Articles | Volume 32, issue 3
https://doi.org/10.5194/npg-32-261-2025
© 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-32-261-2025
© Author(s) 2025. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
On the hydrostatic approximation in rotating stratified flow
LEGI, Univ. Grenoble Alpes, CNRS, 38000 Grenoble, France
Related authors
Sofia Flora, Laura Ursella, and Achim Wirth
EGUsphere, https://doi.org/10.5194/egusphere-2024-3391, https://doi.org/10.5194/egusphere-2024-3391, 2025
Short summary
Short summary
We developed a hierarchy of idealized deterministic-stochastic models to simulate the sea surface currents in the Gulf of Trieste. They include tide-and-wind driven sea surface current components, resolving the slowly varying part of the flow and a stochastic signal, representing the fast-varying small-scale dynamics. The comparison with High Frequency Radar observations shows that the non-Gaussian stochastic model captures the essential dynamics and permits to mimic the observed fat-tailed PDF.
Felipe L. L. Amorim, Julien Le Meur, Achim Wirth, and Vanessa Cardin
Ocean Sci., 20, 463–474, https://doi.org/10.5194/os-20-463-2024, https://doi.org/10.5194/os-20-463-2024, 2024
Short summary
Short summary
Analysis of a high-frequency time series of thermohaline data measured at the EMSO-E2M3A regional facility in the southern Adriatic Pit (SAP) reveals a significant change in the double-diffusive regime in 2017 associated with the intrusion of extremely salty waters into the area, suggesting salt fingering as the dominant regime. The strong heat loss at the surface during this winter allowed deep convection to transport this high-salinity water from the intermediate to deep layers of the pit.
Sofia Flora, Laura Ursella, and Achim Wirth
Nonlin. Processes Geophys., 30, 515–525, https://doi.org/10.5194/npg-30-515-2023, https://doi.org/10.5194/npg-30-515-2023, 2023
Short summary
Short summary
An increasing amount of data allows us to move from low-order moments of fluctuating observations to their PDFs. We found the analytical fat-tailed PDF form (a combination of Gaussian and two-exponential convolutions) for 2 years of sea surface current increments in the Gulf of Trieste, using superstatistics and the maximum-entropy principle twice: on short and longer timescales. The data from different wind regimes follow the same analytical PDF, pointing towards a universal behaviour.
Achim Wirth and Bertrand Chapron
Nonlin. Processes Geophys., 28, 371–378, https://doi.org/10.5194/npg-28-371-2021, https://doi.org/10.5194/npg-28-371-2021, 2021
Short summary
Short summary
In non-equilibrium statistical mechanics, which describes forced-dissipative systems such as air–sea interaction, there is no universal probability density function (pdf). Some such systems have recently been demonstrated to exhibit a symmetry called a fluctuation theorem (FT), which strongly constrains the shape of the pdf. Using satellite data, the mechanical power input to the ocean by air–sea interaction following or not a FT is questioned. A FT is found to apply over specific ocean regions.
Achim Wirth and Florian Lemarié
Earth Syst. Dynam., 12, 689–708, https://doi.org/10.5194/esd-12-689-2021, https://doi.org/10.5194/esd-12-689-2021, 2021
Short summary
Short summary
We show that modern concepts of non-equilibrium statistical mechanics can be applied to large-scale environmental fluid dynamics, where fluctuations are not thermal but come from turbulence. The work theorems developed by Jarzynski and Crooks are applied to air–sea interaction. Rather than looking at the average values of thermodynamic variables, their probability density functions are considered, which allows us to replace the inequalities of equilibrium statistical mechanics with equalities.
Achim Wirth
Ocean Sci. Discuss., https://doi.org/10.5194/os-2019-128, https://doi.org/10.5194/os-2019-128, 2020
Revised manuscript not accepted
Short summary
Short summary
The input of mechanical power to the ocean due to the surface wind-stress is considered using data from satellites observations. Its dependence on the coarse-graining scale of the atmospheric and oceanic velocity in space and time is determined. The power input is found to increase monotonically with shorter coarse-graining in time. Results show that including the dynamics at scales below a few degrees reduces considerably the power input by air-sea interaction.
Achim Wirth
Nonlin. Processes Geophys., 26, 457–477, https://doi.org/10.5194/npg-26-457-2019, https://doi.org/10.5194/npg-26-457-2019, 2019
Short summary
Short summary
The conspicuous feature of the atmosphere–ocean system is the large difference in the masses of the two media. In this respect there is a strong analogy to Brownian motion, with light and fast molecules colliding with heavy and slow Brownian particles. I apply the tools of non-equilibrium statistical mechanics for studying Brownian motion to air–sea interaction.
