Proceedings of the 22nd IAHR-APD Congress 2020, Sapporo, Japan

DEVELOPMENT OF BED VARIATION CALCULATION METHOD CONSIDERING NONEQUILIBRIUM SEDIMENT MOTION AND INTERACTION BETWEEN BED LOAD AND
SUSPENSION: APPLICATION TO RIVER MOUTH SANDBAR WITH LARGE-SCALETOPOGRAPHIC CHANGES DUE TO FLOOD
TAKAHISA GOTOH
Research and Development Initiative, Chuo University, Tokyo, Japan, goto510@tamacc.chuo-u.ac.jp
SHOJI FUKUOKA
Research and Development Initiative, Chuo University, Tokyo, Japan, sfuku@tamacc.chuo-u.ac.jp

ABSTRACT
Large amount of the fine sediments around river mouth are transported in states of suspensions during a flood.
The suspended sediment entrainments are characterized by interactions between bed load motions andsuspensions of the sediments. In general, the suspended sediment entrainments which was the boundary
conditions at the bottom of the advection-diffusion equation of suspended sediment concentrations were
estimated by applying equilibrium conditions of sediment motions. In this study, the quasi-three dimensional
flow and bed variation analysis method considering the non-equilibrium flows and sediment motions with large
amount of suspended sediments was developed. And the interactions between bed load motions and suspensions
was also taken into account the bed variation analysis method. The developed analysis method was applied to
the 2011flood in the Aganogawa River with large-scale topographical changes in the river mouth sandbar. The
calculation results elucidated the enlargements processes of opening widths in the river mouth sandbar andsediment depositions to the downstream toward sea due to the flood.
Keywords: sediment entrainment, suspended load, non-equilibrium sediment motions, quasi-three dimensional flow and bed
variation analysis, river mouth sandbar

1.

INTRODUCTION

The bed materials around the mouth of the lowland river are composed of fine sands, and most of the finesediments are transported in states of suspensions during floods. The suspended sediment entrainments arecaused by translating from bed load to suspensions in high shear stresses of the flow. Therefore, the sedimententrainments are characterized by the interactions between bed load motions and suspended sediments motions.
In general, the bed variation analysis of sandy rivers has been conducted by applying the equilibrium bed load
equations and the two-dimensional or three-dimensional advection-diffusion equation for suspended sediments.
In the advection-diffusion equation of suspended sediment concentrations, the suspended sediment entrainments
which were the boundary conditions of the equation at the bottom have been evaluated by the bed loadconcentration near the bed in equilibrium conditions (van Rijin(1984), Zyserman and Fredsøe(1990), Itakura etal. (1984) etc.). Therefore, the interactions between bed load and suspended sediment motion has not been taken
into account in those conventional bed variation analysis method based on the assumption of equilibrium

Aganogawa
Riv.
Sea ofJapan(a) Before the 2011 flood (2010.11.)(b)

During 2011 flood (7/30 A.M.8:00)

