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August  2017, 22(6): 2447-2463. doi: 10.3934/dcdsb.2017125

## Nonsmooth frameworks for an extended Budyko model

 1 Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand 2 Mathematics and Science Subdivision, University of Hawaii, West Oahu, 91-1001 Farrington Highway, Library Room 203 Kapolei, HI 96707, USA 3 Department of Mathematics, University of Minnesota, 206 Church St SE, Minneapolis, MN 55455, USA

Anna M. Barry, E-mail address: anna.barry@auckland.ac.nz

Received  October 2015 Revised  March 2017 Published  March 2017

In latitude-dependent energy balance models, ice-free and ice-covered conditions form physical boundaries of the system. With carbon dioxide treated as a bifurcation parameter, the resulting bifurcation diagram is nonsmooth with curves of equilibria and boundaries forming corners at points of intersection. Over long time scales, atmospheric carbon dioxide varies dynamically and the nonsmooth diagram becomes a set of quasi-equilibria. However, when introducing carbon dynamics, care must be taken with the physical boundaries and appropriate boundary motion specified. In this article, we extend an energy balance model to include slowly varying carbon dioxide and develop nonsmooth frameworks based on physically relevant boundary dynamics. Within these frameworks, we prove existence and uniqueness of solutions, as well as invariance of the region of phase space bounded by ice-free and ice-covered states.

Citation: Anna M. Barry, Esther WIdiasih, Richard Mcgehee. Nonsmooth frameworks for an extended Budyko model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2447-2463. doi: 10.3934/dcdsb.2017125
##### References:
 [1] D. Abbot, A. Voigt and D. Koll, The Jormungand global climate state and implications for Neoproterozoic glaciations, J. Geophys. Res., 2011. [2] É. Benoît, Chasse au canard. Ⅱ. Tunnels-entonnoirs-peignes, Collect. Math., 32 (1981), 77-97. [3] É. Benoît, Systémes lents-rapides dans $\mathbb{R}^3$ et leurs canards, in Third Schnepfenried Geometry Conference, Vol. 2 (Schnepfenried, 1982), Astérisque, 109, Soc. Math. France, Paris, 1983,159-191. [4] É. Benoît and J.-L. Callot, Chasse au canard. Ⅳ. Annexe numérique, Collect. Math., 32 (1981), 115-119. [5] M. I. Budyko, The effect of solar radiation variations on the climate of the earth, Tellus, 21 (1969), 611-619. [6] R. F. Cahalan and G. R. North, A stability theorem for energy-balance climate models, Journal of the Atmospheric Sciences, 36 (1979), 1178-1188. [7] K. Caldeira and J. F. Kasting, Susceptibility of the early earth to irreversible glaciation caused by carbon dioxide clouds, Nature, 359 (1992), 226-228. [8] J.-L. Callot, Chasse au canard. Ⅲ. Les canards ont la vie bréve, Collect. Math., 32 (1981), 99-114. [9] C. Carathéodory, Vorlesungen Über reelle Funktionen, Leipzig, 1927. [10] M. Desroches, J. Guckenheimer, B. Krauskopf, C. Kuehn, H. Osinga and M. Wechselberger, Mixed-mode oscillations with multiple time scales, SIAM Review, 54 (2012), 211-288. [11] F. Diener and M. Diener, Chasse au canard. Ⅰ. Les canards, Collect. Math., 32 (1981), 37-74. [12] F. Dumortier and R. H. