Editorial Type:
Article
Article Type:
Research Article

The Evolution of Inhomogeneous Wave Statistics through a Variable Medium

P. B. Smit Delft University of Technology, Delft, Netherlands

Search for other papers by P. B. Smit in
Current site
Google Scholar
PubMed Close
and
T. T. Janssen Theiss Research, El Granada, California

Search for other papers by T. T. Janssen in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Aug 2013
DOI:
https://doi.org/10.1175/JPO-D-13-046.1
Page(s):
1741–1758
Restricted access

Abstract

The interaction of ocean waves with variable currents and topography in coastal areas can result in inhomogeneous statistics because of coherent interferences, which affect wave-driven circulation and transport processes. Stochastic wave models, invariably based on some form of the radiative transfer equation (or action balance), do not account for these effects. The present work develops and discusses a generalization of the radiative transfer equation that includes the effects of coherent interferences on wave statistics. Using multiple scales, the study approximates the transport equation for the (complete) second-order wave correlation matrix. The resulting model transports the coupled-mode spectrum (a form of the Wigner distribution) and accounts for the generation and propagation of coherent interferences in a variable medium. The authors validate the model through comparison with analytic solutions and laboratory observations, discuss the differences with the radiative transfer equation and the limitations of this approximation, and illustrate its ability to resolve coherent interference structures in wave fields such as those typically found in refractive focal zones and around obstacles.

Corresponding author address: Pieter Smit, Environmental Fluid Mechanics, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Stevinweg 1, 2628 CN Delft, Netherlands. E-mail: p.b.smit@tudelft.nl

Abstract

The interaction of ocean waves with variable currents and topography in coastal areas can result in inhomogeneous statistics because of coherent interferences, which affect wave-driven circulation and transport processes. Stochastic wave models, invariably based on some form of the radiative transfer equation (or action balance), do not account for these effects. The present work develops and discusses a generalization of the radiative transfer equation that includes the effects of coherent interferences on wave statistics. Using multiple scales, the study approximates the transport equation for the (complete) second-order wave correlation matrix. The resulting model transports the coupled-mode spectrum (a form of the Wigner distribution) and accounts for the generation and propagation of coherent interferences in a variable medium. The authors validate the model through comparison with analytic solutions and laboratory observations, discuss the differences with the radiative transfer equation and the limitations of this approximation, and illustrate its ability to resolve coherent interference structures in wave fields such as those typically found in refractive focal zones and around obstacles.

Corresponding author address: Pieter Smit, Environmental Fluid Mechanics, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Stevinweg 1, 2628 CN Delft, Netherlands. E-mail: p.b.smit@tudelft.nl
Save
Cover Journal of Physical Oceanography
Journal of Physical Oceanography

  • Agarwal, G. S., and E. Wolf, 1970: Calculus for functions of noncommuting operators and general phase–space methods in quantum mechanics. I. Mapping theorems and ordering of functions of noncommuting operators. Phys. Rev., 2D, 2161–2186.

    • Search Google Scholar
    • Export Citation

  • Aiki, H., and R. Greatbatch, 2011: Thickness-weighted mean theory for the effect of surface gravity waves on mean flows in the upper ocean. J. Phys. Oceanogr., 42, 725–747.

    • Search Google Scholar
    • Export Citation

  • Alber, I. E., 1978: The effects of randomness on the stability of two-dimensional surface wavetrains. Proc. Roy. Soc. London,A363, 525–546.

  • Bastiaans, M. J., 1979: Transport equations for the Wigner distribution function. Int. J. Opt., 26, 1265–1272.

  • Bastiaans, M. J., 1997: Application of the Wigner distribution function in optics. The Wigner Distribution—Theory and Applications in Signal Processing, W. Mecklenbrauker and F. Hlawatsch, Eds., Elsevier Science, 375–426.

  • Berkhoff, J., N. Booy, and A. Radder, 1982: Verification of numerical wave propagation models for simple harmonic linear water waves. Coastal Eng., 6, 255–279.

    • Search Google Scholar
    • Export Citation

  • Booij, N., R. C. Ris, and L. H. Holthuijsen, 1999: A third-generation wave model for coastal regions. 1. Model description and validation. J. Geophys. Res., 104 (C4), 7649–7666.

    • Search Google Scholar
    • Export Citation

  • Born, M., and E. Wolf, 1999: Principles of Optics. 7th ed. Cambridge University Press, 836 pp.

  • Bremmer, H., 1972: General remarks concerning theories dealing with scattering and diffraction in random media. Radio Sci., 8, 511–534.

    • Search Google Scholar
    • Export Citation

  • Cohen, L., 1989: Time-frequency distributions—A review. Proc. IEEE, 77, 941–981.

  • Cohen, L., 2010: Phase-space differential equations for modes. Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations, B. Schulze and M. W. Wong, Eds., Birkhäuser Basel, 235–250.

  • Craik, A. D. D., and S. Leibovich, 1976: A rational model for Langmuir circulations. J. Fluid Mech., 73, 401–426.

  • Dingemans, M., 1997: Water Wave Propagation over Uneven Bottoms. Advanced Series on Ocean Engineering, Vol. 23, World Scientific, 471 pp.

  • Dysthe, K. B., 1979: Note on a modification to the nonlinear schrodinger equation for application to deep water waves. Proc. Roy. Soc. London,A369, 105–114.

