Skip to main content
Log in

Modeling ocean wave propagation under sea ice covers

  • Review Paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

Abstract

Operational ocean wave models need to work globally, yet current ocean wave models can only treat ice-covered regions crudely. The purpose of this paper is to provide a brief overview of ice effects on wave propagation and different research methodology used in studying these effects. Based on its proximity to land or sea, sea ice can be classified as: landfast ice zone, shear zone, and the marginal ice zone. All ice covers attenuate wave energy. Only long swells can penetrate deep into an ice cover. Being closest to open water, wave propagation in the marginal ice zone is the most complex to model. The physical appearance of sea ice in the marginal ice zone varies. Grease ice, pancake ice, brash ice, floe aggregates, and continuous ice sheet may be found in this zone at different times and locations. These types of ice are formed under different thermal-mechanical forcing. There are three classic models that describe wave propagation through an idealized ice cover: mass loading, thin elastic plate, and viscous layer models. From physical arguments we may conjecture that mass loading model is suitable for disjoint aggregates of ice floes much smaller than the wavelength, thin elastic plate model is suitable for a continuous ice sheet, and the viscous layer model is suitable for grease ice. For different sea ice types we may need different wave ice interaction models. A recently proposed viscoelastic model is able to synthesize all three classic models into one. Under suitable limiting conditions it converges to the three previous models. The complete theoretical framework for evaluating wave propagation through various ice covers need to be implemented in the operational ocean wave models. In this review, we introduce the sea ice types, previous wave ice interaction models, wave attenuation mechanisms, the methods to calculate wave reflection and transmission between different ice covers, and the effect of ice floe breaking on shaping the sea ice morphology. Laboratory experiments, field measurements and numerical simulations supporting the fundamental research in wave-ice interaction models are discussed. We conclude with some outlook of future research needs in this field.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Komen, G.J., Cavaleri, L., Donelan, M., et al.: Dynamics and Modeling of Ocean Waves. Cambridge University Press, USA (1996)

    Google Scholar 

  2. Thomson, J., Rogers, W.E.: Swell and sea in the emerging Arctic Ocean. Geophysical Research Letters 41, 3136–3140 (2014)

    Article  Google Scholar 

  3. Wadhams, P.: Ice In the Ocean. CRC Press, Amsterdam (2000)

    Google Scholar 

  4. Toyota, T., Takatsuji, S., Nakayama, M.: Characteristics of sea ice floe size distribution in the seasonal ice zone. Geophysical Research Letters 33, L02616 (2006)

    Article  Google Scholar 

  5. Perrie, W., Hu, Y.: Air-ice-ocean momentum exchange. Part II: Ice drift. J. Phys. Oceanogr. 27, 1976–1996 (1997)

    Article  Google Scholar 

  6. Dai, M., Shen, H.H., Hopkins, M.A., et al.: Wave rafting and the equilibrium pancake ice cover thickness. Journal of Geophysical Research: Oceans (1978–2012) 109, C07023 (2004)

    Article  Google Scholar 

  7. Tolman, H.L.: Treatment of unresolved islands and ice in wind wave models. Ocean Modelling 5, 219–231 (2003)

    Article  Google Scholar 

  8. Kohout, A.L., Williams, M.J.M., Dean, S.M., et al.: Storminduced sea-ice breakup and the implications for ice extent. Nature 509, 604–607 (2014)

    Article  Google Scholar 

  9. Wadhams, P., Squire, V.A., Goodman, D.J., et al.: The attenuation rates of ocean waves in the marginal ice zone. Journal of Geophysical Research: Oceans (1978–2012) 93, 6799–6818 (1988)

    Article  Google Scholar 

  10. Squire, V.A., Dugan, J.P., Wadhams, P., et al.: Of ocean waves and sea ice. Annual Review of Fluid Mechanics 27, 115–168 (1995)

    Article  MathSciNet  Google Scholar 

  11. Squire, V.A.: Of ocean waves and sea-ice revisited. Cold Regions Science and Technology 49, 110–133 (2007)

    Article  Google Scholar 

  12. Dumont, D., Kohout, A., Bertino, L.: A wave-based model for the marginal ice zone including a floe breaking parameterization. Journal of Geophysical Research: Oceans (1978–2012) 116, C04001 (2011)

