Articles | Volume 30, issue 4
https://doi.org/10.5194/npg-30-481-2023
© Author(s) 2023. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Special issue:
https://doi.org/10.5194/npg-30-481-2023
© Author(s) 2023. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Research article
| Highlight paper
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03 Nov 2023
Research article | Highlight paper |
| 03 Nov 2023
| 03 Nov 2023
Rate-induced tipping in ecosystems and climate: the role of unstable states, basin boundaries and transient dynamics
Ulrike Feudel
CORRESPONDING AUTHOR
Institute for Chemistry and Biology for the Marine Environment, Carl von Ossietzky University Oldenburg, Oldenburg, Germany
Related authors
Ann Kristin Klose, Jonathan F. Donges, Ulrike Feudel, and Ricarda Winkelmann
Earth Syst. Dynam., 15, 635–652, https://doi.org/10.5194/esd-15-635-2024, https://doi.org/10.5194/esd-15-635-2024, 2024
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We qualitatively study the long-term stability of the Greenland Ice Sheet and AMOC as tipping elements in the Earth system, which is largely unknown given their interaction in a positive–negative feedback loop. Depending on the timescales of ice loss and the position of the AMOC’s state relative to its critical threshold, we find distinct dynamic regimes of cascading tipping. These suggest that respecting safe rates of environmental change is necessary to mitigate potential domino effects.
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.
Ann Kristin Klose, Jonathan F. Donges, Ulrike Feudel, and Ricarda Winkelmann
Earth Syst. Dynam., 15, 635–652, https://doi.org/10.5194/esd-15-635-2024, https://doi.org/10.5194/esd-15-635-2024, 2024
Short summary
Short summary
We qualitatively study the long-term stability of the Greenland Ice Sheet and AMOC as tipping elements in the Earth system, which is largely unknown given their interaction in a positive–negative feedback loop. Depending on the timescales of ice loss and the position of the AMOC’s state relative to its critical threshold, we find distinct dynamic regimes of cascading tipping. These suggest that respecting safe rates of environmental change is necessary to mitigate potential domino effects.
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
Short summary
Short summary
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.
Related subject area
Subject: Bifurcation, dynamical systems, chaos, phase transition, nonlinear waves, pattern formation | Topic: Climate, atmosphere, ocean, hydrology, cryosphere, biosphere | Techniques: Theory
Review article: Interdisciplinary perspectives on climate sciences – highlighting past and current scientific achievements
Variational techniques for a one-dimensional energy balance model
Sensitivity of the polar boundary layer to transient phenomena
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
Review article: Dynamical systems, algebraic topology and the climate sciences
Review article: Large fluctuations in non-equilibrium physics
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
Review article: Hilbert problems for the climate sciences in the 21st century – 20 years later
Anthropocene climate bifurcation
Baroclinic and barotropic instabilities in planetary atmospheres: energetics, equilibration and adjustment
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.
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.
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.
Cited articles
