Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-18T07:53:03.977Z Has data issue: false hasContentIssue false

NONLINEAR SELF-MODULATION OF GRAVITY-CAPILLARY WAVES ON SHEAR CURRENTS IN FINITE DEPTH

Published online by Cambridge University Press:  12 January 2024

TANMOY PAL
Affiliation:
Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103 West Bengal, India; e-mail: tpal2966@gmail.com
ASOKE KUMAR DHAR*
Affiliation:
Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103 West Bengal, India; e-mail: tpal2966@gmail.com

Abstract

A nonlinear evolution equation correct to fourth order is developed for gravity-capillary waves on linear shear currents in finite water depth. Therefore, this equation covers both effects of depth uniform currents and uniform vorticity. Starting from this equation, an instability analysis is then made for narrow banded uniform Stokes waves. The notable feature is that our investigation due to fourth order shows a remarkable improvement compared with the third-order one, and produces an excellent result compatible with the exact result of Longuet-Higgins. We observe that linear shear currents considerably change the modulational instability properties of capillary-gravity waves, such as the growth rate and bandwidth of instability.

Type
Research Article
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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

Benjamin, T. B. and Feir, J. E., “The disintegration of wave trains on deep water. Part 1. Theory”, J. Fluid Mech. 27 (1967) 417430; doi:10.1017/S002211206700045X.CrossRefGoogle Scholar
Bretherton, F. P., Garrett, C. J. R. and Lighthill, M. J., “Wavetrains in inhomogeneous moving media”, Proc. R. Soc. Lond. Ser. A 302 (1968) 529554; doi:10.1098/rspa.1968.0034.Google Scholar
Brevik, I., “Higher-order waves propagating on constant vorticity currents in deep water”, Coastal Engineering 2 (1978) 237259; doi:10.1016/0378-3839(78)90022-4.CrossRefGoogle Scholar
Brinch-Nielsen, U. and Jonsson, I. G., “Fourth order evolution equations and stability analysis for Stokes waves on arbitrary water depth”, Wave Motion 8 (1986) 455472; doi:10.1016/0165-2125(86)90030-2.CrossRefGoogle Scholar
Choi, W., “Nonlinear surface waves interacting with a linear shear current”, Math. Comput. Simulation 80 (2009) 2936; doi:10.1016/j.matcom.2009.06.021.CrossRefGoogle Scholar
Constantin, A., “Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train”, Eur. J. Mech. B Fluids 30 (2011) 1216; doi:10.1016/j.euromechflu.2010.09.008.CrossRefGoogle Scholar
Dalrymple, R. A., “A finite amplitude wave on a linear shear current”, J. Geophys. Res. 79 (1974) 44984504 (1896–1977); doi:10.1029/JC079i030p04498.CrossRefGoogle Scholar
Davey, A. and Stewartson, K., “On three-dimensional packets of surface waves”, Proc. R. Soc. Lond. Ser. A. 338(1613) (1974) 101110; doi:10.1098/rspa.1974.0076.Google Scholar
Dhar, A. K. and Kirby, J. T., “Fourth-order stability analysis for capillary-gravity waves on finite depth currents with constant vorticity”, Phys. Fluids 35 (2023) Article ID: 026601; doi:10.1063/5.0136002.CrossRefGoogle Scholar
Djordjevic, V. D. and Redekopp, L. G., “On two-dimensional packets of capillary-gravity waves”, J. Fluid Mech. 79 (1977) 703714; doi:10.1017/S0022112077000408.CrossRefGoogle Scholar
