Mathematical Model of Cryospheric Response to Climate Changes
Publication: Journal of Cold Regions Engineering
Volume 27, Issue 2
Abstract
This paper focuses on the development of simplified mathematical models of the cryosphere which may be useful in further understanding possible global climate change impacts and in further assessing future impacts captured by global circulation models (GCMs). The mathematical models developed by leveraging the dominating effects of freezing and thawing within the cryosphere to simplify the relevant heat transport equations are tractable to direct solution or numerical modeling. In this paper, the heat forcing function is assumed to be a linear transformation of temperature (assumed to be represented by proxy realizations). The output from the governing mathematical model is total ice volume of the cryosphere. The basic mathematical model provides information as a systems modeling approach that includes sufficient detail to explain ice volume given the estimation of the heat forcing function. A comparison between modeling results in the estimation of ice volume versus ice volume estimates developed from use of proxy data are shown in the demonstration problems presented.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
Acknowledgements are paid to the United States Military Academy, West Point, New York, Department of Mathematical Sciences and the Naval Post Graduate School, Monterey, California, Department of Defense Analysis for their support to the authors during this research. Also acknowledged are the several individuals who have participated in particular tasks in developing this paper including, but by no means limited to, Rene Perez, Laura Hromadka, Bethany Espinosa, and Michael Barton.
References
Alexiades, V., and Solomon, A. D. (1993). Mathematical modeling of melting and freezing processes, Taylor & Francis, London.
Anagnostopoulos, G. G., Koutsoyiannia, D., Christofides, A., Efstratiadis, A., and Mamassis, N. (2010). “A comparison of local and aggregated climate model outputs with observed data.” Hydrol. Sci. J., 55(7), 1094–1110.
Bamber, J. L., and Payne, A. J. (2004). Mass balance of the cryosphere, Cambridge University Press, Cambridge, UK.
Berger, A., Gallee, H., and Tricot, C. (1993). “Glaciation and deglaciation mechanisms in a coupled two-dimensional climate-ice-sheet model.” J. Glaciol., 39(131), 45–49.
Bintanja, R., van de Wal, R. S. W., and Oerlemans, J. (2005). “Modelled atmospheric temperatures and global sea levels over the past million years.” Nature, 437(7055), 125–128.
Bitz, C. M., and Marshall, S. J. (2012). Cryosphere models: Ocean and land, Encyclopedia of sustainability science and technology (section on climate change modeling and methodology) 〈http://www.atmos.washington.edu/~bitz/cryo_chapter.pdf〉.
Bounoua, L., et al. (2010). “Quantifying the negative feedback of vegetation to greenhouse warming: A modeling approach.” Geophys. Res. Lett., 37(23), L23701.
Dyurgerov, M. B., and Meier, M. F. (1999). “Twentieth century climate change: Evidence from small glaciers.”, Institute of Arctic and Alpine Research (INSTAAR), Univ. of Colorado, Boulder, CO.
Guymon, G. L., Berg, R. L., and Hromadka, T. V., II (1993)., U.S. Army Cold Regions Research and Engineering Laboratory, Hanover, NH.
Guymon, G. L., Berg, R. L., Johnson, T., and Hromadka, T. V., II (1981). “Results from a mathematical model of frost heave.”, Transportation Research Board, Washington, DC, 2–6.
Guymon, G. L., Harr, M. E., Berg, R. L., and Hromadka, T. V., II (1981). “A probabilistic-deterministic analysis of one-dimensional ice segregation in a freezing soil column.” Cold Reg. Sci. Technol., 5(2), 127–140.
Guymon, G. L., and Hromadka, T. V., II (1980). “Finite element model of heat conduction with isothermal phase change (two- and three-dimensional).”, U.S. Army Cold Regions Research and Engineering Laboratory, Hanover, NH.
Guymon, G. L., Hromadka, T. V., II, and Berg, R. L. (1980). “A one-dimensional frost heave model based upon simulation of simultaneous heat and water flux.” 1979 Conf. on Soil Water Problems in Cold Regions, Cold Regions Science and Technology, Vol. 3, Elsevier, Amsterdam, 253–262.
Guymon, G. L., Hromadka, T. V., II, and Berg, R. L. (1984). “Two-dimensional model of coupled heat and moisture transport in frost-heaving soils.” J. Energy Res. Technol., 106(3), 336–343.
Hromadka, T. V., II (1986a). “Complex variable boundary elements in engineering.” CRREL Special Rep., U.S. Army Cold Regions Research and Engineering Laboratory, Hanover, NH.
Hromadka, T. V., II (1986b). “A nodal domain integration model of two-dimensional heat and soil-water flow coupled by soil-water phase change.”, U.S. Army Cold Regions Research and Engineering Laboratory, Hanover, NH.
Hromadka, T. V., II (1986c). “Tracking two-dimensional freezing front movement using the complex variable boundary element method.”, U.S. Army Cold Regions Research and Engineering Laboratory, Hanover, NH.
Hromadka, T. V., II (1986d). “Predicting two-dimensional steady-state soil freezing fronts using the CVBEM.” J. Heat Transfer, 108(1), 235–237.
