9.1 Introduction

The 2011 off the Pacific coast of Tohoku Earthquake Tsunami (denoted 2011 Tohoku Earthquake Tsunami hereafter for simplicity) was a tremendous and tragic earthquake-tsunami disaster for Japan. An earthquake of magnitude 9.0 occurred off the Pacific coast of Tohoku, Japan, on March 11, 2011, at 14:46:23 Japan Standard Time (+9 UTC) and the rupture area, assumed to be approximately 450 × 200 km, generated a tsunami that struck Japan from Hokkaido to Kyushu as well as various locations around the Pacific Ocean.

The tsunami first reached the Japanese mainland about 20 min after the earthquake and ultimately affected a 2,000 km stretch of Japan’s Pacific coast (The 2011 Tohoku Earthquake Tsunami Joint Survey Group 2011). The Tohoku region consists of several prefectures ranging from north to south, Aomori Prefecture, Iwate Prefecture, Miyagi Prefecture, and Fukushima Prefecture, which border the Pacific Ocean. Sendai is the largest city in the region. The southern part of Tohoku is relatively flat, especially the Sendai plain. The coastal geomorphology of northern Tohoku features ria coasts, which are steep, narrow bays. The northeastern part of Pacific side of Tohoku is known by the name of Sanriku region starting from North Miyagi Prefecture to South Aomori Prefecture. As of August 8 in 2012, official fatalities were 15,867 with an additional 2,903 missing (National Police Agency of Japan). The major cause of death was the tsunami, and most fatalities occurred in Tohoku, 58 % in Miyagi Prefecture, 33 % in Iwate Prefecture and 9 % in Fukushima Prefecture. There were 126,631 totally damaged and 272,653 partially damaged buildings, along with 116 bridges (National Police Agency 2014).

Before this event, the risk of earthquakes and tsunamis off the Tohoku coast was believed to be high. The Japanese government reported that magnitude 7.5 and 7.7 earthquakes along a 200 km fault offshore of Sendai in southern Sanriku-Oki were expected to occur with 99 % probability and 70–80 % probability within 30 years, respectively (The Headquarters for Earthquake Research Promotion 2005). The 1896 Meiji Sanriku earthquake (Mw 8.2–8.5) and tsunami caused 21,915 deaths, the 1933 Showa Sanriku earthquake (Mw 8.1) and tsunami caused 3,064 deaths, and smaller tsunamis have occurred roughly every 10–50 years. Thus, earthquake and tsunami disaster countermeasures, such as seawalls, gates and offshore tsunami breakwaters, planted trees as a natural tsunami barrier, vertical evacuation buildings, and periodic evacuation training were implemented and practiced in these areas. Therefore, we emphasize that Tohoku was an area highly prepared for tsunamis. Nevertheless, the tsunami disaster countermeasures were insufficient against the 2011 event. Tsunami barriers (onshore and offshore breakwaters, and natural tsunami barriers) were severely damaged, some reinforced concrete buildings were totally destroyed, and the extent of inundation was underestimated in several areas.

This event is important for future tsunami preparation to classify various modern counter measures against mega-tsunami. Therefore, understanding the effectiveness of countermeasures and estimation of error of impact assessments are critical. First, this paper re-analyzed the tsunami inundation heights and run-up heights by the three different numerical models at Kamaishi bay where offshore breakwater was installed before the event. The effectiveness of offshore breakwater and error of numerical model is discussed in comparison with the survey data. The relation between damage and tax revenue is discussed to understand long-term impact on the catastrophic event and its protection.

9.2 Numerical Modeling

9.2.1 Outline of Numerical Models

A series of numerical simulations were performed to estimate the influence of offshore breakwaters on the tsunami inundation heights and to check the uncertainty of numerical modeling for inundation of mega-tsunami. The calculations were conducted by three different methods.

The first numerical model is a conventional nonlinear shallow water equation that assumes irrotational, inviscid and hydrostatic pressure distribution vertically (denoted 2D hereafter). The 2D model uses a standard staggered grid system horizontally and the leapfrog method for time discretization so that the linear terms in the scheme are 2nd-order accurate everywhere. The 2D method was developed in house but it follows standard methodology of tsunami wave modeling. We use the 2D model as a reference to the standard way of impact assessment of tsunami inundation modeling (e.g. Pringle and Yoneyama 2013; Goto et al. 1997).

