In the open ocean, the formation of large waves is non-linear. This non-linearity modifies the shape and structure of steep waves relative to that expected in a linear model. Work by Adcock et al. [“Nonlinear dynamics of wave-groups in random seas: Unexpected walls of water in the open ocean,” Proc. R. Soc. A 471(2184), 20150660 (2015)] used a numerical model to show that, for a wave spectrum representative of that of a storm in the North Sea, non-linear dynamics resulted in the following changes to the shape of extreme wave groups relative to a linear model: expansion of the wave-crest laterally, contraction of the wave-group in the mean wave direction, and a movement of the largest wave to the front of the wave-group. They found only moderate elevation above that expected by the linear model. This paper extends this work to explore the influence of spectral bandwidth and directional spreading on these results. We find that sea-states with low directional spreads are more likely to see significant non-linear changes to extreme wave-groups. However, rather surprisingly, extreme waves formed in seas with broader spectra generally, although not always, see greater non-linear changes for the range of spectra studied here. We also observe that the lateral expansion to the wave-group occurs even for relatively modest sea-states and is a predictable feature. By contrast, the contraction of the wave-group in the mean wave direction only occurs occasionally in the steepest sea-states—although when this is triggered the change can be very dramatic.

1.
C.
Kharif
and
E.
Pelinovsky
, “
Physical mechanisms of the rogue wave phenomenon
,”
Eur. J. Mech. - B/Fluids
22
(
6
),
603
634
(
2003
).
2.
K.
Dysthe
,
H. E.
Krogstad
, and
P.
Müller
, “
Oceanic rogue waves
,”
Annu. Rev. Fluid Mech.
40
,
287
310
(
2008
).
3.
T. A. A.
Adcock
and
P. H.
Taylor
, “
The physics of anomalous (‘rogue’) ocean waves
,”
Rep. Prog. Phys.
77
(
10
),
105901
(
2014
).
4.
J.
Gemmrich
, “
Group characteristics of large waves
,” in
EGU Assembly
(
European Geophysical Union
,
Vienna, Austria
,
2016
), Vol.
18
, p.
8945
.
5.
C.
Swan
and
M.
Latheef
, “
Extreme surface water waves in realistic directionally spread seas
,” in
Extreme Events and Rogue Waves Seminar
,
Bad Honnef, Germany
,
2016
.
6.
O.
Gramstad
and
K.
Trulsen
, “
Influence of crest and group length on the occurrence of freak waves
,”
J. Fluid Mech.
582
,
463
472
(
2007
).
7.
M.
Onorato
,
T.
Waseda
,
A.
Toffoli
,
L.
Cavaleri
,
O.
Gramstad
,
P. A. E. M.
Janssen
,
T.
Kinoshita
,
J.
Monbaliu
,
N.
Mori
, and
A. R.
Osborne
, “
Statistical properties of directional ocean waves: The role of the modulational instability in the formation of extreme events
,”
Phys. Rev. Lett.
102
(
11
),
114502
(
2009
).
8.
I. E.
Alber
, “
The effects of randomness on the stability of two-dimensional surface wavetrains
,”
Proc. R. Soc. A
363
(
1715
),
525
546
(
1978
).
9.
M.
Latheef
and
C.
Swan
, “
A laboratory study of wave crest statistics and the role of directional spreading
,”
Proc. R. Soc. A
469
,
20120696
(
2013
).
10.
R. H.
Gibbs
and
P. H.
Taylor
, “
Formation of wall of water in ‘fully’ nonlinear simulations
,”
Appl. Ocean Res.
27
(
3
),
142
157
(
2005
).
11.
T. A. A.
Adcock
,
P. H.
Taylor
, and
S.
Draper
, “
Nonlinear dynamics of wave-groups in random seas: Unexpected walls of water in the open ocean
,”
Proc. R. Soc. A
471
(
2184
),
20150660
(
2015
).
12.
R. S.
Gibson
, “
Wave interactions and wave statistics in directional seas
,” Ph.D. thesis,
Imperial College, University of London
,
2005
.
13.
R. S.
Gibson
and
C.
Swan
, “
The evolution of large ocean waves: The role of local and rapid spectral changes
,”
Proc. R. Soc. A
463
(
2077
),
21
48
(
2007
).
14.
L.
Shemer
,
E.
Kit
,
H.
Jiao
, and
O.
Eitan
, “
Experiments on nonlinear wave groups in intermediate water depth
,”
J. Waterw., Port, Coastal, Ocean Eng.
124
(
6
),
320
327
(
1998
).
15.
D.
Clamond
,
M.
Francius
,
J.
Grue
, and
C.
Kharif
, “
Long time interaction of envelope solitons and freak wave formations
,”
Eur. J. Mech. - B/Fluids
25
(
5
),
536
553
(
2006
).