Figure 1. Aganogawa River mouth sandbar before and during 2011 flood.
1conditions. The conventional method has estimated the bed load discharge and suspended sedimententrainments as a function of only friction velocity and could not explained the important mechanism of the
non-equilibrium sediment motions for the bed variations.
Nakagawa et. al. (1990) derived the probability density functions per unit time of transitions from saltation
motions to suspensions due to turbulence velocity fluctuations assuming the normal distributions. And Bose et.
al. (2013) obtained the probability distribution of velocity fluctuations theoretically based on the exponential
distributions and investigated the threshold of sediment suspensions from bed load motions. Ota et. al. (2015)
applied the Nakagawa(1990)’s transition rates into the three-dimensional flow and two-dimensional bed
variation analysis method. However, their calculation method assumed that sand particles were moving insliding states. The assumption of their calculations was slightly different from the concept of Nakagawa(1990) ’s method which estimated the transitions from saltation motions to suspensions. And, verifications of
the calculation method were not sufficiently conducted for the sandy rivers where most of fine sediments aretransported in suspensions during floods.
In recent years, the three-dimensional and two-phase model of flow and sediment transport have beendeveloped (Chauchat et al. (2017), etc.). The model was able to calculate the bed load motions and suspensions
without the assumption of the equilibrium conditions. However, it is difficult to apply the calculation model inthe widespread area of rivers having complex bed topographies. The depth integrated model of flood flow andbed variation analysis considering non-equilibrium sediment motion should be constructed to attempt to solve
the issues of practical river managements.
Around the river mouth of the Aganogawa River, the large-scale river mouth sandbar composed of fine sands
is formed as shown in Figure 1(a). In July 2011, the record breaking largest flood occurred, causing large-scale
topographic changes in the river mouth sandbar and enlargements of the opening widths of the river mouth
sandbar as shown in Figure 1(b). Tateyama et. al.(2018) developed the quasi-three-dimensional flow and bed
variation analysis method considering the effects of the steep slope topography over the sandbar of theAganogawa River mouth and explained the enlargement processes of the opening widths of the river mouth
sandbar. However, their analysis method for bed variations was constructed under the equilibrium conditionsof bed load motions and the suspended sediment entrainments, so that the sediments of the eroded river mouth
sandbar could not be transported sufficiently downstream toward the sea. Thus the developments of river mouthterrace due deposition of suspended sediments could not be properly explained in their calculation method.
In this study, the analysis method of the quasi-three dimensional flow and bed variation was developed byconsidering the non-equilibrium flows and sediment motions with large amount of suspended sediments. And
the developed method was applied to the 2011flood in the Aganogawa River with large-scale topographicalchanges in the river mouth.
2. FLOOD FLOW AND BED VARIATION ANALYSIS METHOD CONSIDERING NONEQUILIBRIUM SEDIMENT MOTIONS WITH LARGE AMOUNT OF SUSPENDED
SEDIMENTS

Figure 2. Framework of the calculation method.
Figure 2 shows the framework of the calculation method. The three-dimensional flows and pressure distributions near thebeds were calculated by the quasi-three dimensional flow calculation method (Q3D-FEBS method, Takemura & Fukuoka,2019). The calculation method assumed that the vertical velocity distributions was the third order polynomial in Eq.(1). It
was determined by using the mean velocity Ui, the water surface velocity usi, the bed surface velocities ubi and the verticalgradient of velocity at the water surface. Those velocities were calculated by the depth integrated continuity equation in2

Eq.(2), the depth integrated horizontal momentum equations in Eq.(3) and the horizontal momentum equation on the water
surface and bed surface in Eq.(5) and Eq.(6), where 𝑢𝑖 (𝜂) is i direction flow velocity at any height, Δ𝑢𝑖 = 𝑢𝑠𝑖 − 𝑈𝑖 ,

δ𝑢𝑖 = 𝑢𝑠𝑖 − 𝑢𝑏𝑖 , =(zs-z)/h, Ui is depth averaged flow velocity in i direction, h is water depth, ubi is bed surface
velocity, usi is water surface velocity in i direction, zs is water level, zb is bed elevation, 𝑝′𝑏 is the non-hydrostaticpressure at the bed, t is depth averaged kinematic eddy viscosity coefficient, ts and tb are kinematic eddy
viscosity at water surface and near the bed, respectively. And  denotes fluid density, 𝜏𝑏𝑖 denotes a shear stresses
acting on the river beds and n indicates the Manning's roughness coefficients.
𝑢𝑖 (𝜂) = 𝑈𝑖 + ∆𝑢𝑖 (12𝜂 3 − 12𝜂 2 + 1) − 𝛿𝑢𝑖 (4𝜂 3 − 3𝜂 2 )(1)

𝜕ℎ 𝜕𝑈𝑗 ℎ
+
=0
𝜕𝑡
𝜕𝑥𝑗

(2)

𝜕𝑈𝑖 ℎ 𝜕𝑈𝑖 𝑈𝑗 ℎ𝜕𝑧𝑠 1 𝜕𝑝̅′ ℎ 𝑝′𝑏 𝜕𝑧𝑏 𝜏𝑏𝑖 1 𝜕ℎ𝑇𝑖𝑗
+
= −𝑔ℎ
−
−
−
+
𝜕𝑡
𝜕𝑥𝑗
𝜕𝑥𝑖 𝜌 𝜕𝑥𝑖
𝜌 𝜕𝑥𝑖
𝜌
𝜌 𝜕𝑥𝑗

(3)

