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

Home , Abukuma River, Aomori, Fukushima (city), Fukushima Prefecture, Iwate Prefecture, Kitakami River

Journal of Marine Science and Engineering

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

Hitoshi Tanaka 1,* , Nguyen Xuan Tinh 1 , Nguyen Trong Hiep 1 , Kosuke Kayane 2, Min Roh 3, Makoto Umeda 1, Mikio Sasaki 4, Seiki Kawagoe 5 and Mitsukuni Tsuchiya 6 1 Department of Civil Engineering, Tohoku University, Sendai 980-8579, Japan; [email protected] (N.X.T.); [email protected] (N.T.H.); [email protected] (M.U.) 2 Plant Solution Department, TEPCO Fuel & Power, Inc., Tokyo 100-8560, Japan; [email protected] 3 National Institute of Meteorological Sciences, Jeju 63568, Korea; [email protected] 4 Department of Civil and Environmental Engineering, Hachinohe Institute of Technology, Hachinohe 031-8501, Japan; [email protected] 5 Division of Environment System Management, Fukushima University, Fukushima 960-1296, Japan; [email protected] 6 Formerly Department of Civil and Environmental Engineering, Maebashi Institute of Technology, Maebashi 371-0816, Japan; [email protected] * Correspondence: [email protected]

Received: 14 October 2020; Accepted: 4 November 2020; Published: 5 November 2020

Abstract: 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.

Keywords: Great East Japan Earthquake; tsunami intrusion into rivers; intrusion length; river bed slope; tsunami-induced ﬂow discharge

1. 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 [1–5]. 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

J. Mar. Sci. Eng. 2020, 8, 882; doi:10.3390/jmse8110882 www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2020, 8, 882 2 of 12 rivers and the tsunami peaks have resulted from resonance within the river. Tanaka et al. [9], based on a field investigation along Sri Lankan rivers affected by the 2004 Indian Ocean Tsunami, indicated that the tsunami intrusion into a small river upstream caused flooding and local damage, extending the impact of the tsunami to the upstream cities. Similar conclusions were also obtained by analyzing the 2010 Chilean Tsunami that impacted Japanese rivers [10]. Based on the analysis of video images captured during the 2011 tsunami, Adityawan et al. [11] reported that the tsunami in a river propagates a longer distance upstream than that propagating in a land area. Tolkova and Tanaka [12] analyzed 2011 Tohoku tsunami record in the Kitakami River and showed that the bore kept its shape and identity as it traveled a 10.9 km path along the river. Due to the fact that the tsunami ascended the upstream area, the river embankment and various river management facilities were seriously damaged. More recent tsunami propagation into Japanese rivers during the September 2015 Chilean tsunami has been analyzed by Tolkova [13]. In terms of numerical studies, Viana-Baptista et al. [14] and Yeh et al. [15] made a numerical simulation of tsunami propagation into the Tagus estuary, Lisbon, and a hypothetical case in the Columbia River, respectively. They showed that tsunami propagation in a river causes severe flooding in the upstream area far from the river mouth. Reference [16–23] also used numerical methods to predict tsunami propagation along a river. They concluded that there are various factors contribute to tsunami wave intrusion in rivers, and that for the relatively small tsunami event the intrusion length correlates well with the river mouth characteristics. Generally, estimation of a tsunami inundation zone has been carried out by a depth averaged 2-D numerical simulation in which ocean and land areas are unified in the computational domain, e.g., [5]. However, because the intrusion characteristics of the tsunami into a river greatly reflect the characteristic of river geometry, it may be difficult to fully understand the behavior of tsunami by a unified numerical computation. In addition, model verification is difficult to achieve due to the lack of actual measurement data. Therefore, it is necessary to advance our understanding of the mechanisms of river tsunami intrusion for future tsunami disaster prevention. In this study, the field measurement data of river tsunami intrusion was collected for numerous rivers from three prefectures in Tohoku District, including Iwate, Miyagi, and Fukushima Prefectures, in order to carry out a synthetic investigation of tsunami intrusion into rivers. As a result, the intrusion distance and tsunami-induced flow discharge were determined. The outcome of this study is valuable for the coastal and river authorities in order to reduce the tsunami wave impact on local residents, and on infrastructural damages, in the future.