The second numerical model is the quasi-3D model based on the Euler equation that assumes hydrostatic pressure distribution vertically but allows arbitrary distribution of vertical velocity without irrotational and inviscid assumptions (Yoneyama et al. 2012 and Mori et al. 2013; denoted Q3D hereafter). Turbulence mixing is considered using the k-e model vertically and the parameterized Smagorinsky scheme based on the eddy viscosity model horizontally. The Q3D model uses the Euler equation with curvilinear-sigma coordinates vertically. Therefore, the Q3D model can resolve both tsunami wave motion and bathymetry effects independent from the discretization. These characteristics of Q3D model are expected to improve the inundation modeling compared with the 2D model.

The third numerical model is the full 3D model based on the Navier-Stokes equation that contains no approximation for fluid motion except for turbulence modeling (denoted 3D hereafter). The 3D model uses a standard staggered grid system. The free surface motion is simulated by volume of fluid (VOF) method and bottom bathymetry is approximated by an improved FAVOR method. Turbulence mixing is considered using the k-e model. The tsunami modeling by the 3D model with staggered grid system may encounter difficulties when resolving surface motion due to the vertical scale difference from offshore to near shore. For example, 20 vertical layers at a depth of 100 m describe 5 m resolution but this corresponds to only a few grids near shore. Therefore, the 3D model introduces a non-uniform vertical grid system and two-way coupling between the 2D and 3D model (see detail in Pringle and Yoneyama 2013). Due to the non-uniform grid system, the vertical resolution near the mean water level becomes double near the bottom. The two-way coupling solves the offshore region by the 2D model and the 3D model simulates the near shore region. The hybrid system is not only enhanced due to the reduction in computational time but the vertical resolution of the 3D model is increased near shore.

9.2.2 Numerical Conditions

The horizontal mesh size was set as 50 × 50 m for all models. The vertical discretization of the 3D model were set to 2 m near the free surface and down to 5 m below the water column, while 10 vertical layers were used in the Q3D model. The wet-dry condition and offshore radiation boundary condition were applied in the three models.

The target area for computation is Kamaishi Bay area. Bathymetry for both Kamaishi Bay (southern part) and Ryoishi Bay (northern part) are provided at a resolution of 50 m by the Cabinet Office, Government of Japan. Due to the coarse bathymetry information, land areas are described with a uniform height. The computation was carried out with Dt = 0.1 s time intervals from the time of the earthquake until 2 h later. The astronomical tide was not included in all of the 3D, Q3D and 2D computations. The landside structures and land use are regarded as roughness in terms of Manning’s n coefficients based on the Cabinet Office dataset. The time series of measured tsunami by the Kamaishi GPS buoy (Kawai et al. 2011) was used as a lateral boundary condition for the offshore side. The local inundation heights were validated with the measured inundation and run-up heights (TTJS 2011; Mori et al. 2011, 2012).

9.3 Results and Discussion

9.3.1 Numerical Results

A series of numerical computations was performed for Kamaishi Bay and Ryoishi Bay at Iwate Prefecture together because two bays are surrounded by the two long peninsula at the south and north borders, although they are separated by the short peninsula at the middle (see Fig. 9.1). Kamaishi City in Iwate Prefecture is located in the Sanriku ria coast area. An offshore breakwater for tsunami protection is installed at the mouth of Kamaishi Bay. The construction of offshore tsunami breakwater began in 1978. A pair of offshore breakwaters with lengths of 990 and 670 m was finally completed in 2006 in a water depth of 63 m, making it the deepest caisson breakwater in the world (Tanimoto and Goda 1991).

Fig. 9.1
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Maximum water level simulated by three different models (contour: model, circle: survey data). 2D model (a), (b) Q3D model and (c) 3D/2D hybrid model (blue boxes indicate computational region for 3D model)

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It is important to examine the effectiveness of offshore breakwater for 2011 event and estimate numerical errors dependent on the scheme. We have to note that the water depth of Kamaishi Bay and Ryoishi Bay are 50 and 100 m, approximately, therefore direct comparison of the two bays are physically incorrect.