    Article  Google Scholar 

  13. Masson, D., LeBlond, P.H.: Spectral evolution of wind-generated surface gravity waves in a dispersed ice field. Journal of Fluid Mechanics 202, 43–81 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  14. Perrie, W., Hu, Y.: Air-ice-ocean momentum exchange. Part 1: Energy transfer between waves and ice floes. Journal of Physical Oceanography 26, 1705–1720 (1996)

    Article  Google Scholar 

  15. Toyota, T., Haas, C., Tamura, T.: Size distribution and shape properties of relatively small sea-ice floes in the Antarctic marginal ice zone in late winter. Deep Sea Research Part II: Topical Studies in Oceanography 58, 1182–1193 (2011)

    Article  Google Scholar 

  16. Peters, A.S.: The effect of a floating mat on water waves. Communications on Pure and Applied Mathematics 3, 319–354 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  17. Weitz, M., Keller, J.B.: Reflection of water waves from floating ice in water of finite depth. Communications on Pure and Applied Mathematics 3, 305–318 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  18. Wadhams, P.: The seasonal ice zone//The Geophysics of Sea Ice. 825-991, Springer. US (1986)

    Google Scholar 

  19. Fox, C., Squire, V.A.: Reflection and transmission characteristics at the edge of shore fast sea ice. Journal of Geophysical Research: Oceans (1978–2012) 95, 11629–11639 (1990)

    Article  Google Scholar 

  20. Fox, C., Squire, V.A.: On the oblique reflexion and transmission of ocean waves at shore fast sea ice. Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences 347, 185–218 (1994)

    Article  MATH  Google Scholar 

  21. Barrett, M.D., Squire, V.A.: Ice-coupled wave propagation across an abrupt change in ice rigidity, density, or thickness. Journal of Geophysical Research: Oceans (1978–2012) 101, 20825–20832 (1996)

    Article  Google Scholar 

  22. Keller, J.B.: Gravity waves on ice-covered water. Journal of Geophysical Research: Oceans (1978–2012) 103, 7663–7669 (1998)

    Article  Google Scholar 

  23. De Carolis, G., Desiderio, D.: Dispersion and attenuation of gravity waves in ice: A two-layer viscous fluid model with experimental data validation. Physics Letters A 305, 399–412 (2002)

    Article  Google Scholar 

  24. Wang, R., Shen, H.H.: Gravity waves propagating into an icecovered ocean: A viscoelastic model. Journal of Geophysical Research: Oceans (1978–2012) 115, 399–412 (2010)

    Google Scholar 

  25. Tolman, H.L.: User manual and system documentation of WAVEWATCH III TM version 4.18b, 307 (2014)

    Google Scholar 

  26. Miles, J.W.: On the generation of surface waves by shear flows. Journal of Fluid Mechanics 3, 185–204 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  27. Gent, P.R., Taylor, P.A.: A numerical model of the air flow above water waves. Journal of Fluid Mechanics 77, 105–128 (1976)

    Article  MATH  Google Scholar 

  28. Riley, D.S., Donelan, M.A., Hui, W.H.: An extended Miles’ theory for wave generation by wind. Boundary-Layer Meteorology 22, 209–225 (1982)

    Article  Google Scholar 

  29. Al-Zanaidi, M.A., Hui, W.H.: Turbulent airflow over water waves-a numerical study. Journal of Fluid Mechanics 148, 225–246 (1984)

    Article  MATH  Google Scholar 

  30. Jacobs, S.J.: An asymptotic theory for the turbulent flow over a progressive water wave. Journal of Fluid Mechanics 174, 69–80 (1987)

    Article  MATH  Google Scholar 

  31. Chalikov, D.V., Makin, V.K.: Models of the wave boundary layer. Boundary-Layer Meteorology 56, 83–99 (1991)

    Article  Google Scholar 

  32. Hasselmann, K.: On the non-linear energy transfer in a gravitywave spectrum Part 1: General theory. Journal of Fluid Mechanics 12, 481–500 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  33. Hasselmann, K.: On the non-linear energy transfer in a gravity wave spectrum. Part 2: Conservation theorems; wave-particle analogy; irrevesibility. Journal of Fluid Mechanics 15, 273–281 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  34. Hasselmann, K.: On the non-linear energy transfer in a gravitywave spectrum. Part 3: Evaluation of the energy flux and swell-sea interaction for a Neumann spectrum. Journal of Fluid Mechanics 15, 385–398 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  35. Zakharov, V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. Journal of Applied Mechanics and Technical Physics 9, 190–194 (1968)

