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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 and 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), p25. doi: 10.1029/2011JD015927.

[2]

J.-P. Aubin and A. Cellina, Differential Inclusions, Grundlehren der mathematischen Wissenschaften, 264, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.

[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), 401-419. doi: 10.3934/dcdsb.2008.10.401.

[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), 73-107. doi: 10.3934/dcdsb.2011.16.73.

[5]

M. I. Budyko, The effect of solar radiation variation on the climate of the Earth, Tellus, 5 (1969), 611-619.

[6]

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

[7]

J. Díaz, G. Hetzer and L. Tello, An energy balance climate model with hysteresis, Nonlin. Anal., 64 (2006), 2053-2074. doi: 10.1016/j.na.2005.07.038.

[8]

J. Díaz and L. Tello, Infinitely many solutions for a simple climate model via a shooting method, Math. Meth. Appl. Sci., 25 (2002), 327-334. doi: 10.1002/mma.289.

[9]

J. Díaz and L. Tello, On a climate model with a dynamic nonlinear diffusive boundary condition, Disc. Cont. Dyn. Syst. S, 1 (2008), 253-262. doi: 10.3934/dcdss.2008.1.253.

[10]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. Journal, 21 (1971), 193-226. doi: 10.1512/iumj.1972.21.21017.

[11]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Diff. Eq., 31 (1979), 53-98. doi: 10.1016/0022-0396(79)90152-9.

[12]

A. F. Filippov, Differential equations with discontinuous right-hand side, Amer. Math. Soc. Trans. Ser. 2, 42 (1964), 199-123.

[13]

M. Ghil and S. Childress, Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics, Springer-Verlag, New York, NY, USA, 1987. doi: 10.1007/978-1-4612-1052-8.

[14]

I. Held, Simplicity among complexity, Science, 343 (2014), 1206-1207.

[15]

I. Held and M. Suarez, Simple albedo feedback models of the icecaps, Tellus, 26 (1974), 613-629.

[16]

G. Hetzer, Trajectory attractors of energy balance climate models with bio-feedback, Differ. Equ. Appl., 3 (2011), 565-579. doi: 10.7153/dea-03-35.

[17]

C. K. R. T. Jones, Geometric singular perturbation theory, in Dynamical Systems, Montecatini Terme (ed. L. Arnold), Lecture Notes in Mathematics, 1609, Springer-Verlag, Berlin, 1995, 44-118. doi: 10.1007/BFb0095239.

[18]

R. I. Leine and H. Nijmeijer, Dynamics and Bifurcations of Non-smooth Mechanical Systems, Springer-Verlag, Berlin, Germany and New York, NY, USA, 2004. doi: 10.1007/978-3-540-44398-8.

[19]

R. Q. Lin and G. North, A study of abrupt climate change in a simple nonlinear climate model, Climate Dynamics, 4 (1990), 253-261. doi: 10.1007/BF00211062.

[20]

E. Lorenz, Irregularity: A fundamental property of the atmosphere, Tellus, 36A (1984), 98-110.

[21]

L. Maas, A simple model for the three-dimensional, thermally and wind-driven ocean circulation, Tellus, 46A (1994), 671-680.

[22]

C. Macilwain, A touch of the random, Science, 344 (2014), 1221-1223. doi: 10.1126/science.344.6189.1221.

[23]

, R. McGehee, see http://www-users.math.umn.edu/~mcgehee/Seminars/ClimateChange/presentations/2012-2Fall/20121106Handouts.pdf.

[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), 684-707. doi: 10.1137/10079879X.

[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), 518-536. doi: 10.1137/120871286.

[26]

G. North, Analytic solution to a simple climate with diffusive heat transport, J. Atmos. Sci., 32 (1975), 1301-1307.

[27]

G. North, Theory of energy-balance climate models, J. Atmos. Sci., 32 (1975), 2033-2043. doi: 10.1175/1520-0469(1975)032<2033:TOEBCM>2.0.CO;2.

