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January  2014, 19(1): 257-279. doi: 10.3934/dcdsb.2014.19.257

## A dynamics approach to a low-order climate model

 1 Department of Mathematics, Oberlin College, 10 N. Professor St, Oberlin, OH 44074, United States 2 Department of Mathematics, University of Hawai'i West Oahu, 91-1001 Farrington Hwy, Kapolei, HI, 96707, United States

Received  May 2012 Revised  August 2013 Published  December 2013

Energy Balance Models (EBM) are conceptual models which have proved useful in the study of planetary climate. The focus of EBM is placed on large scale climate components such as incoming solar radiation, albedo, outgoing longwave radiation and heat transport, and their interactions. Until recently, their study has centered on equilibrium solutions of an associated model equation, with no consideration of the dynamical nature of these solutions. In this paper we continue and expand upon recent efforts aimed at placing EBM in a more mathematical, dynamical systems context. In particular, the dynamical behavior of several variants of the Budyko-Sellers model, all but one of which involve the movement of glaciers, is shown to reduce to the study of the system on an attracting one-dimensional invariant manifold in an appropriately defined state space.
Citation: James Walsh, Esther Widiasih. A dynamics approach to a low-order climate model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 257-279. doi: 10.3934/dcdsb.2014.19.257
##### References:
 [1] D. Abbot, A. Viogt and D. Koll, The Jormungand global climate state and implications for Neoproterozoic glaciations,, J. Geophys. Res., 116 (2011). doi: 10.1029/2011JD015927. Google Scholar [2] H. Bao, J. Lyons and C. Zhou, Triple oxygen isotope evidence for elevated CO$_2$ levels after a Neoproterozoic glaciation,, Nature, 453 (2008), 504. Google Scholar [3] P. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space,, Memoirs of the American Mathematical Society, 135 (1998). doi: 10.1090/memo/0645. Google Scholar [4] B. Bodiselitsch, C. Koeberl, S. Master and W. Reimold, Estimating duration and intensity of Neoproterozoic snowball glaciations from Ir anomalies,, Science, 308 (2005), 239. doi: 10.1126/science.1104657. Google Scholar [5] H. Broer, C. Simó and R. Vitolo, Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing,, Nonlinearity, 15 (2002), 1205. doi: 10.1088/0951-7715/15/4/312. Google Scholar [6] H. Broer and R. Vitolo, Dynamical systems modeling of low-frequency variability in low-order atmospheric models,, Disc. Cont. Dyn. Syst. B, 10 (2008), 401. doi: 10.3934/dcdsb.2008.10.401. Google Scholar [7] H. Broer, H. Dijkstra, C. Simó, A. Sterk and R. Vitolo, The dynamics of a low-order model for the Atlantic multidecadal oscillation,, Disc. Cont. Dyn. Syst. B, 16 (2011), 73. doi: 10.3934/dcdsb.2011.16.73. Google Scholar [8] M. I. Budyko, The effect of solar radiation variation on the climate of the Earth,, Tellus, 5 (1969), 611. Google Scholar [9] R. Cahalan and G. North, A stability theorem for energy-balance climate modes,, J. Atmos. Sci., 36 (1979), 1178. doi: 10.1175/1520-0469(1979)036<1178:ASTFEB>2.0.CO;2. Google Scholar [10] P. Chylek and J. A. Coakley, Analytical analysis of a Budyko-type climate model,, J. Atmos. Sci., 32 (1975), 675. doi: 10.1175/1520-0469(1975)032<0675:AAOABT>2.0.CO;2. Google Scholar [11] M. Claussen et al, Earth system models of intermediate complexity: Closing the gap in the spectrum of climate models,, Climate Dynamics, 18 (2002), 579. Google Scholar [12] C. Graves, W-H. Lee and G. North, New parameterizations and sensitivities for simple climate models,, J. Geophys. Res., 198 (1993), 5025. doi: 10.1029/92JD02666. Google Scholar [13] P. Hoffman, A. Kaufman, G. Halverson