Alberti, T., Faranda, D., Lucarini, V., Donner, R., Dubrulle, B., and Daviaud, F.: Scale dependence of fractal dimension in deterministic and stochastic Lorenz-63 systems, Chaos, 33, 023144, https://doi.org/10.1063/5.0106053, 2023. a
Alligood, K. T., Sauer, T. D., and Yorke, J. A.: Chaos: An Introduction to Dynamical Systems, Springer, New York, https://doi.org/10.1007/b97589, 1992. a
Armstrong McKay, D. I., Staal, A., Abrams, J. F., Winkelmann, R., Sakschewski, B., Loriani, S., Fetzner, I., Cornell, S. E., Rockström, J., and Lenton, T. M.: Exceeding 1.5C global warming could trigger multiple climate tipping points, Science, 1171, eabn7950, https://doi.org/10.1126/science.abn7950, 2022. a
Ashwin, P., Perryman, C., and Wieczorek, S.: Parameter shifts for nonautonomous systems in low dimension: bifurcation-and rate-induced tipping, Nonlinearity, 30, 2185, https://doi.org/10.1088/1361-6544/aa675b, 2017. a
Bastiaansen, R., Doelman, A., Eppinga, M. B., and Rietkerk, M.: The effect of climate change on the resilience of ecosystems with adaptive spatial pattern formation, Ecol. Lett., 23, 414–429, https://doi.org/10.1111/ele.13449, 2020. a
Bastiaansen, R., Dijkstra, H. A., and von der Heyd, A. S.: Fragmented tipping in a spatially heterogeneous world, Environ. Res. Lett., 17, 045006, https://doi.org/10.1088/1748-9326/ac59a8, 2022. a, b
Bel, G., Hagberg, A., and Meron, E.: Gradual regime shifts in spatially extended ecosystems, Theor. Ecol., 5, 591–604, https://doi.org/10.1007/s12080-011-0149-6, 2012. a, b
Binzer, A., Guill, C., Brose, U., and Rall, B. C.: The dynamics of food chains under climate change and nutrient enrichment, Philos. T. Roy. Soc. B, 367, 2935–2944, https://doi.org/10.1098/rstb.2012.0230, 2012. a
Boers, N.: Observation-based early-warning signals for a collapse of the Atlantic Meridional Overturning Circulation, Nat. Clim. Change, 11, 680–688, https://doi.org/10.1038/s41558-021-01097-4, 2021. a
Boers, N. and Rypdal, M.: Critical slowing down suggests that the western Greenland Ice Sheet is close to a tipping point, P. Natl. Acad. Sci. USA, 118, e2024192118, https://doi.org/10.1073/pnas.2024192118, 2021. a
Boettiger, C. and Hastings, A.: Early warning signals and the prosecutor's fallacy, P. Roy. Soc. B, 279, 4734–4739, https://doi.org/10.1098/rspb.2012.2085, 2012. a
Boettiger, C., Ross, N., and Hastings, A.: Early warning signals: the charted and uncharted territories, Theor. Ecol., 6, 255–264, https://doi.org/10.1007/s12080-013-0192-6, 2013. a
Boulton, C., Allison, L., and Lenton, T.: Early warning signals of Atlantic Meridional Overturning Circulation collapse, Nat. Commun., 5, 5752, https://doi.org/10.1038/ncomms6752, 2014. a
Boulton, C. A., Lenton, T. M., and Boers, N.: Pronounced loss of Amazon rainforest resilience since the early 2000s, Nat. Clim. Change, 12, 271–278, https://doi.org/10.1038/s41558-022-01287-8, 2022. a
Cameron, M. K.: Finding he quasipotential for nongradient SDEs, Physica D, 241, 1532–1550, https://doi.org/10.1016/j.physd.2012.06.005, 2012. a
Carpenter, S. and Brock, W. A.: Rising variance: a leading indicator of ecological transition, Ecol. Lett., 9, 311–318, https://doi.org/10.1111/j.1461-0248.2005.00877.x, 2006. a
Charo, G. D., Chekroun, M. D., Sciamarella, D., and Ghil, M.: Noise-driven topological changes in chaotic dynamics, Chaos, 31, 103115, https://doi.org/10.1063/5.0059461, 2021. a
