Articles | Volume 29, issue 1
https://doi.org/10.5194/npg-29-93-2022
© Author(s) 2022. 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-29-93-2022
© Author(s) 2022. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Fractional relaxation noises, motions and the fractional energy balance equation
Physics, McGill University, 3600 University St., Montreal, Que. H3A 2T8, Canada
Invited contribution by Shaun Lovejoy, recipient of the EGU Lewis Fry Richardson Medal 2019.
Related authors
Nicolás Acuña Reyes, Elwin van't Wout, Shaun Lovejoy, and Fabrice Lambert
Clim. Past, 20, 1579–1594, https://doi.org/10.5194/cp-20-1579-2024, https://doi.org/10.5194/cp-20-1579-2024, 2024
Short summary
Short summary
This study employs Haar fluctuations to analyse global atmospheric variability over the Last Glacial Cycle, revealing a latitudinal dependency in the transition from macroweather to climate regimes. Findings indicate faster synchronisation between poles and lower latitudes, supporting the pivotal role of poles as climate change drivers.
Shaun Lovejoy
Nonlin. Processes Geophys., 30, 311–374, https://doi.org/10.5194/npg-30-311-2023, https://doi.org/10.5194/npg-30-311-2023, 2023
Short summary
Short summary
How big is a cloud?and
How long does the weather last?require scaling to answer. We review the advances in scaling that have occurred over the last 4 decades: (a) intermittency (multifractality) and (b) stratified and rotating scaling notions (generalized scale invariance). Although scaling theory and the data are now voluminous, atmospheric phenomena are too often viewed through an outdated scalebound lens, and turbulence remains confined to isotropic theories of little relevance.
Roman Procyk, Shaun Lovejoy, and Raphael Hébert
Earth Syst. Dynam., 13, 81–107, https://doi.org/10.5194/esd-13-81-2022, https://doi.org/10.5194/esd-13-81-2022, 2022
Short summary
Short summary
This paper presents a new class of energy balance model that accounts for the long memory within the Earth's energy storage. The model is calibrated on instrumental temperature records and the historical energy budget of the Earth using an error model predicted by the model itself. Our equilibrium climate sensitivity and future temperature projection estimates are consistent with those estimated by complex climate models.
Shaun Lovejoy
Earth Syst. Dynam., 12, 469–487, https://doi.org/10.5194/esd-12-469-2021, https://doi.org/10.5194/esd-12-469-2021, 2021
Short summary
Short summary
Monthly scale, seasonal-scale, and decadal-scale modeling of the atmosphere is possible using the principle of energy balance. Yet the scope of classical approaches is limited because they do not adequately deal with energy storage in the Earth system. We show that the introduction of a vertical coordinate implies that the storage has a huge memory. This memory can be used for macroweather (long-range) forecasts and climate projections.
Shaun Lovejoy
Earth Syst. Dynam., 12, 489–511, https://doi.org/10.5194/esd-12-489-2021, https://doi.org/10.5194/esd-12-489-2021, 2021
Short summary
Short summary
Radiant energy is exchanged between the Earth's surface and outer space. Some of the local imbalances are stored in the subsurface, and some are transported horizontally. In Part 1 I showed how – in a horizontally homogeneous Earth – these classical approaches imply long-memory storage useful for seasonal forecasting and multidecadal projections. In this Part 2, I show how to apply these results to the heterogeneous real Earth.
Shaun Lovejoy and Fabrice Lambert
Clim. Past, 15, 1999–2017, https://doi.org/10.5194/cp-15-1999-2019, https://doi.org/10.5194/cp-15-1999-2019, 2019
Short summary
Short summary
We analyze the statistical properties of the eight past glacial–interglacial cycles as well as subsections of a generic glacial cycle using the high-resolution dust flux dataset from the Antarctic EPICA Dome C ice core. We show that the high southern latitude climate during glacial maxima, interglacial, and glacial inception is generally more stable but more drought-prone than during mid-glacial conditions.
