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

Abstract

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.