1 𝜕𝑈𝑖

′ 𝑢 ′ − 2𝜌𝜈 𝑆 , 𝑆 = (̅̅̅̅̅̅̅
Τ𝑖𝑗 = −𝜌𝑢
𝑖 𝑗
𝑡 𝑖𝑗
𝑖𝑗

2 𝜕𝑥𝑗

+

𝜕𝑈𝑗
𝜕𝑥𝑖

)

𝜕𝑢𝑠𝑖
𝜕𝑢𝑠𝑖
𝜕𝑧𝑠
1 𝜕𝑧𝑠 𝜕𝑝′𝜈𝑡𝑠 𝜕 2 𝑢𝑠𝑖
𝜕 2 𝑢𝑖
+ 𝑢𝑠𝑗
= −𝑔
−
| +
|
,
| = 24∆𝑢𝑖 − 6𝛿𝑢𝑖 , 𝛾𝑡𝑠 = 𝑙𝑑 √𝑘𝑠𝜕𝑡𝜕𝑥𝑗
𝜕𝑥𝑖 𝜌ℎ 𝜕𝑥𝑖 𝜕𝜂 𝑠
𝜌 𝜕𝑧 2 𝑠𝜕𝜂 2 𝑠𝜕𝑢𝑏𝑖
𝜕𝑢𝑏𝑖
𝜕𝑧𝑠 1 𝜕𝑝′ 𝑏
1 𝜕𝑧𝑏 𝜕𝑝′1 𝜏𝑏𝑖 − 𝜏0𝑖
+ 𝑢𝑏𝑗
= −𝑔
−
−
| +
, 𝛿𝑧𝑏 = 𝑐𝑧𝑏 ℎ, 𝑐𝑧𝑏 = 0.03
𝜕𝑡𝜕𝑥𝑗
𝜕𝑥𝑖 𝜌 𝜕𝑥𝑖
𝜌ℎ 𝜕𝑥𝑖 𝜕𝜂 𝑏 𝜌 𝛿𝑧𝑏𝜏𝑏𝑖
𝜐𝑡𝑏 𝜕𝑢̅𝑖=−⌋ , 𝛾𝑡𝑏 = 𝑙𝑑 √𝑘𝑏 ,𝑙𝑑 = 0.07ℎ𝜌ℎ 𝜕𝜂 𝑏𝜏0𝑖
𝐶0𝑔𝑛2= 𝑐𝑏2 𝑢𝑏𝑖 |𝒖𝒃 |, 𝑐𝑏 =1 + 𝑐𝑧𝑏 , 𝐶0 = √ 1⁄3√2𝐶
𝜌ℎ
1 − 0⁄𝜅

(4)

(5)
(6)

(7)

The gradients for vertical directions of the non-hydrostatic pressure at the river bed and water surface in Eq.(5)and in Eq.(6) were calculated by the momentum equations for vertical directions at the beds and water surface
in Eq.(8). The non-equilibrium turbulent intensities near the bed are important to estimate the amount ofsuspended sediments entrainments. The transport equation for the turbulent kinetic energy at the bed 𝑘𝑏 in
Eq.(11) was calculated in addition to the transport equations for the depth integrated turbulent kinetic energy K
in Eq.(9) and the turbulent kinetic energy at the water surface 𝑘𝑠 in Eq.(10). The turbulence production term at
the bottom in Eq.(11) was expressed by using the bed shear stresses.
1 𝜕𝑝′
𝜕𝑤𝑠
| = 𝑢𝑠𝑗,𝜌ℎ 𝜕𝜂 𝑠
𝜕𝑥𝑗