Shaun Lovejoy and Fabrice Lambert
Clim. Past Discuss., https://doi.org/10.5194/cp-2018-110, https://doi.org/10.5194/cp-2018-110, 2018
Manuscript not accepted for further review
Short summary
Short summary
The Holocene has been strikingly long and stable when compared to earlier interglacials, and some have argued that the Holocene's exceptional stability permitted the development of agriculture and the spread of civilization. We characterize the past 800 000 years using a high resolution dust record from an Antarctic ice core. We find that although the Holocene is particularly stable when compared to other interglacials, it is not an outlier and other factors may have kickstarted civilization.
Shaun Lovejoy and Costas Varotsos
Earth Syst. Dynam., 7, 133–150, https://doi.org/10.5194/esd-7-133-2016, https://doi.org/10.5194/esd-7-133-2016, 2016
Short summary
Short summary
We compare the statistical properties of solar, volcanic and combined forcings over the range from 1 to 1000 years to see over which scale ranges they additively combine, a prediction of linear response. The main findings are (a) that the variability in the Zebiac–Cane model and GCMs are too weak at centennial and longer scales; (b) for longer than ≈ 50 years, the forcings combine subadditively; and (c) at shorter scales, strong (intermittency, e.g. volcanic) forcings are nonlinear.
F. Landais, F. Schmidt, and S. Lovejoy
Nonlin. Processes Geophys., 22, 713–722, https://doi.org/10.5194/npg-22-713-2015, https://doi.org/10.5194/npg-22-713-2015, 2015
Short summary
Short summary
In the present study, we investigate the scaling properties of the topography of Mars. Planetary topographic fields are well known to exhibit (mono)fractal behavior. Indeed, fractal formalism is efficient in reproducing the variability observed in topography. Our results suggest a multifractal behavior from the planetary scale down to 10 km. From 10 km to 300 m, the topography seems to be simple monofractal.
S. Lovejoy, L. del Rio Amador, and R. Hébert
Earth Syst. Dynam., 6, 637–658, https://doi.org/10.5194/esd-6-637-2015, https://doi.org/10.5194/esd-6-637-2015, 2015
Short summary
Short summary
Numerical climate models forecast the weather well beyond the deterministic limit. In this “macroweather” regime, they are random number generators. Stochastic models can have more realistic noises and can be forced to converge to the real-world climate. Existing stochastic models do not exploit the very long atmospheric and oceanic memories. With skill up to decades, our new ScaLIng Macroweather Model (SLIMM) exploits this to make forecasts more accurate than GCMs.
C. A. Varotsos, S. Lovejoy, N. V. Sarlis, C. G. Tzanis, and M. N. Efstathiou
Atmos. Chem. Phys., 15, 7301–7306, https://doi.org/10.5194/acp-15-7301-2015, https://doi.org/10.5194/acp-15-7301-2015, 2015
Short summary
Short summary
Varotsos et al. (Theor. Appl. Climatol., 114, 725–727, 2013) found that the solar ultraviolet (UV) wavelengths exhibit 1/f-type power-law correlations. In this study, we show that the residues of the spectral solar incident flux with respect to the Planck law over a wider range of wavelengths (i.e. UV-visible) have a scaling regime too.
