Abstract
The atmospheric \(\textrm{CO}_{2}\) level, global average temperature, and sea level, which are three key metrics characterizing Earth’s surface environments, underwent a series of significant changes over geologic time. Here, I investigate the variation rates of these three variables during the Phanerozoic Eon and show that they systematically exhibit scale-independent behaviors. I then derive a general mathematical form of these scale-independent patterns based on geosystem-specific assumptions and basic physical principles. From the perspective of statistical mechanics, these scale-independent behaviors appearing in the planetary-scale geological system imply that the internal dynamics and interactions of different components in the Earth system have significantly influenced its evolution and stability, which sheds light on Earth’s sustainability and habitability.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abbass K, Qasim MZ, Song H, Murshed M, Mahmood H, Younis I (2022) A review of the global climate change impacts, adaptation, and sustainable mitigation measures. Environ Sci Pollut Res 29(28):42539–42559. https://doi.org/10.1007/s11356-022-19718-6
Alcott LJ, Mills BJW, Poulton SW (2019) Stepwise Earth oxygenation is an inherent property of global biogeochemical cycling. Science 366(6471):1333–1337. https://doi.org/10.1126/science.aax6459
Almeida MP (2003) Thermodynamical entropy (and its additivity) within generalized thermodynamics. Physica A Stat Mech Appl 325(3–4):426–438. https://doi.org/10.1016/S0378-4371(03)00262-0
Alstott J, Bullmore E, Plenz D (2014) Powerlaw: a Python package for analysis of heavy-tailed distributions. PLoS ONE 9(1):e85777. https://doi.org/10.1371/journal.pone.0085777
Anderson TW (1962) On the distribution of the two-sample Cramér–von Mises criterion. Ann Math Stat 33:1148–1159. https://doi.org/10.1214/aoms/1177704477
Anderson PW (1972) More is different: broken symmetry and the nature of the hierarchical structure of science. Science 177(4047):393–396. https://doi.org/10.1126/science.177.4047.393
Bak P (2013) How nature works: the science of self-organized criticality. Springer, Berlin. https://doi.org/10.1007/978-1-4757-5426-1
Bak P, Tang C, Wiesenfeld K (1987) Self-organized criticality: an explanation of the 1/f noise. Phys Rev Lett 59(4):381. https://doi.org/10.1103/PhysRevLett.59.381
Bak P, Christensen K, Danon L, Scanlon T (2002) Unified scaling law for earthquakes. Phys Rev Lett 88(17):178501. https://doi.org/10.1073/pnas.012581099
Bambach RK (2006) Phanerozoic biodiversity mass extinctions. Annu Rev Earth Planet Sci 34:127–155. https://doi.org/10.1146/annurev.earth.33.092203.122654
Berner RA (1990) Atmospheric carbon dioxide levels over Phanerozoic time. Science 249(4975):1382–1386. https://doi.org/10.1126/science.249.4975.138
Bond DPG, Grasby SE (2017) On the causes of mass extinctions. Palaeogeogr Palaeoclimatol Palaeoecol 478:3–29. https://doi.org/10.1016/j.palaeo.2016.11.005
Boulila S, Laskar J, Haq BU, Galbrun B, Hara N (2018) Long-term cyclicities in Phanerozoic sea-level sedimentary record and their potential drivers. Glob Planet Change 165:128–136. https://doi.org/10.1016/j.gloplacha.2018.03.004
Burnham A, Han J, Clark CE, Wang M, Dunn JB, Palou-Rivera I (2012) Life-cycle greenhouse gas emissions of shale gas, natural gas, coal, and petroleum. Environ Sci Technol 46(2):619–627. https://doi.org/10.1021/es201942m
Cannavò F, Nunnari G (2016) On a possible unified scaling law for volcanic eruption durations. Sci Rep 6(1):22289. https://doi.org/10.1038/srep22289
Carlson JM, Doyle J (1999) Highly optimized tolerance: a mechanism for power laws in designed systems. Phys Rev E 60(2):1412. https://doi.org/10.1103/PhysRevE.60.1412