Clarke, J. J., Huntingford, C., Ritchie, P. D. L., and Cox, P. M.: Seeking more robust early warning signals for climate tipping points: the ratio of spectra method (ROSA), Environ. Res. Lett., 18, 035006, https://doi.org/10.1088/1748-9326/acbc8d, 2023. a
Dakos, V., Scheffer, M., van Nes, E., Brovkin, V., Petoukhov, V., and Held, H.: Slowing down as an early warning signal for abrupt climate change, P. Natl. Acad. Sci. USA, 105, 14308–14312, https://doi.org/10.1073/pnas.0802430105, 2008. a
Ditlevsen, P. and Johnsen, S.: Tipping points: early warning and wishful thinking, Geophys. Res. Lett., 37, 2–5, https://doi.org/10.1029/2010GL044486, 2010. a
d'Ovidio, F., Fernandez, V., Hernandez-Garcia, E., and Lopez, C.: Mixing structures in the Mediterranean Sea from finite-size Lyapunov exponents, Geophys. Res. Lett., 31, L17203, https://doi.org/10.1029/2004GL020328, 2004. a
Ebeling, W. and Malchow, H.: Bifurcations in a Bistable Reaction-Diffusion System, Ann. Phys., 36, 121–134, 1979. a
Eisenman, I.: Factors controlling the bifurcation structure of sea ice retreat, J. Geophys. Res.-Atmos., 117, D01111, https://doi.org/10.1029/2011JD016164, 2012. a, b
Eisenman, I. and Wettlaufer, J. S.: Nonlinear threshold behavior during the loss of Arctic sea ice, P. Natl. Acad. Sci. USA, 106, 28–32, https://doi.org/10.1073/pnas.0806887106, 2009. a, b
Fan, J., Meng, J., Ludescher, J., Chen, X., Ashkenazy, Y., Kurths, J., Havlin, S., and Schellnhuber, H. J.: Statistical physics approaches to the complex Earth system, Phys. Rep., 896, 1–84, https://doi.org/10.1016/j.physrep.2020.09.005, 2021. a
Feudel, U., Grebogi, C., Hunt, B. R., and Yorke, J. A.: Map with more than 100 coexisting low-period periodic attractors, Phys. Rev. E, 54, 71–81, https://doi.org/10.1103/PhysRevE.54.71, 1996. a
Ficetola, G. F. and Denoel, M.: Ecological thresholds: an assessment of methods to identify abrupt changes in species-habitat relationships, Ecography, 32, 1075–1084, https://doi.org/10.1111/j.1600-0587.2009.05571.x, 2009. a
Folke, C., Carpenter, S., Walker, B., Scheffer, M., Elmqvist, T., Gunderson, L., and Holling, C.: Regime shifts, resilience, and biodiversity in ecosystem management, Annu. Rev. Ecol. Evol. S., 35, 557–581, https://doi.org/10.1146/annurev.ecolsys.35.021103.105711, 2004. a
Franzke, C. L. E., O'Kane, T. J., Berner, J., Williams, P. D., and Lucarini, V.: Stochastic climate theory and modeling, Wiley Interdisciplinary Reviews-Climate Change, 6, 63–78, https://doi.org/10.1002/wcc.318, 2015. a
Freidlin, M. I. and Wentzell, A. D.: Random perturbations of dynamical systems, Springer, https://doi.org/10.1007/978-3-642-25847-3, 1998. a
Freund, J. A., Mieruch, S., Scholze, B., Wiltshire, K., and Feudel, U.: Bloom dynamics in a seasonally forced phytoplankton-zooplankton model: Trigger mechanisms and timing effects, Ecol. Complex., 3, 126–136, https://doi.org/10.1016/j.ecocom.2005.11.001, 2006. a
Ganapathisubramanian, N. and Showalter, K.: Critical slowing down in the bistable iodate-arsenic(III) reaction, J. Phys. Chem., 87, 1098–1099, https://doi.org/10.1021/j100230a004, 1983. a
Ghil, M. and Lucarini, V.: The physics of climate variability and climate change, Rev. Mod. Phys., 92, 035002, https://doi.org/10.1103/RevModPhys.92.035002, 2020. a
