Numerical Verification of the Hasselmann equation

  • Alexander O. Korotkevich
  • Andrei N. Pushkarev
  • Don Resio
  • Vladimir E. Zakharov


The purpose of this article is numerical verification of the thory of weak turbulence. We performed numerical simulation of an ensemble of nonlinearly interacting free gravity waves (swell) by two different methods: solution of primordial dynamical equations describing potential flow of the ideal fluid with a free surface and, solution of the kinetic Hasselmann equation, describing the wave ensemble in the framework of the theory of weak turbulence. Comparison of the results demonstrates pretty good applicability of the weak turbulent approach. In both cases we observed effects predicted by this theory: frequency downshift, angular spreading as well as formation of Zakharov-Filonenko spectrum I ωω −4. To achieve quantitative coincidence of the results obtained by different methods one has to accomplish the Hasselmann kinetic equation by an empirical dissipation term S diss modeling the coherent effects of white-capping. Adding of the standard dissipation terms used in the industrial wave predicting model (WAM) leads to significant improvement but not resolve the discrepancy completely, leaving the question about optimal choice of S diss open.

Numerical modeling of swell evolution in the framework of the dynamical equations is affected by the side effect of resonances sparsity taking place due to finite size of the modeling domain. We mostly overcame this effect using fine integration grid of 512 × 4096 modes. The initial spectrum peak was located at the wave number k = 300. Similar conditions can be hardly realized in the laboratory wave tanks. One of the results of our article consists in the fact that physical processes in finite size laboratory wave tanks and in the ocean are quite different, and the results of such laboratory experiments can be applied to modeling of the ocean phenomena with extra care. We also present the estimate on the minimum size of the laboratory installation, allowing to model open ocean surface wave dynamics.


