Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-05-20T10:49:39.764Z Has data issue: false hasContentIssue false

Nonlinear and dispersive free surface waves propagating over fluids with weak vertical and horizontal density variation

Published online by Cambridge University Press:  28 April 2014

Sangyoung Son
Department of Civil and Environmental Engineering, University of Ulsan, Ulsan 680-749,Republic of Korea Sonny Astani Department of Civil and Environmental Engineering, University of Southern California, Los Angeles, CA 90089, USA
Patrick J. Lynett*
Sonny Astani Department of Civil and Environmental Engineering, University of Southern California, Los Angeles, CA 90089, USA
Email address for correspondence:


We consider the change in fluid density in a depth-integrated long-wave model. By allowing horizontal and vertical variation of fluid density, a depth-integrated model for long gravity waves over a variable-density fluid has been developed, where density change effects are included as correction terms. In particular, a two-layer fluid system is chosen to represent vertical density variations, where interfacial wave effects on the free surface are accounted for through direct inclusion of the velocity component of the interfacial wave. For the numerical implementation of the model, a finite-volume scheme coupled with an approximate Riemann solver is adopted for leading-order terms while cell-centred finite-volume methods are utilized for others. Numerical tests in which the density field is configured to vary either horizontally or vertically have been performed to verify the model. For horizontal variation of fluid density, a pneumatic breakwater system is simulated and fair agreement is observed between computed and measured data, indicating that the current induced by the upward bubble flux is responsible for wave attenuation to some degree. To investigate the effects of internal motion on the free surface, a two-layer fluid system with monochromatic internal wave motion is tested numerically. Simulated results agree well with the measured and analytical data. Lastly, nonlinear interactions between external- and internal-mode surface waves are studied numerically and analytically, and the model is shown to have nonlinear accuracy limitations similar to existing Boussinesq-type models.

© 2014 Cambridge University Press 

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.)