[28]

G. North, R. Cahalan and J. Coakley, Energy balance climate models, Reviews of Geophysics and Space Physics, 19 (1981), 91-121.

[29]

R. T. Pierrehumbert, D. S. Abbot, A. Voigt and D. Koll, Climate of the Neoproterozoic, Ann. Rev. Earth Planet. Sci., 39 (2011), 417-460. doi: 10.1146/annurev-earth-040809-152447.

[30]

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

[31]

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press, Boca Raton, FL, 1995.

[32]

P. Roebber, Climate variability in a low-order coupled atmosphere-ocean model, Tellus, 47A (1995), 473-494.

[33]

W. Sellers, A global climatic model based on the energy balance of the Earth-Atmosphere system, J. Appl. Meteor., 8 (1969), 392-400. doi: 10.1175/1520-0450(1969)008<0392:AGCMBO>2.0.CO;2.

[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), 1701-1711. doi: 10.1142/S0218127495001253.

[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, Cambridge, United Kingdom and New York, NY, USA, 2013.

[36]

H. E. de Swart, Low-order spectral models of the atmospheric circulation: A survey, Acta Appl. Math., 11 (1988), 49-96. doi: 10.1007/BF00047114.

[37]

L. van Veen, Overturning and wind driven circulation in a low-order ocean-atmosphere model, Dynam. Atmos. Ocean, 37 (2003), 197-221. doi: 10.1016/S0377-0265(03)00032-0.

[38]

L. van Veen, Baroclinic flow and the Lorenz-84 model, Int. J. Bif. Chaos, 13 (2003), 2117-2139. doi: 10.1142/S0218127403007904.

[39]

J. A. Walsh and E. Widiasih, A dynamics approach to a low-order climate model, Disc. Cont. Dyn. Syst. B, 19 (2014), 257-279. doi: 10.3934/dcdsb.2014.19.257.

[40]

E. Widiasih, Dynamics of the Budyko energy balance model, SIAM J. Appl. Dyn. Syst., 12 (2013), 2068-2092. doi: 10.1137/100812306.

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), p25. doi: 10.1029/2011JD015927.

[2]

J.-P. Aubin and A. Cellina, Differential Inclusions, Grundlehren der mathematischen Wissenschaften, 264, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.

[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), 401-419. doi: 10.3934/dcdsb.2008.10.401.

[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), 73-107. doi: 10.3934/dcdsb.2011.16.73.

[5]

M. I. Budyko, The effect of solar radiation variation on the climate of the Earth, Tellus, 5 (1969), 611-619.

[6]

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

[7]

J. Díaz, G. Hetzer and L. Tello, An energy balance climate model with hysteresis, Nonlin. Anal., 64 (2006), 2053-2074. doi: 10.1016/j.na.2005.07.038.

[8]

J. Díaz and L. Tello, Infinitely many solutions for a simple climate model via a shooting method, Math. Meth. Appl. Sci., 25 (2002), 327-334. doi: 10.1002/mma.289.

[9]

J. Díaz and L. Tello, On a climate model with a dynamic nonlinear diffusive boundary condition, Disc. Cont. Dyn. Syst. S, 1 (2008), 253-262. doi: 10.3934/dcdss.2008.1.253.

[10]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. Journal, 21 (1971), 193-226. doi: 10.1512/iumj.1972.21.21017.

[11]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Diff. Eq., 31 (1979), 53-98. doi: 10.1016/0022-0396(79)90152-9.

[12]

A. F. Filippov, Differential equations with discontinuous right-hand side, Amer. Math. Soc. Trans. Ser. 2, 42 (1964), 199-123.

[13]

M. Ghil and S. Childress, Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics, Springer-Verlag, New York, NY, USA, 1987. doi: 10.1007/978-1-4612-1052-8.

[14]

I. Held, Simplicity among complexity, Science, 343 (2014), 1206-1207.

[15]

I. Held and M. Suarez, Simple albedo feedback models of the icecaps, Tellus, 26 (1974), 613-629.

[16]

G. Hetzer, Trajectory attractors of energy balance climate models with bio-feedback, Differ. Equ. Appl., 3 (2011), 565-579. doi: 10.7153/dea-03-35.

