punctuated Equilibrium Analysis of the Climate Evolution of Cenozoic exhibits a Hierarchy of Abrupt Transitions

The Earth's climate has experienced numerous critical transitions during its history, which have often been accompanied by massive and rapid changes in the biosphere. Such transitions are evidenced in various proxy records covering different timescales. The goal is then to identify, date, and rank past critical transitions in terms of importance, thus possibly yielding a more thorough perspective on climatic history. To illustrate such an angle, which inspired the punctuated equilibrium angle on the theory of evolution, we have analyzed 2 key high-resolution datasets: the CENOGRID marine compilation (past 66 Myr), and North Atlantic U1308 record (past 3.3 Myr). By combining recurrence analysis of the individual time series with a multivariate representation of the system based on the theory of the quasi-potential, we identify the key abrupt transitions associated with major regime changes that differentiate various clusters of climate variability. This allows interpreting the time-evolution of the system as a trajectory taking place in a dynamical landscape, whose multiscale features are associated with a hierarchy of tipping points.

physics.ao-ph↗

arXiv2023

Dimensional analysis identifies contrasting dynamics of past climate states and critical transitions

While one can unequivocally identify past climate transitions, we lack comprehensive knowledge about their underlying mechanisms and timescales. Our study employs a dimensional analysis of benthic stable isotope records to uncover, across different timescales, how the climatic fluctuation of the Cenozoic are associated with changes in the number of effective degrees of freedom. Precession timescales dominate the Hothouse and Warmhouse states, while the Icehouse climate is primarily influenced by obliquity and eccentricity timescales. Notably, the Coolhouse state lacks dominant timescales. Our analysis proves effective in objectively identifying abrupt climate shifts and extremes. This is also demonstrated using high-resolution data from the last glacial cycle, revealing abrupt climate shifts within a single climate state. These findings significantly impact our understanding of the inherent stability of each climate state and the evaluation of (paleo-)climate models' ability to replicate key features of past/future climate states and transitions.

physics.geo-ph↗

arXiv2023

Latitudinal storm track shift in a reduced two-level model of the atmosphere

The eddy-driven jet stream and storm tracks in the mid-latitude atmosphere are known to shift in latitude on various timescales, but the physical processes that cause these shifts are still unclear. In this study, we introduce a minimal dynamical system derived from the classical Phillips two-level model with the goal of elucidating the essential mechanisms responsible for the interaction between eddies and mean flow. Specifically, we aim to understand the link between the structure of the eddies and the shift of the latitudinal maximum of the zonal flow. By varying the horizontal shape of the eddies, we find three distinct dynamical regimes whose occurrence depends on the intensity of the external baroclinic forcing: a purely zonal flow, a barotropic eddy regime with net poleward momentum flux, and a baroclinic eddy regime with both net poleward momentum and temperature flux. For weak baroclinic forcing, the classical zonal flow solution with latitudinal maximum at the centre of the beta-channel is found. For strong forcing, if eddies are southwest-northeast tilted and zonally elongated, the system is in the baroclinic eddy regime, resulting in a poleward shift of the jet. The intermediate barotropic eddy regime also features a poleward shifted jet, yet with eddies structurally distinct from the baroclinic regime. Changing the parameters yields transitions between the regimes that can be either continuous or discontinuous in terms of the properties of the atmosphere. The findings of this study also provide insights into the properties of the storm track change between the jet entrance and jet exit regions of the North Atlantic.

physics.flu-dyn↗

arXiv2023

Theoretical tools for understanding the climate crisis from Hasselmann's program and beyond

Klaus Hasselmann's revolutionary intuition in climate science was to take advantage of the stochasticity associated with fast weather processes to probe the slow dynamics of the climate system. This has 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 program has been extremely influential in climate science and beyond. We first summarise the main aspects of such a program using modern concepts and tools of statistical physics and applied mathematics. We then provide an overview of some promising scientific perspectives that might better clarify the science behind the climate crisis and that stem from Hasselmann's ideas. We show how to perform rigorous model reduction by constructing parametrizations in systems that do not necessarily feature a time-scale separation between unresolved and resolved processes. We propose a general framework for explaining the relationship between climate variability and climate change, and for performing climate change projections. This leads 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.

