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A new method valid for highly dispersive and highly nonlinear water waves is presented. It combines a time-stepping of the exact surface boundary conditions with an approximate series expansion solution to the Laplace equation in the interior domain. The starting point is an exact solution to the Laplace equation given in terms of infinite series expansions from an arbitrary z-level. We replace the infinite series operators by finite series (Boussinesq-type) approximations involving up to fifth-derivative operators. The finite series are manipulated to incorporate Padé approximants providing the highest possible accuracy for a given number of terms. As a result, linear and nonlinear wave characteristics become very accurate up to wavenumbers as high as kh = 40, while the vertical variation of the velocity field becomes applicable for kh up to 12. These results represent a major improvement over existing Boussinesq-type formulations in the literature. A numerical model is developed in a single horizontal dimension and it is used to study phenomena such as solitary waves and their impact on vertical walls, modulational instability in deep water involving recurrence or frequency downshift, and shoaling of regular waves up to breaking in shallow water.

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[1] Since the 1970s, solitary waves have commonly been used to model tsunamis especially in experimental and mathematical studies. Unfortunately, the link to geophysical scales is not well established, and in this work, we question the geophysical relevance of this paradigm. In part 1, we simulate the evolution of initial rectangular-shaped humps of water propagating large distances over a constant depth. The objective is to clarify under which circumstances the front of the wave can develop into an undular bore with a leading soliton. In this connection, we discuss and test various measures for the threshold distance necessary for nonlinear and dispersive effects to manifest in a transient wave train. In part 2, we simulate the shoaling of long smooth transient and periodic waves on a mild slope and conclude that these waves are effectively non-dispersive. In this connection, we discuss the relevance of finite amplitude solitary wave theory in laboratory studies of tsunamis. We conclude that order-of-magnitude errors in effective temporal and spatial duration occur when this theory is used as an approximation for long waves on a sloping bottom. In part 3, we investigate the phenomenon of disintegration of long waves into shorter waves, which has been observed, for example, in connection with the Indian Ocean tsunami in 2004. This happens if the front of the tsunami becomes sufficiently steep, and as a result, the front turns into an undular bore. We discuss the importance of these very short waves in connection with breaking and runup and conclude that they do not justify a solitary wave model for the bulk tsunami.

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Boussinesq formulations valid for highly dispersive and highly nonlinear water waves are derived with the objective of improving the accuracy of the vertical variation of the velocity eld as well as the linear and nonlinear properties. First, an exact solution to the Laplace equation is given in terms of in nite-series expansions from an arbitrary z-level which is a constant fraction of the still-water depth. This de nes the fully dispersive and fully nonlinear water-wave problem in terms of ve variables: the free-surface elevation and the horizontal and vertical velocities evaluated at the free surface and at the arbitrary z-level. Next, the in nite series operators are replaced by nite-series (Boussinesq-type) approximations. Three di¬erent approximations are introduced, each involving up to fth-derivative operators, and these formulations are analysed with respect to the linear-velocity pro le, linear dispersion and linear shoaling. Nonlinear characteristics are investigated by a perturbation analysis to third order for regular waves and to second order for bichromatic waves. Finally, a numerical spectral solution is made for highly nonlinear steady waves in deep and shallow water. It can be concluded that the best of the new formulations (method III) allows an accurate description of dispersive nonlinear waves for kh (wavenumber times water depth) as high as 40, while accurate velocity pro les are restricted to kh < 10. These results represent a major improvement over existing Boussinesq formulations from the literature.

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