Simulation of distant tsunami propagation with a radial loading deformation effect
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2013
Received: 29 October 2012
Accepted: 19 March 2013
Published: 17 September 2013
A simple parameterization of the loading deformation of the seafloor is incorporated into a tsunami simulation model in order to realistically calculate tsunami travel time, especially at regions far from the source. The parameterization uses one scalar parameter that is optimized effectively by far-field, deep-sea records of recent giant tsunamis: the 2011 Tohoku and the 2010 Chilean tsunamis. Using this parameterization with the optimal values, the observed tsunamis are realistically simulated in both near and far fields. The optimal values seem equivalent for both giant tsunamis, and are relatively smaller than those previously verified for ocean tide modeling, which is reasonable because of the shorter wavelengths of tsunamis.
Key wordsTsunami travel time self-attraction loading
Tsunami propagation has been simulated by long-wave (non-dispersive) equations, and this approximation has been applied to model observed tsunamis especially in near fields (e.g., Aida, 1969; Satake, 1985; Tsushima et al., 2011). Recently, trans-oceanic tsunami simulations, with travel distances exceeding thousands of kilometers, have reported apparent travel time differences between simulations and observations. The simulated travel times of the leading tsunamis at regions further than 10000 km from their sources were systematically shorter than those of observations by 15–20 minutes for cases of recent giant tsunamis generated by the 2004 Sumatra (Rabinovich et al., 2011), the 2010 Chilean (Kato et al., 2011; Fujii and Satake, 2013), and the 2011 Tohoku (Tang et al., 2012; Grilli et al., 2013) earthquakes. The leading wave of the simulated tsunamis by these ordinary tsunami models propagates with a phase velocity of where g is the gravity acceleration and H is the water depth. We need to provide an appropriate mechanism of the delayed propagation found in the observations and its advanced modeling.
The phase velocity due to the short-wavelength dispersion is well known as where k is the wavenum-ber. This wave dispersion arises as wave trains subsequent to the leading wave (e.g., Lamb, 1932; Takahashi, 1942; Kajiura, 1963; Saito et al., 2010).
Meanwhile, interactions between an elastic Earth and a tsunami with long wavelengths have been theoretically predicted (Ward, 1980; Comer, 1984). The interactions can be recognized as another wave dispersion that arises from the elastic seafloor deformation due to the tsunami loading. The loading deformation produces a feedback on tsunami spatial patterns, and delays propagations of the leading tsunami with the longest wavelengths, as mentioned in the next section. Such a feedback is often called the ocean self-attraction and loading (SAL) effect (e.g., Ray, 1998).
It has been known that the SAL effect must be considered for precise ocean tide modeling, especially for global modeling (e.g., Ray, 1998). Precise calculations of the SAL effect need convolutions of the global distribution of oceanic mass loading using spherical harmonics, Love numbers, and an Earth model (e.g., Farrell, 1972; Kantha and Clayson, 2000; Matsumoto et al., 2000), requiring relatively higher computational costs than ordinary tsunami models. On the other hand, the SAL effect can be modeled by a simple parameterization using one scalar parameter (e.g., Accad and Pekeris, 1978; Parke, 1982) without a substantial increase in computational costs. This parameterization yields less accurate results than the convolution method, but is better than modeling without any loading effects. The SAL effects have been also implemented in modern ocean general circulation modeling beyond the ocean tide (e.g., Stepanov and Hughes, 2004; Tamisiea et al., 2010).
The SAL effects have so far not been taken into account in tsunami propagation modeling. In the present study, a SAL effect is considered in a tsunami propagation model in order to examine the discrepancy of the simulated travel time. We implement a simple parameterization based on Accad and Pekeris (1978), and demonstrate that the parameterization works to simulate realistic travel times and waveforms of near- and far-field tsunamis. The efficiency of the parameterization is examined by considering case studies of two giant tsunamis generated by the 2011 Tohoku and the 2010 Chilean earthquakes. Discussions on the parameterized SAL effect are also made.
2. Tsunami Simulation with SAL Effect
This shows that the tsunami propagation speed of the parameterization is slower than , and is non-dispersive. An empirical value of β = ~ 0.1 has been sometimes used for ocean tide modeling (Ray, 1998; Kantha and Clayson, 2000). In the present study, the optimal β for tsunamis is empirically determined by tsunami records obtained by deep-sea ocean bottom pressure (OBP) gauges.
Tsunami simulations start from initial conditions given by the source models: A model with a moment magnitude of 9.0 based on the inversion analysis by Saito et al. (2011) is used for the 2011 Tohoku tsunami. A rectangular fault model with a moment magnitude of 8.8, the same as that of Kimura et al. (2013), is used for the 2010 Chilean tsunami.