physics.ao-ph↗

arXiv2023

Dimension reduction of noisy interacting systems

We consider a class of models describing an ensemble of identical interacting agents subject to multiplicative noise. In the thermodynamic limit, these systems exhibit continuous and discontinuous phase transitions in a, generally, nonequilibrium setting. We provide a systematic dimension reduction methodology for constructing low dimensional, reduced-order dynamics based on the cumulants of the probability distribution of the infinite system. We show that the low dimensional dynamics returns the correct diagnostic properties since it produces a quantitatively accurate representation of the stationary phase diagram of the system that we compare with exact analytical results and numerical simulations. Moreover, we prove that the reduced order dynamics yields also the prognostic, i.e., time dependent properties, as it provides the correct response of the system to external perturbations. On the one hand, this validates the use of our complexity reduction methodology since it retains information not only of the invariant measure of the system but also of the transition probabilities and time dependent correlation properties of the stochastic dynamics. On the other hand, the breakdown of linear response properties is a key signature of the occurrence of a phase transition. We show that the reduced response operators capture the correct diverging resonant behaviour by quantitatively assessing the singular nature of the susceptibility of the system and the appearance of a pole for real value of frequencies. Hence, this methodology can be interpreted as a low dimensional, reduced order approach to the investigation and detection of critical phenomena in high dimensional interacting systems in settings where order parameters are not known.

cond-mat.stat-mech↗

arXiv2022

Typicality of the 2021 Western North America Summer Heatwave

Elucidating the statistical properties of extreme meteo-climatic events and capturing the physical processes responsible for their occurrence are key steps for improving our understanding of climate variability and climate change and for better evaluating the associated hazards. It has recently become apparent that large deviation theory is very useful for investigating persistent extreme events, and specifically, for flexibly estimating long return periods and for introducing a notion of dynamical typicality. Using a methodological framework based on large deviation theory and taking advantage of long simulations by a state-of-the-art Earth System Model, we investigate the 2021 North America Heatwave. Indeed, our analysis shows that the 2021 event can be seen as an unlikely but possible manifestation of climate variability, whilst its probability of occurrence is greatly amplified by the ongoing climate change. We also clarify the properties of spatial coherence of the 2021 heatwave and elucidate the role played by the Rocky Mountains in modulating hot, dry, and persistent extreme events in the Western Pacific region of North America.

physics.ao-ph↗

arXiv2022

Chameleon attractors in a turbulent flow

Turbulent flows present rich dynamics originating from non-trivial energy fluxes across scales, non-stationary forcings and geometrical constraints. This complexity manifests in non-hyperbolic chaos, randomness, state-dependent persistence and unpredictability. All these features have prevented a full characterization of the underlying turbulent (stochastic) attractor, which will be the key object to unpin this complexity. Here we propose a novel formalism to trace the evolution of the structural characteristics of phase-space trajectories across scales, providing a full characterization of the attractor. We demonstrate that the properties of the dynamically invariant objects depend on the scale we are focusing on. In the case of laboratory experiments on fluids we observe the emergence of an intrinsic timescale, solely determined by nonlinear interactions, controlling the geometrical and topological properties of phase-space trajectories. Given the changing nature of such attractors in time and scales we term them chameleon attractors.

nlin.CD↗

arXiv2022

Emergence of universal scaling in weather extreme events

The frequency and magnitude of weather extreme events have increased significantly during the past few years in response to anthropogenic climate change. However, global statistical characteristics and underlying physical mechanisms are still not fully understood. Here, we adopt a statistical physics and probability theory based method to investigate the nature of extreme weather events, particularly the statistics of the day-to-day air temperature differences. These statistical measurements reveal that the distributions of the magnitudes of the extreme events satisfy a universal \textit{Gumbel} distribution, while the waiting time of those extreme events is governed by a universal \textit{Gamma} function. Further finite-size effects analysis indicates robust scaling behaviours. We additionally unveil that the cumulative distribution of logarithmic waiting times between the record events follows an \textit{Exponential} distribution and that the evolution of this climate system is directional where the underlying dynamics are related to a decelerating release of tension. The universal scaling laws are remarkably stable and unaffected by global warming. Counterintuitively, unlike as expected for record dynamics, we find that the number of quakes of the extreme temperature variability does not decay as one over time but with deviations relevant to large-scale climate extreme events. Our theoretical framework provides a fresh perspective on the linkage of universality, scaling, and climate systems. The findings throw light on the nature of the weather variabilities and could guide us to better forecast extreme events.

