Abstract
Studying the relationship between wave steepness and wave age is important for describing wind wave growth with energy balance equation of significant waves. After invoking the dispersion relation of surface gravity wave in deep water, a new relationship between wave steepness and wave age is revealed based on the “3/2-power law” (Toba, 1972), in which wave steepness is a function of wave age with a drag coefficient as a parameter. With a given wave age, a larger drag coefficient would lead to larger wave steepness. This could be interpreted as the result of interaction between wind and waves. Comparing with previous relationships, the newly proposed one is more consistent with observational data in field and laboratory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Amorocho, J. and J. J. DeVries, 1980. A new evaluation of the wind stress coefficient over water surfaces. J. Geophys. Res. 85(C1): 433–442.
Anctil, F. and M. A. Donelan, 1996. Air-water momentum flux observations over shoaling waves. J. Phys. Oceanogr. 26: 1 344–1 353.
Banner, M. L., W. Chen, E. J. Walsh, J. B. Jensen, S. Lee and C. Fandry, 1999. The Southern Ocean Waves Experiment. Part 1: Overview and mean results. J. Phys. Oceanogr. 29: 2 130–2 145.
Ebuchi, N., Y. Toba and H. Kawamura, 1992. Statistical study on the local equilibrium between wind and wind waves by using data from ocean data buoy stations. J. Oceanogr. Soc. Japan 48: 77–92.
Garratt, J. R., 1977. Review of drag coefficients over oceans and continents. Mon. Wea. Rev. 105: 915–929.
Geernaert, G. L., K. B. Katsaros and K. Richter, 1986. Variations of the drag coefficient and its dependence on sea state. J. Geophys. Res. 91: 7 667–7 679.
Goda, Y. and K. Nagai, 1974. Investigation of the statistical properties of sea waves with fields and simulated data. Rept. Port Harbor Res. Inst. 13: 3–37. (in Japanese)
Guan, C. L. and Q. Sun, 2001. Analyses of wind wave growth relations and their support to the 3/2 power law. J. Ocean Univ. Qingdao 31(5): 633–639. (in Chinese)
Guan, C. L. and Q. Sun, 2002. Analytically derived wind wave growth relationships. China Ocean Engineering 16(3): 359–368.
Guan, C. L. and L. A. Xie, 2004. On the linear parameterization of drag coefficient over sea surface. J. Phys. Oceanogr. 34: 2847–2851.
Hamada, K., 1963. Model experiments on period of surface waves in a medium with a single layer. Zisin (Journal of Seismological Society of Japan) Ser. 2 16(3): 133–144. (in Japanese)
Hasselmann, K., T. P. Barnett, E. Bouws, H. Carlson, D. E. Cartwright, K. Enke, J. A. Ewing, H. Gienapp, D. E. Hasselmann, P. Kruseman, A. Meerburg, P. Müller, D. J. Olbers, K. Richter, W. Sell and H. Walden, 1973. Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP), Dtsch. Hydrogr. Z. A8(12): 1–95.
Hou, Y. J. and T. Wang, 1993. Characteristic parameters of the wind wave spectrum. Oceanologia et Limnologia Sinica 24(2): 126–131. (in Chinese)
Janssen, J. A. M., 1997. Does wind stress depend on sea-state or not? A statistical error analysis of HEXMAX data. Bound-Layer Meteorol. 83: 479–503.
Johnson, H. K., J. Højstrup, H. J. Vested and S. E. Larsen, 1998. On the dependence of sea surface roughness on wind waves. J. Phys. Oceanogr. 28: 1 702–1 716.
Katsaros, K. B. and S. S. Atakturk, 1992. Dependence of wave-breaking statistics on wind stress and wave development. In: Banner, M. L. R. and H. J. Grimshaw ed. Breaking Waves. Springer Press. p. 119–132.
Kawai, S., K. Okada and Y. Toba, 1977. Field data support of three-seconds power law and the gu*σ−4-spectral form for growing wind waves. J. Oceanogr. Soc. Japan 33: 137–150.
Kunishi, H. and N. Imasato, 1966. On the growth of wind waves by high-speed wind flume. Ann. Disaster Prev. Res. Inst. Kyoto Univ. 9: 667–676.
Merzi, N. and W. H. Graf, 1985. Evaluation of the drag coefficient considering the effects of mobility of the roughness elements. Ann. Geophys. 3/4: 473–478.
Smith, S. D., 1980. Wind stress and heat flux over the ocean in gale force winds. J. Phys. Oceanogr. 10: 709–726.
Toba, Y., 1972. Local balance of in the air-sea boundary processes, I: On the growth process of wind waves. J. Oceanogr. Soc. Japan 28: 109–120.
Toba, Y., 1998. Wind-forced strong wave interactions and quasi-local equilibrium between wind and windsea with the friction velocity proportionality. In: W. Perrie ed. Nonlinear Ocean Waves (Advances in Fluid Mechanics, Vol. 17, Series editor: M. Rahman). Computational Mechanics Publications, Southampton and Boston. p. 1–59.
Wang, B. X., 1990. An investigation on the δ-β relationship of ocean waves. J. Ocean Univ. Qingdao 20(3): 1–9 (in Chinese)
Wen, S. C., 1962. Ocean Wave Theory, Shandong People’s Press, Jinan, China. p. 224–265. (in Chinese)
Wen, S. C., D. C. Zhang, P. F. Guo and B. H. Chen, 1989. Parameters in wind wave frequency spectra and their bearing on spectrum forms and growth. Acta Oceanol. Sinica 8(1): 15–39.
Wu, J., 1980. Wind-stress coefficients over sea surface near neutral conditions—a revisit. J. Phys. Oceanogr. 10: 727–740.
Wu, S. P., Y. J. Hou, B. S. Yin, J. B. Song and X. X. Zhao, 2004. Relationship between wave steepness and wave age in the course of wind wave growth. Chin. J. Oceanol. Limnol. 22(4): 340–343.
Yelland, M. J. and P. K. Taylor, 1996. Wind stress measurements from the open ocean. J. Phys. Oceanogr. 26: 541–558.
Zhao, D. L., and Y. Toba, 2001. Dependence of whitecap coverage on wind and wind-wave properties. J. Oceanogr. 57: 603–616.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by Specialized Research Fund for Doctoral Program of Higher Education (No.20040423002) and by National Natural Science Foundation of China (No.40476008)
Rights and permissions
About this article
Cite this article
Liu, B., Ding, Y. & Guan, C. A relationship between wave steepness and wave age for wind waves in deep water. Chin. J. Ocean. Limnol. 25, 36–41 (2007). https://doi.org/10.1007/s00343-007-0036-6
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s00343-007-0036-6