Articles | Volume 22, issue 3
https://doi.org/10.5194/npg-22-275-2015
© Author(s) 2015. This work is distributed under
the Creative Commons Attribution 3.0 License.
the Creative Commons Attribution 3.0 License.
https://doi.org/10.5194/npg-22-275-2015
© Author(s) 2015. This work is distributed under
the Creative Commons Attribution 3.0 License.
the Creative Commons Attribution 3.0 License.
Research article
| 
07 May 2015
Research article |
| 07 May 2015
Oscillations in a simple climate–vegetation model
J. Rombouts
CORRESPONDING AUTHOR
Centre for Complexity Science, University of Warwick, Coventry, UK
Geosciences Department and Environmental Research & Teaching Institute, Ecole Normale Supérieure, Paris, France
Atmospheric & Oceanic Sciences Department and Institute of Geophysics & Planetary Physics, University of California, Los Angeles, CA, USA
Related authors
No articles found.
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.
Keno Riechers, Takahito Mitsui, Niklas Boers, and Michael Ghil
Clim. Past, 18, 863–893, https://doi.org/10.5194/cp-18-863-2022, https://doi.org/10.5194/cp-18-863-2022, 2022
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Building upon Milancovic's theory of orbital forcing, this paper reviews the interplay between intrinsic variability and external forcing in the emergence of glacial interglacial cycles. It provides the reader with historical background information and with basic theoretical concepts used in recent paleoclimate research. Moreover, it presents new results which confirm the reduced stability of glacial-cycle dynamics after the mid-Pleistocene transition.
Denis-Didier Rousseau, Witold Bagniewski, and Michael Ghil
Clim. Past, 18, 249–271, https://doi.org/10.5194/cp-18-249-2022, https://doi.org/10.5194/cp-18-249-2022, 2022
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The study of abrupt climate changes is a relatively new field of research that addresses paleoclimate variations that occur in intervals of tens to hundreds of years. Such timescales are much shorter than the tens to hundreds of thousands of years that the astronomical theory of climate addresses. We revisit several high-resolution proxy records of the past 3.2 Myr and show that the abrupt climate changes are nevertheless affected by the orbitally induced insolation changes.
Eviatar Bach and Michael Ghil
Nonlin. Processes Geophys. Discuss., https://doi.org/10.5194/npg-2021-35, https://doi.org/10.5194/npg-2021-35, 2021
Preprint withdrawn
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Data assimilation (DA) is the process of combining model forecasts with observations in order to provide an optimal estimate of the system state. When models are imperfect, the uncertainty in the forecasts may be underestimated, requiring inflation of the corresponding error covariance. Here, we present a simple method for estimating the magnitude and structure of the model error covariance matrix. We demonstrate the efficacy of this method with idealized experiments.
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.
Denis-Didier Rousseau, Pierre Antoine, Niklas Boers, France Lagroix, Michael Ghil, Johanna Lomax, Markus Fuchs, Maxime Debret, Christine Hatté, Olivier Moine, Caroline Gauthier, Diana Jordanova, and Neli Jordanova
Clim. Past, 16, 713–727, https://doi.org/10.5194/cp-16-713-2020, https://doi.org/10.5194/cp-16-713-2020, 2020
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New investigations of European loess records from MIS 6 reveal the occurrence of paleosols and horizon showing slight pedogenesis similar to those from the last climatic cycle. These units are correlated with interstadials described in various marine, continental, and ice Northern Hemisphere records. Therefore, these MIS 6 interstadials can confidently be interpreted as DO-like events of the penultimate climate cycle.
Stefano Pierini, Mickaël D. Chekroun, and Michael Ghil
Nonlin. Processes Geophys., 25, 671–692, https://doi.org/10.5194/npg-25-671-2018, https://doi.org/10.5194/npg-25-671-2018, 2018
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A four-dimensional nonlinear spectral ocean model is used to study the transition to chaos induced by periodic forcing in systems that are nonchaotic in the autonomous limit. The analysis makes use of ensemble simulations and of the system's pullback attractors. A new diagnostic method characterizes the transition to chaos: this is found to occur abruptly at a critical value and begins with the intermittent emergence of periodic oscillations with distinct phases.
Niklas Boers, Mickael D. Chekroun, Honghu Liu, Dmitri Kondrashov, Denis-Didier Rousseau, Anders Svensson, Matthias Bigler, and Michael Ghil
Earth Syst. Dynam., 8, 1171–1190, https://doi.org/10.5194/esd-8-1171-2017, https://doi.org/10.5194/esd-8-1171-2017, 2017
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We use a Bayesian approach for inferring inverse, stochastic–dynamic models from northern Greenland (NGRIP) oxygen and dust records of subdecadal resolution for the interval 59 to 22 ka b2k. Our model reproduces the statistical and dynamical characteristics of the records, including the Dansgaard–Oeschger variability, with no need for external forcing. The crucial ingredients are cubic drift terms, nonlinear coupling terms between the oxygen and dust time series, and non-Markovian contributions.
Niklas Boers, Bedartha Goswami, and Michael Ghil
Clim. Past, 13, 1169–1180, https://doi.org/10.5194/cp-13-1169-2017, https://doi.org/10.5194/cp-13-1169-2017, 2017
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We introduce a Bayesian framework to represent layer-counted proxy records as probability distributions on error-free time axes, accounting for both proxy and dating errors. Our method is applied to NGRIP δ18O data, revealing that the cumulative dating errors lead to substantial uncertainties for the older parts of the record. Applying our method to the widely used radiocarbon comparison curve derived from varved sediments of Lake Suigetsu provides the complete uncertainties of this curve.
Keroboto B. Z. Ogutu, Fabio D'Andrea, Michael Ghil, and Charles Nyandwi
Earth Syst. Dynam. Discuss., https://doi.org/10.5194/esd-2016-64, https://doi.org/10.5194/esd-2016-64, 2017
Preprint retracted
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The CoCEB model is used to evaluate hypotheses on the long-term effect of investment in emission abatement, and on the comparative efficacy of different approaches to abatement. While many studies in the literature treat abatement costs as an unproductive loss of income, we show that mitigation costs do slow down economic growth over the next few decades, but only up to the mid-21st century or even earlier; growth reduction is compensated later on by having avoided climate negative impacts.