and D. Schrag, A Neoproterozoic snowball Earth,, Science, 281 (1998), 1342. doi: 10.1126/science.281.5381.1342. Google Scholar [14] P. Hoffman and D. Schrag, Snowball Earth,, Sci. Amer., 282 (2000), 68. Google Scholar [15] P. Hoffman and D. Schrag, The snowball Earth hypothesis: Testing the limits of global change,, Terra Nova, 14 (2002), 129. doi: 10.1046/j.1365-3121.2002.00408.x. Google Scholar [16] R. Kerr, Snowball Earth has melted back to a profound wintry mix,, Science, 327 (2010). doi: 10.1126/science.327.5970.1186. Google Scholar [17] J. Kirschivink, Late Proterozoic low-latitude global glaciation: the snowball Earth,, in The Proterozoic Biosphere: A Multidisciplinary Study (eds. J. Schopf and C. Klein), (1992). Google Scholar [18] W. Langford and G. Lewis, Poleward expansion of Hadley cells,, Can. Appl. Math. Quart., 17 (2009), 105. Google Scholar [19] R. Q. Lin and G. North, A study of abrupt climate change in a simple nonlinear climate model,, Climate Dynamics, 4 (1990), 253. doi: 10.1007/BF00211062. Google Scholar [20] R. A. Livermore, A. G. Smith and F. J. Vine, Late Palaeozoic to early mesozoic evolution of Pangaea,, Nature, 322 (1986), 162. doi: 10.1038/322162a0. Google Scholar [21] E. Lorenz, Irregularity: A fundamental property of the atmosphere,, Tellus, 36A (1984), 98. Google Scholar [22] L. Maas, A simple model for the three-dimensional, thermally and wind-driven ocean circulation,, Tellus, 46A (1994), 671. Google Scholar [23] F. Macdonald, M. Schmitz, J. Crowley, C. Root, D. Jones, A. Maloof, J. Strauss, P. Cohen, D. Johnston and D. Schrag, Calibrating the cryogenian,, Science, 327 (2010), 1241. doi: 10.1126/science.1183325. Google Scholar [24] H. Marshall, J. Walker and W. Kuhn, Long-term climate change and the geochemical cycle of carbon,, J. Geophys. Res., 93 (1988), 791. doi: 10.1029/JD093iD01p00791. Google Scholar [25] R., McGehee,, personal communication., (). Google Scholar [26] R. McGehee and C. Lehman, A paleoclimate model of ice-albedo feedback forced by variations in Earth's orbit,, SIAM J. Appl. Dyn. Syst., 11 (2012), 684. doi: 10.1137/10079879X. Google Scholar [27] R. McGehee and E. Widiasih, A finite dimensional version of a dynamic ice-albedo feedback model,, preprint., (). Google Scholar [28] G. North, Theory of energy-balance climate models,, J. Atmos. Sci., 32 (1975), 2033. doi: 10.1175/1520-0469(1975)032<2033:TOEBCM>2.0.CO;2. Google Scholar [29] R. Pierrehumbert, Principles of Planetary Climate,, Cambridge University Press, (2010). Google Scholar [30] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos,, CRC Press, (1995). doi: 10.1117/12.217385. Google Scholar [31] G. Roe and M. Baker, Note on a catastrophe: A feedback analysis of snowball Earth,, J. of Climate, 23 (2010), 4694. doi: 10.1175/2010JCLI3545.1. Google Scholar [32] P. Roebber, Climate variability in a low-order coupled atmosphere-ocean model,, Tellus, 47A (1995), 473. Google Scholar [33] C. Sagan and G. Mullen, Earth and Mars: Evolution of atmospheres and surface temperatures,, Science, 177 (1972), 52. doi: 10.1126/science.177.4043.52. Google Scholar [34] R. Secord, P. Gingerich, K. Lohmann and K. MacLeod, Continental warming preceding the Palaeocene-Eocene thermal maximum,, Nature, 467 (2010), 955. doi: 10.1038/nature09441. Google Scholar [35] W. Sellers, A global climatic model based on the energy balance of the Earth-Atmosphere system,, J. Appl. Meteor., 8 (1969), 392. doi: 10.1175/1520-0450(1969)008<0392:AGCMBO>2.0.CO;2. Google Scholar [36] A. Shil'nikov, G. Nicolis and C. Nicolis, Bifurcation and predictability analysis of a low-order atmospheric circulation model,, Int. J. Bif. Chaos, 5 (1995), 1701. doi: 10.1142/S0218127495001253. Google Scholar [37] H. E. de