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August  2020, 13(4): 703-737. doi: 10.3934/krm.2020024

Strong solutions for the Alber equation and stability of unidirectional wave spectra

1. 

Department of Mathematics, University of Dundee, Dundee DD1 4HN, UK

2. 

School of Naval Architecture and Marine Engineering, National Technical University of Athens, 9 Heroon Polytechniou str., Zographos 157 73, Athens, Greece

3. 

School of Mathematical & Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK

4. 

Department of Mechanical Engineering, Massachusetts Institute of Technology, Room 5-318, 77 Massachusetts Av., Cambridge, MA 02139-4307, USA

* Corresponding author: Agissilaos G. Athanassoulis

Received  June 2019 Revised  January 2020 Published  May 2020

The Alber equation is a moment equation for the nonlinear Schrödinger equation, formally used in ocean engineering to investigate the stability of stationary and homogeneous sea states in terms of their power spectra. In this work we present the first well-posedness theory for the Alber equation with the help of an appropriate equivalent reformulation. Moreover, we show linear Landau damping in the sense that, under a stability condition on the homogeneous background, any inhomogeneities disperse and decay in time. The proof exploits novel $ L^2 $ space-time estimates to control the inhomogeneity and our result applies to any regular initial data (without a mean-zero restriction). Finally, the sufficient condition for stability is resolved, and the physical implications for ocean waves are discussed. Using a standard reference dataset (the "North Atlantic Scatter Diagram") it is found that the vast majority of sea states are stable, but modulationally unstable sea states do appear, with likelihood $ O(1/1000); $ these would be the prime breeding ground for rogue waves.

Citation: Agissilaos G. Athanassoulis, Gerassimos A. Athanassoulis, Mariya Ptashnyk, Themistoklis Sapsis. Strong solutions for the Alber equation and stability of unidirectional wave spectra. Kinetic and Related Models, 2020, 13 (4) : 703-737. doi: 10.3934/krm.2020024
References:
[1]

I. E. Alber, The effects of randomness on the stability of two-dimensional surface wavetrains, Proc. Roy. Soc. London Ser. A, 363 (1978), 525-546.  doi: 10.1098/rspa.1978.0181.

[2]

D. Andrade, R. Stuhlmeier and M. Stiassnie, On the generalized kinetic equation for surface gravity waves, blow-up and its restraint, Fluids, 4 (2018), 2 pp. doi: 10.3390/fluids4010002.

[3]

A. G. Athanassoulis, Exact equations for smoothed Wigner transforms and homogenization of wave propagation, Appl. Comput. Harmon. Anal., 24 (2008), 378-392.  doi: 10.1016/j.acha.2007.06.006.

[4]

A. G. AthanassoulisG. A. Athanassoulis and T. Sapsis, Localized instabilities of the Wigner equation as a model for the emergence of rogue waves, J. Ocean Eng. Mar. Energy, 3 (2017), 353-372.  doi: 10.1007/s40722-017-0095-5.

[5]

A. G. AthanassoulisN. J. Mauser and T. Paul, Coarse-scale representations and smoothed Wigner transforms, J. Math. Pures Appl., 91 (2009), 296-338.  doi: 10.1016/j.matpur.2009.01.001.

[6]

J. BedrossianN. Masmoudi and C. Mouhot, Landau damping in finite regularity for unconfined systems with screened interactions, Comm. Pure Appl. Math., 71 (2018), 537-576.  doi: 10.1002/cpa.21730.

[7]

T. B. Benjamin and J. E. Feir, The disintegration of wave trains on deep water Part 1. Theory, J. Fluid Mech., 27 (1967), 417-430.  doi: 10.1017/S002211206700045X.

[8]

E. M. Bitner-Gregersen and O. Gramstad, DNV GL Strategic Reserach & Innovation position paper 05-2015: rogue waves: Impact on ships and offshore structures, 2015, online article: https://issuu.com/dnvgl/docs/rogue_waves_final/10.

[9]

T. ChenY. Hong and N. Pavlović, Global well-posedness of the NLS system for infinitely many fermions, Arch. Rational Mech. Anal., 224 (2017), 91-123.  doi: 10.1007/s00205-016-1068-x.

[10]

W. Cousins and T. P. Sapsis, Reduced-order precursors of rare events in unidirectional nonlinear water waves, J. Fluid Mech., 790 (2016), 368-388.  doi: 10.1017/jfm.2016.13.