D.-D. Rousseau, M. Ghil, G. Kukla, A. Sima, P. Antoine, M. Fuchs, C. Hatté, F. Lagroix, M. Debret, and O. Moine
Clim. Past, 9, 2213–2230, https://doi.org/10.5194/cp-9-2213-2013, https://doi.org/10.5194/cp-9-2213-2013, 2013
Related subject area
Subject: Bifurcation, dynamical systems, chaos, phase transition, nonlinear waves, pattern formation | Topic: Climate, atmosphere, ocean, hydrology, cryosphere, biosphere
The role of time-varying external factors in the intensification of tropical cyclones
A robust numerical method for the generation and simulation of periodic finite-amplitude internal waves in natural waters
Transformation of internal solitary waves at the edge of ice cover
Review article: Interdisciplinary perspectives on climate sciences – highlighting past and current scientific achievements
Variational techniques for a one-dimensional energy balance model
A new approach to understanding fluid mixing in process-study models of stratified fluids
Sensitivity of the polar boundary layer to transient phenomena
Aggregation of slightly buoyant microplastics in 3D vortex flows
High-frequency Internal Waves, High-mode Nonlinear Waves and K-H Billows on the South China Sea's Shelf Revealed by Marine Seismic Observation
Existence and influence of mixed states in a model of vegetation patterns
Rate-induced tipping in ecosystems and climate: the role of unstable states, basin boundaries and transient dynamics
Review article: Dynamical systems, algebraic topology and the climate sciences
An approach for projecting the timing of abrupt winter Arctic sea ice loss
An adjoint-free algorithm for conditional nonlinear optimal perturbations (CNOPs) via sampling
Review article: Large fluctuations in non-equilibrium physics
On the interaction of stochastic forcing and regime dynamics
Applying dynamical systems techniques to real ocean drifters
Observations of shoaling internal wave transformation over a gentle slope in the South China Sea
Climate bifurcations in a Schwarzschild equation model of the Arctic atmosphere
Effects of rotation and topography on internal solitary waves governed by the rotating Gardner equation
Estimate of energy loss from internal solitary waves breaking on slopes
Regional study of mode-2 internal solitary waves at the Pacific coast of Central America using marine seismic survey data
The effect of strong shear on internal solitary-like waves
Enhanced diapycnal mixing with polarity-reversing internal solitary waves revealed by seismic reflection data
Enhanced internal tidal mixing in the Philippine Sea mesoscale environment
Detecting flow features in scarce trajectory data using networks derived from symbolic itineraries: an application to surface drifters in the North Atlantic
Review article: Hilbert problems for the climate sciences in the 21st century – 20 years later
Anthropocene climate bifurcation
Effects of upwelling duration and phytoplankton growth regime on dissolved-oxygen levels in an idealized Iberian Peninsula upwelling system
Baroclinic and barotropic instabilities in planetary atmospheres: energetics, equilibration and adjustment
Numerical bifurcation methods applied to climate models: analysis beyond simulation
Lyapunov analysis of multiscale dynamics: the slow bundle of the two-scale Lorenz 96 model
Competition between chaotic advection and diffusion: stirring and mixing in a 3-D eddy model
Climatic responses to systematic time variations of parameters: a dynamical approach
Evaluating a stochastic parametrization for a fast–slow system using the Wasserstein distance
Wave propagation in the Lorenz-96 model
Dynamical properties and extremes of Northern Hemisphere climate fields over the past 60 years
On the CCN (de)activation nonlinearities
Detecting changes in forced climate attractors with Wasserstein distance
Insights into the three-dimensional Lagrangian geometry of the Antarctic polar vortex
Subvisible cirrus clouds – a dynamical system approach
Influence of finite-time Lyapunov exponents on winter precipitation over the Iberian Peninsula
Dynamics of the Hadley circulation in an axisymmetric model undergoing stratification periodic forcing
Detecting and tracking eddies in oceanic flow fields: a Lagrangian descriptor based on the modulus of vorticity
A novel method for analyzing the process of abrupt climate change
Equilibrium temperature distribution and Hadley circulation in an axisymmetric model
Samuel Watson and Courtney Quinn
Nonlin. Processes Geophys., 31, 381–394, https://doi.org/10.5194/npg-31-381-2024, https://doi.org/10.5194/npg-31-381-2024, 2024
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The intensification of tropical cyclones (TCs) is explored through a conceptual model derived from geophysical principals. Focus is put on the behaviour of the model with parameters which change in time. The rates of change cause the model to either tip to an alternative stable state or recover the original state. This represents intensification, dissipation, or eyewall replacement cycles (ERCs). A case study which emulates the rapid intensification events of Hurricane Irma (2017) is explored.
Pierre Lloret, Peter J. Diamessis, Marek Stastna, and Greg N. Thomsen
EGUsphere, https://doi.org/10.5194/egusphere-2024-1121, https://doi.org/10.5194/egusphere-2024-1121, 2024
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This study presents a new approach to simulate large ocean density waves that travel long distances without breaking down. This new approach ensures that these waves are depicted more accurately and realistically in our models. This is particularly useful for understanding wave behavior in lakes with distinct water layers, which can help in predicting natural phenomena and their effects on environments like swash zones, where waves meet the shore.
Kateryna Terletska, Vladimir Maderich, and Elena Tobisch
Nonlin. Processes Geophys., 31, 207–217, https://doi.org/10.5194/npg-31-207-2024, https://doi.org/10.5194/npg-31-207-2024, 2024
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The transformation of internal waves at the edge of ice cover can enhance the turbulent mixing and melting of ice in the Arctic Ocean and Antarctica. We studied numerically the transformation of internal solitary waves of depression under smooth ice surfaces compared with the processes beneath the ridged underside of the ice. For large keels, more than 40% of wave energy is lost on the first keel, while for relatively small keels energy losses on the first keel are less than 6%.
