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[HDS13-10] Improvement for Finite-Difference Formula of Non-Linear Longwave Equations to Express Tsunami Decay
Keywords:Far-field tsunami, Numerical simulation, Supercomputer
In this paper, we focused on bottom friction of tsunamis, which is an important factor in decay, and improved the differential formula of nonlinear longwave equations. First, in the friction term in the equation, the position difference of the total water depth (D) for calculating the friction term has been calculated as the arithmetic mean of the two adjacent meshes (central difference). If you use arithmetic averaging, there is a possibility that non-negligible errors may occur compared to when solving continuously. This is because D changes in the friction term on the order of -1/3. Therefore, the positional difference was improved to the one using the average value theorem (fig1). Furthermore, with regard to the time difference, assuming that dt is dt/n within one calculation step dt seconds and assuming that the calculation is repeated n times, it is possible to take the limit at n → ∞, and the difference expression can be transformed as shown in fig 2.
Using these improved difference formula and appropriate coefficients, we calculated in the case of the 2010 Chile earthquake tsunami (Mw 8.8), and as a result its deay is larger than the conventional formula. In the example of fig3, the total energy amount of the later part, which became larger than the observation amount, decreased by about 4% and approached the observation amount. As a calculation condition of the tsunami, all areas including the entire Pacific Ocean (E100° to W90°) are assumed to be 30 seconds mesh (GEBCO 30 sec-grid), and in the area surrounding Hokkaido, Honshu, Shikoku and Kyushu, nesting is performed with a 10 second mesh, and we calculated for a long time (72 hours) in order to see the decay process. For the calculation we use JAGURS (Baba et al., 2015) , considering the elastic loading of the crust and the density effect of seawater, and using the result of the fault parameter (Fujii & Satake, 2013).
By using this formula, the decay of the tsunami becomes larger than the conventional formula by about 5%, and it is 10% or more in large case when the total depth D is small (generally less than 1 m) such as run-up and exposure. Also, by improving the time difference, it is possible to take longer time steps and it is possible to shorten the calculation time.
Although the decay process of the tsunami was not correctly expressed by the previous numerical calculation, it became possible to express the decay process more correctly by this method. In the future, we will continue to study this method and apply this method to various cases to improve the accuracy of the tsunami decay process.