2. Study Area and Data Compilation Figure1 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 Figure1. 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. 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 J. Mar. Sci. Eng. 2020, 8, 882 3 of 12 tsunami in the river. Additionally, the distance from a river mouth to an intrusion end was determined as a tsunami intrusion distance. J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 3 of 13

Figure 1. Location of study areas on the Japan map. (a) Rivers in Fukushima Prefecture. (b) Rivers in Figure 1. Location of study areas on the Japan map. (a) Rivers in Fukushima Prefecture. (b) Rivers in Miyagi Prefecture. (c) Rivers in Iwate Prefecture. Miyagi Prefecture. (c) Rivers in Iwate Prefecture. The tsunami intrusion length was determined based on the field survey immediately after the Moreover,tsunami if occurrence. the mostupstream Locations, such water as a level tsunami station trace and data transported were available debris, were and obtained waterlevel in each changes by tsunami intrusionriver. The could most upstream be detected place then where this the location trace was was left usedbehind to was determine defined as the the tsunami intrusion intrusionend of length. The riverbedthe tsunami slope in the in river. the sectionAdditionally, from the a riverdistance mouth from a to river an intrusionmouth to an end intrusion was usedend was for analysis. determined as a tsunami intrusion distance. The riverbed elevationMoreover, if of the the most Naruse upstream River water is level illustrated station data in were Figure available2 as an and example. water level As changes shown in the ﬁgure, theby riverbed tsunami slopeintrusion from could a riverbe detected mouth then to anthis intrusion location was end used was to estimateddetermine the approximately tsunami as a linear functionintrusion by length. using a least-squares method to obtain the bed slope. J. Mar. Sci. Eng. 2020The, 8 riverbed, x FOR PEER slope REVIEWin the section from a river mouth to an intrusion end was used for analysis. 4 of 13 The riverbed elevation of the Naruse River is illustrated in Figure 2 as an example. As shown in the figure, the riverbed slope from20 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. 15

(m) 10 遡上距離 Upstream distance , xp 5 河床高さ

River bed elevation (m) elevation bed River River傾きを河床勾配に設定 bed slope, S -5 0 10000 20000 30000 40000 Longitudinal河口からの距離 river distance(m) (m)

FigureFigure 2. 2. AnAn example example data data of of riverbed riverbed elevation elevation in in the the Naruse Naruse River, River, Miyagi Prefecture.

3. Tsunami Intrusion Distance

3.1. 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 3. (a) Coastline from Iwate Prefecture to Fukushima Prefecture. (b) Coastal tsunami height. (c) River tsunami intrusion distance.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 4 of 13

(m) 10 遡上距離 Upstream distance , xp 5 河床高さ

River bed elevation (m) elevation bed River River傾きを河床勾配に設定 bed slope, S -5 0 10000 20000 30000 40000 Longitudinal河口からの距離 river distance(m) (m) J. Mar. Sci. Eng. 2020, 8, 882 4 of 12 Figure 2. An example data of riverbed elevation in the Naruse River, Miyagi Prefecture.

3. Tsunami3. Tsunami Intrusion Intrusion Distance Distance

3.1. Relationship3.1. Relationship between between River River Tsunami Tsunami Intrusion Intrusion Distance Distance andand Coastal Tsunami Tsunami Height Height

The obtainedThe obtained tsunami tsunami intrusion intrusion distance, distance, xpp, ,from from all all analyzed analyzed rivers rivers is plotted is plotted together together with the with the measuredmeasured coastal coastal tsunamitsunami height ( (HH) )data data by by the the Joint Joint Survey Survey Group Group [24,25] [24, 25(Figure] (Figure 3), 3and), and the the relationshiprelationship between between river river tsunami tsunami intrusion intrusion distance distance andand coastal tsunami tsunami height height is illustrated is illustrated in in FigureFigure4. Although 4. Although it was it was expected expected that that the the intrusion intrusion distance distance of the tsunami tsunami is isproportional proportional to the to the coastalcoastal tsunami tsunami height, height, they they do do not not have have a stronga strong correlation. correlation.

