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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)
66Symposium Bjørn Gjevik 70 år
BEYOND LINEAR THEORY:
• Nonlinear contributions exist in the (k,)-spectrum
• How do they affect the analysis of data?
77Symposium Bjørn Gjevik 70 år
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:
88Symposium Bjørn Gjevik 70 år
First order Lagrangian solution for a short wave riding on a long wave:
99Symposium Bjørn Gjevik 70 år
1D LINEAR AND LAGRANGIAN WAVESTime series
Co
lou
r sc
ale
in d
B
1010Symposium Bjørn Gjevik 70 år
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
1111Symposium Bjørn Gjevik 70 år
Regular waves
1D CREAMER WAVES
1212Symposium Bjørn Gjevik 70 år
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
1313Symposium Bjørn Gjevik 70 år
Uni-Directional Waves, JONSWAP Spectrum
1st order1st and 2nd order
2LWTk k
S
1414Symposium Bjørn Gjevik 70 år
Wavenumber Distributions, 1st +2nd ord. spectrum First order spectrum: ( , ) JONSWAP ME S D
1515Symposium Bjørn Gjevik 70 år
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)
1616Symposium Bjørn Gjevik 70 år
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
1717Symposium Bjørn Gjevik 70 år
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
1818Symposium Bjørn Gjevik 70 år
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
1919Symposium Bjørn Gjevik 70 år
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:
2121Symposium Bjørn Gjevik 70 år
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
2323Symposium Bjørn Gjevik 70 år
Wavenumber Distributions
2424Symposium Bjørn Gjevik 70 år
Normalized RMS wavenumbers for record in previous slide:
2LWTk k
2525Symposium Bjørn Gjevik 70 år
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
2626Symposium Bjørn Gjevik 70 år
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
2828Symposium Bjørn Gjevik 70 år
WAVELET ”SPECTRUM” AND NORMALIZED DISPERSION RATIO
Long wavelet, = 20Short wavelet, = 5
2LWTk k2
LWTk k
2929Symposium Bjørn Gjevik 70 år
DWM k-DISTRIBUTIONS
(Ekofisk Laser Array 14 Dec. 2003, @1800)
Lin. wave theory
3030Symposium Bjørn Gjevik 70 år
ASARWAM
Buoy
3131Symposium Bjørn Gjevik 70 år
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
3434Symposium Bjørn Gjevik 70 år
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
3636Symposium Bjørn Gjevik 70 år
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