Articles | Volume 32, issue 2
https://doi.org/10.5194/npg-32-189-2025
© Author(s) 2025. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/npg-32-189-2025
© Author(s) 2025. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Research article
| 
27 Jun 2025
Research article |
| 27 Jun 2025
| 27 Jun 2025
Rate-induced transitions and noise-driven resilience in vegetation pattern dynamics
Lilian Vanderveken
CORRESPONDING AUTHOR
Earth and Life Institute, UCLouvain, Louvain-la-Neuve, Belgium
Michel Crucifix
Earth and Life Institute, UCLouvain, Louvain-la-Neuve, Belgium
Related authors
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.
Loïc Sablon, Pierre Maffre, Yves Goddéris, Paul J. Valdes, Justin Gérard, Jarno J. C. Huygh, Anne-Christine Da Silva, and Michel Crucifix
EGUsphere, https://doi.org/10.5194/egusphere-2025-1696, https://doi.org/10.5194/egusphere-2025-1696, 2025
This preprint is open for discussion and under review for Geoscientific Model Development (GMD).
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We propose an innovative climate modelling framework that combines statistical methods with climate simulations to study Earth's environmental systems. The model captures how orbital changes and carbon dioxide levels influence climate atmospheric dynamics, offering a detailed and efficient way to explore long-term processes. This tool provides new opportunities to investigate Earth's climate history and its implications for future changes.
Victor Couplet, Marina Martínez Montero, and Michel Crucifix
Geosci. Model Dev., 18, 3081–3129, https://doi.org/10.5194/gmd-18-3081-2025, https://doi.org/10.5194/gmd-18-3081-2025, 2025
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We present SURFER v3.0, a simple climate model designed to estimate the impact of CO2 and CH4 emissions on global temperatures, sea levels, and ocean pH. We added new carbon cycle processes and calibrated the model to observations and results from more complex models, enabling use over timescales ranging from decades to millions of years. SURFER v3.0 is fast, transparent, and easy to use, making it an ideal tool for policy assessments and suitable for educational purposes.
Justin Gérard, Loïc Sablon, Jarno J. C. Huygh, Anne-Christine Da Silva, Alexandre Pohl, Christian Vérard, and Michel Crucifix
Clim. Past, 21, 239–260, https://doi.org/10.5194/cp-21-239-2025, https://doi.org/10.5194/cp-21-239-2025, 2025
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We used cGENIE, a climate model, to explore how changes in continental configuration, CO2 levels, and orbital configuration affected ocean oxygen levels during the Devonian period (419–359 million years ago). Key factors contributing to ocean anoxia were identified, highlighting the influence of continental configurations, atmospheric conditions, and orbital changes. Our findings offer new insights into the causes and prolonged durations of Devonian ocean anoxic events.
Takahito Mitsui, Peter Ditlevsen, Niklas Boers, and Michel Crucifix
Earth Syst. Dynam. Discuss., https://doi.org/10.5194/esd-2024-39, https://doi.org/10.5194/esd-2024-39, 2024
Preprint under review for ESD
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The late Pleistocene glacial cycles are dominated by a 100-kyr periodicity, rather than other major astronomical periods like 19, 23, 41, or 400 kyr. Various models propose distinct mechanisms to explain this, but their diversity may obscure the key factor behind the 100-kyr periodicity. We propose a time-scale matching hypothesis, suggesting that the ice-sheet climate system responds to astronomical forcing at ~100 kyr because its intrinsic timescale is closer to 100 kyr than to other periods.
Justin Gérard and Michel Crucifix
Earth Syst. Dynam., 15, 293–306, https://doi.org/10.5194/esd-15-293-2024, https://doi.org/10.5194/esd-15-293-2024, 2024
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We used cGENIE, a climate model, to investigate the Atlantic Meridional Overturning Circulation (AMOC) slowdown under a warming scenario. We apply a diagnostic that was used in a previous study (Levang and Schmitt, 2020) to separate the temperature from salinity contribution to this slowdown. We find that, in our model, the initial slowdown of the AMOC was driven by temperature and that salinity takes the lead for the termination of the circulation.