Gossner, M. M., Lewinsohn, T. M., Kahl, T., Grassein, F., Boch, S., Prati, D., Birkhofer, K., Renner, S. C., Sikorski, J., Wubet, T., Arndt, H., Baumgartner, V., Blaser, S., Bluethgen, N., Boerschig, C., Buscot, F., Diekoetter, T., Jorge, L. R., Jung, K., Keyel, A. C., Klein, A.-M., Klemmer, S., Krauss, J., Lange, M., Mueller, J., Overmann, J., Pasalic, E., Penone, C., Perovic, D. J., Purschke, O., Schall, P., Socher, S. A., Sonnemann, I., Tschapka, M., Tscharntke, T., Tuerke, M., Venter, P. C., Weiner, C. N., Werner, M., Wolters, V., Wurst, S., Westphal, C., Fischer, M., Weisser, W. W., and Allan, E.: Land-use intensification causes multitrophic homogenization of grassland communities, Nature, 540, 266–269, https://doi.org/10.1038/nature20575, 2016. a
Goswami, B. N., Venugopal, V., Sengupta, D., Madhusoodanan, M. S., and Xavier, P. K.: Increasing trend of extreme rain events over India in a warming environment, Science, 314, 1442–1445, https://doi.org/10.1126/science.1132027, 2006. a
Graham, R., Hamm, A., and Tel, T.: Nonequilibrium Potentials for Dynamic-Systems with Fractal Attractors or Repellors, Phys. Rev. Lett., 66, 3089–3092, https://doi.org/10.1103/PhysRevLett.66.3089, 1991. a
Grassberger, P.: On Phase-Transitions in Schloegl 2nd Model, Z. Phys. B Con. Mat., 47, 365–374, https://doi.org/10.1007/BF01313803, 1982. a
Guckenheimer, J. and Holmes, P.: Nonlinear Oscillations, Dynamical systems, and Bifurcations of Vector Fields, Springer Verlag, Berlin, Heidelberg, New York, https://doi.org/10.1007/978-1-4612-1140-2, 1986. a, b
Hairer, E., Wanner, G., and Noersett, S. P.: Solving Ordinary Differential Equations I: Nonstiff Problems, Springer, 1993 (code available at: https://doi.org/10.1007/978-3-540-78862-1). a
Halekotte, L. and Feudel, U.: Minimal fatal shocks in multistable complex networks, Sci. Rep., 10, 11783, https://doi.org/10.1038/s41598-020-68805-6, 2020. a, b
Haller, G.: Lagrangian Coherent Structures, in: Annual Review of Fluid Mechanics, edited by: Davis, S. and Moin, P., vol. 47, Annu. Rev. Fluid Mech., 137–162, https://doi.org/10.1146/annurev-fluid-010313-141322, 2015. a
Hasan, C. R., Mac Cárthaigh, R., and Wieczorek, S.: Rate-induced tipping in heterogeneous reacion-diffusion systems: An invariant manifold framework and geographically shifting ecosystems, arXiv [preprint], https://doi.org/10.48550/arXiv.2211.13062, 23 November 2022. a
Heinrichs, M. and Schneider, F.: Relaxation kinetics of steady statea in the continuous flow stirred tank reactor. Response to small and large perturbations: critical slowing down, J. Phys. Chem., 85, 2112–2116, https://doi.org/10.1021/j150614a031, 1981. a
Held, H. and Kleinen, T.: Detection of climate system bifurcations by degenerate fingerprinting, Geophys. Res. Lett., 31, L23207, https://doi.org/10.1029/2004GL020972, 2004. a
Hillebrand, H., Donohue, I., Harpole, W. S., Hodapp, D., Kucera, M., Lewandowska, A. M., Merder, J., Montoya, J. M., and Freund, J. A.: Thresholds for ecological responses to global change do not emerge from empirical data, Nat. Ecol. Evol., 4, 1502–1509, https://doi.org/10.1038/s41559-020-1256-9, 2020. a
Holbrook, S. J., Schmitt, R. J., Adam, T. C., and Brooks, A. J.: Coral Reef Resilience, Tipping Points and the Strength of Herbivory, Sci. Rep., 6, 35817, https://doi.org/10.1038/srep35817, 2016. a
Holling, C. S.: Engineering resilience versus ecological resilience, in: Engineering within ecological constraints, edited by: Schulze, P. C., 31–43, National Academies Press, https://doi.org/10.17226/4919, 1996. a
Horsthemke, W. and Lefever, R.: Noise-induced Transitions, Springer, Berlin, https://doi.org/10.1007/3-540-36852-3, 1984. a
Kai, E. T., Rossi, V., Sudre, J., Weimerskirch, H., Lopez, C., Hernandez-Garcia, E., Marsac, F., and Garcon, V.: Top marine predators track Lagrangian coherent structures, P. Natl. Acad. Sci. USA, 106, 8245–8250, https://doi.org/10.1073/pnas.0811034106, 2009. a
