Sciences in Cold and Arid Regions ›› 2017, Vol. 9 ›› Issue (6): 525-533.doi: 10.3724/SP.J.1226.2017.00525

Previous Articles     Next Articles

Comments on thaw-freeze algorithms for multilayered soil, using the Stefan equation

ChangWei Xie1, William A. Gough2   

  1. 1. Cryosphere Research Station on the Qinghai-Tibet Plateau, State Key Laboratory of Cryospheric Sciences, Northwest Institute of Eco-Environment and Resources, Chinese Academy of Sciences, Lanzhou, Gansu 730000, China;
    2. Department of Physical and Environmental Sciences, University of Toronto at Scarborough, 1265 Military Trail, Toronto, Ontario, Canada
  • Received:2017-05-06 Online:2017-12-01 Published:2018-11-23
  • Contact: ChangWei Xie, Northwest Institute of Eco-Environment and Resources, Chinese Academy of Sciences, No. 320, West Donggang Road, Lanzhou, Gansu 730000, China.
  • Supported by:
    This work was supported by grants from the National Natural Science Foundation of China (41671068, 41421061, and 41771040), the State Key Laboratory of Cryospheric Sciences (SKLCS-ZZ-2017), and the Hundred Talents Program of the Chinese Academy of Sciences granted to ChangWei Xie (51Y551831).

Abstract: The Stefan equation provides a useful and widely used method for predicting the depth of thawing and freezing in a soil where little site-specific information is available. The original Stefan equation was derived for only a homogeneous medium, and some algorithms have been developed for its use in a multilayered system. However, although the Stefan equation was derived more than 100 years ago, there is not a unified understanding for its use in a multilayered system. This paper examines the use of the Stefan equation in multilayered soil, based on comparing three algorithms (JL-algorithm, NM-algorithm, and XG-algorithm). We conclude that the JL and NM algorithms are incorrect, as they arose from flawed mathematical derivations. Both of these algorithms failed to recognize that the thawing depth in a multilayered soil is a piecewise function and not a continuous function of time. This work asserts that the XG-algorithm is a correct and rigorous method to determine the freezing-thawing fronts in multilayered soil.

Key words: Stefan equation, algorithms, thaw depth, multilayered soil

Aldrich HP, 1956. Frost penetration below highway and airfield pavements. Highway Research Board Bulletin, 135: 124-144.
Berggren WS, 1943. Prediction of temperature distribution in frozen soil. Transactions American Geophysical Union, 24(3): 71-77.
Bonan GB, 1989. A computer model of the solar radiation, soil moisture, and soil thermal regimes in boreal forests. Ecological Modeling, 45: 275-306.
Fox JD, 1992. Incorporating freeze-thaw calculations into a water balance model. Water Resource Research, 28: 2229-2244.
Holden JT, Jones IR, Dudek SJ, 1981. Heat and mass flow associated with a freezing front. Engineering Geology, 18: 153-164.
Jumikis AR, 1977. Thermal Geotechnics. New Brunswick: Rutgers University Press, pp. 375.
Kurylyk BK, 2015. Discussion of: A simple thaw-freeze algorithm for a multi-layered soil using the Stefan equation by Xie and Gough, 2013. Permafrost and Periglacial Processes, 200-206. DOI: 10.1002/ppp.1834.
Lunardini VJ, 1981. Heat Transfer in Cold Climates. New York: Van Nostrand Reinhold Publishers, pp. 320-351.
Nelson FE, Outcalt SI, 1987. A computational method for prediction and regionalization of permafrost. Arctic and Alpine Research, 19(3): 279-288.
Nelson FE, Shiklomanov NI, Mueller GR, et al., 1997. Estimating active layer thickness over a large region: Kuparuk River Basin, Alaska, USA. Arctic and Alpine Research, 29(4): 367-378.
Nixon JF, McRoberts EC, 1973. A study of some factors affecting the thawing of frozen soils. Canadian Geotechnical Journal, 10: 439-452, DOI:10.1139/t73-037.
Riseborough D, Shiklomanov N, Etzelmuller B, et al., 2008. Recent Advances in Permafrost Modeling. Permafrost and Periglacial Processes, 19: 137-156, DOI:10.1002/ppp.615.
Vuik C, 1993. Some historical notes on the Stefan problem. South African Journal of Economics, 43(4): 329-335.
Wang CH, Jin SL, Wu ZY, et al., 2009. Evaluation and Application of the Estimation Methods of Frozen (Thawing) Depth over the China. Advances in Earth Science, 24(2): 132-140.
Woo MK, Arain MA, Mollinga M, et al., 2004. A two-directional freeze and thaw algorithm for hydrologic and land surface modeling. Geophysical Research Letters, 31: L12501, DOI:10.1029/2004GL019475.
Xie CW, Gough WA, 2013. A simple thaw-freeze algorithm for a multi-layered soil using the Stefan Equation. Permafrost and Periglacial Processes, 24: 252-260, DOI:10.1002/ppp.1770.
Yi SH, Mcguire AD, Harden J, et al., 2009. Interactions between soil thermal and hydrological dynamics in the response of Alaska ecosystems to fire disturbance. Journal of Geophysical Research, 114: G02015, DOI:10.1029/2008JG000841.
Yi SH, Woo MK, Arain MA, 2007. Impacts of peat and vegetation on permafrost degradation under climate warming. Geophysical Research Letters, 34: L16504, DOI:10.1029/2007GL030550.
No related articles found!
Full text



