Articles | Volume 32, issue 2
https://doi.org/10.5194/npg-32-139-2025
© Author(s) 2025. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/npg-32-139-2025
© Author(s) 2025. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Research article
| 
26 May 2025
Research article |
| 26 May 2025
| 26 May 2025
Finite-size local dimension as a tool for extracting geometrical properties of attractors of dynamical systems
Martin Bonte
CORRESPONDING AUTHOR
Royal Meteorological Institute of Belgium, Brussels, Belgium
Stéphane Vannitsem
Royal Meteorological Institute of Belgium, Brussels, Belgium
Related authors
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Stéphane Vannitsem, X. San Liang, and Carlos A. Pires
Earth Syst. Dynam., 16, 703–719, https://doi.org/10.5194/esd-16-703-2025, https://doi.org/10.5194/esd-16-703-2025, 2025
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Large-scale modes of variability are present in the climate system. These modes are known to have influences on each other but are usually viewed as linear influences. The nonlinear connections among a set of key climate indices are explored here using tools from information theory, which allow us to characterize the causality between indices. It was found that quadratic nonlinear dependencies between climate indices are present at low frequencies, reflecting the complex nature of their dynamics.
Elke Debrie, Jonathan Demaeyer, and Stéphane Vannitsem
Earth Syst. Sci. Data Discuss., https://doi.org/10.5194/essd-2025-149, https://doi.org/10.5194/essd-2025-149, 2025
Preprint under review for ESSD
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In this project, we developed a gridded hourly precipitation dataset for Belgium, covering over 70 years (1940–2016). The data has a spatial resolution of one kilometer, which means it provides highly localized precipitation information. To estimate precipitation for a specific day in the past, we searched for days in the recent radar data period with similar weather patterns, known as the analog method. The median of the produced dataset is available for public use and can be found on Zenodo.
Anupama K. Xavier, Jonathan Demaeyer, and Stéphane Vannitsem
Earth Syst. Dynam., 15, 893–912, https://doi.org/10.5194/esd-15-893-2024, https://doi.org/10.5194/esd-15-893-2024, 2024
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This research focuses on understanding different atmospheric patterns like blocking, zonal, and transition regimes and analyzing their predictability. We used an idealized land–atmosphere coupled model to simulate Earth's atmosphere. Then we identified these blocking, zonal, and transition regimes using Gaussian mixture clustering and studied their predictability using Lyapunov exponents.
David Docquier, Giorgia Di Capua, Reik V. Donner, Carlos A. L. Pires, Amélie Simon, and Stéphane Vannitsem
Nonlin. Processes Geophys., 31, 115–136, https://doi.org/10.5194/npg-31-115-2024, https://doi.org/10.5194/npg-31-115-2024, 2024
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Identifying causes of specific processes is crucial in order to better understand our climate system. Traditionally, correlation analyses have been used to identify cause–effect relationships in climate studies. However, correlation does not imply causation, which justifies the need to use causal methods. We compare two independent causal methods and show that these are superior to classical correlation analyses. We also find some interesting differences between the two methods.
Michel Journée, Edouard Goudenhoofdt, Stéphane Vannitsem, and Laurent Delobbe
Hydrol. Earth Syst. Sci., 27, 3169–3189, https://doi.org/10.5194/hess-27-3169-2023, https://doi.org/10.5194/hess-27-3169-2023, 2023
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The exceptional flood of July 2021 in central Europe impacted Belgium severely. This study aims to characterize rainfall amounts in Belgium from 13 to 16 July 2021 based on observational data (i.e., rain gauge data and a radar-based rainfall product). The spatial and temporal distributions of rainfall during the event aredescribed. In order to document such a record-breaking event as much as possible, the rainfall data are shared with the scientific community on Zenodo for further studies.