Clauset A, Shalizi CR, Newman MEJ (2009) Power-law distributions in empirical data. SIAM Rev 51(4):661–703. https://doi.org/10.1137/070710111
De Boor C (1978) A practical guide to splines, vol 27. Springer, New York
Dewar RC (2005) Maximum entropy production and the fluctuation theorem. J Phys A Math Gen 38(21):L371. https://doi.org/10.1088/0305-4470/38/21/L01
Ehlers E, Krafft T (2006) Earth system science in the Anthropocene. Springer, Berlin. https://doi.org/10.1007/b137853
Folke C, Carpenter SR, Walker B, Scheffer M, Chapin T, Rockström J (2010) Resilience thinking: integrating resilience, adaptability and transformability. Ecol Soc 15(4):9. https://doi.org/10.5751/es-03610-150420
Foster GL, Rohling EJ (2013) Relationship between sea level and climate forcing by CO\(_{2}\) on geological timescales. Proc Natl Acad Sci 110(4):1209–1214. https://doi.org/10.1073/pnas.1216073110
Foster GL, Royer DL, Lunt DJ (2017) Future climate forcing potentially without precedent in the last 420 million years. Nat Commun 8(1):1–8. https://doi.org/10.1038/ncomms14845
Freund RJ, Wilson WJ (2003) Statistical methods. Elsevier, Amsterdam. https://doi.org/10.1016/C2019-0-02521-6
Friedlingstein P, Houghton RA, Marland G, Hackler J, Boden TA, Conway TJ, Canadell JG, Raupach MR, Ciais P, Le Quéré C (2010) Update on CO\(_2\) emissions. Nat Geosci 3(12):811–812. https://doi.org/10.1038/ngeo1022
Gibbs JW (1902) Elementary principles in statistical mechanics: developed with especial reference to the rational foundations of thermodynamics. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511686948
Hallam A, Wignall PB (1999) Mass extinctions and sea-level changes. Earth Sci Rev 48(4):217–250. https://doi.org/10.1016/S0012-8252(99)00055-0
Hannisdal B, Peters SE (2011) Phanerozoic Earth system evolution and marine biodiversity. Science 334(6059):1121–1124. https://doi.org/10.1126/science.1210695
Hayes JM, Waldbauer JR (2006) The carbon cycle and associated redox processes through time. Philos Trans R Soc B Biol Sci 361(1470):931–950. https://doi.org/10.1098/rstb.2006.1840
He Q, Silliman BR (2019) Climate change, human impacts, and coastal ecosystems in the Anthropocene. Curr Biol 29(19):R1021–R1035. https://doi.org/10.1016/j.cub.2019.08.042
Hergarten S (2002) Self organized criticality in Earth systems, vol 2. Springer, Heidelberg. https://doi.org/10.1007/978-3-662-04390-5
Huang Y, Schmitt FG (2014) Time dependent intrinsic correlation analysis of temperature and dissolved oxygen time series using empirical mode decomposition. J Mar Syst 130:90–100. https://doi.org/10.1016/j.jmarsys.2013.06.007
Iversen MH (2023) Carbon export in the ocean: a biologist’s perspective. Annu Rev Mar Sci 15:357–381. https://doi.org/10.1146/annurev-marine-032122-035153
Jablonski D (2001) Lessons from the past: evolutionary impacts of mass extinctions. Proc Natl Acad Sci 98(10):5393–5398. https://doi.org/10.1073/pnas.101092598
Jacobson M, Charlson RJ, Rodhe H, Orians GH (2000) Earth system science: from biogeochemical cycles to global changes. Academic Press, Cambridge. https://doi.org/10.1016/s0074-6142(00)x8104-1
Jaynes ET (1957) Information theory and statistical mechanics. Phys Rev 106(4):620. https://doi.org/10.1103/PhysRev.106.620
Kauffman SA, Johnsen S (1991) Coevolution to the edge of chaos: coupled fitness landscapes, poised states, and coevolutionary avalanches. J Theor Biol 149(4):467–505. https://doi.org/10.1016/S0022-5193(05)80094-3
Kleidon A (2010) A basic introduction to the thermodynamics of the Earth system far from equilibrium and maximum entropy production. Philos Trans R Soc B Biol Sci 365(1545):1303–1315. https://doi.org/10.1098/rstb.2009.0310
