# Highly nonlinear wind waves in Currituck Sound: dense breather turbulence in random ocean waves

- 212 Downloads
- 3 Citations

**Part of the following topical collections:**

## Abstract

We analyze surface wave data taken in Currituck Sound, North Carolina, during a storm on 4 February 2002. Our focus is on the application of *nonlinear Fourier analysis* (NLFA) methods (Osborne 2010) to analyze the data set: The approach spectrally decomposes a nonlinear wave field into *sine waves*, *Stokes waves*, and *phase-locked Stokes waves* otherwise known as *breather trains*. Breathers are nonlinear beats, or packets which “breathe” up and down smoothly over *cycle times* of minutes to hours. The maximum amplitudes of the packets during the cycle have a largest central wave whose properties are often associated with the study of “rogue waves.” The mathematical physics of the nonlinear Schrödinger (NLS) equation is assumed and the methods of algebraic geometry are applied to give the *nonlinear spectral representation*. The distinguishing characteristic of the NLFA method is its ability to spectrally decompose a time series into its *nonlinear coherent structures* (Stokes waves and breathers) rather than just sine waves. This is done by the implementation of *multidimensional, quasi-periodic Fourier series*, rather than ordinary Fourier series. To determine preliminary estimates of nonlinearity, we use the significant wave height *H*_{s}, the peak period *T*_{p}, and the length of the time series *T*. The time series analyzed here have 8192 points and *T* =1677.72 s = 27.96 min. Near the peak of the storm, we find *H*_{s} ≈ 0.55 m, *T*_{p} ≈ 2.4 s so that for the wave steepness of a near Gaussian process, \({S} = \left (\pi ^{5/2}/g\right )H_{s}/{T}_{p}^{2}\), we find *S* ≈ 0.17, quite high for ocean waves. Likewise, we estimate the Benjamin-Feir (BF) parameter for a near Gaussian process, \({I_{BF}} = \left (\pi ^{5/2}/g \right ) H_{s} T/{T}_{p}^{3}\), and we find *I*_{BF} ≈ 119. Since the BF parameter describes the nonlinear behavior of the *modulational instability*, leading to the formation of breather packets in a measured wave train, we find the *I*_{BF} for these storm waves to be a surprisingly high number. This is because *I*_{BF}, as derived here, roughly estimates the number of breather trains in a near Gaussian time series. The BF parameter suggests that there are roughly 119 breather trains in a time series of length 28 min near the peak of the storm, meaning that we would have average breather packets of about 14 s each with about 5-6 waves in each packet. Can these surprising results, estimated from simple parameters, be true from the point of view of the complex nonlinear wave dynamics of the BF instability and the NLS equation? We analyze the data set with the NLFA to verify, from a *nonlinear spectral point of view*, the presence of large numbers of breather trains and we determine many of their properties, including the *rise time* for the breathers to grow to their maximum amplitudes from a quiescent initial state. Energetically, about 95% of the NLFA components are found to consist of breather trains; the remaining small amplitude components are sine and Stokes waves. The presence of a large number of densely packed breather trains suggests an interpretation of the data in terms of *breather turbulence*, highly nonlinear *integrable turbulence* theoretically predicted for the NLS equation, providing an interesting paradigm for the nonlinear wave motion, in contrast to the random phase Gaussian approximation often considered in the analysis of data.

## Keywords

Extreme ocean waves Nonlinear waves Stokes waves Breather packets Solitons Nonlinear stochastic processes Nonlinear Schrödinger equation Riemann theta functions## Notes

### Funding information

This work was supported in part by Dr. Tom Drake under ONR Contract Number N00014-16-C-3001. Andrea Costa was supported through IBS-R028-D1.

