ESDD Earth System Dynamics Discussions ESDD Earth Syst. Dynam. Discuss. 2190-4995 Göttingen, Germany 10.5194/esdd-3-235-2012 Mathematical modelling of positive carbon-climate feedback: permafrost lake methane emission case Sudakov I. A. 1 Vakulenko S. A. 2 St. Petersburg State University of Technology and Design, Mathematics and Informatics Department, St. Petersburg, Russia Institute for Mechanical Engineering Problems, Russian Academy of Sciences, St. Petersburg, Russia 04 04 2012 3 1 235 257 Copyright: © 2012 I. A. Sudakov 2012 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/preprints/3/235/2012/esdd-3-235-2012.html The full text article is available as a PDF file from https://esd.copernicus.org/preprints/3/235/2012/esdd-3-235-2012.pdf

The permafrost methane emission problem is in the focus of attention of different climate models. We present new approach to the permafrost methane emission modeling. The tundra permafrost lakes is potential source of methane emission. Typically, tundra landscape contains a number of small lakes and warming leads to lake extension. We are making use of this process by the nonlinear theory of phase transitions. We find that climate catastrophe possibility depends on a feedback coefficient connecting the methane concentration in atmosphere and temperature, and on the tundra permafrost methane pool.

 References Barles, G., Soner, H. M., and Souganidis, P. E.: Front propagation and phase field theory, SIAM J. Control Optim., 31, 439–469, 1993. Bartlett, K. B. and Harriss, R. C.: Review and assessment of methane emissions from wetlands, Chemosphere, 26, 261–320, 1993. Brakke, K. A.: The Motion of a Surface by its Mean Curvature, Mathematical Notes No. 20, Princeton University Press, Princeton, N.J., 1978. Bronsard, L. and Kohn, R. V.: Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Different. Equat., 90, 211–237, 1991. Buckdahn, R., Cardaliaguet, P., and Quincampoix, M.: A representation formula for the mean curvature motion, SIAM J. Math. Anal., 33, 827–846, 2001. Caginalp, G.:Stefan and Hele-Shaw type problems as asymptotic limits of the phase field equations, Phys. Rev. A 39, 5887–5896, 1989. Cao, M. K., Marshall, S., and Gregson, K.: Global carbon exchange and methane emissions from natural wetlands: Application of a process-based model, J. Geophys. Res., 101, 14399–14414, 1996. Chen, X.: Generation and propagation of interfaces for reaction-diffusion equations, J. Different. Equat., 96, 116–141, 1992. Crandall, M. G. and Lions, P. L.: Viscosity solutions of Hamilton-Jacobi equations, T. Am. Math. Soc., 277, 1–42, 1983. De Mottoni, P. and Schatzman, M.: Geometrical evolution of developed interfaces, T. Am. Math. Soc., 347, 1533–1589, 1995. Downing, J. A., Prairie, Y. T., Cole, J. J., Duarte, C. M., Tranvik, L. J., McDowell, W. H., Kortelainen, P., Caraco, N. F., Melack, J. M., and Middelburg, J. J.: The global abundance and size distribution of lakes, ponds, and impoundments, Limnol. Oceanogr., 51, 2388–2397, 2006. Ecker, K. and Huisken, G.: Mean curvature evolution of entire graphs, Ann. Math., 130, 453–471, 1989. Ecker, K. and Huisken, G.: Interior estimates for hypersurfaces moving by mean curvature, Invent. Math., 105, 547–569, 1991. Evans, L. C., Soner, H. M., and Souganidis, P. E.: Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45, 1097–1123, 1992. Frolking, S. and Grill, P.: Cliruate controls on temporal variability of methane flux from a poor fen in southeastern New Hampshire: measurement and modeling, Global Biogeochem. Cy., 8, 385–387, 1994. Gage, M. and Hamilton, R. S.: The heat equation shrinking convex plane curves, J. Different. Geom., 23, 69–96, 1986. Giga, Y.: Surface Evolution Equations Level Set Method, Vorlesungsreihe, 44, Rheinische Friedrich-Wilhelms-Universität, Bonn, 2002. Huisken, G.: Flow by mean curvature of convex surfaces into spheres, J. Different. Geom., 20, 237–266, 1984. Ilmanen, T.: The level-set flow on a manifold, in: Differential Geometry: Partial Differential Equations on Manifolds, Los Angeles, CA, 1990. Ilmanen, T.: Convergence of the Allen-Cahn equation to Brakke's motion bymean curvature, J. Different. Geom., 38, 417–461, 1993. Isihara, A.: Statistical Physics, Academic Press, New Yor, London, 1971. Kerr, R.: Arctic Armageddon' needs more science, less hype, Science, 329, 620–621, 2010. Kiehl, J. T. and Trenberth, K. E: Earth's annual global mean energy budget, B. Am. Meteorol. Soc., 78, 197–208, 1997. Kirpotin, S. N., Polishchuk, Yu. M., and Bryksina, N. A.: Thermokarst lakes square dynamics of West-Siberian continuous and discontinuous permafrost under impact of global warming, Vestnik of Tomsk State University, 311, 156–159, 2008. Komarov, I.: Thermodynamics and Heatmasstransfer in Disperse Permafrost, Nauchnii mir (Scientific world), Moscow, 2003. Kotsyurbenko, O., China, K., Glagolev, M., Stubner, S., Simankova, M., Nozhevnikova, A., and Conrad, R.: Acetoclastic and hydrogenotrophic methane production and methanogenic populations in an acidic West-Siberian peat bog, Environ. Microbiol., 11, 1159–1173, 2004. \bibitem [Kravtsova and Bystrova(2009)]{Krav09} Kravtsova, V. I. and Bystrova, A. G.: Changes in thermokarst lake size in different regions of Russia for the last 30 years, Earth Cryosphere, XIII, 16–26, 2009. Krylova, A. I., Krupchatnikoff, V. N., and Zinovieva, I. V.: Modelling of methane emission from bog ecosystems, Bull. NCC. Ser. Num. Mod. Atmosphere, Ocean Environ. Stud., 7, 17–26, 2001. \bibitem[Matthews and Fung(1987)] Matt87 Matthews, E. and Fung, I. Y.: Methane emissions from natural wetlands: global distribution, area, and environmental characteristics of sources, Global Biogeochem. Cy., 1, 61–86, 1987. Molotkov, I. and Vakulenko, S.: Nonlinear Localized Waves, Leningrad University Publisher, Leningrad, USSR, 1988. Sudakov, I. A., Bobylev, L. P., and Beresnev, S. A.: Permafrost thermal regime simulation in response to modern climate change, Vestnik St. Petersburg University, Geol. Geogr., 1, 81–88, 2011.