Практический интерес в районах вечной мерзлоты представляет глубина сезонного оттаивания. Построена одномерная (в вертикальном направлении) упрощенная полуэмпирическая модель динамики вечной мерзлоты в “приближении медленных движений границ фазового перехода”, основанная на задаче Стефана и эмпирических соотношениях. Калибровочные параметры модели выбираются для исследуемого района с использованием натурных измерений глубины оттаивания и температуры воздуха. Проверка работоспособности численной модели проведена для района оз. Тулик (Аляска). Получено согласие рассчитанных значений глубины талого слоя и температуры поверхности почвы с результатами измерений
Due to the change in global air temperature, the assessment of permafrost reactions to climate change is of interest. As the climate warms, both the thickness of the thawed soil layer and the period for existence of the talik are increased. The present paper proposes a small-size numerical model of vertical temperature distributions in the thawed and frozen layers when a frozen layer on the soil surface is absent. In the vertical direction, thawed and frozen soils are separated. The theoretical description of the temperature field in soils when they freeze or melt is carried out using the solution of the Stefan problem. The mathematical model is based on thermal conductivity equations for the frozen and melted zones. At the interfacial boundary, the Dirichlet condition for temperature and the Stefan condition are set. The numerical methods for solving of Stefan problems are divided into two classes, namely, methods with explicit division of fronts and methods of end-to-end counting. In the present work, the method with the selection of fronts is implemented. In the one-dimensional Stefan problem, when transformed to new variables, the computational domain in the spatial variable is mapped onto the interval [0 , 1]. In the presented equations, the convective terms characterize the rate of temperature transfer (model 1). A simplified version of the Stefan problem solution is considered without taking into account this rate (“approximation of slow movements of the boundaries of the phase transition”, model 2). The model is tuned to a specific object of research. Model parameter values can vary significantly in different geographic regions. This paper simulates the dynamics of permafrost in the area of Lake Tulik (Alaska) in summer. Test calculations based on the proposed simplified model show its adequacy and consistency with field measurements. The developed model can be used for qualitative studies of the long-term dynamics of permafrost using data of the air temperature, relative air humidity and precipitation
A note on the exact boundary controllability for an imperfect transmission problem
