• Previous Article
    Strong positivity of continuous supersolutions to parabolic equations with rough boundary data
  • DCDS Home
  • This Issue
  • Next Article
    Existence and multiplicity of segregated solutions to a cell-growth contact inhibition problem
April  2015, 35(4): 1503-1519. doi: 10.3934/dcds.2015.35.1503

On a climatological energy balance model with continents distribution

1. 

Dept. Matemática Aplicada y Métodos Informáticos. E.T.S.I. Minas y Energía, Universidad Politécnica de Madrid, Madrid, E-28003, Spain

2. 

Dept. Matemática Aplicada. ETS Arquitectura, Universidad Politécnica de Madrid, Madrid, E-28040, Spain

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 and 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.

[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.

[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.

[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.

[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.

[6]

H. Brezis, Operateurs Maximaux Monotones Et Semigroupes De Contractions Dans Les Espaces De Hilbert, North Holland, Amsterdam, 1973.

[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.

[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.

[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.

[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.

[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.

[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).

[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.

[14]

K. McGuffie and A. Henderson-Sellers, A Climate Modelling Primer, Wiley, 2005.

[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.

[16]

G. Hetzer and L. Tello, On a reaction diffusion system arising in Climatology, Dynamic Systems and Applications, 11 (2002), 381-401.

[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.

[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.

[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.

[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.

[21]

Ch. W. Shu, Total variation diminishing time discretizations, SIAM J. Sci. Stat. Comput., 9 (1998), 1073-1084. doi: 10.1137/0909073.

[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.

[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.

[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.

[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.

[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.

[27]

E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics ($3^{rd}$ ed), Springer-Verlag Berlin Heidelberg. 2009. doi: 10.1007/b79761.

[28]

I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, Pitman Longman. London. 1987.

[29]

R. G. Watts and M. Morantine, Rapid climatic change and the deep ocean, Climatic Change, 16 (1990), 83-97. doi: 10.1007/BF00137347.

[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.

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.

[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.

[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.

[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.

[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.

[6]

H. Brezis, Operateurs Maximaux Monotones Et Semigroupes De Contractions Dans Les Espaces De Hilbert, North Holland, Amsterdam, 1973.

[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.

[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.

[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.

[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.

[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.

[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).

[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.

[14]

K. McGuffie and A. Henderson-Sellers, A Climate Modelling Primer, Wiley, 2005.

[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.

[16]

G. Hetzer and L. Tello, On a reaction diffusion system arising in Climatology, Dynamic Systems and Applications, 11 (2002), 381-401.

[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.

[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.

[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.

[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.

[21]

Ch. W. Shu, Total variation diminishing time discretizations, SIAM J. Sci. Stat. Comput., 9 (1998), 1073-1084. doi: 10.1137/0909073.

[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.

[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.

[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.

[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.

[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.

[27]

E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics ($3^{rd}$ ed), Springer-Verlag Berlin Heidelberg. 2009. doi: 10.1007/b79761.

[28]

I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, Pitman Longman. London. 1987.

[29]

R. G. Watts and M. Morantine, Rapid climatic change and the deep ocean, Climatic Change, 16 (1990), 83-97. doi: 10.1007/BF00137347.

[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.

[1]

Nicholas Long. Fixed point shifts of inert involutions. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297

[2]

Zhihong Xia, Peizheng Yu. A fixed point theorem for twist maps. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 4051-4059. doi: 10.3934/dcds.2022045

[3]

Rongjie Lai, Jiang Liang, Hong-Kai Zhao. A local mesh method for solving PDEs on point clouds. Inverse Problems and Imaging, 2013, 7 (3) : 737-755. doi: 10.3934/ipi.2013.7.737

[4]

Lorena Bociu, Barbara Kaltenbacher, Petronela Radu. Preface: Introduction to the Special Volume on Nonlinear PDEs and Control Theory with Applications. Evolution Equations and Control Theory, 2013, 2 (2) : i-ii. doi: 10.3934/eect.2013.2.2i

[5]

Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017

[6]

Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709

[7]

Yong Ji, Ercai Chen, Yunping Wang, Cao Zhao. Bowen entropy for fixed-point free flows. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6231-6239. doi: 10.3934/dcds.2019271

[8]

Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692

[9]

Luis Hernández-Corbato, Francisco R. Ruiz del Portal. Fixed point indices of planar continuous maps. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2979-2995. doi: 10.3934/dcds.2015.35.2979

[10]

Antonio Garcia. Transition tori near an elliptic-fixed point. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 381-392. doi: 10.3934/dcds.2000.6.381

[11]

Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system. Networks and Heterogeneous Media, 2016, 11 (1) : 1-27. doi: 10.3934/nhm.2016.11.1

[12]

Rafael de la Llave, Jason D. Mireles James. Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4321-4360. doi: 10.3934/dcds.2012.32.4321

[13]

Osama Khalil. Geodesic planes in geometrically finite manifolds. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 881-903. doi: 10.3934/dcds.2019037

[14]

Cleon S. Barroso. The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 467-479. doi: 10.3934/dcds.2009.25.467

[15]

Teck-Cheong Lim. On the largest common fixed point of a commuting family of isotone maps. Conference Publications, 2005, 2005 (Special) : 621-623. doi: 10.3934/proc.2005.2005.621

[16]

Mircea Sofonea, Cezar Avramescu, Andaluzia Matei. A fixed point result with applications in the study of viscoplastic frictionless contact problems. Communications on Pure and Applied Analysis, 2008, 7 (3) : 645-658. doi: 10.3934/cpaa.2008.7.645

[17]

Parin Chaipunya, Poom Kumam. Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays. Conference Publications, 2015, 2015 (special) : 248-257. doi: 10.3934/proc.2015.0248

[18]

Dou Dou, Meng Fan, Hua Qiu. Topological entropy on subsets for fixed-point free flows. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6319-6331. doi: 10.3934/dcds.2017273

[19]

Ruhua Wang, Senjian An, Wanquan Liu, Ling Li. Fixed-point algorithms for inverse of residual rectifier neural networks. Mathematical Foundations of Computing, 2021, 4 (1) : 31-44. doi: 10.3934/mfc.2020024

[20]

Mark S. Gockenbach, Akhtar A. Khan. Identification of Lamé parameters in linear elasticity: a fixed point approach. Journal of Industrial and Management Optimization, 2005, 1 (4) : 487-497. doi: 10.3934/jimo.2005.1.487

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (145)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]