High wind speeds generated during hurricanes result in the formation of extreme waves. Extreme waves by nature are steep meaning that linear wave theory alone is insufficient in understanding and predicting their occurrence. The complex, highly transient nature of the direction of wind and hence of waves generated during hurricanes affects this nonlinear behavior. Herein, we examine how this directionality can affect the second-order nonlinearity of extreme waves generated during hurricanes. This is achieved through both deterministic calculations and experiments based on the observations of Young (2006, “Directional Spectra of Hurricane Wind Waves,” J. Geophys. Res. Oceans, 111(C8), epub). Our calculations show that interactions between the tail and peak of the spectrum can become significant when they travel in different directions, resulting in second-order difference components that exist in the linear range of frequencies. These calculations are generally supported by experimental observations, but we note the difficulty of generating and focusing the high-frequency tail of the spectrum experimentally. Bound second-order difference components or subharmonics typically exist as low frequency infra-gravity waves. Components that exist in the linear range of frequencies may be missed by conventional methods of processing field data where low-pass filtering is used and hence overlooked. In this note, we show that in idealized directional spreading conditions representative of a hurricane, failing to account for second-order difference components may lead to underestimation of extreme wave height.

References

References
1.
Young
,
I. R.
,
2017
, “
A Review of Parametric Descriptions of Tropical Cyclone Wind-Wave Generation
,”
Atmosphere
,
8
(
12
), p.
194
.
2.
Young
,
I. R.
,
2006
, “
Directional Spectra of Hurricane Wind Waves
,”
J. Geophys. Res. Oceans
,
111
(C8), epub.
3.
Esquivel-Trava
,
B.
,
Ocampo-Torres
,
F. J.
, and
Osuna
,
P.
,
2015
, “
Spatial Structure of Directional Wave Spectra in Hurricanes
,”
Ocean Dyn.
,
65
(
1
), pp.
65
76
.
4.
Chen
,
S. S.
, and
Curcic
,
M.
,
2016
, “
Ocean Surface Waves in Hurricane Ike (2008) and Superstorm Sandy (2012): Coupled Model Predictions and Observations
,”
Ocean Modell.
,
103
, pp.
161
176
.
5.
Hu
,
K.
, and
Chen
,
Q.
,
2011
, “
Directional Spectra of Hurricane-Generated Waves in the Gulf of Mexico
,”
Geophys. Res. Lett.
,
38
(19), epub.
6.
Santo
,
H.
,
Taylor
,
P. H.
,
Eatock Taylor
,
R.
, and
Choo
,
Y. S.
,
2013
, “
Average Properties of the Largest Waves in Hurricane Camille
,”
ASME J. Offshore Mech. Arct.
,
135
(
1
), p.
0116021
.
7.
Lindgren
,
G.
,
1970
, “
Some Properties of a Normal Process Near a Local Maximum
,”
Ann. Math. Stat.
,
41
(
6
), pp.
1870
1883
.
8.
Boccotti
,
P.
,
1983
, “
Some New Results on Statistical Properties of Wind Waves
,”
App. Ocean Res.
,
5
(
3
), pp.
134
140
.
9.
Hasselmann
,
K.
,
1962
, “
On the Non-Linear Energy Transfer in a Gravity-Wave Spectrum—Part 1: General Theory
,”
J. Fluid Mech.
,
12
(
4
), pp.
481
500
.
10.
Sharma
,
J. N.
, and
Dean
,
R. G.
,
1981
, “
Second-Order Directional Seas and Associated Wave Forces
,”
Soc. Pet. Eng. J.
,
21
(
1
), pp.
129
140
.
11.
Dalzell
,
J. F.
,
1999
, “
A Note on Finite Depth Second-Order Wave–Wave Interactions
,”
App. Ocean Res.
,
21
(
3
), pp.
105
111
.
12.
Forristall
,
G. Z.
,
2000
, “
Wave Crest Distributions: Observations and Second-Order Theory
,”
J. Phys. Oceanogr.
,
30
(
8
), pp.
1931
1943
.
13.
Longuet-Higgins
,
M. S.
, and
Stewart
,
R. W.
,
1962
, “
Radiation Stress and Mass Transport in Gravity Waves, With Applications to ‘Surf Beats’
,”
J. Fluid Mech.
,
13
(
4
), pp.
481
504
.
14.
Okihiro
,
M.
,
Guza
,
R. T.
, and
Seymour
,
R. J.
,
1992
, “
Bound Infragravity Waves
,”
J. Geophys. Res.
,
97
(C7), pp.