Alam, M. R. 2012 A new triad resonance between co-propagating surface and interfacial waves. J. Fluid Mech. 691, 267278.Google Scholar
Armi, L. & Farmer, D. 1986 Maximal two-layer exchange through a contraction with barotropic net flow. J. Fluid Mech. 164, 2751.Google Scholar
Bulson, P. S. 1963 Large scale bubble breakwater experiments. Dock & Harbour Authority 44 (516), 191197.Google Scholar
Choi, W. & Camassa, R. 1996 Weakly nonlinear internal waves in a two-fluid system. J. Fluid Mech. 313, 83103.CrossRefGoogle Scholar
Choi, W. & Camassa, R. 1999 Fully nonlinear internal waves in a two-fluid system. J. Fluid Mech. 384, 2758.Google Scholar
Clift, R., Grace, J. & Weber, M. 1978 Bubbles, Drops and Particles. Academic Press.Google Scholar
Craig, W., Guyenne, P. & Sulem, C. 2011 Coupling between internal and surface waves. Nat. Hazards 57, 617642.CrossRefGoogle Scholar
Dean, R. G. & Dalrymple, R. A. 1984 Water Wave Mechanics for Engineers and Scientists. Prentice-Hall.Google Scholar
Debsarma, S., Das, K. P. & Kirby, J. T. 2010 Fully nonlinear higher-order model equations for long internal waves in a two-fluid system. J. Fluid Mech. 654, 281303.Google Scholar
Donato, A. N., Peregrine, D. H. & Stocker, J. R. 1999 The focusing of surface waves by internal waves. J. Fluid Mech. 384, 2758.CrossRefGoogle Scholar
Elachi, C. & Apel, J. R. 1976 Internal wave observations made with an airborne synthetic aperture imaging radar. Geophys. Res. Lett. 3 (11), 647650.Google Scholar
Elder, J. W. 1959 The dispersion of marked fluid in turbulent shear flow. J. Fluid Mech. 5, 544560.CrossRefGoogle Scholar
Evans, W. A. B. & Ford, M. J. 1996 An integral equation approach to internal (2-layer) solitary waves. Phys. Fluids 8, 20322047.Google Scholar
Farmer, D. & Armi, L. 1999 The generation and trapping of solitary waves over topography. Science 283, 188190.Google Scholar
Gargett, A. E. & Hughes, B. A. 1972 On the interaction of surface and internal waves. J. Fluid Mech. 52, 179191.Google Scholar
Helfrich, K. & Melville, W. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395425.Google Scholar
Hill, D. F. & Foda, M. A. 1998 Subharmonic resonance of oblique interfacial waves by a progressive surface waves. Proc. R. Soc. Lond. A 454, 11291144.Google Scholar
Hwung, H.-H., Yang, R.-Y. & Shugan, I. V. 2009 Exposure of internal waves on the sea surface. J. Fluid Mech. 626, 120.CrossRefGoogle Scholar
Joyce, T. M. 1974 Nonlinear interactions among standing surface and internal gravity waves. J. Fluid Mech. 63, 801825.Google Scholar
Kanarsk, Y. & Maderich, V. 2003 A non-hydrostatic numerical model for calculation free-surface stratified flows. Ocean Dyn. 53, 176185.CrossRefGoogle Scholar
Kao, T., Pan, F.-S. & Renouard, D. 1985 Internal solitons on the pycnocline: generation, propagation, and shoaling and breaking over a slope. J. Fluid Mech. 159, 1953.Google Scholar
Kennedy, A. B., Kirby, J. T., Chen, Q. & Dalrymple, R. A. 2001 Boussinesq-type equations with improved nonlinear performance. Wave Motion 33, 225243.Google Scholar
Kim, D.-H., Lynett, P. & Socolofsky, S. 2009 A depth-integrated model for weakly dispersive, turbulent, and rotational fluid flows. Ocean Model. 27 (3–4), 198214.Google Scholar
Koop, C. G. & Butler, G. 1981 An investigation of internal solitary waves in a two-fluid system. J. Fluid Mech. 112, 225251.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Leighton, F., Borthwick, A. & Taylor, P. 2010 1-D numerical modelling of shallow flows with variable horizontal density. Intl J. Numer. Meth. Fluids 62, 12091231.Google Scholar
Lewis, J. E., Lake, B. M. & Ko, D. R. S. 1974 On the interaction of internal waves and surface gravity waves. J. Fluid Mech. 64, 773800.Google Scholar
Liu, C.-M. 2006 Second-order random internal and surface waves in a two-fluid system. Geophys. Res. Lett. 33, L06610.Google Scholar
Liu, P. L.-F. & Wang, X. 2012 A multilayer model for nonlinear internal wave propagation in shallow water. J. Fluid Mech. 695, 341365.CrossRefGoogle Scholar
