Intrusion Distance and Flow Discharge in Rivers during the 2011 Tohoku Tsunami

On 11 March 2011, the Great East Japan Earthquake generated huge tsunami waves. Then, tsunami propagation occurred in rivers, resulting in further expansion of the flooded area. Full data sets of tsunami characteristics such as tsunami inland and river propagation distances as well as the river geometries from all rivers within the Tohoku District were compiled and analyzed. It was found that tsunami propagation distance in rivers was about 1.2 to 4.5 times that of the inland area. There was a good correlation between propagation distance in rivers and the river bed slope. Furthermore, new empirical formulae for calculating the damping coefficient of tsunami wave height and tsunami intrusion length were successfully derived based on the current comprehensive data sets covering a wide range of river geometry and bottom slope. Moreover, tsunami-induced flow discharge was evaluated by using measured water level variation.


Introduction
At 14:46 on 11 March 2011, the Great East Japan Earthquake occurred off Sanriku Coast near the oceanic trench located on the boundary between the Pacific Plate and North American Plate. This earthquake with a maximum scale of Mw.9.0 (Japan Meteorological Agency) caused a massive tsunami and serious damages along the Pacific Ocean Coast from Hokkaido to Kanto District, especially in Tohoku District, which is located in the Northeast of Japan and consists of Iwate Prefecture, Miyagi, and Fukushima Prefectures. Tsunami waves with a height of over 10m attacked the coastal area, resulting in serious damage in these regions. One of the important factors causing such devastating damage was tsunami run-up into rivers and subsequent overflow into the adjacent land area [1][2][3][4][5].
There have been various researchers studying tsunami propagation in the land area such as [6,7]. However, studies dealing with a synthetic investigation on tsunami intrusion distance in a river are limited. Abe [8] analyzed measured data of water level in five large rivers during the Central Sea of Japan Earthquake in 1983 and concluded that tsunami waves can reach several kilometers upstream of Figure 1 shows the location of rivers investigated in the present study. There are in total 39 rivers covering wide a range of river geometries in Iwate, Miyagi, and Fukushima Prefectures in the Tohoku District of Japan. In the Japanese river classification system, the rivers are divided into Class-A and Class-B based on their magnitude and social importance. Class-A rivers are relatively larger in terms of catchment and river morphology than Class-B rivers. Class-A rivers are governed by the national government, while Class-B are governed by the prefectural government. The river morphology data were obtained from the corresponding river authorities of each river. In total, data from 21 rivers in Iwate Prefecture, 9 rivers in Miyagi Prefecture, and 9 rivers in Fukushima Prefecture were collected and analyzed. It is noted that this study did not take into account the data from rivers that have been affected by an upstream weir, such as the New Kitakami River and the Abukuma River. The location of the rivers in each prefecture is plotted over the elevation map of Japan at the bottom of Figure 1. It is understood that that the topography of Iwate and Fukushima Prefectures has a steeper slope and relatively short distance than Miyagi Prefecture. On the other hand, rivers that flow through Miyagi Prefecture have larger catchment area and gentler slope.

Study Area and Data Compilation
The tsunami intrusion length was determined based on the field survey immediately after the tsunami occurrence. Locations, such as a tsunami trace and transported debris, were obtained in each river. The most upstream place where the trace was left behind was defined as the intrusion end of the tsunami in the river. Additionally, the distance from a river mouth to an intrusion end was determined as a tsunami intrusion distance. The tsunami intrusion length was determined based on the field survey immediately after the tsunami occurrence. Locations, such as a tsunami trace and transported debris, were obtained in each river. The most upstream place where the trace was left behind was defined as the intrusion end of the tsunami in the river. Additionally, the distance from a river mouth to an intrusion end was determined as a tsunami intrusion distance.
Moreover, if the most upstream water level station data were available and water level changes by tsunami intrusion could be detected then this location was used to determine the tsunami intrusion length.
The riverbed slope in the section from a river mouth to an intrusion end was used for analysis. The riverbed elevation of the Naruse River is illustrated in Figure 2 as an example. As shown in the figure, the riverbed slope from a river mouth to an intrusion end was estimated approximately as a linear function by using a least-squares method to obtain the bed slope. Moreover, if the most upstream water level station data were available and water level changes by tsunami intrusion could be detected then this location was used to determine the tsunami intrusion length.
The riverbed slope in the section from a river mouth to an intrusion end was used for analysis. The riverbed elevation of the Naruse River is illustrated in Figure 2 as an example. As shown in the figure, the riverbed slope from a river mouth to an intrusion end was estimated approximately as a linear function by using a least-squares method to obtain the bed slope.

