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doi: 10.3934/dcdss.2022093
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On a global climate model with non-monotone multivalued coalbedo

 1 Dept. Ingeniería Geológica y Minera. E.T.S.I. Minas y Energía, Center for Computational Simulation. Universidad Politécnica de Madrid, Calle Ríos Rosas, 21. 28003 Madrid, Spain 2 Dept. Matemática Aplicada. ETS Arquitectura, Center for Computational Simulation. Universidad Politécnica de Madrid, Av. Juan de Herrera, 4. 28040 Madrid, Spain

Dedicated to Professor Georg Hetzer on the occasion of his 70th birthday

Received  December 2021 Revised  February 2022 Early access April 2022

We are concerned with a global energy balance climate model formulated through a parabolic equation whose space domain is a manifold which simulates the Earth surface. The climate energy balance model includes the effect of coalbedo as one of the mean temperature feedback. We extend some mathematical results proved for maximal monotone coalbedo to the case where the coalbedo has not a monotone dependency on temperature. Numerical approximation is performed by the Finite Volume Method which allows to obtain and compare numerical solutions with different values of the coalbedo.

Citation: Arturo Hidalgo, Lourdes Tello. On a global climate model with non-monotone multivalued coalbedo. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022093
References:
 [1] D. Arcoya, J. I. Diaz and L. Tello, S-shaped bifurcation branch in a quasilinear multivalued model arising in climatology, J. Differential Equations, 150 (1998), 215-225.  doi: 10.1006/jdeq.1998.3502. [2] T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations, Springer-Verlag (New York), 1982. doi: 10.1007/978-1-4612-5734-9. [3] M. Badii and J. I. Díaz, Time periodic solutions for a diffusive energy balance model in climatology, J. Math. Anal. Appl., 233 (1999), 713-729.  doi: 10.1006/jmaa.1999.6335. [4] H. Brézis, Operateurs Maximaux Monotones et Semigroupes de Contractions dans les Espaces de Hilbert. North Holland, Amsterdam, 1973. [5] M. I. Budyko, The effects of solar radiation variations on the climate of the earth, Tellus, 21 (1969), 611-619. [6] 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, 28–56, 1993. [7] J. I. Díaz, J. Hernández and L. Tello, On the multiplicity of equilibrium solutions to a nonlinear diffusion equation on a manifold arising in climatology, J. Math. Anal. Appl., 216 (1997), 593-613.  doi: 10.1006/jmaa.1997.5691. [8] J. I. Díaz and G. Hetzer, A functional quasilinear reaction - Diffusion equation arising in climatology, In the book Équations aux Dérivées Partielles et Applications. Articles dedies a J. L. Lions. Elsevier Paris, (1998), 461–480. [9] J. I. Díaz, G. Hetzer and L. Tello, An energy balance model climate model with hysteresis, Nonlinear Analysis, 64 (2006), 2053-2074.  doi: 10.1016/j.na.2005.07.038. [10] J. I. Díaz, A. Hidalgo and L. Tello, Multiple solutions and numerical analysis to the dynamic and stationary models coupling a delayed energy balance model involving latent heat and discontinuous albedo with a deep ocean, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), no. 2170, 20140376, 20 pp. doi: 10.1098/rspa.2014.0376. [11] J. I. Díaz and L. Tello, A nonlinear parabolic problem on a Riemannian manifold without boundary arising in climatology, Collect. Math., 50 (1999), 19-51. [12] J. I. Díaz and L. Tello, Infinitely many stationary solutions for a simple climate model via a shooting method, Math. Methods Appl. Sci., 25 (2002), 327-334.  