Skip to main content
    • Aa
    • Aa



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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Available formats
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      Available formats
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      Available formats
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[2] A. C. M. Beljaars , “The parameterization of surface fluxes in large-scale models under free convection”, Quart. J. Roy. Met. Soc. 121 (1995) 255270.

[5] J. W. Deardorff , “Convective velocity and temperature scales for the unstable planetary boundary layer and for Rayleigh convection”, J. Atmos. Sci. 27 (1970) 12111213.

[6] H. Eicken , “From the microscopic, to the macroscopic, to the regional scale: Growth, microstructure and properties of sea ice”, in: Sea ice: an introduction to its physics, chemistry, biology and geology, Chapter 2 (eds D. N. Thomas and G. D. Dieckmann), (Blackwell Publishing, Oxford, 2003).

[7] A. C. Fowler , “The formation of freckles in binary alloys”, IMA J. Appl. Math. 35 (1985) 159174.

[8] A. A. Grachev , C. W. Fairall and S. E. Larsen , “On the determination of the neutral drag coefficient in the convective boundary layer”, Bound. Layer Meteorol. 86 (1998) 257278.

[9] D. M. Holland and J. A. Jenkins , “Modelling thermodynamic ice-ocean interactions at the base of an ice shelf”, J. Phys. Oceanogr. 29 (1999) 17871800.

[10] R. A. Lake and E. L. Lewis , “Salt rejection by sea ice during growth”, J. Geophys. Res. 75 (1970) 583598.

[11] G. H. Leonard , C. R. Purdie , P. J. Langhorne , T. G. Haskell , M. J. M. Williams and R. D. Frew , “Observations of platelet ice growth and oceanographic conditions during the winter of 2003 in McMurdo Sound, Antarctica”, J. Geophys. Res. 111 (2006) C04012.

[12] M. Leppäranta , “A review of analytical models of sea-ice growth”, Atmosphere-Ocean 31 (1993) 123138.

[14] L. Mahrt , D. Vickers , J. Edson , J. Sun , J. Højstrup , J. Hare and J. M. Wilczak , “Heat flux in the coastal zone”, Bound. Layer Meteorol. 86 (1998) 421446.

[15] S. Martin , “Frazil ice in rivers and oceans”, Annu. Rev. Fluid Mech. 13 (1981) 379397.

[16] G. A. Maykut , “The surface heat and mass balance”, in: The geophysics of sea ice, Chapter 5 (ed. N. Untersteiner) (Plenum, New York, 1986) 395465.

[17] G. A. Maykut and N. Untersteiner , “Some results from a time-dependent, thermodynamic model of sea ice”, J. Geophys. Res. 76 (1971) 15501575.

[19] M. G. McPhee , C. Kottmeier and J. H. Morison , “Ocean heat flux in the central Weddell Sea during winter”, J. Phys. Oceanogr. 29 (1999) 11661179.

[20] M. G. McPhee , G. A. Maykut and J. H. Morison , “Dynamics and thermodynamics of the ice/upper ocean system in the marginal ice zone of the Greenland sea”, J. Geophys. Res. 92 (1987) 70177031.

[21] M. G. McPhee and J. H. Morison , “Under-ice boundary layer”, in: Encyclopedia of ocean sciences (Academic Press, London, 2001) 30713078.

[22] D. Notz , M. G. McPhee , M. G. Worster , G. A. Maykut , K. H. Schlünzen and H. Eicken , “Impact of underwater-ice evolution on Arctic summer sea ice”, J. Geophys. Res. 108 (2003) 32233238.

[24] D. Pringle , J. H. Trodahl and T. Haskell , “Direct measurement of sea ice thermal conductivity: No surface reduction”, J. Geophys. Res. 111 (2006) C05020.

[25] D. J. Pringle , H. J. Eicken , H. J. Trodahl and L. G. E. Backstrom , “Thermal conductivity of landfast Antarctic and Arctic sea ice”, J. Geophys. Res. 112 (2007) C04017.

[26] C. Purdie , P. Langhorne , G. Leonard and T. Haskell , “Growth of first year land-fast Antarctic sea ice determined from winter temperature measurements”, Ann. Glaciol. 44 (2006) 170176.

[27] G. A. Schmidt , C. A. Bitz , U. Mikolajewicz and L. Tremblay , “Ice-ocean boundary conditions for coupled models”, Ocean Modelling 7 (2004) 5974.

[29] I. J. Smith , P. J. Langhorne , H. J. Trodahl , T. G. Haskell , R. Frew and R. Vennell , “Platelet ice and the land-fast sea ice of McMurdo Sound, Antarctica”, Ann. Glaciol. 33 (2001) 2127.

[31] B. J. J. M. Van den Hurk and A. A. Holtslag , “On the bulk parameterization of surface fluxes for various conditions and parameter ranges”, Bound. Layer Meteorol. 82 (1997) 119134.

[32] J. E. Weber , “Heat and salt transfer associated with formation of sea-ice”, Tellus 29 (1977) 151160.

[35] M. G. Worster , “Convection in mushy layers”, Annu. Rev. Fluid Mech. 29 (1997) 91122.

[36] M. G. Worster and J. S. Wettlaufer , “Natural convection, solute trapping, and channel formation during solidification of saltwater”, J. Phys. Chem. 101 (1997) 61326136.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The ANZIAM Journal
  • ISSN: 1446-1811
  • EISSN: 1446-8735
  • URL: /core/journals/anziam-journal
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 22 *
Loading metrics...

Abstract views

Total abstract views: 38 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 25th June 2017. This data will be updated every 24 hours.