American Institute of Mathematical Sciences

September  2017, 22(7): 2687-2715. doi: 10.3934/dcdsb.2017131

Diffusive heat transport in Budyko's energy balance climate model with a dynamic ice line

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

Received  July 2016 Revised  September 2016 Published  April 2017

Fund Project: The author recognizes and appreciates the support of the Mathematics and Climate Research Network (http://www.mathclimate.org).

M. Budyko and W. Sellers independently introduced seminal energy balance climate models in 1969, each with a goal of investigating the role played by positive ice albedo feedback in climate dynamics. In this paper we replace the relaxation to the mean horizontal heat transport mechanism used in the models of Budyko and Sellers with diffusive heat transport. We couple the resulting surface temperature equation with an equation for movement of the edge of the ice sheet (called the ice line), recently introduced by E. Widiasih. We apply the spectral method to the temperature-ice line system and consider finite approximations. We prove there exists a stable equilibrium solution with a small ice cap, and an unstable equilibrium solution with a large ice cap, for a range of parameter values. If the diffusive transport is too efficient, however, the small ice cap disappears and an ice free Earth becomes a limiting state. In addition, we analyze a variant of the coupled diffusion equations appropriate as a model for extensive glacial episodes in the Neoproterozoic Era. Although the model equations are no longer smooth due to the existence of a switching boundary, we prove there exists a unique stable equilibrium solution with the ice line in tropical latitudes, a climate event known as a Jormungand or Waterbelt state. As the systems introduced here contain variables with differing time scales, the main tool used in the analysis is geometric singular perturbation theory.