    Article  Google Scholar 

  36. Janssen, P.: The Interaction of Ocean Waves and Wind. Cambridge University Press, USA (2004)

    Book  Google Scholar 

  37. Hasselmann, K.: On the spectral dissipation of ocean waves due to white capping. Boundary-Layer Meteorology 6, 107–127 (1974)

    Article  Google Scholar 

  38. Phillips, O.M.: Spectral and statistical properties of the equilibrium range in wind-generated gravity waves. Journal of Fluid Mechanics 156, 505–531 (1985)

    Article  MATH  Google Scholar 

  39. Jenkins, A.D.: A Lagrangian model for wind- and wave-induced near-surface currents. Coastal Engineering 11, 513–526 (1987)

    Article  Google Scholar 

  40. Rogers, W.E., Orzech, M.D.: Implementation and testing of ice and mud source functions in WAVEWATCH III. NRL Memorandum Report. NRL/MR/7320-13-9462, 31 (2013)

    Google Scholar 

  41. Gaster, M.: A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability. Journal of Fluid Mechanics 14, 222–224 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  42. Liu, A.K., Mollo-Christensen, E.: Wave propagation in a solid ice pack. Journal of Physical Oceanography 18, 1702–1712 (1988)

    Article  Google Scholar 

  43. Meylan, M.H., Squire, V.A., Fox, C.: Toward realism in modeling ocean wave behavior in marginal ice zones. Journal of Geophysical Research: Oceans (1978–2012) 102, 22981–22991 (1997)

    Article  Google Scholar 

  44. Meylan, M.H., Masson, D.: A linear Boltzmann equation to model wave scattering in the marginal ice zone. Ocean Modelling 11, 417–427 (2006)

    Article  Google Scholar 

  45. Liu, A.K., Holt, B., Vachon, P.W.: Wave propagation in the marginal ice zone: Model predictions and comparisons with buoy and synthetic aperture radar data. Journal of Geophysical Research: Oceans (1978–2012) 96, 4605–4621 (1991)

    Article  Google Scholar 

  46. Squire, V.A., Allan, A.J.: Propagation of flexural gravity waves in sea ice, in Sea Ice Processe and Models. In: Proceedings of the Arctic Ice Dynamics Joint Experiment, Pritchard, R. S. edn. 327–338, University of Washington Press, Seattle (1980)

    Google Scholar 

  47. Shen, H.H., Squire, V.A.: Wave damping in compact pancake ice fields due to interactions between pancakes. Antarctic Sea Ice: Physical Processes, Interactions and Variability 74, 325–341 (1998)

    Google Scholar 

  48. Wang, R.X.: Water Wave Propagation in Ice-Covered Oceans, [Ph.D. Thesis]. Clarkson University, USA (2010)

    Google Scholar 

  49. Lange, M.A., Ackley, S.F., Wadhams, P., et al.: Development of sea ice in the Weddell Sea. Ann. Glaciol. 12, 92–96 (1989)

    Google Scholar 

  50. Squire, V.A.: A comparison of the mass-loading and elastic plate models of an ice field. Cold Regions Science and Technology 21, 219–229 (1993)

    Article  Google Scholar 

  51. Sakai, S., Hanai, K.: Empirical formula of dispersion relation of waves in sea ice. In: Ice in the Environment. Proceedings of the 16th IAHR International Symposium on Ice 327–335 (2002)

    Google Scholar 

  52. Newyear, K., Martin, S.: A comparison of theory and laboratory measurements of wave propagation and attenuation in grease ice. Journal of Geophysical Research: Oceans (1978–2012) 102, 25091–25099 (1997)

    Article  Google Scholar 

  53. Newyear, K., Martin, S.: Comparison of laboratory data with a viscous two layer model of wave propagation in grease ice. Journal of Geophysical Research: Oceans (1978–2012) 104, 7837–7840 (1999)

    Article  Google Scholar 

  54. Bennetts, L.G., Squire, V.A.: On the calculation of an attenuation coefficient for transects of ice-covered ocean. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 468, 136–162 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  55. Bennetts, L.G., Squire, V.A.: Model sensitivity analysis of scattering-induced attenuation of ice-coupled waves. Ocean Modelling 45, 1–13 (2012)