[11]

G. DematteisT. Grafke and E. Vanden-Eijnden, Rogue waves and large deviations in deep sea, Proc. Natl. Acad. Sci. U. S. A., 115 (2018), 855-860.  doi: 10.1073/pnas.1710670115.

[12]

R. Dubertrand and S. Müller, Spectral statistics of chaotic many-body systems, New J. Phys., 18 (2016), 033009. doi: 10.1088/1367-2630/18/3/033009.

[13]

O. Gramstad, Modulational instability in JONSWAP sea states using the Alber equation, in Proceedings of the ASME 2017 36th International Conference on Ocean, Offshore and Arctic Engineering, Vol. 7B, Trondheim, Norway, 2017, 9 pp. doi: 10.1115/OMAE2017-61671.

[14]

O. GramstadE. Bitner-GregersenK. Trulsen and J. C. Nieto Borge, Modulational instability and rogue waves in crossing sea states, J. Phys. Oceanogr., 48 (2018), 1317-1331.  doi: 10.1175/JPO-D-18-0006.1.

[15]

J. HanH. LiuN. Huang and Z. Wang, Stochastic resonance based on modulation instability in spatiotemporal chaos, Opt. Express, 25 (2017), 8306-8314.  doi: 10.1364/OE.25.008306.

[16]

K. Hasselmann, T. P. Barnett, E. Bouws, H. Carlson, D. E. Cartwright, K. Enke, J. A. Ewing, H. Gienapp, D. E. Hasselmann, P. Kruseman and A. Meerburg, Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP), Deut. Hydrogr. Z., 8 (1973).

[17]

P. A. E. M. Janssen, Nonlinear four-wave interactions and freak waves, J. Phys. Oceanogr., 33 (2003), 863-884. 

[18] G. J. KomenL. CavaleriM. DonelanK. HasselmannS. Hasselmann and P. A. E. M. Janssen, Dynamics and Modelling of Ocean Waves, Cambridge University Press, Cambridge, 1994.  doi: 10.1017/CBO9780511628955.
[19]

M. Lewin and J. Sabin, The Hartree equation for infinitely many particles I. Well-posedness theory, Comm. Math. Phys., 334 (2015), 117-170.  doi: 10.1007/s00220-014-2098-6.

[20]

M. Lewin and J. Sabin, The Hartree equation for infinitely many particles, Ⅱ: Dispersion and scattering in 2D, Anal. PDE, 7 (2014), 1339-1363.  doi: 10.2140/apde.2014.7.1339.

[21]

C. Mouhot and C. Villani, On Landau damping, Acta Math., 207 (2011), 29-201.  doi: 10.1007/s11511-011-0068-9.

[22] M. K. Ochi, Ocean Waves: The Stochastic Approach, Cambridge University Press, 1998.  doi: 10.1017/CBO9780511529559.
[23]

M. Onorato, A. Osborne, R. Fedele and M. Serio, Landau damping and coherent structures in narrow-banded 1 + 1 deep water gravity waves, Phys. Rev. E, 67 (2003), 046305. doi: 10.1103/PhysRevE.67.046305.

[24]

M. Onorato, A. R. Osborne and M. Serio, Modulational instability in crossing sea states: A possible mechanism for the formation of freak waves, Phys. Rev. Lett., 96 (2006), 014503. doi: 10.1103/PhysRevLett.96.014503.

[25]

M. OnoratoS. ResidoriU. BortolozzoA. Montina and F. T. Arecchi, Rogue waves and their generating mechanisms in different physical contexts, Phys. Rep., 528 (2013), 47-89.  doi: 10.1016/j.physrep.2013.03.001.

[26]

O. Penrose, Electrostatic instabilities of a uniform non-Maxwellian plasma, Phys. Fluids, 3 (1960), 258-265.  doi: 10.1063/1.1706024.

[27]

I. S. Reed, On a moment theorem for complex gaussian processes, IRE Trans. Inf. Theory, 8 (1962), 194-195.  doi: 10.1109/TIT.1962.1057719.

[28]

A. Ribal, On the Alber equation for random water waves, Ph.D thesis, Swinburne University of Technology, Melbourne, Australia, 2013.

[29]

A. RibalA. V. BabaninI. YoungA. Toffoli and M. Stiassnie, Recurrent solutions of the Alber equation initialized by Joint North Sea Wave Project spectra, J. Fluid Mech., 719 (2013), 314-344.  doi: 10.1017/jfm.2013.7.