physics.ao-ph↗

arXiv2022

On Some Aspects of the Response to Stochastic and Deterministic Forcings

The perturbation theory of operator semigroups is used to derive response formulas for a variety of combinations of acting forcings and reference background dynamics. In the case of background stochastic dynamics, we decompose the response formulas using the Koopman operator generator eigenfunctions and the corresponding eigenvalues, thus providing a functional basis towards identifying relaxation timescales and modes in physically relevant systems. To leading order, linear response gives the correction to expectation values due to extra deterministic forcings acting on either stochastic or chaotic dynamical systems. When considering the impact of weak noise, the response is linear in the intensity of the (extra) noise for background stochastic dynamics, while the second order response given the leading order correction when the reference dynamics is chaotic. In this latter case we clarify that previously published diverging results can be brought to common ground when a suitable interpretation - Stratonovich vs. Ito - of the noise is given. Finally, the response of two-point correlations to perturbations is studied through the resolvent formalism via a perturbative approach. Our results allow, among other things, to estimate how the correlations of a chaotic dynamical system changes as a results of adding stochastic forcing.

nlin.CD↗

arXiv2022

Meridional energy transport extremes and the general circulation of Northern Hemisphere mid-latitudes: dominant weather regimes and preferred zonal wavenumbers

The extratropical meridional energy transport in the atmosphere is fundamentally intermittent in nature, having extremes large enough to affect the net seasonal transport. Here, we investigate how these extreme transports are associated with the dynamics of the atmosphere at multiple spatial scales, from planetary to synoptic. We use the ERA5 reanalysis data to perform a wavenumber decomposition of meridional energy transport in the Northern Hemisphere mid-latitudes during winter and summer. We then relate extreme transport events to atmospheric circulation anomalies and dominant weather regimes, identified by clustering 500hPa geopotential height fields. In general, planetary-scale waves determine the strength and meridional position of the synoptic-scale baroclinic activity with their phase and amplitude, but important differences emerge between seasons. During winter, large wavenumbers ($k=2-3$) are key drivers of the meridional energy transport extremes, and planetary and synoptic-scale transport extremes virtually never co-occur. In summer, extremes are associated with higher wavenumbers ($k=4-6$), identified as synoptic-scale motions. We link these waves and the transport extremes to recent results on exceptionally strong and persistent co-occurring summertime heat waves across the Northern Hemisphere mid-latitudes. We show that the weather regime structures associated with these heat wave events are typical for extremely large poleward energy transport events.

physics.ao-ph↗

arXiv2022

A novel concept of fractal dimension in deterministic and stochastic Lorenz-63 systems

Many natural systems show emergent phenomena at different scales, leading to scaling regimes with signatures of chaos at large scales and an apparently random behavior at small scales. These features are usually investigated quantitatively by studying the properties of the underlying attractor, the compact object asymptotically hosting the trajectories of the system with their invariant density in the phase-space. This multi-scale nature of natural systems makes it practically impossible to get a clear picture of the attracting set as it spans over a wide range of spatial scales and may even change in time due to non-stationary forcing. Here we combine an adaptive decomposition method with extreme value theory to study the properties of the instantaneous scale-dependent dimension, which has been recently introduced to characterize such temporal and spatial scale-dependent attractors in turbulence and astrophysics. To provide a quantitative analysis of the properties of this metric, we test it on the well-known low-dimensional deterministic Lorenz-63 system perturbed with additive or multiplicative noise. We demonstrate that the properties of the invariant set depend on the scale we are focusing on and that the scale-dependent dimensions can discriminate between additive and multiplicative noise, despite the fact that the two cases exhibit very similar stochastic attractors at large scales. The proposed formalism can be generally helpful to investigate the role of multi-scale fluctuations within complex systems, allowing us to deal with the problem of characterizing the role of stochastic fluctuations across a wide range of physical systems.

nlin.CD↗

arXiv2022

Non-equilibrium Ensembles for the three-dimensional Navier-Stokes Equations

At the molecular level fluid motions are, by first principles, described by time reversible laws. On the other hand, the coarse grained macroscopic evolution is suitably described by the Navier-Stokes equations, which are inherently irreversible, due to the dissipation term. Here, a reversible version of three-dimensional Navier-Stokes is studied, by introducing a fluctuating viscosity constructed in such a way that enstrophy is conserved, along the lines of the paradigm of microcanonical versus canonical treatment in equilibrium statistical mechanics. Through systematic simulations we attack two important questions: (a) What are the conditions that must be satisfied in order to have a statistical equivalence between the two non-equilibrium ensembles? (b) What is the empirical distribution of the fluctuating viscosity observed by changing the Reynolds number and the number of modes used in the discretization of the evolution equation? The latter point is important also to establish regularity conditions for the reversible equations. We find that the probability to observe negative values of the fluctuating viscosity becomes very quickly extremely small when increasing the effective Reynolds number of the flow in the fully resolved hydro dynamical regime, at difference from what was observed previously.

physics.flu-dyn↗

arXiv2022

Lévy-noise versus Gaussian-noise-induced Transitions in the Ghil-Sellers Energy Balance Model