3

1 𝜕𝑝′
𝜕𝑤𝑏
| = 𝑢𝑏𝑗𝜌ℎ 𝜕𝜂 𝑏
𝜕𝑥𝑗

𝜕𝐾
𝜕𝐾
𝐾 ⁄2
𝜐𝑡 𝜕 2 𝐾
1 𝜐𝑡𝑏 𝜕𝑘𝑏
+ 𝑈𝑗
= 𝑃𝑘 − 𝐶𝑑
+
+
𝜕𝑡
𝜕𝑥𝑗
𝑙𝑑
ℎ𝜎𝑘 𝜕𝑥 2𝑗 ℎ2 𝜎𝑘 𝜕𝜂2
1 𝜕𝑈𝑖 𝜕𝑈𝑗
12
(2Δ𝑢𝑖 Δ𝑢𝑖 − 7𝛿 𝑢𝑖 Δ𝑢𝑖 + 8𝛿 𝑢𝑖 𝛿 𝑢𝑖 )}𝑃𝑘 = 0.5ℎ𝜐𝑡 { (+) −2 𝜕𝑥𝑗 𝜕𝑥𝑖
15ℎ2
𝜕𝑘𝑏
= 12 𝛥𝑘 − 6 𝛿𝑘, 𝛥𝑘 = 𝑘𝑠 − 𝐾, 𝛿𝑘 = 𝑘𝑠 − 𝑘𝑏𝜕𝜂𝑘(𝜂) = 𝛥𝑘(12𝜂 3 − 12𝜂 2 + 1) + 𝛿𝑘(−4𝜂 3 + 3𝜂 2 ) + 𝐾
3
𝜕𝑘𝑠
𝜕𝑘𝑠
𝑘𝑠 ⁄2 𝜐𝑡𝑠 𝜕 2 𝑘𝑠 1 𝜐𝑡𝑠 𝜕 2 𝑘𝑠
𝜕 2 𝑘𝑠
+ 𝑢𝑠𝑗
= −𝐶𝑑
+
+
,
= −24𝛥𝑘 + 6𝛿𝑘, 𝐶𝑑 = 0.08, 𝜎𝑘 = 1.0
𝜕𝑡
𝜕𝑥𝑗
𝑙𝑑
𝜎𝑘 𝜕𝑥 2𝑗 ℎ2 𝜎𝑘 𝜕𝜂 2
𝜕𝜂 2
3

𝜕𝑘𝑏
𝜕𝑘𝑏
𝑘𝑏 ⁄2 𝜐𝑡𝑏 𝜕 2 𝑘𝑏 1 𝜐𝑡𝑏 𝜕 2 𝑘𝑏
+ 𝑢𝑏𝑗
= 𝑃𝑘𝑏 − 𝐶𝑑
+
+
,
𝜕𝑡
𝜕𝑥𝑗
𝑙𝑑
𝜎𝑘 𝜕𝑥 2𝑗 ℎ2 𝜎𝑘 𝜕𝜂 2
𝜕𝑢̅𝑖
𝜏𝑏𝑖 2
𝑃𝑘𝑏 ≈ 𝜏𝑏𝑖⌋ =𝜕𝑧 𝑏 𝜌𝜐𝑡𝑏

𝜕 2 𝑘𝑏= 48𝛥𝑘 − 12𝛿𝑘
𝜕𝜂 2

[
(8)

[
(9)

(10)

(11)

The bed variations were calculated by the momentum equations and the continuity equations considering thenon-equilibrium motions of bed load and suspended sediments and the interactions between them as follows.
The velocities of the sand particles ̅̅̅̅̅̅
𝑢𝑝𝑖𝑘 of the bed load in the horizontal direction are evaluated by thesaltation analysis (Osada & Fukuoka, 2013) using the momentum equation of a sand particle in Eq.(12)regarding the three dimensional flows calculated by the quasi-three dimensional flow analysis, where 𝐴2 and𝐴3 are a two-dimensional and three-dimensional shape factor, respectively. s is sand particle density. k indicatesparticle numbers and dk is particle diameter.
3