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.
Samuel George Hartharn-Evans, Marek Stastna, and Magda Carr
Nonlin. Processes Geophys., 31, 61–74, https://doi.org/10.5194/npg-31-61-2024, https://doi.org/10.5194/npg-31-61-2024, 2024
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Across much of the ocean, and the world's lakes, less dense water (either because it is warm or fresh) overlays denser water, forming stratification. The mixing of these layers affects the distribution of heat, nutrients, plankton, sediment, and buoyancy, so it is crucial to understand. We use small-scale numerical experiments to better understand these processes, and here we propose a new analysis tool for understanding mixing within those models, looking at where two variables intersect.
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.
Irina I. Rypina, Lawrence J. Pratt, and Michael Dotzel
Nonlin. Processes Geophys., 31, 25–44, https://doi.org/10.5194/npg-31-25-2024, https://doi.org/10.5194/npg-31-25-2024, 2024
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This paper investigates the aggregation of small, spherical, slightly buoyant, rigid particles in a simple 3D vortex flow. Our goal was to gain insights into the behaviour of slightly buoyant marine microplastics in a flow that qualitatively resembles ocean eddies. Attractors are mapped out for the steady, axisymmetric; steady, asymmetric; and nonsteady, asymmetric vortices over a range of flow and particle parameters. Simple theoretical arguments are used to interpret the results.
Linghan Meng, Haibin Song, Yongxian Guan, Shun Yang, Kun Zhang, and Mengli Liu
EGUsphere, https://doi.org/10.5194/egusphere-2024-92, https://doi.org/10.5194/egusphere-2024-92, 2024
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In the seismic section, high-frequency and mode-2 internal waves, along with shear instability, were identified in the ocean. Strong nonlinear high-frequency waves, believed to be from shoaling, Behind them are larger mode-2 internal solitary waves. These waves show instability, notably the second mode-2 internal waves with distinct K-H billows. Seismic data revealed that diapycnal mixing from these events in the shelf area is 3.5 times greater than than that in the slope area.
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.
Camille Hankel and Eli Tziperman
Nonlin. Processes Geophys., 30, 299–309, https://doi.org/10.5194/npg-30-299-2023, https://doi.org/10.5194/npg-30-299-2023, 2023
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We present a novel, efficient method for identifying climate
tipping pointthreshold values of CO2 beyond which rapid and irreversible changes occur. We use a simple model of Arctic sea ice to demonstrate the method’s efficacy and its potential for use in state-of-the-art global climate models that are too expensive to run for this purpose using current methods. The ability to detect tipping points will improve our preparedness for rapid changes that may occur under future climate change.
Bin Shi and Guodong Sun
Nonlin. Processes Geophys., 30, 263–276, https://doi.org/10.5194/npg-30-263-2023, https://doi.org/10.5194/npg-30-263-2023, 2023
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We introduce a sample-based algorithm to obtain the conditional nonlinear optimal perturbations. Compared with the classical adjoint-based method, it is easier to implement and reduces the required storage for the basic state. When we reduce the number of samples to some extent, it reduces the computation markedly more when using the sample-based method, which can guarantee that the CNOP obtained is nearly consistent with some minor fluctuating errors oscillating in spatial distribution.
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.
Joshua Dorrington and Tim Palmer
Nonlin. Processes Geophys., 30, 49–62, https://doi.org/10.5194/npg-30-49-2023, https://doi.org/10.5194/npg-30-49-2023, 2023
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Atmospheric models often include random forcings, which aim to replicate the impact of processes too small to be resolved. Recent results in simple atmospheric models suggest that this random forcing can actually stabilise certain slow-varying aspects of the system, which could provide a path for resolving known errors in our models. We use randomly forced simulations of a
toychaotic system and theoretical arguments to explain why this strange effect occurs – at least in simple models.
Irina I. Rypina, Timothy Getscher, Lawrence J. Pratt, and Tamay Ozgokmen
Nonlin. Processes Geophys., 29, 345–361, https://doi.org/10.5194/npg-29-345-2022, https://doi.org/10.5194/npg-29-345-2022, 2022
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Techniques from dynamical systems theory have been widely used to study transport in ocean flows. However, they have been typically applied to numerically simulated trajectories of water parcels. This paper applies different dynamical systems techniques to real ocean drifter trajectories from the massive release in the Gulf of Mexico. To our knowledge, this is the first comprehensive comparison of the performance of different dynamical systems techniques with application to real drifters.
Steven R. Ramp, Yiing Jang Yang, Ching-Sang Chiu, D. Benjamin Reeder, and Frederick L. Bahr
Nonlin. Processes Geophys., 29, 279–299, https://doi.org/10.5194/npg-29-279-2022, https://doi.org/10.5194/npg-29-279-2022, 2022
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Earlier work in the vicinity of the shelf and slope in the northeastern South China Sea serendipitously revealed the presence of large, stunning bed forms (sand dunes) whose height (>15 m) and length (>350 m) are quite unique and unusual. We hypothesize that the dunes formed due to shoaling very large-amplitude nonlinear internal waves that scour the bottom and resuspend and redistribute the sediments. As a first step, the wave characteristics are observed and described in detail.
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.
Kateryna Terletska and Vladimir Maderich
Nonlin. Processes Geophys., 29, 161–170, https://doi.org/10.5194/npg-29-161-2022, https://doi.org/10.5194/npg-29-161-2022, 2022
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Internal solitary waves (ISWs) emerge in the ocean and seas in various forms and break on the shelf zones in a variety of ways. This results in intensive mixing that affects processes such as biological productivity and sediment transport. Mechanisms of wave interaction with slopes are related to breaking and changing polarity. Our study focuses on wave transformation over idealized shelf-slope topography using a two-layer stratification. Four types of ISW transformation over slopes are shown.