Figure 9.1 shows the maximum water surface elevation for computations with the offshore breakwater for three different methods. The amplification of the tsunami can be seen for several steep valleys in both bays. The influence of the offshore tsunami breakwater can be seen clearly around the breakwater in Fig. 9.1. Within the bay, the maximum surface elevation is reduced from about 12–15 m to 9–11 m depending on the location. This corresponds to a 20–40 % reduction of inundation height along the shores of Kamaishi Bay (Mori et al. 2013). The reduced tsunami energy in Kamaishi Bay gives different damage characteristics and post event impact compared to Otsuchi Bay that will be described in the next section. Regarding the differences between three models, the results outside of Kamaishi bay are quite similar between the 2D, Q3D and 3D models, respectively. The maximum run-up is reached at 30 m maximally at the Ryoishi area. The run-up height is reduced significantly in Kamaishi bay for three models, although they are different at the onshore side. The run-up height of Kamaishi Bay is initially 22 m at the bay mouth, drops to 10 m near the offshore breakwater, and remains roughly constant at 10 m to the shoreline. The 2D or 3D results are higher than the Q3D results at the deep region inside of Kamaishi bay. This is due to reduction of tsunami height near the offshore breakwater. In addition, the 3D model gives slightly higher run-up/inundation height than the 2D model on the landside. This difference is due to the difference in the governing equations and modeling of bottom roughness. Overall, the Q3D model gives the best results with respect to the survey data. The relative computational costs of 3D and Q3D are 20 and 5 times longer respectively, than those of the 2D simulation under an OpenMP environment.

The effects of the offshore breakwater can be verified by the difference between simulations with or without the breakwater. Figure 9.2 shows comparisons of surface elevation on the land between numerical results by the Q3D model with or without the offshore breakwater and survey data (the locations are denoted in Fig. 9.1a). The maximum values of inundation height by the Q3D show relatively good agreement with the survey data except far from coastal line. These locations are strongly influenced by on-land structures that are not considered in the models. Through the comparison of two different runs, it is found that the offshore breakwater reduced the tsunami height by about 25–40 % and significantly lessened the damage at Kamaishi area. The arrival time of the maximum height was delayed 1.8–2.5 min due to the offshore breakwater.

Fig. 9.2
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Comparison of surface elevation on the land between numerical results by Q3D model with/without offshore breakwater and survey data (locations: a–d from top to bottom denoted in Fig. 9.1a, solid line: with breakwater, dashed line: without breakwater, dashed horizontal line: survey data by TTJS 2011 and Mori et al. 2011)

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Figure 9.3 and Table 9.1 indicate the direct comparison of maximum inundation height between three models and survey data along the coast (location numbering is indicated in Fig. 9.1). The location numbers from 1 to 53 are located at Kamaishi Bay and the others are located at Ryoishi Bay . The amplification of the tsunami can be seen for several steep valleys in both bays. Both the 2D model and Q3D model are overestimated in Kamaishi Bay but three models are slightly underestimated in Ryoishi Bay. The root-mean-square error (RMSE), maximum absolute error (MAE) and standard deviation (STD) indicate 2.4–3.9 m errors. These numerical results agree with the observed survey observations within 20 % accuracy and the Q3D model are 15 % better than the other methods. The Q3D model uses terrain following coordinate considering vertical velocity profile which has an advantage over the structured grid system used in the 2D and 3D models. The validation of the numerical model is required for velocity because the damage of structure depends on the momentum of fluids that is proportional to water depth times the square of the velocity. The relative differences in computed peak velocities are approximately 50–250 % onshore. The velocities of the 2D model are smaller than the 3D and Q3D models overall. As a result, the depth-averaged momentum can vary quite significantly depending on the numerical model. Due to page limitation, there is no space to discuss the accuracy of velocity but Fig. 9.4 shows the maximum velocity distribution by the Q3D model. The maximum velocity is reached at 12 m/s and the high velocity regions are located near shore and landside. The numerical modelings of these regions are sensitive to local acceleration of fluids and are highly dependent on the numerical scheme.

Fig. 9.3
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Comparison of maximum inundation height along the coast between three models and survey data (bar: survey data (TTJS 2011; Mori et al. 2011), red line: 3D model, green line: Q3D model, black line with circle: 2D model, position of location number is denoted in Fig. 9.1)

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Table 9.1 The accuracy of maximum inundation height hindcast (unit: m)
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Fig. 9.4
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Maximum depth integrated horizontal velocity by the Q3D model

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Concluding the above discussion, the tsunami wave reduction by offshore breakwater can be estimated in the range of 20 % error and it can be improved by further development of the numerical scheme. The reduced tsunami energy in Kamaishi Bay gives different damage characteristics compared to Otsuchi Bay, which will be discussed in the next section.