## References

- Ablowitz MJ, Segur H (1981) Solitons and the inverse scattering transform. SIAM, PhiladelphiaCrossRefGoogle Scholar
- Akhmediev N, Elconskii VM, Kulagin NE (1987) Exact first order solutions of the nonlinear Schrödinger equation. Theor Math Phys 72:809CrossRefGoogle Scholar
- Ardag D, Resio DT (2017) Inconsistencies in spectral evolution produced by operational models due to inaccurate estimates of nonlinear interactions. Accepted by Journal of Physical OceanographyGoogle Scholar
- Babanin AV, Young IR, Banner ML (2001) Breaking probabilities for dominant surface waves on water of finite constant depth. J Geophys Res 106(11):659–11676Google Scholar
- Badulin SI, Pushkarev AN, Resio DT, Zakharov VE (2005) Self-similarity of wind driven seas. Nonlinear Process Geophys 12(6):891–945CrossRefGoogle Scholar
- Badulin S, Babanin A, Zakharov V, Resio DT (2007) Weakly turbulent laws of wind-wave growth. Journal of Fluid Mechanics - J. Fluid Mech. 591:339–378Google Scholar
- Belokolos ED, Bobenko AI, Enol’skii BZ, Its AR, Matveev VB (1994) Algebro-geometric approach to nonlinear integrable equations. Springer, BerlinGoogle Scholar
- Benjamin TB, Feir JE (1967) The disintegration of wave trains on deep water. Part 1 J Fluid Mech 27:417430Google Scholar
- Bitner-Gregersen EM, Gramstad O (2015) ROGUE WAVES: impact on ships and offshore structures. DNV GL Strategic Research and Innovation Position Paper, 0502015Google Scholar
- Costa A, Osborne AR, Resio DT, Alessio S, Chrivì E, Saggese E, Bellomo K, Long C (2014) Soliton turbulence in shallow water ocean surface waves. PRL 113:108501CrossRefGoogle Scholar
- Davis RE, Regier LA (1977) Methods for estimating directional wave spectra from multi-element arrays. J Mar Res 35:453–477Google Scholar
- Dean RG (1990) Freak waves: a possible explanation. In: Dean R G (ed) A Torum and OT Gudmestad. Kluwer, Dordrecht, pp 609–612Google Scholar
- Dean RG, Dalrymple RA (1991) Water wave mechanics for engineers and scientists. World Scientific, SingaporeCrossRefGoogle Scholar
- El GA, Kamchatnov AM (2005) Kinetic equation for dense soliton gas. Phys Rev Lett 95:204101CrossRefGoogle Scholar
- Goda Y (2010) Random seas and design of maritime structures. World Scientific, SingaporeCrossRefGoogle Scholar
- Hasimoto H, Ono H (1972) Nonlinear modulation of gravity waves. J Phys Soc Jpn 33:805–811CrossRefGoogle Scholar
- Hasselmann K (1962) On the non-linear energy transfer in a gravity wave spectrum, Part 1. General theory. J Fluid Mech 12:481500CrossRefGoogle Scholar
- Hasselmann K (1963a) On the non-linear transfer in a gravity wave spectrum, Part 2, Conservation theory, wave-particle correspondence, irreversibility. J Fluid Mech 15:273281Google Scholar
- Hasselmann K (1963b) On the non-linear transfer in a gravity wave spectrum, Part 3. Evaluation of energy flux and sea-swell interactions for a Neumann spectrum. J Fluid Mech 15:385398Google Scholar
- Holthuijsen LH (2007) Waves in oceanic and coastal waters. Cambridge University Press, Cambridge, pp ISBN0-521-86028-8CrossRefGoogle Scholar
- Janssen PAEM (2004) The interaction of ocean waves and wind. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Johnson RS (1997) A modern introduction to the mathematical theory of water waves. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Kharif CE, Pelinovsky E, Slunyaev AS (2009) Rogue waves in the ocean. Springer, BerlinGoogle Scholar
- Kinsman B (1965) Wind waves: their generation and propagation on the ocean surface. Dover, New YorkGoogle Scholar
- Komen GJ, Cavaleri L, Donelan M, Hasselmann K, Hasselmann S, Janssen PAEM (1994) Dynamics and modelling of ocean waves. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Korotkevich AO, Pushkarev AN, Resio DT, Zakharov VE (2007) Numerical verification of the Hasselmann equation. In: Kundu A (ed) Tsunami and nonlinear waves, 135–172. Springer, xi,316, pp ISBN: 978-3-540-71255-8; physics/0702034Google Scholar
- Kotljarov VP, Its AR (1976) Dopov. Akad. Nauk. Ukr. RSR. A 11, 9650968 (in Ukranian)Google Scholar
