A new Korteweg-de Vries type of equation for uni-directional waves over slowly varying bottom has been derived in Part I. The equation retains the Hamiltonian structure of the underlying complete set of equations for surface waves. For flat bottom it reduces to the standard Korteweg-de Vries equation. Uniform travelling waves (solitary and cnoidal waves) that exist when the bottom is flat will distort over a varying bottom. In this paper, the distortion of periodic and solitary travelling waves will be studied. The distortion is in the first instant approximated by a quasi-homogeneous succession of uniform waves, each one being determined by specifying the horizontal momentum (and hence the amplitude) at the location of the wave. The changing value of the momentum with position is found first from energy conservation. For periodic, cnoidal waves, for which the mass vanishes, the change of wavelength has to be taken into account; some numerical results are given. Solitary waves carry a mass that depends on the amplitude (momentum) and the quasi-homogeneous approximation has to be modified to satisfy mass-conservation. This is achieved by introducing an additional parameter in the base functions with which the distortion is approximated. Instead of using pure solitary waves, one modification consists of adding a tail of finite, but varying length and amplitude. When the bottom decreases sufficiently fast far away from the wave, an alternative description of the distortion will be presented as a succession of solitary waves above a varying, non-flat equilibrium elevation of the surface. In both cases, the dynamic equations obtained from energy and mass conservation differ in essential order from the result without modification.

The exact equations for surface waves over an uneven bottom can be formulated as a Hamiltonian system, with the total energy of the fluid as Hamiltonian. If the bottom variations are over a length scale L that is longer than the characteristic wavelength e, approximating the kinetic energy for the case of "rather long and rather low" waves gives Boussinesq type of equations. If in the case of an even bottom one restricts further to uni-directional waves, the Korteweg-de Vries (KdV) is obtained. For slowly varying bottom this uni-directionalization will be studied in detail in this pan I, in a very direct way which is simpler than other derivations found in the literature. The surface elevation is shown to be described by a forced KdV-type of equation. The modification of the obtained KdV-equation shares the property of the standard KdV-equation that it has a Hamiltonian structure, but now the structure map depends explicitly on the spatial variable through the bottom topography. The forcing is derived explicitly, and the order of the forcing, compared to the first order contributions of dispersion and nonlinearity in KdV, is shown to depend on the ratio between ~ and L; for very mild bottom variations, the forcing is negligible. For localized topography the effect of this forcing is investigated. In part II the distortion of solitary waves will be studied.

No abstract

We discuss the implementation of the finite volume method on a staggered grid to solve the full shallow water equations with a conservative approximation for the advection term. Stelling & Duinmeijer [15] noted that the advection approximation may be energy-head or momentum conservative, and if suitable which of these to implement depends upon the particular flow being considered. The momentum conservative scheme pursued here is shown to be suitable for 1D problems such as transcritical flow with a shock and dam break over a rectangular bed, and we also found that our simulation of dam break over a dry sloping bed is in good agreement with the exact solution. Further, the results obtained using the generalised momentum conservative approximation for 2D shallow water equations to simulate wave run up on a conical island are in good agreement with benchmark experimental data.

We consider the solitary wave solutions of a Korteweg-de Vries equation, where the coefficients in the equation vary with time over a certain region. When these coefficients vary rapidly compared with the solitary wave, then it is well known that the solitary wave may fission into two or more solitary waves. On the other hand, when these coefficients vary slowly, the solitary wave deforms adiabatically with the production of a trailing shelf. In this paper we re-examine this latter case, and show that the trailing shelf, on a very long time-scale, can lead to the generation of small secondary solitary waves. This result thus provides a connection between the adiabatic deformation regime and the fission regime.