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Mathematical Geoscience pp 65–138Cite as

Climate Dynamics

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Part of the book series: Interdisciplinary Applied Mathematics ((IAM,volume 36))

Abstract

Chapter 2 treats simple models of climate dynamics. It begins by describing radiative transfer in the atmosphere, with particular attention to the simple case of a grey, one-dimensional atmosphere. Following this, the convective temperature structure of the troposphere is discussed. The latter part of the chapter deals with energy balance models, in particular with a view to understanding ice age causes. This introduces successively the ice-albedo feedback mechanism, the carbon cycle in atmosphere and ocean, and the rôle of bicarbonate buffering in the ocean.

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Notes

  1. 1.

    This overly simple description is inaccurate in one main respect, which is that the hemispheric polewards circulation actually consists of three cells, not one: a tropical cell, a mid-latitude cell and a polar cell. The prevailing winds are westerly (from the west) only in the mid-latitude cells; tropical winds (the trade winds), for example, are easterlies (from the east).

  2. 2.

    The analogy is probably rather loose, since it is more the absence of convective (rather than radiative) cooling of the greenhouse which causes its elevated temperature.

  3. 3.

    This now specifically assumes that no other energy transport processes occur.

  4. 4.

    The molecular weight is effectively the weight of a molecule of a substance. Equivalently, it is determined by the weight of a fixed number of molecules, known as a mole, and equal to Avogadro’s number 6×1023 molecules. For air, a mixture predominantly of nitrogen (78%), oxygen (21%) and argon (0.9%), the molecular weight is given by the equivalent quantity for the mixture. It has the value M a =28.8×10−3 kg mole−1. Useful references for such quantities and their units are Kaye and Laby (1960) and Massey (1986).

  5. 5.

    This is something of a simplification. Net addition of radiant energy to the atmosphere can cause changes in sensible heat (via temperature), latent heat (via moisture) or gravitational potential energy (via thermal expansion); we thus implicitly neglect the latter two; see also Eq. (3.25) in Sect. 3.2.3, and the next footnote.

  6. 6.

    For a moist, saturated atmosphere, we may take the moisture mixing ratio m to be a function of T, and in this case the latent heat ρ a Lm (L being latent heat) simply modifies the heat capacity coefficient. Question 2.11 shows how to calculate m(T). See also Sect. 3.2.7.

  7. 7.

    The value of A assumes T is measured in degrees Celsius.

  8. 8.

    This is not the only possible mechanism. Another is the North Atlantic salt oscillator, discussed in Sect. 2.5.7.

  9. 9.

    The problems of plate tectonics are discussed in Chap. 8.

  10. 10.

    The units here are in terms of silica, SiO2. If we suppose that weathering is described by the reaction (2.129), then one mole of CO2 (of weight 44 grams) is used to produce one mole of SiO2 (of weight 60 grams). So to convert units of kg(SiO2) m−2 y−1 to units of kg(CO2) m−2 y−1, multiply by 44/60≈0.73.

  11. 11.

    The current net annual addition of CO2 to the atmosphere because of fossil fuel consumption and deforestation is about 3.5 Gt carbon, or 1.3×1013 kgCO2 y−1; this is forty times larger than the volcanic production rate. (The actual rate of addition is more than twice as large again, but is compensated by net absorption by the oceans and in photosynthesis.)

  12. 12.

    In fact the corrected approach is to assume charge neutrality, but allowing for the net negative charge of the conservative ions: chloride, sodium, etc.

References

  • Agustin L et al. (EPICA community) (2004) Eight glacial cycles from an Antarctic ice core. Nature 429:623–628

    Article  Google Scholar 

  • Alley RB, Mayewski PA, Sowers T, Stuiver M, Taylor KC, Clark PU (1997) Holocene climatic instability: a prominent, widespread event 8200 years ago. Geology 25:483–486

    Article  Google Scholar 

  • Andrews DG (2000) An introduction to atmospheric physics. Cambridge University Press, Cambridge

    Google Scholar 

  • Arrhenius S (1896) On the influence of carbonic acid in the air upon the temperature of the ground. Philos Mag 41:237–275

    Google Scholar 

  • Berner RA, Lasaga AC, Garrels RM (1983) The carbonate–silicate geochemical cycle and its effect on atmospheric carbon dioxide over the past 100 million years. Am J Sci 283:641–683

    Article  Google Scholar 

  • Bigg G (2003) The oceans and climate, 2nd edn. Cambridge University Press, Cambridge