[1] Mohan Bahadur Chand,Rijan Bhakta Kayastha. Study of thermal properties of supraglacial debris and degree-day factors on Lirung Glacier, Nepal[J]. Sciences in Cold and Arid Regions, 2018, 10(5): 357 -368 .
[2] AiHong Xie,ShiMeng Wang,YiCheng Wang,ChuanJin Li. Comparison of temperature extremes between Zhongshan Station and Great Wall Station in Antarctica[J]. Sciences in Cold and Arid Regions, 2018, 10(5): 369 -378 .
[3] YanZai Wang,YongQiu Wu,MeiHui Pan,RuiJie Lu. Comparison of two classification methods to identify grain size fractions of aeolian sediment[J]. Sciences in Cold and Arid Regions, 2018, 10(5): 413 -420 .
[4] YinHuan Ao,ShiHua Lyu,ZhaoGuo Li,LiJuan Wen,Lin Zhao. Numerical simulation of the climate effect of high-altitude lakes on the Tibetan Plateau[J]. Sciences in Cold and Arid Regions, 2018, 10(5): 379 -391 .
[5] Zhuo Ga,Za Dui,Duodian Luozhu,Jun Du. Comparison of precipitation products to observations in Tibet during the rainy season[J]. Sciences in Cold and Arid Regions, 2018, 10(5): 392 -403 .
[6] Rong Yang,JunQia Kong,ZeYu Du,YongZhong Su. Altitude pattern of carbon stocks in desert grasslands of an arid land region[J]. Sciences in Cold and Arid Regions, 2018, 10(5): 404 -412 .
[7] Yang Qiu,ZhongKui Xie,XinPing Wang,YaJun Wang,YuBao Zhang,YuHui He,WenMei Li,WenCong Lv. Effect of slow-release iron fertilizer on iron-deficiency chlorosis, yield and quality of Lilium davidii var. unicolor in a two-year field experiment[J]. Sciences in Cold and Arid Regions, 2018, 10(5): 421 -427 .
[8] Ololade A. Oyedapo,Joseph M. Agbedahunsi,H. C Illoh,Akinwumi J. Akinloye. Comparative foliar anatomy of three Khaya species (Meliaceae) used in Nigeria as antisickling agent[J]. Sciences in Cold and Arid Regions, 2018, 10(4): 279 -285 .
[9] YuMing Wei,XiaoFei Ma,PengShan Zhao. Transcriptomic comparison to identify rapidly evolving genes in Braya humilis[J]. Sciences in Cold and Arid Regions, 2018, 10(5): 428 -435 .
[10] FangLei Zhong,AiJun Guo,XiaoJuan Yin,JinFeng Cui,Xiao Yang,YanQiong Zhang. Sociodemographic characteristics, cultural biases, and environmental attitudes: An empirical application of grid-group cultural theory in Northwestern China[J]. Sciences in Cold and Arid Regions, 2018, 10(5): 436 -446 .