Wenhao Fan, Haibin Song, Yi Gong, Shun Yang, and Kun Zhang
Nonlin. Processes Geophys., 29, 141–160, https://doi.org/10.5194/npg-29-141-2022, https://doi.org/10.5194/npg-29-141-2022, 2022
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Compared with mode-1 internal solitary waves (ISWs), mode-2 ISWs in the ocean require further study. A mass of mode-2 ISWs developing at the Pacific coast of Central America have been imaged using seismic reflection data. We find that the relationship between the mode-2 ISW propagation speed and amplitude is diverse. It is affected by seawater depth, pycnocline depth, and pycnocline thickness. The ISW vertical amplitude structure is affected by the ISW nonlinearity and the pycnocline deviation.
Marek Stastna, Aaron Coutino, and Ryan K. Walter
Nonlin. Processes Geophys., 28, 585–598, https://doi.org/10.5194/npg-28-585-2021, https://doi.org/10.5194/npg-28-585-2021, 2021
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Large-amplitude waves in the interior of the ocean-internal waves in the ocean propagate in a dynamic, highly variable environment with changes in background current, local depth, and stratification. These waves have a well-known mathematical theory that, despite considerable progress, has some gaps. In particular, waves have been observed in situations that preclude an application of the mathematical theory. We present numerical simulations of the spontaneous generation of such waves.
Yi Gong, Haibin Song, Zhongxiang Zhao, Yongxian Guan, Kun Zhang, Yunyan Kuang, and Wenhao Fan
Nonlin. Processes Geophys., 28, 445–465, https://doi.org/10.5194/npg-28-445-2021, https://doi.org/10.5194/npg-28-445-2021, 2021
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When the internal solitary wave propagates to the continental shelf and slope, the polarity reverses due to the shallower water depth. In this process, the internal solitary wave dissipates energy and enhances diapycnal mixing, thus affecting the local oceanic environment. In this study, we used reflection seismic data to evaluate the spatial distribution of the diapycnal mixing around the polarity-reversing internal solitary waves.
Jia You, Zhenhua Xu, Qun Li, Robin Robertson, Peiwen Zhang, and Baoshu Yin
Nonlin. Processes Geophys., 28, 271–284, https://doi.org/10.5194/npg-28-271-2021, https://doi.org/10.5194/npg-28-271-2021, 2021
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Turbulent mixing in the ocean is mainly attributed to internal wave breaking, but the modulation of the mesoscale environment is unclear. The spatially inhomogeneous and seasonally variable diapycnal diffusivities in the upper Philippine Sea were estimated from Argo float data using a strain-based, fine-scale parameterization. Internal tides contributed significant diapycnal mixing here, with the mesoscale environment greatly regulating the intensity and spatial inhomogeneity of tidal mixing.
David Wichmann, Christian Kehl, Henk A. Dijkstra, and Erik van Sebille
Nonlin. Processes Geophys., 27, 501–518, https://doi.org/10.5194/npg-27-501-2020, https://doi.org/10.5194/npg-27-501-2020, 2020
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The surface transport of heat, nutrients and plastic in the North Atlantic Ocean is organized into large-scale flow structures. We propose a new and simple method to detect such features in ocean drifter data sets by identifying groups of trajectories with similar dynamical behaviour using network theory. We successfully detect well-known regions such as the Subpolar and Subtropical gyres, the Western Boundary Current region and the Caribbean Sea.
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.
João H. Bettencourt, Vincent Rossi, Lionel Renault, Peter Haynes, Yves Morel, and Véronique Garçon
Nonlin. Processes Geophys., 27, 277–294, https://doi.org/10.5194/npg-27-277-2020, https://doi.org/10.5194/npg-27-277-2020, 2020
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The oceans are losing oxygen, and future changes may worsen this problem. We performed computer simulations of an idealized Iberian Peninsula upwelling system to identify the main fine-scale processes driving dissolved oxygen variability as well as study the response of oxygen levels to changes in wind patterns and phytoplankton species. Our results suggest that oxygen levels would decrease if the wind blows for long periods of time or if phytoplankton is dominated by species that grow slowly.
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.
Henk A. Dijkstra
Nonlin. Processes Geophys., 26, 359–369, https://doi.org/10.5194/npg-26-359-2019, https://doi.org/10.5194/npg-26-359-2019, 2019
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I provide a personal view on the role of bifurcation analysis of climate models in the development of a theory of variability in the climate system. By outlining the state of the art of the methodology and by discussing what has been done and what has been learned from a hierarchy of models, I will argue that there are low-order phenomena of climate variability, such as El Niño and the Atlantic Multidecadal Oscillation.
Mallory Carlu, Francesco Ginelli, Valerio Lucarini, and Antonio Politi
Nonlin. Processes Geophys., 26, 73–89, https://doi.org/10.5194/npg-26-73-2019, https://doi.org/10.5194/npg-26-73-2019, 2019
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We explore the nature of instabilities in a well-known meteorological toy model, the Lorenz 96, to unravel key mechanisms of interaction between scales of different resolutions and time scales. To do so, we use a mathematical machinery known as Lyapunov analysis, allowing us to capture the degrees of chaoticity associated with fundamental directions of instability. We find a non-trivial group of such directions projecting significantly on slow variables, associated with long term dynamics.
Genevieve Jay Brett, Larry Pratt, Irina Rypina, and Peng Wang
Nonlin. Processes Geophys., 26, 37–60, https://doi.org/10.5194/npg-26-37-2019, https://doi.org/10.5194/npg-26-37-2019, 2019
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The relative importance of chaotic stirring and smaller-scale turbulent mixing for the distribution of dye in an idealized ocean flow feature is quantified using three different methods. We find that stirring is the dominant process in large areas with fast stirring, while mixing dominates in small fast-stirring regions and all slow-stirring regions. This quantification of process dominance can help oceanographers think about when to model stirring accurately, which can be costly.
