Hostname: page-component-5cf477f64f-r2nwp Total loading time: 0 Render date: 2025-04-08T11:04:14.270Z Has data issue: false hasContentIssue false
  • English
  • Français

Can bottom friction suppress ‘freak wave’ formation?

Published online by Cambridge University Press:  14 May 2008

VIACHESLAV V. VORONOVICH ,
VICTOR I. SHRIRA  and
GARETH THOMAS
VIACHESLAV V. VORONOVICH
Affiliation:
School of Mathematics, University College Cork, Cork, Ireland
VICTOR I. SHRIRA
Affiliation:
Department of Mathematics, Keele University, Keele, ST5 5BGUK
GARETH THOMAS
Affiliation:
School of Mathematics, University College Cork, Cork, Ireland
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper examines the effect of the bottom stress on the weakly nonlinear evolution of a narrow-band wave field, as a potential mechanism of suppression of ‘freak’ wave formation in water of moderate depth. Relying upon established experimental studies the bottom stress is modelled by the quadratic drag law with an amplitude/bottom roughness-dependent drag coefficient. The asymptotic analysis yields Davey–Stewartson-type equations with an added nonlinear complex friction term in the envelope equation. The friction leads to a power-law decay of the spatially uniform wave amplitude. It also affects the modulational (Benjamin–Feir) instability, e.g. alters the growth rates of sideband perturbations and the boundaries of the linearized stability domains in the modulation wavevector space. Moreover, the instability occurs only if the amplitude of the background wave exceeds a certain threshold. Since the friction is nonlinear and increases with wave amplitude, its effect on the formation of nonlinear patterns is more dramatic. Numerical experiments show that even when the friction is small compared to the nonlinear term, it hampers formation of the Akhmediev/Ma-type breathers (believed to be weakly nonlinear ‘prototypes’ of freak waves) at the nonlinear stage of instability. The specific predictions for a particular location depend on the bottom roughness ks in addition to the water depth and wave field characteristics.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 604 , 10 June 2008 , pp. 263 - 296
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Ablowitz, M. J. & Herbst, B. M. 1990 On homoclinic structure and numerically induced chaos for the nonlinear Schrödinger equation. SIAM J Appl. Maths 50, 339–351.CrossRefGoogle Scholar
Ablowitz, M. J. & Ladik, J. F. 1976 A nonlinear difference scheme and inverse scattering. Stud. Appl. Maths 55, 213–229.CrossRefGoogle Scholar
Akhmediev, N. N., Eleonskii, V. M. & Kulagin, N. E. 1987 Exact first-order solutions of the nonlinear Schrödinger equation. Theor. Math. Phys. 72, 809–818.CrossRefGoogle Scholar
Badulin, S. I., Pushkarev, A. N., Resio, D. & Zakharov, V. E. 2005 Self-similarity of wind-driven seas. Nonlin. Proc. Geophys. 12, 891–945.CrossRefGoogle Scholar
Benney, D. J. & Roskes, G. J. 1969 Wave instabilities. Stud. Appl. Maths 48, 377–385.CrossRefGoogle Scholar
Besse, C. 2004 A relaxation scheme for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 42, 934–952.CrossRefGoogle Scholar
Caulliez, G. 2002 Self-similarity of near-breaking short gravity wind waves. Phys. Fluids 14, 8, 2917–2920.CrossRefGoogle Scholar
Colin, T., Dias, F. & Ghidaglia, J. M. 1995 On the rotational effects in the modulations of weakly nonlinear water waves over finite depth. Eur. J. Mech. B/Fluids 14, 775–793.Google Scholar
Davey, A. & Stewartson, K. 1974 On three-dimensional packets of surface waves. Proc. R. Soc. Lond. A 338, 101–110.Google Scholar