3. Basin-Wide Tsunami Data
4. Validation of SAL Effect
Significant RMS reduction is basically identified with maximum correlation coefficients with a zero lag between the observation and simulation. The loading factor (β) is then optimized by the maximum RMS reduction.
5. Discussions on the Optimal β
5.1 β and spatial scale of loading deformation
The difference between the phase velocities with, and without, the SAL effect is , being 0.8–1.0% with the optimal β. Meanwhile, the simulation without the SAL effect shows an apparent linear discrepancy for the travel time commonly for both tsunamis (Fig. 5(a)). The linear discrepancy agrees with the difference of the phase velocities. The optimal β common for both the tsunamis is also verified from Fig. 5(a), because the phase velocity represented by Eq. (4) indicates a linear deviation from .
5.2 Effect of ocean density stratification on β
It has been mentioned above that the distant tsunami propagation is basically explained by the long-wave approximation, i.e., a one-layer ocean model, and the elastic loading deformation. In a strict sense, it is known that the density structure of the real ocean contributes to the slow propagation of ocean gravity waves. So the optimal β to some extent includes the effects of the density structure of the ocean as well. This effect is evaluated in this subsection.
Here, a deviation factor is re-evaluated. For example, a two-layer model for investigating the Kuroshio current, by Isobe and Imawaki (2002) used parameters of h1 = 600 m, h2 = 2400 m, and ε = 0.0020, and give a deviation factor of 0.99984. Parameters found in other several two-layer models on western boundary currents (Hurlburt, 1986; Yoon and Yasuda, 1987; Endoh and Hibiya, 2000) give a deviation factor 0.99981–0.99990, being a 0.01–0.02% deviation from . Therefore, the delay of the distant tsunami propagation behind (0.8–1.0% deviation) arises mostly from the SAL effect, and weakly from ocean density stratification. Similar considerations have recently been conducted also by Watada et al. (2012) and Tsai et al. (2013).
6. Summary and Perspective
A simple parameterization of radial seafloor deformation due to tsunami loading is implemented in a global tsunami propagation model. This loading effect can well account for observed leading tsunamis especially in terms of the far-field travel time from their source regions. Optimal values of the scalar parameter are β = 0.015–0.020, and equivalent for both cases of the 2011 Tohoku and the 2010 Chilean tsunamis. The optimal β indicates that the spatial scales of giant tsunamis are 400–700 km. A tsunami with such spatial scales generates an elastic, vertical seafloor deformation: for example, a 1.5–2.0 cm subsidence of the seafloor is generated in response to a 1-m tsunami loading. As a consequence, the phase velocity of a giant tsunami is slower than .
We recommend that the loading effect is considered for simulating distant tsunami propagations with travel times in excess of several hours where discrepancies of simulation without loading effects result in tsunami arrivals of more than several minutes earlier than the observed tsunami (Fig. 5). Though a smaller β may be preferred for tsunamis with smaller spatial scales, whose seismic magnitudes are probably smaller (M < 7–8), such smaller tsunamis become invisible in far fields. Also, changing β barely deteriorates simulated results, including tsunami waveforms, for near-field calculations (Figs. 4(e)–(f)). Thus, from the viewpoint of practical purposes, tsunami calculations using parameterization with the optimal value will also be useful for moderate tsunamis, as well as giant ones. Owing to an easy implementation and no increase in computational costs compared to standard long-wave calculations, the proposed method will immediately work for the early prediction/warning of future great tsunamis in both near and far fields (e.g., Wei et al., 2008; Tang et al., 2009; Tsushima et al., 2011), and possibly enable source inversions with Green’s functions including far-field data (e.g., Fujii and Satake, 2013).
The loading parameterization proposed in this study is the same method as used for classical ocean tide modeling. As already noted by ocean tide modelers, convolutions of the whole global oceanic mass loading, rather than a simplified parameterization, are preferred to more precisely calculate the loading deformation and its feedback (Ray, 1989; Matsumoto et al., 2000). Also, the interaction may be appropriately solved by a recent advanced simulation scheme of Maeda and Furumura (2013) to collectively calculate seismic waves, ocean acoustic waves, and tsunamis. More powerful computer facilities than current supercomputer systems will be required to enable these precise tsunami calculations.
The OBP data were downloaded from the websites of DART (http://www.ndbc.noaa.gov/dart.shtml) and JAMSTEC (http://www.jamstec.go.jp/scdc/top_e.html). Numerical simulations were carried out by the Altix 4700 supercomputer system of the National Institute for Earth Science and Disaster Prevention. We thank two reviewers for their valuable comments.
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