Dysthe, K. B., “Note on a modification to the nonlinear Schr $\ddot{\mathrm{o}}$ dinger equation for application to deep water waves”, Proc. R. Soc. Lond., Ser. A 369(1736) (1979) 105114; doi:10.1098/rspa.1979.0154.Google Scholar
Harrison, W. J., “The influence of viscosity and capillarity on waves of finite amplitude”, Proc. Lond. Math. Soc. s2–7(1) (1909) 107121; doi:10.1112/plms/s2-7.1.107.CrossRefGoogle Scholar
Hjelmervik, K. B. and Trulsen, K., “Freak wave statistics on collinear currents”, J. Fluid Mech. 637 (2009) 267284; doi:10.1017/S0022112009990607.CrossRefGoogle Scholar
Hogan, S. J., “The fourth-order evolution equation for deep-water gravity-capillary waves”, Proc. R. Soc. Lond. Ser. A 402(1823) (1985) 359372; doi:10.1098/rspa.1985.0122.Google Scholar
Hsu, H.-C., Francius, M., Montalvo, P. and Kharif, C., “Gravity–capillary waves in finite depth on flows of constant vorticity”, Proc. R. Soc. Lond. Ser. A 472(2195) (2016), Article ID: 20160363; doi:10.1098/rspa.2016.0363.Google ScholarPubMed
Hsu, H. C., Kharif, C., Abid, M. and Chen, Y. Y., “A nonlinear Schrödinger equation for gravity–capillary water waves on arbitrary depth with constant vorticity. Part 1”, J. Fluid Mech. 854 (2018) 146163; doi:10.1017/jfm.2018.627.CrossRefGoogle Scholar
Huang, Z. and Mei, C. C., “Effects of surface waves on a turbulent current over a smooth or rough seabed”, J. Fluid Mech. 497 (2003), 253287; doi:10.1017/S0022112003006657.CrossRefGoogle Scholar
Janssen, P. A. E. M., “On a fourth-order envelope equation for deep-water waves”, J. Fluid Mech. 126 (1983) 111; doi:10.1017/S0022112083000014.CrossRefGoogle Scholar
Johnson, R. S. and Stewartson, K., “On the modulation of water waves on shear flows”, Proc. R. Soc. Lond. Ser. A 347(1651) (1976) 537546; doi:10.1098/rspa.1976.0015.Google Scholar
Kantardgi, I., “Effect of depth current profile on wave parameters”, Coastal Engineering 26 (1995) 195206; doi:10.1016/0378-3839(95)00021-6.CrossRefGoogle Scholar
Kishida, N. and Sobey, R. J., “Stokes theory for waves on linear shear current”, J. Eng. Mech. 114 (1988) 13171334; doi:10.1061/(ASCE)0733-9399(1988)114:8(1317).Google Scholar
Liao, B., Dong, G., Ma, Y. and Gao, J. L., “Linear-shear-current modified Schr $\ddot{\mathrm{o}}$ dinger equation for gravity waves in finite water depth”, Phys. Rev. E 96 (2017) Article ID 043111; doi:10.1103/PhysRevE.96.043111.CrossRefGoogle Scholar
Liu, P. L.-F., Dingemans, M. W. and Kostense, J. K., “Long-wave generation due to the refraction of short-wave groups over a shear current”, J. Phys. Oceanogr. 20 (1990) 5359; doi:10.1175/1520-0485(1990)020<0053:LWGDTT>2.0.CO;2.2.0.CO;2>CrossRefGoogle Scholar
Longuet-Higgins, M. S., “The instabilities of gravity waves of finite amplitude in deep water II. Subharmonics”, Proc. R. Soc. Lond. Ser. A 360(1703) (1978) 489505; doi:10.1098/rspa.1978.0081.Google Scholar
Longuet-Higgins, M. S. and Stewart, R. W., “The changes in amplitude of short gravity waves on steady non-uniform currents”, J. Fluid Mech. 10 (1961) 529549; doi:10.1017/S0022112061000342.CrossRefGoogle Scholar
Ma, Y., Ma, X., Perlin, M. and Dong, G., “Extreme waves generated by modulational instability on adverse currents”, Phys. Fluids 25 (2013) Article ID: 114109; doi:10.1063/1.4832715.CrossRefGoogle Scholar
