scholarly journals Lyapunov analysis of multiscale dynamics: The slow manifold of the two-scale Lorenz '96 model

2018 ◽  
Author(s):  
Mallory Carlu ◽  
Francesco Ginelli ◽  
Valerio Lucarini ◽  
Antonio Politi
Keyword(s):  
Degrees Of Freedom ◽  
Climate Models ◽  
Finite Size ◽  
Slow Variable ◽  
Slow Manifold ◽  
Size Analysis ◽  
Lyapunov Analysis ◽  
Full Spectrum ◽  
Multiscale Dynamics ◽  
Lorenz 96

Abstract. We investigate the geometrical structure of instabilities in the two-scales Lorenz '96 model through the prism of Lyapunov analysis. Our detailed study of the full spectrum of covariant Lyapunov vectors reveals the presence of a slow manifold in tangent space, composed by a set of vectors with a significant projection on the slow degrees of freedom; they correspond to the smallest (in absolute sense) Lyapunov exponents and thereby to the longer time scales. We show that the dimension of this manifold is extensive in the number of both slow and fast degrees of freedom, and discuss its relationship with the results of a finite-size analysis of instabilities, supporting the conjecture that the slow-variable behavior is effectively determined by a non-trivial subset of degrees of freedom. More precisely, we show that the slow manifold corresponds to the Lyapunov spectrum region where fast and slow instability rates overlap, mixing their evolution into a set of vectors which simultaneously carry information on both scales. We suggest these results may pave the way for future applications to ensemble forecasting and data assimilation in weather and climate models.

scholarly journals Lyapunov analysis of multiscale dynamics: the slow bundle of the two-scale Lorenz 96 model

2019 ◽  
Vol 26 (2) ◽  
pp. 73-89 ◽  
Author(s):  
Mallory Carlu ◽  
Francesco Ginelli ◽  
Valerio Lucarini ◽  
Antonio Politi
Keyword(s):  
Degrees Of Freedom ◽  
Climate Models ◽  
Finite Size ◽  
Slow Variable ◽  
Size Analysis ◽  
Lyapunov Analysis ◽  
Full Spectrum ◽  
Multiscale Dynamics ◽  
Spectrum Region ◽  
Lorenz 96

Abstract. We investigate the geometrical structure of instabilities in the two-scale Lorenz 96 model through the prism of Lyapunov analysis. Our detailed study of the full spectrum of covariant Lyapunov vectors reveals the presence of a slow bundle in tangent space, composed by a set of vectors with a significant projection onto the slow degrees of freedom; they correspond to the smallest (in absolute value) Lyapunov exponents and thereby to the longer timescales. We show that the dimension of the slow bundle is extensive in the number of both slow and fast degrees of freedom and discuss its relationship with the results of a finite-size analysis of instabilities, supporting the conjecture that the slow-variable behavior is effectively determined by a nontrivial subset of degrees of freedom. More precisely, we show that the slow bundle corresponds to the Lyapunov spectrum region where fast and slow instability rates overlap, “mixing” their evolution into a set of vectors which simultaneously carry information on both scales. We suggest that these results may pave the way for future applications to ensemble forecasting and data assimilations in weather and climate models.

Finite-size analysis and the influence of the cross-section shape of the granular column collapses

2021 ◽  
Author(s):  
Teng Man ◽  
Herbert Huppert ◽  
Ling Li ◽  
Sergio Galindo-Torres
Keyword(s):  
Size Effect ◽  
Cross Section ◽  
Finite Size ◽  
Size Analysis ◽  
Cross Section Shape ◽  
Section Shape ◽  
The Cross ◽  
Shape Influence ◽  
Deposition Morphology ◽  
Granular Column

