1 1 Symposium Bjørn Gjevik 70 år Harald E. Krogstad, Department of Mathematical Sciences, NTNU,...

11Symposium Bjørn Gjevik 70 år

Harald E. Krogstad, Department of Mathematical Sciences,

NTNU, Trondheim

and work in progress withKarsten Trulsen,

Department of Mathematics,University of Oslo, Oslo

OCEANIC WAVES – OBSERVATIONS AND INTERPRETATIONS

22Symposium 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

33Symposium Bjørn Gjevik 70 år

THE ”STANDARD MODEL”

Random surface:

-spectrum:

),(),(,

)(

kxk

kx dZet ti

),(),(),( kkk dZdZEd ,k

44Symposium Bjørn Gjevik 70 år

WAVE SPECTRA

Dispersionsurface

k

, k

xk

yk

2

2

( , ) ( ) ( )

( ) ( ) ( , )

k kd d k

d k S D d d

k k k

k

)tanh(khgk,

S

Linear Theory:

55Symposium 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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BEYOND LINEAR THEORY:

• Nonlinear contributions exist in the (k,)-spectrum

• How do they affect the analysis of data?

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LINEAR, RANDOM LAGRANGIAN MODEL

0

0

0 0,

0 0,

, ,

, ,

i t

i t

z t e dZ

t i e dZk

kx

k

kx

k

x k

kd x k

(deep water)

0

0 0

, ,

,

t z t

x x

x x d x

Elevation:

Horizontaldisplacement:

Spectral amplitude is located on thedispersion surface.

0 ,Z k

0 0, ,t z t x xEuler:

Lagrange:

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

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1D LINEAR AND LAGRANGIAN WAVESTime series

Co

lou

r sc

ale

in d

B

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1D form:

CREAMER et al. TRANSFORMATION

2 3 4 2 4

, , , , , ,

H , H , H , H , 0 H ,

t t t t

x x x x

Hilbert transformH

DB Creamer et al. J. Fluid Mech., 1989

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Regular waves

1D CREAMER WAVES

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3rd order Perturbation Expansion

1 1

2

1 1 1 1 1 1 1 1 1

2

, ,

, ,

q qW q q H q q q q d q W q F q q d q q

q W q gk

k

1

12 1 1 1 12

,q

H q qq q q d q

W q

1

1 1 1 1, 0q

W q F q q d q q

2nd order spectrum:

(4th in steepness)

Dispersion surface shift:

H. Mitsuyasu et al., J. Fluid Mech., 1979

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Uni-Directional Waves, JONSWAP Spectrum

1st order1st and 2nd order

2LWTk k

S

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Wavenumber Distributions, 1st +2nd ord. spectrum First order spectrum: ( , ) JONSWAP ME S D

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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)

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N,,j),,(dZ),(Tee)t(Y j,

itij

j 1

kkk

kx

The Inverse Problem: Obtain from estimates of !

, , ,H d kΣ T k T k kCross Spectrum:

ANALYSIS OF MEASUREMENTS

Transfer functions

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1arg max log Tr ,ML

ΣΣ Σ Σ Σ

D

Measurements

; , , ,H d kΣ Σ T k T k kD

1st step:

2nd step:

Obtain the best spectrum in accordance with MLΣ

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Standard Linear Wave Theory Approach:

, ,S D k

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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2

2

1 x y

x x x y

y x y y

i k i k

S i k k k k

i k k k k

Σ

( ) ( , ) ( , ) ( , )g g k d d k kk k

, , , , ,

, 1, , '

x y

x y

t t t

ik ik

0 0 0

T k

ELEVATION/SLOPE TRIPLET

Five integral properties of k:

2 2, , , ,x y x y x yk k k k k k

Measurements:

Transfer Functions:

2020Symposium Bjørn Gjevik 70 år

2 2, , 1d S d kd d k kk k k

(B) Estimated Dispersion Relation (Standard Method)

1/22 2, | | ,c c x yD k k k k k k

(C) No Dispersion Relation:

1/ 2 1

22

22

1 1exp '

2 2

x x x y x y

x y x y y y

M M

k k k k k kM

k k k k k k

k k k k k

, | | LWTS D k k k

(A) Forced Dispersion Relation:

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WADIC, Field observations (Wavescan buoy)

Hm0 > 6m, 22 records

Directional Spread (degrees)“Check Ratio’’ =

1/22

LWT

k

k

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Conventional Analysis from the Ekofisk laser array

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Wavenumber Distributions

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Normalized RMS wavenumbers for record in previous slide:

2LWTk k

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THE DIRECTIONAL WAVELET METHOD (DWM)

m t k xDirectional Morlet wavelet moving in direction k:

(k,)

Probes

M. Donelan et al., J. Phys. Ocean., 1996

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MORLET WAVELET:

2

221/42

1t

itm t e e

-20 -15 -10 -5 0 5 10 15 20

-0.1

-0.05

0

0.05

0.1

0.15

Time

Re

al a

nd

Ima

gin

ary

pa

rts

= 5

-50 0 50-0.05

-0.025

0

0.025

0.05

Time

Re

al a

nd

Ima

gin

ary

pa

rts

= 20

Real partImaginary part

Real partImaginary part

2727Symposium Bjørn Gjevik 70 år

• a wavelet matched filter analysis

• provides a detailed (t,,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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WAVELET ”SPECTRUM” AND NORMALIZED DISPERSION RATIO

Long wavelet, = 20Short wavelet, = 5

2LWTk k2

LWTk k

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DWM k-DISTRIBUTIONS

(Ekofisk Laser Array 14 Dec. 2003, @1800)

Lin. wave theory

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ASARWAM

Buoy

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ASAR Buoy WAM

3232Symposium 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

3333Symposium Bjørn Gjevik 70 år

APPENDIX: EXTRA SLIDES

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Depression Track 21 – 23 January 1882

Return

3535Symposium Bjørn Gjevik 70 år

0 50 100 150 200 250 300

-3

-2

-1

0

1

2

3

Linear (blue) / Lagrange (red), S=0.095, Time:13.2Tp

0 50 100 150 200 250 300

-3

-2

-1

0

1

2

3

Linear (blue) / Creamer (red)

Return

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Return

3737Symposium Bjørn Gjevik 70 år

-50

5

A

-505

B

-505

C

0 200 400 600 800 1000 1200-505

D

Design: Mark A. Donelan, RSMAS, US, Anne Karin Magnusson, DNMI, Norway

2.6m

~20

m

Sampling frequency = 5Hz, 4 channels – continuous sampling

EKOFISK LASER ARRAYReturn