Jonas Van Breedam, Philippe Huybrechts, and Michel Crucifix
Clim. Past, 19, 2551–2568, https://doi.org/10.5194/cp-19-2551-2023, https://doi.org/10.5194/cp-19-2551-2023, 2023
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We investigated the different boundary conditions to allow ice sheet growth and ice sheet decline of the Antarctic ice sheet when it appeared ∼38–34 Myr ago. The thresholds for ice sheet growth and decline differ because of the different climatological conditions above an ice sheet (higher elevation and higher albedo) compared to a bare topography. We found that the ice–albedo feedback and the isostasy feedback respectively ease and delay the transition from a deglacial to glacial state.
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
Short summary
Short summary
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.
Mikhail Y. Verbitsky and Michel Crucifix
Clim. Past, 19, 1793–1803, https://doi.org/10.5194/cp-19-1793-2023, https://doi.org/10.5194/cp-19-1793-2023, 2023
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Are phenomenological dynamical paleoclimate models physically similar to Nature? We demonstrated that though they may be very accurate in reproducing empirical time series, this is not sufficient to claim physical similarity with Nature until similarity parameters are considered. We suggest that the diagnostics of physical similarity should become a standard procedure before a phenomenological model can be utilized for interpretations of historical records or future predictions.
Marina Martínez Montero, Michel Crucifix, Victor Couplet, Nuria Brede, and Nicola Botta
Geosci. Model Dev., 15, 8059–8084, https://doi.org/10.5194/gmd-15-8059-2022, https://doi.org/10.5194/gmd-15-8059-2022, 2022
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We present SURFER, a lightweight model that links CO2 emissions and geoengineering to ocean acidification and sea level rise from glaciers, ocean thermal expansion and Greenland and Antarctic ice sheets. The ice sheet module adequately describes the tipping points of both Greenland and Antarctica. SURFER is understandable, fast, accurate up to several thousands of years, capable of emulating results obtained by state of the art models and well suited for policy analyses.
Jonas Van Breedam, Philippe Huybrechts, and Michel Crucifix
Geosci. Model Dev., 14, 6373–6401, https://doi.org/10.5194/gmd-14-6373-2021, https://doi.org/10.5194/gmd-14-6373-2021, 2021
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Ice sheets are an important component of the climate system and interact with the atmosphere through albedo variations and changes in the surface height. On very long timescales, it is impossible to directly couple ice sheet models with climate models and other techniques have to be used. Here we present a novel coupling method between ice sheets and the atmosphere by making use of an emulator to simulate ice sheet–climate interactions for several million years.
Mikhail Y. Verbitsky and Michel Crucifix
Earth Syst. Dynam., 12, 63–67, https://doi.org/10.5194/esd-12-63-2021, https://doi.org/10.5194/esd-12-63-2021, 2021
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We demonstrate here that a single physical phenomenon, specifically, a naturally changing balance between intensities of temperature advection and diffusion in the viscous ice media, may influence the entire spectrum of the Pleistocene variability from orbital to millennial timescales.
Mikhail Y. Verbitsky and Michel Crucifix
Earth Syst. Dynam., 11, 281–289, https://doi.org/10.5194/esd-11-281-2020, https://doi.org/10.5194/esd-11-281-2020, 2020
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Using the central theorem of dimensional analysis, the π theorem, we show that the relationship between the amplitude and duration of glacial cycles is governed by a property of scale invariance that does not depend on the physical nature of the underlying positive and negative feedbacks incorporated by the system. It thus turns out to be one of the most fundamental properties of the Pleistocene climate.
Mikhail Y. Verbitsky, Michel Crucifix, and Dmitry M. Volobuev
Earth Syst. Dynam., 10, 257–260, https://doi.org/10.5194/esd-10-257-2019, https://doi.org/10.5194/esd-10-257-2019, 2019
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We demonstrate here that nonlinear character of ice sheet dynamics, which was derived naturally from the conservation laws, is an effective means for propagating high-frequency forcing upscale.
Mikhail Y. Verbitsky, Michel Crucifix, and Dmitry M. Volobuev
Earth Syst. Dynam., 9, 1025–1043, https://doi.org/10.5194/esd-9-1025-2018, https://doi.org/10.5194/esd-9-1025-2018, 2018
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Using a dynamical climate model purely reduced from the conservation laws of ice-moving media, we show that ice-sheet physics coupled with a linear climate temperature feedback conceal enough dynamics to satisfactorily explain the system response over the full Pleistocene. There is no need, a priori, to call for a nonlinear response of, for example, the carbon cycle.
