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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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

M. I. Budyko, The effect of solar radiation variation on the climate of the Earth,, Tellus, 5 (1969), 611.

[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.

[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.

[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.

[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.

[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.

[14]

P. Hoffman and D. Schrag, Snowball Earth,, Sci. Amer., 282 (2000), 68.

[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.

[16]

R. Kerr, Snowball Earth has melted back to a profound wintry mix,, Science, 327 (2010). doi: 10.1126/science.327.5970.1186.

[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).

[18]

W. Langford and G. Lewis, Poleward expansion of Hadley cells,, Can. Appl. Math. Quart., 17 (2009), 105.

[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.

[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.

[21]

E. Lorenz, Irregularity: A fundamental property of the atmosphere,, Tellus, 36A (1984), 98.

[22]

L. Maas, A simple model for the three-dimensional, thermally and wind-driven ocean circulation,, Tellus, 46A (1994), 671.

[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.

[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.

[25]

R., McGehee,, personal communication., ().

[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.

[27]

R. McGehee and E. Widiasih, A finite dimensional version of a dynamic ice-albedo feedback model,, preprint., ().

[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.

[29]

R. Pierrehumbert, Principles of Planetary Climate,, Cambridge University Press, (2010).

[30]

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos,, CRC Press, (1995). doi: 10.1117/12.217385.

[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.

[32]

P. Roebber, Climate variability in a low-order coupled atmosphere-ocean model,, Tellus, 47A (1995), 473.

[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.

[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.

[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.

[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.

[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.

[38]

K. K. Tung, Topics in Mathematical Modeling,, Princeton University Press, (2007).

[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.

[40]

L. van Veen, Baroclinic flow and the Lorenz-84 model,, Int. J. Bif. Chaos, 13 (2003), 2117. doi: 10.1142/S0218127403007904.

[41]

E. Widiasih, Instability of the ice free earth: Dynamics of a discrete time energy balance model,, preprint , ().

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

M. I. Budyko, The effect of solar radiation variation on the climate of the Earth,, Tellus, 5 (1969), 611.

[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.

[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.

[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.

[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.

[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.

[14]

P. Hoffman and D. Schrag, Snowball Earth,, Sci. Amer., 282 (2000), 68.

[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.

[16]

R. Kerr, Snowball Earth has melted back to a profound wintry mix,, Science, 327 (2010). doi: 10.1126/science.327.5970.1186.

[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).

[18]

W. Langford and G. Lewis, Poleward expansion of Hadley cells,, Can. Appl. Math. Quart., 17 (2009), 105.

[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.

[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.

[21]

E. Lorenz, Irregularity: A fundamental property of the atmosphere,, Tellus, 36A (1984), 98.

[22]

L. Maas, A simple model for the three-dimensional, thermally and wind-driven ocean circulation,, Tellus, 46A (1994), 671.

[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.

[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.

[25]

R., McGehee,, personal communication., ().

[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.

[27]

R. McGehee and E. Widiasih, A finite dimensional version of a dynamic ice-albedo feedback model,, preprint., ().

[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.

[29]

R. Pierrehumbert, Principles of Planetary Climate,, Cambridge University Press, (2010).

[30]

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos,, CRC Press, (1995). doi: 10.1117/12.217385.

[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.

[32]

P. Roebber, Climate variability in a low-order coupled atmosphere-ocean model,, Tellus, 47A (1995), 473.

[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.

[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.

[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.

[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.

[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.

[38]

K. K. Tung, Topics in Mathematical Modeling,, Princeton University Press, (2007).

[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.

[40]

L. van Veen, Baroclinic flow and the Lorenz-84 model,, Int. J. Bif. Chaos, 13 (2003), 2117. doi: 10.1142/S0218127403007904.

[41]

E. Widiasih, Instability of the ice free earth: Dynamics of a discrete time energy balance model,, preprint , ().

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