Relationship between River Tsunami Intrusion Distance and Coastal Tsunami Height
The obtained tsunami intrusion distance, x p , from all analyzed rivers is plotted together with the measured coastal tsunami height (H) data by the Joint Survey Group [24,25] (Figure 3), and the relationship between river tsunami intrusion distance and coastal tsunami height is illustrated in Figure 4. Although it was expected that the intrusion distance of the tsunami is proportional to the coastal tsunami height, they do not have a strong correlation.

Relationship between River Tsunami Intrusion Distance and Coastal Tsunami Height
The obtained tsunami intrusion distance, xp, from all analyzed rivers is plotted together with the measured coastal tsunami height (H) data by the Joint Survey Group [24,25] (Figure 3), and the relationship between river tsunami intrusion distance and coastal tsunami height is illustrated in Figure 4. Although it was expected that the intrusion distance of the tsunami is proportional to the coastal tsunami height, they do not have a strong correlation.    Figure 5 shows the relationships between the river tsunami intrusion distance, , and land tsunami propagation distance, , for three different prefectures. The tsunami propagation distance inland region was determined from the inundation map by Haraguchi and Iwamatsu [26,27]. Based on this result, the river tsunami intrusion distance was 1.2 to 4.5 times that of the tsunami intrusion in land area. Remarkably, intrusion in the Old Kitakami River, the Yoshida River, and the Naruse River in Miyagi Prefecture and Fujiwara River in Fukushima Prefecture followed the straight line of = 4.5 . In other words, the river tsunami intrusion has occurred over a much longer distance than on land regions in these rivers. During the present tsunami, the ascended tsunami to the upstream area caused river embankment overflow and then flooded into the land in many places. Therefore, it is necessary to fully understand these intrusion characteristics and to take into consideration future tsunami disaster prevention.   Figure 5 shows the relationships between the river tsunami intrusion distance, x p , and land tsunami propagation distance, x L , for three different prefectures. The tsunami propagation distance inland region was determined from the inundation map by Haraguchi and Iwamatsu [26,27]. Based on this result, the river tsunami intrusion distance was 1.2 to 4.5 times that of the tsunami intrusion in land area. Remarkably, intrusion in the Old Kitakami River, the Yoshida River, and the Naruse River in Miyagi Prefecture and Fujiwara River in Fukushima Prefecture followed the straight line of x p = 4.5x L . In other words, the river tsunami intrusion has occurred over a much longer distance than on land regions in these rivers. During the present tsunami, the ascended tsunami to the upstream area caused river embankment overflow and then flooded into the land in many places. Therefore, it is necessary to fully understand these intrusion characteristics and to take into consideration future tsunami disaster prevention.

Relationship of Tsunami Intrusion Distance in River and Land Area
inland region was determined from the inundation map by Haraguchi and Iwamatsu [26,27]. Based on this result, the river tsunami intrusion distance was 1.2 to 4.5 times that of the tsunami intrusion in land area. Remarkably, intrusion in the Old Kitakami River, the Yoshida River, and the Naruse River in Miyagi Prefecture and Fujiwara River in Fukushima Prefecture followed the straight line of = 4.5 . In other words, the river tsunami intrusion has occurred over a much longer distance than on land regions in these rivers. During the present tsunami, the ascended tsunami to the upstream area caused river embankment overflow and then flooded into the land in many places. Therefore, it is necessary to fully understand these intrusion characteristics and to take into consideration future tsunami disaster prevention.

