Abstract
The seasonal change in depths of the frozen and thawed soils within their active layer is reduced to a moving boundary problem, which describes the dynamics of the total ice content using an independent mass balance equation and treats the soil frost/thaw depths as moving (sharp) interfaces governed by some Stefan-type moving boundary conditions, and hence simultaneously describes the liquid water and solid ice states as well as the positions of the frost/thaw depths in soil. An adaptive mesh method for the moving boundary problem is adopted to solve the relevant equations and to determine frost/thaw depths, water content and temperature distribution. A series of sensitivity experiments by the numerical model under the periodic sinusoidal upper boundary condition for temperature are conducted to validate the model, and to investigate the effects of the model soil thickness, ground surface temperature, annual amplitude of ground surface temperature and thermal conductivity on frost/thaw depths and soil temperature. The simulated frost/thaw depths by the model with a periodical change of the upper boundary condition have the same period as that of the upper boundary condition, which shows that it can simulate the frost/thaw depths reasonably for a periodical forcing.
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References
Woo M K, Winter T C. The role of permafrost and seasonal frost in the hydrology of northern wetlands in North America. J Hydrol, 141: 5–31 (1993)
Luo L, Robock A, Vinnikov K Y, et al. Effects of frozen soil on soil temperature, spring infiltration and runoff: Results from the PILPS 2 (d) experiment at Valdai, Russia. J Hydrometeorology, 4: 334–351 (2003)
Niu G Y, Yang Z L. Effects of frozen soil on snowmelt runoff and soil water storage at a continental scale. J Hydrometeorology, 7: 937–952 (2006)
Cogley J G, Pitman A J, Henderson-Sellers A. A land surface for large-scale climate models. Tech Note 90-1. Peterborough, Ont: Trent Univ, 1990
Muller S W. Permafrost or Permanently Frozen Ground and Related Engineering Problems. Ann Arbor, Mich: Edwards, 1947
Javierre E, Vuik C, Vermolen F J, Zwaag S van der. A comparison of numerical models for one-dimensional Stefan problems. J Comput Appl Math, 192: 445–459 (2006)
Liu Q, Shui H S, Zhang X Y. Advances in numerical simulation of interfacial/free-surfaces flows (in Chinese). Adv Mech, 32(2): 259–274 (2002)
Wang Z F, Liu R X. Front tracing methods (in Chinese). Mechanics and Practice, 22: 1–14 (2000)
Crank J Free and Moving Boundary Problems. Oxford: Clarendon Press, 1984
Segal G, Vuik C, Vermolen F. A conserving discretization for the free boundary in a two-dimensional Stefan problem. J Compute Phys, 141: 1–21 (1998)
Liu R X, Wang Z F. Methods of Numerical Simulation and Tracking for Moving Interfaces (in Chinese). Hefei: University of Science and Technology of China Press, 2001
Minkowycz W J, Sparrow E M. Advances in Numerical Heat Transfer 1. Washington: Taylor & Francis, 1997
Hirt C W, Nichols B D. Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys, 39: 201–225 (1981)
Gueyfier D, Li J. Volume-of-fluid interface tracking with smoothed surface stress methods for 3D flows. J Comput Phys, 152(2): 423–456 (1999)
Osher S, Sethian J A. Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J Comput Phys, 79: 12–49 (1988)
Vermolen F J, Javierre E, Vuik C, Zha L, Zwaag S van der. A three-dimensional model for particle dissolution in binary alloys. Comput Materials Science, 39: 767–774 (2007)
Wheeler A A, Boettinger W J, McFadden G B. Phase field model for isothermal phase transitions in binary alloys. Phys Rev A, 45: 7424–7439 (1992)
Mackenzie J A, Robertson M L. A moving mesh method for the solution of the one-dimensional phase field equations. J Comput Phys, 181: 526–544 (2002)
Nan Z T, Li S X, Cheng G D. Prediction of permafrost distribution on the Qinghai-Tibet Plateau in the next 50 and 100 years. Sci China Ser D-Earth Sci, 48(6): 797–804 (2005)
Clapp R B, Hornberger G M. Empirical equations for some soil hydraulic properties. Water Resource Res, 14: 601–604 (1978)
Sun S F. Parameterization Study of Physical and Biochemical Mechanism in Land Surface Process (in Chinese). Beijing: China Meteorology Press, 2005
Gardner W R. Relation of root distribution to water uptake and availability. Agron J, 56: 35–41 (1964)
Molz F J, Remson I. Extraction term models of soil moisture use by transpiring plants. Water Resource Res, 6: 1346–1356 (1970)
Lei Z D, Yang S X, Xie S H. Soil Water Dynamics (in Chinese). Beijing: Tsinghua University Press, 1998
Visintin. Models of phase transitions. Progress in Nonlinear Differential Equations and Their Applications. Boston: Birkhäuser, 1996
Bangerth W, Rannacher R. Adaptive Finite Element Methods for Differential Equations. New York: Birkhäuser, 2003
Li X, Koike T. Frozen soil parameterization in SiB2 and its validation with GAME-Tibet observations. Cold Regions Science and Technology, 36(1–3): 165–182 (2003)
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This work was supported by the National Basic Research Program (Grant No. 2005CB321703), the Knowledge Innovation Project of Chinese Academy of Sciences (Grant Nos. KZCX2-yw-126-2, KZCX2-yw-217), and the Chinese Coordinated Observation and Prediction of the Earth System project (Grant No. GYHY20070605)
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Xie, Z., Song, L. & Feng, X. A moving boundary problem derived from heat and water transfer processes in frozen and thawed soils and its numerical simulation. Sci. China Ser. A-Math. 51, 1510–1521 (2008). https://doi.org/10.1007/s11425-008-0096-x
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DOI: https://doi.org/10.1007/s11425-008-0096-x