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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
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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
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3 3 Symposium Bjørn Gjevik 70 år THE ”STANDARD MODEL” Random surface: -spectrum:
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4 4 Symposium Bjørn Gjevik 70 år WAVE SPECTRA Dispersion surface Linear Theory:
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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)
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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?
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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:
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8 8 Symposium Bjørn Gjevik 70 år First order Lagrangian solution for a short wave riding on a long wave:
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9 9 Symposium Bjørn Gjevik 70 år 1D LINEAR AND LAGRANGIAN WAVES Time series Colour scale in dB
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10 Symposium Bjørn Gjevik 70 år 1D form: CREAMER et al. TRANSFORMATION DB Creamer et al. J. Fluid Mech., 1989
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11 Symposium Bjørn Gjevik 70 år Regular waves 1D CREAMER WAVES
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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
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13 Symposium Bjørn Gjevik 70 år Uni-Directional Waves, JONSWAP Spectrum 1 st order 1 st and 2 nd order
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14 Symposium Bjørn Gjevik 70 år Wavenumber Distributions, 1 st +2 nd ord. spectrum
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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)
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16 Symposium Bjørn Gjevik 70 år The Inverse Problem: Obtain c from estimates of ! Cross Spectrum: ANALYSIS OF MEASUREMENTS Transfer functions
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17 Symposium Bjørn Gjevik 70 år Measurements 1 st step: 2 nd step: Obtain the best spectrum in accordance with
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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
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19 Symposium Bjørn Gjevik 70 år ELEVATION/SLOPE TRIPLET Five integral properties of k: Measurements: Transfer Functions:
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20 Symposium Bjørn Gjevik 70 år (B) Estimated Dispersion Relation (Standard Method) (C) No Dispersion Relation: (A) Forced Dispersion Relation:
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21 Symposium Bjørn Gjevik 70 år WADIC, Field observations (Wavescan buoy) Hm0 > 6m, 22 records Directional Spread (degrees) “Check Ratio’’ =
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22 Symposium Bjørn Gjevik 70 år Conventional Analysis from the Ekofisk laser arrayEkofisk laser array
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23 Symposium Bjørn Gjevik 70 år Wavenumber Distributions
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24 Symposium Bjørn Gjevik 70 år Normalized RMS wavenumbers for record in previous slide:
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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
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26 Symposium Bjørn Gjevik 70 år MORLET WAVELET:
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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
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28 Symposium Bjørn Gjevik 70 år WAVELET ”SPECTRUM” AND NORMALIZED DISPERSION RATIO Long wavelet, s = 20Short wavelet, s = 5
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29 Symposium Bjørn Gjevik 70 år DWM k-DISTRIBUTIONS (Ekofisk Laser Array 14 Dec. 2003, @1800) Lin. wave theory
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30 Symposium Bjørn Gjevik 70 år ASAR WAM Buoy
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31 Symposium Bjørn Gjevik 70 år ASAR Buoy WAM
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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
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33 Symposium Bjørn Gjevik 70 år APPENDIX: EXTRA SLIDES
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34 Symposium Bjørn Gjevik 70 år Depression Track 21 – 23 January 1882 Return
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35 Symposium Bjørn Gjevik 70 år Return
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36 Symposium Bjørn Gjevik 70 år Return
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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
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