ESD Earth System Dynamics ESD Earth Syst. Dynam. 2190-4987 Copernicus Publications Göttingen, Germany 10.5194/esd-2-13-2011 Entropy production and multiple equilibria: the case of the ice-albedo feedback Herbert C. 1 2 Paillard D. 1 Dubrulle B. 2 Laboratoire des Sciences du Climat et de l'Environnement, IPSL, CEA-CNRS-UVSQ, UMR 8212, Gif-sur-Yvette, France Service de Physique de l'Etat Condensé, DSM, CEA Saclay, CNRS URA 2464, Gif-sur-Yvette, France 23 02 2011 2 1 13 23 Copyright: © 2011 C. Herbert et al. 2011 This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/ This article is available from https://esd.copernicus.org/articles/2/13/2011/esd-2-13-2011.html The full text article is available as a PDF file from https://esd.copernicus.org/articles/2/13/2011/esd-2-13-2011.pdf

Nonlinear feedbacks in the Earth System provide mechanisms that can prove very useful in understanding complex dynamics with relatively simple concepts. For example, the temperature and the ice cover of the planet are linked in a positive feedback which gives birth to multiple equilibria for some values of the solar constant: fully ice-covered Earth, ice-free Earth and an intermediate unstable solution. In this study, we show an analogy between a classical dynamical system approach to this problem and a Maximum Entropy Production (MEP) principle view, and we suggest a glimpse on how to reconcile MEP with the time evolution of a variable. It enables us in particular to resolve the question of the stability of the entropy production maxima. We also compare the surface heat flux obtained with MEP and with the bulk-aerodynamic formula.

 References Arnold, V.: Ordinary Differential Equations, Springer, New York, USA, 1984. Bruers, S.: A discussion on maximum entropy production and information theory, J. Phys.&nbsp;A, 40, 7441–7450, 2007. Budyko, M.: The effect of solar radiation variations on the climate of the {E}arth, Tellus, 21, 611–619, 1969. Cahalan, R. and North, G.: A Stability Theorem for Energy-Balance Climate Models, J. Atmos. Sci., 36, 1178–1188, 1979. Chylek, P. and Coakley, J.: Analytical analysis of a Budyko-type climate model, J. Atmos. Sci., 32, 675–679, 1975. Crafoord, C. and K{ä}ll{é}n, E.: A Note on the Condition for Existence of More than One Steady-State Solution in Budyko-Sellers Type Models, J. Atmos. Sci., 35, 1123–1124, 1978. Dewar, R.: Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states, J. Phys.&nbsp;A, 36, 631–641, 2003. Dewar, R.: Maximum entropy production and the fluctuation theorem, J. Phys.&nbsp;A, 38, 371–381, 2005. Dufresne, J.-L., Fournier, R., Hourdin, C., and Hourdin, F.: Net exchange reformulation of radiative transfer in the CO<sub>2</sub> 15 μm band on {M}ars, J. Atmos. Sci., 62, 3303–3319, 2005. Faegre, A.: An intransitive model of the {E}arth-atmosphere-ocean system, J. Appl. Meteorol., 11, 4–6, 1972. Filyukov, A.: Compatibility property of steady systems, J. Eng. Phys. Thermophys., 14, 814–819, 1968. Filyukov, A. and Karpov, V.: Description of steady transport processes by the method of the most probable path of evolution, J. Eng. Phys. Thermophys., 13, 624–630, 1967a. Filyukov, A. and Karpov, V.: Method of the most probable path of evolution in the theory of stationary irreversible processes, J. Eng. Phys. Thermophys., 13, 798–804, 1967b. Fraedrich, K.: Structural and stochastic analysis of a zero-dimensional climate system, Q. J. Roy. Meteor. Soc., 104, 461–474, 1978. Fraedrich, K., Jansen, H., Kirk, E., Luksch, U., and Lunkeit, F.: The Planet Simulator: Towards a user friendly model, Meteorol. Z., 14, 299–304, 2005. Frederiksen, J.: Nonlinear albedo-temperature coupling in climate models, J. Atmos. Sci., 33, 2267–2272, 1976. Gal-Chen, T. and Schneider, S.: Energy balance climate modeling: Comparison of radiative and dynamic feedback mechanisms, Tellus, 28, 108–121, 1976. Gerard, J., Delcourt, D., and Francois, L.: The maximum entropy production principle in climate models: application to the faint young sun paradox, Q. J. Roy. Meteor. Soc., 116, 1123–1132, 1990. Ghil, M.: Climate stability for a Sellers-type model, J. Atmos. Sci., 33, 3–20, 1976. Grassl, H.: The climate at maximum entropy production by meridional atmospheric and oceanic heat fluxes, Q. J. Roy. Meteor. Soc., 107, 153–166, 1981. Grinstein, G. and Linsker, R.: Comments on a derivation and application of the "maximum entropy production" principle, J. Phys.&nbsp;A, 40, 9717–9720, 2007. Guckenheimer, J. and Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol.&nbsp;42 of Applied Mathematical Sciences, Springer, New York, USA, 1983. Held, I. and Suarez, M.: Simple albedo feedback models of the icecaps, Tellus, 26, 613–629, 1974. Herbert, C., Paillard, D., Kageyama, M., and Dubrulle, B.: Present and Last Glacial Maximum climates as maximum entropy production states, Q. J. Roy. Meteor. Soc., submitted, 2010. Jaynes, E.: The minimum entropy production principle, Ann. Rev. Phys. Chem., 31, 579–601, 1980. Jupp, T.