Effect of static compression on near-field tsunami waves: Three-dimensional solution

Abstract

An analytical solution of three-dimensional surface wave profiles due to arbitrary spatio-temporal disturbance of a circular ocean bottom in a compressible ocean is obtained by incorporating the influence of the static ocean background compression under the assumptions of linearized water wave theory. Time-domain simulations of the surface profile in three dimensions and the pressure distribution within the water column for a circular uniform rise and tilt are shown. The corresponding animation movies depict the temporal evolution of the surface profile and pressure field inside the water column eloquently, which was not shown in earlier literature. The impact of static compression is also discussed through the simulations. A novel analytical expression of the potential function for a generic tilted motion ($r^{m}cos\left(m\theta \right),m\in Z$) of the circular ocean floor is derived. An efficient numerical code is developed to find surface elevation and pressure distribution, implementing the inverse Fourier integral as matrix multiplication. Validation is performed for the specific case of a rising flat ocean floor, showing the oscillations due to acoustic-gravity modes. Initially, a simplified problem of a flat rising ocean bottom is solved using the eigenfunction matching method, which involves finding a particular solution for the nonhomogeneous ocean bottom condition and the solution for its homogeneous counterpart. Solutions are obtained using a newly developed inner product between the depth-dependent functions. Later, a Green function technique is used to incorporate the impact of the arbitrary spatio-temporal motion of the circular portion of the ocean bed. The solution obtained from the eigenfunction matching method is utilized to obtain the analytical form of Green’s function and, eventually, an expression of surface elevation and pressure distribution inside the ocean water column.