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 & Continuous Dynamical Systems - B, 2017, 22 (7) : 2687-2715. doi: 10.3934/dcdsb.2017131
References:
[1]

D. AbbotA. 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]

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

[4]

H. BroerH. DijkstraC. SimóA. 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. Google Scholar

[5]

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

[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. Google Scholar

[7]

J. DíazG. 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. 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), 327-334. 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), 253-262. doi: 10.3934/dcdss.2008.1.253. Google Scholar

[10] M. di BernardoC. J. BuddA. 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. Google Scholar
[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. Google Scholar

[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. Google Scholar

[13]

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

[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. Google Scholar
[15]

C. GravesW.-H. Lee and G. North, New parameterizations and sensitivities for simple climate models, J. Geophys. Res., 98 (1993), 5025-5036. doi: 10.1029/92JD02666. Google Scholar

[16]

I. Held, Simplicity among complexity, Science, 343 (2014), 1206-1207. doi: 10.1126/science.1248447. Google Scholar

[17]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

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

[22]

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

[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. Google Scholar

[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. Google Scholar

[25]

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

[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. Google Scholar

[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. Google Scholar

[28]

G. NorthR. Cahalan and J. Coakley, Energy balance climate models, Reviews of Geophysics and Space Physics, 19 (1981), 91-121. doi: 10.1029/RG019i001p00091. Google Scholar

[29]

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

[30]

A. RobertsJ. GuckenheimerE. WidiasihA. Timmerman and C. K. R. T. Jones, Mixed-mode oscillations of El Niño-Southern Oscillation, J. Atmos. Sci., 73 (1995), 1755-1766. Google Scholar

[31]

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

[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. Google Scholar

[33]

A. Shil'nikovG. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[36]

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

[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. Google Scholar

[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. Google Scholar

[39]

J. A. WalshE. WidiasihJ. 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. Google Scholar

[40]

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

show all references

References:
[1]

D. AbbotA. 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]

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

[4]

H. BroerH. DijkstraC. SimóA. 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. Google Scholar

[5]

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

[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. Google Scholar

[7]

J. DíazG. 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. 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), 327-334. 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), 253-262. doi: 10.3934/dcdss.2008.1.253. Google Scholar

[10] M. di BernardoC. J. BuddA. 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. Google Scholar
[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. Google Scholar

[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. Google Scholar

[13]

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

[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. Google Scholar
[15]

C. GravesW.-H. Lee and G. North, New parameterizations and sensitivities for simple climate models, J. Geophys. Res., 98 (1993), 5025-5036. doi: 10.1029/92JD02666. Google Scholar

[16]

I. Held, Simplicity among complexity, Science, 343 (2014), 1206-1207. doi: 10.1126/science.1248447. Google Scholar

[17]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

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

[22]

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

[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. Google Scholar

[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. Google Scholar

[25]

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

[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. Google Scholar

[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. Google Scholar

[28]

G. NorthR. Cahalan and J. Coakley, Energy balance climate models, Reviews of Geophysics and Space Physics, 19 (1981), 91-121. doi: 10.1029/RG019i001p00091. Google Scholar

[29]

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

[30]

A. RobertsJ. GuckenheimerE. WidiasihA. Timmerman and C. K. R. T. Jones, Mixed-mode oscillations of El Niño-Southern Oscillation, J. Atmos. Sci., 73 (1995), 1755-1766. Google Scholar

[31]

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

[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. Google Scholar

[33]

A. Shil'nikovG. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[36]

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

[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. Google Scholar

[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. Google Scholar

[39]

J. A. WalshE. WidiasihJ. 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. Google Scholar

[40]

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

Figure 1.  Solid: Plot of (3) with $\beta=24.5^\circ$. Dashed: Quadratic approximation (6)
Figure 2.  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: $N=5$
Figure 3.  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).
Figure 4.  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)
Figure 5.  Bifurcation plot for the Jormungand model with diffusive heat transport, with $N=5$ in (15)
Figure 6.  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), $C=3.09$
Figure 7.  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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