1. 1 1 Symposium Bjørn Gjevik 70 år Harald E. Krogstad, Department of Mathematical Sciences, NTNU, Trondheim and work in progress with Karsten Trulsen, Department of Mathematics, University of Oslo, Oslo OCEANIC WAVES – OBSERVATIONS AND INTERPRETATIONS

2. 2 2 Symposium Bjørn Gjevik 70 år 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. 3 3 Symposium Bjørn Gjevik 70 år THE ”STANDARD MODEL” Random surface: -spectrum:

4. 4 4 Symposium Bjørn Gjevik 70 år WAVE SPECTRA Dispersion surface Linear Theory:

5. 5 5 Symposium Bjørn Gjevik 70 år 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. 6 6 Symposium Bjørn Gjevik 70 år BEYOND LINEAR THEORY: Nonlinear contributions exist in the (k,w)-spectrum How do they affect the analysis of data?

7. 7 7 Symposium Bjørn Gjevik 70 år LINEAR, RANDOM LAGRANGIAN MODEL (deep water) Elevation: Horizontal displacement: Spectral amplitude is located on the dispersion surface. Euler: Lagrange:

8. 8 8 Symposium Bjørn Gjevik 70 år First order Lagrangian solution for a short wave riding on a long wave:

9. 9 9 Symposium Bjørn Gjevik 70 år 1D LINEAR AND LAGRANGIAN WAVES Time series Colour scale in dB

10. 10 Symposium Bjørn Gjevik 70 år 1D form: CREAMER et al. TRANSFORMATION DB Creamer et al. J. Fluid Mech., 1989

11. 11 Symposium Bjørn Gjevik 70 år Regular waves 1D CREAMER WAVES

12. 12 Symposium Bjørn Gjevik 70 år 3 rd order Perturbation Expansion 2 nd order spectrum: (4 th in steepness) Dispersion surface shift: H. Mitsuyasu et al., J. Fluid Mech., 1979

13. 13 Symposium Bjørn Gjevik 70 år Uni-Directional Waves, JONSWAP Spectrum 1 st order 1 st and 2 nd order

14. 14 Symposium Bjørn Gjevik 70 år Wavenumber Distributions, 1 st +2 nd ord. spectrum

15. 15 Symposium Bjørn Gjevik 70 år Next step (in progress): Spectra from unidirectional and directional wave fields simulated by Modified Nonlinear Schrödinger Equations Dynamic development of 1 st order k-spectrum: (K. B. Dysthe, K. Trulsen, HEK, et al., J. Fluid Mech., 2003)

16. 16 Symposium Bjørn Gjevik 70 år The Inverse Problem: Obtain c from estimates of  ! Cross Spectrum: ANALYSIS OF MEASUREMENTS Transfer functions

17. 17 Symposium Bjørn Gjevik 70 år Measurements 1 st step: 2 nd step: Obtain the best spectrum in accordance with

18. 18 Symposium Bjørn Gjevik 70 år 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. 19 Symposium Bjørn Gjevik 70 år ELEVATION/SLOPE TRIPLET Five integral properties of k: Measurements: Transfer Functions:

20. 20 Symposium Bjørn Gjevik 70 år (B) Estimated Dispersion Relation (Standard Method) (C) No Dispersion Relation: (A) Forced Dispersion Relation:

21. 21 Symposium Bjørn Gjevik 70 år WADIC, Field observations (Wavescan buoy) Hm0 > 6m, 22 records Directional Spread (degrees) “Check Ratio’’ =

22. 22 Symposium Bjørn Gjevik 70 år Conventional Analysis from the Ekofisk laser arrayEkofisk laser array

23. 23 Symposium Bjørn Gjevik 70 år Wavenumber Distributions

24. 24 Symposium Bjørn Gjevik 70 år Normalized RMS wavenumbers for record in previous slide:

25. 25 Symposium Bjørn Gjevik 70 år THE DIRECTIONAL WAVELET METHOD (DWM) Directional Morlet wavelet moving in direction k: (k,w)(k,w) Probes M. Donelan et al., J. Phys. Ocean., 1996

26. 26 Symposium Bjørn Gjevik 70 år MORLET WAVELET:

27. 27 Symposium Bjørn Gjevik 70 år a wavelet matched filter analysis provides a detailed (t,w,k)-representation of the energy in the signals uses no predefined dispersion relation provides reduced (averaged) wavenumber/frequency spectra from the full representation THE DIRECTIONAL WAVELET METHOD

28. 28 Symposium Bjørn Gjevik 70 år WAVELET ”SPECTRUM” AND NORMALIZED DISPERSION RATIO Long wavelet, s = 20Short wavelet, s = 5

29. 29 Symposium Bjørn Gjevik 70 år DWM k-DISTRIBUTIONS (Ekofisk Laser Array 14 Dec. 2003, @1800) Lin. wave theory

30. 30 Symposium Bjørn Gjevik 70 år ASAR WAM Buoy

31. 31 Symposium Bjørn Gjevik 70 år ASAR Buoy WAM

32. 32 Symposium Bjørn Gjevik 70 år http://www.boost-technologies.com/esa/images/ thanks to: Fabrice Collard, BOOST Technologies/CLS, Brest Fabrice Ardhuin, Service Hydrographique et Océanographique, Brest

33. 33 Symposium Bjørn Gjevik 70 år APPENDIX: EXTRA SLIDES

34. 34 Symposium Bjørn Gjevik 70 år Depression Track 21 – 23 January 1882 Return

35. 35 Symposium Bjørn Gjevik 70 år Return

36. 36 Symposium Bjørn Gjevik 70 år Return

37. 37 Symposium Bjørn Gjevik 70 år Design: Mark A. Donelan, RSMAS, US, Anne Karin Magnusson, DNMI, Norway 2.6m ~20 m Sampling frequency = 5Hz, 4 channels – continuous sampling EKOFISK LASER ARRAY Return