Jonathan Demaeyer, Jonas Bhend, Sebastian Lerch, Cristina Primo, Bert Van Schaeybroeck, Aitor Atencia, Zied Ben Bouallègue, Jieyu Chen, Markus Dabernig, Gavin Evans, Jana Faganeli Pucer, Ben Hooper, Nina Horat, David Jobst, Janko Merše, Peter Mlakar, Annette Möller, Olivier Mestre, Maxime Taillardat, and Stéphane Vannitsem
Earth Syst. Sci. Data, 15, 2635–2653, https://doi.org/10.5194/essd-15-2635-2023, https://doi.org/10.5194/essd-15-2635-2023, 2023
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A benchmark dataset is proposed to compare different statistical postprocessing methods used in forecasting centers to properly calibrate ensemble weather forecasts. This dataset is based on ensemble forecasts covering a portion of central Europe and includes the corresponding observations. Examples on how to download and use the data are provided, a set of evaluation methods is proposed, and a first benchmark of several methods for the correction of 2 m temperature forecasts is performed.
David Docquier, Stéphane Vannitsem, and Alessio Bellucci
Earth Syst. Dynam., 14, 577–591, https://doi.org/10.5194/esd-14-577-2023, https://doi.org/10.5194/esd-14-577-2023, 2023
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The climate system is strongly regulated by interactions between the ocean and atmosphere. However, many uncertainties remain in the understanding of these interactions. Our analysis uses a relatively novel approach to quantify causal links between the ocean surface and lower atmosphere based on satellite observations. We find that both the ocean and atmosphere influence each other but with varying intensity depending on the region, demonstrating the power of causal methods.
Stéphane Vannitsem
Nonlin. Processes Geophys., 30, 1–12, https://doi.org/10.5194/npg-30-1-2023, https://doi.org/10.5194/npg-30-1-2023, 2023
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The impact of climate change on weather pattern dynamics over the North Atlantic is explored through the lens of information theory. These tools allow the predictability of the succession of weather patterns and the irreversible nature of the dynamics to be clarified. It is shown that the predictability is increasing in the observations, while the opposite trend is found in model projections. The irreversibility displays an overall increase in time in both the observations and the model runs.
David Docquier, Stéphane Vannitsem, Alessio Bellucci, and Claude Frankignoul
EGUsphere, https://doi.org/10.5194/egusphere-2022-1340, https://doi.org/10.5194/egusphere-2022-1340, 2022
Preprint withdrawn
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Understanding whether variations in ocean heat content are driven by air-sea heat fluxes or by ocean dynamics is of crucial importance to enhance climate projections. We use a relatively novel causal method to quantify interactions between ocean heat budget terms based on climate models. We find that low-resolution models overestimate the influence of ocean dynamics in the upper ocean, and that changes in ocean heat content are dominated by air-sea fluxes at high resolution.
Nicolas Ghilain, Stéphane Vannitsem, Quentin Dalaiden, Hugues Goosse, Lesley De Cruz, and Wenguang Wei
Earth Syst. Sci. Data, 14, 1901–1916, https://doi.org/10.5194/essd-14-1901-2022, https://doi.org/10.5194/essd-14-1901-2022, 2022
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Modeling the climate at high resolution is crucial to represent the snowfall accumulation over the complex orography of the Antarctic coast. While ice cores provide a view constrained spatially but over centuries, climate models can give insight into its spatial distribution, either at high resolution over a short period or vice versa. We downscaled snowfall accumulation from climate model historical simulations (1850–present day) over Dronning Maud Land at 5.5 km using a statistical method.
Tommaso Alberti, Reik V. Donner, and Stéphane Vannitsem
Earth Syst. Dynam., 12, 837–855, https://doi.org/10.5194/esd-12-837-2021, https://doi.org/10.5194/esd-12-837-2021, 2021
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We provide a novel approach to diagnose the strength of the ocean–atmosphere coupling by using both a reduced order model and reanalysis data. Our findings suggest the ocean–atmosphere dynamics presents a rich variety of features, moving from a chaotic to a coherent coupled dynamics, mainly attributed to the atmosphere and only marginally to the ocean. Our observations suggest further investigations in characterizing the occurrence and spatial dependency of the ocean–atmosphere coupling.