Achim Wirth
Geosci. Model Dev. Discuss., https://doi.org/10.5194/gmd-2018-300, https://doi.org/10.5194/gmd-2018-300, 2019
Revised manuscript not accepted
Short summary
Short summary
The dynamics of three local linear models of air-sea-interaction commonly employed in climate or ocean simulations is compared. The models differ by whether or not the ocean velocity is included in the shear calculation applied to the ocean and the atmosphere. Analytic calculations for the models with deterministic and random forcing (white and colored) are presented.The fluctuation-dissipation-relation, the fluctuation-dissipation-theorem and the fluctuation-theorem is discussed.
C. Q. C. Akuetevi and A. Wirth
Ocean Sci., 11, 471–481, https://doi.org/10.5194/os-11-471-2015, https://doi.org/10.5194/os-11-471-2015, 2015
C. Q. C. Akuetevi and A. Wirth
Ocean Sci. Discuss., https://doi.org/10.5194/osd-11-753-2014, https://doi.org/10.5194/osd-11-753-2014, 2014
Revised manuscript not accepted
A. Wirth
Nonlin. Processes Geophys., 20, 25–34, https://doi.org/10.5194/npg-20-25-2013, https://doi.org/10.5194/npg-20-25-2013, 2013
Sofia Flora, Laura Ursella, and Achim Wirth
EGUsphere, https://doi.org/10.5194/egusphere-2024-3391, https://doi.org/10.5194/egusphere-2024-3391, 2025
Short summary
Short summary
We developed a hierarchy of idealized deterministic-stochastic models to simulate the sea surface currents in the Gulf of Trieste. They include tide-and-wind driven sea surface current components, resolving the slowly varying part of the flow and a stochastic signal, representing the fast-varying small-scale dynamics. The comparison with High Frequency Radar observations shows that the non-Gaussian stochastic model captures the essential dynamics and permits to mimic the observed fat-tailed PDF.
Felipe L. L. Amorim, Julien Le Meur, Achim Wirth, and Vanessa Cardin
Ocean Sci., 20, 463–474, https://doi.org/10.5194/os-20-463-2024, https://doi.org/10.5194/os-20-463-2024, 2024
Short summary
Short summary
Analysis of a high-frequency time series of thermohaline data measured at the EMSO-E2M3A regional facility in the southern Adriatic Pit (SAP) reveals a significant change in the double-diffusive regime in 2017 associated with the intrusion of extremely salty waters into the area, suggesting salt fingering as the dominant regime. The strong heat loss at the surface during this winter allowed deep convection to transport this high-salinity water from the intermediate to deep layers of the pit.
Sofia Flora, Laura Ursella, and Achim Wirth
Nonlin. Processes Geophys., 30, 515–525, https://doi.org/10.5194/npg-30-515-2023, https://doi.org/10.5194/npg-30-515-2023, 2023
Short summary
Short summary
An increasing amount of data allows us to move from low-order moments of fluctuating observations to their PDFs. We found the analytical fat-tailed PDF form (a combination of Gaussian and two-exponential convolutions) for 2 years of sea surface current increments in the Gulf of Trieste, using superstatistics and the maximum-entropy principle twice: on short and longer timescales. The data from different wind regimes follow the same analytical PDF, pointing towards a universal behaviour.
Achim Wirth and Bertrand Chapron
Nonlin. Processes Geophys., 28, 371–378, https://doi.org/10.5194/npg-28-371-2021, https://doi.org/10.5194/npg-28-371-2021, 2021
Short summary
Short summary
In non-equilibrium statistical mechanics, which describes forced-dissipative systems such as air–sea interaction, there is no universal probability density function (pdf). Some such systems have recently been demonstrated to exhibit a symmetry called a fluctuation theorem (FT), which strongly constrains the shape of the pdf. Using satellite data, the mechanical power input to the ocean by air–sea interaction following or not a FT is questioned. A FT is found to apply over specific ocean regions.