The bed load volume per unit width was calculated by the continuity equation of the bed load considering theinteractions between bed load motions and suspensions in Eq.(13). Those interactions were estimated by thetransition term from bed load to suspensions and sedimentations term from suspensions to bed load motions in
the equations. The amount of sedimentation 𝑉𝑠𝑑𝑘 from suspensions to bed load motions were described as the
products of the particle settling velocity wsk and the suspended sediment concentration near the river bed 𝐶𝑠𝑘 asshown in Eq. (14). The suspended sediment concentration near the river bed 𝐶𝑠𝑘 was evaluated by the threedimensional advection-diffusion equations described later. The deposit rate to the river bed per unit time 𝑉𝑏𝑑𝑘
were also evaluated by the products of the particle settling velocity and the bed load concentration 𝐶𝑏𝑘 . The bed
load concentration 𝐶𝑏𝑘 was calculated by dividing the bed load volume per unit width by mean saltation
heights as shown in Eq. (14). The mean saltation heights were evaluated as three times of each particle size by
the saltation analysis. And, the pick-up rate 𝑉𝑝𝑏𝑘 from the river bed was obtained from Eq.(15) (Osada &Fukuoka, 2013).
The suspended sediment entrainments 𝑉𝑠𝑢𝑘 from the bed load motions were calculated by using the verticalparticle velocity 𝑤𝑝𝑘 near the beds and the bed load concentrations 𝐶𝑏𝑘 as shown in Eq. (16). The verticalparticle velocities 𝑤𝑝𝑘 were obtained from the vertical momentum equation of a sand particle considering thefluid forces generated by turbulence fluctuations 𝐹 ′ 𝑧 in addition to the fluid forces by mean flow velocities
𝐹𝑧 near the bed (Eq.(17)). The fluid forces due to the turbulence fluctuations 𝐹 ′ 𝑧 were assumed that the
turbulence was isotropic and the velocity fluctuations followed the normal distributions. On the other hand,
Bose et.al. (2013) indicated that velocity fluctuations obeyed the Gram-Charlier-based twosided exponential or
Laplace distributions. It is necessary to investigate the effects of differences of the probability distribution of
turbulent velocity fluctuations in the future.
The suspended sediment transports were calculated by the three-dimensional advection-diffusion equation
using those suspended sediment entrainments as the boundary condition at the bed (Eq.(19)). And the temporalchanges in the bed elevations was obtained from the continuous equation of the river bed shown in Eq.(20).
𝜕𝑢𝑝𝑖𝑘 1
= 𝜌𝐶𝐷 𝐴2 𝑑𝑘2 (𝑢𝑏𝑖 − 𝑢𝑝𝑖 )|𝒖𝒃 − 𝒖𝒑𝒌 | − 𝐴3 (𝜌𝑠 − 𝜌)𝑔𝑖 𝑑𝑘3
𝜕𝑡2
1 𝑇𝑠𝑎𝑙̅̅̅̅̅̅
𝑢𝑝𝑖𝑘 =∫ 𝑢𝑝𝑖𝑘 𝑑𝑡 , 𝑇𝑠𝑎𝑙 = 10
𝑇𝑠𝑎𝑙 0𝜕𝑉𝑚𝑘 𝜕𝑞𝑏𝑘𝑗
+= 𝑉𝑏𝑝𝑘 − 𝑉𝑏𝑑𝑘 − 𝑉𝑠𝑢𝑘 + 𝑉𝑠𝑑𝑘 ,𝑞𝑏𝑘𝑗 = 𝑉𝑚𝑘 ̅̅̅̅̅̅
𝑢𝑝𝑗𝑘
𝜕𝑡
𝜕𝑥𝑗
𝑉𝑠𝑑𝑘 = 𝐶𝑠𝑘 𝑤𝑠𝑘 , 𝑉𝑏𝑑𝑘 = 𝐶𝑏𝑘 𝑤𝑠𝑘 , 𝐶𝑏𝑘 = 𝑉𝑚𝑘 /(3𝑑𝑘 )𝑝𝑘
𝐴3 𝑑𝑘3
𝑉𝑏𝑝𝑘 = 𝜀𝑝 (2) ( 𝑇 )𝐴2 𝑑𝑘
𝑝
𝑉𝑠𝑝𝑘 = 𝛼𝐶𝑏𝑘 𝑤𝑝𝑘 , 𝑤𝑝𝑘 > 0, 𝛼 = 0.2

𝐴3 (𝜌𝑠 + 𝜌𝐶𝑀 )𝑑𝑘3

𝐴3 (𝜌𝑠 + 𝜌𝐶𝑀 )𝑑𝑘3

𝑑𝑤𝑝𝑘
= 𝐹𝑧 + 𝐹 ′ 𝑧 − 𝐴3 (𝜌𝑠 − 𝜌)𝑔𝑧 𝑑𝑘3
𝑑𝑡
1= 𝐴2 𝐶𝐷 𝜌d2 (𝑤𝑏 + 𝑤𝑏 ′ − 𝑤𝑝𝑘 )|𝒖𝒃 + 𝒖′𝒃 − 𝒖𝒑𝒌 | − 𝐴3 (𝜌𝑠 − 𝜌)𝑔𝑧 𝑑𝑘3