    Article  Google Scholar 

  56. Meylan M., Squire, V.A.: The response of ice floes to ocean waves. Journal of Geophysical Research: Oceans (1978–2012) 99, 891–900 (1994)

    Article  Google Scholar 

  57. Kohout, A.L.: Water wave scattering by floating elastic plates with application to sea ice, [Ph.D Thesis]. University of Auckland, USA (2008)

    Google Scholar 

  58. Kohout, A.L., Meylan, M.H., Sakai, S., et al.: Linear water wave propagation through multiple floating elastic plates of variable properties. Journal of fluids and structures, 23, 649–663 (2007)

    Article  Google Scholar 

  59. Wang, R., Shen, H.H.: A continuum model for the linear wave propagation in ice-covered oceans: An approximate solution. Ocean Modelling 38, 244–250 (2011)

    Article  Google Scholar 

  60. Zhao, X., Shen, H.H.: Ocean wave transmission and reflection between two connecting viscoelastic ice covers: An approximate solution. Ocean Modelling 71, 102–113 (2013)

    Article  Google Scholar 

  61. Sarpkaya, T., Isaacson, M.: Mechanics of Wave Forces on Off-shore Structures. Van Nostrand Reinhold Company, New York (1981)

    Google Scholar 

  62. Meylan, M.H., Squire, V.A.: Response of a circular ice floe to ocean waves. Journal of Geophysical Research: Oceans (1978–2012) 101, 8869–8884 (1996)

    Article  Google Scholar 

  63. Zhao, X., Shen, H.H.; Ocean wave transmission and reflection between two connecting viscoelastic ice covers: A variational approach, to be submitted (2014)

    Google Scholar 

  64. Meylan, M.H.: Wave response of an ice floe of arbitrary geometry. Journal of Geophysical Research: Oceans (1978–2012) 107, 5-1–5-11 (2002)

    Article  Google Scholar 

  65. Kohout, A.L., Meylan, M.H.: An elastic plate model for wave attenuation and ice floe breaking in the marginal ice zone. Journal of Geophysical Research: Oceans (1978–2012) 113, C09016 (2008)

    Article  Google Scholar 

  66. Dumont, D., Kohout, A., Bertino, L.: A wave-based model for the marginal ice zone including a floe breaking parameterization. Journal of Geophysical Research: Oceans (1978–2012) 116, C04001 (2011)

    Article  Google Scholar 

  67. Williams, T.D., Bennetts, L.G., Squire, V.A., et al.: Waveice interactions in the marginal ice zone. Part 1: Theoretical foundations. Ocean Modelling 71, 81–91 (2013)

    Article  Google Scholar 

  68. Williams, T.D., Bennetts, L.G., Squire, V.A., et al.: Waveice interactions in the marginal ice zone. Part 2: Numerical implementation and sensitivity studies along 1D transects of the ocean surface. Ocean Modelling 71, 92–101 (2013)

    Article  Google Scholar 

  69. Harms, V.W.: Steady wave-drift of modeled ice floes. Journal of Waterway, Port, Coastal, and Ocean Engineering 113, 606–622 (1987)

    Article  Google Scholar 

  70. Huang, G., Law, A.W.K., Huang Z.: Wave-induced drift of small floating objects in regular waves. Ocean Engineering 38, 712–718 (2011)

    Article  Google Scholar 

  71. Huang, G., Law, A.W.K.: Wave-induced drift of large floating objects in regular waves. Journal of Waterway, Port, Coastal, and Ocean Engineering 139, 535–542 (2013)

    Article  Google Scholar 

  72. Montiel, F., Bonnefoy, F., Ferrant, P., et al.: Hydroelastic response of floating elastic discs to regular waves. Part 1: Wave basin experiments. Journal of Fluid Mechanics 723, 604–628 (2013)

    Article  MATH  Google Scholar 

  73. Montiel, F., Bennetts, L.G., Squire, V.A., et al.: Hydroelastic response of floating elastic discs to regular waves. Part 2: Modal analysis. Journal of Fluid Mechanics 723, 629–652 (2013)

    Article  MATH  Google Scholar 

  74. Toyota, T.: A study on growth processes of sea ice in the southern region of the Okhotsk sea, evaluated from heat budget and sea ice sample analysis, [Ph.D. Thesis]. Hokkaido University, Japan (1998)