Stephan Hemri, Sebastian Lerch, Maxime Taillardat, Stéphane Vannitsem, and Daniel S. Wilks
Nonlin. Processes Geophys., 27, 519–521, https://doi.org/10.5194/npg-27-519-2020, https://doi.org/10.5194/npg-27-519-2020, 2020
Jonathan Demaeyer and Stéphane Vannitsem
Nonlin. Processes Geophys., 27, 307–327, https://doi.org/10.5194/npg-27-307-2020, https://doi.org/10.5194/npg-27-307-2020, 2020
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Postprocessing schemes used to correct weather forecasts are no longer efficient when the model generating the forecasts changes. An approach based on response theory to take the change into account without having to recompute the parameters based on past forecasts is presented. It is tested on an analytical model and a simple model of atmospheric variability. We show that this approach is effective and discuss its potential application for an operational environment.
Michiel Van Ginderachter, Daan Degrauwe, Stéphane Vannitsem, and Piet Termonia
Nonlin. Processes Geophys., 27, 187–207, https://doi.org/10.5194/npg-27-187-2020, https://doi.org/10.5194/npg-27-187-2020, 2020
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A generic methodology is developed to estimate the model error and simulate the model uncertainty related to a specific physical process. The method estimates the model error by comparing two different representations of the physical process in otherwise identical models. The found model error can then be used to perturb the model and simulate the model uncertainty. When applying this methodology to deep convection an improvement in the probabilistic skill of the ensemble forecast is found.
Emmanuel Roulin and Stéphane Vannitsem
Nonlin. Processes Geophys. Discuss., https://doi.org/10.5194/npg-2019-45, https://doi.org/10.5194/npg-2019-45, 2019
Preprint withdrawn
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We need seasonal predictions of temperature and precipitation to prepare hydrological outlooks. Since the skill is limited, statistical correction and combination of outputs from multiple models are necessary. We use the forecasts of past situations from the EUROSIP multi-model system for 6 case studies in Western Europe and the Mediterranean Region. We identify skill for spring temperature in most areas and winter precipitation in Sweden and Greece. Sample size for training appears crucial.
Jonathan Demaeyer and Stéphane Vannitsem
Nonlin. Processes Geophys., 25, 605–631, https://doi.org/10.5194/npg-25-605-2018, https://doi.org/10.5194/npg-25-605-2018, 2018
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We investigate the modeling of the effects of the unresolved scales on the large scales of the coupled ocean–atmosphere model MAOOAM. Two different physically based stochastic methods are considered and compared, in various configurations of the model. Both methods show remarkable performances and are able to model fundamental changes in the model dynamics. Ways to improve the parameterizations' implementation are also proposed.
Stéphane Vannitsem and Pierre Ekelmans
Earth Syst. Dynam., 9, 1063–1083, https://doi.org/10.5194/esd-9-1063-2018, https://doi.org/10.5194/esd-9-1063-2018, 2018
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The El Niño–Southern Oscillation phenomenon is a slow dynamics present in the coupled ocean–atmosphere tropical Pacific system which has important teleconnections with the northern extratropics. These teleconnections are usually believed to be the source of an enhanced predictability in the northern extratropics at seasonal to decadal timescales. This question is challenged by investigating the causality between these regions using an advanced technique known as convergent cross mapping.
Lesley De Cruz, Sebastian Schubert, Jonathan Demaeyer, Valerio Lucarini, and Stéphane Vannitsem
Nonlin. Processes Geophys., 25, 387–412, https://doi.org/10.5194/npg-25-387-2018, https://doi.org/10.5194/npg-25-387-2018, 2018
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The predictability of weather models is limited largely by the initial state error growth or decay rates. We have computed these rates for PUMA, a global model for the atmosphere, and MAOOAM, a more simplified, coupled model which includes the ocean. MAOOAM has processes at distinct timescales, whereas PUMA surprisingly does not. We propose a new programme to compute the natural directions along the flow that correspond to the growth or decay rates, to learn which components play a role.