Relationship between Tsunami Intrusion Distance and Riverbed Slope
For tsunami intrusion process into rivers, riverbed slope is expected to be the most important factor. Based on the current comprehensive data sets from three prefectures, Figure 6 is obtained for the relationship between tsunami propagation distance and river bed slope. By applying the leastsquare method to all data of the tsunami intrusion distance, a new empirical equation is obtained as follows.

Relationship between Tsunami Intrusion Distance and Riverbed Slope
For tsunami intrusion process into rivers, riverbed slope is expected to be the most important factor. Based on the current comprehensive data sets from three prefectures, Figure 6 is obtained for the relationship between tsunami propagation distance and river bed slope. By applying the least-square method to all data of the tsunami intrusion distance, a new empirical equation is obtained as follows.  Tanaka et al. [3] have already proposed another empirical formula based on data sets from Miyagi Prefecture alone. In order to derive a formula for tsunami propagation distance in a river, they assumed the following general exponential relation for the wave attenuation in a channel, as used by [28,29]: where x is the upstream coordinate from a river mouth, H is the wave height, H0 corresponds to the initial incoming wave height at x = 0, and k is the damping coefficient. Based on the measured data from six rivers including the Abukuma River, the Kitakami River, the Old Kitakami River, the Sunaoshi River, the Natori River, and the Naruse River in Miyagi Prefecture, Tanaka et al. [3] obtained an empirical equation for this damping coefficient k and the riverbed slope S as follows: Eq. (4) Eq. (1) Figure 6. Relationship between intrusion distance and riverbed slope.
Tanaka et al. [3] have already proposed another empirical formula based on data sets from Miyagi Prefecture alone. In order to derive a formula for tsunami propagation distance in a river, they assumed the following general exponential relation for the wave attenuation in a channel, as used by [28,29]: where x is the upstream coordinate from a river mouth, H is the wave height, H 0 corresponds to the initial incoming wave height at x = 0, and k is the damping coefficient.
Based on the measured data from six rivers including the Abukuma River, the Kitakami River, the Old Kitakami River, the Sunaoshi River, the Natori River, and the Naruse River in Miyagi Prefecture, Tanaka et al. [3] obtained an empirical equation for this damping coefficient k and the riverbed slope S as follows: Figure 7 indicates an example of tsunami height damping in the Natori River during the 2011 Tohoku Tsunami. From the slope of the regression line, they obtained the damping coefficient for rivers in Miyagi Prefecture.
Tanaka et al. [3] have already proposed another empirical formula based on data sets from Miyagi Prefecture alone. In order to derive a formula for tsunami propagation distance in a river, they assumed the following general exponential relation for the wave attenuation in a channel, as used by [28,29]: where x is the upstream coordinate from a river mouth, H is the wave height, H0 corresponds to the initial incoming wave height at x = 0, and k is the damping coefficient. Based on the measured data from six rivers including the Abukuma River, the Kitakami River, the Old Kitakami River, the Sunaoshi River, the Natori River, and the Naruse River in Miyagi Prefecture, Tanaka et al. [3] obtained an empirical equation for this damping coefficient k and the riverbed slope S as follows:  By assuming that the tsunami height at the most upstream end equal to 5% of the initial tsunami height (H/H 0 = 0.05), the following empirical equation between the tsunami intrusion distance, x p, and riverbed slope, S, is derived: x p = 6.28x10 4 e −4.67×10 3 S (m) In Figure 6, Equation (4) thus obtained is also plotted, which gives an underestimated value for the propagation distance in rivers as compared with the newly derived Equation (1). It is noted that the empirical coefficient in Equation (4) was determined using the data sets from Miyagi Prefecture alone, whereas the new Equation (1) covers a wide range of the riverbed slope by compiling data sets from three prefectures. The new equation enables us to estimate the tsunami intrusion distance in a river more accurately.