[30]

P. K. ShuklaM. Marklund and L. Stenflo, Modulational instability of nonlinearly interacting incoherent sea states, JETP Lett., 84 (2006), 645-649.  doi: 10.1134/S0021364006240039.

[31]

J. N. SteerM. L. McallisterA. G. L. Borthwick and T. S. van der Bremer, Experimental observation of modulational instability in crossing surface gravity wavetrains, Fluids, 4 (2019), 1-15.  doi: 10.3390/fluids4020105.

[32]

M. StiassnieA. Regev and Y. Agnon, Recurrent solutions of Alber's equation for random water-wave fields, J. Fluid Mech., 598 (2008), 245-266.  doi: 10.1017/S0022112007009998.

[33]

R. Stuhlmeier and M. Stiassnie, Evolution of statistically inhomogeneous degenerate water wave quartets, Philos. Trans. Roy. Soc. A, 376 (2018), 20170101. doi: 10.1098/rsta.2017.0101.

[34]

A.-S. de Suzzoni, An equation on random variables and systems of fermions, preprint, 2016, arXiv: 1507.06180.

[35]

P. Wahlberg, The random Wigner distribution of Gaussian stochastic processes with covariance in S0($\mathbb{R}$2d), J. Funct. Spaces Appl., 3 (2005), 163-181.  doi: 10.1155/2005/252415.

[36]

H. C. Yuen and B. M. Lake, Nonlinear dynamics of deep-water gravity waves, Adv. in Appl. Mech., 22 (1982), 67-229.  doi: 10.1016/S0065-2156(08)70066-8.

[37]

V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968), 190-194.  doi: 10.1007/BF00913182.

[38]

V. E. Zakharov and L. A. Ostrovsky, Modulation instability: The beginning, Phys. D, 238 (2009), 540-548.  doi: 10.1016/j.physd.2008.12.002.

[39]

DNVGL-RP-C205: Environmental conditions and environmental loads, Recommended Practice report, 2017. Available from: https://oilgas.standards.dnvgl.com/download/dnvgl-rp-c205-environmental-conditions-and-environmental-loads.

show all references

References:
[1]

I. E. Alber, The effects of randomness on the stability of two-dimensional surface wavetrains, Proc. Roy. Soc. London Ser. A, 363 (1978), 525-546.  doi: 10.1098/rspa.1978.0181.

[2]

D. Andrade, R. Stuhlmeier and M. Stiassnie, On the generalized kinetic equation for surface gravity waves, blow-up and its restraint, Fluids, 4 (2018), 2 pp. doi: 10.3390/fluids4010002.

[3]

A. G. Athanassoulis, Exact equations for smoothed Wigner transforms and homogenization of wave propagation, Appl. Comput. Harmon. Anal., 24 (2008), 378-392.  doi: 10.1016/j.acha.2007.06.006.

[4]

A. G. AthanassoulisG. A. Athanassoulis and T. Sapsis, Localized instabilities of the Wigner equation as a model for the emergence of rogue waves, J. Ocean Eng. Mar. Energy, 3 (2017), 353-372.  doi: 10.1007/s40722-017-0095-5.

[5]

A. G. AthanassoulisN. J. Mauser and T. Paul, Coarse-scale representations and smoothed Wigner transforms, J. Math. Pures Appl., 91 (2009), 296-338.  doi: 10.1016/j.matpur.2009.01.001.

[6]

J. BedrossianN. Masmoudi and C. Mouhot, Landau damping in finite regularity for unconfined systems with screened interactions, Comm. Pure Appl. Math., 71 (2018), 537-576.  doi: 10.1002/cpa.21730.

[7]

T. B. Benjamin and J. E. Feir, The disintegration of wave trains on deep water Part 1. Theory, J. Fluid Mech., 27 (1967), 417-430.  doi: 10.1017/S002211206700045X.

[8]

E. M. Bitner-Gregersen and O. Gramstad, DNV GL Strategic Reserach & Innovation position paper 05-2015: rogue waves: Impact on ships and offshore structures, 2015, online article: https://issuu.com/dnvgl/docs/rogue_waves_final/10.

[9]

T. ChenY. Hong and N. Pavlović, Global well-posedness of the NLS system for infinitely many fermions, Arch. Rational Mech. Anal., 224 (2017), 91-123.  doi: 10.1007/s00205-016-1068-x.