Dynamical Equation Probability Distribution Function Angular Spreading Spectral Maximum Weak Turbulence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Nordheim LW (1928) Proc.R.Soc. A119:689CrossRefGoogle Scholar
  2. [2]
    Peierls R (1929) Ann. Phys. (Leipzig) 3:1055CrossRefGoogle Scholar
  3. [3]
    Zakharov VE, Falkovich G, Lvov VS (1992) Kolmogorov Spectra of Turbulence I. Springer-Verlag, BerlinGoogle Scholar
  4. [4]
    Hasselmann K (1962) J.Fluid Mech. 12:1CrossRefGoogle Scholar
  5. [5]
    Zakharov VE, Filonenko NN (1966) Doklady Acad. Nauk SSSR 160:1292Google Scholar
  6. [6]
    Pushkarev AN, Zakharov VE (1996) Phys. Rev. Lett. 76:3320CrossRefGoogle Scholar
  7. [7]
    Pushkarev AN (1999) European Journ. of Mech. B/Fluids 18:345CrossRefGoogle Scholar
  8. [8]
    Pushkarev AN, Zakharov VE (2000) Physica D 135:98CrossRefGoogle Scholar
  9. [9]
    Tanaka M (2001) Fluid Dyn. Res. 28:41CrossRefGoogle Scholar
  10. [10]
    Onorato M, Osborne AR, Serio M, at al. (2002) Phys. Rev. Lett. 89:144501 arXiv:nlin.CD/0201017CrossRefGoogle Scholar
  11. [11]
    Dysthe KB, Trulsen K, Krogstad HE, Socquet-Juglard H (2003) J. Fluid Mech. 478:1–10CrossRefGoogle Scholar
  12. [12]
    Dyachenko AI, Korotkevich AO, Zakharov VE (2003) JETP Lett. 77:546 arXiv:physics/0308101CrossRefGoogle Scholar
  13. [13]
    Dyachenko AI, Korotkevich AO, Zakharov VE (2004) Phys. Rev. Lett. 92:134501 arXiv:physics/0308099CrossRefGoogle Scholar
  14. [14]
    Yokoyama N (2004) J. Fluid Mech. 501:169CrossRefGoogle Scholar
  15. [15]
    Zakharov VE, Korotkevich AO, Pushkarev AN, Dyachenko AI (2005) JETP Lett. 82:487 arXiv:physics/0508155.CrossRefGoogle Scholar
  16. [16]
    Dysthe K, Socquet-Juglard H, Trulsen K, at al. (2005) In “Rogue waves”, Proceedings of the 14th’ Aha Huliko’a Hawaiian Winter Workshop 91Google Scholar
  17. [17]
    Lvov Yu, Nazarenko SV, Pokorni B (2006) Physica D 218:24 arXiv:mathph/0507054CrossRefGoogle Scholar
  18. [18]
    Nazarenko SV (2006) J. Stat. Mech. L02002 arXiv:nlin.CD/0510054Google Scholar
  19. [19]
    Annenkov SYu, Shrira VI (2006) Phys. Rev. Lett. 96:204501CrossRefGoogle Scholar
  20. [20]
    Dyachenko AI, Newell AC, Pushkarev AN, Zakharov VE (1992) Physica D 57:96CrossRefGoogle Scholar
  21. [21]
    Korotkevich AO (2003) Numerical Simulation of Weak Turbulence of Surface Waves. PhD thesis, L.D. Landau Institute for Theoretical Physics RAS, Moscow, RussiaGoogle Scholar
  22. [22]
    Dyachenko AI, Korotkevich AO, Zakharov VE (2003) JETP Lett. 77:477 arXiv:physics/0308100CrossRefGoogle Scholar
  23. [23]
    Zakharov VE (1968) J. Appl. Mech. Tech. Phys. 2:190Google Scholar
  24. [24]
    Zakharov VE (1999) Eur. J. Mech. B/Fluids 18:327CrossRefGoogle Scholar
  25. [25]
    Kolmogorov A (1941) Dokl. Akad. Nauk SSSR 30:9 [Proc. R. Soc. London A434, 9 (1991)].Google Scholar
  26. [26]
    V. E. Zakharov (1967) PhD thesis, Budker Institute for Nuclear Physics, Novosibirsk, USSRGoogle Scholar
  27. [27]
    Zakharov VE, Zaslavskii MM (1982) Izv. Atm. Ocean. Phys. 18:747Google Scholar
  28. [28]
    Toba Y (1973) J. Oceanogr. Soc. Jpn. 29:209CrossRefGoogle Scholar
  29. [29]
    Donelan MA, Hamilton J, Hui WH (1985) Phil. Trans. R. Soc. London A315:509CrossRefGoogle Scholar
  30. [30]
    Hwang PA, at al. (2000) J. Phys. Oceanogr 30:2753CrossRefGoogle Scholar
  31. [31]
    Dyachenko AI, Zakharov VE (1994) Phys. Lett., A190:144Google Scholar
  32. [32]
    Hasselmann S, Hasselmann K, Barnett TP (1985) J. Phys. Oceanogr. 15:1378CrossRefGoogle Scholar
  33. [33]
    Dungey JC, Hui WH (1985) Proc. R. Soc. A368:239Google Scholar
  34. [34]
    Masuda A (1981) J. Phys. Oceanogr. 10:2082CrossRefGoogle Scholar
  35. [35]
    Masuda A (1986) in Phillips OM, Hasselmann K, Waves Dynamics and Radio Probing of the Ocean Surface. Plenum Press, New YorkGoogle Scholar
  36. [36]
    Lavrenov IV (1998) Mathematical modeling of wind waves at non-uniform ocean. Gidrometeoizdat, St.Petersburg, RussiaGoogle Scholar
  37. [37]
    Polnikov VG (2001) Wave Motion 1008:1Google Scholar
  38. [38]
    Webb DJ (1978) Deep-Sea Res. 25:279CrossRefGoogle Scholar
  39. [39]
    Resio D., Tracy B (1982) Theory and calculation of the nonlinear energy transfer between sea waves in deep water. Hydraulics Laboratory, US Army Engineer Waterways Experiment Station, WIS Report 11Google Scholar
  40. [40]
    Resio D, Perrie W (1991) J.Fluid Mech. 223:603CrossRefGoogle Scholar
  41. [41]
    Pushkarev A, Resio D, Zakharov VE (2003) Physica D 184:29CrossRefGoogle Scholar
  42. [42]
    Pushkarev A, Zakharov VE (2000) 6th International Workshop on Wave Hind-casting and Forecasting, November 6–10, Monterey, California, USA), 456 (published by Meteorological Service of Canada)Google Scholar
  43. [43]
    Badulin SI, Pushkarev A, Resio D, Zakharov VE (2005) Nonlinear Processes in GeophysicsGoogle Scholar
  44. [44]
    SWAN Cycle III user manual, Scholar
  45. [45]
    Zakharov VE, Guyenne P, Dias F (2001) Wave turbulence in one-dimensional models. Physica D, 152–153:573CrossRefGoogle Scholar
  46. [46]
    Dias F, Pushkarev A, Zakharov VE (2004) One-Dimensional Wave Turbulence. Physics Reports, 398:1 (2004).CrossRefGoogle Scholar
  47. [47]
    Dias F, Guyenne P, Pushkarev A, Zakharov VE (2000) Wave turbulence in one-dimensional models. Preprint N2000-4, Centre de Methematiques et del leur Appl., E.N.S de CACHAN, 1Google Scholar
  48. [48]
    Guyenne P, Zakharov VE, Dias F (2001) Turbulence of one-dimensional weakly nonlinear dispersive waves. Contemporary Mathematics, 283:107Google Scholar
  49. [49]
    Frigo M, Johnson SG (2005) Proc. IEEE 93:216 Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Alexander O. Korotkevich
    • 1
  • Andrei N. Pushkarev
    • 2
    • 3
  • Don Resio
    • 4
  • Vladimir E. Zakharov
    • 5
    • 2
    • 1
    • 3
  1. 1.Landau Institute for Theoretical Physics RASMoscowRussian Federation
  2. 2.Lebedev Physical Institute RASMoscowRussian Federation
  3. 3.Waves and Solitons LLCPhoenixUSA
  4. 4.Coastal and Hydraulics LaboratoryU.S. Army Engineer Research and Development CenterVicksburgUSA
  5. 5.Department of MathematicsUniversity of ArizonaTucsonUSA

Personalised recommendations