プレート沈込み型地震に対する半特異面数学モデルの構築

Abstract

In 1750, great mathematician Leonhard Euler developed a comprehensive theory of the motion of a point mass, a massive rigid body, a massive elastic body, and massive fluid (with no viscosity), under a given force [2, 3]. It is amazing that, even now, Euler's system for the earthquake waves requires no change, although it is pretty complicated. Unfortunately, Euler's system has no mechanism of causing the earthquake; the plate tectonics was not known in Euler's era. This paper clarifies the limitations of Euler's system and searches for the origin of big earthquakes. The earthquake of plate-subduction type occurs between a “continental plate” and a “ocean plate”, where the latter subducts the former. We image that the ocean plate is the Pacific plate and the continental plate is the North American plate which covers the north half of Japan. In this case, the ocean plate approaches the continental plate at a speed of about 8 cm/year. In the subduction, both plates touch each other and they slide, without breaking others largely; see Figure 1 in the text. Hence, two plates facing each other have a thin “fault”(断層 in Japanese) between them. First, we express the above situation mathematically. As the 0-th approximation, we assume that the fault width is zero in subduction zone. Then, the plate boundary, let it be 𝐵(𝑥, 𝑦, 𝑧), is settled uniquely. The plates above and below 𝐵(𝑥, 𝑦, 𝑧) are of different speeds, hence 𝐵(𝑥, 𝑦, 𝑧) is a singular plane. Actually, both plates are apart from each other a little. The gap between two plates was measured to be about 8 cm at a point near the Japan trench; see the text around Figure 2. So, we introduce a gap-function 𝐺(𝑥, 𝑦, 𝑧) which simulates the gap, and we replace 𝐵(𝑥, 𝑦, 𝑧) by 𝐺(𝑥, 𝑦, 𝑧). The second point of this paper is the “asperity” proposed by Theme Lay and Hiroo Kanamori in 1980. Asperities are big bumps on the surface of ocean plate. Lay and Kanamori insist that the asperities make the ocean and continental plates “locked”(固着した in Japanese), which is the main reason of big earthquakes. The surface of Pacific plate became clear after 1990s by development of the method of investigating structure of the underground of sea, see Figure 2 in the text. Setting the surface of ocean plate by imitating Figure 2, we propose a realistic model of the plate-locking. Finally, we must say that Euler's system will be refined again and again in future, because we have so far considered neither thermodynamics nor big destruction of the plate itself.