Catherine Nicolis
Nonlin. Processes Geophys., 25, 649–658, https://doi.org/10.5194/npg-25-649-2018, https://doi.org/10.5194/npg-25-649-2018, 2018
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Ordinarily the climatic impact of systematic variations of parameters arising from anthropogenic effects is addressed on the basis of large numerical models, where parameters are set to a prescribed level and the system is subsequently left to relax. We have revisited the problem from a nonlinear dynamics perspective in which the time variation of parameters is fully incorporated into the evolution laws. Some universal trends of the response have been identified.
Gabriele Vissio and Valerio Lucarini
Nonlin. Processes Geophys., 25, 413–427, https://doi.org/10.5194/npg-25-413-2018, https://doi.org/10.5194/npg-25-413-2018, 2018
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Constructing good parametrizations is key when studying multi-scale systems. We consider a low-order model and derive a parametrization via a recently developed statistical mechanical approach. We show how the method allows for seamlessly treating the case when the unresolved dynamics is both faster and slower than the resolved one. We test the skill of the parametrization by using the formalism of the Wasserstein distance, which allows for measuring how different two probability measures are.
Dirk L. van Kekem and Alef E. Sterk
Nonlin. Processes Geophys., 25, 301–314, https://doi.org/10.5194/npg-25-301-2018, https://doi.org/10.5194/npg-25-301-2018, 2018
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In this paper we investigate the spatiotemporal properties of waves in the Lorenz-96 model. In particular, we explain how these properties are related to the presence of Hopf and pitchfork bifurcations. We also explain bifurcation scenarios by which multiple stable waves can coexist for the same parameter values.
Davide Faranda, Gabriele Messori, M. Carmen Alvarez-Castro, and Pascal Yiou
Nonlin. Processes Geophys., 24, 713–725, https://doi.org/10.5194/npg-24-713-2017, https://doi.org/10.5194/npg-24-713-2017, 2017
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We study the dynamical properties of the Northern Hemisphere atmospheric circulation by analysing the sea-level pressure, 2 m temperature, and precipitation frequency field over the period 1948–2013. The metrics are linked to the predictability and the persistence of the atmospheric flows. We study the dependence on the seasonal cycle and the fields corresponding to maxima and minima of the dynamical indicators.
Sylwester Arabas and Shin-ichiro Shima
Nonlin. Processes Geophys., 24, 535–542, https://doi.org/10.5194/npg-24-535-2017, https://doi.org/10.5194/npg-24-535-2017, 2017
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The paper bridges cloud/aerosol modelling with bifurcation analysis. It identifies two nonlinear peculiarities in the differential equations describing formation of atmospheric clouds through vapour condensation on a population of aerosol particles. A key finding of the paper is an analytic estimate for the timescale of the process. The study emerged from discussions on the causes of hysteretic behaviour of the system that we observed in the results of numerical simulations.
Yoann Robin, Pascal Yiou, and Philippe Naveau
Nonlin. Processes Geophys., 24, 393–405, https://doi.org/10.5194/npg-24-393-2017, https://doi.org/10.5194/npg-24-393-2017, 2017
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If climate is viewed as a chaotic dynamical system, its trajectories yield on an object called an attractor. Being perturbed by an external forcing, this attractor could be modified. With Wasserstein distance, we estimate on a derived Lorenz model the impact of a forcing similar to climate change. Our approach appears to work with small data sizes. We have obtained a methodology quantifying the deformation of well-known attractors, coherent with the size of data available.
Jezabel Curbelo, Víctor José García-Garrido, Carlos Roberto Mechoso, Ana Maria Mancho, Stephen Wiggins, and Coumba Niang
Nonlin. Processes Geophys., 24, 379–392, https://doi.org/10.5194/npg-24-379-2017, https://doi.org/10.5194/npg-24-379-2017, 2017
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Lagrangian coherent structures have supported the description of transport processes in fluid dynamics. In this work we use the M function to provide new insights into the 3-D Lagrangian structure of the southern stratosphere. Dynamical systems concepts appropriate to 3-D, such as normally hyperbolic invariant curves, are discussed and applied to describe the vertical extension of the stratospheric polar vortex and its evolution.
Elisa Johanna Spreitzer, Manuel Patrik Marschalik, and Peter Spichtinger
Nonlin. Processes Geophys., 24, 307–328, https://doi.org/10.5194/npg-24-307-2017, https://doi.org/10.5194/npg-24-307-2017, 2017
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We developed a simple analytical model for describing subvisible cirrus clouds qualitatively. Using theory of dynamical systems we found two different states for the long-term behaviour of subvisible cirrus clouds, i.e. an attractor case (stable equilibrium point) and a limit cycle scenario. The transition between the states constitutes a Hopf bifurcation and is determined by environmental conditions such as vertical updraughts and temperature.
Daniel Garaboa-Paz, Nieves Lorenzo, and Vicente Pérez-Muñuzuri
Nonlin. Processes Geophys., 24, 227–235, https://doi.org/10.5194/npg-24-227-2017, https://doi.org/10.5194/npg-24-227-2017, 2017
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This paper evaluates the connection between winter precipitation over the Iberian Peninsula and the large-scale tropospheric mixing over the eastern Atlantic Ocean. Finite-time Lyapunov exponents (FTLEs) have been calculated from 1979 to 2008 to evaluate this mixing. Our study suggests that significant negative correlations exist between summer FTLE anomalies and winter precipitation over Portugal and Spain.
Nazario Tartaglione
Nonlin. Processes Geophys., 24, 167–178, https://doi.org/10.5194/npg-24-167-2017, https://doi.org/10.5194/npg-24-167-2017, 2017
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This paper aims to show how the tropical circulation responds to changes of the vertical stratification of the imposed temperature that drives the model. These changes mimic the presence of water vapor cycles. Thus, for simplicity's sake we impose a periodic change of this stratification with variable periods of 10–90 days. The model responds with quasi-periodic oscillations having two or more dominant frequencies. After a long forcing time period, chaotic behavior starts to appear cyclically.
