[1] A new two-dimensional finite difference numerical scheme is developed to simulate the propagation of distant tsunamis over slowly varying topography with improved dispersion effect of waves. The new scheme solves the shallow water equations on a uniform grid system. However, the actual computation is made on a hidden grid system whose grid size is adjustable according to the condition required to satisfy local dispersion relationships of waves for varying water depth. The present model is tested for the cases of two-dimensional propagation of an initial Gaussian hump over various uniform water depth regions. For a varying topography, a one-dimensional test is conducted. The tests show that accuracy of the numerical model is improved significantly over that of the conventional finite difference models. Finally, the present model is applied to the simulation of a historical tsunami event. The numerical result shows that the submerged rise and ridge play an important role of a waveguide for the tsunamis propagating over that topography.
The parabolic approximation is developed to study the combined refraction/diffraction of weakly nonlinear shallow-water waves. Two methods of approach are used. In the first method Boussinesq equations are used to derive evolution equations for spectral-wave components in a slowly varying two-dimensional domain. The second method modifies the K-P equation (Kadomtsev & Petviashvili 1970) to include varying depth in two dimensions. Comparisons are made between present numerical results, experimental data (Whalin 197 1) and previous numerical calculations (Madsen & Warren 1984).
Two-dimensional Boussinesq-type depth-averaged equations are derived for describing the interactions of weakly nonlinear shallow-water waves with slowly varying topography and currents. The current velocity varies appreciably within a characteristic wavelength. The effects of vorticity in the current field are considered. The wave field is decomposed into Fourier time harmonics. A set of evolution equations for the wave amplitude functions of different harmonics is derived by adopting the parabolic approximation. Numerical solutions are obtained for shallow-water waves propagating over rip currents on a plane beach and an isolated vortex ring. Numerical results show that the wave diffraction and nonlinearity are important in the examples considered.
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