Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-18T11:41:38.294Z Has data issue: false hasContentIssue false
  • English
  • Français

Observations of bispectra of shoaling surface gravity waves

Published online by Cambridge University Press:  21 April 2006

Steve Elgar  and
R. T. Guza
Steve Elgar
Affiliation:
Scripps Institution of Oceanography, Mail Code A-022, University of California, La Jolla, California 92093 Present address. Engineering Science Department, College of Engineering, University of Idaho, Moscow, Idaho 83843, USA.
R. T. Guza
Affiliation:
Scripps Institution of Oceanography, Mail Code A-022, University of California, La Jolla, California 92093
Get access Rights & Permissions [Opens in a new window]

Abstract

Aspects of the nonlinear dynamics of waves shoaling between 9 and 1 m water depths are elucidated via the bispectrum. Bispectral-signal levels are generally high, indicating significant nonlinear coupling. In 9 m depth, the biphases of interactions involving frequencies at, and higher than, the peak of the energy spectra are suggestive of Stokes-like nonlinearities (Hasselman, Munk & MacDonald 1963). Further shoaling gradually modifies these biphases to values consistent with a wave profile that is pitched shoreward, relative to a vertical axis. Bicoherence and biphase observations with a double-peaked (swell and wind-wave) power spectrum provide evidence for excitation of modes at intermediate frequencies via difference interactions, as well as the sum interactions responsible for harmonic growth. Shoreward-propagating low-frequency (surf-beat) energy is shown to have statistically significant coupling to higher-frequency modes within the power-spectral peak. In 18 m depth, the biphase of these interactions is close to 180°, a value consistent with the classical concept of bound long waves. In shallower water, however, substantial biphase evolution occurs, and there is no longer a unique phase relationship between surf beat and the envelope of high-frequency waves. The contributions to sea-surface-elevation skewness and asymmetry (with respect to a vertical axis) from interactions among various wave triads are given by the real and imaginary parts of the bispectrum, respectively. In very shallow water, coupling between surf beat and higher-frequency waves results in a skewness with sign opposite to, and about 40% of the magnitude of, the skewness resulting from interactions between the power-spectral-peak frequency and higher frequencies.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 161 , December 1985 , pp. 425 - 448
Copyright
© 1985 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barnett, T. P., Johnson, L. C., Naitoh, P., Hicks, N. & Nute, C. 1971 Bispectrum analysis of electroencephalogram signals during waking and sleeping. Science 172, 401–402.Google Scholar
Brillinger, D. R. 1965 An introduction to polyspectra. Ann. Math. Statist. 36, 1351–1374.Google Scholar
Brillinger, D. R. & Rosenblatt, M. 1967a Asymptotic theory of estimates of k -th order spectra. In Advanced Seminar on Spectral Analysis of Time Series (ed. B. Harris), pp. 153–188. Wiley.
Brillinger, D. R. & Rosenblatt, M. 1967b Computation and interpretation of k-th order spectra. In Advanced Seminar on Spectral Analysis of Time Series (ed. B. Harris), pp. 189–232. Wiley.
Elgar, S. & Guza, R. T. 1985a Shoaling gravity waves: comparisons between field observations, linear theory, and a nonlinear model. J. Fluid Mech, 158, 47–70.Google Scholar
Elgar, S. & Guza, R. T. 1985b Nonlinear model predictions of shallow water gravity wave bispectra. Submitted to J. Fluid Mech.Google Scholar
Elgar, S., Guza, R. T. & Seymour, R. J. 1984 Groups of waves in shallow water. J. Geophys. Res. 89, 3623–3634.Google Scholar