(12)

(13)
(14)
(15)
(16)
(17)

2

′
′
′
′
′ ′
′ ′
𝑤𝑏′ ≈ 𝑁(𝜇, 𝜎), 𝜇 = 𝑤𝑏 , 𝜎 = √̅̅̅̅̅̅̅̅̅
𝑤𝑏𝑖
𝑤𝑏𝑖, ̅̅̅̅̅̅̅̅̅
𝑤𝑏𝑖
𝑤𝑏𝑖= ̅̅̅̅̅̅̅̅
𝑣𝑏𝑖
𝑣𝑏𝑖 = ̅̅̅̅̅̅̅̅̅
𝑢𝑏𝑖
𝑢𝑏𝑖 = 2/3𝑘𝑏

𝜕𝐶𝑠𝑘 ∆𝑧 𝜕𝐶𝑠𝑘 𝑢∆𝑧 𝜕𝐶𝑠𝑘 𝑣∆𝑧 𝜕𝐶𝑠𝑘 𝑤∆𝑧
+
+
+
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕
𝜕𝐶𝑠𝑘
𝜕
𝜕𝐶𝑠𝑘
𝜕
𝜕𝐶𝑠𝑘
=
(𝜈 ∆𝑧
)+
(𝜈 ∆𝑧
) + (𝜈𝑡 ∆𝑧) + (𝑉𝑝𝑠𝑘 − 𝑉𝑑𝑠𝑘 )|𝑧
𝑏
𝜕𝑥 𝑡
𝜕𝑥
𝜕𝑦 𝑡
𝜕𝑦
𝜕𝑧
𝜕𝑧

(18)
(19)

𝑘𝑚𝑎𝑥

𝜕𝑧𝑏
𝐴2
=− ∑
(𝑉 − 𝑉𝑏𝑑𝑘 )𝜕𝑡
𝐴3 𝑏𝑝𝑘

(20)

𝑘=1

3.

OBSERVATIONS AND CALCULATION CONDITION

Figure 3 shows the water level and discharge observation points and the values of the Manning's roughness
coefficients. The Manning's roughness coefficients were determined so that the time series of calculated water
surface profiles reproduced the observed water surface profiles in the following chapter. The detailed
observations of time series of the water surface profiles were conducted between 0.2km to 3.5km points
upstream of the river mouth.
The upstream boundary condition of the calculation was the observed water level hydrograph at the
Yokogoshi observation station (13.5km point). And the downstream boundary condition was the observed sea
level hydrograph at the Niigata Nishi Port observation station.
Figure 4 indicates the observed particle size distributions of bed materials in the objective area. It was foundthat the bed materials were composed of almost uniform fine sands. Thus, the bed materials in the calculation4

was set as a grain size of 0.7mm uniformly at the upstream reaches of 1km point. And the grain size of 0.4mm
was given uniformly downstream reaches of 1.0 km point.

Figure 3. Water level and discharge observation points and the Manning's roughness coefficients.
Third peak
Second peakFirst peakFigure 4. Observed sediment particle sizes andcalculation condition.

Figure 5. Observed and calculated dischargehydrograph at Yokogoshi(13.5km)
observation stations.

Third peak
Second peakFirst peakFigure 6. Observed and calculated water surface profiles in the flood rising period.

(a) 0.2km(b) 0.4km
Figure 7. Observed and calculated water level hydrographs at 0.2km and 0.4km points.
5(a) 0.2km(b) -0.45km

Figure 8. Observed and calculated cross-sectional bed profiles of sandbars at the river mouth before and after the flood.

(a) 7/28 22:00 (First peak)(b) 7/29 15:00 (Second peak)(c) 7/30 8:00 (Third peak)(d) After flood

Figure 9. the contours of calculated river bed topography around the river mouth sandbar during the flood.

4.