FigureFigure 3. (a) 3. Coastline (a) Coastline from from Iwate Iwate Prefecture Prefecture to Fukushima Fukushima Prefecture. Prefecture. (b) Coastal (b) Coastal tsunami tsunami height. height.(c) River tsunami intrusion distance. J. Mar.(c) Sci. River Eng. tsunami 2020, 8, x intrusion FOR PEER distance. REVIEW 5 of 13

40000 Miyagi岩手県 Pref. Iwate宮城県 Pref.

(m) 30000 Fukushima福島県 Pref. P x

(m) 20000

p (m) x P X

10000 河川遡上距離

0 0 5 10 15 20 25 30 津波高さH ((m)m)H (m) FigureFigure 4. 4.Relationship Relationship between between river river tsunami tsunami intrusion intrusion distance distance and and coastal coastal tsunami tsunami height. height.

3.2.3.2. Relationship Relationship of of Tsunami Tsunami Intrusion Intrusion Distance Distance in in River River and and Land Land Area Area Figure5 shows the relationships between the river tsunami intrusion distance, x , and land Figure 5 shows the relationships between the river tsunami intrusion distance, 𝑥p, and land tsunami propagation distance, x , for three diﬀerent prefectures. The tsunami propagation distance tsunami propagation distance, L𝑥, for three different prefectures. The tsunami propagation distance inlandinland region region was was determined determined from from the the inundation inundation map map by Haraguchiby Haraguchi and and Iwamatsu Iwamatsu [26,27 [26,27].]. Based Based on thison result,this result, the riverthe river tsunami tsunami intrusion intrusion distance distance was was 1.2 1.2 to to 4.5 4.5 times times that that of of the the tsunami tsunami intrusion intrusion inin land land area. area. Remarkably, Remarkably, intrusion intrusion in in the the Old Old Kitakami Kitakami River, River, the the Yoshida Yoshida River, River, and and the the Naruse Naruse RiverRiver in in Miyagi Miyagi Prefecture Prefecture and and Fujiwara Fujiwara River River in in Fukushima Fukushima Prefecture Prefecture followed followed the the straight straight line line of of x = 4.5x . In other words, the river tsunami intrusion has occurred over a much longer distance than p𝑥= 4.5𝑥L . In other words, the river tsunami intrusion has occurred over a much longer distance than onon land land regions regions in in these these rivers. rivers. During During the the present present tsunami, tsunami, the the ascended ascended tsunami tsunami to to the the upstream 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.

40000 Iwate Pref. Miyagi Pref. 30000 Fukushima Pref.

(m) 20000 p x

10000

0 0 2000 4000 6000 8000 10000

xL (m)

Figure 5. Relationship of tsunami intrusion distance inside of river and land area.

3.3. 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.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 5 of 13

40000 Miyagi岩手県 Pref. Iwate宮城県 Pref.

(m) 30000 Fukushima福島県 Pref. P x

(m) 20000

p (m) x P X

10000 河川遡上距離

0 0 5 10 15 20 25 30 津波高さH ((m)m)H (m) Figure 4. Relationship between river tsunami intrusion distance and coastal tsunami height.

3.2. Relationship of Tsunami Intrusion Distance in River and Land Area

Figure 5 shows the relationships between the river tsunami intrusion distance, 𝑥, and land tsunami propagation distance, 𝑥, 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

𝑥J. Mar.= 4.5 Sci.𝑥. Eng.In other2020, 8words,, 882 the river tsunami intrusion has occurred over a much longer distance 5than of 12 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 area caused river embankment overﬂow and then ﬂooded into the land in many places. Therefore, is necessary to fully understand these intrusion characteristics and to take into consideration future it is necessary to fully understand these intrusion characteristics and to take into consideration future tsunami disaster prevention. tsunami disaster prevention.

40000 Iwate Pref. Miyagi Pref. 30000 Fukushima Pref.

(m) 20000 p x

10000

0 0 2000 4000 6000 8000 10000

xL (m)

FigureFigure 5. 5. RelationshipRelationship of of tsunami tsunami intrusion intrusion dist distanceance inside inside of of river river and and land land area. area.

