The Cryosphere Discussions www.the-cryosphere-discuss.net 1994-0432 1994-0440 4 3 2010 10.5194/tcd-4-1243-2010 http://www.the-cryosphere-discuss.net/4/1243/2010/ http://www.the-cryosphere-discuss.net/4/1243/2010/tcd-4-1243-2010.html http://www.the-cryosphere-discuss.net/4/1243/2010/tcd-4-1243-2010.pdf 1243 1276 2010-08-11 An energy-conserving model of freezing variably-saturated soil M. Dall'Amico matteo@mountain-eering.com S. Endrizzi S. Gruber R. Rigon Department of Civil and Environmental Engineering, University of Trento, Trento, Italy Environment Canada – National Hydrology Research Centre, Saskatoon, Saskatchewan, Canada Department of Geography, University of Zurich, Winterthurerstrasse 190 Zurich, Switzerland now at: Mountain-eering srl, Via Siemens 19 Bolzano, Italy In this paper we provide a method for solving the energy equation in freezing soil. The solver is linked with the solution of Richards equation, and therefore able to approximate water movement near the liquid-solid phase transition. The equations show non-linear characteristics causing oscillatory behavior in the solution close to the phase transition, when normal methods of iterative integration, as Newton or Picard, are used. Thus, a globally convergent Newton method has been implemented to achieve convergence. The method is tested by comparison with an analytical solution to the Stefan problem and by comparison with experimental data derived from literature. Anisimov, O., Shiklomanov, N., and Nelson, E.: Variability of seasonal thaw depth in permafrost regions: A stochastic modeling approach, Ecol. Model., 153, 217–227, 2002. Brooks,~R H. and Corey,~A T.: Hydraulic properties of porous media, Hydrology paper, 3, Colorado State University, Fort Collins, 1964. Cainelli,~O., Tarantino,~A., Costisella,~C., and Bellin,~A.: Does the static water retention curve reproduce correctly subsurface water dynamics?, Submitted to Water Resour. Res., submitted, 2010. Carslaw,~H. and Jaeger,~J.: Conduction of Heat in Solids, Clarendon Press Oxford, 1959. Christoffersen,~P. and Tulaczyk,~S.: Response of subglacial sediments to basal freeze-on: 1. Theory and comparison to observations from beneath the West Antarctica Ice Sheet,~J. Geophys. Res., 108, 2222, doi:10.1029/2002JB001935, 2003. Clapp,~R B. and Hornberger,~G M.: Empirical equations for some hydraulic properties, Water Resour. Res., 14, 601–605, 1978. Crank, J. and Nicolson, P.: A practical method for numeri- cal evaluation of solutions of partial differential equations of the heat-conduction type, Advances in Computational Mathematics, 6(1), 207–226, 1996. Daanen,~R., Misra,~D., and Epstein,~H.: Active-layer hydrology in nonsorted circle ecosystems of the Arctic Tundra, Vadose Zone J., 6, 694–704, 2007. Dall'Amico,~M.: Coupled water and heat transfer in permafrost modeling, Ph.D. thesis, Institute of Civil and Environmental Engineering, Università degli Studi di Trento, Trento, available from http://eprints-phd.biblio.unitn.it/335/, 2010. Endrizzi,~S., Quinton,~W L B., Marsh,~P., and Dall'Amico,~M.: Investigating the energy-based runoff generation theory for organic-covered permafrost using spatially distributed modelling, to be submitted to Hydrol. Earth Syst. Sci., 2010. Farouki,~O T.: The thermal properties of soils in cold regions, Cold Reg. Sci. Technol., 5, 67–75, 1981. Fredlund,~D. and Xing,~A.: Equations for the soil-water characteristic curve, Can. Geotech. J., 31, 521–532, 1994. Fuchs,~M., Campbell,~G., and Papendick,~R.: An analysis of sensible and latent heat flow in a partially frozen unsaturated soil, Soil Sci. Soc. Am. J., 42, 379–385, 1978. Gardner,~W.: Some steady-state solutions of the unsaturated moisture flow equation with application to evaporation from a water table, Soil Sci., 85, 228–232, 1958. Hansson,~K., Simunek,~J., Mizoguchi,~M., Lundin,~L., and van Genuchten,~M.: Water flow and heat transport in frozen soil numerical solution and freeze-thaw applications, Vadose Zone J., 3, 693–704, 2004. Harlan,~R.: Analysis of coupled heat-fluid transport in partially frozen soil, Water Resour. Res., 9, 1314–1323, 1973. Jame,~Y. and Norum,~D.: Heat