Inferring the instability of a dynamical system from the skill of data assimilation exercises

Data assimilation (DA) aims at optimally merging observational data and model outputs to create a coherent statistical and dynamical picture of the system under investigation. Indeed, DA aims at minimizing the effect of observational and model error and at distilling the correct ingredients of its dynamics. DA is of critical importance for the analysis of systems featuring sensitive dependence on the initial conditions, as chaos wins over any finitely accurate knowledge of the state of the system, even in absence of model error. Clearly, the skill of DA is guided by the properties of dynamical system under investigation, as merging optimally observational data and model outputs is harder when strong instabilities are present. In this paper we reverse the usual angle on the problem and show that it is indeed possible to use the skill of DA to infer some basic properties of the tangent space of the system, which may be hard to compute in very high-dimensional systems. Here, we focus our attention on the first Lyapunov exponent and the Kolmogorov–Sinai entropy and perform numerical experiments on the Vissio–Lucarini 2020 model, a recently proposed generalization of the Lorenz 1996 model that is able to describe in a simple yet meaningful way the interplay between dynamical and thermodynamical variables.

Chaos in the Rulkov Neuron Model Based on Marotto’s Theorem

The existence of chaos in the Rulkov neuron model is proved based on Marotto’s theorem. Firstly, the stability conditions of the model are briefly renewed through analyzing the eigenvalues of the model, which are very important preconditions for the existence of a snap-back repeller. Secondly, the Rulkov neuron model is decomposed to a one-dimensional fast subsystem and a one-dimensional slow subsystem by the fast–slow dynamics technique, in which the fast subsystem has sensitive dependence on the initial conditions and its snap-back repeller and chaos can be verified by numerical methods, such as waveforms, Lyapunov exponents, and bifurcation diagrams. Thirdly, for the two-dimensional Rulkov neuron model, it is proved that there exists a snap-back repeller under two iterations by illustrating the existence of an intersection of three surfaces, which pave a new way to identify the existence of a snap-back repeller.

Chaos arising from the hydrological behaviour of a floodplain river during the last century

The hydrological regime is the main factor governing the functioning of floodplain rivers. A full comprehension of its dynamic leads to a better understanding of the system’s behaviour and of the proper methods that must be used. We analysed the daily water level of the Paraná River during the last century at three gauge stations using linear and non-linear tools to characterise the hydrological dynamic and to analyse to what extent chaotic behaviour prevails. The three water level time series were characterised as non-linear and non-stationary by power spectrum, autocorrelation function, and surrogate test analyses. A strange attractor was developed when the phase space was reconstructed, having a low dimensional chaos supported by correlation dimension, positive maximum Lyapunov exponents, and recurrence quantification analysis. In line with this, the system resulted unpredictable with a threshold by sample entropy, and with an intermediate hydrological complexity, while Hurst exponent characterised the system as persistent and with sensitive dependence on initial conditions. In a general overview, all the evidence obtained indicates that the Paraná River’s behaviour is at the edge of chaos. A latitudinal gradient of decreasing chaoticity was observed as the floodplain extent increased, whereas complexity was highest at the intermediate river station due to the inflow of tributaries with different hydrology. This paper attempts to offer some additional insights for understanding the hydrological behaviour of floodplain rivers and the proper methods to understand their complexity.

Coupled models of magma oceans and their primordial atmospheres: volatile speciation, cooling history, and impact of condensables

<p>The earliest atmospheres of rocky planets originate from extensive volatile release during magma ocean epochs that occur during assembly of the planet. These establish the initial distribution of the major volatile elements between different chemical reservoirs that subsequently evolve via geological cycles. Current theoretical techniques are limited in exploring the anticipated range of compositional and thermal scenarios of early planetary evolution. However, these are of prime importance to aid astronomical inferences on the environmental context and geological history of extrasolar planets. In order to advance the potential synergies between exoplanet observations and inferrences on the earliest history and climate state of the solar system terrestial planets, I will present a novel numerical framework that links an evolutionary, vertically-resolved model of the planetary silicate mantle with a radiative-convective model of the atmosphere. Numerical simulations using this framework illustrate the sensitive dependence of mantle crystallization and atmosphere build-up on volatile speciation and predict variations in atmospheric spectra with planet composition that may be detectable with future observations of exoplanets. Magma ocean thermal sequences fall into three general classes of primary atmospheric volatile with increasing cooling timescale: CO, N<sub>2</sub>, and O<sub>2</sub> with minimal effect on heat flux, H<sub>2</sub>O, CO<sub>2</sub>, and CH<sub>4</sub> with intermediate influence, and H<sub>2</sub> with several orders of magnitude increase in solidification time and atmosphere vertical stratification. In addition to these time-resolved results, I will present a novel formulation and application of a multi-species moist-adiabat for condensable-rich magma ocean and archean earth analog atmospheres, and outline how the cooling of such atmospheres can lead to exotic climate states that provide testable predictions for terrestrial exoplanets.</p>

Inferring the instability of a dynamical system from the skill of data assimilation exercises

Data assimilation (DA) aims at optimally merging observational data and model outputs to create a coherent statistical and dynamical picture of the system under investigation. Indeed, DA aims at minimizing the effect of observational and model error, and at distilling the correct ingredients of its dynamics. DA is of critical importance for the analysis of systems featuring sensitive dependence on the initial conditions, as chaos wins over any finitely accurate knowledge of the state of the system, even in absence of model error. Clearly, the skill of DA is guided by the properties of dynamical system under investigation, as merging optimally observational data and model outputs is harder when strong instabilities are present. In this paper we reverse the usual angle on the problem and show that it is indeed possible to use the skill of DA to infer some basic properties of the tangent space of the system, which may be hard to compute in very high-dimensional systems. Here, we focus our attention on the first Lyapunov exponent and the Kolmogorov-Sinai entropy, and perform numerical experiments on the Vissio-Lucarini 2020 model, a recently proposed generalisation of the Lorenz 1996 model that is able to describe in a simple yet meaningful way the interplay between dynamical and thermodynamical variables.

