Harald E. Krogstad, Department of Mathematical Sciences, NTNU, Trondheim

1. OCEANIC WAVES – OBSERVATIONS AND INTERPRETATIONS Harald E. Krogstad, Department of Mathematical Sciences, NTNU, Trondheim and work in progress with Karsten Trulsen, Department of Mathematics, University of Oslo, Oslo

2. The ANODA Swell Study (~1985) Depression Track B. Gjevik, H. Krogstad, A. Lygre and O. Rygg: Long period swell wave events on the Norwegian shelf, J. Phys. Ocean. 18 (1988) pp. 724–737

3. THE ”STANDARD MODEL” Random surface: -spectrum:

4. WAVE SPECTRA Dispersion surface Linear Theory:

5. EU COST Action 714: ”Measurements and Use of Directional Spectra of Ocean Waves” EDITORS: Kimmo Kahma, Danièle Hauser, Harald E. Krogstad, Susanne Lehner, Jaak A.J. Monbaliu, Lucy R. Wyatt + 32 other contributors Ref: EUR 21367 (2005) Freely available as a PDF-file on the Internet, 465 p. (~ 30Mb)

6. BEYOND LINEAR THEORY: • Nonlinear contributions existin the (k,w)-spectrum • How do they affect the analysis of data?

7. LINEAR, RANDOM LAGRANGIAN MODEL Elevation: Horizontal displacement: (deep water) Spectral amplitude is located on the dispersion surface. Euler: Lagrange:

8. First order Lagrangian solution for a short wave riding on a long wave:

9. 1D LINEAR AND LAGRANGIAN WAVES Time series Colour scale in dB

10. CREAMER et al. TRANSFORMATION 1D form: DB Creamer et al.J. Fluid Mech., 1989

11. Regular waves 1D CREAMER WAVES

12. 3rd order Perturbation Expansion 2nd order spectrum: (4th in steepness) Dispersion surface shift: H. Mitsuyasu et al., J. Fluid Mech., 1979

13. 1st order 1st and 2nd order Uni-Directional Waves, JONSWAP Spectrum

14. Wavenumber Distributions, 1st +2nd ord. spectrum

15. Next step (in progress): Spectra from unidirectional and directional wave fields simulated by Modified Nonlinear Schrödinger Equations Dynamic development of 1st order k-spectrum: (K. B. Dysthe, K. Trulsen, HEK, et al. , J. Fluid Mech., 2003)

16. Cross Spectrum: ANALYSIS OF MEASUREMENTS Transfer functions The Inverse Problem: Obtain c from estimates ofS!

17. Obtain the best spectrum in accordance with Measurements 1st step: 2nd step:

18. Standard Linear Wave Theory Approach: Many methods for obtaining D: Truncated Fourier series Maximum Likelihood methods Maximum Entropy (Burg and Shannon) Bayesian techniques … However, in some cases the transfer functions are independent of LWT

19. ELEVATION/SLOPE TRIPLET Measurements: Transfer Functions: Five integral properties of k:

20. (A) Forced Dispersion Relation: (B) Estimated Dispersion Relation (Standard Method) (C) No Dispersion Relation:

21. Directional Spread (degrees) “Check Ratio’’ = WADIC, Field observations (Wavescan buoy) Hm0 > 6m, 22 records

22. Conventional Analysis from the Ekofisk laser array

23. Wavenumber Distributions

24. Normalized RMS wavenumbers for record in previous slide:

25. Probes (k,w) THE DIRECTIONAL WAVELET METHOD (DWM) Directional Morlet wavelet moving in direction k: M. Donelan et al., J. Phys. Ocean., 1996

26. MORLET WAVELET:

27. THE DIRECTIONAL WAVELET METHOD a wavelet matched filter analysis uses no predefined dispersion relation • provides a detailed (t,w,k)-representation of the energy in the signals • provides reduced (averaged) wavenumber/frequency spectra from the full representation

28. Short wavelet, s = 5 Long wavelet, s = 20 WAVELET ”SPECTRUM” AND NORMALIZED DISPERSION RATIO

29. DWM k-DISTRIBUTIONS (Ekofisk Laser Array 14 Dec. 2003, @1800) Lin. wave theory

30. WAM Buoy ASAR

31. Buoy ASAR WAM

32. http://www.boost-technologies.com/esa/images/ thanks to: Fabrice Collard, BOOST Technologies/CLS, Brest Fabrice Ardhuin, Service Hydrographique et Océanographique, Brest

33. APPENDIX: EXTRA SLIDES

34. Depression Track 21 – 23 January 1882 Return

35. Return

36. Return

37. 5 2.6m A 0 -5 ~20 m 5 B 0 -5 5 C 0 -5 5 D 0 -5 0 200 400 600 800 1000 1200 EKOFISK LASER ARRAY Return Design: Mark A. Donelan, RSMAS, US, Anne Karin Magnusson, DNMI, Norway Sampling frequency= 5Hz, 4 channels – continuous sampling