This paper presents a theory of tsunami generation and propagation on a spherically symmetric, self-gravitating, elastic Earth in terms of normal modes. We predict the character of newborn tsunamis at regional far field distances from a simply parameterized moment tensor point source with step function time history. Tsunami eigenfunctions are shown to penetrate the Earth from tens of kilometers at 2, 000 second period to tens of meters at 20 second period. This behavior explains why the longest tsunami periods are the only ones influenced by crust and mantle structure and why they are preferentially excited by submarine earthquakes. We find that large tsunamis need sizable parent earthquakes because over 96% of their energy is concentrated in the ocean. This makes the entire solid Earth virtually a node for tsunami generation. The excitation of tsunami modes is strongly dependent upon the moment, mechanism and depth of faulting. Calculated tsunami energy, E_T, can vary by a factor of 100 for sources of equal moment within 30 kilometers of the sea floor. Maximum E_T for dip slip and strike slip faulting with moment M₀=10²⁰N·m is 1.5×10¹³ and 1.1×10¹² joules. With mechanism and depth fixed, this source model predicts that E_T is proportional to M₀². The ratio of tsunami to radiated seismic energy is less than 1% for all but the largest events. We believe that this theory coupled with a seismic source recovery technique could be a realistic basis for the forecasting of potentially dangerous tsunamis in real time.

View full abstract

Elastic constants of single-crystalline Fe_1-xO (x=0.08), MnO and CoO have been measured by the rectangular parallelepiped resonance (RPR) method at temperatures between 30°C and Néel point T_N (T_N≅-87°C for Fe_0.92O, ≅ -150°C for MnO and ≅16°C for CoO). In Fe_0.92O and MnO, the shear constant C₄₄ (or Cs for CoO) shows a drastic change at and around the Néel temperature, while the bulk modulus Ks and another shear constant Cs (or C₄₄ for CoO) continue to show a linear temperature dependence even in the vicinity of the Néel point. The anomalous temperature variation of C₄₄ in Fe_0.92O persists above room temperatures. By using several empirical rules, the elastic properties of stoichiometric wüstite and magnesiowüstite are also evaluated. The elastic constants (Mbar) and their temperature coefficients (kbar/deg) at room temperature are:

View full abstract

A theoretical study is made on the scattering of a Rayleigh wave which is incident on a rough semi-circular surface. Partition of energy fluxes of the waves scattered by the semi-circular rough surface is obtained. It is then revealed that the resonances of P and S waves occurring in the semi-circular body considerably suppress the transmission of the incident Rayleigh wave through the semi-circular rough surface, while the Rayleigh wave resonance appearing along the curved free surface of the semi-circular body strengthens the transmission of the incident Rayleigh wave. Directivity of the energy flux of the waves scattered into the semi-infinite elastic body is also discussed; partition of the energy fluxes of the waves scattered into two quarter spaces which are on the incoming and leaving sides of the incident wave greatly depends on two factors, the relative size of the semi-circular body to the wavelength of the incident Rayleigh wave and the appearance of the resonance of the Rayleigh wave in the semicircular body. In the semi-infinite elastic body, the zone of high energy density of the scattered S waves runs at an angle of 45° to the plane free surface; this zone is named breathing zone, since the zone is produced as the results of the interchange of energy between the generated Rayleigh waves and the scattered S waves. As for the phase lag of the transmitted Rayleigh wave, it is retarded in the range of kb_??_1, 0 (k: wave number of the incident wave, b: radius of the semi-circular rough surface) owing to the generation of retarded waves in the semi-circular body, while it is advanced in the range of kb_??_1.0 due to the appearance of the P and S waves which are transmitted through the medium below the semi-circular body directly from its front root to the rear root.

View full abstract

Wave trains immediately following the initial motion of P waves are analyzed assuming that they are scattered P waves by heterogeneities in the earth medium. Three-dimensional partition of the energy density of scattered P waves is theoretically studied, where PP single isotropic scattering and homogeneous and random distribution of scatterers are assumed. The theoretical analysis shows that the three-dimensional mean trajectory for scattered P waves in velocity space is represented by a spheroid, which is prolate to the propagation direction of the direct P wave, and the spheroid tends to become spherical as time goes on. This model is applied to an analysis of short-period (1-30Hz) records of small local earthquakes observed by a three-component velocity seismometer installed at the bottom of a deep borehole. The three-dimensional partition of energy density is calculated by using the covariance tensor for particle velocity observed. The observed horizontal component of energy density of scattered P waves agrees well with the theoretical one, although the observed vertical component is larger than and the observed radial component is smaller than the theoretical one. Our theoretical model is proved to be good in accounting for horizontal component. In order to explain changes in vertical plane, it is suggested that refraction and reflection at horizontal lateral boundaries and the earth's surface, and/or PS conversion scattering should be considered.

View full abstract

An edge wave traveling along a continental shelf with a sloping region sandwitched by a flat shelf and an ocean bottom is investigated theoretically by estimating the dispersion relation and the amplitude distribution versus the distance from the coast. The results are discussed in comparison with a step shelf model. It is demonstrated that dispersion curves shifted toward a long period range with decrease of a dip angle of a sloping region and that the minimum point of a group velocity curve becomes less clearer in the case of a sloping model. It is found that the maximum amplitude of the Kamchatka tsunami observed in Japan can be accounted for by an edge wave propagated along the shelf. The result is useful for predicting arrival time of the maximum amplitude for a distant tsunami with an oblique incidence to the continental shelf.

View full abstract