J. Pinel and S. Lovejoy
Atmos. Chem. Phys., 14, 3195–3210, https://doi.org/10.5194/acp-14-3195-2014, https://doi.org/10.5194/acp-14-3195-2014, 2014
G. A. Schmidt, J. D. Annan, P. J. Bartlein, B. I. Cook, E. Guilyardi, J. C. Hargreaves, S. P. Harrison, M. Kageyama, A. N. LeGrande, B. Konecky, S. Lovejoy, M. E. Mann, V. Masson-Delmotte, C. Risi, D. Thompson, A. Timmermann, L.-B. Tremblay, and P. Yiou
Clim. Past, 10, 221–250, https://doi.org/10.5194/cp-10-221-2014, https://doi.org/10.5194/cp-10-221-2014, 2014
S. Lovejoy, D. Schertzer, and D. Varon
Earth Syst. Dynam., 4, 439–454, https://doi.org/10.5194/esd-4-439-2013, https://doi.org/10.5194/esd-4-439-2013, 2013
A. Gires, I. Tchiguirinskaia, D. Schertzer, and S. Lovejoy
Nonlin. Processes Geophys., 20, 343–356, https://doi.org/10.5194/npg-20-343-2013, https://doi.org/10.5194/npg-20-343-2013, 2013
Nicolás Acuña Reyes, Elwin van't Wout, Shaun Lovejoy, and Fabrice Lambert
Clim. Past, 20, 1579–1594, https://doi.org/10.5194/cp-20-1579-2024, https://doi.org/10.5194/cp-20-1579-2024, 2024
Short summary
Short summary
This study employs Haar fluctuations to analyse global atmospheric variability over the Last Glacial Cycle, revealing a latitudinal dependency in the transition from macroweather to climate regimes. Findings indicate faster synchronisation between poles and lower latitudes, supporting the pivotal role of poles as climate change drivers.
Shaun Lovejoy
Nonlin. Processes Geophys., 30, 311–374, https://doi.org/10.5194/npg-30-311-2023, https://doi.org/10.5194/npg-30-311-2023, 2023
Short summary
Short summary
How big is a cloud?and
How long does the weather last?require scaling to answer. We review the advances in scaling that have occurred over the last 4 decades: (a) intermittency (multifractality) and (b) stratified and rotating scaling notions (generalized scale invariance). Although scaling theory and the data are now voluminous, atmospheric phenomena are too often viewed through an outdated scalebound lens, and turbulence remains confined to isotropic theories of little relevance.
Roman Procyk, Shaun Lovejoy, and Raphael Hébert
Earth Syst. Dynam., 13, 81–107, https://doi.org/10.5194/esd-13-81-2022, https://doi.org/10.5194/esd-13-81-2022, 2022
Short summary
Short summary
This paper presents a new class of energy balance model that accounts for the long memory within the Earth's energy storage. The model is calibrated on instrumental temperature records and the historical energy budget of the Earth using an error model predicted by the model itself. Our equilibrium climate sensitivity and future temperature projection estimates are consistent with those estimated by complex climate models.
Shaun Lovejoy
Earth Syst. Dynam., 12, 469–487, https://doi.org/10.5194/esd-12-469-2021, https://doi.org/10.5194/esd-12-469-2021, 2021
Short summary
Short summary
Monthly scale, seasonal-scale, and decadal-scale modeling of the atmosphere is possible using the principle of energy balance. Yet the scope of classical approaches is limited because they do not adequately deal with energy storage in the Earth system. We show that the introduction of a vertical coordinate implies that the storage has a huge memory. This memory can be used for macroweather (long-range) forecasts and climate projections.
Shaun Lovejoy
Earth Syst. Dynam., 12, 489–511, https://doi.org/10.5194/esd-12-489-2021, https://doi.org/10.5194/esd-12-489-2021, 2021
Short summary
Short summary
Radiant energy is exchanged between the Earth's surface and outer space. Some of the local imbalances are stored in the subsurface, and some are transported horizontally. In Part 1 I showed how – in a horizontally homogeneous Earth – these classical approaches imply long-memory storage useful for seasonal forecasting and multidecadal projections. In this Part 2, I show how to apply these results to the heterogeneous real Earth.
Shaun Lovejoy and Fabrice Lambert
Clim. Past, 15, 1999–2017, https://doi.org/10.5194/cp-15-1999-2019, https://doi.org/10.5194/cp-15-1999-2019, 2019
Short summary
Short summary
We analyze the statistical properties of the eight past glacial–interglacial cycles as well as subsections of a generic glacial cycle using the high-resolution dust flux dataset from the Antarctic EPICA Dome C ice core. We show that the high southern latitude climate during glacial maxima, interglacial, and glacial inception is generally more stable but more drought-prone than during mid-glacial conditions.
