Solution of the one-dimensional Stefan problem with two transitions for modelling of the water freezing in a glacial crevasse

• Main Authors: S. Popov V., С. Попов В.
• Other Authors: The author is grateful to A.S. Boronina and two anonymous reviewers for fruitful criticism. The study was financially supported by the Russian Science Foundation No. 22-27-00266., Автор выражает благодарность А.С. Борониной и двум анонимным рецензентам за конструктивную критику настоящей работы. Исследование выполнено при финансовой поддержке Российского научного фонда № 22-27- 00266.
• Format: Article in Journal/Newspaper
• Language: Russian
• Published: IGRAS 2023
• Subjects: mathematical modelling;numerical solution;Stefan problem;finite difference schemes;ice crevasses;Antarctica; математическое моделирование;численное решение;задача Стефана;конечноразностные схемы;ледниковые трещины;Антарктида; Antarctic; East Antarctica; Mirny; Annals of Glaciology; Antarc*; Antarctica; Антарктида
• Online Access: https://ice-snow.igras.ru/jour/article/view/1157; https://doi.org/10.31857/S2076673423010131

This article presents a numerical solution of the one-dimensional Stefan problem with two phase transitions, which is implemented on a non-uniform grid. The system of equations is written in a general form, i.e. it includes not only conductive, but also convective and dissipative terms. The problem is solved numerically by the front-fixing method on a non-uniform grid using an implicit finite-difference scheme, which is implemented by the sweep method. This algorithm can also be used to create more complex mathematical models of heat and mass transfer, as well as to describe glacial and subglacial processes. The mathematical apparatus proposed in the article was used to solve a specific problem of water freezing in a glacial crevasse. The presence and progression of crevasses, in turn, is a demonstrative factor indicating the dynamic activity of the glacier. Crevasses formed in one way or another can not only expand, but also decrease in size until they completely disappear. One of the reasons for their closure is the freezing of near-surface meltwater in the crevasse. Such a process was observed on glaciers near Mirny and Novolazarevskaya stations (East Antarctica). This process is modeled as an example of solving the Stefan problem. It is believed that all media are homogeneous and isotropic. The temperature of the water in the crevasse corresponds to the melting temperature of the ice. Modeling has shown that for the coastal part of the cold Antarctic glacier with an average temperature of –10°C and below, crevasses 5–10 cm of width freeze in less than a week. Wider ones freeze a little longer. 30 cm wide crevasses close in about two to three weeks, depending on the temperature of the glacier. Представлено численное решение одномерной задачи Стефана с двумя фазовыми границами в виде конечно-разностных схем, реализованных на неравномерной сетке. Уравнения записаны в наиболее общей форме, то есть включают в себя не только кондуктивный, но также конвективный и диссипативный члены. В качестве примера выполнено ...