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996), x+100 pp. [13] J. M. Edmond, M. R. Palmer, E. T. Brown and Y. Huh, Fluvial geochemistry of the eastern slope of the northeastern andes and its foredeep in the drainage of the orinoco in colombia and venezuela, Geochimica et cosmochimica acta, 60 (1996), 2949-2974. [14] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, Journal of Differential Equations, 31 (1979), 53-98. [15] A. F. Filippov, Differential equations with discontinuous right-hand side, American Mathematical Society Translations, 2 (1964), 199-231. [16] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publ. Dortrecht, 1988. [17] C. E. Graves, W.-H. Lee and G. R. North, New parameterizations and sensitivities for simple climate models, Journal of Geophysical Research, 98 (1993), 5025-5036. [18] P. Hoffman and D. Schrag, The snowball earth hypothesis: Testing the limits of global change, Terra Nova, 14 (2002), 129-155. [19] A. M. Hogg, Glacial cycles and carbon dioxide: A conceptual model, Geophysical Research Letters, 35 (2008), L01701. [20] M. R. Jeffrey, Hidden dynamics in models of discontinuity and switching, Physica D: Nonlinear Phenomena, 273 (2014), 34-45. [21] C. K. R. T. Jones, Geometric singular perturbation theory, Dynamical Systems, 1609 (1995), 44-118. [22] J. Kirschvink, Late proterozoic low-latitude global glaciation: The snowball earth, The Proterozoic Biosphere: A Multidisciplinary Study, (1992), 51-52. [23] M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions, SIAM Journal on Mathematical Analysis, 33 (2001), 286-314. [24] L. R. Kump, S. L. Brantley and M. A. Arthur, Chemical weathering, atmospheric co2, and climate, Annual Review of Earth and Planetary Sciences, 28 (2000), 611-667. [25] R. McGehee and C. Lehman, A paleoclimate model of ice-albedo feedback forced by variations in earth's orbit, SIAM Journal on Applied Dynamical Systems, 11 (2012), 684-707. [26] 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. [27] G. R. North, Theory of energy-balance climate models, J. Atmos. Sci, 32 (1975), 2033-2043. [28] R. T. Pierrehumbert, D. S. Abbot, A. Voigt and D. Koll, Climate of the Neoproterozoic, Annual Review of Earth and Planetary Sciences, 39 (2011), 417-460. [29] D. Pollard and J. F. Kasting, Snowball Earth: A thin-ice solution with flowing sea glaciers, Journal of Geophysical Research: Oceans (1978-2012), 110(C7), 2005. [30] W. D. Sellers, A global climatic model based on the energy balance of the earth-atmosphere system, Journal of Applied Meteorology, 8 (1969), 392-400. [31] J. Sieber and P. Kowalczyk, Small-scale instabilities in dynamical systems with sliding, Physica D: Nonlinear Phenomena, 239 (2010), 44-57. [32] P. Szmolyan and M. Wechselberger, Canards in $\mathbb{R}^3$, Journal of Differential Equations, 177 (2001), 419-453. [33] K. K. Tung, Topics in Mathematical Modelling, Princeton University Press, 2007. [34] J. Walsh and E. Widiasih, A dynamics approach to a low-order climate model, Discrete & Continuous Dynamical Systems-Series B, 19 (2014), 257-279. [35] M. Wechselberger, Existence and bifurcation of canards in $\mathbb{R}^3$ in the case of a folded node, SIAM Journal on Applied Dynamical Systems, 4 (2005), 101-139. [36] E. R. Widiasih, Dynamics of the Budyko energy balance model, SIAM Journal on Applied Dynamical Systems, 12 (2013), 2068-2092.