Rahel Vortmeyer-Kley, Ulf Gräwe, and Ulrike Feudel
Nonlin. Processes Geophys., 23, 159–173, https://doi.org/10.5194/npg-23-159-2016, https://doi.org/10.5194/npg-23-159-2016, 2016
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Since eddies play a major role in the dynamics of oceanic flows, it is of great interest to gain information about their tracks, lifetimes and shapes. We develop an eddy tracking tool based on structures in the flow with collecting (attracting) or separating (repelling) properties. In test cases mimicking oceanic flows it yields eddy lifetimes close to the analytical ones. It even provides a detailed view of the dynamics that can be useful to gain more insight into eddy dynamics in oceanic flows.
P. C. Yan, G. L. Feng, and W. Hou
Nonlin. Processes Geophys., 22, 249–258, https://doi.org/10.5194/npg-22-249-2015, https://doi.org/10.5194/npg-22-249-2015, 2015
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A novel method is created to detect the process of the abrupt change, which has not been mentioned yet in traditional research. By building an ideal time series with a transition process, the results show that the process could be detected clearly. When applied to a climate index, this method detects five processes, and all of them have reappeared via the “start-end states” phase diagram. Additionally, it is detectable that the persist time of the process is related to global warming.
N. Tartaglione
Nonlin. Processes Geophys., 22, 173–185, https://doi.org/10.5194/npg-22-173-2015, https://doi.org/10.5194/npg-22-173-2015, 2015
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At the Equator, where the heating is larger than that at other latitudes, air rises and diverges poleward in the upper troposphere, descending more or less at 30° latitude; this circulation is the Hadley cell.
We studied the impact of different meridional and vertical temperature distributions on a few features of the Hadley cell. Some parameters show a regular dependence on these distributions; others remain rather stable with distributions, but when they change, they do it in an abrupt way.
Cited articles
Adams, B., Carr, J., Lenton, T. M., and White, A.: One-dimensional daisyworld: spatial interactions and pattern formation, J. Theor. Biol., 223, 505–513, https://doi.org/10.1016/S0022-5193(03)00139-5, 2003.
Aleina, F. C., Baudena, M., D'Andrea, F., and Provenzale, A.: Multiple equilibria on planet Dune: climate–vegetation dynamics on a sandy planet, Tellus B, 65, 17662, https://doi.org/10.3402/tellusb.v65i0.17662, 2013.
Andronov, A., Vitt, A., and Khaikin, A.: Theory of Oscillators, Pergamon, Oxford, 1966.
Ayers, G. P. and Cainey, J. M.: The CLAW hypothesis: a review of the major developments, Environ. Chem., 4, 366–374, 2007.
Bhattacharya, K., Ghil, M., and Vulis, I. L.: Internal variability of an energy-balance model with delayed albedo effects, J. Atmos. Sci., 39, 1747–1773, 1982.
Biton, E. and Gildor, H.: The seasonal effect in one-dimensional Daisyworld, J. Theor. Biol., 314, 145–156, https://doi.org/10.1016/j.jtbi.2012.08.043, 2012.
Bonan, G. B.: Forests and climate change: forcings, feedbacks, and the climate benefits of forests, Science, 320, 1444–1449, https://doi.org/10.1126/science.1155121, 2008.
Brovkin, V., Claussen, M., Petoukhov, V., and Ganopolski, A.: On the stability of the atmosphere–vegetation system in the Sahara/Sahel region, J. Geophys. Res.-Atmos., 103, 31613–31624, https://doi.org/10.1029/1998JD200006, 1998.
Brovkin, V., Levis, S., Loutre, M.-F., Crucifix, M., Claussen, M., Ganopolski, A., Kubatzki, C., and Petoukhov, V.: Stability analysis of the climate–vegetation system in the northern high latitudes, Climatic Change, 57, 119–138, https://doi.org/10.1023/A:1022168609525, 2003.
Budyko, M. I.: The effect of solar radiation variations on the climate of the Earth, Tellus, 21, 611–619, 1969.
Charlson, R., Lovelock, J., Andreae, M., and Warren, S.: Oceanic phytoplankton, atmospheric sulphur, cloud albedo and climate, Nature, 326, 655–661, https://doi.org/10.1038/326655A0, 1987.
Charney, J. G.: Dynamics of deserts and drought in the Sahel, Q. J. Roy. Meteorol. Soc., 101, 193–202, https://doi.org/10.1002/qj.49710142802, 1975.
Charney, J. G., Stone, P. H., and Quirk, W. J.: Drought in the Sahara: a biogeophysical feedback mechanism, Science, 187, 434–435, https://doi.org/10.1126/science.187.4175.434, 1975.
Claussen, M.: On multiple solutions of the atmosphere–vegetation system in present-day climate, Global Change Biol., 4, 549–559, https://doi.org/10.1046/j.1365-2486.1998.t01-1-00122.x, 1998.
Claussen, M.: Late Quaternary vegetation-climate feedbacks, Clim. Past, 5, 203–216, https://doi.org/10.5194/cp-5-203-2009, 2009.
Claussen, M., Kubatzki, C., Brovkin, V., Ganopolski, A., Hoelzmann, P., and Pachur, H.-J.: Simulation of an abrupt change in Saharan vegetation in the Mid-Holocene, Geophys. Res. Lett., 26, 2037–2040, https://doi.org/10.1029/1999GL900494, 1999.
Claussen, M., Mysak, L. A., Weaver, A. J., Crucifix, M., Fichefet, T., Loutre, M. F., Weber, S. L., Alcamo, J., Alexeev, V. A., Berger, A., Calov, R., Ganopolski, A., Goosse, H., Lohman, G., Lunkeit, F., Mokhov, I. I., Petoukhov, V., Stone, P., and Wang, Z.: Earth system models of intermediate complexity: closing the gap in the spectrum of climate system models, Clim. Dynam., 18, 579–586, https://doi.org/10.1007/s00382-001-0200-1, 2002.
Crucifix, M.: Oscillators and relaxation phenomena in Pleistocene climate theory, Philos. T. Roy. Soc. A, 370, 1140–1165, https://doi.org/10.1098/rsta.2011.0315, 2012.