&nbsp;E. and Cox, P.: MEP and planetary climates: insights from a two-box climate model containing atmospheric dynamics, Philos. T. Roy. Soc.&nbsp;B, 365, 1355–1365, 2010. Kleidon, A. and Lorenz, R.: Non-equilibrium {T}hermodynamics and the {P}roduction of {E}ntropy: {L}ife, {E}arth, and {B}eyond, Springer, Berlin, Germany, 2005. le&nbsp;Hir, G., Donnadieu, Y., Krinner, G., and Ramstein, G.: Toward the snowball earth deglaciation, Clim. Dynam., 35, 285–297, 2010. Lian, M. and Cess, R.: Energy balance climate models: A reappraisal of ice-albedo feedback, J. Atmos. Sci., 34, 1058–1062, 1977. Lorenz, E.&nbsp;N.: Climatic determinism, Meteorol. Monogr., 8, 1–3, 1968. Lorenz, E.&nbsp;N.: Climatic change as a mathematical problem, J. Appl. Meteorol., 9, 325–329, 1970. Lorenz, R., Lunine, J., Withers, P., and McKay, C.: Titan, {M}ars and {E}arth: Entropy production by latitudinal heat transport, Geophys. Res. Lett., 28, 415–418, 2001. Lucarini, V.: Thermodynamic efficiency and entropy production in the climate system, Phys. Rev.&nbsp;E, 80, 021118, <a href="http://dx.doi.org/10.1103/PhysRevE.80.021118">https://doi.org/10.1103/PhysRevE.80.021118</a>, 2009. Lucarini, V., Fraedrich, K., and Lunkeit, F.: Thermodynamic analysis of snowball {E}arth hysteresis experiment: Efficiency, entropy production and irreversibility, Q. J. Roy. Meteor. Soc., 136, 2–11, 2010. Martyushev, L. and Seleznev, V.: Maximum entropy production principle in physics, chemistry and biology, Physics Reports, 426, 1–45, 2006. Monthus, C.: Non-equilibrium steady states: maximization of the Shannon entropy associated to the distribution of dynamical trajectories in the presence of constraints, arXiv:1011.1342v3, 2010. Nicolis, C.: Comment on the connection between stability and entropy production, Q. J. Roy. Meteor. Soc., 129, 3501–3504, 2003. Nicolis, C. and Nicolis, G.: Stability, complexity and the maximum dissipation conjecture, Q. J. Roy. Meteor. Soc., 136, 1161–1169, 2010. North, G.: Analytical solution to a simple climate model with diffusive heat transport, J. Atmos. Sci., 32, 1301–1307, 1975a. North, G.: Theory of energy-balance climate models, J. Atmos. Sci., 32, 2033–2043, 1975b. North, G., Howard, L., Pollard, D., and Wielicki, B.: Variational formulation of Budyko-Sellers climate models, J. Atmos. Sci., 36, 255–259, 1979. North, G., Cahalan, R., and Coakley, J.: Energy Balance Climate Models, Rev. Geophys. Space Phys., 19, 91–121, 1981. Oerlemans, J. and van&nbsp;den Dool, H.: Energy balance climate models: Stability experiments with a refined albedo and updated coefficients for infrared emission, J. Atmos. Sci., 35, 371–381, 1978. Ozawa, H. and Ohmura, A.: Thermodynamics of a global-mean state of the atmosphere – a state of maximum entropy increase, J. Climate, 10, 441–445, 1997. Ozawa, H., Ohmura, A., Lorenz, R., and Pujol, T.: The second law of thermodynamics and the global climate system: A review of the maximum entropy production principle, Rev. Geophys., 41, 1018, 2003. Paltridge, G.: Global dynamics and climate-a system of minimum entropy exchange, Q. J. Roy. Meteor. Soc., 101, 475–484, 1975. Paltridge, G.: The steady-state format of global climate, Q. J. Roy. Meteor. Soc., 104, 927–945, 1978. Paltridge, G.: Climate and thermodynamic systems of maximum dissipation, Nature, 279, 630–631, 1979. Peixoto, J.&nbsp;P. and Oort, A.&nbsp;H.: Physics of {C}limate, Springer, New York, USA, 1992. Pierrehumbert, R.: High levels of atmospheric carbon dioxide necessary for the termination of global glaciation, Nature, 429, 646–649, 2004. Pierrehumbert, R.: Climate dynamics of a hard snowball Earth, J. Geophys. Res., 110, D01111, <a href="http://dx.doi.org/10.1029/2004JD005162">https://doi.org/10.1029/2004JD005162</a>, 2005. Pujol, T. and Fort, J.: States of maximum entropy production in a one-dimensional vertical model with convective adjustment, Tellus, 54, 363–369, 2002. Robert, R. and Sommeria, J.: Relaxation towards a statistical equilibrium state in two-dimensional perfect fluid dynamics, Phys. Rev. Lett., 69, 2776–2779, 1992. Schneider, S. and Gal-Chen, T.: Numerical experiments in climate stability, J. Geophys. Res., 78, 6182–6194, 1973. Sellers, W.: A global climatic model based on the energy balance of the earth-atmosphere system, J. Appl. Meteorol., 8, 392–400, 1969. Shimokawa, S. and Ozawa, H.: On the thermodynamics of the oceanic general circulation: entropy increase rate of an open dissipative system and its surroundings, Tellus&nbsp;A, 53, 266–277, 2001. Shimokawa, S. and Ozawa, H.: On the thermodynamics of the oceanic general circulation: Irreversible transition to a state with higher rate of entropy production, Q. J. Roy. Meteor. Soc., 128, 2115–2128, 2002. Su, C. and Hsieh, D.: Stability of the Budyko climate model, J. Atmos. Sci., 33, 2273–2275, 1976.