Abstract
Depth-integrated long-wave models, such as the shallow-water and Boussinesq equations, are standard fare in the study of small amplitude surface waves in shallow water. While the shallow-water theory features conservation of mass, momentum and energy for smooth solutions, mechanical balance equations are not widely used in Boussinesq scaling, and it appears that the expressions for many of these quantities are not known. This work presents a systematic derivation of mass, momentum and energy densities and fluxes associated with a general family of Boussinesq systems. The derivation is based on a reconstruction of the velocity field and the pressure in the fluid column below the free surface, and the derivation of differential balance equations which are of the same asymptotic validity as the evolution equations. It is shown that all these mechanical quantities can be expressed in terms of the principal dependent variables of the Boussinesq system: the surface excursion η and the horizontal velocity w at a given level in the fluid.
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Agnon, Y., Madsen, P.A., Schäffer, H.A.: A new approach to high-order Boussinesq models. J. Fluid Mech. 399, 319–333 (1999)
Alazman, A.A., Albert, J.P., Bona, J.L., Chen, M., Wu, J.: Comparison between the BBM equation and a Boussinesq system. Adv. Differ. Equ. 11, 121–166 (2006)
Ali, A., Kalisch, H.: Energy balance for undular bores. C. R., Méc. 338, 67–70 (2010)
Alvarez-Samaniego, B., Lannes, D.: Large time existence for 3D water-waves and asymptotics. Invent. Math. 171, 485–541 (2008)
Amick, C.J.: Regularity and uniqueness of solutions to the Boussinesq system of equations. J. Differ. Equ. 54, 231–247 (1984)
Benjamin, T.B., Lighthill, M.J.: On cnoidal waves and bores. Proc. R. Soc. Lond. Ser. A 224, 448–460 (1954)
Benjamin, T.B., Bona, J.B., Mahony, J.J.: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. Ser. A 272, 47–78 (1972)
Bjørkavåg, M., Kalisch, H.: Wave breaking in Boussinesq models for undular bores. Phys. Lett. A 375, 157–1578 (2011)
Bona, J.L., Chen, M.: A Boussinesq system for two-way propagation of nonlinear dispersive waves. Physica D 116, 191–224 (1998)
Bona, J.L., Pritchard, W.G., Scott, L.R.: An evaluation of a model equation for water waves. Philos. Trans. R. Soc. Lond. Ser. A 302, 457–510 (1981)
Bona, J.L., Chen, M., Saut, J.-C.: Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: Derivation and linear theory. J. Nonlinear Sci. 12, 283–318 (2002)
Bona, J.L., Chen, M., Saut, J.-C.: Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II: The nonlinear theory. Nonlinearity 17, 925–952 (2004)
Bona, J.L., Colin, T., Lannes, D.: Long wave approximations for water waves. Arch. Ration. Mech. Anal. 178, 373–410 (2005)
Bona, J.L., Dougalis, V.A., Mitsotakis, D.E.: Numerical solution of KdV-KdV systems of Boussinesq equations. I. The numerical scheme and generalized solitary waves. Math. Comput. Simul. 74, 214–228 (2007)
Bona, J.L., Grujić, Z., Kalisch, H.: A KdV-type Boussinesq system: From the energy level to analytic spaces. Discrete Contin. Dyn. Syst. 26, 1121–1139 (2010)
Boussinesq, J.: Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 17, 55–108 (1872)
Chazel, F., Benoit, M., Ern, A., Piperno, S.: A double-layer Boussinesq-type model for highly nonlinear and dispersive waves. Proc. R. Soc. Lond. Ser. A 465, 2319–2346 (2009)
Chen, M.: Exact solution of various Boussinesq systems. Appl. Math. Lett. 11, 45–49 (1998)
Chen, M.: Numerical investigation of a two-dimensional Boussinesq system. Discrete Contin. Dyn. Syst. 23, 1169–1190 (2009)
Christov, C.I.: An energy-consistent dispersive shallow-water model. Wave Motion 34, 161–174 (2001)
Craig, W.: An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits. Commun. Partial Differ. Equ. 10, 787–1003 (1985)
Craig, W., Groves, M.D.: Hamiltonian long-wave approximations to the water-wave problem. Wave Motion 19, 367–389 (1994)
Craig, W., Sulem, C.: Numerical simulation of gravity waves. J. Comput. Phys. 108, 73–83 (1993)
Craig, W., Guyenne, P., Kalisch, H.: Hamiltonian long-wave expansions for free surfaces and interfaces. Commun. Pure Appl. Math. 58, 1587–1641 (2005)
