### Abstract

In this research, we consider a mathematical model of permafrost lake growth and then using this model we propose a simple phenomenological equation that allows us to evaluate the impact of the Siberian permafrost on climate. Mathematically, permafrost thawing can be described by the classical Stefan approach. We can use a modified approach based on the phase transition theory. This takes into account that thawing layer has a small but non-zero width. The transition from the frozen state to the thawing state is a microscopic process, while lakes are great macroscopic objects. Thus we can assume that locally a lake boundary is a sphere of a large radius of curvature. Moreover, the growth is a slow process. Under such assumptions, thawing front velocity can be investigated. Indeed, there are possible asymptotic approaches based on so-called mean curvature motion. As a result, we obtain a deterministic equation that serves as an extremely simplified model of lake growth. We can, therefore, propose here a simple method to compute methane emission into the atmosphere using natural assumption that the horizontal dimensions of the lakes are much larger than the lake depth. We note that the permafrost lake model that we developed for the methane emission positive feedback loop problem is a conceptual climate model.