show all references

##### References:
 [1] D. Abbot, A. Voigt and D. Koll, The Jormungand global climate state and implications for Neoproterozoic glaciations, J. Geophys. Res., 2011. [2] É. Benoît, Chasse au canard. Ⅱ. Tunnels-entonnoirs-peignes, Collect. Math., 32 (1981), 77-97. [3] É. Benoît, Systémes lents-rapides dans $\mathbb{R}^3$ et leurs canards, in Third Schnepfenried Geometry Conference, Vol. 2 (Schnepfenried, 1982), Astérisque, 109, Soc. Math. France, Paris, 1983,159-191. [4] É. Benoît and J.-L. Callot, Chasse au canard. Ⅳ. Annexe numérique, Collect. Math., 32 (1981), 115-119. [5] M. I. Budyko, The effect of solar radiation variations on the climate of the earth, Tellus, 21 (1969), 611-619. [6] R. F. Cahalan and G. R. North, A stability theorem for energy-balance climate models, Journal of the Atmospheric Sciences, 36 (1979), 1178-1188. [7] K. Caldeira and J. F. Kasting, Susceptibility of the early earth to irreversible glaciation caused by carbon dioxide clouds, Nature, 359 (1992), 226-228. [8] J.-L. Callot, Chasse au canard. Ⅲ. Les canards ont la vie bréve, Collect. Math., 32 (1981), 99-114. [9] C. Carathéodory, Vorlesungen Über reelle Funktionen, Leipzig, 1927. [10] M. Desroches, J. Guckenheimer, B. Krauskopf, C. Kuehn, H. Osinga and M. Wechselberger, Mixed-mode oscillations with multiple time scales, SIAM Review, 54 (2012), 211-288. [11] F. Diener and M. Diener, Chasse au canard. Ⅰ. Les canards, Collect. Math., 32 (1981), 37-74. [12] F. Dumortier and R. H. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996), x+100 pp. [13] J. M. Edmond, M. R. Palmer, E. T. Brown and Y. Huh, Fluvial geochemistry of the eastern slope of the northeastern andes and its foredeep in the drainage of the orinoco in colombia and venezuela, Geochimica et cosmochimica acta, 60 (1996), 2949-2974. [14] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, Journal of Differential Equations, 31 (1979), 53-98. [15] A. F. Filippov, Differential equations with discontinuous right-hand side, American Mathematical Society Translations, 2 (1964), 199-231. [16] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publ. Dortrecht, 1988. [17] C. E. Graves, W.-H. Lee and G. R. North, New parameterizations and sensitivities for simple climate models, Journal of Geophysical Research, 98 (1993), 5025-5036. [18] P. Hoffman and D. Schrag, The snowball earth hypothesis: Testing the limits of global change, Terra Nova, 14 (2002), 129-155. [19] A. M. Hogg, Glacial cycles and carbon dioxide: A conceptual model, Geophysical Research Letters, 35 (2008), L01701. [20] M. R. Jeffrey, Hidden dynamics in models of discontinuity and switching, Physica D: Nonlinear Phenomena, 273 (2014), 34-45. [21] C. K. R. T. Jones, Geometric singular perturbation theory, Dynamical Systems, 1609 (1995), 44-118. [22] J. Kirschvink, Late proterozoic low-latitude global glaciation: The snowball earth, The Proterozoic Biosphere: A Multidisciplinary Study, (1992), 51-52. [23] M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions, SIAM Journal on Mathematical Analysis, 33 (2001), 286-314. [24] L. R. Kump, S. L. Brantley and M. A. Arthur, Chemical weathering, atmospheric co2, and climate, Annual Review of Earth and Planetary Sciences, 28 (2000), 611-667. [25] R. McGehee and C. Lehman, A paleoclimate model of ice-albedo feedback forced by variations in earth's orbit, SIAM Journal on Applied Dynamical Systems, 11 (2012), 684-707. [26] 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. [27] G. R. North, Theory of energy-balance climate models, J. Atmos. Sci, 32 (1975), 2033-2043. [28] R. T. Pierrehumbert, D. S. Abbot, A. Voigt and D. Koll, Climate of the Neoproterozoic, Annual Review of Earth and Planetary Sciences, 39 (2011), 417-460. [29] D. Pollard and J. F. Kasting, Snowball Earth: A thin-ice solution with flowing sea glaciers, Journal of Geophysical Research: Oceans (1978-2012), 110(C7), 2005. [30] W. D. Sellers, A global climatic model based on the energy balance of the earth-atmosphere system, Journal of Applied Meteorology, 8 (1969), 392-400. [31] J. Sieber and P. Kowalczyk, Small-scale instabilities in dynamical systems with