Kaszás, B., Feudel, U., and Tél, T.: Death and revival of chaos, Phys. Rev. E, 94, 062221, https://doi.org/10.1103/PhysRevE.94.062221, 2016. a, b
Kaszás, B., Feudel, U., and Tél, T.: Tipping phenomena in typical dynamical systems subjected to parameter drift, Sci. Rep., 9, 8654, https://doi.org/10.1038/s41598-019-44863-3, 2019. a, b, c, d
Khovanov, I. A., Luchinsky, D. G., McClintock, P. V. E., and Silchenko, A. N.: Fluctuational escape from chaotic attractors in multistable systems, Int. J. Bifurcat. Chaos, 18, 1727–1739, https://doi.org/10.1142/S0218127408021312, 2008. a
Klinshov, V. V., Nekorkin, V. I., and Kurths, J.: Stability threshold approach for complex dynamical systems, New J. Phys., 18, 013004, https://doi.org/10.1088/1367-2630/18/1/013004, 2015. a
Klose, A. K., Karle, V., Winkelmann, R., and Donges, J. F.: Emergence of cascading dynamics in interacting tipping elements of ecology and climate, Roy. Soc. Open Sci., 7, 200599, https://doi.org/10.1098/rsos.200599, 2020. a, b
Klose, A. K., Donges, J. F., Feudel, U., and Winkelmann, R.: Rate-induced tipping cascades arising from interactions between the Greenland Ice Sheet and the Atlantic Meridional Overturning Circulation, Earth Syst. Dynam. Discuss. [preprint], https://doi.org/10.5194/esd-2023-20, in review, 2023. a, b
Kouvaris, N. E., Kori, H., and Mikhailov, A. S.: Travelling and pinned fronts in bistable reaction diffusion systems on networks, Plos ONE, 7, e45029, https://doi.org/10.1371/journal.pone.0045029, 2012. a
Kraut, S. and Feudel, U.: Noise-induced Escape through a Chaotic Saddle: Lowering of the Activation Energy, Physica D, 181, 222–234, https://doi.org/10.1016/S0167-2789(03)00098-8, 2003. a
Kroenke, J., Wunderling, N., Winkelmann, R., Staal, A., Stumpf, B., Tuinenburg, O. A., and Donges, J. F.: Dynamics of tipping cascades on complex networks, Phys. Rev. E, 101, 042311, https://doi.org/10.1103/PhysRevE.101.042311, 2020. a, b, c
Kuehn, C.: A mathematical framework for critical transitions: Bifurcations, fast–slow systems and stochastic dynamics, Physica D, 240, 1020–1035, https://doi.org/10.1016/j.physd.2011.02.012, 2011. a
Kuhlbrodt, T., Titz, S., Feudel, U., and Rahmstorf, S.: A simple model of seasonal open ocean convection. Part II: Labrador Sea stability and stochastic forcing, Ocean Dynam., 52, 36–49, 2002. a
Kuznetsov, Y. A.: Elements of applied bifurcation theory, Springer, New York, https://doi.org/10.1007/978-1-4757-3978-7, 1995. a
Lenderink, G. and Haarsma, R.: Variability and Multiple Equilibria of the Thermohaline Circulation Associated with Deep-Water Formation, J. Phys. Oceanogr., 24, 1480–1493, https://doi.org/10.1175/1520-0485(1994)024<1480:VAMEOT>2.0.CO;2, 1994. a
Lenton, T. M.: Early warning of climate tipping points, Nat. Clim. Change, 1, 201–209, https://doi.org/10.1038/NCLIMATE1143, 2011. a
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. a
Lenton, T. M., Livina, V. N., Dakos, V., van Nes, E. H., and Scheffer, M.: Early warning of climate tipping points from critical slowing down: comparing methods to improve robustness, Philos. T. Roy. Soc. A, 370, 1185–1204, https://doi.org/10.1098/rsta.2011.0304, 2012. a
Lohmann, J., Castellana, D., Ditlevsen, P. D., and Dijkstra, H. A.: Abrupt climate change as a rate-dependent cascading tipping point, Earth Syst. Dynam., 12, 819–835, https://doi.org/10.5194/esd-12-819-2021, 2021. a, b, c