Djordjevic, V. D. & Redekopp, L. G. 1977 On two-dimensional packets of capillary-gravity waves. J Fluid Mech. 79, 703–714.CrossRefGoogle Scholar
Dyachenko, A. I. & Zakharov, V. E. 2005 Modulational instability of Stokes waves → freak wave. JETP Lett. 81, 255–259.CrossRefGoogle Scholar
Dysthe, K. B. & Trulsen, K. 1999 Note on breather type solutions of the NLS as models for freak-waves. Phys. Scripta T 82, 48–52.CrossRefGoogle Scholar
Grant, W. D. & Madsen, O. S. 1979 Combined wave and current interaction with a rough bottom. J. Geophys. Res. C 84, 1797–1808.CrossRefGoogle Scholar
Grant, W. D., Williams, A. J. & Glenn, S. M. 1984 Bottom stress estimates and their prediction on the Northern California continental shelf during CODE-1: The importance of wave-current interaction. J. Phys. Oceanogr. 14, 506–526.2.0.CO;2>CrossRefGoogle Scholar
Hasegawa, A. & Tai, K. 1989 Effects of modulational instability on coherent transmission systems. Opt. Lett. 14, 512–514.CrossRefGoogle ScholarPubMed
Haver, S. 2000 On the existence of freak waves. In Rogue Waves 2000 (ed. Olagnon, M. & Athanassulis, G.) IFREMEr, Brest, France. ISBN 2-84433-063-0.Google Scholar
Henderson, K. L., Peregrine, D. H. & Dold, J. W. 1999 Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equation. Wave Motion 29, 341–361.CrossRefGoogle Scholar
Janssen, P. A. E. M. 2003 Nonlinear four-wave interactions and freak waves. J. Phys. Oceanogr. 33, 863–884.2.0.CO;2>CrossRefGoogle Scholar
Janssen, P. A. E. M. 2004 The Interaction of Ocean Waves and Wind. Cambridge University Press.CrossRefGoogle Scholar
Jensen, B. L., Sumer, B. M. & Fredsøe, J. 1989 Turbulent oscillatory boundary laters at high Reynolds numbers. J. Fluid Mech. 206, 265–297.CrossRefGoogle Scholar
Jonsson, I. G. 1980 A new approach to oscillatory rough turbulent boundary layers. Ocean Engng 7, 109–152.CrossRefGoogle Scholar
Kajiura, K. 1964 On the bottom friction in an oscillatory current. Bull. Earthq. Res. Inst. 42, 147–174.Google Scholar
Kajiura, K. 1968 A model of the bottom boundary layer in water waves. Bull. Earthq. Res. Inst. 46, 75–123.Google Scholar
Karlsson, M. 1995 Modulational instability in lossy optical fibers. J. Opt. Soc. Am. B 12, 2071–2077.CrossRefGoogle Scholar
Kawata, T. & Inoue, H. 1978 Inverse scattering method for the nonlinear evolution equations under nonvanishing conditions. J. Phys. Soc. Japan 44, 1722–1729.CrossRefGoogle Scholar
Kharif, C. & Pelinovsky, E. 2003 Physical mechanisms of the rogue wave phenomenon. Eur. J. Fluid. Mech. B/Fluids 22, 603–634.CrossRefGoogle Scholar
Klopman, G. 1994 Vertical structure of the flow due to waves and currents. Progr. Rep. H 840, Part II. Delft Hydraulics.Google Scholar
Komen, G. J., Cavaleri, L., Donelan, M., Hasselmann, K., Hasselmann, S. & Janssen, P. A. E. M. 1994 Dynamics and Modelling of Ocean Waves. Cambridge University Press.CrossRefGoogle Scholar
Kuznetsov, E. A. 1977 On solitons in parametrically unstable plasma. Dokl. USSR 236, 575–577 (in Russian) (English transl. Sov. Phys. Dokl. 22, 9, 507–508).Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Lowe, R. J., Falter, J. L., Bandet, M. D., Pawlak, G., Atkinson, M. J., Monismith, S. G. & Koseff, J. R. 2005 Spectral wave dissipation over a barrier reef. J. Geophys. Res. 110, C04001.Google Scholar
Ma, Ya.-C. 1979 The perturbed plane-wave solutions of the cubic Schrödinger equation. Stud. Appl. Maths 60, 43–58.CrossRefGoogle Scholar
McLaughlin, D. W. & Shatah, J. 1998 Homoclinic orbits for pde's. In Recent Advances in Partial Differential Equations, Venice 1996. Proc. Symp. Appl. Maths, vol. 54, pp. 281–299. Am. Math. Soc., Providence, R.I.CrossRefGoogle Scholar