sliding, Physica D: Nonlinear Phenomena, 239 (2010), 44-57. [32] P. Szmolyan and M. Wechselberger, Canards in $\mathbb{R}^3$, Journal of Differential Equations, 177 (2001), 419-453. [33] K. K. Tung, Topics in Mathematical Modelling, Princeton University Press, 2007. [34] J. Walsh and E. Widiasih, A dynamics approach to a low-order climate model, Discrete & Continuous Dynamical Systems-Series B, 19 (2014), 257-279. [35] M. Wechselberger, Existence and bifurcation of canards in $\mathbb{R}^3$ in the case of a folded node, SIAM Journal on Applied Dynamical Systems, 4 (2005), 101-139. [36] E. R. Widiasih, Dynamics of the Budyko energy balance model, SIAM Journal on Applied Dynamical Systems, 12 (2013), 2068-2092.
Bifurcation diagrams from energy balance models illustrating hysteresis in the climate system. In each figure, solid lines correspond to stable steady states while dashed lines correspond to unstable steady states. The positive horizontal axis can be thought of as increasing atmospheric carbon dioxide, and the vertical axis is the latitude of the ice line. Stability of snowball and ice-free states is inferred; these are physical boundaries and not true equilibria of the equations
The physical region of the phase space and possible fixed points of the system given by the $\eta$-nullcline, $h=0$. The location of the equilibrium is determined by the critical effective area of exposed land $0<\eta_c<1$. Solid black portions of the curve represent stable equilibria while the dashed lines denote unstable equilibria. Solid black portions of the boundary are attractive sliding regions and dashed boundaries are crossing regions
Attractors of the system when (a) $\eta_c=0.85$ and (b) $\eta_c=0.6$. The $+$ symbol marks the initial condition and the horizontal long-dashed line is the $A$-nullcline. In (a), the orbit reaches the ice-free state and slides until it reaches the intersection of the folded curve with this boundary. It then enters the physical region and approaches the small ice cap equilibrium. In (b), the fixed point is unstable and the orbit oscillates between the ice-free and ice-covered boundaries. Parameters are as in Table 1 and $\delta=0.01$. Simulations were performed using Mathematica 9
Periodic orbits of the Jormungand system when (a) $\eta_c=0.8$ and (b) $\eta_c=0.15$. The folded curve is the $\eta$-nullcline $h_J(A, \eta)=0$ and dashing is as in Figures 2 and 3. Parameters are as in Table 3 and $\delta=0.01$
Parameter values as in [1]
 Parameters Value Units $Q$ 321 $\text{W}\text{m}^{-2}$ $s_1$ 1 dimensionless $s_2$ -0.482 dimensionless $B$ 1.5 $\text{W}\text{m}^{-2}\text{K}^{-1}$ $C$ 2.5B $\text{W}\text{m}^{-2}\text{K}^{-1}$ $\alpha_1$ 0.32 dimensionless $\alpha_2$ 0.62 dimensionless $T_c$ $-10$ ℃
 Parameters Value Units $Q$ 321 $\text{W}\text{m}^{-2}$ $s_1$ 1 dimensionless $s_2$ -0.482 dimensionless $B$ 1.5 $\text{W}\text{m}^{-2}\text{K}^{-1}$ $C$ 2.5B $\text{W}\text{m}^{-2}\text{K}^{-1}$ $\alpha_1$ 0.32 dimensionless $\alpha_2$ 0.62 dimensionless $T_c$ $-10$ ℃
Functions as in [26]
 Functions $s(y) = 1 - \frac{0.482}{2} (3 y^2 - 1)$ $h(A, \eta)=\rho\left(112.88+56.91\eta-24.31\eta^2-11.05\eta^3-\frac{A}{1.5}\right)$ $g(A, \eta)=\delta(\eta-\eta_c)$ $\alpha(\eta, y)=\begin{cases} &\alpha_1 \text { when } y< \eta \\ & \frac{\alpha_1+\alpha_2}{2} \text{ when } y=\eta\\ &\alpha_2 \text{ when } y>\eta \end{cases}$
 Functions $s(y) = 1 - \frac{0.482}{2} (3 y^2 - 1)$ $h(A, \eta)=\rho\left(112.88+56.91\eta-24.31\eta^2-11.05\eta^3-\frac{A}{1.5}\right)$ $g(A, \eta)=\delta(\eta-\eta_c)$ $\alpha(\eta, y)=\begin{cases} &\alpha_1 \text { when } y< \eta \\ & \frac{\alpha_1+\alpha_2}{2} \text{ when } y=\eta\\ &\alpha_2 \text{ when } y>\eta \end{cases}$
Parameter values as in Table 1 unless specified above. Additional values taken from [1]
 Parameters Value Units $T_c$ 0 ℃ $M$ 25 dimensionless $\alpha_w$ $0.35$ dimensionless $\alpha_i$ $0.45$ dimensionless $\alpha_s$ $0.8$ dimensionless
 Parameters Value Units $T_c$ 0 ℃ $M$ 25 dimensionless $\alpha_w$ $0.35$ dimensionless $\alpha_i$ $0.45$ dimensionless $\alpha_s$ $0.8$ dimensionless
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