Lesley De Cruz, Jonathan Demaeyer, and Stéphane Vannitsem
Geosci. Model Dev., 9, 2793–2808, https://doi.org/10.5194/gmd-9-2793-2016, https://doi.org/10.5194/gmd-9-2793-2016, 2016
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Large-scale weather patterns such as the North Atlantic Oscillation, which dictates the harshness of European winters, vary over the course of years. By recreating it in a simple ocean-atmosphere model, we hope to understand what drives this slow, hard-to-predict variability. MAOOAM is such a model, in which the resolution and included physical processes can easily be modified. The modular system allowed us to show the robustness of the slow variability against changes in model resolution.
S. Vannitsem and L. De Cruz
Geosci. Model Dev., 7, 649–662, https://doi.org/10.5194/gmd-7-649-2014, https://doi.org/10.5194/gmd-7-649-2014, 2014
Related subject area
Subject: Bifurcation, dynamical systems, chaos, phase transition, nonlinear waves, pattern formation | Topic: Climate, atmosphere, ocean, hydrology, cryosphere, biosphere | Techniques: Theory
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Baroclinic and barotropic instabilities in planetary atmospheres: energetics, equilibration and adjustment
Xueqiang Gou, Chao Xin, Liwen Xu, Ping Yuan, Yijun Zhang, and Mingli Chen
Nonlin. Processes Geophys., 32, 107–115, https://doi.org/10.5194/npg-32-107-2025, https://doi.org/10.5194/npg-32-107-2025, 2025
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We investigate the stability characteristics of lightning discharge channels through their differential resistance properties. Our analysis reveals that lightning channels exhibit bistable behavior, transitioning between low- and high-conductivity states depending on channel length and electric-field conditions. This work provides new insights into lightning channel dynamics and could contribute to improved lightning protection strategies.
Aksel Samuelsberg and Per Kristen Jakobsen
Nonlin. Processes Geophys., 32, 23–33, https://doi.org/10.5194/npg-32-23-2025, https://doi.org/10.5194/npg-32-23-2025, 2025
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We explored a simplified climate model based on Earth's energy budget. One advantage of such models is that they are easier to study mathematically. Using a mathematical technique known as boundary integral methods, we present a new way to solve these climate models. This method is particularly useful for modeling climates very different from Earth's current state, such as those on other planets or during past ice ages.
Alexander G. Voronovich
Nonlin. Processes Geophys., 31, 559–569, https://doi.org/10.5194/npg-31-559-2024, https://doi.org/10.5194/npg-31-559-2024, 2024
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A derivation of discrete dynamical equations for the dry atmosphere without dissipative processes based on the least action principle is presented. For a given set of generally irregularly spaced grid points and a given mode of interpolation, through the minimization of action, the algorithm ensures maximal closeness of the evolution of the discrete system to the motion of the continuous atmosphere. The spatial resolution can be adjusted while executing calculations.
Linghan Meng, Haibin Song, Yongxian Guan, Shun Yang, Kun Zhang, and Mengli Liu
Nonlin. Processes Geophys., 31, 477–495, https://doi.org/10.5194/npg-31-477-2024, https://doi.org/10.5194/npg-31-477-2024, 2024
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With seismic data, we observed high-frequency internal waves (HIWs) with amplitudes of around 10 m. A shoaling thermocline and gentle slope suggest that HIWs result from fission. Remote sensing data support this. Strong shear caused Ri below 0.25 over 20–30 km, indicating instability. HIWs enhance mixing, averaging 10-4 m2s-1, revealing a new energy cascade from shoaling waves to turbulence, and enhancing our understanding of energy dissipation and mixing in the northern South China Sea.