Crucifix, M. and Hewitt, C. D.: Impact of vegetation changes on the dynamics of the atmosphere at the Last Glacial Maximum, Clim. Dynam., 25, 447–459, https://doi.org/10.1007/s00382-005-0013-8, 2005.
De Gregorio, S., Pielke, R. A., and Dalu, G. A.: A delayed biophysical system for the Earth's climate, J. Nonlin. Sci., 2, 293–318, https://doi.org/10.1007/BF01208927, 1992.
Dekker, S. C., Rietkerk, M., and Bierkens, M. F. P.: Coupling microscale vegetation-soil water and macroscale vegetation-precipitation feedbacks in semiarid ecosystems, Global Change Biol., 13, 671–678, https://doi.org/10.1111/j.1365-2486.2007.01327.x, 2007.
Dijkstra, H. A.: Characterization of the multiple equilibria regime in a global ocean model, Tellus A, 59, 695–705, https://doi.org/10.1111/j.1600-0870.2007.00267.x, 2007.
Dijkstra, H. A. and Ghil, M.: Low-frequency variability of the large-scale ocean circulation: a dynamical systems approach, Rev. Geophys., 43, RG3002, https://doi.org/10.1029/2002RG000122, 2005.
Ermentrout, B.: Simulating, Analyzing, and Animating Dynamical Systems: a Guide to XPPAUT for Researchers and Students, SIAM, Philadelphia, USA, available at: http://www.math.pitt.edu/ bard/xpp/xpp.html (last access: 20 August 2014), 2002.
Fernando Salazar, J. and Poveda, G.: Role of a simplified hydrological cycle and clouds in regulating the climate-biota system of Daisyworld, Tellus B, 61, 483–497, https://doi.org/10.1111/j.1600-0889.2008.00411.x, 2009.
Ganopolski, A. and Calov, R.: The role of orbital forcing, carbon dioxide and regolith in 100 kyr glacial cycles, Clim. Past, 7, 1415–1425, https://doi.org/10.5194/cp-7-1415-2011, 2011.
Ghil, M.: Climate stability for a Sellers-type model, J. Atmos. Sci., 33, 3–20, 1976.
Ghil, M.: Cryothermodynamics: the chaotic dynamics of paleoclimate, Physica D, 77, 130–159, https://doi.org/10.1016/0167-2789(94)90131-7, 1994.
Ghil, M.: Hilbert problems for the geosciences in the 21st century, Nonlin. Processes Geophys., 8, 211–211, https://doi.org/10.5194/npg-8-211-2001, 2001.
Ghil, M.: A mathematical theory of climate sensitivity or, How to deal with both anthropogenic forcing and natural variability?, in: Ch. 2 in Climate Change: Multidecadal and Beyond, edited by: Chang, C. P., Ghil, M., Latif, M., and Wallace, J. M., World Scientific Publ. Co./Imperial College Press, Singapore and London, UK, 21 pp., in press, 2015.
Ghil, M. and Childress, S.: Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics, Applied Mathematical Sciences, vol. 60, Springer-Verlag, New York, 1987.
Ghil, M. and Robertson, A. W.: Solving problems with GCMs: general circulation models and their role in the climate modeling hierarchy, in: General Circulation Model Development: Past, Present and Future, edited by: Randall, D., Academic Press, San Diego, 285–325, 2000.
Ghil, M. and Tavantzis, J.: Global Hopf bifurcation in a simple climate model, SIAM J. Appl. Math., 43, 1019–1041, https://doi.org/10.1137/0143067, 1983.
Ghil, M., Zaliapin, I., and Thompson, S.: A delay differential model of ENSO variability: parametric instability and the distribution of extremes, Nonlin. Processes Geophys., 15, 417–433, https://doi.org/10.5194/npg-15-417-2008, 2008.
Gildor, H. and Tziperman, E.: A sea ice climate switch mechanism for the 100-kyr glacial cycles, J. Geophys. Res.-Oceans, 106, 9117–9133, https://doi.org/10.1029/1999JC000120, 2001.
Horton, D. E., Poulsen, C. J., and Pollard, D.: Influence of high-latitude vegetation feedbacks on late Palaeozoic glacial cycles, Nat. Geosci., 3, 572–577, https://doi.org/10.1038/ngeo922, 2010.
Imbrie, J. and Imbrie, J. Z.: Modeling the climatic response to orbital variations, Science, 207, 943–954, https://doi.org/10.1126/science.207.4434.943, 1980.
IPCC – Intergovernmental Panel on Climate Change: Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the IPCC, edited by: Stocker, T. F., Qin, D., Plattner, G. K., Tignor, M., Allen, S. K., Boschung, J., Nauels, A., Xia, Y., Bex, V., and Midgley, P. M., Cambridge University Press, Cambridge, UK, and New York, USA, 1535 pp., 2013.
Irizarry-Ortiz, M. M., Wang, G., and Eltahir, E. A. B.: Role of the biosphere in the mid-Holocene climate of West Africa, J. Geophys. Res.-Atmos., 108, 4042, https://doi.org/10.1029/2001JD000989, 2003.
Janssen, R. H. H., Meinders, M. B. J., Van Nes, E. H., and Scheffer, M.: Microscale vegetation-soil feedback boosts hysteresis in a regional vegetation-climate system, Global Change Biol., 14, 1104–1112, https://doi.org/10.1111/j.1365-2486.2008.01540.x, 2008.
Jin, F.-F., Neelin, J. D., and Ghil, M.: El Niño on the Devil's Staircase: annual subharmonic steps to chaos, Science, 264, 70–72, 1994.
Källèn, E., Crafoord, C., and Ghil, M.: Free Oscillations in a Climate Model with Ice-Sheet Dynamics, J. Atmos. Sci., 36, 2292–2303, 1979.
Lenton, T. M., Held, H., Kriegler, E., Hall, J. W., Lucht, W., Rahmstorf, S., and Schellnhuber, H. J.: Tipping elements in the Earth's climate system, P. Natl. Acad. Sci. USA, 105, 1786–1793, https://doi.org/10.1073/pnas.0705414105, 2008.
