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April  2015, 35(4): 1503-1519. doi: 10.3934/dcds.2015.35.1503

## On a climatological energy balance model with continents distribution

Received  September 2013 Revised  May 2014 Published  November 2014

We present some results on the mathematical treatment of a global two-dimensional diffusive climate model with land - sea distribution. The model is based on a long time averaged energy balance and leads to a nonlinear parabolic equation for the averaged surface temperature. The spatial domain is a compact two-dimensional Riemannian manifold without boundary simulating the Earth surface with land - sea configuration. In the oceanic areas the model is coupled with a deep ocean model. The coupling is given by a dynamic and diffusive boundary condition. We study the existence of a bounded weak solution and its numerical approximation.
Citation: Arturo Hidalgo, Lourdes Tello. On a climatological energy balance model with continents distribution. Discrete & Continuous Dynamical Systems, 2015, 35 (4) : 1503-1519. doi: 10.3934/dcds.2015.35.1503
##### References:
 [1] T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampere Equations, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5734-9.  Google Scholar [2] D. S. Balsara and Ch. W. Shu, Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. Comp. Phys., 160 (2000), 405-452. doi: 10.1006/jcph.2000.6443.  Google Scholar [3] W. H. Berger, S. Burker and E. Vincent, Glacial-Holocene transition: Climate Pulsations and Sporadic Shutdown of NADW production, in Abrupt Climatic Change - Evidence and Implications, (eds. W.H. Berger, L.D. Labeyrie), Reidel Publishing Co. Dordrecht Holland, 216 (1987), 279-297. doi: 10.1007/978-94-009-3993-6_25.  Google Scholar [4] R. Bermejo , J. Carpio, J. I. Díaz and L. Tello, Mathematical and numerical analysis of a nonlinear diffusive climate energy balance Model, Math. Comp. Model., 49 (2009), 1180-1210. doi: 10.1016/j.mcm.2008.04.010.  Google Scholar [5] R. Bermejo, J. Carpio, J. I. Díaz and P. Galán, A finite element algorithm of a nonlinear climate energy balance model Pure and Appl. Geophysics, 165 (2008), 1025-1047. Google Scholar [6] H. Brezis, Operateurs Maximaux Monotones Et Semigroupes De Contractions Dans Les Espaces De Hilbert, North Holland, Amsterdam, 1973. Google Scholar [7] M. I. Budyko, The effects of solar radiation variations on the climate of the Earth, Tellus, 21 (1969), 611-619. doi: 10.1111/j.2153-3490.1969.tb00466.x.  Google Scholar [8] J. Casper and H. Atkins, A finite volume high order ENO scheme for two dimensional hyperbolic systems, J. Comp. Phys., 106 (1993), 62-76. doi: 10.1006/jcph.1993.1091.  Google Scholar [9] J. I. Diaz, Mathematical analysis of some diffusive energy balance climate models, in the book Mathematics, Climate and Environment, (J. I. Diaz and J. L. Lions, eds. Masson, Paris, (1993), 28-56.  Google Scholar [10] J. I. Díaz and L. Tello, A nonlinear parabolic problem on a Riemannian manifold without boundary arising in Climatology, Collectanea Mathematica, 50 (1999), 19-51.  Google Scholar [11] J. I. Díaz and L. Tello, Sobre un modelo climático de balance de energía superficial acoplado con un océano profundo, Actas XVII CEDYA/ VI CMA. Univ. Salamanca, 2001. Google Scholar [12] J. I. Díaz and L. Tello, A 2D climate energy balance model coupled with a 3D deep ocean model, Electronic Journal of Differential Equations, Conf. 16, (2007). Google Scholar [13] M. Dumbser, C. Enaux and E. F. Toro, Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws, J. Comp. Phys., 227 (2008), 3971-4001. doi: 10.1016/j.jcp.2007.12.005.  