Fluctuations, response, and resonances in a simple atmospheric model

We study the response of a simple quasi-geostrophic barotropic model of the atmosphere to various classes of perturbations affecting its forcing and its dissipation using the formalism of the Ruelle response theory. We investigate the geometry of such perturbations by constructing the covariant Lyapunov vectors of the unperturbed system and discover in one specific case–orographic forcing–a substantial projection of the forcing onto the stable directions of the flow. This results into a resonant response shaped as a Rossby-like wave that has no resemblance to the unforced variability in the same range of spatial and temporal scales. Such a climatic surprise corresponds to a violation of the fluctuation–dissipation theorem, in agreement with the basic tenets of nonequilibrium statistical mechanics. The resonance can be attributed to a specific group of rarely visited unstable periodic orbits of the unperturbed system. Our results reinforce the idea of using basic methods of nonequilibrium statistical mechanics and high-dimensional chaotic dynamical systems to approach the problem of understanding climate dynamics

A new mathematical framework for atmospheric blocking events

We use a simple yet Earth-like hemispheric atmospheric model to propose a new framework for the mathematical properties of blocking events. Using finite-time Lyapunov exponents, we show that the occurrence of blockings is associated with conditions featuring anomalously high instability. Longer-lived blockings are very rare and have typically higher instability. In the case of Atlantic blockings, predictability is especially reduced at the onset and decay of the blocking event, while a relative increase of predictability is found in the mature phase. The opposite holds for Pacific blockings, for which predictability is lowest in the mature phase. Blockings are realised when the trajectory of the system is in the neighbourhood of a specific class of unstable periodic orbits (UPOs), natural modes of variability that cover the attractor the system. UPOs corresponding to blockings have, indeed, a higher degree of instability compared to UPOs associated with zonal flow. Our results provide a rigorous justification for the classical Markov chains-based analysis of transitions between weather regimes. The analysis of UPOs elucidates that the model features a very severe violation of hyperbolicity, due to the presence of a substantial variability in the number of unstable dimensions, which explains why atmospheric states can differ a lot in term of their predictability. The resulting lack of robustness might be a fundamental cause contributing to the difficulty in representing blockings in numerical models and in predicting how their statistics will change as a result of climate change. This corresponds to fundamental issues limiting our ability to construct very accurate numerical models of the atmosphere, in term of predictability of the both the first and of the second kind in the sense of Lorenz

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.Comment: 28 pages, 11 figures, final accepted versio

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 neighborhood of the various UPOs. Each UPO and its neighborhood 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. The attractor of a chaotic system is densely populated by an infinite number of unstable periodic orbits (UPOs), which are exact periodic solutions of the evolution equations. UPOs can be used to decompose the complex phenomenology of a chaotic flow into elementary components and have shown great potential for the understanding of macroscopic features in turbulent fluid flows. Here, we investigate how a long forward trajectory of the celebrated Lorenz 1963 model featuring the classical parameters’ value can be seen as a scattering process where the scatterers are the UPOs. This process helps elucidate how a generic ensemble of initial conditions converges to the invariant measure through diffusion and provides a new interpretation of quasi-invariant sets of the system in terms of UPOs

Heterogeneity of the attractor of the Lorenz '96 model: Lyapunov analysis, unstable periodic orbits, and shadowing properties

It is well known that the predictability of weather and climate is strongly state-dependent. Special, easily recognisable, and extremely relevant atmospheric states like blockings are associated with anomalous instability. This reflects the general property that the attractors of chaotic dynamical systems can feature considerable heterogeneity in terms of dynamical properties, and specifically, of their instability. The attractor of a chaotic dynamical systems is 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 - has considerable implications in terms of the structural stability of the system and of the possibility to model accurately its behaviour through numerical models. As a step in the direction of better understanding the properties of high-dimensional chaotic systems, here we provide an extensive numerical investigation of the variability of the dynamical properties across the attractor of the much studied Lorenz '96 model. By combining the Lyapunov analysis of the tangent space with the study of the shadowing of the chaotic trajectory performed by a very large set of unstable periodic orbits, we show that the observed variability in the number of unstable dimensions, which is a serious breakdown of hyperbolicity, is associated with the presence of a substantial number of 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 is associated with a significant drop of the quality of the shadowing. By performing a coarse graining based on the shadowing unstable periodic orbits, we are able to characterise 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

Vectors, molecular epidemiology and phylogeny of TBEV in Kazakhstan and central Asia

BACKGROUND: In the South of Kazakhstan, Almaty Oblastʼ (region) is endemic for tick-borne encephalitis, with 0.16–0.32 cases/100,000 population between 2016–2018. The purpose of this study was to determine the prevalence and circulating subtypes of tick-borne encephalitis virus (TBEV) in Almaty Oblastʼ and Kyzylorda Oblastʼ. METHODS: In 2015 we investigated 2341 ticks from 7 sampling sites for the presence of TBEV. Ticks were pooled in 501 pools and isolated RNA was tested for the presence of TBEV by RT-qPCR. For the positive samples, the E gene was amplified, sequenced and a phylogenetic analysis was carried out. RESULTS: A total of 48 pools were TBEV-positive by the RT-qPCR. TBEV-positive ticks were only detected in three districts of Almaty Oblastʼ and not in Kyzylorda Oblastʼ. The positive TBEV pools were found within Ixodes persulcatus, Haemaphysalis punctata and Dermacentor marginatus. These tick species prevailed only in Almaty Oblastʼ whereas in Kyzylorda Oblastʼ Hyalomma asiaticum and D. marginatus are endemic. The minimum infection rates (MIR) in the sampling sites were 4.4% in Talgar, 2.8% in Tekeli and 1.1% in Yenbekshikazakh, respectively. The phylogenetic analysis of the generated sequences indicates that TBEV strains found in Almaty Oblastʼ clusters in the Siberian subtype within two different clades. CONCLUSIONS: We provided new data about the TBEV MIR in ticks in Almaty Oblastʼ and showed that TBEV clusters in the Siberian Subtype in two different clusters at the nucleotide level. These results indicate that there are different influences on the circulating TBEV strains in south-eastern Kazakhstan. These influences might be caused by different routes of the virus spread in ticks which might bring different genetic TBEV lineages to Kazakhstan. The new data about the virus distribution and vectors provided here will contribute to an improvement of monitoring of tick-borne infections and timely anti-epidemic measures in Kazakhstan