    Google Scholar 

  • Bolshakov VA (2003) Modern climatic data for the Pleistocene: implications for a new concept of the orbital theory of paleoclimate. Russ J Earth Sci 5:125–143

    Article  Google Scholar 

  • Bond GC, Showers W, Cheseby M, Lotti R, Almasi P, de Menocal P, Priore P, Cullen H, Hajdas I, Bonani G (1997) A pervasive millennial-scale cycle in North Atlantic Holocene and glacial climates. Science 278:1257–1266

    Article  Google Scholar 

  • Bond GC, Showers W, Elliott M, Evans M, Lotti R, Hajdas I, Bonani G, Johnson S (1999) The North Atlantic’s 1–2 kyr climate rhythm: relation to Heinrich events, Dansgaard/Oeschger cycles and Little Ice Age. In: Clark PU, Webb RS, Keigwin LD (eds) Mechanisms of global climate change. Geophys monogr, vol 112. AGU, Washington, pp 35–58

    Google Scholar 

  • Broecker WS, Bond G, Klas M (1990) A salt oscillator in the glacial Atlantic? I: the concept. Paleoceanography 5:469–477

    Article  Google Scholar 

  • Budyko MI (1969) The effect of solar radiation variations on the climate of the Earth. Tellus 21:611–619

    Article  Google Scholar 

  • Calov R, Ganopolski A, Petoukhov V, Claussen M, Greve R (2002) Large-scale instabilities of the Laurentide ice sheet simulated in a fully coupled climate-system model. Geophys Res Lett 29(24):69. doi:10.1029/2002GL016078

    Article  Google Scholar 

  • Chandler MA, Sohl LE (2000) Climate forcings and the initiation of low-latitude ice sheets during the Neoproterozoic Varanger glacial interval. J Geophys Res D 105(10):20737–20756

    Article  Google Scholar 

  • Chandrasekhar S (1960) Radiative transfer. Dover, New York

    Google Scholar 

  • Chapman S (1930) A theory of upper atmospheric ozone. Mem R Meteorol Soc 3:103–125

    Google Scholar 

  • Crowley TJ, Baum SK (1993) Effect of decreased solar luminosity on late Precambrian ice extent. J Geophys Res 98(D9):16723–16732

    Article  Google Scholar 

  • Emerson SR, Hedges JI (2008) Chemical oceanography and the marine carbon cycle. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Evatt G, Fowler AC, Clark CD, Hulton N (2006) Subglacial floods beneath ice sheets. Philos Trans R Soc 364:1769–1794

    Article  MathSciNet  Google Scholar 

  • Fowler AC, Schiavi E (1998) A theory of ice sheet surges. J Glaciol 44:104–118

    Google Scholar 

  • Fricker HA, Scambos T, Bindschadler R, Padman L (2007) An active subglacial water system in West Antarctica mapped from space. Science 315:1544–1548

    Article  Google Scholar 

  • Ganopolski A, Rahmstorf S (2001) Rapid changes of glacial climate simulated in a coupled climate model. Nature 409:153–158

    Article  Google Scholar 

  • Gill AE (1982) Atmosphere-ocean dynamics. Academic Press, San Diego

    Google Scholar 

  • Goodwin ID (1988) The nature and origin of a jökulhlaup near Casey Station, Antarctica. J Glaciol 34:95–101

    Google Scholar 

  • Harland WB (1964) Critical evidence for a great infra-Cambrian glaciation. Geol Rundsch 54:45–61

    Article  Google Scholar 

  • Harland WB (2007) Origins and assessment of snowball Earth hypotheses. Geol Mag 144:633–642

    Article  Google Scholar 

  • Heinrich H (1988) Origin and consequences of cyclic ice rafting in the Northeast Atlantic Ocean during the past 130000 years. Quat Res 29:142–152

    Article  Google Scholar 

  • Hoffman PF, Kaufman AJ, Halverson GP, Schrag DP (1998) A Neoproterozoic snowball Earth. Science 281:1342–1346

    Article  Google Scholar 

  • Houghton JT (2002) The physics of atmospheres, 3rd edn. Cambridge University Press, Cambridge

    Google Scholar 

  • Houghton JT, Meira Filho LG, Callander BA, Harris N, Kattenberg A, Maskell K (eds) (1996) Climate change 1995: the science of climate change. Cambridge University Press, Cambridge