    Google Scholar 

  75. Shen, H.H., Ackley, S.F., Yuan, Y.: Limiting diameter of pancake ice. Journal of Geophysical Research: Oceans (1978–2012) 109, C12035 (2004)

    Article  Google Scholar 

  76. Wang, R., Shen, H.H., Evers, K.U.: An experimental study of wave induced ice production. In: Proceedings of the 19th IAHR International Symposium on Ice, Vancouver, Canada, July 6–11 (2008)

    Google Scholar 

  77. Callinan, C., Evers, K.U., Wilkinson, J., et al.: Wave propagation in frazil/pancake and fragmented ice covers Part I: Description of the laboratory study. In: Proceedings of the 22nd IAHR Ice Symposium, Aug. 11–15, Singapore (2014)

    Google Scholar 

  78. Zhao, X., Callinan, C., Shen, H.H.: Wave propagation in frazil/pancake and fragmented ice covers Part II: Preliminary data analysis. Proceedings of the 22nd IAHR Ice Symposium, Aug. 11–15, Singapore (2014)

    Google Scholar 

  79. Lansing, A.: Endurance: Shackletons Incredible Voyage, (2nd ed.) Carroll and Graf Publishers. ISBN 0-7867-0621-X, 288p (1959)

    Google Scholar 

  80. Squire, V.A., Moore, S.C.: Direct measurement of the attenuation of ocean waves by pack ice. Nature 283, 365–368 (1980)

    Article  Google Scholar 

  81. Wadhams, P., Squire, V.A., Ewing, J.A., et al.: The effect of the marginal ice zone on the directional wave spectrum of the ocean. Journal of Physical Oceanography 16, 358–376 (1986)

    Article  Google Scholar 

  82. Frankenstein, S., Loset, S., Shen, H.H.: Wave-ice interactions in Barents Sea marginal ice zone. Journal of Cold Regions Engineering 15, 91–102 (2001)

    Article  Google Scholar 

  83. McKenna, R.F., Crocker, G.B.: Ice floe collisions interpreted from acceleration data during LIMEX 89. Atmosphere-Ocean 30, 246–269 (1992)

    Article  Google Scholar 

  84. Rottier, P.J.: Floe pair interaction event rates in the marginal ice zone. Journal of Geophysical Research: Oceans (1978–2012) 97, 9391–9400 (1992)

    Article  Google Scholar 

  85. Campbell, A.J., Bechle, A.J., Wu, C.H.: Observations of surface waves interacting with ice using stereo imaging. Journal of Geophysical Research: Oceans 119, 266–284 (2014)

    Google Scholar 

  86. Ailliot, P., Baxevani, A., Cuzol, A., et al.: Spacetime models for moving fields with an application to significant wave height fields. Environmetrics 22, 354–369 (2011)

    Article  MathSciNet  Google Scholar 

  87. He, G.W., Zhang, J.B.: Elliptic model for space-time correlations in turbulent shear flows. Physical Review E 73, 055303 (2006)

    Article  Google Scholar 

  88. Zhao, X., He, G.W.: Space-time correlations of fluctuating velocities in turbulent shear flows. Physical Review E 79, 046316 (2009)

    Article  Google Scholar 

  89. He, X., He, G., Tong, P.: Small-scale turbulent fluctuations beyond Taylors frozen-flow hypothesis. Physical Review E 81, 065303 (2010)

    Article  Google Scholar 

  90. Luding, S.: Introduction to discrete element methods: Basic of contact force models and how to perform the micro-macro transition to continuum theory. European Journal of Environmental and Civil Engineering 12, 785–826 (2008)

    Article  Google Scholar 

  91. Sun, S., Shen, H.H.: Simulation of pancake ice load on a circular cylinder in a wave and current field. Cold Regions Science and Technology 78, 31–39 (2012)

    Article  Google Scholar 

  92. Zhao, X., Shen, H.H.: Numerical simulation of gravity waves in viscous and viscoelastic materials. In: Proceedings of the 7th International Symposium on Environmental Hydraulics 2014 ISEH VII, Jan. 7–9, Singapore (2014)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hayley H. Shen.

Additional information

The project was supported by the US Office of Naval Research (N00014-13-1-0294).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, X., Shen, H.H. & Cheng, S. Modeling ocean wave propagation under sea ice covers. Acta Mech Sin 31, 1–15 (2015). https://doi.org/10.1007/s10409-015-0017-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10409-015-0017-5

Keywords

Navigation