Mei, C. C. & Hancock, M. J. 2003 Weakly nonlinear surface waves over a random seabed. J. Fluid. Mech. 475, 247–268.CrossRefGoogle Scholar
Mei, C. C. & Li, Y. 2004 Evolution of solitons over a randomly rough seabed. Phys. Rev. E 70, 016302-1–11.Google Scholar
Mei, C. C. & Stiassnie, M. & Yue, D. K.-P. 2005 Theory and Applications of Ocean Surface Waves. World Scientific.Google Scholar
Myrhaug, D. 1989 A rational approach to wave friction coefficients for rough, smooth and transitional turbulent flow. Coastal. Engng 13, 11–21.CrossRefGoogle Scholar
Myrhaug, D., Staattelid, O. H. & Lambrakos, K. F. 1998 Seabed shear stresses under random waves: Predictions vs estimates from field measurements. Ocean Engng 25, 907–916.CrossRefGoogle Scholar
Nielsen, P. 1992 Coastal Bottom Boundary Layers and Sediment Transport. World Scientific.CrossRefGoogle Scholar
Onorato, M., Osborne, A. R., Serio, M. & Bertone, S. 2001 Freak waves in random oceanic sea states. Phys. Rev. Lett. 86, 5831–5834.CrossRefGoogle ScholarPubMed
Osborne, A. R., Onorato, M. & Serio, M. 2000 The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains. Phys. Lett. A 275, 386–393.CrossRefGoogle Scholar
Peregrine, D. H. 1983 Water waves, nonlinear Schrödinger equations and their solutions. J Austal. Math. Soc. B 25, 16–43.CrossRefGoogle Scholar
Sedletsky, Yu. V. 2003 The fourth-order nonlinear Schrödinger equation for the envelope of Stokes waves on the surface of finite-depth fluid. JETP 97, 180–193 (10.1134/1.1600810).CrossRefGoogle Scholar
Segur, H., Henderson, D., Carter, J., Hammack, J., Li, G.-M., Pheiff, D. & Socha, K. 2005 Stabilizing the Benjamin-Feir instability. J. Fluid Mech. 539, 229–271.CrossRefGoogle Scholar
Sleath, J. F. A. 1987 Turbulent oscillatory flow over rough beds. J. Fluid Mech. 182, 369–409.CrossRefGoogle Scholar
Slunyaev, A., Kharif, C., Pelinovsky, E. & Talipova, T. 2002 Nonlinear wave focusing on water of finite depth. Physica D 173, 77–96.Google Scholar
Smith, R. 1976 Giant waves. J Fluid Mech. 77, 693–697.CrossRefGoogle Scholar
Song, J.-B. & Banner, M. L. 2002 On determining the onset and strength of breaking for deep water waves. Part I: Unforced irrotational wave groups. J. Phys. Oceanogr. 32, 2541–2558.CrossRefGoogle Scholar
Soulsby, R. L. 1990 Tidal-current boundary layers. In The Sea (ed. Mehaute, B. Le & Hanes, D. M.). Ocean Engineering Science, vol. 9, pp. 523–566. Wiley-Interscience.Google Scholar
Soulsby, R. L. 1998 Dynamics of Marine Sands. Thomas Telford Ltd.CrossRefGoogle Scholar
Soulsby, R. L., Hamm, L., Klopman, G., Myrhaug, D., Simons, R. R. & Thomas, G. P. 1993 Wave-current interactions within and outside the bottom boundary layer. Coastal Engng 21, 41–69.CrossRefGoogle Scholar
Swart, D. H. 1974 Offshore sediment transport and equilibrium beach profiles. Delft Hydraulics, Publ. 131.Google Scholar
Tanaka, M. 1990 Maximum amplitude of modulated wave train. Wave Motion 12, 559–568.CrossRefGoogle Scholar
Terray, E. A., Donelan, M. A., Agrawal, Y. C., Drennan, W. M., Kahma, K. K., Hwang, P. A. & Kitaigorodskii, S. A. 1996 Estimates of kinetic energy under breaking waves. J. Phys. Oceanogr. 26, 792–807.2.0.CO;2>CrossRefGoogle Scholar
Thais, L., Chapalain, G., Simons, R. R., Klopman, G. & Thomas, G. P. 2001 Estimates of decay rates in the presence of turbulent currents. Appl. Ocean Res. 23, 125–137.CrossRefGoogle Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of deep water. J. Appl. Mech. Tech. Phys. 9, 190–194.CrossRefGoogle Scholar
Zakharov, V. E. & Shabat, A. B. 1971 Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Zh. Eksp. Teor. Fiz. 61, 118–134.Google Scholar