We study the impact of applying stochastic forcing to the Ghil-Sellers energy balance climate model in the form of a fluctuating solar irradiance. Through numerical simulations, we explore the noise-induced transitions between the competing warm and snowball climate states. We consider multiplicative stochastic forcing driven by Gaussian and $α$-stable Lévy - $α\in(0,2)$ - noise laws, examine the statistics of transition times, and estimate most probable transition paths. While the Gaussian noise case has been carefully studied in a plethora of investigations on metastable systems, much less is known about the Lévy case, especially in the case of high- and infinite-dimensional systems. In the weak noise limit, the expected residence time in each metastable state scales in a fundamentally different way in the Gaussian vs. Lévy noise case with respect to the intensity of the noise. In the former case, the classical Kramers-like exponential law is recovered. In the latter case, power laws are found, with the exponent equal to $-α$, in apparent agreement with rigorous results obtained for additive noise in a related - yet different - reaction-diffusion equation as well as in simpler models. This can be better understood by treating the Lévy noise as a compound Poisson process. The transition paths are studied in a projection of the state space and remarkable differences are observed between the two different types of noise. The snowball-to-warm and the warm-to-snowball most probable transition path cross at the single unstable edge state on the basin boundary. In the case of Lévy noise, the most probable transition paths in the two directions are wholly separated, as transitions apparently take place via the closest basin boundary region to the outgoing attractor. This property can be better elucidated by considering singular perturbations to the solar irradiance.

cond-mat.stat-mech↗

arXiv2022

Decomposing the Dynamics of the Lorenz 1963 model using Unstable Periodic Orbits: Averages, Transitions, and Quasi-Invariant Sets

Unstable periodic orbits (UPOs) are a valuable tool for studying chaotic dynamical systems, as they allow one to distill their dynamical structure. We consider here the Lorenz 1963 model with the classic parameters' value. We investigate how a chaotic trajectory can be approximated using a complete set of UPOs up to symbolic dynamics' period 14. At each instant, we rank the UPOs according to their proximity to the position of the orbit in the phase space. We study this process from two different perspectives. First, we find that longer period UPOs overwhelmingly provide the best local approximation to the trajectory. Second, we construct a finite-state Markov chain by studying the scattering of the orbit between the neighbourhood of the various UPOs. Each UPO and its neighbourhood are taken as a possible state of the system. Through the analysis of the subdominant eigenvectors of the corresponding stochastic matrix we provide a different interpretation of the mixing processes occurring in the system by taking advantage of the concept of quasi-invariant sets.

nlin.CD↗

arXiv2022

Robustness of competing climatic states

The climate is a non-equilibrium system undergoing the continuous action of forcing and dissipation. Under the effect of a spatially inhomogeneous absorption of solar energy, all the climate components dynamically respond until an approximate steady state (or attractor) is reached. However, multiple steady states can co-exist for a given forcing and with the same boundary conditions. Here, we apply the Thermodynamic Diagnostic Tool (TheDiaTo) to investigate the statistical properties of five co-existing climates, ranging from a snowball to an ice-free aquaplanet, obtained in MITgcm coupled simulations. The aim is to explore the multistability of the climate model setup by highlighting differences in competing steady states and their characteristic signatures regarding the meridional transport of heat and water mass, the Lorenz energy cycle and the material entropy production. We also investigate how such attractors change when the model configuration is varied. We consider, in particular, the effect of changing the representation of the cloud albedo, and of implementing an improved closure of the energy budget. We find that, even if the dynamics remains on the same attractor, state variables are modified. The set of metrics in TheDiaTo quantifies such modifications and represents a valuable tool for model evaluation.

physics.ao-ph↗

arXiv2021

Eddy saturation in a reduced two-level model of the atmosphere

Eddy saturation describes the nonlinear mechanism in geophysical flows whereby, when average conditions are considered, direct forcing of the zonal flow increases the eddy kinetic energy, while the energy associated with the zonal flow does not increase. Here we present a minimal baroclinic model that exhibits complete eddy saturation. Starting from Phillips' classical quasi-geostrophic two-level model on the beta channel of the mid-latitudes, we derive a reduced order model comprising of six ordinary differential equations including parameterised eddies. This model features two physically realisable steady state solutions, one a purely zonal flow and one where, additionally, finite eddy motions are present. As the baroclinic forcing in the form of diabatic heating is increased, the zonal solution loses stability and the eddy solution becomes attracting. After this bifurcation, the zonal components of the solution are independent of the baroclinic forcing, and the excess of heat in the low latitudes is efficiently transported northwards by finite eddies, in the spirit of baroclinic adjustment.

physics.ao-ph↗