Citation: James Walsh. Diffusive heat transport in Budyko's energy balance climate model with a dynamic ice line. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2687-2715. doi: 10.3934/dcdsb.2017131
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] P. Ashwin and P. Ditlevsen, The middle Pleistocene transition as a generic bifurcation on a slow manifold, Climate Dynamics, 45 (2015), 2683-2695.  doi: 10.1007/s00382-015-2501-9. [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] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications, Springer-Verlag, London, UK, 2008.  doi: 10.1007/978-1-84628-708-4. [11] 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. [12] 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. [13] A. F. Filippov, Differential equations with discontinuous right-hand side, Amer. Math. Soc. Trans. Ser. 2, 42 (1964), 199-123. [14] 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. [15] C. Graves, W.-H. Lee and G. North, New parameterizations and sensitivities for simple climate models, J. Geophys. Res., 98 (1993), 5025-5036.  doi: 10.1029/92JD02666. [16] I. Held, Simplicity among complexity, Science, 343 (2014), 1206-1207.  doi: 10.1126/science.1248447. [17] I. Held and M. Suarez, Simple albedo feedback models of the icecaps, Tellus, 26 (1974), 613-629. [18] 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. [19] 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. [20] 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. [21] E. Lorenz, Irregularity: A fundamental property of the atmosphere, Tellus, 36 (1984), 98-110. [22] L. Maas, A simple model for the three-dimensional, thermally and wind-driven ocean circulation, Tellus, 46 (1994), 671-680. [23] 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. [24] 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. [25] G. North, Analytic solution to a simple climate with diffusive heat transport, J. Atmos. Sci., 32 (1975), 1301-1307. [26] 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. [27] G. North, The small ice cap instability in diffusive climate models, J. Atmos. Sci., 41 (1984), 3390-3395.  doi: 10.1175/1520-0469(1984)041<3390:TSICII>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.  doi: 10.1029/RG019i001p00091. [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] A. Roberts, J. Guckenheimer, E. Widiasih, A. Timmerman and C. K. R. T. Jones, Mixed-mode oscillations of El Niño-Southern Oscillation, J. Atmos. Sci., 73 (1995), 1755-1766. [31] P. Roebber, Climate variability in a low-order coupled atmosphere-ocean model, Tellus, 47 (1995), 473-494. [32] 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. [33] 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. [34] 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. [35] 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. [36] L. van Veen, Baroclinic flow and the Lorenz-84 model, Int. J. Bif. Chaos, 13 (2003), 2117-2139.  doi: 10.1142/S0218127403007904. [37] J. A. Walsh and C. Rackauckas, On the Budyko-Sellers energy balance climate model with ice line coupling, Disc. Cont. Dyn. Syst. B, 20 (2015), 2187-2216.  doi: 10.3934/dcdsb.2015.20.2187. [38] 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. [39] J. A. Walsh, E. Widiasih, J. Hahn and R. McGehee, Periodic orbits for a discontinuous vector field arising from a conceptual model of glacial cycles, Nonlinearity, 29 (2016), 1843-1864.  doi: 10.1088/0951-7715/29/6/1843. [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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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] P. Ashwin and P. Ditlevsen, The middle Pleistocene transition as a generic bifurcation on a slow manifold, Climate Dynamics, 45 (2015), 2683-2695.  doi: 10.1007/s00382-015-2501-9. [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] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications, Springer-Verlag, London, UK, 2008.  doi: 10.1007/978-1-84628-708-4. [11] 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. [12] 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. [13] A. F. Filippov, Differential equations with discontinuous right-hand side, Amer. Math. Soc. Trans. Ser. 2, 42 (1964), 199-123. [14] 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. [15] C. Graves, W.-H. Lee and G. North, New parameterizations and sensitivities for simple climate models, J. Geophys. Res., 98 (1993), 5025-5036.  doi: 10.1029/92JD02666. [16] I. Held, Simplicity among complexity, Science, 343 (2014), 1206-1207.  doi: 10.1126/science.1248447. [17] I. Held and M. Suarez, Simple albedo feedback models of the icecaps, Tellus, 26 (1974), 613-629. [18] 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. [19] 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. [20] 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. [21] E. Lorenz, Irregularity: A fundamental property of the atmosphere, Tellus, 36 (1984), 98-110. [22] L. Maas, A simple model for the three-dimensional, thermally and wind-driven ocean circulation, Tellus, 46 (1994), 671-680. [23] 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. [24] 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. [25] G. North, Analytic solution to a simple climate with diffusive heat transport, J. Atmos. Sci., 32 (1975), 1301-1307. [26] 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. [27] G. North, The small ice cap instability in diffusive climate models, J. Atmos. Sci., 41 (1984), 3390-3395.  doi: 10.1175/1520-0469(1984)041<3390:TSICII>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.  doi: 10.1029/RG019i001p00091. [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] A. Roberts, J. Guckenheimer, E. Widiasih, A. Timmerman and C. K. R. T. Jones, Mixed-mode oscillations of El Niño-Southern Oscillation, J. Atmos. Sci., 73 (1995), 1755-1766. [31] P. Roebber, Climate variability in a low-order coupled atmosphere-ocean model, Tellus, 47 (1995), 473-494. [32] 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. [33] 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. [34] 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. [35] 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. [36] L. van Veen, Baroclinic flow and the Lorenz-84 model, Int. J. Bif. Chaos, 13 (2003), 2117-2139.  doi: 10.1142/S0218127403007904. [37] J. A. Walsh and C. Rackauckas, On the Budyko-Sellers energy balance climate model with ice line coupling, Disc. Cont. Dyn. Syst. B, 20 (2015), 2187-2216.  doi: 10.3934/dcdsb.2015.20.2187. [38] 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. [39] J. A. Walsh, E. Widiasih, J. Hahn and R. McGehee, Periodic orbits for a discontinuous vector field arising from a conceptual model of glacial cycles, Nonlinearity, 29 (2016), 1843-1864.  doi: 10.1088/0951-7715/29/6/1843. [40] E. Widiasih, Dynamics of the Budyko energy balance model, SIAM J. Appl. Dyn. Syst., 12 (2013), 2068-2092.  doi: 10.1137/100812306.
Solid: Plot of (3) with $\beta=24.5^\circ$. Dashed: Quadratic approximation (6)
Plots of function (25). (a) $N=1$ in (15), $D=0.45$. (b) $N=1, D=0.35$. (c) D=0.35. Solid: $N=2$. Dashed: $N=5$
Plots of function (25) with $D=0.394$. (a) and (b) $N=1$ in (15). (c) Solid: $N=2$ in (15). Dashed: $N=5$ in (15).
Plots of the Jormungand diffusion model functions $h^-(\eta) \,$ ($\eta<\rho) \,$ and $h^+(\eta) \,$ $\, (\eta\geq \rho$) for $D=0.25$. (a) $N=1$ in (15). (b) Solid: $N=2$ in (15). Dashed: $N=5$ in (15)
Bifurcation plot for the Jormungand model with diffusive heat transport, with $N=5$ in (15)
Plots of function $h(\eta)$ given by (52). (a) $N=1$ in (15). (b) Including higher-order terms in (4). Solid: $N=2$ ($s_4=-0.044$). Dashed: $N=5$ ($s_6=0.006, s_8=0.016, s_{10}=0.006$). Parameters as in (26), $C=3.09$
Plots of functions $h^-(\eta)$ (64) (for $\eta<\rho$) and $h^+(\eta)$ (for $\eta> \rho$) (a) $N=1$ in (15). (b) $N=5$ ($s_4, ... s_{10}$ as in Figure 6). Parameters for (a) and (b) as in Figure 13 in [37]. (c) $N=1$, parameters as in (40)
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