    Google Scholar 

  • Hyde WT, Crowley TJ, Baum SK, Peltier WR (2000) Neoproterozoic ‘snowball Earth’ simulations with a coupled climate/ice sheet model. Nature 405:425–429

    Article  Google Scholar 

  • Johnsen SJ, Clausen HB, Dansgaard W, Fuhrer K, Gundestrup N, Hammer CU, Iversen P, Jouzel J, Stauffer B, Steffensen JP (1992) Irregular glacial interstadials recorded in a new Greenland ice core. Nature 359:311–313

    Article  Google Scholar 

  • Kasting JF (1989) Long-term stability of the Earth’s climate. Palaeogeogr Palaeoclimatol Palaeoecol 75:83–95

    Article  Google Scholar 

  • Kasting JF, Ackermann TP (1986) Climatic consequences of very high carbon dioxide levels in Earth’s early atmosphere. Science 234:1383–1385

    Article  Google Scholar 

  • Kaye GWC, Laby TH (1960) Physical and chemical constants, 12th edn. Longman, Harlow

    Google Scholar 

  • Köhler P, Fischer H (2006) Simulating low frequency changes in atmospheric CO2 during the last 740000 years. Clim Past 2:57–78

    Article  Google Scholar 

  • Köhler P, Fischer H, Munhoven G, Zeebe RE (2005) Quantitative interpretation of atmospheric carbon records over the last glacial termination. Glob Biogeochem Cycles 19:GB4020. doi:10.1029/2004GB002345

    Article  Google Scholar 

  • Krauskopf KB, Bird DK (1995) Introduction to geochemistry. McGraw-Hill, New York

    Google Scholar 

  • Lang C, Leuenberger M, Schwander J, Johnsen S (1999) 16°C rapid temperature variation in central Greenland 70000 years ago. Science 286:934–937

    Article  Google Scholar 

  • Lasaga AC, Berner RA, Garrels RM (1985) An improved geochemical model of atmospheric CO2 fluctuations over the past 100 million years. In: Sundquist ET, Broecker WS (eds) The carbon cycle and atmospheric CO2: natural variations Archaen to present. AGU, Washington, pp 397–411

    Google Scholar 

  • Leuenberger MC, Lang C, Schwander J (1999) Delta 15N measurements as a calibration tool for the paleothermometer and gas–ice age differences: a case study for the 8200 BP event on GRIP ice. J Geophys Res 104(D18):22163–22170

    Article  Google Scholar 

  • Liou KN (2002) An introduction to atmospheric radiation, 2nd edn. Academic Press, San Diego

    Google Scholar 

  • MacAyeal DR (1993) Binge/purge oscillations of the Laurentide ice sheet as a cause of the North Atlantic’s Heinrich events. Paleoceanography 8:775–784

    Article  Google Scholar 

  • Manabe S, Stouffer RJ (1995) Simulation of abrupt climate change induced by freshwater input to the North Atlantic Ocean. Nature 378:165–167

    Article  Google Scholar 

  • Marshall HG, Walker JCG, Kuhn WR (1988) Long term climate change and the geochemical cycle of carbon. J Geophys Res 93:791–801

    Article  Google Scholar 

  • Massey BS (1986) Measures in science and engineering. Ellis Horwood, Chichester

    MATH  Google Scholar 

  • Millero FJ (1995) Thermodynamics of the carbon dioxide system in the oceans. Geochim Cosmochim Acta 59:661–677

    Article  Google Scholar 

  • Munhoven G, François LM (1996) Glacial–interglacial variability of atmospheric CO2 due to changing continental silicate rock weathering: a model study. J Geophys Res 101:21423–21437

    Article  Google Scholar 

  • North GR (1975a) Analytical solution to a simple climate model with diffusive heat transport. J Atmos Sci 32:1301–1307

    Article  Google Scholar 

  • North GR (1975b) Theory of energy-balance climate models. J Atmos Sci 32:2033–2043

    Article  Google Scholar 

  • North GR, Mengel JG, Short DA (1983) Simple energy balance model resolving the season and continents: applications to astronomical theory of ice ages. J Geophys Res 88:6576–6586

    Article  Google Scholar 

  • Petit JR, Jouzel J, Raynaud D, Barkov NI, Barnola J-M, Basile I, Bender M, Chappellaz J, Davis M, Delaygue G, Delmotte M, Kotlyakov VM, Legrand M, Lipenkov VY, Lorius C, Pépin L, Ritz C, Saltzman E, Stievenard M (1999) Climate and atmospheric history of the past 420000 years from the Vostok ice core, Antarctica. Nature 399:429–436

