A High Resolution Normal Mode Solution of Japan Sea

Abstract:

Normal mode calculation of a semi-closed or completely closed bay or ocean basin helps us to understand the oscillation characteristics including those excited by incoming tsunamis. In addition, tsunami propagation can be synthesized by superposition of normal modes.

Japan Sea is an almost closed ocean basin where many large tsunamigenic earthquakes occurred (fig. 1). Satake and Shimazaki (1988) calculated the normal modes using a 20km grid (~10’ or about 2,000 ocean grids), compared the observed and calculated normal modes from the 1964 Niigata and 1983 Japan Sea earthquakes, and discussed the their different excitation characteristics . Because of development of computer and numerical computation techniques, it is worthwhile to revisit this problem.

Starting from Laplace’s tidal equations and ignoring the rotation of the earth, Loomis (1975) discretized the problem into the eigenvalue problem of a symmetric sparse matrix, which was solved by Householder transformations. This method is used by Satake and Shimazaki (1988) for Japan Sea and Aida (1996) for Tokyo Bay. However, this method needs O(n^3) operation in time and O(n^2) in memory (n is the total number of water grid. e.g., for Japan Sea in 30 sec grid, n~10^6), which would require a super computer.

To overcome this disadvantage, we first introduce a recent iteration method called Implicitly Restarted Arnoldi Method (Lehoucq et al., 1997), which itself speeds up the calculation a bit. Then after we develop a sparse version of matrix storage and multiplication, the operation count in time and memory reduced dramatically to O(n^1.5) (including about 0.5 for iteration process) and O(n) respectively, utilizing the special property of the matrix and the iteration method. This means any current computer can easily solve a large eigenvalue problem.

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