MacIver, R. D., Simons, R. R. and Thomas, G. P., “Gravity waves interacting with a narrow jet-like current”, J. Geophys. Res.: Oceans 111(C3) (2006) C03009; doi:10.1029/2005JC003030.CrossRefGoogle Scholar
McGoldrick, L. F., “On Wilton’s ripples: a special case of resonant interactions”, J. Fluid Mech. 42 (1970) 193200; doi:10.1017/S0022112070001179.CrossRefGoogle Scholar
Mei, C. C. and Lo, E., “The effects of a jet-like current on gravity waves in shallow water”, J. Phys. Oceanogr. 14 (1984) 471477; doi:10.1175/1520-0485(1984)014<0471:TEOAJL>2.0.CO;2.2.0.CO;2>CrossRefGoogle Scholar
Oikawa, M., Chow, K. and Benney, D. J., “The propagation of nonlinear wave packets in a shear flow with a free surface”, Stud. Appl. Math. 76 (1987) 6992; doi:10.1002/sapm198776169.CrossRefGoogle Scholar
Okamura, M. and Oikawa, M., “The linear stability of finite amplitude surface waves on a linear shearing flow”, J. Phys. Soc. Japan 58 (1989) 23862396; doi:10.1143/JPSJ.58.2386.CrossRefGoogle Scholar
Onorato, M., Osborne, A. R., Serio, M. and Bertone, S., “Freak waves in random oceanic sea states”, Phys. Rev. Lett. 86 (2001) 58315834; doi:10.1103/PhysRevLett.86.5831.CrossRefGoogle ScholarPubMed
Onorato, M., Osborne, A. R., Serio, M., Cavaleri, L., Brandini, C. and Stansberg, C. T., “Extreme waves, modulational instability and second order theory: wave flume experiments on irregular waves”, Eur. J. Mech. B Fluids 25 (2006) 586601. doi:10.1016/j.euromechflu.2006.01.002.CrossRefGoogle Scholar
Pak, O. S. and Chow, K. W., “Free surface waves on shear currents with non-uniform vorticity: third-order solutions”, Fluid Dyn. Res. 41 (2009) Article ID: 035511; doi:10.1088/0169-5983/41/3/035511.CrossRefGoogle Scholar
Peregrine, D. H., “Interaction of water waves and currents”, Adv. Appl. Mech. 16 (1976) 9117; doi:10.1016/S0065-2156(08)70087-5.CrossRefGoogle Scholar
Sedletsky, Y. V., “The modulational instability of Stokes waves on the surface of finite-depth fluid”, Phys. Lett. A 343 (2005) 293299; doi:10.1016/j.physleta.2005.04.076.CrossRefGoogle Scholar
Simmen, J. and Saffman, P. G., “Steady deep-water waves on a linear shear current”, Stud. Appl. Math. 73 (1985) 3557; doi:10.1002/sapm198573135.CrossRefGoogle Scholar
Teles da Silva, A. F. and Peregrine, D. H., “Nonlinear perturbations on a free surface induced by a submerged body: a boundary integral approach”, Eng. Anal. Bound. Elem. 7 (1990) 214222; doi:10.1016/0955-7997(90)90007-V.CrossRefGoogle Scholar
Thomas, R., Kharif, C. and Manna, M., “A nonlinear Schr $\ddot{\mathrm{o}}$ dinger equation for water waves on finite depth with constant vorticity”, Phys. Fluids 24 (2012) Article ID: 127102; doi:10.1063/1.4768530.CrossRefGoogle Scholar
Tsao, S., “Behaviour of surface waves on a linearly varying flow”, Tr. Mosk. Fiz.-Tekh. Inst. Issled. Mekh. Prikl. Mat. 3 (1959) 6684.Google Scholar
Wilton, J. R., “On ripples”, London, Edinburgh, Dublin Philos. Mag. J. Sci. 29 (1915) 688700; doi:10.1080/14786440508635350.CrossRefGoogle Scholar
Yuen, H. C. and Lake, B. M., “Nonlinear deep water waves: theory and experiment”, Phys. Fluids 18 (1975) 956960; doi:10.1063/1.861268.CrossRefGoogle Scholar