Journal of Nonlinear Science

, Volume 22, Issue 3, pp 371-398

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Mechanical Balance Laws for Boussinesq Models of Surface Water Waves

  • Alfatih AliAffiliated withDepartment of Mathematics, University of Bergen
  • , Henrik KalischAffiliated withDepartment of Mathematics, University of Bergen Email author 


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.


Water waves Boussinesq systems Conservation laws Pressure

Mathematics Subject Classification (2000)

35Q35 35Q53 76B15 76M45