- Lamb H (1916) Hydrodynamics. Cambridge University Press, CambridgeGoogle Scholar
- Long CE, Resio DT (2004) Directional wave observations. In: Currituck Sound, North Carolina, conference: proceedings, 8th Int. workshop of wave hindcasting and forecasting, HawaiiGoogle Scholar
- Long CE, Resio DT (2007) Wind wave spectral observations in Currituck Sound, North Carolina. J Geophys Res 112:1–21, C05001. https://doi.org/10.1029/2006JC003835 CrossRefGoogle Scholar
- Longuet-Higgins MS (1952) On the statistical distribution of the heights of sea waves. J Marine Res 11 (3):245–266Google Scholar
- Longuet-Higgins MS, Stewart RW (1960) Changes in the form of short gravity waves on long waves and tidal currents. J Fluid Mech 8:565583Google Scholar
- Ma YC (1979) The perturbed plane-wave solutions of the cubic Schrödinger equation. Stud Appl Math 60:43CrossRefGoogle Scholar
- Mei CC (1983) The applied dynamics of ocean surface waves. Wiley, New YorkGoogle Scholar
- Osborne AR (1982) The simulation and measurement of random ocean wave statistics. In: Osborne AR, Malanotte-Rizzoli P (eds) Topics in ocean physics. North- Holland, AmsterdamGoogle Scholar
- Osborne AR (1993) The behavior of solitons in random-function solutions of the periodic Korteweg-deVries equation. Phys Rev Lett 71(19):31153118CrossRefGoogle Scholar
- Osborne AR (2010) Nonlinear ocean waves and the inverse scattering transform. Academic Press, BostonGoogle Scholar
- Osborne AR (2017) (2018) Nonlinear Fourier methods for ocean waves. Procedia IUTAM, IUTAM Symposium Wind Waves, 48, LondonGoogle Scholar
- Pelinovsky E, Kharif C (2008) Extreme ocean waves. Springer, BerlinCrossRefGoogle Scholar
- Peregrine DH (1983) Water waves, nonlinear Schrödinger equations and their solutions. J Austral Math Soc Ser B 25:1643CrossRefGoogle Scholar
- Pierson WJ Jr, Moskowitz LA (1962) Proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigorodskii. J Geophys Res 69:5181–5190CrossRefGoogle Scholar
- Pushkarev A, Resio DT, Zakharov VE (2003) Weak turbulent approach to the wind-generated gravity sea waves. Physica D 184(1-4):29–63CrossRefGoogle Scholar
- Pushkarev A, Resio DT, Zakharov VE (2004) Second generation diffusion model of interacting gravity waves on the surface of deep fluid. Nonlin Process Geophys 11(3):329–342CrossRefGoogle Scholar
- Resio DT, Long CE, Vincent CL (2004) Equilibrium-range constant in wind-generated wave spectra. J Geophys Res 109:C01018CrossRefGoogle Scholar
- Resio DT, Long CE, Perrie W (2011) The role of nonlinear momentum fluxes on the evolution of directional wind-wave spectra. J Phys Oceanogr 41:781–801. https://doi.org/10.1175/2010JPO4545.1 CrossRefGoogle Scholar
- Resio DT, Vincent CL, Ardag D (2016) Characteristics of directional wave spectra and implications for detailed-balance wave modeling. Ocean Model 103:38–52CrossRefGoogle Scholar
- Tracy ER, Chen HH (1988) Nonlinear self-modulation: an exactly solvable model. Phys Rev A 37:815839CrossRefGoogle Scholar
- Whitham GB (1974) Linear and nonlinear waves. Wiley, New YorkGoogle Scholar
- Yuen HC, Lake BM (1982) Nonlinear dynamics of deep-water gravity waves. Adv Appl Mech 22:67229Google Scholar
- Young IR (1999) Wind generated ocean waves. Elsevier, OxfordGoogle Scholar
- Zakharov VE (1968) Stability of periodic waves of finite amplitude on the surface of a deep fluid. J Appl Mech Tech Phys USSR 2:190Google Scholar
- Zakharov VE, Filonenko NN (1967) Energy spectrum for stochastic oscillations of the surface of a liquid. JETP 11:10Google Scholar
- Zakharov VE (1999) Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid. Eur J Mech B 18(3):327–344CrossRefGoogle Scholar
- Zakharov VE (2009) Turbulence in integrable systems. Stud Appl Math 122:219–234CrossRefGoogle Scholar
- Zakharov VE, Korotkevich AO, Pushkarev AN, Resio DT (2007) Coexistence of weak and strong wave turbulence in a swell propagation. Phys Rev Lett 99:164501CrossRefGoogle Scholar
- Zakharov V, Resio DT, Pushkarev A (2017) Balanced source terms for wave generation within the Hasselmann equation. Nonlin Processes Geophys 24:581–597CrossRefGoogle Scholar