Parameter estimation for a chaotic dynamical system with partial observations

Parameter estimation in chaotic dynamical systems is an important and practical issue. Nevertheless, the high-dimensionality and the sensitive dependence on initial conditions typically makes the problem difficult to solve. In this paper, we propose an innovative parameter estimation approach, utilizing numerical differentiation for observation data preprocessing. Given plenty of noisy observations on a portion of dependent variables, numerical differentiation allows them and their derivatives to be accurately approximated. Substituting those approximations into the original system can effectively simplify the parameter estimation problem. As an example, we consider parameter estimation in the well-known Lorenz model given partial noisy observations. According to the Lorenz equations, the estimated parameters can be simply given by least squares regression using the approximated functions provided by data preprocessing. Numerical examples show the effectiveness and accuracy of our method. We also prove the uniqueness and stability of the solution.

Calculation of Electron Impact Single Ionization TDCS of Tungsten Atoms at 200, 500 and 1000 eV

We report triple differential cross-sections (TDCSs) for the electron impact single ionization of tungsten atoms for the ionization taking place from the outer sub shells of tungsten atoms, viz. W (6s), W (5d), W (5p) and W (4f). The study of the electron-induced processes such as ionization, excitation, autoionization from tungsten and its charged states is strongly required to diagnose and model the fusion plasma in magnetic devices such as Tokamaks. Particularly, the cross-section data are important to understand the electron spectroscopy involved in the fusion plasma. In the present study, we report TDCS results for the ionization of W atoms at 200, 500 and 1000 eV projectile energy at different values of scattered electron angles. It was observed that the trends of TDCSs for W (5d) are significantly different from the trends of TDCSs for W (6s), W (5p) and W (4f). It was further observed that the TDCS for W atoms has sensitive dependence on value of momentum transfer and projectile energy.

Migration of Low-Mass Planets

Planet migration is the variation over time of a planet’s semimajor axis, leading to either a contraction or an expansion of the orbit. It results from the exchange of energy and angular momentum between the planet and the disk in which it is embedded during its formation and can cause the semimajor axis to change by as much as two orders of magnitude over the disk’s lifetime. The migration of forming protoplanets is an unavoidable process, and it is thought to be a key ingredient for understanding the variety of extrasolar planetary systems. Although migration occurs for protoplanets of all masses, its properties for low-mass planets (those having up to a few Earth masses) differ significantly from those for high-mass planets. The torque that is exerted by the disk on the planet is composed of different contributions. While migration was first thought to be invariably inward, physical processes that are able to halt or even reverse migration were later uncovered, leading to the realization that the migration path of a forming planet has a very sensitive dependence on the underlying disk parameters. There are other processes that go beyond the case of a single planet experiencing smooth migration under the disk’s tide. This is the case of planetary migration in low-viscosity disks, a fashionable research avenue because protoplanetary disks are thought to have very low viscosity, if any, over most of their planet-forming regions. Such a process is generally significantly chaotic and has to be tackled through high-resolution numerical simulations. The migration of several low-mass planets is also is a very fashionable topic, owing to the discovery by the Kepler mission of many multiple extrasolar planetary systems. The orbital properties of these systems suggest that at least some of them have experienced substantial migration. Although there have been many studies to account for the orbital properties of these systems, there is as yet no clear picture of the different processes that shaped them. Finally, some recently unveiled processes could be important for the migration of low-mass planets. One process is aero-resonant migration, in which a swarm of planetesimals subjected to aerodynamic drag push a planet inward when they reach a mean-motion resonance with the planet, while another process is based on so-called thermal torques, which arise when thermal diffusion in the disk is taken into account, or when the planet, heated by accretion, releases heat into the ambient gas.

Sensitive dependence of trajectories on tracer seeding positions – coherent structures in German Bight backward drift simulations

Backward drift simulations can aid the interpretation of in situ monitoring data. In some cases, however, trajectories are very sensitive to even small changes in the tracer release position. A corresponding spread of backward simulations implies attraction in the forward passage of time and, hence, uncertainty about the probed water body's origin. This study examines surface drift simulations in the German Bight (North Sea). Lines across which drift behaviour changes non-smoothly are obtained as ridges in the fields of the finite-time Lyapunov exponent (FTLE), a parameter used in dynamical systems theory to identify Lagrangian coherent structures (LCSs). Results closely resemble those obtained considering two-particle relative dispersion. It is argued that simulated FTLE fields might be used in support of the interpretation of monitoring data, indicating when simulations of backward trajectories are unreliable because of their high sensitivity to tracer seeding positions.