Lohmann, J., Dijkstra, H. A., Jochum, M., Lucarini, V., and Ditlevsen, P. D.: Multistability and Intermediate Tipping of the Atlantic Ocean Circulation, arXiv [preprint], https://doi.org/10.48550/arXiv.2304.05664, 12 April 2023. a
Lorenz, E. N.: Deterministic Nonperiodic Flow, J. Atmos. Sci., 20, 130–141, https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2, 1963. a
Lovejoy, S. and Schertzer, D.: Stochastic and scaling climate sensitivities: Solar, volcanic and orbital forcings, Geophys. Res. Lett., 39, L11702, https://doi.org/10.1029/2012GL051871, 2012. a
Maier, R. and Stein, D.: Transition-Rate Theory for Nongradient Drift Fields, Phys. Rev. Lett., 69, 3691–3695, https://doi.org/10.1103/PhysRevLett.69.3691, 1992. a
Mancho, A. M., Wiggins, S., Curbelo, J., and Mendoza, C.: Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems, Communi. Nonlinear Sci., 18, 3530–3557, https://doi.org/10.1016/j.cnsns.2013.05.002, 2013. a
Mehling, O., Bellomo, K., Angeloni, M., Pasquero, C., and von Hardenberg, J.: High-Latitude precipitation as a driver of multicentennial variability of the AMOC in a climate model of intermediate complexity, Clim. Dynam., 61, 1519–1534, https://doi.org/10.1007/s00382-022-06640-3, 2022. a, b
Mitra, C., Kurths, J., and Donner, R. V.: An integrative quantifier of multistability in complex systems based on ecological resilience, Sci. Rep., 5, 16196,https://doi.org/10.1038/srep16196, 2015. a
Mou, C., Luo, J., and Nicolis, G.: Stochastic Thermodynamics of Nonequilibrium Steady-States in Chemical-Reaction Systems, J. Chem. Phys., 84, 7011–7017, https://doi.org/10.1063/1.450623, 1986. a
Mumby, P. J., Hastings, A., and Edwards, H. J.: Thresholds and the resilience of Caribbean coral reefs, Nature, 450, 98–101, https://doi.org/10.1038/nature06252, 2007. a
Notz, D.: The future of ice sheets and sea ice: Between reversible retreat and unstoppable loss, P. Natl. Acad. Sci. USA, 106, 20590–20595, https://doi.org/10.1073/pnas.0902356106, 2009. a, b
O'Keeffe, P. E. and Wieczorek, S.: Tipping Phenomena and Points of No Return in Ecosystems: Beyond Classical Bifurcations, SIAM J. Appl.Dyn. Syst., 19, 2371–2402, https://doi.org/10.1137/19M1242884, 2020. a, b, c
Ott, E.: Chaos in Dynamical Systems, Cambridge University Press, Cambridge, https://doi.org/10.1017/CBO9780511803260, 1992. a
Pierini, S. and Ghil, M.: Tipping points induced by parameter drift in an excitable ocean model, Sci. Rep., 11, 11126, https://doi.org/10.1038/s41598-021-90138-1, 2021. a
Rahmstorf, S.: Multiple convection patterns and thermohaline flow in an idealized OGCM, J. Climate, 8, 3028–3039, https://doi.org/10.1175/1520-0442(1995)008<3028:MCPATF>2.0.CO;2, 1995. a, b
Rahmstorf, S.: On the freshwater forcing and transport of the Atlantic thermohaline circulation, Clim. Dynam., 12, 799–811, https://doi.org/10.1007/s003820050144, 1996. a
Ritchie, P. and Sieber, J.: Probability of noise- and rate-induced tipping, Phys. Rev. E, 95, 052209, https://doi.org/10.1103/PhysRevE.95.052209, 2017. a
Ritchie, P. D. L., Clarke, J. J., Cox, P. M., and Huntingford, C.: Overshooting tipping point thresholds in a changing climate, Nature, 592, 517–523, https://doi.org/10.1038/s41586-021-03263-2, 2021. a
Ritchie, P. D. L., Alkhayuon, H., Cox, P. M., and Wieczorek, S.: Rate-induced tipping in natural and human systems, Earth Syst. Dynam., 14, 669–683, https://doi.org/10.5194/esd-14-669-2023, 2023. a
Rooth, C.: Hydrology and Ocean Circulation, Prog. Oceanogr., 11, 131–149, https://doi.org/10.1016/0079-6611(82)90006-4, 1982. a
