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.
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 Although scaling theory and the data are now voluminous, atmospheric phenomena are too often viewed through an outdated scalebound lens, and turbulence remains confined to isotropic theories of little relevance.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.
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.
Cited articles
Atanackovic, M., Pilipovic, S., Stankovic, B., and Zorica, D.: Fractional Calculus with applications in mechanics: variations and diffusion processes, Wiley, 313 pp., 2014.
Babenko, Y. I.: Heat and Mass Transfer, Khimiya, Leningrad, 1986 (in Russian).
Bender, C. M. and Orszag, S. A.: Advanced mathematical methods for scientists and engineers, Mc Graw Hill, 1978.
Biagini, F., Hu, Y., Øksendal, B., and Zhang, T.: Stochastic Calculus for Fractional Brownian Motion and Applications, Springer-Verlag, https://doi.org/10.1007/978-1-84628-797-8, 2008.
Budyko, M. I.: The effect of solar radiation variations on the climate of the earth, Tellus, 21, 611–619, 1969.
Buizza, R., Miller, M., and Palmer, T. N.: Stochastic representation of model uncertainties in the ECMWF Ensemble Prediction System, Q. J. Roy. Meteor. Soc., 125, 2887–2908, 1999.
Chekroun, M. D., Simonnet, E., and Ghil, M.: Stochastic Climate Dynamics: Random Attractors and Time-dependent Invariant Measures, Physica D, 240, 1685–1700, 2010.
Coffey, W. T., Kalmykov, Y. P., and Titov, S. V.: Characteristic times of anomalous diffusion in a potential, in: Fractional Dynamics: Recent Advances, edited by: Klafter, J., Lim, S., and Metzler, R., World Scientific, 51–76, 2012.
Del Rio Amador, L. and Lovejoy, S.: Predicting the global temperature with the Stochastic Seasonal to Interannual Prediction System (StocSIPS), Clim. Dynam., 53, 4373–4411, https://doi.org/10.1007/s00382-019-04791-4, 2019.
Del Rio Amador, L. and Lovejoy, S.: Using regional scaling for temperature forecasts with the Stochastic Seasonal to Interannual Prediction System (StocSIPS), Clim. Dynam., 57, 727–756, https://doi.org/10.1007/s00382-021-05737-5, 2021a.
Del Rio Amador, L. and Lovejoy, S.: Long-range Forecasting as a Past Value Problem: Untangling Correlations and Causality with scaling, Geophys. Res. Lett., 48, e2020GL092147, 2021b.
Dijkstra, H.: Nonlinear Climate Dynamics, Cambridge University Press, 357 pp., https://doi.org/10.1017/CBO9781139034135, 2013.
Franzke, C. and O'Kane, T. (Eds.): Nonlinear and Stochastic Climate Dynamics, Cambridge University Press, Cambridge, https://doi.org/10.1017/9781316339251, 2017.
Gripenberg, G. and Norros, I.: On the Prediction of Fractional Brownian Motion, J. Appl. Probab., 33, 400–410, 1996.
Hasselmann, K.: Stochastic Climate models, part I: Theory, Tellus, 28, 473–485, 1976.
Hébert, R.: A Scaling Model for the Forced Climate Variability in the Anthropocene, MSc thesis, McGill University, Montreal, 2017.
Hébert, R. and Lovejoy, S.: The runaway Green's function effect: Interactive comment on “Global warming projections derived from an observation-based minimal model” by K. Rypdal, Earth System Dyn. Disc., 6, C944–C953, 2015.
Hébert, R., Lovejoy, S., and Tremblay, B.: An Observation-based Scaling Model for Climate Sensitivity Estimates and Global Projections to 2100, Clim. Dynam., 56, 1105–1129 https://doi.org/10.1007/s00382-020-05521-x, 2021.
Herrmann, R.: Fractional Calculus: an Introduction for Physicists, World Scientific, ISBN: 139789814340243, 2011.
Hilfer, R. (Ed.): Applications of Fractional Calculus in Physics, World Scientific, ISBN: 9810234570, 2000.
Hipel, K. W. and McLeod, A. I.: Time series modelling of water resources and environmental systems, 1st edn., Elsevier, ISBN: 9780080870366, 1994.
IPCC: Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge University Press, Cambridge, ISBN: 9781107661820, 2013.
Jumarie, G.: Stochastic differential equations with fractional Brownian motion inputs, Int. J. Syst. Sci., 24, 1113–1131, 1993.
Karczewska, A. and Lizama, C.: Solutions to stochastic fractional relaxation equations, Phys. Scripta, T136, 7 pp., https://doi.org/10.1088/0031-8949/2009/T136/014030, 2009.
Kou, S. C. and Sunney Xie, X.: Generalized Langevin Equation with Fractional Gaussian Noise: Subdiffusion within a Single Protein Molecule, Phys. Rev. Lett., 93, 4, https://doi.org/10.1103/PhysRevLett.93.180603, 2004.
Lovejoy, S.: What is climate?, EOS, 94, 1–2, 2013.