Dougalis, V.A., Mitsotakis, D.E., Saut, J.-C.: On some Boussinesq systems in two space dimensions: theory and numerical analysis. Modél. Math. Anal. Numér. 41, 825–854 (2007)
Dougalis, V.A., Mitsotakis, D.E., Saut, J.-C.: On initial-boundary value problems for a Boussinesq system of BBM-BBM type in a plane domain. Discrete Contin. Dyn. Syst. 23, 1191–1204 (2009)
Dutykh, D., Dias, F.: Energy of tsunami waves generated by bottom motion. Proc. R. Soc. Lond. Ser. A 465, 725–744 (2009)
Fokas, A.S., Pelloni, B.: Boundary value problems for Boussinesq type systems. Math. Phys. Anal. Geom. 8, 59–96 (2005)
Green, A.E., Naghdi, P.M.: A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78, 237–246 (1976)
Kaup, D.J.: A higher-order wave equation and the method for solving it. Prog. Theor. Phys. 54, 396–408 (1975)
Kennedy, A.B., Kirby, J.T., Chen, Q., Dalrymple, R.A.: Boussinesq-type equations with improved nonlinear performance. Wave Motion 33, 225–243 (2001)
Keulegan, G.H., Patterson, G.W.: Mathematical theory of irrotational translation waves. Natl. Bur. Stand. J. Res. 24, 47–101 (1940)
Kim, G., Lee, C., Suh, K.-D.: Extended Boussinesq equations for rapidly varying topography. Ocean Eng. 33, 842–851 (2009)
Kirby, J.: A general wave equation for waves over rippled beds. J. Fluid Mech. 162, 171–186 (1986)
Kundu, P.K., Cohen, I.M.: Fluid Mechanics. Academic Press, New York (2008)
Lannes, D.: Well-posedness of the water-waves equations. J. Am. Math. Soc. 18, 605–654 (2005)
Lannes, D., Bonneton, P.: Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation. Phys. Fluids 21, 016601 (2009)
Madsen, P.A., Schäffer, H.A.: Higher-order Boussinesq-type equations for surface gravity waves: derivation and analysis. Philos. Trans. R. Soc. Lond. Ser. A 356, 3123–3184 (1998)
Madsen, P.A., Furman, D.R., Wang, B.: A Boussinesq-type method for fully nonlinear waves interacting with rapidly varying bathymetry. Coast. Eng. 53, 487–504 (2006)
Nachbin, A., Choi, W.: Nonlinear waves over highly variable topography. Eur. Phys. J. 147, 113–132 (2007)
Peregrine, D.H.: Calculation of the development of an undular bore. J. Fluid Mech. 25, 321–330 (1966)
Peregrine, D.H.: Equations for water waves and the approximation behind them. In: Waves on Beaches and Resulting Sediment Transport, pp. 95–121. Academic Press, New York (1972)
Schneider, G., Wayne, C.E.: The long-wave limit for the water wave problem. I. The case of zero surface tension. Commun. Pure Appl. Math. 53, 1475–1535 (2000)
Schonbek, M.E.: Existence of solutions for the Boussinesq system of equations. J. Differ. Equ. 42, 325–352 (1981)
Shi, F., Dalrymple, R.A., Kirby, J.T., Chen, Q., Kennedy, A.B.: A fully nonlinear Boussinesq model in generalized curvilinear coordinates. Coast. Eng. 42, 337–358 (2001)
Stoker, J.J.: Water Waves: The Mathematical Theory with Applications. Pure and Applied Mathematics, vol. IV. Interscience Publishers, New York (1957)
Su, C.H., Gardner, C.S.: Korteweg-de Vries equation and generalizations. III: Derivation of the Korteweg–de Vries equation and Burgers equation. J. Math. Phys. 10, 536–539 (1969)
Teng, M.H., Wu, T.Y.: Effects of channel cross-sectional geometry on long wave generation and propagation. Phys. Fluids 9, 3368–3377 (1997)
Wahlen, E.: Hamiltonian long-wave approximations of water waves with constant vorticity. Phys. Lett. A 372, 2597–2602 (2008)
Wei, G., Kirby, J.T., Grilli, S.T., Subramanya, R.: A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves. J. Fluid Mech. 294, 71–92 (1995)
Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)
Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130, 39–72 (1997)
Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Am. Math. Soc. 12, 445–495 (1999)
Zakharov, V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190–194 (1968)
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Ali, A., Kalisch, H. Mechanical Balance Laws for Boussinesq Models of Surface Water Waves. J Nonlinear Sci 22, 371–398 (2012). https://doi.org/10.1007/s00332-011-9121-2
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DOI: https://doi.org/10.1007/s00332-011-9121-2