Shaun Lovejoy and Fabrice Lambert
Clim. Past Discuss., https://doi.org/10.5194/cp-2018-110, https://doi.org/10.5194/cp-2018-110, 2018
Manuscript not accepted for further review
Short summary
Short summary
The Holocene has been strikingly long and stable when compared to earlier interglacials, and some have argued that the Holocene's exceptional stability permitted the development of agriculture and the spread of civilization. We characterize the past 800 000 years using a high resolution dust record from an Antarctic ice core. We find that although the Holocene is particularly stable when compared to other interglacials, it is not an outlier and other factors may have kickstarted civilization.
Shaun Lovejoy and Costas Varotsos
Earth Syst. Dynam., 7, 133–150, https://doi.org/10.5194/esd-7-133-2016, https://doi.org/10.5194/esd-7-133-2016, 2016
Short summary
Short summary
We compare the statistical properties of solar, volcanic and combined forcings over the range from 1 to 1000 years to see over which scale ranges they additively combine, a prediction of linear response. The main findings are (a) that the variability in the Zebiac–Cane model and GCMs are too weak at centennial and longer scales; (b) for longer than ≈ 50 years, the forcings combine subadditively; and (c) at shorter scales, strong (intermittency, e.g. volcanic) forcings are nonlinear.
F. Landais, F. Schmidt, and S. Lovejoy
Nonlin. Processes Geophys., 22, 713–722, https://doi.org/10.5194/npg-22-713-2015, https://doi.org/10.5194/npg-22-713-2015, 2015
Short summary
Short summary
In the present study, we investigate the scaling properties of the topography of Mars. Planetary topographic fields are well known to exhibit (mono)fractal behavior. Indeed, fractal formalism is efficient in reproducing the variability observed in topography. Our results suggest a multifractal behavior from the planetary scale down to 10 km. From 10 km to 300 m, the topography seems to be simple monofractal.
S. Lovejoy, L. del Rio Amador, and R. Hébert
Earth Syst. Dynam., 6, 637–658, https://doi.org/10.5194/esd-6-637-2015, https://doi.org/10.5194/esd-6-637-2015, 2015
Short summary
Short summary
Numerical climate models forecast the weather well beyond the deterministic limit. In this “macroweather” regime, they are random number generators. Stochastic models can have more realistic noises and can be forced to converge to the real-world climate. Existing stochastic models do not exploit the very long atmospheric and oceanic memories. With skill up to decades, our new ScaLIng Macroweather Model (SLIMM) exploits this to make forecasts more accurate than GCMs.
C. A. Varotsos, S. Lovejoy, N. V. Sarlis, C. G. Tzanis, and M. N. Efstathiou
Atmos. Chem. Phys., 15, 7301–7306, https://doi.org/10.5194/acp-15-7301-2015, https://doi.org/10.5194/acp-15-7301-2015, 2015
Short summary
Short summary
Varotsos et al. (Theor. Appl. Climatol., 114, 725–727, 2013) found that the solar ultraviolet (UV) wavelengths exhibit 1/f-type power-law correlations. In this study, we show that the residues of the spectral solar incident flux with respect to the Planck law over a wider range of wavelengths (i.e. UV-visible) have a scaling regime too.