3.3.3.3. Relationship Relationship between between Tsunami Tsunami In Intrusiontrusion Distance Distance and and Riverbed Riverbed Slope Slope ForFor tsunami tsunami intrusion intrusion process process into into rivers, rivers, riverbed riverbed slope slope is is expected expected to to be be the the most most important important factor.factor. Based Based on on the current comprehensivecomprehensive data data sets sets from from three three prefectures, prefectures, Figure Figure6 is 6 obtained is obtained for thefor therelationship relationship between between tsunami tsunami propagation propagation distance distan andce riverand river bed slope.bed slope. By applying By applying the least-square the least- J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 6 of 13 squaremethod method to all data to all of data the tsunamiof the tsunami intrusion intrusion distance, distance, a new empiricala new empirical equation equation is obtained is obtained as follows. as follows. 0.71 xp = 48.4S−.(m) (1) 𝑥 = 48.4𝑆 (m) (1)

40000 Miyagi宮城県 Miyagi Pref.Pref. Iwate岩手県 Iwater Pref. Pref. (m)

p Fukushima福島県 Fukushima Pref. x (m) 30000 P Eq. 式Kayane (3)(1) et al., 2012 x =×43 −× Eq.xSp (4)6.33 10 exp() 4.67 10

(m) 式Present(4) study −0.71 p xS= 48.4 x 20000 p

津波遡上距離 Range of S in 10000 Range of S in Iwate Pref. Fukushima Pref. Range of S in Miyagi Pref.

Tsunami intrusion distance, distance, intrusion Tsunami 0 1E-4 1E-3 0.01 River河床勾配S slope,SS

Figure 6. Relationship between intrusion distance and riverbed slope. 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 Tanaka et al. [3] have already proposed another empirical formula based on data sets from Prefecture alone. In order to derive a formula for tsunami propagation distance in a river, they assumed Miyagi Prefecture alone. In order to derive a formula for tsunami propagation distance in a river, the following general exponential relation for the wave attenuation in a channel, as used by [28,29]: they assumed the following general exponential relation for the wave attenuation in a channel, as used by [28,29]: kx H = H0e− (2) 𝐻=𝐻𝑒 (2) where x is the upstream coordinate from a river mouth, H is the wave height, H0 corresponds to the whereinitial x incoming is the upstream wave height coordinate at x = 0,from and ak riveris the mouth, damping H is coe theﬃ wavecient. 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:

𝑘 = 4.77 × 10𝑒. (m−1) (3)

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.

10 2011 - Survey/Trace H = 9.76 × e-0.00028 × x

1 (m) H

0.1 0 5000 10000 15000 2200005000 x (m)

Figure 7. Tsunami damping in the Natori River [3].

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 6 of 13

. 𝑥 = 48.4𝑆 (m) (1)

40000 Miyagi宮城県 Miyagi Pref.Pref. Iwate岩手県 Iwater Pref. Pref. (m)

p Fukushima福島県 Fukushima Pref. x (m) 30000 P Eq. 式Kayane (3)(1) et al., 2012 x =×43 −× Eq.xSp (4)6.33 10 exp() 4.67 10

(m) 式Present(4) study −0.71 p xS= 48.4 x 20000 p

津波遡上距離 Range of S in 10000 Range of S in Iwate Pref. Fukushima Pref. Range of S in Miyagi Pref.

Tsunami intrusion distance, distance, intrusion Tsunami 0 1E-4 1E-3 0.01 River河床勾配S slope,SS

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]:

J. Mar. Sci. Eng. 2020, 8, 882 𝐻=𝐻𝑒 (2)6 of 12

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 Based on the measured data from six rivers including the Abukuma River, the Kitakami River, Old Kitakamithe Old Kitakami River, the River, Sunaoshi the Sunaoshi River, the River, Natori the River,Natori andRiver, the and Naruse the Naruse River inRiver Miyagi in Miyagi Prefecture, TanakaPrefecture, et al. [3] Tanaka obtained et al. an [3] empirical obtained equation an empirical for thisequation damping for this coe dampingﬃcient kcoefficientand the riverbedk and the slope S as follows:riverbed slope S as follows: 5 4.67x103S 1 k = 4.77 10− e (m− ) (3) 𝑘 = 4.77× × 10𝑒. (m−1) (3) Figure7 indicates an example of tsunami height damping in the Natori River during the 2011 Figure 7 indicates an example of tsunami height damping in the Natori River during the 2011 TohokuTohoku Tsunami. Tsunami. From From the slopethe slope of the of regressionthe regression line, line, they they obtained obtained the the damping damping coe coefficientﬃcient for for rivers in Miyagirivers Prefecture.in Miyagi Prefecture.

10 2011 - Survey/Trace H = 9.76 × e-0.00028 × x

1 (m) H

0.1 0 5000 10000 15000 2200005000 x (m)

FigureFigure 7. 7.Tsunami Tsunami dampingdamping in the the Natori Natori River River [3]. [3 ].

