Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T19:27:14.005Z Has data issue: false hasContentIssue false
  • English
  • Français

MODELLING SEA ICE GROWTH

Part of: Parabolic equations and systems

Published online by Cambridge University Press:  03 November 2009

MARK J. MCGUINNESS
MARK J. MCGUINNESS*
Affiliation:
Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Taejon 305-701, South Korea (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Stefan problemfreezing brinesalt transportmodelling sea ice growth

MSC classification

Secondary: 35K05: Heat equation
Type
Research Article
Information
The ANZIAM Journal , Volume 50 , Issue 3: This Special Issue is dedicated to Dr Stephen White , January 2009 , pp. 306 - 319
Copyright
Copyright © Australian Mathematical Society 2009

References

[1]Anderson, D. L., “Growth rate of sea ice”, J. Glaciol. 3 (1961) 1170–1172.CrossRefGoogle Scholar
[2]Beljaars, A. C. M., “The parameterization of surface fluxes in large-scale models under free convection”, Quart. J. Roy. Met. Soc. 121 (1995) 255–270.Google Scholar
[3]Carslaw, H. S. and Jaeger, J. C., Conduction of heat in solids (Oxford University Press, Oxford, 1959).Google Scholar
[4]Daly, S. F., Frazil Ice Dynamics, CRREL Monograph 84-1, Hanover NH, 1984.Google Scholar
[5]Deardorff, J. W., “Convective velocity and temperature scales for the unstable planetary boundary layer and for Rayleigh convection”, J. Atmos. Sci. 27 (1970) 1211–1213.2.0.CO;2>CrossRefGoogle Scholar
[6]Eicken, H., “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).Google Scholar
[7]Fowler, A. C., “The formation of freckles in binary alloys”, IMA J. Appl. Math. 35 (1985) 159–174.CrossRefGoogle Scholar
[8]Grachev, A. A., Fairall, C. W. and Larsen, S. E., “On the determination of the neutral drag coefficient in the convective boundary layer”, Bound. Layer Meteorol. 86 (1998) 257–278.CrossRefGoogle Scholar
[9]Holland, D. M. and Jenkins, J. A., “Modelling thermodynamic ice-ocean interactions at the base of an ice shelf”, J. Phys. Oceanogr. 29 (1999) 1787–1800.2.0.CO;2>CrossRefGoogle Scholar
[10]Lake, R. A. and Lewis, E. L., “Salt rejection by sea ice during growth”, J. Geophys. Res. 75 (1970) 583–598.CrossRefGoogle Scholar
[11]Leonard, G. H., Purdie, C. R., Langhorne, P. J., Haskell, T. G., Williams, M. J. M. and Frew, R. D., “Observations of platelet ice growth and oceanographic conditions during the winter of 2003 in McMurdo Sound, Antarctica”, J. Geophys. Res. 111 (2006) C04012.Google Scholar
[12]Leppäranta, M., “A review of analytical models of sea-ice growth”, Atmosphere-Ocean 31 (1993) 123–138.CrossRefGoogle Scholar
[13]Lock, G. S. H., The growth and decay of ice (Cambridge University Press, Cambridge, 2005).Google Scholar
[14]Mahrt, L., Vickers, D., Edson, J., Sun, J., Højstrup, J., Hare, J. and Wilczak, J. M., “Heat flux in the coastal zone”, Bound. Layer Meteorol. 86 (1998) 421–446.CrossRefGoogle Scholar
[15]Martin, S., “Frazil ice in rivers and oceans”, Annu. Rev. Fluid Mech. 13 (1981) 379–397.CrossRefGoogle Scholar
[16]Maykut, G. A., “The surface heat and mass balance”, in: The geophysics of sea ice, Chapter 5 (ed. N. Untersteiner) (Plenum, New York, 1986) 395–465.CrossRefGoogle Scholar
[17]Maykut, G. A. and Untersteiner, N., “Some results from a time-dependent, thermodynamic model of sea ice”, J. Geophys. Res. 76 (1971) 1550–1575.CrossRefGoogle Scholar