Le Treut, H. and Ghil, M.: Orbital forcing, climatic interactions, and glaciation cycles, J. Geophys. Res.-Oceans, 88, 5167–5190, https://doi.org/10.1029/JC088iC09p05167, 1983.
Lorenz, E. N.: Deterministic nonperiodic flow, J. Atmos. Sci., 20, 130–141, 1963.
Lovelock, J. E. and Kump, L. R.: Failure of climate regulation in a geophysiological model, Nature, 369, 732–734, https://doi.org/10.1038/369732a0, 1994.
Meir, P., Cox, P., and Grace, J.: The influence of terrestrial ecosystems on climate, Trends Ecol. Evol., 21, 254–260, https://doi.org/10.1016/j.tree.2006.03.005, 2006.
Meissner, K. J., Weaver, A. J., Matthews, H. D., and Cox, P. M.: The role of land surface dynamics in glacial inception: a study with the UVic Earth System Model, Clim. Dynam., 21, 515–537, https://doi.org/10.1007/s00382-003-0352-2, 2003.
Nevison, C., Gupta, V., and Klinger, L.: Self-sustained temperature oscillations on Daisyworld, Tellus B, 51, 806–814, https://doi.org/10.1034/j.1600-0889.1999.t01-3-00005.x, 1999.
North, G. R., Cahalan, R. F., and Coakley, J.: Energy balance climate models, Rev. Geophys. Space GE, 19, 91–121, 1981.
Otterman, J.: Baring high-albedo soils by overgrazing: A hypothesized desertification mechanism, Science, 186, 531–533, https://doi.org/10.1126/science.186.4163.531, 1974.
Otterman, J., Chou, M. D., and Arking, A.: Effects of nontropical forest cover on climate, J. Clim. Appl. Meteorol., 23, 762–767, 1984.
Popper, K.: The Open Universe: An Argument for Indeterminism, Reprinted 1991 by Routledge, Abingdon, New York, 1982.
Popper, K.: The Logic of Scientific Discovery, 1959; reprinted by Routledge Classics, London, New York, 2002.
Renssen, H., Brovkin, V., Fichefet, T., and Goosse, H.: Holocene climate instability during the termination of the African Humid Period, Geophys. Res. Lett., 30, 1184, https://doi.org/10.1029/2002GL016636, 2003.
Roques, L., Chekroun, M. D., Cristofol, M., Soubeyrand, S., and Ghil, M.: Parameter estimation for energy balance models with memory, P. Roy. Soc. A, 470, 20140349, https://doi.org/10.1098/rspa.2014.0349, 2014.
Saltzman, B.: Climatic systems analysis, Adv. Geophys., 25, 173–233, 1983.
Schneider, S. H. and Dickinson, R. E.: Climate modeling, Rev. Geophys. Space GE, 12, 447–493, 1974.
Sellers, W. D.: A global climatic model based on the energy balance of the earth–atmosphere system, J. Appl. Meteorol., 8, 392–400, https://doi.org/10.1175/1520-0450(1969)008<0392:AGCMBO>2.0.CO;2, 1969.
Shepon, A. and Gildor, H.: The lightning-biota climatic feedback, Global Change Biol., 14, 440–450, https://doi.org/10.1111/j.1365-2486.2007.01501.x, 2008.
Stommel, H.: Thermohaline convection with two stable regimes of flow, Tellus, 13, 224–230, 1961.
Svirezhev, Y. M. and von Bloh, W.: A minimal model of interaction between climate and vegetation: qualitative approach, Ecol. Model., 92, 89–99, https://doi.org/10.1016/0304-3800(95)00198-0, 1996.
Svirezhev, Y. M. and von Bloh, W.: Climate, vegetation, and global carbon cycle: the simplest zero-dimensional model, Ecol. Model., 101, 79–95, https://doi.org/10.1016/S0304-3800(97)01973-X, 1997.
Tziperman, E., Stone, L., Cane, M., and Jarosh, H.: El Niño chaos: overlapping of resonances between the seasonal cycle and the Pacific ocean–atmosphere oscillator, Science, 264, 72–74, 1994.
Watson, A. J. and Lovelock, J. E.: Biological homeostasis of the global environment: the parable of Daisyworld, Tellus B, 35, 284–289, https://doi.org/10.1111/j.1600-0889.1983.tb00031.x, 1983.
Weaver, I. S. and Dyke, J. G.: The importance of timescales for the emergence of environmental self-regulation, J. Theor. Biol., 313, 172–180, https://doi.org/10.1016/j.jtbi.2012.07.034, 2012.
Wood, A. J., Ackland, G. J., Dyke, J. G., Williams, H. T. P., and Lenton, T. M.: Daisyworld: a review, Rev. Geophys., 46, RG1001, https://doi.org/10.1029/2006RG000217, 2008.
Zaliapin, I. and Ghil, M.: Another look at climate sensitivity, Nonlin. Processes Geophys., 17, 113–122, https://doi.org/10.5194/npg-17-113-2010, 2010.
Zeng, N. and Neelin, J. D.: The role of vegetation–climate interaction and interannual variability in shaping the African savanna, J. Climate, 13, 2665–2670, 2000.
Zeng, N., Neelin, J. D., Lau, K.-M., and Tucker, C. J.: Enhancement of interdecadal climate variability in the Sahel by vegetation interaction, Science, 286, 1537–1540, https://doi.org/10.1126/science.286.5444.1537, 1999.
Short summary
Our conceptual model describes global temperature and vegetation extent. We use elements from Daisyworld and classical energy balance models and add an ocean with sea ice. The model exhibits oscillatory behavior within a plausible range of parameter values.
Its periodic solutions have sawtooth behavior that is characteristic of relaxation oscillations, as well as suggestive of Quaternary glaciation cycles. The model is one of the simplest of its kind to produce such oscillatory behavior.
Our conceptual model describes global temperature and vegetation extent. We use elements from...