[17]

C. K. R. T. Jones, Geometric singular perturbation theory, in Dynamical Systems, Montecatini Terme (ed. L. Arnold), Lecture Notes in Mathematics, 1609, Springer-Verlag, Berlin, 1995, 44-118. doi: 10.1007/BFb0095239.

[18]

R. I. Leine and H. Nijmeijer, Dynamics and Bifurcations of Non-smooth Mechanical Systems, Springer-Verlag, Berlin, Germany and New York, NY, USA, 2004. doi: 10.1007/978-3-540-44398-8.

[19]

R. Q. Lin and G. North, A study of abrupt climate change in a simple nonlinear climate model, Climate Dynamics, 4 (1990), 253-261. doi: 10.1007/BF00211062.

[20]

E. Lorenz, Irregularity: A fundamental property of the atmosphere, Tellus, 36A (1984), 98-110.

[21]

L. Maas, A simple model for the three-dimensional, thermally and wind-driven ocean circulation, Tellus, 46A (1994), 671-680.

[22]

C. Macilwain, A touch of the random, Science, 344 (2014), 1221-1223. doi: 10.1126/science.344.6189.1221.

[23]

, R. McGehee, see http://www-users.math.umn.edu/~mcgehee/Seminars/ClimateChange/presentations/2012-2Fall/20121106Handouts.pdf.

[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), 684-707. doi: 10.1137/10079879X.

[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), 518-536. doi: 10.1137/120871286.

[26]

G. North, Analytic solution to a simple climate with diffusive heat transport, J. Atmos. Sci., 32 (1975), 1301-1307.

[27]

G. North, Theory of energy-balance climate models, J. Atmos. Sci., 32 (1975), 2033-2043. doi: 10.1175/1520-0469(1975)032<2033:TOEBCM>2.0.CO;2.

[28]

G. North, R. Cahalan and J. Coakley, Energy balance climate models, Reviews of Geophysics and Space Physics, 19 (1981), 91-121.

[29]

R. T. Pierrehumbert, D. S. Abbot, A. Voigt and D. Koll, Climate of the Neoproterozoic, Ann. Rev. Earth Planet. Sci., 39 (2011), 417-460. doi: 10.1146/annurev-earth-040809-152447.

[30]

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

[31]

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press, Boca Raton, FL, 1995.

[32]

P. Roebber, Climate variability in a low-order coupled atmosphere-ocean model, Tellus, 47A (1995), 473-494.

[33]

W. Sellers, A global climatic model based on the energy balance of the Earth-Atmosphere system, J. Appl. Meteor., 8 (1969), 392-400. doi: 10.1175/1520-0450(1969)008<0392:AGCMBO>2.0.CO;2.

[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), 1701-1711. doi: 10.1142/S0218127495001253.

[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, Cambridge, United Kingdom and New York, NY, USA, 2013.

[36]

H. E. de Swart, Low-order spectral models of the atmospheric circulation: A survey, Acta Appl. Math., 11 (1988), 49-96. doi: 10.1007/BF00047114.

[37]

L. van Veen, Overturning and wind driven circulation in a low-order ocean-atmosphere model, Dynam. Atmos. Ocean, 37 (2003), 197-221. doi: 10.1016/S0377-0265(03)00032-0.

[38]

L. van Veen, Baroclinic flow and the Lorenz-84 model, Int. J. Bif. Chaos, 13 (2003), 2117-2139. doi: 10.1142/S0218127403007904.

[39]

J. A. Walsh and E. Widiasih, A dynamics approach to a low-order climate model, Disc. Cont. Dyn. Syst. B, 19 (2014), 257-279. doi: 10.3934/dcdsb.2014.19.257.

[40]

E. Widiasih, Dynamics of the Budyko energy balance model, SIAM J. Appl. Dyn. Syst., 12 (2013), 2068-2092. doi: 10.1137/100812306.

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