Masa Kageyama, Pascale Braconnot, Sandy P. Harrison, Alan M. Haywood, Johann H. Jungclaus, Bette L. Otto-Bliesner, Jean-Yves Peterschmitt, Ayako Abe-Ouchi, Samuel Albani, Patrick J. Bartlein, Chris Brierley, Michel Crucifix, Aisling Dolan, Laura Fernandez-Donado, Hubertus Fischer, Peter O. Hopcroft, Ruza F. Ivanovic, Fabrice Lambert, Daniel J. Lunt, Natalie M. Mahowald, W. Richard Peltier, Steven J. Phipps, Didier M. Roche, Gavin A. Schmidt, Lev Tarasov, Paul J. Valdes, Qiong Zhang, and Tianjun Zhou
Geosci. Model Dev., 11, 1033–1057, https://doi.org/10.5194/gmd-11-1033-2018, https://doi.org/10.5194/gmd-11-1033-2018, 2018
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The Paleoclimate Modelling Intercomparison Project (PMIP) takes advantage of the existence of past climate states radically different from the recent past to test climate models used for climate projections and to better understand these climates. This paper describes the PMIP contribution to CMIP6 (Coupled Model Intercomparison Project, 6th phase) and possible analyses based on PMIP results, as well as on other CMIP6 projects.
Guillaume Lenoir and Michel Crucifix
Nonlin. Processes Geophys., 25, 145–173, https://doi.org/10.5194/npg-25-145-2018, https://doi.org/10.5194/npg-25-145-2018, 2018
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We develop a general framework for the frequency analysis of irregularly sampled time series. We also design a test of significance against a general background noise which encompasses the Gaussian white or red noise. Our results generalize and unify methods developed in the fields of geosciences, engineering, astronomy and astrophysics. All the analysis tools presented in this paper are available to the reader in the Python package WAVEPAL.
Guillaume Lenoir and Michel Crucifix
Nonlin. Processes Geophys., 25, 175–200, https://doi.org/10.5194/npg-25-175-2018, https://doi.org/10.5194/npg-25-175-2018, 2018
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There is so far no general framework for handling the continuous wavelet transform when the time sampling is irregular. Here we provide such a framework with the Morlet wavelet, based on the results of part I of this study. We also design a test of significance against a general background noise which encompasses the Gaussian white or red noise. All the analysis tools presented in this article are available to the reader in the Python package WAVEPAL.
Natalie S. Lord, Michel Crucifix, Dan J. Lunt, Mike C. Thorne, Nabila Bounceur, Harry Dowsett, Charlotte L. O'Brien, and Andy Ridgwell
Clim. Past, 13, 1539–1571, https://doi.org/10.5194/cp-13-1539-2017, https://doi.org/10.5194/cp-13-1539-2017, 2017
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We present projections of long-term changes in climate, produced using a statistical emulator based on climate data from a state-of-the-art climate model. We use the emulator to model changes in temperature and precipitation over the late Pliocene (3.3–2.8 million years before present) and the next 200 thousand years. The impact of the Earth's orbit and the atmospheric carbon dioxide concentration on climate is assessed, and the data for the late Pliocene are compared to proxy temperature data.
Paul J. Valdes, Edward Armstrong, Marcus P. S. Badger, Catherine D. Bradshaw, Fran Bragg, Michel Crucifix, Taraka Davies-Barnard, Jonathan J. Day, Alex Farnsworth, Chris Gordon, Peter O. Hopcroft, Alan T. Kennedy, Natalie S. Lord, Dan J. Lunt, Alice Marzocchi, Louise M. Parry, Vicky Pope, William H. G. Roberts, Emma J. Stone, Gregory J. L. Tourte, and Jonny H. T. Williams
Geosci. Model Dev., 10, 3715–3743, https://doi.org/10.5194/gmd-10-3715-2017, https://doi.org/10.5194/gmd-10-3715-2017, 2017
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In this paper we describe the family of climate models used by the BRIDGE research group at the University of Bristol as well as by various other institutions. These models are based on the UK Met Office HadCM3 models and here we describe the various modifications which have been made as well as the key features of a number of configurations in use.