Lo, J.-M. 1991 Air bubble barrier effect on neutrally buoyant objects. J. Hydraul Res. 29 (4), 437455.Google Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1960 Changes in the form of short gravity waves on long waves and tidal currents. J. Fluid Mech. 8, 565583.Google Scholar
Lowe, R. J., Rottman, J. W. & Linden, P. F. 2005 The non-Boussinesq lock-exchange problem. Part 1. Theory and experiments. J. Fluid Mech. 537, 101124.Google Scholar
Lynett, P. & Liu, P. L.-F. 2002a A numerical study of submarine landslide generated waves and runup. Proc. R. Soc. Lond. A 458, 28852910.Google Scholar
Lynett, P. & Liu, P. L.-F. 2002b A two-dimensional, depth-integrated model for internal wave propagation. Wave Motion 36, 221240.Google Scholar
Lynett, P. & Liu, P. L.-F. 2004 A two-layer approach to wave modelling. Proc. R. Soc. Lond. A 460, 26372669.Google Scholar
Ma, Y.-C. 1982 Effect of long waves on the evolution of deep-water surface gravity waves. Phys. Fluids 25 (3), 411419.Google Scholar
Ma, G., Shi, F. & Kirby, J. T. 2011 A polydisperse two-fluid model for surf zone bubble simulation. J. Geophys. Res. 116, C05010.Google Scholar
Mellor, G. L. 1991 An equation of state for numerical models of oceans and estuaries. J. Atmos. Ocean. Technol. 8, 609611.Google Scholar
Milgram, J. H. 1983 Mean flow in round bubble plumes. J. Fluid Mech. 133, 345376.Google Scholar
Nguyen, H. Y. & Dias, F. 2008 A Boussinesq system for two-way propagation of interfacial waves. Physica D 237, 23652389.CrossRefGoogle Scholar
Nwogu, O. 1993 Alternative form of Boussinesq equations for nearshore wave propagation. ASCE J. Waterway Port Coastal Ocean Engng 119 (6), 618638.Google Scholar
Părău, E. & Dias, F. 2001 Interfacial periodic waves of permanent form with free-surface boundary conditions. J. Fluid Mech. 437, 325336.CrossRefGoogle Scholar
Phillips, O. M. 1977 Dynamics of the Upper Ocean. 2nd edn. Cambridge University Press.Google Scholar
Schäffer, H. A. 1996 Second-order wavemaker theory for irregular waves. Ocean Engng 23 (1), 4788.Google Scholar
Segur, H. & Hammack, J. L. 1982 Soliton models of long internal waves. J. Fluid Mech. 118, 285305.Google Scholar
Selezov, I. T., Avrameko, O. V., Gurtovyi, Y. V. & Naradovyi, V. V. 2010 Nonlinear interaction of internal and surface gravity waves in a two-layer fluid with free surface. J. Math. Sci. 168 (4), 590602.Google Scholar
Sharma, J. N. & Dean, R. G. 1981 Second-order directional seas and associated wave forces. Soc. Petrol. Engng J. 4, 129140.Google Scholar
Son, S. & Lynett, P.2014 Interaction of dispersive water waves with weakly sheared currents of arbitrary profile. Coast. Engng (submitted).Google Scholar
Straub, L. G., Bowers, C. E. & Tarapore, Z. S.1959 Experimental studies of pneumatic and hydraulic breakwaters. Technical Paper No. 25, Series B, St. Anthony Falls Hydraulic Laboratory, University of Minnesota, Minneapolis, USA.Google Scholar
Taylor, G. 1955 The action of a surface current used as a breakwater. Proc. R. Soc. Lond. A 231, 446478.Google Scholar
Toro, E. F. 2002 Shock-Capturing Methods for Free-Surface Shallow Flows. Wiley.Google Scholar
Umeyama, M. 2002 Experimental and theoretical analyses of internal waves of finite amplitude. ASCE J. Waterway Port Coastal Ocean Engng 128 (3), 133141.Google Scholar
Wei, G., Kirby, J. T., Grilli, S. T. & Subramanya, R. 1995 A fully nonlinear Boussinesq model for surface waves. I. Highly nonlinear unsteady waves. J. Fluid Mech. 294, 7192.Google Scholar
Wood, I. R. 1970 A lock exchange flow. J. Fluid Mech. 42, 671687.CrossRefGoogle Scholar
Yamamoto, S. & Daiguji, H. 1993 Higher-order-accurate upwind schemes for solving the compressible Euler and Navier–Stokes equations. Comput. Fluids 22, 259270.Google Scholar
Zhang, C.-X., Wang, Y.-X., Wang, G.-Y. & Yu, L.-M. 2010 Wave dissipating performance of air bubble breakwaters with different layouts. J. Hydrodyn. B 22 (5), 671680.Google Scholar
Zhou, J. G., Causon, D. M., Mingham, C. G. & Ingram, D. M. 2001 The surface gradient method for the treatment of source terms in the shallow water equations. J. Comput. Phys. 168 (2), 125.Google Scholar
Supplementary material: PDF

Son and Lynett supplementary material

Supplementary material

Download Son and Lynett supplementary material(PDF)
PDF 113 KB