Freilich, M. H. & Guza, R. T. Nonlinear effects on shoaling suoaling surface gravity waves. Phil. Trans. R. Soc. Lond. A 311, 1–41.
Gable, C. G. (ed.) 1981 Report on data from the Nearshore Sediment Transport Study experiment at Leadbetter Beach, Santa Barbara, California, January-February, 1980. IMR Ref. 80–5., University of California., Inst. of Marine Resources, La Jolla, CA.
Godfrey, M. D. 1965 An exploratory study of the bispectrum of economic time series. Appl. Statist. 14, 48–69.Google Scholar
Guza, R. T. & Thornton, E. B. 1985 Observations of surf beat. J. Geophys. Res. 90, 3161–3172.Google Scholar
Guza, R. T., Thornton, E. B. & Holman, R. A. 1984 Swash on steep and shallow beaches. In Proc.19th Coastal Enging Conf., pp. 708–723. Houston, Texas, American Society of Civil Engineers.
Hasselman, K., Munk, W. & MacDonald, G. 1963 Bispectra of ocean waves. In Time Series Analysis (ed. M. Rosenblatt), pp. 125–139. Wiley.
Haubrich, R. A. 1965 Earth noises, 5 to 500 millicycles per second, 1. J. Geophys. Res. 70, 1415–1427.Google Scholar
Helland, K. N., Van Atta, C. W. & Stegun, G. N. 1977 Spectral energy transfer in high Reynolds number turbulence. J. Fluid Mech. 79, 337–359.Google Scholar
Hinich, M. J. & Clay, C. S. 1968 The application of the discrete Fourier transform in the estimation of power spectra, coherence, and bispectra of geophysical data. Rev. Geophys. 6, 347–363.Google Scholar
Huber, P. J., Kleiner, B., Gasser, T. & Dumermath, G. 1971 Statistical methods for investigating phase relations in stationary stochastic processes. IEEE Trans. Audio and Electroacoustics 19, 78–86.Google Scholar
Jenkins, G. M. & Watts, D. G. 1968 Spectral Analysis and its Applications. Holden-Day.
Kim, Y. C., Beall, J. M., Powers, E. J. & Miksad, R. W. 1980 Bispectrum and nonlinear wave coupling. Phys. Fluids 23, 250–263.Google Scholar
Kim, Y. C. & Powers, E. J. 1978 Digital bispectral analysis of self-excited fluctuation spectra. Phys. Fluids 21, 1452–1453.Google Scholar
Kim, Y. C. & Powers, E. J. 1979 Digital bispectral analysis and its application to nonlinear wave interactions. IEEE Trans. Plasma Science 1, 120–131.Google Scholar
Lii, K. S., Rosenblatt, M. & Van Atta, C. 1976 Bispectral measurements in turbulence. J. Fluid Mech. 77, 45–62.Google Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1962 Radiation stress and mass transport in gravity waves, with application to ‘surf beats’. J. Fluid Mech. 13, 481–504.Google Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1964 Radiation stresses in water waves; a physical discussion with applications. Deep-Sea Res. 11, 529–562.Google Scholar
Masuda, A. & Kuo, Y. Y. 1981 A note on the imaginary part of bispectra. Deep-Sea Res. 28, 213–222.Google Scholar
Mccomas, C. H. & Briscoe, M. G. 1980 Bispectra of internal waves. J. Fluid Mech. 97, 205–213.Google Scholar
Munk, W. H. 1949 Surf beats. Trans. Am. Geophys. Un. 30, 849–854.Google Scholar
Peregrine, D. H. 1967 Long waves on a beach. J. Fluid Mech. 27, 815–827.Google Scholar
Rosenblatt, M. & Van Ness, J. W. 1965 Estimation of the bispectrum. Ann. Math. Statist. 36, 1120–1136.Google Scholar
Sato, T., Sasaki, K. & Taketani, M. 1980 Bispectral passive velocimeter of a moving noisy machine. J. Accoust. Soc. Am. 68, 1729–1735.Google Scholar
Suhayda, J. N. 1972 Experimental study of the shoaling transformation of waves on a sloping bottom. Ph.D. dissertation, Scripps Institution of Oceanography, University of California, La Jolla, CA.
Suhayda, J. N. 1974 Standing waves on beaches. J. Geophys. Res. 72, 3065–3071.Google Scholar
Van Atta, C. W. 1979 Inertial range bispectra in turbulence. Phys. Fluids 22, 1440–1443.Google Scholar
Yeh, T. T. & Van Atta, C. W. 1973 Spectral transfer of scales and velocity fields in heated-grid turbulence. J. Fluid Mech. 58, 233–261.Google Scholar