• Stefan equation;
  • algorithm;
  • thaw-freeze depth;
  • multi-layered soil;
  • permafrost


The Stefan equation is one of the simplest approximate analytical solutions for the thaw-freeze problem. It provides a useful method for predicting the depth of thawing/freezing in soils when little site-specific information is available. The limited number of parameters in the Stefan equation makes possible its application in a multi-layered system. We demonstrate that a widely used algorithm (JL-algorithm), which has been frequently used in permafrost regions, was derived by an incorrect mathematical method. It will inevitably result in systematic errors in the simulation if this algorithm is used in a multi-layered soil.

We present another simple thaw-freeze algorithm (XG-algorithm) for multi-layered soils. The new algorithm can be used to determine the freeze/thaw front in multi-layered soils no matter how thick each layer is and how many layers the soil profile contains. Simulation results of the JL-algorithm and the XG-algorithm are compared using hypothetical soil profiles, and the XG-algorithm is also used to simulate the thaw depth at three permafrost monitoring sites on the Qinghai-Tibet Plateau and one on the Loess Plateau, China. These applications show that the XG-algorithm could be readily used to analyse the factors that affect active-layer thickness. It can also be coupled with hydrological or land surface models to simulate the freeze-thaw cycles in permafrost regions and for related engineering applications. Copyright © 2013 John Wiley & Sons, Ltd.