[10]

W. Cousins and T. P. Sapsis, Reduced-order precursors of rare events in unidirectional nonlinear water waves, J. Fluid Mech., 790 (2016), 368-388.  doi: 10.1017/jfm.2016.13.

[11]

G. DematteisT. Grafke and E. Vanden-Eijnden, Rogue waves and large deviations in deep sea, Proc. Natl. Acad. Sci. U. S. A., 115 (2018), 855-860.  doi: 10.1073/pnas.1710670115.

[12]

R. Dubertrand and S. Müller, Spectral statistics of chaotic many-body systems, New J. Phys., 18 (2016), 033009. doi: 10.1088/1367-2630/18/3/033009.

[13]

O. Gramstad, Modulational instability in JONSWAP sea states using the Alber equation, in Proceedings of the ASME 2017 36th International Conference on Ocean, Offshore and Arctic Engineering, Vol. 7B, Trondheim, Norway, 2017, 9 pp. doi: 10.1115/OMAE2017-61671.

[14]

O. GramstadE. Bitner-GregersenK. Trulsen and J. C. Nieto Borge, Modulational instability and rogue waves in crossing sea states, J. Phys. Oceanogr., 48 (2018), 1317-1331.  doi: 10.1175/JPO-D-18-0006.1.

[15]

J. HanH. LiuN. Huang and Z. Wang, Stochastic resonance based on modulation instability in spatiotemporal chaos, Opt. Express, 25 (2017), 8306-8314.  doi: 10.1364/OE.25.008306.

[16]

K. Hasselmann, T. P. Barnett, E. Bouws, H. Carlson, D. E. Cartwright, K. Enke, J. A. Ewing, H. Gienapp, D. E. Hasselmann, P. Kruseman and A. Meerburg, Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP), Deut. Hydrogr. Z., 8 (1973).

[17]

P. A. E. M. Janssen, Nonlinear four-wave interactions and freak waves, J. Phys. Oceanogr., 33 (2003), 863-884. 

[18] G. J. KomenL. CavaleriM. DonelanK. HasselmannS. Hasselmann and P. A. E. M. Janssen, Dynamics and Modelling of Ocean Waves, Cambridge University Press, Cambridge, 1994.  doi: 10.1017/CBO9780511628955.
[19]

M. Lewin and J. Sabin, The Hartree equation for infinitely many particles I. Well-posedness theory, Comm. Math. Phys., 334 (2015), 117-170.  doi: 10.1007/s00220-014-2098-6.

[20]

M. Lewin and J. Sabin, The Hartree equation for infinitely many particles, Ⅱ: Dispersion and scattering in 2D, Anal. PDE, 7 (2014), 1339-1363.  doi: 10.2140/apde.2014.7.1339.

[21]

C. Mouhot and C. Villani, On Landau damping, Acta Math., 207 (2011), 29-201.  doi: 10.1007/s11511-011-0068-9.

[22] M. K. Ochi, Ocean Waves: The Stochastic Approach, Cambridge University Press, 1998.  doi: 10.1017/CBO9780511529559.
[23]

M. Onorato, A. Osborne, R. Fedele and M. Serio, Landau damping and coherent structures in narrow-banded 1 + 1 deep water gravity waves, Phys. Rev. E, 67 (2003), 046305. doi: 10.1103/PhysRevE.67.046305.

[24]

M. Onorato, A. R. Osborne and M. Serio, Modulational instability in crossing sea states: A possible mechanism for the formation of freak waves, Phys. Rev. Lett., 96 (2006), 014503. doi: 10.1103/PhysRevLett.96.014503.

[25]

M. OnoratoS. ResidoriU. BortolozzoA. Montina and F. T. Arecchi, Rogue waves and their generating mechanisms in different physical contexts, Phys. Rep., 528 (2013), 47-89.  doi: 10.1016/j.physrep.2013.03.001.

[26]

O. Penrose, Electrostatic instabilities of a uniform non-Maxwellian plasma, Phys. Fluids, 3 (1960), 258-265.  doi: 10.1063/1.1706024.

[27]

I. S. Reed, On a moment theorem for complex gaussian processes, IRE Trans. Inf. Theory, 8 (1962), 194-195.  doi: 10.1109/TIT.1962.1057719.

[28]

A. Ribal, On the Alber equation for random water waves, Ph.D thesis, Swinburne University of Technology, Melbourne, Australia, 2013.