<p>The collapse of granular columns, which sheds light on the kinematics, dynamics, and deposition morphology of mass-driven flows, is crucial for understanding complex flows in both natural and engineering systems, such as debris flows and landslides. However, our research shows that a strong size effect and cross-section shape influence exist in this test. Thus, it is essential to better understand these effects. In this study, we explore the influence of both relative column sizes and cross-section shapes on the run-out behavior of collapsed granular columns and analyze their influence on the deposition morphology with the discrete element method (DEM) with Voronoi-based spheropolyhedron particles. We link the size effect that occurs in granular column collapse problems to the finite-size scaling functions and investigate the characteristic correlation length associated with the granular column collapses. The collapsing behavior of granular columns with different cross-section shapes is also studied, and we find that particles tend to accumulate in the direction normal to the edge of the cross-section instead of the vertex of it. The differences in the run-out behavior in different directions when the cross-section is no longer a circle can also be explained by the finite-size analysis we have performed in this study. We believe that such a study is crucial for us to better understand how granular material flows, how it deposits, and how to consider the size effect in the rheology of granular flows.</p>

scholarly journals Variational data assimilation with superparameterization

2015 ◽  
Vol 2 (2) ◽  
pp. 513-536 ◽  
Author(s):  
I. Grooms ◽  
Y. Lee
Keyword(s):  
Data Assimilation ◽  
Large Scale ◽  
Climate Models ◽  
Low Cost ◽  
Ocean Model ◽  
Variational Data Assimilation ◽  
Scale Model ◽  
Small Scale ◽  
New Applications ◽  
Lorenz 96

Abstract. Superparameterization (SP) is a multiscale computational approach wherein a large scale atmosphere or ocean model is coupled to an array of simulations of small scale dynamics on periodic domains embedded into the computational grid of the large scale model. SP has been successfully developed in global atmosphere and climate models, and is a promising approach for new applications. The authors develop a 3D-Var variational data assimilation framework for use with SP; the relatively low cost and simplicity of 3D-Var in comparison with ensemble approaches makes it a natural fit for relatively expensive multiscale SP models. To demonstrate the assimilation framework in a simple model, the authors develop a new system of ordinary differential equations similar to the two-scale Lorenz-'96 model. The system has one set of variables denoted {Yi}, with large and small scale parts, and the SP approximation to the system is straightforward. With the new assimilation framework the SP model approximates the large scale dynamics of the true system accurately.

scholarly journals Lewis fry Richardson Medal Lecture – How many modes are neededto predict climate bifurcations?: Lessons from an experiment

2021 ◽  
Author(s):  
Bérengère Dubrulle ◽  
François Daviaud ◽  
Davide Faranda ◽  
Louis Marié ◽  
Brice Saint-Michel
Keyword(s):  
Degrees Of Freedom ◽  
Climate Models ◽  
Environmental Policies ◽  
Climate Projections ◽  
Finite Resolution ◽  
The Pacific ◽  
The Pacific Ocean ◽  
Lewis Fry Richardson ◽  
Do So

Abstract. According to everyone’s experience, predicting the weather reliably over more than 8 days seems an impossible taskfor our best weather agencies. At the same time, politicians and citizens are asking scientists for climate projections severaldecades into the future to guide economic and environmental policies, especially regarding the maximum admissible emissions of CO2. To what extent is this request scientifically admissible? In this lecture we will investigate this question, focusing on the topic of predictions of transitions between metastable statesof the atmospheric or oceanic circulations. Two relevant exemples are the switching between zonal and blocked atmosphericcirculation at midlatitudes and the alternance of El Niño and La Niña phases in the Pacific ocean. The main issue is whetherpresent climate models, that necessarily have a finite resolution and a smaller number of degrees of freedom than the actualterrestrial system, are able to reproduce such spontaneous or forced transitions. To do so, we will draw an analogy betweenclimate observations and results obtained in our group on a laboratory-scale, turbulent, von Kármán flow, in which spontaneoustransitions between different states of the circulation take place. We will detail the analogy, and investigate the nature of thetransitions, the number of degrees of freedom that characterizes the latter and discuss the effect of reducing the number ofdegrees of freedom in such systems. We will also discuss the role of fluctuations and their origin, and stress the importance ofdescribing very small scales to capture fluctuations of correct intensity and scale.

