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Hopf bifurcation in a conceptual climate model with ice-albedo and precipitation-temperature feedbacks. (English) Zbl 1430.37107
Summary: In this paper we analyse a dynamical system based on the so-called KCG (Källén, Crafoord, Ghil) conceptual climate model. This model describes an evolution of the globally averaged temperature and the average extent of the ice sheets. In the nondimensional form the equations can be simplified facilitating the subsequent analysis. We consider the limiting case of a stationary snow line for which the phase plane can be completely analysed and the type of each critical point can be determined. One of them can exhibit the Hopf bifurcation and we find sufficient conditions for its existence. Those, in turn, have a straightforward physical meaning and indicate that the model predicts internal oscillations of the climate. Using the typical real-world values of appearing parameters we conclude that the obtained results are in the same ballpark as the conditions on our planet during the quaternary ice ages. Our analysis is a rigorous justification of a generalization of some previous results by KCG and other authors.
MSC:
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
86A08 Climate science and climate modeling
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[1] Pierrehumbert, R. T.; Abbot, D. S.; Voigt, A.; Koll, D., Climate of the neoproterozoic, Ann. Rev. Earth Planet. Sci., 39 (2011)
[2] Hyde, William T.; Crowley, Thomas J.; Baum, Steven K.; Peltier, W. Richard, Neoproterozoic ‘snowball earth’ simulations with a coupled climate/ice-sheet model, Nature, 405, 6785, 425-429 (2000)
[3] Milankovič, Milutin, Canon of Insolation and the Ice-Age Problem (1998), Zavod za udžbenike i nastavna sredstva
[4] Hays, James D.; Imbrie, John; Shackleton, Nicholas J., Variations in the earth’s orbit: pacemaker of the ice ages, Science, 194, 4270, 1121-1132 (1976)
[5] Lisiecki, Lorraine E.; Raymo, Maureen E., A pliocene-pleistocene stack of 57 globally distributed benthic \(\delta 18\) o records, Paleoceanography, 20, 1 (2005)
[6] McGehee, Richard; Lehman, Clarence, A paleoclimate model of ice-albedo feedback forced by variations in earth’s orbit, SIAM J. Appl. Dyn. Syst., 11, 2, 684-707 (2012) · Zbl 1254.86032
[7] Berger, André, Milankovitch theory and climate, Rev. Geophys., 26, 4, 624-657 (1988)
[8] Clark, Peter U.; Archer, David; Pollard, David; Blum, Joel D.; Rial, Jose A.; Brovkin, Victor; Mix, Alan C.; Pisias, Nicklas G.; Roy, Martin, The middle pleistocene transition: characteristics, mechanisms, and implications for long-term changes in atmospheric pco 2, Quat. Sci. Rev., 25, 23, 3150-3184 (2006)
[9] Le Treut, Hervé; Ghil, Michael, Orbital forcing, climatic interactions, and glaciation cycles, J. Geophys. Res.: Oceans, 88, C9, 5167-5190 (1983)
[10] Tziperman, Eli; Raymo, Maureen E.; Huybers, Peter; Wunsch, Carl, Consequences of pacing the pleistocene 100 kyr ice ages by nonlinear phase locking to milankovitch forcing, Paleoceanography, 21, 4 (2006)
[11] Fowler, A. C.; Rickaby, R. E.M.; Wolff, E. W., Exploration of a simple model for ice ages, GEM-Int. J. Geomath., 4, 2, 227-297 (2013) · Zbl 1277.86005
[12] De Saedeleer, Bernard; Crucifix, Michel; Wieczorek, Sebastian, Is the astronomical forcing a reliable and unique pacemaker for climate? A conceptual model study, Clim. Dynam., 40, 1-2, 273-294 (2013)
[13] Hogg, Andrew McC, Glacial cycles and carbon dioxide: A conceptual model, Geophys. Res. Lett., 35, 1 (2008)
[14] Saltzman, Barry; Hansen, Anthony R.; Maasch, Kirk A., The late quaternary glaciations as the response of a three-component feedback system to earth-orbital forcing, J. Atmos. Sci., 41, 23, 3380-3389 (1984)
[15] Paillard, Didier, Glacial cycles: toward a new paradigm, Rev. Geophys., 39, 3, 325-346 (2001)
[17] Ashwin, Peter; Ditlevsen, Peter, The middle pleistocene transition as a generic bifurcation on a slow manifold, Clim. Dyn., 45, 9-10, 2683-2695 (2015)
[18] Crucifix, Michel, Oscillators and relaxation phenomena in pleistocene climate theory, Phil. Trans. R. Soc. A, 370, 1962, 1140-1165 (2012)
[19] Claussen, Martin; Mysak, L.; Weaver, A.; Crucifix, Michel; Fichefet, Thierry; Loutre, M-F; Weber, Shlomo; Alcamo, Joseph; Alexeev, Vladimir; Berger, André, Earth system models of intermediate complexity: closing the gap in the spectrum of climate system models, Clim. Dyn., 18, 7, 579-586 (2002)
[20] Maasch, Kirk A.; Saltzman, Barry, A low-order dynamical model of global climatic variability over the full pleistocene, J. Geophys. Res.: Atmos., 95, D2, 1955-1963 (1990)
[21] Saltzman, Barry, Dynamical Paleoclimatology: Generalized Theory of Global Climate Change, Vol. 80 (2002), Academic Press