453
469
.https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/92JC00270
15.
Herbers
,
T. H. C.
,
Elgar
,
S.
, and
Guza
,
R. T.
,
1994
, “
Infragravity-Frequency (0.005–0.05 Hz) Motions on the Shelf—Part I: Forced Waves
,”
J. Phys. Oceanogr.
,
24
(
5
), pp.
917
927
.
16.
Toffoli
,
A.
,
Onorato
,
M.
, and
Monbaliu
,
J.
,
2006
, “
Wave Statistics in Unimodal and Bimodal Seas From a Second-Order Model
,”
Eur. J. Mech. B
,
25
(
5
), pp.
649
661
.
17.
Christou
,
M.
,
Tromans
,
P.
,
Vanderschuren
,
L.
, and
Ewans
,
K.
,
2009
, “
Second-Order Crest Statistics of Realistic Sea States
,”
11th International Workshop on Wave Hindcasting and Forecasting
, Halifax, NS, Canada, Oct. 18–23, pp.
18
23
.http://www.waveworkshop.org/11thWaves/Papers/Christou_et_al_second_order_crest_statistics.pdf
18.
Walker
,
D. A. G.
,
Taylor
,
P. H.
, and
Eatock Taylor
,
R.
,
2004
, “
The Shape of Large Surface Waves on the Open Sea and the Draupner New Year Wave
,”
Appl. Ocean Res.
,
26
(
3–4
), pp.
73
83
.
19.
Toffoli
,
A.
,
Monbaliu
,
J.
,
Onorato
,
M.
,
Osborne
,
A. R.
,
Babanin
,
A. V.
, and
Bitner-Gregersen
,
E. M.
,
2007
, “
Second-Order Theory and Setup in Surface Gravity Waves: A Comparison With Experimental Data
,”
J. Phys. Oceanogr.
,
37
(
11
), pp.
2726
2739
.
20.
McAllister
,
M. L.
,
Adcock
,
T. A. A.
,
Taylor
,
P. H.
, and
van den Bremer
,
T. S.
,
2018
, “
The Set-Down and Set-Up of Directionally Spread and Crossing Surface Gravity Wave Groups
,”
J. Fluid Mech.
,
835
, pp. 131–169.
21.
Donelan
,
M. A.
,
Hamilton
,
J.
, and
Hui
,
W.
,
1985
, “
Directional Spectra of Wind-Generated Ocean Waves
,”
Philos. Trans. R. Soc. London, Ser. A
,
315
(
1534
), pp.
509
562
.
22.
Tucker
,
M. J.
, and
Pitt
,
E. G.
,
2001
,
Waves in Ocean Engineering
, Vol.
5
, Elsevier, Amsterdam, The Netherlands.
23.
Fitzgerald
,
C. J.
,
Taylor
,
P. H.
,
Eatock Taylor
,
R.
,
Grice
,
J.
, and
Zang
,
J.
,
2014
, “
Phase Manipulation and the Harmonic Components of Ringing Forces on a Surface-Piercing Column
,”
Proc. R. Soc. A.
,
470
(2168), p. 20130847.
24.
Longuet-Higgins
,
M. S.
, and
Stewart
,
R. W.
,
1960
, “
Changes in the Form of Short Gravity Waves on Long Waves and Tidal Currents
,”
J. Fluid Mech.
,
8
(
4
), pp.
565
583
.
25.
Barrick
,
D. E.
, and
Weber
,
B. L.
,
1977
, “
On the Nonlinear Theory for Gravity Waves on the Ocean's Surface—Part II: Interpretation and Applications
,”
J. Phys. Oceanogr.
,
7
(
1
), pp.
11
21
.
26.
Jonathan
,
P.
, and
Taylor
,
P. H.
,
1997
, “
On Irregular, Nonlinear Waves in a Spread Sea
,”
ASME J. Offshore Mech. Arct. Eng.
,
119
(
1
), pp.
37
41
.
27.
Taylor
,
P. H.
, and
Williams
,
B. A.
,
2002
, “
Wave Statistics for Intermediate Depth Water: New Waves and Symmetry
,”
ASME
Paper No. OMAE2002-28554.
28.
Whittaker
,
C. N.
,
Raby
,
A. C.
,
Fitzgerald
,
C. J.
, and
Taylor
,
P. H.
,
2016
, “
The Average Shape of Large Waves in the Coastal Zone
,”
Coastal Eng.
,
114
, pp.
253
264
.
You do not currently have access to this content.