Lovejoy, S.: Mathematica software for simulation and analysis of scaling and multifractals, Department of Physics, McGill University, http://www.physics.mcgill.ca/~gang/software/doc/mathematicasoftware.17.9.14.nb.zip (last access: 14 February 2022), 2014.
Lovejoy, S.: A voyage through scales, a missing quadrillion and why the climate is not what you expect, Clim. Dynam., 44, 3187–3210, https://doi.org/10.1007/s00382-014-2324-0, 2015a.
Lovejoy, S.: Using scaling for macroweather forecasting including the pause, Geophys. Res. Lett., 42, 7148–7155, https://doi.org/10.1002/2015GL065665, 2015b.
Lovejoy, S.: The spectra, intermittency and extremes of weather, macroweather and climate, Nature Scientific Reports, 8, 1–13, https://doi.org/10.1038/s41598-018-30829-4, 2018.
Lovejoy, S.: Weather, Macroweather and Climate: our random yet predictable atmosphere, Oxford University Press, 334 pp., ISBN: 978-0-19-086421-7, 2019.
Lovejoy, S.: The half-order energy balance equation – Part 1: The homogeneous HEBE and long memories, Earth Syst. Dynam., 12, 469–487, https://doi.org/10.5194/esd-12-469-2021, 2021a.
Lovejoy, S.: The half-order energy balance equation – Part 2: The inhomogeneous HEBE and 2D energy balance models, Earth Syst. Dynam., 12, 489–511, https://doi.org/10.5194/esd-12-489-2021, 2021b.
Lovejoy, S. and Schertzer, D.: The Weather and Climate: Emergent Laws and Multifractal Cascades, Cambridge University Press, 496 pp., ISBN: 978-1-107-01898-3, 2013.
Lovejoy, S., Del Rio Amador, L., and Hébert, R.: Harnessing butterflies: theory and practice of the Stochastic Seasonal to Interannual Prediction System (StocSIPS), in: Nonlinear Advances in Geosciences, edited by: Tsonis, A. A., Springer Nature, 305–355, https://doi.org/10.1007/978-3-319-58895-7_17, 2017.
Lovejoy, S., del Rio Amador, L., and Hébert, R.: The ScaLIng Macroweather Model (SLIMM): using scaling to forecast global-scale macroweather from months to decades, Earth Syst. Dynam., 6, 637–658, https://doi.org/10.5194/esd-6-637-2015, 2015.
Lovejoy, S., Procyk, R., Hébert, R., and del Rio Amador, L.: The Fractional Energy Balance Equation, Q. J. Roy. Meteor. Soc., 1–25, https://doi.org/10.1002/qj.4005, 2021.
Lutz, E.: Fractional Langevin equation, Phys. Rev. E, 64, 4, https://doi.org/10.1103/PhysRevE.64.051106, 2001.
Magin, R., Sagher, Y., and Boregowda, S.: Application of fractional calculus in modeling and solving the bioheat equation, in: Design and Nature II, edited by: Collins, M. W. and Brebbia, C. A., WIT Press, 207–216, ISBN: 1-85312-721-3, 2004.
Mainardi, F. and Pironi, P.: The Fractional Langevin Equation: Brownian Motion Revisited, Extracta Mathematicae, 10, 140–154, 1996.
Mandelbrot, B. B.: The Fractal Geometry of Nature, Freeman, ISBN-10.0716711869, 1982.
Mandelbrot, B. B. and Van Ness, J. W.: Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10, 422–450, 1968.
Mandelbrot, B. B. and Wallis, J. R.: Computer Experiments with fractional gaussian noises: part 3, mathematical appendix, Water Resour. Res., 5, 260–267, https://doi.org/10.1029/WR005i001p00260, 1969.
Mathews, J. and Walker, R. L.: Mathematical methods of Physics, W. A. Benjamin, ISBN: 8053-7002-1, 1973.
Metzler, R. and Klafter, J.: The Random Walks Guide To Anomalous Diffusion: A Fractional Dynamics Approach, Phys. Rep., 339, 1–77, 2000.
Newman, M.: An Empirical Benchmark for Decadal Forecasts of Global Surface Temperature Anomalies, J. Climate, 26, 5260–5269, https://doi.org/10.1175/JCLI-D-12-00590.1, 2013.
Nonnenmacher, T. F. and Metzler, R.: Applications of fractional calculus techniques to problems in biophysics, in: Fractional Calculus in Physics, edited by: Hilfer, R., World Scientific, 377–427, ISBN: 9810234570, 2000.
North, G. R. and Kim, K. Y.: Energy Balance Climate Models, Wiley-VCH, 369 pp., ISBN: 978-3-527-41132-0, 2017.
Oldham, K. B.: Diffusive transport to planar, cylindrical and spherical electrodes, J. Electroanal. Chem. Interfacial Electrochem., 41, 351–358, 1973.
Oldham, K. B. and Spanier, J.: A general solution of the diffusion equation for semi infinite geometries, J. Math. Anal. Appl., 39, 665–669, 1972.
