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The ANZIAM Journal (2009), 50:306-319 Cambridge University Press
Copyright © Australian Mathematical Society 2009

Research Article



a1 Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Taejon 305-701, South Korea (email:
Article author query
mcguinness mj [Google Scholar]


The freezing of water to ice is a classic problem in applied mathematics, involving the solution of a diffusion equation with a moving boundary. However, when the water is salty, the transport of salt rejected by ice introduces some interesting twists to the tale. A number of analytic models for the freezing of water are briefly reviewed, ranging from the famous work by Neumann and Stefan in the 1800s, to the mushy zone models coming out of Cambridge and Oxford since the 1980s. The successes and limitations of these models, and remaining modelling issues, are considered in the case of freezing sea-water in the Arctic and Antarctic Oceans. A new, simple model which includes turbulent transport of heat and salt between ice and ocean is introduced and solved analytically, in two different cases—one where turbulence is given by a constant friction velocity, and the other where turbulence is buoyancy-driven and hence depends on ice thickness. Salt is found to play an important role, lowering interface temperatures, increasing oceanic heat flux, and slowing ice growth.

(Received December 09 2007)

(Revised November 24 2008)

2000 Mathematics subject classificationprimary 35K05; secondary 86A05

Keywords and phrasesStefan problem; freezing brine; salt transport; modelling sea ice growth