Characterizing Fractal and Hierarchical Morphologies Beyond the Fractal Dimension

1994 ◽  
Vol 367 ◽  
Author(s):  
Raphael Blumenfeld ◽  
Robin C. Ball
Keyword(s):  
Asymptotic Behavior ◽  
Fractal Dimension ◽  
Finite Size ◽  
Size Analysis ◽  
Corrections To Scaling ◽  
Diffusion Limited Aggregation ◽  
Diffusion Limited ◽  
Markovian Processes ◽  
Branching Patterns ◽  
Admissible Solutions

AbstractWe present a novel correlation scheme to characterize the morphology of fractal and hierarchical patterns beyond traditional scaling. The method consists of analysing correlations between more than two-points in logarithmic coordinates. This technique has several advantages: i) It can be used to quantify the currently vague concept of morphology; ii) It allows to distinguish between different signatures of structures with similar fractal dimension but different morphologies already for relatively small systems; iii) The method is sensitive to oscillations in logarithmic coordinates, which are both admissible solutions for renormalization equations and which appear in many branching patterns (e.g., noise-reduced diffusion-limited-aggregation and bronchial structures); iv) The methods yields information on corrections to scaling from the asymptotic behavior, which is very useful in finite size analysis. Markovian processes are calculated exactly and several structures are analyzed by this method to demonstrate its advantages.

Finite-size analysis of continuous-variable quantum key distribution with entanglement in the middle

2019 ◽  
Vol 28 (1) ◽  
pp. 010305 ◽  
Author(s):  
Ying Guo ◽  
Yu Su ◽  
Jian Zhou ◽  
Ling Zhang ◽  
Duan Huang
Keyword(s):  
Quantum Key Distribution ◽  
Finite Size ◽  
Key Distribution ◽  
Continuous Variable ◽  
Size Analysis

scholarly journals Mechanics and thermodynamics of a new minimal model of the atmosphere

2020 ◽  
Vol 135 (10) ◽  
Author(s):  
Gabriele Vissio ◽  
Valerio Lucarini
Keyword(s):  
Numerical Models ◽  
Preliminary Evidence ◽  
Chaotic Motions ◽  
One Dimensional ◽  
Fundamental Properties ◽  
Multiscale Dynamics ◽  
Energy Cycle ◽  
The One ◽  
Lorenz 96 ◽  
Model Features

AbstractThe understanding of the fundamental properties of the climate system has long benefitted from the use of simple numerical models able to parsimoniously represent the essential ingredients of its processes. Here, we introduce a new model for the atmosphere that is constructed by supplementing the now-classic Lorenz ’96 one-dimensional lattice model with temperature-like variables. The model features an energy cycle that allows for energy to be converted between the kinetic form and the potential form and for introducing a notion of efficiency. The model’s evolution is controlled by two contributions—a quasi-symplectic and a gradient one, which resemble (yet not conforming to) a metriplectic structure. After investigating the linear stability of the symmetric fixed point, we perform a systematic parametric investigation that allows us to define regions in the parameters space where at steady-state stationary, quasi-periodic, and chaotic motions are realised, and study how the terms responsible for defining the energy budget of the system depend on the external forcing injecting energy in the kinetic and in the potential energy reservoirs. Finally, we find preliminary evidence that the model features extensive chaos. We also introduce a more complex version of the model that is able to accommodate for multiscale dynamics and that features an energy cycle that more closely mimics the one of the Earth’s atmosphere.

Finite Element Model Based Full Spectrum Response Analysis of a Cracked Rotor With Internal and External Damping

2019 ◽  
Author(s):  
Dipendra Kumar Roy ◽  
Rajiv Tiwari
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Element Model ◽  
Rotor System ◽  
Response Analysis ◽  
Stable Operation ◽  
Internal Damping ◽  
Full Spectrum ◽  
Fault Parameters ◽  
Spectrum Response