[22] Engler, Hans; Kaper, Hans G.; Kaper, Tasso J.; Vo, Theodore, Dynamical systems analysis of the Maasch-Saltzman model for glacial cycles, Physica D, 359, 1-20 (2017) · Zbl 1378.37123
[23] Gildor, Hezi; Tziperman, Eli, Sea ice as the glacial cycles’ climate switch: Role of seasonal and orbital forcing, Paleoceanography, 15, 6, 605-615 (2000)
[24] Paillard, Didier; Parrenin, Frédéric, The antarctic ice sheet and the triggering of deglaciations, Earth Planet. Sci. Lett., 227, 3, 263-271 (2004)
[25] Crucifix, Michel, How can a glacial inception be predicted?, Holocene, 21, 5, 831-842 (2011)
[26] Roberts, Andrew; Widiasih, Esther; Wechselberger, Martin; Jones, Christopher KRT, Mixed mode oscillations in a conceptual climate model, Physica D, 292, 70-83 (2015) · Zbl 1364.86018
[27] Budyko, Mikhail I., The effect of solar radiation variations on the climate of the earth, Tellus, 21, 5, 611-619 (1969)
[28] Sellers, William D., A global climatic model based on the energy balance of the earth-atmosphere system, J. Appl. Meteorol., 8, 3, 392-400 (1969)
[29] North, Gerald R., Theory of energy-balance climate models, J. Atmos. Sci., 32, 11, 2033-2043 (1975)
[30] North, Gerald R.; Cahalan, Robert F.; Coakley, James A., Energy balance climate models, Rev. Geophys., 19, 1, 91-121 (1981)
[31] North, G. R.; Mengel, J. G.; Short, D. A., Simple energy balance model resolving the seasons and the continents: application to the astronomical theory of the ice ages, J. Geophys. Res.: Oceans, 88, C11, 6576-6586 (1983)
[32] McGehee, Richard; Widiasih, Esther, A quadratic approximation to Budyko’s ice-albedo feedback model with ice line dynamics, SIAM J. Appl. Dyn. Syst., 13, 1, 518-536 (2014) · Zbl 1301.34106
[33] Widiasih, Esther R., Dynamics of the Budyko energy balance model, SIAM J. Appl. Dyn. Syst., 12, 4, 2068-2092 (2013) · Zbl 1290.37033
[34] Walsh, James; Widiasih, Esther; Hahn, Jonathan; McGehee, Richard, Periodic orbits for a discontinuous vector field arising from a conceptual model of glacial cycles, Nonlinearity, 29, 6, 1843 (2016) · Zbl 1364.37172
[35] Diaz, J. I.; Hetzer, G.; Tello, L., An energy balance climate model with hysteresis, Nonlinear Anal. TMA, 64, 9, 2053-2074 (2006) · Zbl 1098.35090
[36] Díaz, J. I.; Tello, L., Infinitely many stationary solutions for a simple climate model via a shooting method, Math. Methods Appl. Sci., 25, 4, 327-334 (2002) · Zbl 1181.35186
[37] Fowler, Andrew, A simple thousand-year prognosis for oceanic and atmospheric carbon change, Pure Appl. Geophys., 172, 1, 49-56 (2015)
[38] Källén, E.; Crafoord, C.; Ghil, M., Free oscillations in a climate model with ice-sheet dynamics, J. Atmos. Sci., 36, 12, 2292-2303 (1979)
[39] Ghil, Michael; Le Treut, Hervé, A climate model with cryodynamics and geodynamics, J. Geophys. Res.: Oceans, 86, C6, 5262-5270 (1981)
[40] Ghil, Michael; Tavantzis, John, Global hopf bifurcation in a simple climate model, SIAM J. Appl. Math., 43, 5, 1019-1041 (1983) · Zbl 0573.76047
[41] Nadeau, Alice; McGehee, Richard, A simple formula for a planet’s mean annual insolation by latitude, Icarus, 291, 46-50 (2017)
[42] Fowler, Andrew, Mathematical Geoscience, Vol. 36 (2011), Springer Science & Business Media · Zbl 1219.86001
[43] Pierrehumbert, Raymond T., Principles of Planetary Climate (2010), Cambridge University Press · Zbl 1244.86002
[44] Walsh, James; McGehee, Richard, Modeling climate dynamically, College Math. J., 44, 5, 350-363 (2013)
[45] Paillard, Didier, The timing of pleistocene glaciations from a simple multiple-state climate model, Nature, 391, 6665, 378-381 (1998)
[46] Ditlevsen, Peter D.; Ashwin, Peter, Complex climate response to astronomical forcing: The middle-pleistocene transition in glacial cycles and changes in frequency locking., Front. Phys., 6, 62 (2018)
[47] Lenton, Timothy M., Early warning of climate tipping points, Nature Clim. Change, 1, 4, 201-209 (2011)
[48] Weertman, Johannes, Milankovitch solar radiation variations and ice age ice sheet sizes, Nature, 261, 17-20 (1976)
[49] Oerlemans, J., Model experiments on the 100,000-yr glacial cycle, Nature, 287, 5781, 430-432 (1980)
[50] Van der Veen, Cornelis J., Fundamentals of Glacier Dynamics (2013), CRC Press
[51] Oerlemans, Johannes; Van Der Veen, Cornelis J., Ice Sheets and Climate, Vol. 21 (1984), Springer
[52] Fowler, Andrew, Glaciers and ice sheets, (The Mathematics of Models for Climatology and Environment (1997), Springer), 301-336 · Zbl 0897.73054
[53] Guckenheimer, John; Holmes, Philip, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Vol. 42 (2013), Springer Science & Business Media · Zbl 0515.34001
[54] Kuznetsov, Yuri A., Elements of Applied Bifurcation Theory, Vol. 112 (2013), Springer Science & Business Media
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