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September  2015, 20(7): 2187-2216. doi: 10.3934/dcdsb.2015.20.2187

On the Budyko-Sellers energy balance climate model with ice line coupling

James Walsh 1, and  Christopher Rackauckas 2,

1. 

Department of Mathematics, Oberlin College, 10 N. Professor St, Oberlin, OH 44074
2. 

Department of Mathematics, University of California{Irvine, Irvine, CA 92697, United States

Received  September 2014 Revised  January 2015 Published  July 2015

Over 40 years ago, M. Budyko and W. Sellers independently introduced low-order climate models that continue to play an important role in the mathematical modeling of climate. Each model has one spatial variable, and each was introduced to investigate the role ice-albedo feedback plays in influencing surface temperature. This paper serves in part as a tutorial on the Budyko-Sellers model, with particular focus placed on the coupling of this model with an ice sheet that is allowed to respond to changes in temperature, as introduced in recent work by E. Widiasih. We review known results regarding the dynamics of this coupled model, with both continuous (``Sellers-type") and discontinuous (``Budyko-type") equations. We also introduce two new Budyko-type models that are highly effective in modeling the extreme glacial events of the Neoproterozoic Era. We prove in each case the existence of a stable equilibrium solution for which the ice sheet edge rests in tropical latitudes. Mathematical tools used in the analysis include geometric singular perturbation theory and Filippov's theory of differential inclusions.
Keywords: ice-albedo feedback, singular perturbations, Climate modeling, differential inclusions..
Mathematics Subject Classification: Primary: 34K19; Secondary: 34K26, 34A60, 86-0.
Citation: James Walsh, Christopher Rackauckas. On the Budyko-Sellers energy balance climate model with ice line coupling. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2187-2216. doi: 10.3934/dcdsb.2015.20.2187
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]

J.-P. Aubin and A. Cellina, Differential Inclusions,, Grundlehren der mathematischen Wissenschaften, (1984). doi: 10.1007/978-3-642-69512-4. Google Scholar

[3]

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

[4]

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

[5]

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

[6]

J. Díaz, ed., The Mathematics of Models for Climatology and Environment,, Springer-Verlag (published in cooperation with NATO Scientific Affairs Division), (1997). doi: 10.1007/978-3-642-60603-8. Google Scholar

[7]

J. Díaz, G. Hetzer and L. Tello, An energy balance climate model with hysteresis,, Nonlin. Anal., 64 (2006), 2053. doi: 10.1016/j.na.2005.07.038. Google Scholar

[8]

J. Díaz and L. Tello, Infinitely many solutions for a simple climate model via a shooting method,, Math. Meth. Appl. Sci., 25 (2002), 327. doi: 10.1002/mma.289. Google Scholar

[9]

J. Díaz and L. Tello, On a climate model with a dynamic nonlinear diffusive boundary condition,, Disc. Cont. Dyn. Syst. S, 1 (2008), 253. doi: 10.3934/dcdss.2008.1.253. Google Scholar

[10]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows,, Indiana Univ. Math. Journal, 21 (1971), 193. doi: 10.1512/iumj.1972.21.21017. Google Scholar

[11]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations,, J. Diff. Eq., 31 (1979), 53. doi: 10.1016/0022-0396(79)90152-9. Google Scholar

[12]

A. F. Filippov, Differential equations with discontinuous right-hand side,, Amer. Math. Soc. Trans. Ser. 2, 42 (1964), 199. Google Scholar

[13]

M. Ghil and S. Childress, Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics,, Springer-Verlag, (1987). doi: 10.1007/978-1-4612-1052-8. Google Scholar

[14]

I. Held, Simplicity among complexity,, Science, 343 (2014), 1206. Google Scholar

[15]

I. Held and M. Suarez, Simple albedo feedback models of the icecaps,, Tellus, 26 (1974), 613. Google Scholar

[16]

G. Hetzer, Trajectory attractors of energy balance climate models with bio-feedback,, Differ. Equ. Appl., 3 (2011), 565. doi: 10.7153/dea-03-35. Google Scholar

[17]

C. K. R. T. Jones, Geometric singular perturbation theory,, in Dynamical Systems, (1609), 44. doi: 10.1007/BFb0095239. Google Scholar

[18]

R. I. Leine and H. Nijmeijer, Dynamics and Bifurcations of Non-smooth Mechanical Systems,, Springer-Verlag, (2004). doi: 10.1007/978-3-540-44398-8. 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]

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

[21]

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

[22]

C. Macilwain, A touch of the random,, Science, 344 (2014), 1221. doi: 10.1126/science.344.6189.1221. Google Scholar

[23]

, R. McGehee,, see , (): 2012. Google Scholar

[24]

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

[25]

R. McGehee and E. Widiasih, A quadratic approximation to Budyko's ice-albedo feedback model with ice line dynamics,, SIAM J. Appl. Dyn. Syst., 13 (2014), 518. doi: 10.1137/120871286. Google Scholar

[26]

G. North, Analytic solution to a simple climate with diffusive heat transport,, J. Atmos. Sci., 32 (1975), 1301. Google Scholar

[27]

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

[28]

G. North, R. Cahalan and J. Coakley, Energy balance climate models,, Reviews of Geophysics and Space Physics, 19 (1981), 91. Google Scholar

[29]

R. T. Pierrehumbert, D. S. Abbot, A. Voigt and D. Koll, Climate of the Neoproterozoic,, Ann. Rev. Earth Planet. Sci., 39 (2011), 417. doi: 10.1146/annurev-earth-040809-152447. Google Scholar

[30]

C. J. Poulsen and R. L. Jacob, Factors that inhibit snowball Earth simulation,, Paleoceanography, 19 (2004). doi: 10.1029/2004PA001056. Google Scholar

[31]

