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A study of abrupt climate change in a simple nonlinear climate model


We use a seasonal energy balance climate model to study the behavior of the snowline cycle as a function of external parameters such as the solar constant. Our studies are confined in this study to cases with zonally symmetric land-sea distributions (bands or caps of land). The model is nonlinear in that the seasonally varying snow/sea ice line modifies the energy receipt through its different albedo from open land or water. The repeating steady-state seasonal cycle of the model is solved by a truncated Fourier series in time. This method is several thousand times faster than a time stepping approach. The results are interesting in that a number of bifurcations in the snowline behavior are found and studied for various geographies. Polar land caps and land bands positioned near the poles exhibit a variety of discontinuous summer snow cover behaviors (abrupt transitions as a parameter such as solar constant is slowly varied), which may be relevant to the inception and decay of continental ice sheets.

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  1. Berger WH, Labeyrie LD (eds) (1987) Abrupt climate change. Reidel, Dordrecht

  2. Broecker WS, Peteet DM, Rind D (1985) Does the ocean-atmosphere system have more than one stable mode of operation? Nature 315: 21–26

  3. Burden RL, Faires JD (1985) Numerical analysis, 3rd edn. Prindle, Weber & Schmidt, Boston, pp 496–505

  4. Budyko MI (1969) The effect of solar radiation variations on the climate of the Earth. Tellus 21: 611–619

  5. Cahalan RF, North GR (1979) A stability theorem for energy balance climate models. J Atmos Sci 36: 1205–1216

  6. Crowley TJ (1983) The geologic record of climate change. Rev Geophys Space Phys 1: 828–877

  7. Crowley TJ, North GR (1988) Abrupt climate change and extinction events in Earth history. Science 240: 996–1002

  8. Crowley TJ, Short DA, Mengel JG, North GR (1986) Role of seasonality in the evolution of climate over the past 100 million years. Science 231: 579–584

  9. Crowley TJ, Mengel JG, Short DA (1987) Gondwanaland's seasonal cycle. Nature 329: 803–807

  10. Drazin PG, Griffel DH (1977) On the branching structure of diffusive climatological models. J Atmos Sci 34: 1696–1706

  11. Ghil M (1976) Climate stability for a Sellers-type model. J Atmos Sci 33: 3–20

  12. Hyde WT, Crowley TJ, Kim KY, North GR (1989) Comparison of GCM and energy balance model simulations of seasonal temperature changes over the past 18 000 years. J Clim 2: 864–887

  13. Iooss G, Joseph DD (1980) Elementary stability and bifurcation theory. Springer, New York Heidelberg Berlin, pp 126–133

  14. Lin RQ (1988) A hierarchy of perturbative models for solving nonlinear problems in geophysical fluid dynamics: systematic use of symbolic manipulation. PhD thesis, UCLA

  15. Mengel JG, Short DA, North GR (1988) Seasonal snowline instability in an energy balance model. Clim Dyn 2: 127–131

  16. North GR (1975a) Analytical solution to a simple climate model with diffusive heat transport. J Atmos Sci 32: 1301–1307

  17. North GR (1975b) Theory of energy-balance climate model. J Atmos Sci 32: 2033–2043

  18. North GR (1984) The small ice cap instability in diffusive climate model. J Atmos Sci 41: 3390–3395

  19. North GR, Coakley JA Jr (1979) Differences between seasonal and mean annual energy balance model calculations of climate and climate sensitivity. J Atmos Sci 36: 1189–1204

  20. North GR, Crowley TJ (1985) Application of a seasonal climate model to Cenozoic glaciation. J Geol Soc Lond 142: 475–482

  21. North GR, Mengel JG, Short DA (1983) Simple energy balance model resolving the seasons and the continents: application to the astronomical theory of the ice ages. J Geophys Res 88: 6576–6586

  22. Ramanathan V, Cess RD, Harrison EF, Minnis P, Barkstrom BR, Ahmad E, Hartmann D (1989) Cloud-radiative forcing and climate: results from the Earth Radiation Budget Experiment. Science 243: 57–62

  23. Savin SM (1977) The history of the Earth's surface temperature during the past 100 million years. Annu Rev Earth Planet Sci 5: 319–355

  24. Sellers WD (1969) A climate model based on the energy balance of the earth-atmosphere system. J Clim Appl Meteorol 8: 392–400

  25. Short DA, North GR, Bess TD, Smith GL (1984) Infrared parameterization and simple climate models. J Clim Appl Meteorol 23: 1222–1233

  26. Suarez MJ, Held IM (1979) The sensitivity of an energy balance model to variations in orbital parameters. J Geophys Res 84: 4825–4836

  27. Watts R, Hayder M (1984) The effect of land-sea distribution on ice sheet formation. Ann Glaciol 56: 234–236

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Correspondence to G R North.

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Lin, R.Q., North, G.R. A study of abrupt climate change in a simple nonlinear climate model. Climate Dynamics 4, 253–261 (1990). https://doi.org/10.1007/BF00211062

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  • Open Land
  • Solar Constant
  • Abrupt Climate
  • Abrupt Climate Change
  • Truncate Fourier Series