Google Scholar [14] K. McGuffie and A. Henderson-Sellers, A Climate Modelling Primer, Wiley, 2005. Google Scholar [15] G. Hetzer, The structure of the principal component for semilinear diffusion equations from energy balance climate models, Houston Journal of Math., 16 (1990), 203-216.  Google Scholar [16] G. Hetzer and L. Tello, On a reaction diffusion system arising in Climatology, Dynamic Systems and Applications, 11 (2002), 381-401.  Google Scholar [17] A. Hidalgo and L. Tello, A Finite Volume Scheme for simulating the coupling between deep ocean and an atmospheric energy balance model, In the book Modern Mathematical Tools and Techniques in Capturing Complexity. Springer Series in Synergetics, (2011), 239-255. doi: 10.1007/978-3-642-20853-9_17.  Google Scholar [18] G. R. North, J. G. Mengel and A. A. Short, Simple energy balance models resolving the seasons and the continents: Application to the astronomical theory of the ice ages, Journal of Geophysical Research, 88 (1983), 6576-6586. doi: 10.1029/JC088iC11p06576.  Google Scholar [19] S. P. Ritz, T. F. Stocker and F. Joos, A coupled dynamical ocean-energy balance atmospheric model of paleoclimate studies, Journal of Climate, 1 (2011), 349-375. doi: 10.1175/2010JCLI3351.1.  Google Scholar [20] W. D. Sellers, A global climatic model based on the energy balance of the earth-atmosphere system, J. Appl. Meteorol, 8 (1969), 392-400. doi: 10.1175/1520-0450(1969)008<0392:AGCMBO>2.0.CO;2.  Google Scholar [21] Ch. W. Shu, Total variation diminishing time discretizations, SIAM J. Sci. Stat. Comput., 9 (1998), 1073-1084. doi: 10.1137/0909073.  Google Scholar [22] T. F. Stocker, D. G. Wright and L. A. Mysak, A zonally averaged, coupled ocean-atmospheric model for paleoclimate studies, Journal of Climate, 5 (1992), 773-797. doi: 10.1175/1520-0442(1992)005<0773:AZACOA>2.0.CO;2.  Google Scholar [23] P. H. Stone, A simplified radiative-dynamical model for the static stability of rotating atmospheres, J. Atmos. Sci., 29 (1972), 405-418. doi: 10.1175/1520-0469(1972)029<0405:ASRDMF>2.0.CO;2.  Google Scholar [24] V. A. Titarev and E. F. Toro, Finite volume WENO schemes for three-dimensional conservation laws, J. Comp. Phys., 201 (2004), 238-260. doi: 10.1016/j.jcp.2004.05.015.  Google Scholar [25] V. A. Titarev and E. F. Toro, ADER schemes for three-dimensional non-linear hyperbolic systems, J. Comp. Phys., 204 (2005), 715-736. doi: 10.1016/j.jcp.2004.10.028.  Google Scholar [26] E. F. Toro and A. Hidalgo, ADER finite volume schemes for nonlinear reaction-diffusion equations, Appl. Num. Math., 59 (2009), 73-100. doi: 10.1016/j.apnum.2007.12.001.  Google Scholar [27] E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics ($3^{rd}$ ed), Springer-Verlag Berlin Heidelberg. 2009. doi: 10.1007/b79761.  Google Scholar [28] I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, Pitman Longman. London. 1987.  Google Scholar [29] R. G. Watts and M. Morantine, Rapid climatic change and the deep ocean, Climatic Change, 16 (1990), 83-97. doi: 10.1007/BF00137347.  Google Scholar [30] R. G. Watts and E. Hayder, A two-dimensional, seasonal, energy balance climate model with continents and ice sheets: testing the Milankovitch theory, Tellus, 36 (1984), 120-131. doi: 10.1111/j.1600-0870.1984.tb00232.x.  Google Scholar