Abstract The ratio of internal and external damping is one of the important fault parameters and it leads to instability of a rotor shaft at higher spin speeds. The crack in a rotor is one of the sources of its instability due to the crack internal damping. A rotor with crack internal damping that originates from the rubbing action between the two crack faces. For a sustained stable operation of the rotor, it is imperative to analyze rotor parameters such as the internal and external damping and other parameters, like the additive crack stiffness and disc eccentricity. Therefore, the present work considers a full spectrum response analysis of a transverse cracked shaft based on the finite element method. The rotary and translations of inertia are considered including of gyroscopic effect in the rotor system. The transverse crack is modeled based on the switching crack assumption. The crack in the rotor gives forcing with multiple harmonics with the forward and backward. The equation of motion has been developed for the rotor system having four degrees of freedom at each node and using MATLAB™ Simulink the responses are generated for a numerical example.

scholarly journals A Modal Rendition of ENSO Diversity

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Rajib Chattopadhyay ◽  
Shivsai Ajit Dixit ◽  
B. N. Goswami
Keyword(s):  
El Niño ◽  
Climate Models ◽  
El Nino ◽  
Southern Oscillation ◽  
Slow Manifold ◽  
Disaster Mitigation ◽  
Coupled Models ◽  
Decadal Modulation ◽  
Enso Diversity ◽  
Modal Description

Abstract The El Nino and Southern Oscillation (ENSO) ‘diversity’ has been considered as a major factor limiting its predictability, a critical need for disaster mitigation associated with the trademark climatic swings of the ENSO. Improving climate models for ENSO forecasts relies on deeper understanding of the ENSO diversity but currently at a nascent stage. Here, we show that the ENSO diversity thought previously as ‘complex,’ arises largely as varied contributions from three leading modes of the ENSO to a given event. The ENSO ‘slow manifold’ can be fully described by three leading predictable modes, a quasi-quadrennial mode (QQD), a quasi-biennial (QB) mode and a decadal modulation of the quasi-biennial (DQB). The modal description of ENSO provides a framework for understanding the predictability of and global teleconnections with the ENSO. We further demonstrate it to be a useful framework for understanding biases of climate models in simulating and predicting the ENSO. Therefore, skillful prediction of all shades of ENSO depends critically on the coupled models’ ability to simulate the three modes with fidelity, providing basis for optimism for future of ENSO forecasts.

scholarly journals The Observed Hemispheric Symmetry in Reflected Shortwave Irradiance

2013 ◽  
Vol 26 (2) ◽  
pp. 468-477 ◽  
Author(s):  
Aiko Voigt ◽  
Bjorn Stevens ◽  
Jürgen Bader ◽  
Thorsten Mauritsen
Keyword(s):  
Northern Hemisphere ◽  
Atmospheric Aerosols ◽  
Degrees Of Freedom ◽  
Climate Models ◽  
Radiant Energy ◽  
Energy System ◽  
Hemispheric Differences ◽  
Satellite Measurements ◽  
Planetary Albedo ◽  
Accurate Observation

Abstract While the concentration of landmasses and atmospheric aerosols on the Northern Hemisphere suggests that the Northern Hemisphere is brighter than the Southern Hemisphere, satellite measurements of top-of-atmosphere irradiances found that both hemispheres reflect nearly the same amount of shortwave irradiance. Here, the authors document that the most precise and accurate observation, the energy balanced and filled dataset of the Clouds and the Earth’s Radiant Energy System covering the period 2000–10, measures an absolute hemispheric difference in reflected shortwave irradiance of 0.1 W m−2. In contrast, the longwave irradiance of the two hemispheres differs by more than 1 W m−2, indicating that the observed climate system exhibits hemispheric symmetry in reflected shortwave irradiance but not in longwave irradiance. The authors devise a variety of methods to estimate the spatial degrees of freedom of the time-mean reflected shortwave irradiance. These are used to show that the hemispheric symmetry in reflected shortwave irradiance is a nontrivial property of the Earth system in the sense that most partitionings of Earth into two random halves do not exhibit hemispheric symmetry in reflected shortwave irradiance. Climate models generally do not reproduce the observed hemispheric symmetry, which the authors interpret as further evidence that the symmetry is nontrivial. While the authors cannot rule out that the observed hemispheric symmetry in reflected shortwave irradiance is accidental, their results motivate a search for mechanisms that minimize hemispheric differences in reflected shortwave irradiance and planetary albedo.

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