[18]McGuinness, M. J., Collins, K., Trodahl, H. J. and Haskell, T. G., “Nonlinear thermal transport and brine convection in first year sea ice”, Ann. Glaciol. 27 (1998) 471–476.CrossRefGoogle Scholar
[19]McPhee, M. G., Kottmeier, C. and Morison, J. H., “Ocean heat flux in the central Weddell Sea during winter”, J. Phys. Oceanogr. 29 (1999) 1166–1179.2.0.CO;2>CrossRefGoogle Scholar
[20]McPhee, M. G., Maykut, G. A. and Morison, J. H., “Dynamics and thermodynamics of the ice/upper ocean system in the marginal ice zone of the Greenland sea”, J. Geophys. Res. 92 (1987) 7017–7031.CrossRefGoogle Scholar
[21]McPhee, M. G. and Morison, J. H., “Under-ice boundary layer”, in: Encyclopedia of ocean sciences (Academic Press, London, 2001) 3071–3078.CrossRefGoogle Scholar
[22]Notz, D., McPhee, M. G., Worster, M. G., Maykut, G. A., Schlünzen, K. H. and Eicken, H., “Impact of underwater-ice evolution on Arctic summer sea ice”, J. Geophys. Res. 108 (2003) 3223–3238.Google Scholar
[23]Ono, N., “Thermal properties of sea ice, IV. Thermal constants of sea ice”, Low Temperature Sci. A26 (1968) 329–349.Google Scholar
[24]Pringle, D., Trodahl, J. H. and Haskell, T., “Direct measurement of sea ice thermal conductivity: No surface reduction”, J. Geophys. Res. 111 (2006) C05020.Google Scholar
[25]Pringle, D. J., Eicken, H. J., Trodahl, H. J. and Backstrom, L. G. E., “Thermal conductivity of landfast Antarctic and Arctic sea ice”, J. Geophys. Res. 112 (2007) C04017.Google Scholar
[26]Purdie, C., Langhorne, P., Leonard, G. and Haskell, T., “Growth of first year land-fast Antarctic sea ice determined from winter temperature measurements”, Ann. Glaciol. 44 (2006) 170–176.CrossRefGoogle Scholar
[27]Schmidt, G. A., Bitz, C. A., Mikolajewicz, U. and Tremblay, L., “Ice-ocean boundary conditions for coupled models”, Ocean Modelling 7 (2004) 59–74.CrossRefGoogle Scholar
[28]Smith, I. J., Langhorne, P. J., Trodahl, H. J., Haskell, T. G. and Cole, D. M., “Platelet ice – the McMurdo Sound debate”, in: Ice in Surface Waters – Proceedings of the 14th IAHR Symposium on Ice (ed. H. T. Shen), (A.A. Balkema, Rotterdam, Netherlands, 1999) 371–378.Google Scholar
[29]Smith, I. J., Langhorne, P. J., Trodahl, H. J., Haskell, T. G., Frew, R. and Vennell, R., “Platelet ice and the land-fast sea ice of McMurdo Sound, Antarctica”, Ann. Glaciol. 33 (2001) 21–27.CrossRefGoogle Scholar
[30]Turner, J. S., Buoyancy effects in fluids (Cambridge University Press, Cambridge, 1979).Google Scholar
[31]Van den Hurk, B. J. J. M. and Holtslag, A. A., “On the bulk parameterization of surface fluxes for various conditions and parameter ranges”, Bound. Layer Meteorol. 82 (1997) 119–134.CrossRefGoogle Scholar
[32]Weber, J. E., “Heat and salt transfer associated with formation of sea-ice”, Tellus 29 (1977) 151–160.CrossRefGoogle Scholar
[33]Woods, A. W. and Huppert, H. E., “The growth of a compositionally stratified solid above a horizontal boundary”, J. Fluid Mech. 199 (1989) 29–53.CrossRefGoogle Scholar
[34]Worster, M. G., “Solidification of an alloy from a cooled boundary”, J. Fluid Mech. 167 (1986) 481–501.CrossRefGoogle Scholar
[35]Worster, M. G., “Convection in mushy layers”, Annu. Rev. Fluid Mech. 29 (1997) 91–122.CrossRefGoogle Scholar
[36]Worster, M. G. and Wettlaufer, J. S., “Natural convection, solute trapping, and channel formation during solidification of saltwater”, J. Phys. Chem. 101 (1997) 6132–6136.CrossRefGoogle Scholar