Vera Melinda Galfi, Tommaso Alberti, Lesley De Cruz, Christian L. E. Franzke, and Valerio Lembo
Nonlin. Processes Geophys., 31, 185–193, https://doi.org/10.5194/npg-31-185-2024, https://doi.org/10.5194/npg-31-185-2024, 2024
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In the online seminar series "Perspectives on climate sciences: from historical developments to future frontiers" (2020–2021), well-known and established scientists from several fields – including mathematics, physics, climate science and ecology – presented their perspectives on the evolution of climate science and on relevant scientific concepts. In this paper, we first give an overview of the content of the seminar series, and then we introduce the written contributions to this special issue.
Gianmarco Del Sarto, Jochen Bröcker, Franco Flandoli, and Tobias Kuna
Nonlin. Processes Geophys., 31, 137–150, https://doi.org/10.5194/npg-31-137-2024, https://doi.org/10.5194/npg-31-137-2024, 2024
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We consider a one-dimensional model for the Earth's temperature. We give sufficient conditions to admit three asymptotic solutions. We connect the value function (minimum value of an objective function depending on the greenhouse gas (GHG) concentration) to the global mean temperature. Then, we show that the global mean temperature is the derivative of the value function and that it is non-decreasing with respect to GHG concentration.
Amandine Kaiser, Nikki Vercauteren, and Sebastian Krumscheid
Nonlin. Processes Geophys., 31, 45–60, https://doi.org/10.5194/npg-31-45-2024, https://doi.org/10.5194/npg-31-45-2024, 2024
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Current numerical weather prediction models encounter challenges in accurately representing regimes in the stably stratified atmospheric boundary layer (SBL) and the transitions between them. Stochastic modeling approaches are a promising framework to analyze when transient small-scale phenomena can trigger regime transitions. Therefore, we conducted a sensitivity analysis of the SBL to transient phenomena by augmenting a surface energy balance model with meaningful randomizations.
Lilian Vanderveken, Marina Martínez Montero, and Michel Crucifix
Nonlin. Processes Geophys., 30, 585–599, https://doi.org/10.5194/npg-30-585-2023, https://doi.org/10.5194/npg-30-585-2023, 2023
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In semi-arid regions, hydric stress affects plant growth. In these conditions, vegetation patterns develop and effectively allow for vegetation to persist under low water input. The formation of patterns and the transition between patterns can be studied with small models taking the form of dynamical systems. Our study produces a full map of stable and unstable solutions in a canonical vegetation model and shows how they determine the transitions between different patterns.
Ulrike Feudel
Nonlin. Processes Geophys., 30, 481–502, https://doi.org/10.5194/npg-30-481-2023, https://doi.org/10.5194/npg-30-481-2023, 2023
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Many systems in nature are characterized by the coexistence of different stable states for given environmental parameters and external forcing. Examples can be found in different fields of science, ranging from ecosystems to climate dynamics. Perturbations can lead to critical transitions (tipping) from one stable state to another. The study of these transitions requires the development of new methodological approaches that allow for modeling, analyzing and predicting them.
Michael Ghil and Denisse Sciamarella
Nonlin. Processes Geophys., 30, 399–434, https://doi.org/10.5194/npg-30-399-2023, https://doi.org/10.5194/npg-30-399-2023, 2023
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The problem of climate change is that of a chaotic system subject to time-dependent forcing, such as anthropogenic greenhouse gases and natural volcanism. To solve this problem, we describe the mathematics of dynamical systems with explicit time dependence and those of studying their behavior through topological methods. Here, we show how they are being applied to climate change and its predictability.