and mass transfer in a freezing unsaturated porous medium, Water Resour. Res., 16, 811–819, 1980. Johansen,~O.: Thermal Conductivity of Soils, Ph.D. thesis, Norwegian Technical University, Trondheim, 1975. Kelley,~C.: Solving Nonlinear Equations with Newton's Method, Society for Industrial Mathematics, 2003. Loch,~J.: Thermodynamic equilibrium between ice and water in porous media, Soil Sci., 126, 77–80, 1978. Lukyanov,~V. and Golovko,~M.: Calculations of the Depth of Freeze in Soils, Union of Transp. Constr., Moscow, 23, 1957, (in russian). Lunardini,~V.: Heat Transfer in Cold Climates, Van Nostrand Rheinhold, New York, 1981. Lunardini,~V.: Freezing of soil with phase change occurring over a Þnite temperature difference, in: Proceedings of the 4th offshore mechanics and arctic engineering symposium, The American Society of Mechanical Engineers, 1985. Luo,~L., Robock,~A., Vinnikov,~K., Schlosser,~C., Slater,~A., Boone,~A., Braden,~H., Cox,~P., de~Rosnay,~P., Dickinson,~R. et~al.: Effects of frozen soil on soil temperature, spring infiltration, and runoff: Results from the PILPS 2 (d) experiment at Valdai, Russia, J. Hydrometeorol., 4, 334–351, 2003. McKenzie,~J., Voss,~C. and Siegel,~D.: Groundwater flow with energy transport and water–ice phase change: Numerical simulations, benchmarks, and application to freezing in peat bogs, Adv. Water Resour., 30, 966–983, 2007. Miller,~R.: Phase equilibria and soil freezing, in Permafrost: Proceedings of the Second International Conference, Washington DC: National Academy of Science-National Research Council, vol. 287, 193–197, 1965. Mualem,~Y.: A new model for predicting the hydraulic conductivity of unsaturated porous media, Water Resour. Res., 12, 513–522, 1976. Nakano,~Y. and Brown,~J.: Effect of a freezing zone of finite width on thermal regime of soils, Water Resour. Res., 5, 1226–1233, 1971. Newman,~G. and Wilson,~G.: Heat and mass transfer in unsaturated soils during freezing, Can. Geotech. J., 34, 63–70, 1997. Phukan,~A.: Frozen Ground Engineering, Prentice-Hall, Englewood Cliffs, NJ, 1985. Rigon,~R., Bertoldi,~G., and Over,~T M.: GEOtop: a distributed hydrological model with coupled water and energy budgets, J. Hydrometeorol., 7, 371–388, 2006. Schaap,~M., Leij,~F., and van Genuchten,~M.: Rosetta: a computer program for estimating soil hydraulic parameters with hierarchical pedotransfer functions, J. Hydrol., 251, 163–176, 2001. Shoop,~S. and Bigl,~S.: Moisture migration during freeze and thaw of unsaturated soils: modeling and large scale experiments, Cold Reg. Sci. Technol., 25, 33–45, 1997. Smirnova,~T., Brown,~J., Benjamin,~S., and Kim,~D.: Parameterization of cold-season processes in the MAPS land-surface scheme, J. Geophys. Res., 105, 4077–4086, 2000. Spaans,~E. and Baker,~J.: The soil freezing characteristic: its measurement and similarity to the soil moisture characteristic, Soil Sci. Soc. Am. J., 60, 13–19, 1996. Tomita,~H.: Analysis of spurious surface temperature at the atmosphere–land interface and a new method to solve the surface energy balance equation, J. Hydrometeorol., 10, 833–844, 2009. Van Genuchten,~M T.: A closed-form equation for predicting the hydraulic conductivity of unsaturated soils., Soil Sci. Soc. Am J., 44, 892–898, 1980. Viterbo,~P., Beljaars,~A., Mahfouf,~J., and Teixeira,~J.: The representation of soil moisture freezing and its impact on the stable boundary layer, Q. J. Roy. Meteor. Soc., 125, 2401–2426, 1999. Watanabe,~K.: Water and heat flow in a directionally frozen silty soil, Proceedings of the third HYDRUS Workshop, Jun 28, 2008, Tokyo University of Agriculture and Technology, Tokyo, Japan, ISBN:978-4-9901192, 15–22, 2008. Williams,~P. and Smith,~M.: The Frozen Earth: Fundamentals of Geocryology, Cambridge University Press, 1989. Zhang,~X., Sun,~S., and Xue,~Y.: Development and testing of a frozen soil parameterization for cold region studies, J. Hydrometeorol., 8, 690–701, 2007. Zhang,~Y., Carey,~S., and Quinton,~W.: Evaluation of the algorithms and parameterizations for ground thawing and freezing simulation in permafrost regions,~J. Geophys. Res, 113, D17116, doi:10.1029/2007JD009343, 2008.