    Article  Google Scholar 

  • Pierrehumbert RT (2004) High atmospheric carbon dioxide necessary for the termination of global glaciation. Nature 429:646–648

    Article  Google Scholar 

  • Rahmstorf S (1995) Bifurcations of the Atlantic thermohaline circulation in response to changes in the hydrological cycle. Nature 378:145–149

    Article  Google Scholar 

  • Rahmstorf S (2002) Ocean circulation and climate during the past 120000 years. Nature 419:207–214

    Article  Google Scholar 

  • Ruddiman WF (2001) Earth’s climate: past and future. Freeman, New York

    Google Scholar 

  • Schmidt MW, Vautravers MJ, Spero HJ (2006) Rapid subtropical North Atlantic salinity oscillations across Dansgaard–Oeschger cycles. Nature 443:561–564

    Article  Google Scholar 

  • Sellers WD (1969) A climate model based on the energy balance of the Earth-atmosphere system. J Appl Meteorol 8:392–400

    Article  Google Scholar 

  • Severinghaus JP, Brook EJ (1999) Abrupt climate change at the end of the last glacial period inferred from trapped air in polar ice. Science 286:930–934

    Article  Google Scholar 

  • Solomon S (1999) Stratospheric ozone depletion: a review of concepts and history. Rev Geophys 37:275–316

    Article  Google Scholar 

  • Sparrow EM, Cess RD (1978) Radiation heat transfer. Hemisphere, Belmont

    Google Scholar 

  • Stocker TF, Johnsen SJ (2003) A minimum thermodynamic model for the bipolar seesaw. Paleoceanography 18:1087. doi:10.1029/2003PA000920

    Article  Google Scholar 

  • Stommel H (1961) Thermohaline convection with two stable régimes of flow. Tellus 13:224–230

    Article  Google Scholar 

  • Strutt JW (Lord Rayleigh) (1871) On the light from the sky, its polarization and colour. Phil Mag 41:107–120

    Google Scholar 

  • Takahashi T, Sutherland SC, Sweeney C, Poisson A, Metzl N, Tilbrook B, Bates N, Wanninkhof R, Feely RA, Sabine C, Olafsson J, Nojiri Y (2002) Global sea–air CO2 flux based on climatological surface ocean pCO2, and seasonal biological and temperature effects. Deep-Sea Res II 49:1601–1622

    Article  Google Scholar 

  • Taylor KC, Lamorey GW, Doyle GA, Alley RB, Grootes PM, Mayewski PA, White JWC, Barlow LK (1993) The ‘flickering switch’ of late Pleistocene climate change. Nature 361:432–436

    Article  Google Scholar 

  • Taylor KC, Mayewski PA, Alley RB, Brook EJ, Gow AJ, Grootes PM, Meese DA, Saltzman ES, Severinghaus JP, Twickler MS, White JWC, Whitlow S, Zielinski GA (1997) The Holocene–Younger Dryas transition recorded at Summit, Greenland. Science 278:825–827

    Article  Google Scholar 

  • Thomas GE, Stamnes K (1999) Radiative transfer in the atmosphere and ocean. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Toggweiler JR, Russell JL, Carson SR (2006) Mid-latitude westerlies, atmospheric CO2, and climate change during the ice ages. Paleoceanography 21:PA2005. doi:10.1029/2005PA001154

    Article  Google Scholar 

  • Walker JCG, Hays PB, Kasting JF (1981) A negative feedback mechanism for the long-term stabilization of Earth’s surface temperature. J Geophys Res 86(C10):9776–9782

    Article  Google Scholar 

  • Wingham DJ, Siegert MJ, Shepherd A, Muir AS (2006) Rapid discharge connects Antarctic subglacial lakes. Nature 440:1033–1037

    Article  Google Scholar 

  • Zeebe RE, Wolf-Gladrow D (2001) CO2 in seawater: equilibrium, kinetics, isotopes. Elsevier, Amsterdam

    Google Scholar 

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  1. MACSI, Department of Mathematics & Statistics, University of Limerick, Limerick, Ireland

    Andrew Fowler

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Fowler, A. (2011). Climate Dynamics. In: Mathematical Geoscience. Interdisciplinary Applied Mathematics, vol 36. Springer, London. https://doi.org/10.1007/978-0-85729-721-1_2

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