Research Papers: Ocean Engineering

A Note on the Second-Order Contribution to Extreme Waves Generated During Hurricanes

[+] Author and Article Information
Mark L. McAllister

Department of Engineering Science,
University of Oxford,
Oxford OX1 3PJ, UK
e-mail: mark.mcallister@eng.ox.ac.uk

Thomas A. A. Adcock

Department of Engineering Science,
University of Oxford,
Oxford OX1 3PJ, UK
e-mail: thomas.adcock@eng.ox.ac.uk

Paul H. Taylor

Faculty of Engineering and Mathematical
University of Western Australia,
Crawley, WA 6009, Australia
e-mail: paul.taylor@eng.ox.ac.uk

Ton S. van den Bremer

Department of Engineering Science,
University of Oxford,
Oxford OX1 3PJ, UK
e-mail: ton.vandenbremer@eng.ox.ac.uk

Contributed by the Ocean, Offshore, and Arctic Engineering Division of ASME for publication in the JOURNAL OF OFFSHORE MECHANICS AND ARCTIC ENGINEERING. Manuscript received September 26, 2018; final manuscript received January 11, 2019; published online February 21, 2019. Assoc. Editor: Puneet Agarwal.

J. Offshore Mech. Arct. Eng 141(4), 041102 (Feb 21, 2019) (7 pages) Paper No: OMAE-18-1163; doi: 10.1115/1.4042540 History: Received September 26, 2018; Revised January 11, 2019

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.

Copyright © 2019 by ASME
Topics: Waves , Filtration
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Young, I. R. , 2017, “ A Review of Parametric Descriptions of Tropical Cyclone Wind-Wave Generation,” Atmosphere, 8(12), p. 194.
Young, I. R. , 2006, “ Directional Spectra of Hurricane Wind Waves,” J. Geophys. Res. Oceans, 111(C8), epub.
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.
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.
Hu, K. , and Chen, Q. , 2011, “ Directional Spectra of Hurricane-Generated Waves in the Gulf of Mexico,” Geophys. Res. Lett., 38(19), epub.
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.
Lindgren, G. , 1970, “ Some Properties of a Normal Process Near a Local Maximum,” Ann. Math. Stat., 41(6), pp. 1870–1883.
Boccotti, P. , 1983, “ Some New Results on Statistical Properties of Wind Waves,” App. Ocean Res., 5(3), pp. 134–140.
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.
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.
Dalzell, J. F. , 1999, “ A Note on Finite Depth Second-Order Wave–Wave Interactions,” App. Ocean Res., 21(3), pp. 105–111.
Forristall, G. Z. , 2000, “ Wave Crest Distributions: Observations and Second-Order Theory,” J. Phys. Oceanogr., 30(8), pp. 1931–1943.
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.
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
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.
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.
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
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.
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.
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.
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.
Tucker, M. J. , and Pitt, E. G. , 2001, Waves in Ocean Engineering, Vol. 5, Elsevier, Amsterdam, The Netherlands.
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.
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.
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.
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.
Taylor, P. H. , and Williams, B. A. , 2002, “ Wave Statistics for Intermediate Depth Water: New Waves and Symmetry,” ASME Paper No. OMAE2002-28554.
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.


Grahic Jump Location
Fig. 1

Illustration of expected wave-averaged surface elevation under extreme waves generated during tropical storms. Panels ad show examples of directional spreading distributions D(f, θ) measured in each quadrant of a hurricane taken from Y06 (see Table 1 for full details); contours are drawn at 0.9, 0.8, 0.7, 0.6, and 0.5, and vertical lines show the direction local wind (solid) and dominant waves (dashed). Panels eh show the parameterization of spreading distributions above proposed in Sec. 2. Panels il show the linear NewWave free-surface elevation for each of the proposed directional distributions and a JONSWAP spectrum D(f,θ)η̂(f). Panels mp show the predicted wave-averaged surface elevation for each NewWave group: unfiltered (black lines) and low-pass filtered at 0.5fp (red lines, gray in print version). The right-hand side blue axes for rows il and mp illustrate our results at a field scale of 50:1 (a0 = 7.5 m, d =100 m, and f0 = 0.1 Hz), whereas the left-hand side axes correspond to lab scale (see Sec. 4).

Grahic Jump Location
Fig. 2

Second-order difference interaction kernel B (from Ref. [11], see Appendix) shown as a function of normalized frequency f/fp and crossing angle θ1θ2 of interacting wave pairs for fp = 0.7 Hz and d =2.0 m

Grahic Jump Location
Fig. 3

The effects of water depth on formation of setup and corresponding second-order contribution to overall crest height; predictions shown for water depths kd = 0.5, 1, 10 left to right and directional skew Δθ = 0–180 deg in 10 deg increments from black to gray. Panels ac show second-order difference waves; panels df show second-order sum waves; panels gi show total second-order contribution. Calculations are carried out for linear wave groups η(1) based on the parameters in Table 2.

Grahic Jump Location
Fig. 4

Predicted amplitude of second-order contribution to crest height as a function of high-frequency cutoff for a JONSWAP spectrum with the parameters detailed in Table 2 and directional distribution (3) for Δθ = 0, 90, 180 deg

Grahic Jump Location
Fig. 5

Schematic diagram, showing wave tank setup and measurement gauge locations

Grahic Jump Location
Fig. 6

Results of laboratory experiments with frequency-dependent directional skew (2) for different directional skews Δθ = 90, 135, 180, 225 deg, and frequency-independent spreading σθ = 20 deg. Panels ad show input directional distributions D(f, θ) with contours drawn at 0.9, 0.8, 0.7, 0.6, and 0.5. Panels ef show the linear free surface elevation η(1) and panels ij show second-order difference waves, where experiments are shown as black lines and predictions based on inputs to the tank are shown as red lines (gray in print version).



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