Rossi, V., Ser-Giacomi, E., Lopez, C., and Hernandez-Garcia, E.: Hydrodynamic provinces and oceanic connectivity from a transport network help designing marine reserves, Geophys. Res. Lett., 41, 2883–2891, https://doi.org/10.1002/2014GL059540, 2014. a
Sandulescu, M., López, C., Hernández-García, E., and Feudel, U.: Plankton blooms in vortices: the role of biological and hydrodynamic timescales, Nonlin. Processes Geophys., 14, 443–454, https://doi.org/10.5194/npg-14-443-2007, 2007. a
Scheffer, M., Hosper, S., Meijer, M., Moss, B., and Jeppesen, E.: Alternative Equilibria in Shallow Lakes, Trends Ecol. Evol., 8, 275–279, https://doi.org/10.1016/0169-5347(93)90254-M, 1993. a
Scheffer, M., Bascompte, J., Brock, W., Brovkin, V., Carpenter, S., Dakos, V., Held, H., van Nes, E., Rietkerk, M., and Sugihara, G.: Early warning signals for critical transitions, Nature, 461, 53–59, https://doi.org/10.1038/nature08227, 2009. a
Schellnhuber, H. J., Rahmstorf, S., and Winkelmann, R.: Commentary: Why the right climate target was agreed in Paris, Nat. Clim. Change, 6, 649–653, https://doi.org/10.1038/nclimate3013, 2016. a
Schertzer, D. and Lovejoy, S.: Multifractals, Generalized Scale Invaiance and Complexity In Geophysics, Int. J. Bifurcat. Chaos, 21, 3417–3456, https://doi.org/10.1142/S0218127411030647, 2011. a
Schlögl, F.: Chemical reaction models for non-equilibrium phase transitions, Z. Physik, 253, 147–161, 1972. a
Schoenmakers, S. and Feudel, U.: A resilience concept based on system functioning: A dynamical systems perspective, Chaos, 31, 053126, https://doi.org/10.1063/5.0042755, 2021. a
Sharma, Y., Abbott, K. C., Dutta, P. S., and Gupta, A. K.: Stochasticity and bistability in insect outbreak dynamics, Theor. Ecol., 8, 163–174, https://doi.org/10.1007/s12080-014-0241-9, 2015. a
Siteur, K., Siero, E., Eppinga, M. B., Rademacher, J. D. M., Doelman, A., and Rietkerk, M.: Beyond Turing: The response of patterned ecosystems to environmental change, Ecol. Complex., 20, 81–96, https://doi.org/10.1016/j.ecocom.2014.09.002, 2014. a
Siteur, K., Eppinga, M. B., Doelman, A., Siero, E., and Rietkerk, M.: Ecosystems off track: rate-induced critical transitions in ecological models, Oikos, 125, 1689–1699, https://doi.org/10.1111/oik.03112, 2016. a
Smith, L., Ziehmann, C., and Fraedrich, K.: Uncertainty dynamics and predictability in chaotic systems, Q. J. Roy. Meteor. Soc., 125, 2855–2886, https://doi.org/10.1256/smsqj.56004, 1999. a
Stephens, P., Sutherland, W., and Freckleton, R.: What is the Allee effect?, Oikos, 87, 185–190, https://doi.org/10.2307/3547011, 1999. a
Stommel, H.: Thermohaline Convection with 2 Stable Regimes of Flow, Tellus, 13, 224–230, https://doi.org/10.1111/j.2153-3490.1961.tb00079.x, 1961. a
Surovyatkina, E.: Prebifurcation noise amplification and noise-dependent hysteresis as indicators of bifurcations in nonlinear geophysical systems, Nonlin. Processes Geophys., 12, 25–29, https://doi.org/10.5194/npg-12-25-2005, 2005. a
Tel, T., Bodai, T., Drotos, G., Haszpra, T., Herein, M., Kaszas, B., and Vincze, M.: The Theory of Parallel Climate Realizations A New Framework of Ensemble Methods in a Changing Climate: An Overview, J. Stat. Phys., 179, 1496–1530, https://doi.org/10.1007/s10955-019-02445-7, 2020. a
Tredicce, J., Lippi, G., P.Mandel, Charasse, B., Chevalier, A., and Picque, B.: Critical slowing down at a bifurcation, Am. J. Phys., 72, 799–809, https://doi.org/10.1119/1.1688783, 2004. a