By assuming that the tsunami height at the most upstream end equal to 5% of the initial tsunami height (H/H0 = 0.05), the following empirical equation between the tsunami intrusion distance, xp, and riverbed slope, S, is derived: 4 4.67 103S xp = 6.28x10 e− × (m) (4) In Figure6, 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.

3.4. 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.

2 0.71 1 k = 6.19x10− S (m− ) (5)

Figure8 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.

H n 2 0.71 o = exp 6.19 10− S x (m unit) (6) H0 − × × − J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 7 of 13

.× 𝑥 = 6.28𝑥10 𝑒 (m) (4)

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.

3.4. 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.

𝑘 = 6.19𝑥10𝑆. (m−1) (5)

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.

J. Mar. Sci. Eng. 2020, 8, 882 =𝑒𝑥𝑝−6.19 × 10𝑆. ×𝑥 (m-unit) 7 of(6) 12

0.005 Eq. (3) Eq. (5) 0.004

) 0.003 -1 (m k 0.002 Range of S in Fukushima Pref. Range of S in Iwate Pref. 0.001 Range of S in Miyagi Pref.

0.000 1E-4 1E-3 0.01 S

FigureFigure 8.8. RelationshipRelationship ofof dampingdamping coecoefficientfficient andand riverbedriverbed slope.slope. 4. Evaluation of Tsunami-Induced Flow Discharge and Velocity in a River 4. Evaluation of Tsunami-Induced Flow Discharge and Velocity in a River 4.1. Methodology 4.1. Methodology Due to increased opportunities for capturing real tsunamis by a video camera in recent years, studiesDue have to increased been carried opportunities out to assess for flowcapturing velocity real from tsunamis video by recordings a video camera [30–33]. in Inrecent this study,years, astudies distinctly have di beenfferent carried approach out to utilizing assess flow measured velocity water from level video data recordings along a river[30–33]. will In be this applied study, in a orderdistinctly to evaluate different river approach discharge utilizing and velocity measured induced water by level tsunami data along wave propagation.a river will be For applied this purpose, in order Adityawanto evaluate etriver al. [ 10discharge] used a continuityand velocity equation induced for by a one-dimensionaltsunami wave propagation. flow analysis For as this purpose, Adityawan et al. [10] used a continuity equation for a one-dimensional flow analysis as ∂A ∂Q + = 0 (7) ∂t ∂x 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: Z xN ∂η Qi QN = B dx (8) − xi ∂t where η is the water level, B is the river width, Qi is the discharge at cross-section i located at xi, and QN is the upstream discharge at xN (upstream end). Equation (8) is solved numerically using the finite difference method to obtain discharge and velocity at the measurement point as follows: B ηt ηt ∆t + B ηt ηt ∆t XN 1 j+1 j+1 j−+1 j j j− t − − − Q = xj+ xj + QN (9) i j=i 2∆t 1 − 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 Figure9a, and share the river mouth immediately before it pours to the sea (Figure9b). The water level variation has been obtained along these rivers during the 2011 Tohoku tsunami at the measuring stations summarized in Table1. 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 QN is used in Equation (9). J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 8 of 13

𝜕𝐴 𝜕𝑄 =0 (7) 𝜕𝑡 𝜕𝑥

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:

𝑄 −𝑄 = 𝐵 𝑑𝑥 (8)

where η is the water level, B is the river width, 𝑄 is the discharge at cross-section i located at xi, and 𝑄 is the upstream discharge at 𝑥 (upstream end). Equation (8) is solved numerically using the finite difference method to obtain discharge and velocity at the measurement point as follows:

𝑄 = ∑ 𝑥 −𝑥𝑄 (9)

where Δt 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 Δt = 10 min are illustrated in Figure 10, which J.can Mar. be Sci. substituted Eng. 2020, 8, 882 into Equation (9) for numerical integration. Moreover, measured discharge8 at of 12the upstream boundary 𝑄 is used in Equation (9).

FigureFigure 9. 9.( a()a) Location Location of of water water level level measuring measuring stations stations along along Naruse Naruse and and Yoshida Yoshida Rivers; Rivers; (b ()b Naruse) Naruse RiverRiver mouth mouth morphology. morphology.