doi: 10.1002/mma.289. [13] G. Hetzer, Forced periodic oscillations in the climate system via an energy balance model, Comment. Math. Uni. Carolin., 28 (1987), 593-601. [14] G. Hetzer, The structure of the principal component for semilinear diffusion equations from energy balance climate models, Houston J. Math., 16 (1990), 203-216. [15] G. Hetzer, A parameter dependent time-periodic reaction-diffusion equation from climate modeling; S-shapedness of the principal branch of fixed points of the time-$1$-map, Differential Integral Equations, 7 (1994), 1419-1425. [16] G. Hetzer, The shift-semiflow of a multi-valued evolution equation from climate modelling, Nonlinear Analysis: Theory, Methods & Applications, 47 (2001), 2905-2916.  doi: 10.1016/S0362-546X(01)00412-6. [17] G. Hetzer, The number of stationary solutions for a one-dimensional Budyko-type climatemodel, Nonlinear Anal. Real World Appl., 2 (2001), 259-272.  doi: 10.1016/S0362-546X(00)00103-6. [18] G. Hetzer, Global existence for a functional reaction-diffusion problem from climate modeling, Discrete Contin. Dyn. Syst., 2011 (2011), 660-671.  doi: 10.3934/proc.2011.2011.660. [19] 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. [20] G. Hetzer and P. G. Schmidt, A global attractor and stationary solutions for a reaction diffusion system arising from climate modeling, Nonlinear Anal., 14 (1990), 915-926.  doi: 10.1016/0362-546X(90)90109-T. [21] G. Hetzer and P. G. Schmidt, Global existence and asymptotic behavior for a quasilinear reaction diffusion system from climate modeling, J. Math. Anal. Appl., 160 (1991), 250-262.  doi: 10.1016/0022-247X(91)90303-H. [22] G. Hetzer and L. Tello, On a reaction diffusion system arising in Climatology, Dynamic Syst. Appl., 11 (2002), 381-402. [23] A. Hidalgo and L. Tello, On a climatological energy balance model with continents distribution, Discrete Contin. Dyn. Syst., 35 (2015), 1503-1519.  doi: 10.3934/dcds.2015.35.1503. [24] A. Hidalgo and L. Tello, Numerical approach of the equilibrium solutions of a global climate model, Mathematics, 8 (2020), 1542.  doi: 10.3390/math8091542. [25] G. North and K. Kim, Energy Balance Climate Models, Wiley-VCH Verlag, Weinheim, Germany 2017. [26] J. Rombouts and M. Ghil, Oscillations in a simple climate-vegetation model, Nonlin. Processes Geophys., 22 (2015), 275-288.  doi: 10.5194/npg-22-275-2015. [27] B. E. Schmidt, On a nonlinear eigenvalue problem arising from climate modeling, Nonlinear Anal., 30 (1997), 3645-3656.  doi: 10.1016/S0362-546X(96)00239-8. [28] B. E. Schmidt, Bifurcation from S-shaped solution curves in a class of Sturm Liouville problems related to climate modeling, Adv. Math. Sci. Appl., 10 (2000), 513-537. [29] B. E. Schmidt, Bifurcation of stationary solutions for Legendre-type boundary value problems arising from energy balance climate models, Thesis (Ph.D.)-Auburn University, 1994. [30] 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. [31] P. H. Stone, A simplified radiative-dynamical model for the static stability of rotating atmospheres, Journal of Atmospheric Sciences, 29 (1972), 405-418. [32] I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, Pitman Longman. London, 1986, 1995.