CALCULATION RESULTS

Figure 5 indicates the comparison between observed and calculated discharge hydrographs in the Yokogoshi
observation station at 13.5km point. And Figure 6 shows the comparison between the observed and calculated
water surface profiles in the flood rising period. In 2011 flood, the discharge hydrograph had three peaks shownin Figure 5. The first peak occurred on July 28th at 22:00, the second peak occurred on 29th at 15:00 and the third
peak occurred on 30th 8:00. The calculation results could almost reproduce the time series of observed water
surface profiles around those peak periods and the observed discharge hydrograph.
Figure 7 indicates the observed and calculated water level hydrographs at 0. 2km and 0.4km points where thewater levels tended to be raised by the existing river mouth sandbar, respectively. The calculation results almost
6explained the observed water level hydrographs around the times of each peak discharge, although thereproducibility of those during the low water levels was slightly low.
Figure 8 shows the observed cross-sectional bed profiles of sandbars before and after the flood and temporalchanges in the calculated cross-sectional bed profiles on the river mouth (0.2km point) and the downstream ofthe sandbar (-0.45km point). And Figure 9 indicates the contours of time series of the calculated bed topographyaround the river mouth sandbar during the flood. The calculation results clarified that the flood flows overflowed
the river mouth sandbar around the time of the third peak (7/30 8:00). After the third peak, it was found that thecross-sectional bed profile of the river mouth sandbar at 0.2km point was greatly scoured due to the overflows.
The scoured sediments on the sandbar were transported and deposited longitudinally in the downstream of thesandbar (see Figure 9(d)). And those sediment depositions caused to slightly shift the directions of the main
flows toward the left bank. Thus the river bed near the left bank around -0.45km point tended to be scoured.
Figure 10 indicates the longitudinal distributions of the amount of the calculated bed load and suspended load
at 7/29 15:00 (the second peak) and 7/30 8:00 (the third peak discharge), respectively. The calculation results
revealed that most of the sediments were transported in states of suspensions around the river mouth of theAganogawa River. Figure 11 shows the differences between sediment entrainments from bed load motions tosuspensions and the sedimentations from suspensions to bed load motions at the time of the third peak. The Red
color means suspended sediment entrainments from bedload and the blue color means sedimentation fromsuspension to bedload. The large amount of sediment entrainments from bed load motions to suspensions
occurred on the sandbar of river mouth around 0.2km point. Therefore the amount of suspended load increased
longitudinally due to the increasing transitions from bed load motions on the river mouth sandbar around 0.2km
as shown in Figure 10, while the amount of bed load decreased longitudinally around there. These calculation
results demonstrated the interactions between bed load motions and suspensions were calculated properly in thedeveloped calculation method.

Figure 10. Longitudinal distributions of amount of thecalculated bed load and suspended load at
7/29 15:00 (Second peak) and 7/30 8:00
(Third peak discharge).

Figure 11. Differences between sediment entrainmentsfrom bed load motions to suspensions and thesedimentations from suspensions to bed loadmotions.

(a) 7/29 15:00 (Second peak)(b) 7/30 8:00 (Third peak)Figure 12. the contours of suspended sediment concentrations near the beds around the river mouth
sandbar at 7/29 15:00 (Second peak) and 7/30 8:00 (Third peak discharge).
7

Figure 12 shows the contours of the suspended sediment concentrations near the beds around the river mouth
sandbar at 7/29 15:00 (the second peak) and 7/30 8:00 (the third peak discharge). The high concentrations ofsuspended sediments occurred on the river mouth sandbar due to the strong sediment entrainments by the
overflows. And those high concentrations and the amount of suspended load shown in Figure 10 were gradually
decreasing due to the sedimentations for the longitudinal direction toward sea. The sedimentations induced thedevelopments of the river mouth delta as shown in Figure 9(d).
5.

CONCLUSIONSThe quasi-three dimensional flow and bed variation analysis method was constructed by considering nonequilibrium motions and interactions between bed load motions and suspensions of sediments and the changes
in the river mouth sandbar in the Aganogawa River was investigated by this method.
The developed flood flow and bed variation analysis method elucidated processes of the opening widths
enlargements in the river mouth sandbar and developments of the river mouth terrace by sediment depositionsin the sea due to the 2011 flood. The calculation results indicated that the large amount of sediments weretransported in states of suspensions and the transitions from bed load motions to suspensions largely occurredon the river mouth sandbar due to the overflow.
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