Turing, A. M.: The Chemical Basis of Morphogenesis, Philos. T. R. Soc. Lond. B, 237, 37–72, https://doi.org/10.1098/rstb.1952.0012, 1952. a
Van Nes, E. H., Amaro, T., Scheffer, M., and Duineveld, G. C.: Possible mechanisms for a marine benthic regime shift in the North Sea, Mar. Ecol. Prog. Ser., 330, 39–47, https://doi.org/10.3354/meps330039, 2007. a
Vanselow, A., Wieczorek, S., and Feudel, U.: When very slow is too fast – collapse of a predator-prey system, J. Theor. Biol., 479, 64–72, https://doi.org/10.1016/j.jtbi.2019.07.008, 2019. a, b, c
Vanselow, A., Halekotte, L., Pal, P., Wieczorek, S., and Feudel, U.: Rate-induced tipping can trigger plankton blooms, arXiv [preprint], https://doi.org/10.48550/arXiv.2212.01244, 17 November 2022. a, b, c
Wagner, T. J. W. and Eisenman, I.: False alarms: How early warning signals falsely predict abrupt sea ice loss, Geophys. Res. Lett., 42, 10333–10341, https://doi.org/10.1002/2015GL066297, 2015. a
Weijer, W., Maltrud, M. E., Hecht, M. W., Dijkstra, H. A., and Kliphuis, M. A.: Response of the Atlantic Ocean circulation to Greenland Ice Sheet melting in a strongly-eddying ocean model, Geophys. Res. Lett., 39, L09606, https://doi.org/10.1029/2012GL051611, 2012. a
Weijer, W., Cheng, W., Drijfhout, S. S., Fedorov, A. V., Hu, A., Jackson, L. C., Liu, W., McDonagh, E. L., Mecking, J. V., and Zhang, J.: Stability of the Atlantic Meridional Overturning Circulation: A Review and Synthesis, J. Geophys. Res.-Oceans, 124, 5336–5375, https://doi.org/10.1029/2019JC015083, 2019. a
Wieczorek, S., Ashwin, P., Luke, C. M., and Cox, P. M.: Excitability in ramped systems: the compost-bomb instability, P. Roy. Soc. A, 467, 1243–1269, https://doi.org/10.1098/rspa.2010.0485, 2011. a, b, c
Wiggins, S.: The dynamical systems approach to Lagrangian transport in oceanic flows, Annu. Rev. Fluid Mech., 37, 295–328, https://doi.org/10.1146/annurev.fluid.37.061903.175815, 2005. a
Wood, R. A., Rodriguez, J. M., Smith, R. S., Jackson, L. C., and Hawkins, E.: Observable, low-order dynamical controls on thresholds of the Atlantic meridional overturning circulation, Clim. Dynam., 53, 6815–6834, https://doi.org/10.1007/s00382-019-04956-1, 2019. a
Wunderling, N., Donges, J. F., Kurths, J., and Winkelmann, R.: Interacting tipping elements increase risk of climate domino effects under global warming, Earth Syst. Dynam., 12, 601–619, https://doi.org/10.5194/esd-12-601-2021, 2021. a, b
Zelnik, Y., Kinast, S., Yizhaq, H., Bel, G., and Meron, E.: Regime shifts in models of dryland vegetation, Philos. T. Roy. Soc. A, 371, 20120358, https://doi.org/10.1098/rsta.2012.0358, 2013. a
Zelnik, Y., Gandhi, P., Knobloch, E., and Meron, E.: Implications of tristability in pattern-forming ecosystems, Chaos, 28, 033609, https://doi.org/10.1063/1.5018925, 2018. a
Executive editor
This study suggests a necessary shift in the paradigm of nonlinear dynamical systems analysis in climate science, ecology, and beyond. The author delves into a relatively unexplored facet of critical transitions, which is of great importance in examples within the Earth and environmental sciences. In particular, she explains mechanisms based on these ideas in examples of systems that describe population dynamics and climate transitions.
This study suggests a necessary shift in the paradigm of nonlinear dynamical systems analysis in...
Short summary
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.
Many systems in nature are characterized by the coexistence of different stable states for given...