Table 1. Measuring stations of water level along Naruse and Yoshida Rivers. The number in () refers to Table 1. Measuring stations of water level along Naruse and Yoshida Rivers. The number in () refers the distance from the river mouth. to the distance from the river mouth. Naruse River Yoshida River Naruse River Yoshida River NobiruNobiru (0.5 (0.5 km) km) Ono-Naruse (4.04 km) Ono-Yoshida (4.04 km) Ono-Naruse (4.04 km) Ono-Yoshida (4.04 km) Kashimadai-Naruse (8.99 km) Kashimadai-Yoshida (8.99 km) Kashimadai-NaruseTakeya (17.24 (8.99 km) km) Kashimadai-Yoshida Hataya (13.6 km) (8.99 km) J. Mar. Sci. Eng. 2020, 8, x FORTakeya PEER REVIEW (17.24 km) Hataya (13.6 km) 9 of 13

8 Nobiru (0.50 km) 7 Ono-Naruse (4.04 km) Ono-Yoshida (4.18 km) Kashimadai-Naruse (8.99 km) Kashimadai-Yoshida (8.99 km) Takeya (17.24 km) Hataya (13.60 km) 6

2 Water level (m)

-1 15:00 16:00 17:00 18:00 19:00 20:00 21:00 Time (hh:mm) Figure 10. Measured water level data along Naruse and Yoshida Rivers (11 March 2011). Figure 10. Measured water level data along Naruse and Yoshida Rivers (11 March 2011).

4.2. Results and Discussion Based on on Equation Equation (9), (9), tempor temporalal variation variation of ofthe the discharge discharge Q andQ and velocity velocity V at anV atarbitrary an arbitrary point pointcan be can estimated be estimated as shown as shownin Figure in Figure11. Figure 11. 11 Figurea shows 11a the shows river the discharge river discharge induced inducedby the tsunami by the wavetsunami in Naruse wave in and Naruse Yoshida and Rivers. Yoshida The Rivers. discharge The discharge at the river at mouth the river (Nobiru) mouth (Nobiru)was as large was as as 12,000 large mas3 12,000/s with ma 3distinct/s with reduction a distinct reductionover time. overThe magnit time. Theude magnitude is reduced isas reduced the wave as propagates the wave propagates upstream. Theupstream. calculated The discharge calculated was discharge converted was to convertedvelocity by to dividing velocity its by value dividing with itsthe value flow area with as the shown ﬂow in Figure 11b. As compared with the past evaluations such as [30–33], the estimated tsunami flow velocity shown in the figure using the conservation equation is of a similar magnitude and pattern.

12000 Naruse-Yoshida Rivers

9000 Nobiru (0.50 km) Ono-Naruse (4.04 km) Kashimadai-Naruse (8.99 km ) 6000 Ono-Yoshida (4.18 km) Kashimadai-Yoshida (8.99 km) /s)

3 3000 (m Q 0

-3000

-6000 15:00 16:00 17:00 18:00 19:00 20:00 21:00 Time (hh:mm) (a)

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 9 of 13

8 Nobiru (0.50 km) 7 Ono-Naruse (4.04 km) Ono-Yoshida (4.18 km) Kashimadai-Naruse (8.99 km) Kashimadai-Yoshida (8.99 km) 6 Takeya (17.24 km) Hataya (13.60 km)

2 Water level (m)

-1 15:00 16:00 17:00 18:00 19:00 20:00 21:00 Time (hh:mm) Figure 10. Measured water level data along Naruse and Yoshida Rivers (11 March 2011).

4.2. Results and Discussion Based on Equation (9), temporal variation of the discharge Q and velocity V at an arbitrary point canJ. Mar. be Sci. estimated Eng. 2020 ,as8, 882shown in Figure 11. Figure 11a shows the river discharge induced by the tsunami9 of 12 wave in Naruse and Yoshida Rivers. The discharge at the river mouth (Nobiru) was as large as 12,000 m3/s with a distinct reduction over time. The magnitude is reduced as the wave propagates upstream. Thearea calculated as shown discharge in Figure 11wasb. converted As compared to velocity with the by past dividing evaluations its value such with as the [30 flow–33], area the estimatedas shown intsunami Figure ﬂow 11b. velocityAs compared shown with in the the ﬁgure past usingevalua thetions conservation such as [30–33], equation the isestimated of a similar tsunami magnitude flow velocityand pattern. shown in the figure using the conservation equation is of a similar magnitude and pattern.