Hromadka, T. V., II, and Guymon, G. L. (1982). “Application of boundary integral equation to prediction of freezing fronts in soil.” Cold Reg. Sci. Technol., 6(2), 115–121.
Hromadka, T. V., II, and Guymon, G. L. (1984). “Simple model of ice segregation using an analytic function to model heat and soil-water flow.” J. Energy Res. Technol., 106(4), 515–520.
Hromadka, T. V., II, and Guymon, G. L. (1985). “An algorithm to reduce approximation error from the complex-variable boundary-element method applied to soil freezing.” Numer. Heat Transfer, 8(1), 115–130.
Imbrie, J., et al. (1984). The orbital theory of Pleistocene climate: Support from a revised chronology of the marine d18O record, in Milankovitch and Climate, Part 1, A. Berger, ed., Springer, New York, 269–305.
Imbrie, J., and Imbrie, J. Z. (1980). “Modeling the climatic response to orbital variations.” Sci., 207(4434), 943–953.
Imbrie, J. Z., Imbrie-Moore, A., and Lisiecki, L. (2011). “A phase-space model for Pleistocene ice volume.” Earth Planet. Sci. Lett., 307(1–2), 94–102.
Jouzel, J., et al. (1987). “Vostok ice core: A continuous isotope temperature record over the last climatic cycle (160,000 years).” Nature, 329(6138), 403–408.
Jouzel, J., et al. (1993). “Extending the Vostok ice-core record of Palaeoclimate to the penultimate glacial period.” Nature, 364(6436), 407–412.
Jouzel, J., et al. (1996). “Climatic interpretation of the recently extended Vostok ice records.” Clim. Dyn., 12(8), 513–521.
Jouzel, J., et al. (2007a). “Orbital and millennial Antarctic climate variability over the past 800,000 years.” Sci., 317(5839), 793–797.
Jouzel, J., et al. (2007b). EPICA Dome C ice core 800KYr deuterium data and temperature estimates, NOAA/NCDC Paleoclimatology Program, Boulder, CO.
Kawamura, K., et al. (2007a). Dome Fuji ice core 340KYr (2500m) d18O data, NOAA/NCDC Paleoclimatology Program, Boulder, CO.
Kawamura, K., et al. (2007b). Dome Fuji ice core preliminary temperature reconstruction, 0-340 kyr, NOAA/NCDC Paleoclimatology Program, Boulder, CO.
Kirkby, J., Mangini, A., and Muller, R. (2004). “The Glacial Cycles and Cosmic Rays.” submitted to Earth and Planetary Science Letters 〈http://arxiv.org/abs/physics/0407005v1〉 (Jul. 4, 2001).
Lisiecki, L. E., and Raymo, M. E. (2005a). LR04 global Pliocene-Pleistocene Benthic d18O stack, NOAA/NGDC Paleoclimatology Program, Boulder, CO.
Lisiecki, L., and Raymo, M. (2005b). “A Pliocene-Pleistocene stack of 57 globally distributed Benthic d18O Records.” Paleoceanography, 20(1), PA1003.
Loague, K. M., and Freeze, R. A. (1985). “A comparison of rainfall-runoff modeling techniques on small upland catchments.” Water Resour. Res., 21(2), 229–248.
Loulergue, L., et al. (2007). “New constraints on the gas age-ice age difference along the EPICA ice cores, 0-50 kyr.” Climate of the Past, 3(3), 527–540.
Liu, H. S. (1992). “Frequency variations of the earth’s obliquity and the 100-kyr ice-age cycles.” Nature, 358(6385), 397.
Mix, A. C., and Ruddiman, W. F. (1984). “Oxygen–isotope analysis and Pleistocene ice volumes.” Quat. Res., 21(1), 1–20.
Petit, J. R., et al. (1999). “Climate and atmospheric history of the past 420,000 years from the Vostok Ice Core, Antarctica.” Nature, 399(6735), 429–436.
Petit, J. R., et al. (2001). Vostok Ice Core Data for 420,000 years, NOAA/NGDC Paleoclimatology Program, Boulder, CO.
Randall, D. A., et al. (2007). “Climate Models and Their Evaluation.” Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on climate Change, Cambridge University Press, Cambridge, UK.
Raper, S. C. B., and Braithwaite, R. J. (2009). “Glacier volume response time and its links to climate and topography based on a conceptual model of glacier hypsometry.” Cryosphere Discuss., 3(1), 243–275.
Ruddiman, W. F. (2001). Earth’s climate: Past and future, W.H. Freeman & Sons, New York.
Wigley, T. M. L., and Raper, S. C. B. (2005). “Extended scenarios for glacier melt due to anthropogenic forcing.” Geophys. Res. Lett., 32(5), L05704.
Zachos, J., Pagani, M., Sloan, L., Thomas, E., and Billups, K. (2001). “Trends, rhythms, and aberrations in global climate change 65 Ma to present.” Sci., 292(5517), 686–693.
Information & Authors
Information
Published In
Copyright
© 2013 American Society of Civil Engineers.
History
Received: Jul 7, 2011
Accepted: Oct 4, 2012
Published online: Oct 5, 2012
Published in print: Jun 1, 2013
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.