Evaluation of Tsunami Damping Coefficient
In the same way as Tanaka et al. [3], a new expression for tsunami damping coefficient can be derived by combining the new empirical Equations (1) and (2) and assuming that tsunami height at the intrusion endpoint is equal to 5% of the tsunami height at the river month. Figure 8 shows comparison results between the old and new equations of the tsunami damping coefficient. The old Equation (3) overestimates the k-value as compared with Equation (5). This is due to the limited data from Miyagi Prefecture alone to obtained Equation (3) in the previous study. It is recommended to use the new Equation (5) to calculate the tsunami height and intrusion distance into rivers. Finally, substituting Equation (5) into Equation (2), a dimensionless exponential relation is obtained for tsunami height attenuation.
to the limited data from Miyagi Prefecture alone to obtained Equation (3) in the previous study. It is recommended to use the new Equation (5) to calculate the tsunami height and intrusion distance into rivers. Finally, substituting Equation (5) into Equation (2), a dimensionless exponential relation is obtained for tsunami height attenuation.

Methodology
Due to increased opportunities for capturing real tsunamis by a video camera in recent years, studies have been carried out to assess flow velocity from video recordings [30][31][32][33]. In this study, a distinctly different approach utilizing measured water level data along a river will be applied in order to evaluate river discharge and velocity induced by tsunami wave propagation. For this purpose, Adityawan et al.

Methodology
Due to increased opportunities for capturing real tsunamis by a video camera in recent years, studies have been carried out to assess flow velocity from video recordings [30][31][32][33]. In this study, a distinctly different approach utilizing measured water level data along a river will be applied in order to evaluate river discharge and velocity induced by tsunami wave propagation. For this purpose, Adityawan et al. [10] used a continuity equation for a one-dimensional flow analysis as where A is the cross-sectional area, t is time, Q is the discharge, and x is the horizontal distance from a reference point (going upstream). Equation (7) is integrated to obtain the following: where η is the water level, B is the river width, Q i is the discharge at cross-section i located at x i , and Q N is the upstream discharge at x N (upstream end). Equation (8) is solved numerically using the finite difference method to obtain discharge and velocity at the measurement point as follows: where ∆t is the time interval according to the measurement. Velocity estimation was conducted in the Naruse and Yoshida Rivers located in Miyagi Prefecture. These two rivers flow nearly parallel, as depicted in Figure 9a, and share the river mouth immediately before it pours to the sea (Figure 9b). The water level variation has been obtained along these rivers during the 2011 Tohoku tsunami at the measuring stations summarized in Table 1. The raw data measured in the rivers with the time interval ∆t = 10 min are illustrated in Figure 10, which can be substituted into Equation (9) for numerical integration. Moreover, measured discharge at the upstream boundary Q N is used in Equation (9).
Prefecture. These two rivers flow nearly parallel, as depicted in Figure 9a, and share the river mouth immediately before it pours to the sea (Figure 9b). The water level variation has been obtained along these rivers during the 2011 Tohoku tsunami at the measuring stations summarized in Table 1. The raw data measured in the rivers with the time interval Δt = 10 min are illustrated in Figure 10, which can be substituted into Equation (9) for numerical integration. Moreover, measured discharge at the upstream boundary is used in Equation (9).

Results and Discussion
Based on Equation (9), temporal variation of the discharge Q and velocity V at an arbitrary point can be estimated as shown in Figure 11. Figure 11a shows the river discharge induced by the tsunami wave in Naruse and Yoshida Rivers. The discharge at the river mouth (Nobiru) was as large as 12,000 m 3 /s with a distinct reduction over time. The magnitude is reduced as the wave propagates upstream. The calculated discharge was converted to velocity by dividing its value with the flow area as shown in Figure 11b. As compared with the past evaluations such as [30][31][32][33], the estimated tsunami flow velocity shown in the figure using the conservation equation is of a similar magnitude and pattern.