J. Pinel and S. Lovejoy
Atmos. Chem. Phys., 14, 3195–3210, https://doi.org/10.5194/acp-14-3195-2014, https://doi.org/10.5194/acp-14-3195-2014, 2014
G. A. Schmidt, J. D. Annan, P. J. Bartlein, B. I. Cook, E. Guilyardi, J. C. Hargreaves, S. P. Harrison, M. Kageyama, A. N. LeGrande, B. Konecky, S. Lovejoy, M. E. Mann, V. Masson-Delmotte, C. Risi, D. Thompson, A. Timmermann, L.-B. Tremblay, and P. Yiou
Clim. Past, 10, 221–250, https://doi.org/10.5194/cp-10-221-2014, https://doi.org/10.5194/cp-10-221-2014, 2014
S. Lovejoy, D. Schertzer, and D. Varon
Earth Syst. Dynam., 4, 439–454, https://doi.org/10.5194/esd-4-439-2013, https://doi.org/10.5194/esd-4-439-2013, 2013
A. Gires, I. Tchiguirinskaia, D. Schertzer, and S. Lovejoy
Nonlin. Processes Geophys., 20, 343–356, https://doi.org/10.5194/npg-20-343-2013, https://doi.org/10.5194/npg-20-343-2013, 2013
Related subject area
Subject: Scaling, multifractals, turbulence, complex systems, self-organized criticality | Topic: Climate, atmosphere, ocean, hydrology, cryosphere, biosphere | Techniques: Simulation
Clustering of settling microswimmers in turbulence
Phytoplankton retention mechanisms in estuaries: a case study of the Elbe estuary
On dissipation time scales of the basic second-order moments: the effect on the Energy and Flux-Budget (EFB) turbulence closure for stably stratified turbulence
Comparing estimation techniques for temporal scaling in palaeoclimate time series
Jingran Qiu, Zhiwen Cui, Eric Climent, and Lihao Zhao
Nonlin. Processes Geophys., 31, 229–236, https://doi.org/10.5194/npg-31-229-2024, https://doi.org/10.5194/npg-31-229-2024, 2024
Short summary
Short summary
The swimming of settling microswimmers in a fluid flow is found to induce a gyrotactic torque, causing them to swim against gravity. A Lagrangian model of the swimmer under this effect is used in the analysis of small-scale clustering in turbulence. The intensity and location of clustering under this swimming-induced gyrotactic torque are found to depend on not only the swimming velocity but also the settling speed, indicating the importance of the settling effect on gyrotaxis.
Laurin Steidle and Ross Vennell
Nonlin. Processes Geophys., 31, 151–164, https://doi.org/10.5194/npg-31-151-2024, https://doi.org/10.5194/npg-31-151-2024, 2024
Short summary
Short summary
Phytoplankton are key in estuaries, as they form the ecosystem's base. Despite being washed out by river flow and facing a large range of different salinities, they persist. Our Lagrangian simulation of the Elbe estuary shows that buoyancy helps them to be retained. Riverbanks and tidal flats offer refuges from strong currents. Our findings emphasize the need for careful ecosystem management in estuaries.
Evgeny Kadantsev, Evgeny Mortikov, Andrey Glazunov, Nathan Kleeorin, and Igor Rogachevskii
EGUsphere, https://doi.org/10.5194/egusphere-2023-3164, https://doi.org/10.5194/egusphere-2023-3164, 2024
Short summary
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Our study investigates how turbulence behaves in stable conditions using direct numerical simulations. We found that rethinking how energy dissipates in these situations is crucial. By revising existing models, we uncovered limitations in understanding how temperature is transported vertically in very stable conditions. We focused on how turbulence works in extreme stability offering new insights that could improve our understanding of natural phenomena affected by stable atmospheric conditions.
Raphaël Hébert, Kira Rehfeld, and Thomas Laepple
Nonlin. Processes Geophys., 28, 311–328, https://doi.org/10.5194/npg-28-311-2021, https://doi.org/10.5194/npg-28-311-2021, 2021
Short summary
Short summary
Paleoclimate proxy data are essential for broadening our understanding of climate variability. There remain, however, challenges for traditional methods of variability analysis to be applied to such data, which are usually irregular. We perform a comparative analysis of different methods of scaling analysis, which provide variability estimates as a function of timescales, applied to irregular paleoclimate proxy data.
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Short summary
The difference between the energy that the Earth receives from the Sun and the energy it emits as black-body radiation is stored in a scaling hierarchy of structures in the ocean, soil and hydrosphere. The simplest scaling storage model leads to the fractional energy balance equation (FEBE). We examine the statistical properties of FEBE when it is driven by random fluctuations. In this paper, we explore the statistical properties of this mathematically simple yet neglected equation.
The difference between the energy that the Earth receives from the Sun and the energy it emits...