show all references

References:
 [1] D. Arcoya, J. I. Diaz and L. Tello, S-shaped bifurcation branch in a quasilinear multivalued model arising in climatology, J. Differential Equations, 150 (1998), 215-225.  doi: 10.1006/jdeq.1998.3502. [2] T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations, Springer-Verlag (New York), 1982. doi: 10.1007/978-1-4612-5734-9. [3] M. Badii and J. I. Díaz, Time periodic solutions for a diffusive energy balance model in climatology, J. Math. Anal. Appl., 233 (1999), 713-729.  doi: 10.1006/jmaa.1999.6335. [4] H. Brézis, Operateurs Maximaux Monotones et Semigroupes de Contractions dans les Espaces de Hilbert. North Holland, Amsterdam, 1973. [5] M. I. Budyko, The effects of solar radiation variations on the climate of the earth, Tellus, 21 (1969), 611-619. [6] 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, 28–56, 1993. [7] J. I. Díaz, J. Hernández and L. Tello, On the multiplicity of equilibrium solutions to a nonlinear diffusion equation on a manifold arising in climatology, J. Math. Anal. Appl., 216 (1997), 593-613.  doi: 10.1006/jmaa.1997.5691. [8] J. I. Díaz and G. Hetzer, A functional quasilinear reaction - Diffusion equation arising in climatology, In the book Équations aux Dérivées Partielles et Applications. Articles dedies a J. L. Lions. Elsevier Paris, (1998), 461–480. [9] J. I. Díaz, G. Hetzer and L. Tello, An energy balance model climate model with hysteresis, Nonlinear Analysis, 64 (2006), 2053-2074.  doi: 10.1016/j.na.2005.07.038. [10] J. I. Díaz, A. Hidalgo and L. Tello, Multiple solutions and numerical analysis to the dynamic and stationary models coupling a delayed energy balance model involving latent heat and discontinuous albedo with a deep ocean, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), no. 2170, 20140376, 20 pp. doi: 10.1098/rspa.2014.0376. [11] J. I. Díaz and L. Tello, A nonlinear parabolic problem on a Riemannian manifold without boundary arising in climatology, Collect. Math., 50 (1999), 19-51. [12] J. I. Díaz and L. Tello, Infinitely many stationary solutions for a simple climate model via a shooting method, Math. Methods Appl. Sci., 25 (2002), 327-334.  doi: 10.1002/mma.289. [13] G. Hetzer, Forced periodic oscillations in the climate system via an energy balance model, Comment. Math. Uni. Carolin., 28 (1987), 593-601. [14] G. Hetzer, The structure of the principal component for semilinear diffusion equations from energy balance climate models, Houston J. Math., 16 (1990), 203-216. [15] G. Hetzer, A parameter dependent time-periodic reaction-diffusion equation from climate modeling; S-shapedness of the principal branch of fixed points of the time-$1$-map, Differential Integral Equations, 7 (1994), 1419-1425. [16] G. Hetzer, The shift-semiflow of a multi-valued evolution equation from climate modelling, Nonlinear Analysis: Theory, Methods & Applications, 47 (2001), 2905-2916.  doi: 10.1016/S0362-546X(01)00412-6. [17] G. Hetzer, The number of stationary solutions for a one-dimensional Budyko-type climatemodel, Nonlinear Anal. Real World Appl., 2 (2001), 259-272.  doi: 10.1016/S0362-546X(00)00103-6. [18] G. Hetzer, Global existence for a functional reaction-diffusion problem from climate modeling, Discrete Contin. Dyn. Syst., 2011 (2011), 660-671.  doi: 10.3934/proc.2011.2011.660. [19] 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. [20] G. Hetzer and P. G. Schmidt, A global attractor and stationary solutions for a reaction diffusion system arising from climate modeling, Nonlinear Anal., 14 (1990), 915-926.  doi: 10.1016/0362-546X(90)90109-T. [21] G. Hetzer and P. G. Schmidt, Global existence and asymptotic behavior for a quasilinear reaction diffusion system from climate modeling, J. Math. Anal. Appl., 160 (1991), 250-262.  doi: 10.1016/0022-247X(91)90303-H. [22] G. Hetzer and L. Tello, On a reaction diffusion system arising in Climatology, Dynamic Syst. Appl., 11 (2002), 381-402. [23] A. Hidalgo and L. Tello, On a climatological energy balance model with continents distribution, Discrete Contin. Dyn. Syst., 35 (2015), 1503-1519.  doi: 10.3934/dcds.2015.35.1503. [24] A. Hidalgo and L. Tello, Numerical approach of the equilibrium solutions of a global climate model, Mathematics, 8 (2020), 1542.  doi: 10.3390/math8091542. [25] G. North and K. Kim, Energy Balance Climate Models, Wiley-VCH Verlag, Weinheim, Germany 2017. [26] J. Rombouts and M. Ghil, Oscillations in a simple climate-vegetation model, Nonlin. Processes Geophys., 22 (2015), 275-288.  doi: 10.5194/npg-22-275-2015. [27] B. E. Schmidt, On a nonlinear eigenvalue problem arising from climate modeling, Nonlinear Anal., 30 (1997), 3645-3656.  doi: 10.1016/S0362-546X(96)00239-8. [28] B. E. Schmidt, Bifurcation from S-shaped solution curves in a class of Sturm Liouville problems related to climate modeling, Adv. Math. Sci. Appl., 10 (2000), 513-537. [29] B. E. Schmidt, Bifurcation of stationary solutions for Legendre-type boundary value problems arising from energy balance climate models, Thesis (Ph.D.)-Auburn University, 1994. [30] 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. [31] P. H. Stone, A simplified radiative-dynamical model for the static stability of rotating atmospheres, Journal of Atmospheric Sciences, 29 (1972), 405-418. [32] I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, Pitman Longman. London, 1986, 1995.
Spatial distribution of the temperature for an output time $t = 5$. Solid red line: numerical solution when considering the ocean continents distribution. Dashed blue line: numerical solution when only ocean is taken into account in the whole domain. Dotted black line: initial condition. The labels: Continent, Ocean apply for the Ocean-Continents case
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