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos,, CRC Press, (1995). Google Scholar

[32]

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

[33]

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

[34]

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

[35]

T. Stocker, D. Qin, G.-K. Plattner, M. Tignor, S. Allen, J. Boschung, A. Nauels, Y. Xia, V. Bex and P. Midgley, eds., Climate Change 2013 The Physical Science Basis, Chapter 1,, Cambridge University Press, (2013). Google Scholar

[36]

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

[37]

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

[38]

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

[39]

J. A. Walsh and E. Widiasih, A dynamics approach to a low-order climate model,, Disc. Cont. Dyn. Syst. B, 19 (2014), 257. doi: 10.3934/dcdsb.2014.19.257. Google Scholar

[40]

E. Widiasih, Dynamics of the Budyko energy balance model,, SIAM J. Appl. Dyn. Syst., 12 (2013), 2068. doi: 10.1137/100812306. 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]

J.-P. Aubin and A. Cellina, Differential Inclusions,, Grundlehren der mathematischen Wissenschaften, (1984). doi: 10.1007/978-3-642-69512-4. Google Scholar

[3]

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

[4]

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

[5]

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

[6]

J. Díaz, ed., The Mathematics of Models for Climatology and Environment,, Springer-Verlag (published in cooperation with NATO Scientific Affairs Division), (1997). doi: 10.1007/978-3-642-60603-8. Google Scholar

[7]

J. Díaz, G. Hetzer and L. Tello, An energy balance climate model with hysteresis,, Nonlin. Anal., 64 (2006), 2053. doi: 10.1016/j.na.2005.07.038. Google Scholar

[8]

J. Díaz and L. Tello, Infinitely many solutions for a simple climate model via a shooting method,, Math. Meth. Appl. Sci., 25 (2002), 327. doi: 10.1002/mma.289. Google Scholar

[9]

J. Díaz and L. Tello, On a climate model with a dynamic nonlinear diffusive boundary condition,, Disc. Cont. Dyn. Syst. S, 1 (2008), 253. doi: 10.3934/dcdss.2008.1.253. Google Scholar

[10]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows,, Indiana Univ. Math. Journal, 21 (1971), 193. doi: 10.1512/iumj.1972.21.21017. Google Scholar

[11]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations,, J. Diff. Eq., 31 (1979), 53. doi: 10.1016/0022-0396(79)90152-9. Google Scholar

[12]

A. F. Filippov, Differential equations with discontinuous right-hand side,, Amer. Math. Soc. Trans. Ser. 2, 42 (1964), 199. Google Scholar

[13]

M. Ghil and S. Childress, Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics,, Springer-Verlag, (1987). doi: 10.1007/978-1-4612-1052-8. Google Scholar

[14]

I. Held, Simplicity among complexity,, Science, 343 (2014), 1206. Google Scholar

[15]

I. Held and M. Suarez, Simple albedo feedback models of the icecaps,, Tellus, 26 (1974), 613. Google Scholar

[16]

G. Hetzer, Trajectory attractors of energy balance climate models with bio-feedback,, Differ. Equ. Appl., 3 (2011), 565. doi: 10.7153/dea-03-35. Google Scholar

[17]

C. K. R. T. Jones, Geometric singular perturbation theory,, in Dynamical Systems, (1609), 44. doi: 10.1007/BFb0095239. Google Scholar

[18]

R. I. Leine and H. Nijmeijer, Dynamics and Bifurcations of Non-smooth Mechanical Systems,, Springer-Verlag, (2004). doi: 10.1007/978-3-540-44398-8. 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]

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

[21]

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

[22]

C. Macilwain, A touch of the random,, Science, 344 (2014), 1221. doi: 10.1126/science.344.6189.1221. Google Scholar

[23]

, R. McGehee,, see , (): 2012. Google Scholar

[24]

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

[25]

R. McGehee and E. Widiasih, A quadratic approximation to Budyko's ice-albedo feedback model with ice line dynamics,, SIAM J. Appl. Dyn. Syst., 13 (2014), 518. doi: 10.1137/120871286. Google Scholar

[26]

G. North, Analytic solution to a simple climate with diffusive heat transport,, J. Atmos. Sci., 32 (1975), 1301. Google Scholar

[27]

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

[28]

G. North, R. Cahalan and J. Coakley, Energy balance climate models,, Reviews of Geophysics and Space Physics, 19 (1981), 91. Google Scholar

[29]

R. T. Pierrehumbert, D. S. Abbot, A. Voigt and D. Koll, Climate of the Neoproterozoic,, Ann. Rev. Earth Planet. Sci., 39 (2011), 417. doi: 10.1146/annurev-earth-040809-152447. Google Scholar

[30]

C. J. Poulsen and R. L. Jacob, Factors that inhibit snowball Earth simulation,, Paleoceanography, 19 (2004). doi: 10.1029/2004PA001056. Google Scholar

[31]

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos,, CRC Press, (1995). Google Scholar

[32]

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

[33]

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

[34]

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

[35]

T. Stocker, D. Qin, G.-K. Plattner, M. Tignor, S. Allen, J. Boschung, A. Nauels, Y. Xia, V. Bex and P. Midgley, eds., Climate Change 2013 The Physical Science Basis, Chapter 1,, Cambridge University Press, (2013). Google Scholar

[36]

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

[37]

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

[38]

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

[39]

J. A. Walsh and E. Widiasih, A dynamics approach to a low-order climate model,, Disc. Cont. Dyn. Syst. B, 19 (2014), 257. doi: 10.3934/dcdsb.2014.19.257. Google Scholar

[40]

E. Widiasih, Dynamics of the Budyko energy balance model,, SIAM J. Appl. Dyn. Syst., 12 (2013), 2068. doi: 10.1137/100812306. Google Scholar

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