Achim Wirth and Florian Lemarié
Earth Syst. Dynam., 12, 689–708, https://doi.org/10.5194/esd-12-689-2021, https://doi.org/10.5194/esd-12-689-2021, 2021
Short summary
Short summary
We show that modern concepts of non-equilibrium statistical mechanics can be applied to large-scale environmental fluid dynamics, where fluctuations are not thermal but come from turbulence. The work theorems developed by Jarzynski and Crooks are applied to air–sea interaction. Rather than looking at the average values of thermodynamic variables, their probability density functions are considered, which allows us to replace the inequalities of equilibrium statistical mechanics with equalities.
Achim Wirth
Ocean Sci. Discuss., https://doi.org/10.5194/os-2019-128, https://doi.org/10.5194/os-2019-128, 2020
Revised manuscript not accepted
Short summary
Short summary
The input of mechanical power to the ocean due to the surface wind-stress is considered using data from satellites observations. Its dependence on the coarse-graining scale of the atmospheric and oceanic velocity in space and time is determined. The power input is found to increase monotonically with shorter coarse-graining in time. Results show that including the dynamics at scales below a few degrees reduces considerably the power input by air-sea interaction.
Achim Wirth
Nonlin. Processes Geophys., 26, 457–477, https://doi.org/10.5194/npg-26-457-2019, https://doi.org/10.5194/npg-26-457-2019, 2019
Short summary
Short summary
The conspicuous feature of the atmosphere–ocean system is the large difference in the masses of the two media. In this respect there is a strong analogy to Brownian motion, with light and fast molecules colliding with heavy and slow Brownian particles. I apply the tools of non-equilibrium statistical mechanics for studying Brownian motion to air–sea interaction.
Achim Wirth
Geosci. Model Dev. Discuss., https://doi.org/10.5194/gmd-2018-300, https://doi.org/10.5194/gmd-2018-300, 2019
Revised manuscript not accepted
Short summary
Short summary
The dynamics of three local linear models of air-sea-interaction commonly employed in climate or ocean simulations is compared. The models differ by whether or not the ocean velocity is included in the shear calculation applied to the ocean and the atmosphere. Analytic calculations for the models with deterministic and random forcing (white and colored) are presented.The fluctuation-dissipation-relation, the fluctuation-dissipation-theorem and the fluctuation-theorem is discussed.
C. Q. C. Akuetevi and A. Wirth
Ocean Sci., 11, 471–481, https://doi.org/10.5194/os-11-471-2015, https://doi.org/10.5194/os-11-471-2015, 2015
C. Q. C. Akuetevi and A. Wirth
Ocean Sci. Discuss., https://doi.org/10.5194/osd-11-753-2014, https://doi.org/10.5194/osd-11-753-2014, 2014
Revised manuscript not accepted
A. Wirth
Nonlin. Processes Geophys., 20, 25–34, https://doi.org/10.5194/npg-20-25-2013, https://doi.org/10.5194/npg-20-25-2013, 2013
Related subject area
Subject: Bifurcation, dynamical systems, chaos, phase transition, nonlinear waves, pattern formation | Topic: Climate, atmosphere, ocean, hydrology, cryosphere, biosphere | Techniques: Simulation
Rate-induced transitions and noise-driven resilience in vegetation pattern dynamics
The stochastic skeleton model for the Madden-Julian Oscillation with time-dependent observation-based forcing
A robust numerical method for the generation and propagation of periodic finite-amplitude internal waves in natural waters using high-accuracy simulations
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
Lilian Vanderveken and Michel Crucifix
Nonlin. Processes Geophys., 32, 189–200, https://doi.org/10.5194/npg-32-189-2025, https://doi.org/10.5194/npg-32-189-2025, 2025
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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.
Noemie Ehstand, Reik V. Donner, Cristobal Lopez, Marcelo Barreiro, and Emilio Hernandez-Garcia
EGUsphere, https://doi.org/10.5194/egusphere-2025-343, https://doi.org/10.5194/egusphere-2025-343, 2025
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The Madden-Julian Oscillation (MJO) is a large-scale tropical wave of enhanced and suppressed rainfalls, slowly moving eastward at the equator, influencing the weather and climate globally. We study the MJO using a simplified model designed to capture its large-scale features. We introduce new, more realistic, inputs into the model, show that this enhanced model successfully replicates key characteristics of the MJO, and identify some of its limitations.
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.
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.
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Short summary
The hydrostatic approximation is the basis of most simulations of ocean and climate dynamics. It is evaluated here by using a projection method in the 4D Fourier space. The evaluation is analytic.
The hydrostatic approximation is the basis of most simulations of ocean and climate dynamics. It...