  • Hlawatsch, F., and P. Flandrin, 1997: The interference structure of the wigner distribution and related time–frequency signal representations. The Wigner Distribution—Theory and Applications in Signal Processing, W. Mecklenbrauker and F. Hlawatsch, Eds., Elsevier, 59–133.

  • Hoefel, F., and S. Elgar, 2003: Wave-induced sediment transport and sandbar migration. Science, 299, 1885–1887.

  • Janssen, A., and T. Claasen, 1985: On positivity of time–frequency distributions. IEEE Trans. Acoust. Speech Signal Process.,33, 1029–1032.

  • Janssen, P., 2009: The Interaction of Ocean Waves and Wind. Cambridge University Press, 300 pp.

  • Janssen, T. T., and T. H. C. Herbers, 2009: Nonlinear wave statistics in a focal zone. J. Phys. Oceanogr., 39, 1948–1964.

  • Janssen, T. T., T. H. C. Herbers, and J. A. Battjes, 2008: Evolution of ocean wave statistics in shallow water: Refraction and diffraction over seafloor topography. J. Geophys. Res.,113, C03024, doi:10.1029/2007JC004410.

  • Komen, G., L. Cavaleri, M. Donelan, K. Hasselmann, S. Hasselmann, and P. Janssen, 1994: Dynamics and Modelling of Ocean Waves. Cambridge University Press, 532 pp.

  • Kuik, A., G. van Vledder, and L. Holthuijsen, 1988: A method for the routine analysis of pitch-and-roll buoy wave data. J. Phys. Oceanogr., 18, 1020–1034.

    • Search Google Scholar
    • Export Citation

  • Magne, R., K. A. Belibassakis, T. H. C. Herbers, F. Ardhuin, W. C. O'Reilly, and V. Rey, 2007: Evolution of surface gravity waves over a submarine canyon. J. Geophys. Res.,112, C01002, doi:10.1029/2005JC003035.

  • Mandel, L., and E. Wolf, 1995: Optical Coherence and Quantum Optics. Cambridge University Press, 1166 pp.

  • McWilliams, J. C., and J. M. Restrepo, 1999: The wave-driven ocean circulation. J. Phys. Oceanogr., 29, 2523–2540.

  • Mei, C., M. Stiassnie, and D. Yue, 2005: Theory and Applications of Ocean Surface Waves. Advanced Series on Ocean Engineering, Vol. 21, World Scientific, 503 pp.

  • Mesiah, A., 1961: Quantum Mechanics: Vol. I. North-Holland Publishing Company, 1136 pp.

  • O'Reilly, W. C., and R. T. Guza, 1991: Comparison of spectral refraction and refraction–diffraction wave models. J. Waterw. Port Coastal Ocean Eng., 117, 199–215.

    • Search Google Scholar
    • Export Citation

  • Penney, W. G., and A. T. Price, 1952: The diffraction theory of sea waves and the shelter afforded by breakwaters. Part I. Proc. Roy. Soc. London,A244, 236–253.

  • Salmon, R., 1998: Lectures on Geophysical Fluid Dynamics. Oxford University Press, 400 pp.

  • Smith, J. A., 2006: Observed variability of ocean wave Stokes drift, and the Eulerian response to passing groups. J. Phys. Oceanogr., 36, 1381–1402.

    • Search Google Scholar
    • Export Citation

  • Stamnes, J., 1986: Waves in Focal Regions: Propagation, Diffraction, and Focusing of Light, Sound, and Water Waves. A. Hilger, 618 pp.

  • Strichartz, R., 1993: A Guide to Distribution Theory and Fourier Transforms. CRC Press, 224 pp.

  • Svendsen, I. A., 2006: Introduction to Nearshore Hydrodynamics. Advanced Series on Ocean Engineering, Vol. 24, World Scientific, 744 pp.

  • The WAMDI Group, 1988: The WAM model—A third generation ocean wave prediction model. J. Phys. Oceanogr., 18, 1775–1810.

  • Tolman, H. L., 1991: A third-generation model for wind waves on slowly varying, unsteady, and inhomogeneous depths and currents. J. Phys. Oceanogr., 21, 782–797.

    • Search Google Scholar
    • Export Citation

  • Torre, A., 2005: Linear Ray and Wave Optics in Phase Space: Bridging Ray and Wave Optics via the Wigner Phase-Space. Elsevier, 540 pp.

  • Ville, J., 1948: Thorie et applications de la notion de signal analytique. Cables Transm.,2A, 61–74.

  • Vincent, C. L., and M. J. Briggs, 1989: Refraction–diffraction of irregular waves over a mound. J. Waterw. Port Coastal Ocean Eng., 115, 269–284.

    • Search Google Scholar
    • Export Citation

  • Walther, A., 1968: Radiometry and coherence. J. Opt. Soc. Amer., 58, 1256–1259.

  • Wigner, E., 1932: On the quantum correction for thermodynamic equilibrium. Phys. Rev., 40, 749–759.

  • Wise Group, 2007: Wave modeling—The state of the art. Prog. Oceanogr., 75, 603–674.

  • Zijlema, M., G. Stelling, and P. Smit, 2011: SWASH: An operational public domain code for simulating wave fields and rapidly varied flows in coastal waters. Coastal Eng., 58, 992–1012.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 1255 779 30
PDF Downloads 434 102 11