show all references

##### References:
 [1] T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampere Equations, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5734-9.  Google Scholar [2] D. S. Balsara and Ch. W. Shu, Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. Comp. Phys., 160 (2000), 405-452. doi: 10.1006/jcph.2000.6443.  Google Scholar [3] W. H. Berger, S. Burker and E. Vincent, Glacial-Holocene transition: Climate Pulsations and Sporadic Shutdown of NADW production, in Abrupt Climatic Change - Evidence and Implications, (eds. W.H. Berger, L.D. Labeyrie), Reidel Publishing Co. Dordrecht Holland, 216 (1987), 279-297. doi: 10.1007/978-94-009-3993-6_25.  Google Scholar [4] R. Bermejo , J. Carpio, J. I. Díaz and L. Tello, Mathematical and numerical analysis of a nonlinear diffusive climate energy balance Model, Math. Comp. Model., 49 (2009), 1180-1210. doi: 10.1016/j.mcm.2008.04.010.  Google Scholar [5] R. Bermejo, J. Carpio, J. I. Díaz and P. Galán, A finite element algorithm of a nonlinear climate energy balance model Pure and Appl. Geophysics, 165 (2008), 1025-1047. Google Scholar [6] H. Brezis, Operateurs Maximaux Monotones Et Semigroupes De Contractions Dans Les Espaces De Hilbert, North Holland, Amsterdam, 1973. Google Scholar [7] M. I. Budyko, The effects of solar radiation variations on the climate of the Earth, Tellus, 21 (1969), 611-619. doi: 10.1111/j.2153-3490.1969.tb00466.x.  Google Scholar [8] J. Casper and H. Atkins, A finite volume high order ENO scheme for two dimensional hyperbolic systems, J. Comp. Phys., 106 (1993), 62-76. doi: 10.1006/jcph.1993.1091.  Google Scholar [9] J. I. Diaz, Mathematical analysis of some diffusive energy balance climate models, in the book Mathematics, Climate and Environment, (J. I. Diaz and J. L. Lions, eds. Masson, Paris, (1993), 28-56.  Google Scholar [10] J. I. Díaz and L. Tello, A nonlinear parabolic problem on a Riemannian manifold without boundary arising in Climatology, Collectanea Mathematica, 50 (1999), 19-51.  Google Scholar [11] J. I. Díaz and L. Tello, Sobre un modelo climático de balance de energía superficial acoplado con un océano profundo, Actas XVII CEDYA/ VI CMA. Univ. Salamanca, 2001. Google Scholar [12] J. I. Díaz and L. Tello, A 2D climate energy balance model coupled with a 3D deep ocean model, Electronic Journal of Differential Equations, Conf. 16, (2007). Google Scholar [13] M. Dumbser, C. Enaux and E. F. Toro, Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws, J. Comp. Phys., 227 (2008), 3971-4001. doi: 10.1016/j.jcp.2007.12.005.  Google Scholar [14] K. McGuffie and A. Henderson-Sellers, A Climate Modelling Primer, Wiley, 2005. Google Scholar [15] G. Hetzer, The structure of the principal component for semilinear diffusion equations from energy balance climate models, Houston Journal of Math., 16 (1990), 203-216.  Google Scholar [16] G. Hetzer and L. Tello, On a reaction diffusion system arising in Climatology, Dynamic Systems and Applications, 11 (2002), 381-401.  Google Scholar [17] A. Hidalgo and L. Tello, A Finite Volume Scheme for simulating the coupling between deep ocean and an atmospheric energy balance model, In the book Modern Mathematical Tools and Techniques in Capturing Complexity. Springer Series in Synergetics, (2011), 239-255. doi: 10.1007/978-3-642-20853-9_17.  Google Scholar [18] G. R. North, J. G. Mengel and A. A. Short, Simple energy balance models resolving the seasons and the continents: Application to the astronomical theory of the ice ages, Journal of Geophysical Research, 88 (1983), 6576-6586. doi: 10.1029/JC088iC11p06576.  Google Scholar [19] S. P. Ritz, T. F. Stocker and F. Joos, A coupled dynamical ocean-energy balance atmospheric model of paleoclimate studies, Journal of Climate, 1 (2011), 349-375. doi: 10.1175/2010JCLI3351.1.  Google Scholar [20] W. D. Sellers, A global climatic model based on the energy balance of the earth-atmosphere system, J. Appl. Meteorol, 8 (1969), 392-400. doi: 10.1175/1520-0450(1969)008<0392:AGCMBO>2.0.CO;2.  Google Scholar [21] Ch. W. Shu, Total variation diminishing time discretizations, SIAM J. Sci. Stat. Comput., 9 (1998), 1073-1084. doi: 10.1137/0909073.  Google Scholar [22] T. F. Stocker, D. G. Wright and L. A. Mysak, A zonally averaged, coupled ocean-atmospheric model for paleoclimate studies, Journal of Climate, 5 (1992), 773-797. doi: 10.1175/1520-0442(1992)005<0773:AZACOA>2.0.CO;2.  Google Scholar [23] P. H. Stone, A simplified radiative-dynamical model for the static stability of rotating atmospheres, J. Atmos. Sci., 29 (1972), 405-418. doi: 10.1175/1520-0469(1972)029<0405:ASRDMF>2.0.CO;2.  Google Scholar [24] V. A. Titarev and E. F. Toro, Finite volume WENO schemes for three-dimensional conservation laws, J. Comp. Phys., 201 (2004), 238-260. doi: 10.1016/j.jcp.2004.05.015.  Google Scholar [25] V. A. Titarev and E. F. Toro, ADER schemes for three-dimensional non-linear hyperbolic systems, J. Comp. Phys., 204 (2005), 715-736. doi: 10.1016/j.jcp.2004.10.028.  Google Scholar [26] E. F. Toro and A. Hidalgo, ADER finite volume schemes for nonlinear reaction-diffusion equations, Appl. Num. Math., 59 (2009), 73-100. doi: 10.1016/j.apnum.2007.12.001.  Google Scholar [27] E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics ($3^{rd}$ ed), Springer-Verlag Berlin Heidelberg. 2009. doi: 10.1007/b79761.  Google Scholar [28] I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, Pitman Longman. London. 1987.  Google Scholar [29] R. G. Watts and M. Morantine, Rapid climatic change and the deep ocean, Climatic Change, 16 (1990), 83-97. doi: 10.1007/BF00137347.  Google Scholar [30] R. G. Watts and E. Hayder, A two-dimensional, seasonal, energy balance climate model with continents and ice sheets: testing the Milankovitch theory, Tellus, 36 (1984), 120-131. doi: 10.1111/j.1600-0870.1984.tb00232.x.  Google Scholar
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