Results and Discussion
Based on Equation (9), temporal variation of the discharge Q and velocity V at an arbitrary point can be estimated as shown in Figure 11. Figure 11a shows the river discharge induced by the tsunami wave in Naruse and Yoshida Rivers. The discharge at the river mouth (Nobiru) was as large as 12,000 m 3 /s with a distinct reduction over time. The magnitude is reduced as the wave propagates upstream. The calculated discharge was converted to velocity by dividing its value with the flow area as shown in Figure 11b. As compared with the past evaluations such as [30][31][32][33], the estimated tsunami flow velocity shown in the figure using the conservation equation is of a similar magnitude and pattern.

Results and Discussion
Based on Equation (9), temporal variation of the discharge Q and velocity V at an arbitrary point can be estimated as shown in Figure 11. Figure 11a shows the river discharge induced by the tsunami wave in Naruse and Yoshida Rivers. The discharge at the river mouth (Nobiru) was as large as 12,000 m 3 /s with a distinct reduction over time. The magnitude is reduced as the wave propagates upstream. The calculated discharge was converted to velocity by dividing its value with the flow area as shown in Figure 11b. As compared with the past evaluations such as [30][31][32][33], the estimated tsunami flow velocity shown in the figure using the conservation equation is of a similar magnitude and pattern.  Figure 12 shows a comparison result between the design flood discharge and maximum river discharge induced by the tsunami in the Naruse and Yoshida Rivers. The return period of the design flood is 100 years [34]. The maximum discharge due to the tsunami propagation at the river mouth was computed as more than two times that of the design flood discharge. In fact, this area was affected severely by the tsunami propagation and inundation due to the extreme discharge much bigger than the design flood discharge [35,36].  Figure 12 shows a comparison result between the design flood discharge and maximum river discharge induced by the tsunami in the Naruse and Yoshida Rivers. The return period of the design flood is 100 years [34]. The maximum discharge due to the tsunami propagation at the river mouth was computed as more than two times that of the design flood discharge. In fact, this area was affected severely by the tsunami propagation and inundation due to the extreme discharge much bigger than the design flood discharge [35,36].  Figure 12 shows a comparison result between the design flood discharge and maximum river discharge induced by the tsunami in the Naruse and Yoshida Rivers. The return period of the design flood is 100 years [34]. The maximum discharge due to the tsunami propagation at the river mouth was computed as more than two times that of the design flood discharge. In fact, this area was affected severely by the tsunami propagation and inundation due to the extreme discharge much bigger than the design flood discharge [35,36].

Conclusions
A comprehensive study of river tsunami intrusion distance and tsunami-induced flow discharge in Tohoku District was carried out by analyzing the collected data sets after the 2011 Tsunami in Japan. The main conclusions are as follows.
(1) There is no clear relationship between the coastal tsunami height and tsunami intrusion distance in the river. (2) The tsunami propagation distance inside of the river is approximately 1.2 to 4.5 times longer than the travel distance over inland areas. (3) Based on the comprehensive data compilation, new empirical equations were obtained for calculating the damping coefficient and tsunami intrusion distance using the riverbed slope.
The new equation results compared well with the measured data in three prefectures in Tohoku District. The new equation forms are similar to the equations by Tanaka et al. [3], but the present equations cover a wide range of river morphologies and geometries. Therefore, as compared with the previously proposed equation by Tanaka et al. [3], the applicability of the present equations will be higher. However, further verification of will be required by adding more data based on field observations, numerical experiments, and laboratory experiments. (4) The continuity equation is applied to estimate the flow discharge during the 2011 tsunami.
The magnitude of the velocity is similar to those obtained from video analysis by other researchers.
Author Contributions: The first three authors, H.T., N.X.T., N.T.H., and K.K., mainly contributed to this article. The first author led group discussion and checked the contents, whereas the second and third authors collected and analyzed the field measurement data sets. The rest of the co-authors, M.R., M.U., M.S., S.K. and M.T., made contribution in field investigation to collect the tsunami propagation distance. All authors have read and agreed to the published version of the manuscript.
Funding: This work was funded by the TAISEI Research Foundation.