Fourier, Scattering, and Wavelet Transforms: Applications to Internal Gravity Waves with Comparisons to Linear Tidal Data

  • James A. Hawkins
  • Alex Warn-Varnas
  • Ivan Christov
Part of the Lecture Notes in Earth Sciences book series (LNEARTH, volume 112)

Abstract

Analysis of tides and internal waves from model studies in the South China Sea is done using three techniques. We summarize results from standard Fourier methods, continuous wavelet analysis and the direct scattering transform. Because the Fourier and wavelet analysis are inherently linear methods their utility in application to nonlinear dynamics is often questioned. Nevertheless, they have shown to be useful in delineating first order dynamics (for example finding fundamental modes). On the other hand the scattering transform, sometimes described as a ‘nonlinear Fourier’ technique, can in some cases succeed in elucidating non-linear dynamics where linear methods have proven less successful. We apply these procedures to model results from Lamb’s 2D non-hydrostatic model applied to the South China Sea and in some cases the multi-component tides used to force the Lamb model.

Keywords

Discrete fourier transform Continuous wavelet transform Direct scattering transform Luyon strait Internal gravity waves 

References

  1. 1.
    J. R. Apel: 2003. A new analytical model for internal solitons in the ocean. J. Phys. Oceanogr. 33, 2247–2269.CrossRefGoogle Scholar
  2. 2.
    K. G. Lamb: 1994. Numerical experiments of internal wave generation by strong tidal flow across a finite amplitude bank edge. J. Geophys. Res. 99(C1), 843–864.CrossRefGoogle Scholar
  3. 3.
    A. C. Warn-Varnas, S. A. Chin-Bing, D. B. King, Z. Hallock, and J. A. Hawkins: 2003. Ocean-acoustic solitary wave studies and predictions. Surveys in Geophysics. 24 39–79.CrossRefGoogle Scholar
  4. 4.
    A. Grinsted, J. Moore, and S. Jevrejeva: 2004. Application of the cross wavelet transform and wavelet coherence to geophysical time series, Nonlinear Processes in Geophysics 11, 561–566.Google Scholar
  5. 5.
    T. F. Duda, J. F. Lynch, J. D. Irish, R. C. Beardsley, S. R. Ramp, and C.-S. Chiu: 2004. Internal tide and nonlinear wave behavior at the Continental Slope in the North China Sea. IEEE J. Ocean Eng. 29, 1105–1130.CrossRefGoogle Scholar
  6. 6.
    S. R. Ramp, 2006. Private communication.Google Scholar
  7. 7.
    S.-Y. Chao, D.-S. Ko, R.-C. Lien, and P.-T. Shaw: 2007. Assessing the West Ridge of Luzon Strait as an internal wave mediator. J. Oceanogr. 63 (No.6), 897–911.CrossRefGoogle Scholar
  8. 8.
    Signal Processing Toolbox User’s Guide for use with MATLAB. The MathWorks, Inc. (2002).Google Scholar
  9. 9.
    A.R. Osborne, T.L. Burch: 1980. Internal solitons in the Andaman Sea. Science 208, 451–460.CrossRefGoogle Scholar
  10. 10.
    A.R. Osborne: 1994. Automatic algorithm for the numerical inverse scattering transform of the Korteweg–de Vries equation. Math. Comput. Simul. 37, 431–450.CrossRefGoogle Scholar
  11. 11.
    A.R. Osborne, M. Serio, L. Bergamasco, and L. Cavaleri: 1998. Solitons, cnoidal waves and nonlinear interactions in shallow-water ocean surface waves. Physica D 123, 64–81.CrossRefGoogle Scholar
  12. 12.
    A.R. Osborne and E. Segre: 1991. Numerical solutions of the Korteweg–de Vries equation using the periodic scattering transform μ-representation. Physica D 44, 575–604.CrossRefGoogle Scholar
  13. 13.
    A.R. Osborne, E. Segre, G. Boffetta, and L. Calaveri: 1991. Soliton basis states in shallow-water ocean surface waves. Phys. Rev. Lett. 67, 592–595.CrossRefGoogle Scholar
  14. 14.
    A.R. Osborne and M. Petti: 1994. Laboratory-generated, shallow-water surface waves: analysis using the periodic, inverse scattering transform. Phys. Fluids 6, 1727–1744.CrossRefGoogle Scholar
  15. 15.
    A.R. Osborne and L. Bergamasco: The solitons of Zabusky and Kruskal revisited: perspective in terms of the periodic spectral transform. Physica D 18, 26–46.Google Scholar
  16. 16.
    W.B. Zimmerman and G.W. Haarlemmer: 1999. Internal gravity waves: analysis using the the periodic, inverse scattering transform. Nonlin. Process. Geophys. 6, 11–26.CrossRefGoogle Scholar
  17. 17.
    I. Christov: 2008. Internal solitary waves in the ocean: analysis using the periodic, inverse scattering transform. Math. Comput. Simul., arXiv:0708.3421, (in press).Google Scholar
  18. 18.
    S. Jevrejeva, J. C. Moore, and A. Grinsted: 2003. Influence of the artic oscillation and El Niño-Southern Oscillation (ENSO) on ice condition in the Baltic Sea: The wavelet approach. J. Geophys. Res. 108, D21, 4677–4688.CrossRefGoogle Scholar
  19. 19.
    P. Kumar and E. Foufoula-Geogiou: 1994. Wavelet analysis in geophysics: an introduction. Wavelets in Geophysics, E. Foufoula-Georgiou and P. Kumar, eds. Academic Press, San Diego. pp. 1–45.Google Scholar
  20. 20.
    L.H. Kantha and C.A. Clayson: 2000. Appendix B: Wavelet Transforms. Numerical Models of Oceans and Oceanic Processes. L.H. Kantha and C.A. Clayson, eds. Academic Press, San Diego. pp. 786–818.Google Scholar
  21. 21.
    C. Torrence, G.P. Compo: 1998. A practical guide to wavelet analysis. Bull. Am. Met. Soc. 79, 61–78.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • James A. Hawkins
    • 1
  • Alex Warn-Varnas
    • 2
  • Ivan Christov
    • 2
    • 3
  1. 1.Planning Systems Inc.SlidellUSA
  2. 2.Naval Research Laboratory Stennis Space CenterUSA
  3. 3.Northwestern UniversityEvanstonUSA

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