9.3.2 Damage and Recovery Relations Based on Tax Revenue

Otsuchi village in Iwate Prefecture is located just north of Kamaishi city in the Sanriku ria coast area. The water depth in the middle of Otsuchi Bay is similar to that of Kamaishi, so that the previous Showa Sanriku tsunami in 1933 and expected tsunami heights were similar for the two locations. For example, the measured inundation heights from the Showa Sanriku tsunami at Otsuchi Bay and Kamaishi Bay were 5.4 m and 6.0 m, respectively. However, the offshore tsunami breakwater was constructed at Kamaishi Bay after that thus the difference of damage can be mainly regarded offshore breakwater effects. A 6.4 m high onshore breakwater protected Otsuchi village. An offshore breakwater protected Kamaishi city.

These two locations show large differences as shown in Fig. 9.5. Two pictures of Kamaishi port and Otsuchi port were taken at April 2011. The damage of Kamaishi port is significantly smaller than Otsuchi port, qualitatively. Collapse of small structures can be seen but larger buildings were relatively undamaged on the Kamaishi waterfront. On the other hand, the most of structures and onshore breakwaters near the port in Otsuchi were totally destroyed as shown in Fig. 9.5b. The fatality ratios of Otsuchi area and Kamaishi area are 8.12 and 2.63 %, respectively. Although the damage to Kamaishi port was severe, it was not at the level of destruction observed at Otsuchi, reinforcing the conclusion that the offshore breakwater significantly reduced the extent of damage in this area.

Fig. 9.5
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Difference of damage at two different bays (April, 2011); Kamaishi port (left; a) and Otsuchi port (right, b)

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Based on analysis of numerical calculations, it can be seen that the influence of offshore tsunami breakwaters significantly reduced the tsunami impact on onshore damage, in comparison with other similar areas such as Otsuchi. Not considering the cost of construction, effectiveness of tsunami mitigation through the use of breakwaters was verified for the first time by the experience of the Tohoku Earthquake tsunami. The degree of damage to the residential area strongly influence on the recovery of the devastated area. Table 9.2 and Fig. 9.6 show the time history of local tax at Kamaishi city and Otsuchi village based on public release, respectively. Kamaishi city includes the southern part of Otsuchi bay, Ryoishi Bay and Kamaishi Bay . Therefore, some part of the damage in the Otsuchi Bay is included in Kamaishi city. The local tax revenue is the sum of city tax, property tax, light vehicle tax and tobacco tax and is one of representative values of local activity. The local tax revenue dropped 28 % from 2008 to 2012 in Kamaishi city but such reduction was more severe in Otsuchi village. The local tax revenue decreased 59 % from 2008 to 2012. The loss of local tax revenue mainly caused by reduction to city tax and property tax (80 %). Once residence disappears and the tax revenue is decreased, it is difficult for the local economy to recover. The minimum protection of the area is necessary even for severe hazards.

Table 9.2 Changes of local tax revenue at Otsuchi and Kamaishi from 2008 to 2013
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Fig. 9.6
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Local tax revenue at Otsuchi and Kamaishi normalized at 2008 (Source: White Papers issued by Otsuchi and Kamaishi cities; financial data available in http://www.town.otsuchi.iwate.jp/bunya/zaisei_jyoho/ and http://www.city.kamaishi.iwate.jp/index.cfm/10,0,79,html respectively)

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9.4 Conclusion

The 2011 Tohoku Earthquake Tsunami was the first case where modern, well-developed tsunami countermeasures were put to the test for such an extreme event. One of the most important issues in natural science, engineering, and social science is to understand the relationships among tsunami forces, local damage, and the community resilience . For future improvement of tsunami disaster countermeasures much can be learned from this catastrophic event.

The damage to coastal structures, ports, houses, buildings, bridges and other infrastructure was strongly dependent on location and protection methods. Here we selected one typical area in the ria coastal region of Sanriku for study, Kamaishi, where an offshore tsunami breakwater was installed as an expensive hardware protection. The offshore tsunami breakwaters, which were partially destroyed, were still effective in mitigating the level of destruction in the Kamaishi port area. We quantitatively discussed the estimated error of the effectiveness of offshore breakwater protection by three different numerical methods, 2D, Q3D and 3D models. The validation results of maximum inundation height on the landside against survey results gave 20 % error of impact assessment of offshore breakwater depends on the numerical method. It indicates that further investigation of numerical modeling on the landside behavior of tsunami is necessary for understanding tsunami dissipation onshore. Furthermore, our analyses of numerical results indicate that such structural protection may have resulted in lower overall inundation heights and could avoid catastrophic destruction of the city. The local tax revenue indicates more than 30 % difference of economic impact due to offshore breakwater and its effect continues a few years after disaster.

From important lessons learned from the 2011 Tohoku tsunami, we have the opportunity to further vital preparation against tsunamis in future.