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EVOLUTION OF A PAIR OF RANDOM INHOMOGENEOUS WAVE SYSTEMS OVER INFINITE-DEPTH WATER

Part of: Incompressible inviscid fluids Hydrodynamic stability

Published online by Cambridge University Press:  15 May 2019

S. DEBSARMA Open the ORCID record for S. DEBSARMA [Opens in a new window]  and
D. CHOWDHURY Open the ORCID record for D. CHOWDHURY [Opens in a new window]
S. DEBSARMA*
Affiliation:
Department of Applied Mathematics, University of Calcutta, 92 A.P.C. Road, Kolkata 700009, India email suma_debsarma@rediffmail.com, dipankar.chowdhury05@rediffmail.com
D. CHOWDHURY
Affiliation:
Department of Applied Mathematics, University of Calcutta, 92 A.P.C. Road, Kolkata 700009, India email suma_debsarma@rediffmail.com, dipankar.chowdhury05@rediffmail.com
*
✉suma_debsarma@rediffmail.com
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Abstract

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A system of two coupled nonlinear spectral transport equations is derived for two obliquely interacting narrowband Gaussian random surface wavetrains, slowly varying in space and time. Using these two equations, stability analysis is performed for two initially homogeneous wave spectra, subject to unidirectional perturbations. We observe that the effect of randomness produces a decrease in the growth rate of instability, but it is higher than the growth for a single wavetrain. The growth rate of instability is observed to decrease with the increase in spectral width.

Keywords

crossing sea statesevolution equationinstabilityrandomness

MSC classification

Primary: 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Secondary: 76B07: Free-surface potential flows 76E99: None of the above, but in this section
Type
Research Article
Information
The ANZIAM Journal , Volume 61 , Issue 2 , April 2019 , pp. 233 - 247
Copyright
© 2019 Australian Mathematical Society 

References

Abramowitz, M. and Stegun, I., Handbook of mathematical functions, National Bureau of Standards, Applied Mathematics Series 55 (reprinted with corrections 1965 (US Government Printing Office, Washington, DC, 1965).Google Scholar
Alber, I. E., “The effect of randomness on stability of two dimensional surface wave trains”, Proc. R. Soc. Lond. A 363 (1978) 525–546; doi:10.1098/rspa.1978.0181.Google Scholar
Benjamin, T. B. and Feir, J. E., “The disintegration of wave trains on deep water, part I. Theory”, J. Fluid Mech. 27 (1967) 417–430; doi:10.1017/S002211206700045X.Google Scholar
Benney, D. J. and Saffman, P. G., “Nonlinear interaction of random waves in a dispersive medium”, Proc. R. Soc. Lond. A 289 (1966) 301–320; doi:10.1098/rspa.1966.0013.Google Scholar
Crawford, D. R., Saffman, P. G. and Yuen, H. C., “Evolution of a random inhomogeneous field of nonlinear deep-water gravity waves”, Wave Motion 2 (1980) 1–16;doi:10.1016/0165-2125(80)90029-3.Google Scholar
Davey, A. and Stewartson, K., “On three dimensional packets of surface waves”, Proc. R. Soc. Lond. A 338 (1974) 97–114; doi:10.1098/rspa.1974.0076.Google Scholar
Dhar, A. K. and Das, K. P., “The effect of randomness on stability of surface gravity waves from fourth order nonlinear evolution equation”, Int. J. Appl. Mech. Eng. 6 (2001) 11–34.Google Scholar
Gramstad, O. and Trulsen, K., “Fourth-order coupled nonlinear Schrödinger equations for gravity waves on deep water”, Phys. Fluids 23 (2011) 062102 (1-9); doi:10.1063/1.3598316.Google Scholar
Hasselmann, K. et al. , “Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP)”, Ergänzungsheft zur Deutschen Hydrographischen Zeitschrift Reihe A 8(12) (1973) 1–95; https://www.researchgate.net/publication/256197895.Google Scholar
Laine-Pearson, F. E., “Instability growth rates of crossing sea states”, Phys. Rev. E81 (2010) 036316 (1-7); doi:10.1103/PhysRevE.81.036316.Google Scholar
Onorato, M., Osborne, A. R., Serio, M. and Bertone, S., “Freak waves in random oceanic sea states”, Phys. Rev. Lett. 86 (2001) 5831–5834; doi:10.1103/PhysRevLett.86.5831.Google Scholar
Onorato, M., Osborne, A. R. and Serio, M., “Modulational instability in crossing sea states: A possible mechanism for the formation of freak waves”, Phys. Rev. Lett. 96 (2006) 014503 (1-4); doi:10.1103/PhysRevLett.96.014503.Google Scholar
Onorato, M. et al. , “Statistical properties of directional ocean waves: the royal of the modulational instability in the formation of extreme events”, Phys. Rev. Lett. 102 (2009) 114502 (1-4);doi:10.1103/PhysRevLett.102.114502.Google Scholar
Ruban, V. P., “Giant waves in weakly crossing sea states”, J. Exp. Theor. Phys. 110 (2010) 529–536; doi:10.1134/S1063776110030155.Google Scholar
Senapati, S., Debsarma, S. and Das, K. P., “Evolution of a random field of surface gravity waves in a two fluid domain”, Int. J. Appl. Mech. Eng. 17 (2012) 481–493.Google Scholar
Shukla, P. K., Kourakis, I., Eliasson, B., Marklund, M. and Stenflo, L., “Instability and evolution of nonlinearly interacting water waves”, Phys. Rev. Lett. 97 (2006) 094501 (1-4);doi:10.1103/PhysRevLett.97.094501.Google Scholar
Spiegel, M. R., Theory and problems of probability and statistics (McGraw-Hill, New York, 1992) 109–115.Google Scholar
Toffoli, A., Bitner Gregersen, E. M., Osborne, A. R., Serio, M., Monbaliu, J. and Onorato, M., “Extreme waves in random crossing seas: Laboratory experiments and numerical simulation”, Geograph. Res. Lett. 38 (2011) L06605,1-5; doi:10.1029/2011GLO46827.Google Scholar
Zakharov, V. E., “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.Google Scholar