Kleidon A, Lorenz RD (2004) Non-equilibrium thermodynamics and the production of entropy: life, Earth, and beyond. Springer, Heidelberg. https://doi.org/10.1007/b12042
Kocsis ÁT, Scotese CR (2021) Mapping paleocoastlines and continental flooding during the Phanerozoic. Earth Sci Rev 213:103463. https://doi.org/10.1016/j.earscirev.2020.103463
Konhauser KO, Planavsky NJ, Hardisty DS, Robbins LJ, Warchola TJ, Haugaard R, Lalonde SV, Partin CA, Oonk PBH, Tsikos H, Lyons TW, Bekkee A, Johnson CM (2017) Iron formations: a global record of Neoarchaean to Palaeoproterozoic environmental history. Earth Sci Rev 172:140–177
Laakso TA, Schrag DP (2017) A theory of atmospheric oxygen. Geobiology 15(3):366–384. https://doi.org/10.1111/gbi.12230
Lenton TM (1998) Gaia and natural selection. Nature 394(6692):439–447. https://doi.org/10.1038/28792
Lenton T, Watson A (2013) Revolutions that made the Earth. Oxford University Press, Oxford
Lin J (1991) Divergence measures based on the Shannon entropy. IEEE Trans Inf Theory 37(1):145–151. https://doi.org/10.1109/18.61115
Lovelock J (2016) Gaia: a new look at life on Earth. Oxford University Press, Oxford
Ludwig JA, Reynolds JF (1988) Statistical ecology: a primer in methods and computing, vol 1. Wiley, Hoboken
Macreadie PI, Costa MDP, Atwood TB, Friess DA, Kelleway JJ, Kennedy H, Lovelock CE, Serrano O, Duarte CM (2021) Blue carbon as a natural climate solution. Nat Rev Earth Environ 2(12):826–839. https://doi.org/10.1038/s43017-021-00224-1
Martyushev LM (2010) The maximum entropy production principle: two basic questions. Philos Trans R Soc B Biol Sci 365(1545):1333–1334. https://doi.org/10.1098/rstb.2009.0295
Massey FJ Jr (1951) The Kolmogorov–Smirnov test for goodness of fit. J Am Stat Assoc 46(253):68–78. https://doi.org/10.1080/01621459.1951.10500769
Mayhew PJ, Bell MA, Benton TG, McGowan AJ (2012) Biodiversity tracks temperature over time. Proc Natl Acad Sci 109(38):15141–15145. https://doi.org/10.1073/pnas.120084410
McInerney FA, Wing SL (2011) The paleocene–eocene thermal maximum: a perturbation of carbon cycle, climate, and biosphere with implications for the future. Annu Rev Earth Planet Sci 39:489–516. https://doi.org/10.1146/annurev-earth-040610-133431
Middelburg JJ (1989) A simple rate model for organic matter decomposition in marine sediments. Geochim Cosmochim Acta 53(7):1577–1581. https://doi.org/10.1016/0016-7037(89)90239-1
Newman MEJ (1997) A model of mass extinction. J Theor Biol 189(3):235–252. https://doi.org/10.1006/jtbi.1997.0508
Peters SE (2005) Geologic constraints on the macroevolutionary history of marine animals. Proc Natl Acad Sci 102(35):12326–12331. https://doi.org/10.1073/pnas.0502616102
Peters SE (2007) The problem with the Paleozoic. Paleobiology 33(2):165–181. https://doi.org/10.1666/06067.1
Pielou EC (1966) Shannon’s formula as a measure of specific diversity: its use and misuse. Am Nat 100(914):463–465. https://doi.org/10.1086/282439
Plotnick RE, McKinney ML (1993) Ecosystem organization and extinction dynamics. Palaios 8:202–212. https://doi.org/10.2307/3515172
Prigogine I (1980) From being to becoming: time and complexity in the physical sciences. Freeman, New York
Raup DM (1986) Biological extinction in Earth history. Science 231(4745):1528–1533. https://doi.org/10.1126/science.11542058
Raupach MR, Canadell JG (2010) Carbon and the Anthropocene. Curr Opin Environ Sustain 2(4):210–218. https://doi.org/10.1073/pnas.0907765106
Royer DL, Berner RA, Beerling DJ (2001) Phanerozoic atmospheric CO\(_2\) change: evaluating geochemical and paleobiological approaches. Earth Sci Rev 54(4):349–392. https://doi.org/10.1016/S0012-8252(00)00042-8