N. Bounceur, M. Crucifix, and R. D. Wilkinson
Earth Syst. Dynam., 6, 205–224, https://doi.org/10.5194/esd-6-205-2015, https://doi.org/10.5194/esd-6-205-2015, 2015
P. A. Araya-Melo, M. Crucifix, and N. Bounceur
Clim. Past, 11, 45–61, https://doi.org/10.5194/cp-11-45-2015, https://doi.org/10.5194/cp-11-45-2015, 2015
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By using a statistical tool termed emulator, we study the sensitivity of the Indian monsoon during the the Pleistocene. The originality of the present work is to consider, as inputs, several elements of the climate forcing that have varied in the past, and then use the emulator as a method to quantify the link between forcing variability and climate variability. The methodology described here may naturally be applied to other regions of interest.
Q. Z. Yin, U. K. Singh, A. Berger, Z. T. Guo, and M. Crucifix
Clim. Past, 10, 1645–1657, https://doi.org/10.5194/cp-10-1645-2014, https://doi.org/10.5194/cp-10-1645-2014, 2014
M. N. A. Maris, B. de Boer, S. R. M. Ligtenberg, M. Crucifix, W. J. van de Berg, and J. Oerlemans
The Cryosphere, 8, 1347–1360, https://doi.org/10.5194/tc-8-1347-2014, https://doi.org/10.5194/tc-8-1347-2014, 2014
M. Crucifix
Clim. Past, 9, 2253–2267, https://doi.org/10.5194/cp-9-2253-2013, https://doi.org/10.5194/cp-9-2253-2013, 2013
Related subject area
Subject: Bifurcation, dynamical systems, chaos, phase transition, nonlinear waves, pattern formation | Topic: Climate, atmosphere, ocean, hydrology, cryosphere, biosphere | Techniques: Simulation
A robust numerical method for the generation and propagation of periodic finite-amplitude internal waves in natural waters using high-accuracy simulations
On the hydrostatic approximation in rotating stratified flow
The role of time-varying external factors in the intensification of tropical cyclones
Transformation of internal solitary waves at the edge of ice cover
A new approach to understanding fluid mixing in process-study models of stratified fluids
Aggregation of slightly buoyant microplastics in 3D vortex flows
An approach for projecting the timing of abrupt winter Arctic sea ice loss
On the interaction of stochastic forcing and regime dynamics
Estimate of energy loss from internal solitary waves breaking on slopes
The effect of strong shear on internal solitary-like waves
Enhanced diapycnal mixing with polarity-reversing internal solitary waves revealed by seismic reflection data
Effects of upwelling duration and phytoplankton growth regime on dissolved-oxygen levels in an idealized Iberian Peninsula upwelling system
Pierre Lloret, Peter J. Diamessis, Marek Stastna, and Greg N. Thomsen
Nonlin. Processes Geophys., 31, 515–533, https://doi.org/10.5194/npg-31-515-2024, https://doi.org/10.5194/npg-31-515-2024, 2024
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This study presents a new approach to simulating 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 predict natural phenomena and their effects on environments like swash zones, where waves meet the shore.
Achim Wirth
EGUsphere, https://doi.org/10.5194/egusphere-2024-3307, https://doi.org/10.5194/egusphere-2024-3307, 2024
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The hydrostatic approximation is the basis of most simulations of ocean and climate dynamics. It is here evaluated by using a projection method in the 4D Fourier space. The evaluation is analytic.
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.
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%.
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.
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.
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.
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.
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.
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.
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.