[29]

A. RibalA. V. BabaninI. YoungA. Toffoli and M. Stiassnie, Recurrent solutions of the Alber equation initialized by Joint North Sea Wave Project spectra, J. Fluid Mech., 719 (2013), 314-344.  doi: 10.1017/jfm.2013.7.

[30]

P. K. ShuklaM. Marklund and L. Stenflo, Modulational instability of nonlinearly interacting incoherent sea states, JETP Lett., 84 (2006), 645-649.  doi: 10.1134/S0021364006240039.

[31]

J. N. SteerM. L. McallisterA. G. L. Borthwick and T. S. van der Bremer, Experimental observation of modulational instability in crossing surface gravity wavetrains, Fluids, 4 (2019), 1-15.  doi: 10.3390/fluids4020105.

[32]

M. StiassnieA. Regev and Y. Agnon, Recurrent solutions of Alber's equation for random water-wave fields, J. Fluid Mech., 598 (2008), 245-266.  doi: 10.1017/S0022112007009998.

[33]

R. Stuhlmeier and M. Stiassnie, Evolution of statistically inhomogeneous degenerate water wave quartets, Philos. Trans. Roy. Soc. A, 376 (2018), 20170101. doi: 10.1098/rsta.2017.0101.

[34]

A.-S. de Suzzoni, An equation on random variables and systems of fermions, preprint, 2016, arXiv: 1507.06180.

[35]

P. Wahlberg, The random Wigner distribution of Gaussian stochastic processes with covariance in S0($\mathbb{R}$2d), J. Funct. Spaces Appl., 3 (2005), 163-181.  doi: 10.1155/2005/252415.

[36]

H. C. Yuen and B. M. Lake, Nonlinear dynamics of deep-water gravity waves, Adv. in Appl. Mech., 22 (1982), 67-229.  doi: 10.1016/S0065-2156(08)70066-8.

[37]

V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968), 190-194.  doi: 10.1007/BF00913182.

[38]

V. E. Zakharov and L. A. Ostrovsky, Modulation instability: The beginning, Phys. D, 238 (2009), 540-548.  doi: 10.1016/j.physd.2008.12.002.

[39]

DNVGL-RP-C205: Environmental conditions and environmental loads, Recommended Practice report, 2017. Available from: https://oilgas.standards.dnvgl.com/download/dnvgl-rp-c205-environmental-conditions-and-environmental-loads.

Figure 3.  Numerical investigation of the stability condition for a stable JONSWAP spectrum, cf. Section 8 for more details. We are using a target of $ 1/4\pi $ as in equation 54. Left: Plots of the curve $ \Gamma_X $ on the complex plane for different values of $ X. $ Since $ 1/4\pi $ is always outside the $ \Gamma_X, $ this spectrum is stable. Right: The span of the real parts of $ \Gamma_X $ for different values of $ X $
Figure 4.  Numerical investigation of the stability condition for an unstable JONSWAP spectrum. Left: Plots of the curve $ \Gamma_X $ on the complex plane for different values of $ X. $ Since $ 1/4\pi $ is contained in some curves $ \Gamma_X, $ the spectrum is unstable. Right: The span of the real parts of $ \Gamma_X $ for different values of $ X. $ In this case it highlights clearly the bandwidth of unstable wavenumbers $ X $
Figure 1.  The domains of integration for the integrals $ I_j, $ $ j = 1,\dots,6 $
Figure 2.  Some common profiles of JONSWAP spectra
Figure 5.  A number of points on the $ (\gamma,\alpha) $ plane are tested for stability of the corresponding JONSWAP spectrum, cf. equation 52. $ \alpha $ controls the power of the sea state (larger $ \alpha $ means larger significant wave height) and $ \gamma $ controls the effective bandwidth (larger $ \gamma $ means more narrowly peaked spectrum). The carrier wavenumber $ k_0 $ can easily be seen not to affect the (in)stability of the spectrum. $ (\gamma,\alpha) $ points found to be stable are marked with a full square, while points found to be unstable are marked with an empty square. For reference the proposed separatrices of [29] and [13] are shown (they are of the form $ \alpha\cdot\gamma/\beta = C, $ where $ \beta $ is the mean wave steepness and $ C = 0.77 $ [13] or $ C = 0.974 $ [29]). More details can be found in Section 8. Top: Linear scaling in both axes. Bottom: Log scaling in the $ \alpha $ (vertical) axis
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