Time-domain wave response of a compressible ocean due to an arbitrary ocean bottom motion

Abstract

The time-domain motion of a finite depth ocean subject to an arbitrary (in both time and space) imposed displacement of the bottom is studied under the assumption of linear theory. This solution provides results for this limiting case which may be helpful for benchmarking. The focus is on the numerical simulation of the near-field waves with application to the simulation of tsunami waves. The fluid domain is assumed two-dimensional, and the effect of compressibility is included. The time-domain solution is built from the frequency domain solution taking a Fourier series expansion of the bottom motion. This expansion allows complex displacements to be simulated. The solution in the frequency domain is expressed as a sum over modes. The time-domain solution is calculated by numerical evaluation of the Fourier transform in time, allowing arbitrary time-dependent motion. This code is extremely efficient and highly accurate, and there is no time–stepping so that errors do not accumulate in time. The eigenfunction expansion method to obtain the velocity potential for a flat ocean bottom case is independently derived. A shallow water limit for all the above cases is provided, giving a method to check the correctness of the numerical solution. Separate treatment for all the situations under the compressible assumption is also performed. The horizontal and vertical particle velocities are graphically presented for the time-harmonic oscillation. Time-dependent surface wave propagation is computed to show the initiation of tsunami waves in the deep ocean and their subsequent propagation. The calculations presented here allow for the simulation of tsunami wave generation and to investigate various effects, including the role of acoustic gravity waves. It is shown that the compressibility is not always significant, but that when the water is either sufficiently deep or the rise sufficiently rapid, acoustic gravity waves are produced. It is shown that, in this case, the ocean surface undergoes a rapid oscillation and that this may be a method to detect tsunamis.