Palma, W.: Long-memory time series, Wiley, ISBN: 9780470114025, 2007.
Palmer, T. N. and Williams, P. (Eds.): Stochastic physics and Climate models, Cambridge University Press, Cambridge, 480 pp., ISBN: 9780521761055, 2010.
Papoulis, A.: Probability, Random Variables and Stochastic Processes, Mc Graw Hill, ISBN-10: 0070484481, 1965.
Penland, C.: A stochastic model of IndoPacific sea surface temperature anomalies, Phys. D, 98, 534–558, 1996.
Penland, C. and Magorian, T.: Prediction of Nino 3 sea surface temperatures using linear inverse modeling, J. Climate, 6, 1067–1076, 1993.
Podlubny, I.: Fractional Differential Equations, Academic Press, 340 pp., ISBN 9780125588409, 1999.
Procyk, R.: The Fractional Energy Balance Equation: the Unification of Externally Forced and Internal Variability, MSc thesis, McGill University, Montreal, Canada, 111 pp., 2021.
Procyk, R., Lovejoy, S., and Hébert, R.: The fractional energy balance equation for climate projections through 2100, Earth Syst. Dynam., 13, 81–107, https://doi.org/10.5194/esd-13-81-2022, 2020.
Procyk, R., Lovejoy, S., and Hébert, R.: The fractional energy balance equation for climate projections through 2100, Earth Syst. Dynam., 13, 81–107, https://doi.org/10.5194/esd-13-81-2022, 2022.
Rypdal, K.: Global temperature response to radiative forcing: Solar cycle versus volcanic eruptions, J. Geophys. Res., 117, D06115, https://doi.org/10.1029/2011JD017283, 2012.
Rypdal, K.: Global warming projections derived from an observation-based minimal model, Earth Syst. Dynam., 7, 51–70, https://doi.org/10.5194/esd-7-51-2016, 2016.
Sardeshmukh, P., Compo, G. P., and Penland, C.: Changes in probability assoicated with El Nino, J. Climate, 13, 4268–4286, 2000.
Sardeshmukh, P. D. and Sura, P.: Reconciling non-gaussian climate statistics with linear dynamics, J. Climate, 22, 1193–1207, 2009.
Schertzer, D., Larchevíque, M., Duan, J., Yanovsky, V. V., and Lovejoy, S.: Fractional Fokker-Planck equation for nonlinear stochastic differential equation driven by non-Gaussian Levy stable noises, J. Math. Phys., 42, 200–212, 2001.
Schiessel, H., Friedrich, C., and Blumen, A.: Applications to problems in polymer physics and rheology, in: Fractional Calculus in physics, edited by: Hilfer, R., World Scientific, 331–376, ISBN: 9810234570, 2000.
Sellers, W. D.: A global climate model based on the energy balance of the earth-atmosphere system, J. Appl. Meteorol., 8, 392–400, 1969.
Sierociuk, D., Dzielinski, A., Sarwas, G., Petras, I., Podlubny, I., and Skovranek, T.: Modelling heat transfer in heterogeneous media using fractional calculus, Philos. T. R. Soc. A, 371, 20120146, https://doi.org/10.1098/rsta.2012.0146, 2013.
van Hateren, J. H.: A fractal climate response function can simulate global average temperature trends of the modern era and the past millennium, Clim. Dynam., 40, 2651, https://doi.org/10.1007/s00382-012-1375-3, 2013.
Vojta, T., Skinner, S., and Metzler, R.: Probability density of the fractional Langevin equation with reflecting walls, Phys. Rev. E, 100, 042142, https://doi.org/10.1103/PhysRevE.100.042142, 2019.
Watkins, N.: Fractional Stochastic Models for Heavy Tailed, and Long-Range Dependent, Fluctuations in Physical Systems, in: Nonlinear and Stochastic Climate Dynamics, edited by: Franzke, C. and O'Kane, T., Cambridge University Press, 340–368, ISBN: 9781316339251, 2017.
Watkins, N., Chapman, S., Klages, R., Chechkin, A., Ford, I., and Stainforth, D.: Generalised Langevin Equations and the Climate Response Problem, Earth and Space Science Open Archive, https://doi.org/10.1002/essoar.10501367.1, 2019.
Watkins, N. W., Chapman, S. C., Chechkin, A., Ford, I., Klages, R., and Stainforth, D. A.: On Generalized Langevin Dynamics and the Modelling of Global Mean Temperature, arXiv [preprint], arXiv:2007.06464v1, 4 December 2020.
West, B. J., Bologna, M., and Grigolini, P.: Physics of Fractal Operators, Springer, 354 pp., ISBN: 0-387-95554-2, 2003.
Ziegler, E. and Rehfeld, K.: TransEBM v. 1.0: description, tuning, and validation of a transient model of the Earth's energy balance in two dimensions, Geosci. Model Dev., 14, 2843–2866, https://doi.org/10.5194/gmd-14-2843-2021, 2021.
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...