Scarano FR (2017) Ecosystem-based adaptation to climate change: concept, scalability and a role for conservation science. Perspect Ecol Conserv 15(2):65–73. https://doi.org/10.1016/j.pecon.2017.05.003
Schroeder M (2009) Fractals, chaos, power laws: minutes from an infinite paradise. Courier Corporation, Massachusetts
Scotese CR, Song H, Mills BJW, van der Meer DG (2021) Phanerozoic paleotemperatures: the Earth’s changing climate during the last 540 million years. Earth Sci Rev 215:103503. https://doi.org/10.1016/j.earscirev.2021.103503
Scott DW (2009) Sturges’ rule. Wiley Interdiscip Rev Comput Stat 1(3):303–306. https://doi.org/10.1002/wics.35
Shang H (2023a) Dichotomous effects of oxidative metabolisms: a theoretical perspective on the dolomite problem. Glob Planet Change 222:104041. https://doi.org/10.1016/j.gloplacha.2023.104041
Shang H (2023b) A generic hierarchical model of organic matter degradation and preservation in aquatic systems. Commun Earth Environ 4:16. https://doi.org/10.1038/s43247-022-00667-4
Shang H (2023c) Mineral evolution facilitated Earth’s oxidation. Commun Earth Environ 4:213. https://doi.org/10.1038/s43247-023-00824-3
Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27(3):379–423. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x
Skilling J (1989) Classic maximum entropy. In: Skilling J (ed) Maximum entropy and Bayesian methods. Springer, Berlin, pp 45–52. https://doi.org/10.1007/978-94-017-2219-3
Solow AR (2005) Power laws without complexity. Ecol Lett 8(4):361–363. https://doi.org/10.1111/j.1461-0248.2005.00738.x
Song H, Wignall PB, Song H, Dai X, Chu D (2019) Seawater temperature and dissolved oxygen over the past 500 million years. J Earth Sci 30(2):236–243. https://doi.org/10.1007/s12583-018-1002-2
Song H, Kemp DB, Tian L, Chu D, Song H, Dai X (2021) Thresholds of temperature change for mass extinctions. Nat Commun 12(1):1–8. https://doi.org/10.1038/s41467-021-25019-2
Steffen W, Rockström J, Richardson K, Lenton TM, Folke C, Liverman D, Summerhayes CP, Barnosky AD, Cornell SE, Crucifix M, Donges JF (2018) Trajectories of the Earth system in the Anthropocene. Proc Natl Acad Sci 115(33):8252–8259. https://doi.org/10.1073/pnas.1810141115
Summerhayes CP, Zalasiewicz J (2018) Global warming and the Anthropocene. Geol Today 34(5):194–200. https://doi.org/10.1111/gto.12247
Vallino JJ, Algar CK (2016) The thermodynamics of marine biogeochemical cycles: Lotka revisited. Annu Rev Mar Sci 8:333–356. https://doi.org/10.1146/annurev-marine-010814-015843
Vérard C, Veizer J (2019) On plate tectonics and ocean temperatures. Geology 47(9):881–885. https://doi.org/10.1130/G46376.1
Vuong QH (1989) Likelihood ratio tests for model selection and non-nested hypotheses. Econom J Econom Soc 57:307–333. https://doi.org/10.2307/1912557
Yoro KO, Daramola MO (2020) CO\(_2\) emission sources, greenhouse gases, and the global warming effect. In: Rahimpour MR, Farsi M, Makarem MA (eds) Advances in carbon capture. Elsevier, Amsterdam, pp 3–28. https://doi.org/10.1016/B978-0-12-819657-1.00001-3
Acknowledgements
I thank Editor Roussos Dimitrakopoulos for handling this manuscript and two anonymous reviewers for thoughtful and constructive comments.
Author information
Authors and Affiliations
Contributions
HS conceived the project, performed the research, and wrote the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The author declares no conflict of interest.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Shang, H. Scale-Independent Variation Rates of Phanerozoic Environmental Variables and Implications for Earth’s Sustainability and Habitability. Math Geosci 56, 1469–1485 (2024). https://doi.org/10.1007/s11004-024-10135-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11004-024-10135-8