16.
L.
Shemer
and
B.
Dorfman
, “
Experimental and numerical study of spatial and temporal evolution of nonlinear wave groups
,”
Nonlinear Processes Geophys.
15
(
6
),
931
942
(
2008
).
17.
A.
Slunyaev
, “
On the wave group asymmetry caused by nonlinear evolution
,” in
EGU General Assembly Conference Abstracts
(
European Geophysical Union
,
2015
), Vol.
17
, p.
7079
.
18.
T. A. A.
Adcock
and
P. H.
Taylor
, “
Non-linear evolution of uni-directional focussed wave-groups on a deep water: A comparison of models
,”
Appl. Ocean Res.
59
,
147
152
(
2016
).
19.
T. A. A.
Adcock
and
P. H.
Taylor
, “
Non-linear evolution of large waves in deep water – the influence of directional spreading and spectral bandwidth
,” in
Twenty-sixth (2016) International Ocean and Polar Engineering Conference
,
Rhodes (Rodos)
,
2016
.
20.
M. J.
Tucker
,
P. G.
Challenor
, and
D. J. T.
Carter
, “
Numerical simulation of a random sea: A common error and its effect upon wave group statistics
,”
Appl. Ocean Res.
6
(
2
),
118
122
(
1984
).
21.
Y.
Goda
,
Random Seas and Design of Maritime Structures
(
World Scientific
,
2010
).
22.
K. B.
Dysthe
, “
Note on a modification to the nonlinear Schrodinger equation for application to deep water waves
,”
Proc. R. Soc. A
369
(
1736
),
105
114
(
1979
).
23.
K.
Trulsen
and
K. B.
Dysthe
, “
A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water
,”
Wave Motion
24
(
3
),
281
289
(
1996
).
24.
K.
Trulsen
and
K. B.
Dysthe
, “
Frequency down-shift through self modulation and breaking
,” in
Water Wave Kinematics
(
Springer
,
1990
), pp.
561
572
.
25.
L.
Shemer
,
K.
Goulitski
, and
E.
Kit
, “
Evolution of wide-spectrum unidirectional wave groups in a tank: An experimental and numerical study
,”
Eur. J. Mech. - B/Fluids
26
(
2
),
193
219
(
2007
).
26.
T. A. A.
Adcock
and
P. H.
Taylor
, “
Fast and local non-linear evolution of steep wave-groups on deep water: A comparison of approximate models to fully non-linear simulations
,”
Phys. Fluids
28
(
1
),
016601
(
2016
).
27.
H.
Socquet-Juglard
,
K.
Dysthe
,
K.
Trulsen
,
H. E.
Krogstad
, and
J.
Liu
, “
Probability distributions of surface gravity waves during spectral changes
,”
J. Fluid Mech.
542
,
195
216
(
2005
).
28.
R. H.
Gibbs
, “
Walls of Water on the Open Ocean
,” D.Phil. thesis,
University of Oxford
,
2004
, Trinty term.
29.
P. A. E. M.
Janssen
, “
Nonlinear four-wave interactions and freak waves
,”
J. Phys. Oceanogr.
33
(
4
),
863
884
(
2003
).
30.
N.
Mori
and
P. A. E. M.
Janssen
, “
On kurtosis and occurrence probability of freak waves
,”
J. Phys. Oceanogr.
36
(
7
),
1471
1483
(
2006
).
31.
T.
Waseda
, “Impact of directionality on the extreme wave occurrence in a discrete random wave system,” in Proceedings of 9th International Workshop on Wave Hindcasting and Forecasting, Victoria, Canada, 2006.
32.
C. T.
Stansberg
, “
Effects from directionality & spectral bandwidth on non-linear spatial modulations of deep-water surface gravity wave trains
,” in
Coastal Engineering Conference
(
American Society of Civil Engineers
,
1995
), Vol.
1
, p.
579
.
33.
T. A. A.
Adcock
,
R. H.
Gibbs
, and
P. H.
Taylor
, “
The nonlinear evolution and approximate scaling of directionally spread wave groups on deep water
,”
Proc. R. Soc. A
468
(
2145
),
2704
2721
(
2012
).
34.
F. M.
Monaldo
, “
Measurement of wave coherence properties using spaceborne synthetic aperture radar
,”
Mar. Struct.
13
(
4
),
349
366
(
2000
).
35.
H. E.
Krogstad
,
A. K.
Magnusson
, and
M. A.
Donelan
, “
Wavelet and local directional analysis of ocean waves
,” in
The Sixteenth International Offshore and Polar Engineering Conference
(
International Society of Offshore and Polar Engineers
,
2006
).
36.
J.
Slocum
,
Sailing Alone Around the World
(
Centuary Company
,
1900
).
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