12000 Naruse-Yoshida Rivers

9000 Nobiru (0.50 km) Ono-Naruse (4.04 km) Kashimadai-Naruse (8.99 km ) 6000 Ono-Yoshida (4.18 km) Kashimadai-Yoshida (8.99 km) /s)

3 3000 (m Q 0

-3000

-6000 15:00 16:00 17:00 18:00 19:00 20:00 21:00 Time (hh:mm) J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW (a) 10 of 13

10 Naruse-Yoshida Rivers 8 Nobiru (0.50 km) Ono-Naruse (4.04 km) Ono-Yoshida (4.18 km) Kashimadai-Naruse (8.99 km ) Kashimadai-Yoshida (8.99 km) 6

2 (m/s)

V 0

-2

-4

-6 15:00 16:00 17:00 18:00 19:00 20:00 21:00 Time (hh:mm) (b)

Figure 11. EstimatedEstimated discharge discharge and and flow ﬂow velocity velocity of ofNaruse Naruse and and Yoshida Yoshida Rivers Rivers (11 (11March March 2011). 2011). (a) (Rivera) River discharge discharge variation, variation, and and (b) (flowb) ﬂow velocity. velocity.

Figure 12 showsshows a comparisoncomparison result betweenbetween the designdesign floodflood dischargedischarge andand maximummaximum riverriver discharge induced byby thethe tsunami in the Naruse and Yoshida Rivers.Rivers. The return period of the design floodflood is 100100 yearsyears [[34].34]. TheThe maximummaximum discharge due to thethe tsunamitsunami propagationpropagation atat thethe riverriver mouthmouth waswas computedcomputed as as more more than than two two times times that that of the of designthe design flood flood discharge. discharge. In fact, In this fact, area this was area affected was affected severely by the tsunami propagation and inundation due to the extreme discharge much bigger than the design flood discharge [35,36].

12000 Nobiru Naruse River Calculated tsunami discharge 10000 Design flood discharge

Yoshida River 8000 Calculated tsunami discharge Design flood discharge /s) 3 6000 Ono-Naruse (m max

Q 4000

2000 Ono-Yoshida Kashimadai-Naruse Kashimadai-Yoshida 0 0 5 10 15 20 25 30 35 40

Distance from river mouth (km) Figure 12. Comparison of maximum discharge induced by tsunami and design flood discharge of Naruse and Yoshida Rivers.

5. Conclusions

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 10 of 13

10 Naruse-Yoshida Rivers 8 Nobiru (0.50 km) Ono-Naruse (4.04 km) Ono-Yoshida (4.18 km) Kashimadai-Naruse (8.99 km ) Kashimadai-Yoshida (8.99 km) 6

2 (m/s)

V 0

-2

-4

-6 15:00 16:00 17:00 18:00 19:00 20:00 21:00 Time (hh:mm) (b)

Figure 11. Estimated discharge and flow velocity of Naruse and Yoshida Rivers (11 March 2011). (a) River discharge variation, and (b) flow velocity.

Figure 12 shows a comparison result between the design flood discharge and maximum river dischargeJ. Mar. Sci. Eng.induced2020, 8 by, 882 the tsunami in the Naruse and Yoshida Rivers. The return period of the design10 of 12 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 affectedseverely severely by the tsunami by the tsunami propagation propagation and inundation and inundation due to the due extreme to the discharge extreme muchdischarge bigger much than biggerthe design than ﬂoodthe design discharge flood [ 35discharge,36]. [35,36].

12000 Nobiru Naruse River Calculated tsunami discharge 10000 Design flood discharge

Yoshida River 8000 Calculated tsunami discharge Design flood discharge /s) 3 6000 Ono-Naruse (m max

Q 4000

2000 Ono-Yoshida Kashimadai-Naruse Kashimadai-Yoshida 0 0 5 10 15 20 25 30 35 40

Distance from river mouth (km) FigureFigure 12. 12. ComparisonComparison ofof maximummaximum discharge discharge induced induced by tsunamiby tsunami and and design design flood flood discharge discharge of Naruse of Naruseand Yoshida and Yoshida Rivers. Rivers.

5.5. Conclusions 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. J. Mar. Sci. Eng. 2020, 8, 882 11 of 12

Acknowledgments: We received valuable river water level data from various organizations such as the Ministry of Land, Infrastructure, Transport (MLIT), Iwate, Miyagi, and Fukushima Prefectures. The authors would like to gratefully acknowledge all the supports. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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