Giovanni Jona-Lasinio
Nonlin. Processes Geophys., 30, 253–262, https://doi.org/10.5194/npg-30-253-2023, https://doi.org/10.5194/npg-30-253-2023, 2023
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Non-equilibrium is dominant in geophysical and climate phenomena. Most of the processes that characterize energy flow occur far from equilibrium. These range from very large systems, such as weather patterns or ocean currents that remain far from equilibrium, owing to an influx of energy, to biological structures. In the last decades, progress in non-equilibrium physics has come from the study of very rare fluctuations, and this paper provides an introduction to these theoretical developments.
Kolja L. Kypke, William F. Langford, Gregory M. Lewis, and Allan R. Willms
Nonlin. Processes Geophys., 29, 219–239, https://doi.org/10.5194/npg-29-219-2022, https://doi.org/10.5194/npg-29-219-2022, 2022
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Climate change is causing rapid temperature increases in the polar regions. A fundamental question is whether these temperature increases are reversible. If we control carbon dioxide emissions, will the temperatures revert or will we have passed a tipping point beyond which return to the present state is impossible? Our mathematical model of the Arctic climate indicates that under present emissions the Arctic climate will change irreversibly to a warm climate before the end of the century.
Karl R. Helfrich and Lev Ostrovsky
Nonlin. Processes Geophys., 29, 207–218, https://doi.org/10.5194/npg-29-207-2022, https://doi.org/10.5194/npg-29-207-2022, 2022
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Internal solitons are an important class of nonlinear waves commonly observed in coastal oceans. Their propagation is affected by the Earth's rotation and the variation in the water depth. We consider an interplay of these factors using the corresponding extension of the Gardner equation. This model allows a limiting soliton amplitude and the corresponding increase in wavelength, making the effects of rotation and topography on a shoaling wave especially significant.
Michael Ghil
Nonlin. Processes Geophys., 27, 429–451, https://doi.org/10.5194/npg-27-429-2020, https://doi.org/10.5194/npg-27-429-2020, 2020
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The scientific questions posed by the climate sciences are central to socioeconomic concerns today. This paper revisits several crucial questions, starting with
What can we predict beyond 1 week, for how long, and by what methods?, and ending with
Can we achieve enlightened climate control of our planet by the end of the century?We review the progress in dealing with the nonlinearity and stochasticity of the Earth system and emphasize major strides in coupled climate–economy modeling.
Kolja Leon Kypke, William Finlay Langford, and Allan Richard Willms
Nonlin. Processes Geophys., 27, 391–409, https://doi.org/10.5194/npg-27-391-2020, https://doi.org/10.5194/npg-27-391-2020, 2020
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The climate of Earth is governed by nonlinear processes of geophysics. This paper presents energy balance models (EBMs) embracing these nonlinear processes which lead to positive feedback, amplifying the effects of anthropogenic forcing and leading to bifurcations. We define bifurcation as a change in the topological equivalence class of the system. We initiate a bifurcation analysis of EBMs of Anthropocene climate, which shows that a catastrophic climate change may occur in the next century.
Peter Read, Daniel Kennedy, Neil Lewis, Hélène Scolan, Fachreddin Tabataba-Vakili, Yixiong Wang, Susie Wright, and Roland Young
Nonlin. Processes Geophys., 27, 147–173, https://doi.org/10.5194/npg-27-147-2020, https://doi.org/10.5194/npg-27-147-2020, 2020
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Baroclinic and barotropic instabilities are well known as the processes responsible for the production of the most important energy-containing eddies in the atmospheres and oceans of Earth and other planets. Linear and nonlinear instability theories provide insights into when such instabilities may occur, grow to a large amplitude and saturate, with examples from the laboratory, simplified numerical models and planetary atmospheres. We conclude with a number of open issues for future research.
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Short summary
In recent years, there have been more and more floods due to intense precipitation, such as the July 2021 event in Belgium. Predicting precipitation is a difficult task, even just for the next few hours. This study focuses on a tool that assesses whether a given situation is stable or not (i.e., whether it is likely to stay as it is or could evolve in an unpredictable manner).
In recent years, there have been more and more floods due to intense precipitation, such as the...
