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