Cited articles
Ashwin, P., Wieczorek, S., Vitolo, R., and Cox, P.: Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system, Philos. T. Roy. Soc. A, 370, 1166–1184, https://doi.org/10.1098/rsta.2011.0306, 2012. a
Bastiaansen, R. and Doelman, A.: The dynamics of disappearing pulses in a singularly perturbed reaction–diffusion system with parameters that vary in time and space, Physica D, 388, 45–72, https://doi.org/10.1016/j.physd.2018.09.003, 2019. 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, b, c
Chen, Y., Kolokolnikov, T., Tzou, J., and Gai, C.: Patterned vegetation, tipping points, and the rate of climate change, Eur. J. Appl. Math., 26, 945–958, https://doi.org/10.1017/S0956792515000261, 2015. a, b
Deblauwe, V., Barbier, N., Couteron, P., Lejeune, O., and Bogaert, J.: The global biogeography of semi-arid periodic vegetation patterns, Global Ecol. Biogeogr., 17, 715–723, https://doi.org/10.1111/j.1466-8238.2008.00413.x, 2008. a
Dijkstra, H. A.: Vegetation pattern formation in a semi-arid climate, Int. J. Bifurcat. Chaos., 21, 3497–3509, https://doi.org/10.1142/S0218127411030696, 2011. a
Gilad, E., von Hardenberg, J., Provenzale, A., Shachak, M., and Meron, E.: A mathematical model of plants as ecosystem engineers, J. Theor. Biol., 244, 680–691, https://doi.org/10.1016/j.jtbi.2006.08.006, 2007. a
Hastings, A., Abbott, K. C., Cuddington, K., Francis, T., Gellner, G., Lai, Y. C., Morozov, A., Petrovskii, S., Scranton, K., and Zeeman, M. L.: Transient phenomena in ecology, Science, 361, eaat6412, https://doi.org/10.1126/science.aat6412, 2018. a
Kéfi, S., Domínguez-García, V., Donohue, I., Fontaine, C., Thébault, E., and Dakos, V.: Advancing our understanding of ecological stability, Ecol. Lett., 22, 1349–1356, https://doi.org/10.1111/ele.13340, 2019. a, b
Klausmeier, C. A.: Regular and irregular patterns in semiarid vegetation, Science, 284, 1826–1828, https://doi.org/10.1126/science.284.5421.1826, 1999. a, b
Lefever, R. and Lejeune, O.: On the origin of tiger bush, B. Math. Biol., 59, 263–294, https://doi.org/10.1007/BF02462004, 1997. a
Meron, E.: Nonlinear Physics of Ecosystems, CRC Press, https://doi.org/10.1201/b18360, 2015. a
Morozov, A., Abbott, K., Cuddington, K., Francis, T., Gellner, G., Hastings, A., Lai, Y. C., Petrovskii, S., Scranton, K., and Zeeman, M. L.: Long transients in ecology: theory and applications, Phys. Life Rev., 32, 1–40, https://doi.org/10.1016/j.plrev.2019.09.004, 2020. a
Murray, J. D.: Mathematical Biology II: Spatial Models and Biomedical Applications, in: 3rd Edn., Springer, New York, NY, ISBN 978-1-4757-7870-0, https://doi.org/10.1007/b98869, 2003. a
Rietkerk, M. and van de Koppel, J.: Regular pattern formation in real ecosystems, Trends Ecol. Evol., 23, 169–175, https://doi.org/10.1016/j.tree.2007.10.013, 2008. a
Sherratt, J. A.: History-dependent patterns of whole ecosystems, Ecol. Complex., 14, 8–20, https://doi.org/10.1016/j.ecocom.2012.12.002, 2013. a
Siteur, K., Siero, E., Eppinga, M. B., Rademacher, J. D., 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, b
Siteur, K., Liu, Q.-X., Rottschäfer, V., van der Heide, T., Rietkerk, M., Doelman, A., Boström, C., and van de Koppel, J.: Phase-separation physics underlies new theory for the resilience of patchy ecosystems, P. Natl. Acad. Sci. USA, 120, 2017, https://doi.org/10.1073/pnas.2202683120, 2023. a
van de Koppel, J., Rietkerk, M., Dankers, N., and Herman, P. M.: Scale-dependent feedback and regular spatial patterns in young mussel beds, Am. Nat., 165, https://doi.org/10.1086/428362, 2005. a
Vanderveken, L.: Slvanderveken/Riet_Stoch: v1.0.0 (v1.0.0), Zenodo [code], https://doi.org/10.5281/zenodo.13739304, 2024. a
Vanderveken, L., Martínez Montero, M., and Crucifix, M.: Existence and influence of mixed states in a model of vegetation patterns, Nonlin. Processes Geophys., 30, 585–599, https://doi.org/10.5194/npg-30-585-2023, 2023. a, b, c
von Hardenberg, J., Meron, E., Shachak, M., and Zarmi, Y.: Diversity of vegetation patterns and desertification, Phys. Rev. Lett., 87, 198101-1–198101-4, https://doi.org/10.1103/PhysRevLett.87.198101, 2001. a
Zelnik, Y. R., 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, b
Short summary
Vegetation patterns in semi-arid regions arise from interactions between plants and environmental factors. This study uses a numerical model to explore how vegetation responds to changes in rainfall and random disturbances. We identify key timescales that influence resilience, showing that ecosystems rely on both stable and unstable states to adapt. These findings offer insights into the resilience mechanisms that help ecosystems maintain stability under environmental stress.
Vegetation patterns in semi-arid regions arise from interactions between plants and...