Swart, Low-order spectral models of the atmospheric circulation: A survey,, Acta Appl. Math., 11 (1988), 49. doi: 10.1007/BF00047114. Google Scholar [38] K. K. Tung, Topics in Mathematical Modeling,, Princeton University Press, (2007). Google Scholar [39] L. van Veen, Overturning and wind driven circulation in a low-order ocean-atmosphere model,, Dynam. Atmos. Ocean, 37 (2003), 197. doi: 10.1016/S0377-0265(03)00032-0. Google Scholar [40] L. van Veen, Baroclinic flow and the Lorenz-84 model,, Int. J. Bif. Chaos, 13 (2003), 2117. doi: 10.1142/S0218127403007904. Google Scholar [41] E. Widiasih, Instability of the ice free earth: Dynamics of a discrete time energy balance model,, preprint , (). Google Scholar

show all references

##### References:
 [1] D. Abbot, A. Viogt and D. Koll, The Jormungand global climate state and implications for Neoproterozoic glaciations,, J. Geophys. Res., 116 (2011). doi: 10.1029/2011JD015927. Google Scholar [2] H. Bao, J. Lyons and C. Zhou, Triple oxygen isotope evidence for elevated CO$_2$ levels after a Neoproterozoic glaciation,, Nature, 453 (2008), 504. Google Scholar [3] P. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space,, Memoirs of the American Mathematical Society, 135 (1998). doi: 10.1090/memo/0645. Google Scholar [4] B. Bodiselitsch, C. Koeberl, S. Master and W. Reimold, Estimating duration and intensity of Neoproterozoic snowball glaciations from Ir anomalies,, Science, 308 (2005), 239. doi: 10.1126/science.1104657. Google Scholar [5] H. Broer, C. Simó and R. Vitolo, Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing,, Nonlinearity, 15 (2002), 1205. doi: 10.1088/0951-7715/15/4/312. Google Scholar [6] H. Broer and R. Vitolo, Dynamical systems modeling of low-frequency variability in low-order atmospheric models,, Disc. Cont. Dyn. Syst. B, 10 (2008), 401. doi: 10.3934/dcdsb.2008.10.401. Google Scholar [7] H. Broer, H. Dijkstra, C. Simó, A. Sterk and R. Vitolo, The dynamics of a low-order model for the Atlantic multidecadal oscillation,, Disc. Cont. Dyn. Syst. B, 16 (2011), 73. doi: 10.3934/dcdsb.2011.16.73. Google Scholar [8] M. I. Budyko, The effect of solar radiation variation on the climate of the Earth,, Tellus, 5 (1969), 611. Google Scholar [9] R. Cahalan and G. North, A stability theorem for energy-balance climate modes,, J. Atmos. Sci., 36 (1979), 1178. doi: 10.1175/1520-0469(1979)036<1178:ASTFEB>2.0.CO;2. Google Scholar [10] P. Chylek and J. A. Coakley, Analytical analysis of a Budyko-type climate model,, J. Atmos. Sci., 32 (1975), 675. doi: 10.1175/1520-0469(1975)032<0675:AAOABT>2.0.CO;2. Google Scholar [11] M. Claussen et al, Earth system models of intermediate complexity: Closing the gap in the spectrum of climate models,, Climate Dynamics, 18 (2002), 579. Google Scholar [12] C. Graves, W-H. Lee and G. North, New parameterizations and sensitivities for simple climate models,, J. Geophys. Res., 198 (1993), 5025. doi: 10.1029/92JD02666. Google Scholar [13] P. Hoffman, A. Kaufman, G. Halverson and D. Schrag, A Neoproterozoic snowball Earth,, Science, 281 (1998), 1342. doi: 10.1126/science.281.5381.1342. Google Scholar [14] P. Hoffman and D. Schrag, Snowball Earth,, Sci. Amer., 282 (2000), 68. Google Scholar [15] P. Hoffman and D. Schrag, The snowball Earth hypothesis: Testing the limits of global change,, Terra Nova, 14 (2002), 129. doi: 10.1046/j.1365-3121.2002.00408.x. Google Scholar [16] R. Kerr, Snowball Earth has melted back to a profound wintry mix,, Science, 327 (2010). doi: 10.1126/science.327.5970.1186. Google Scholar [17] J. Kirschivink, Late Proterozoic low-latitude global glaciation: the snowball Earth,, in The Proterozoic Biosphere: A Multidisciplinary Study (eds. J. Schopf and C. Klein), (1992). Google Scholar [18] W. Langford and G. Lewis, Poleward expansion of Hadley cells,, Can. Appl. Math. Quart., 17 (2009), 105. Google Scholar [19] R. Q. Lin and G. North, A study of abrupt climate change in a simple nonlinear climate model,, Climate Dynamics, 4 (1990), 253. doi: 10.1007/BF00211062. Google Scholar [20] R. A. Livermore, A. G. Smith and F. J. Vine, Late Palaeozoic to early mesozoic evolution of Pangaea,, Nature, 322 (1986), 162. doi: 10.1038/322162a0. Google Scholar [21] E. Lorenz, Irregularity: A fundamental property of the atmosphere,, Tellus, 36A (1984), 98. Google Scholar [22] L. Maas, A simple model for the three-dimensional, thermally and wind-driven ocean circulation,, Tellus, 46A (1994), 671. Google Scholar [23] F. Macdonald, M. Schmitz, J. Crowley, C. Root, D. Jones, A. Maloof, J. Strauss, P. Cohen, D. Johnston and D. Schrag, Calibrating the cryogenian,, Science, 327 (2010), 1241. doi: 10.1126/science.1183325. Google Scholar [24] H. Marshall, J. Walker and W. Kuhn, Long-term climate change and the geochemical cycle of carbon,, J. Geophys. Res., 93 (1988), 791. doi: 10.1029/JD093iD01p00791. Google Scholar [25] R., McGehee,, personal communication., (). Google Scholar [26] R. McGehee and C. Lehman, A paleoclimate model of ice-albedo feedback forced by variations in Earth's orbit,, SIAM J. Appl. Dyn. Syst., 11 (2012), 684. doi: 10.1137/10079879X. Google Scholar [27] R. McGehee and E. Widiasih, A finite dimensional version of a dynamic ice-albedo feedback model,, preprint., (). Google Scholar [28] G. North, Theory of energy-balance climate models,, J. Atmos. Sci., 32 (1975), 2033. doi: 10.1175/1520-0469(1975)032<2033:TOEBCM>2.0.CO;2. Google Scholar [29] R. Pierrehumbert, Principles of Planetary Climate,, Cambridge University Press, (2010). Google Scholar [30] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos,, CRC Press, (1995). doi: 10.1117/12.217385. Google Scholar [31] G. Roe and M. Baker, Note on a catastrophe: A feedback analysis of snowball Earth,, J. of Climate, 23 (2010), 4694. doi: 10.1175/2010JCLI3545.1. Google Scholar [32] P. Roebber, Climate variability in a low-order coupled atmosphere-ocean model,, Tellus, 47A (1995), 473. Google Scholar [33] C. Sagan and G. Mullen, Earth and Mars: Evolution of atmospheres and surface temperatures,, Science, 177 (1972), 52. doi: 10.1126/science.177.4043.52. Google Scholar [34] R. Secord, P. Gingerich, K. Lohmann and K. MacLeod, Continental warming preceding the Palaeocene-Eocene thermal maximum,, Nature, 467 (2010), 955. doi: 10.1038/nature09441. Google Scholar [35] W. Sellers, A global climatic model based on the energy balance of the Earth-Atmosphere system,, J. Appl. Meteor., 8 (1969), 392. doi: 10.1175/1520-0450(1969)008<0392:AGCMBO>2.0.CO;2. Google Scholar [36] A. Shil'nikov, G. Nicolis and C. Nicolis, Bifurcation and predictability analysis of a low-order atmospheric circulation model,, Int. J. Bif. Chaos, 5 (1995), 1701. doi: 10.1142/S0218127495001253. Google Scholar [37] H. E. de Swart, Low-order spectral models of the atmospheric circulation: A survey,, Acta Appl. Math., 11 (1988), 49. doi: 10.1007/BF00047114. Google Scholar [38] K. K. Tung, Topics in Mathematical Modeling,, Princeton University Press, (2007). Google Scholar [39] L. van Veen, Overturning and wind driven circulation in a low-order ocean-atmosphere model,, Dynam. Atmos. Ocean, 37 (2003), 197. doi: 10.1016/S0377-0265(03)00032-0. Google Scholar [40] L. van Veen, Baroclinic flow and the Lorenz-84 model,, Int. J. Bif. Chaos, 13 (2003), 2117. doi: 10.1142/S0218127403007904. Google Scholar [41] E. Widiasih, Instability of the ice free earth: Dynamics of a discrete time energy balance model,, preprint , (). Google Scholar
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