Advances in River Sediment Research – Fukuoka et al. (eds)© 2013 Taylor & Francis Group, London, ISBN 978-1-138-00062-9

Growing and decaying processes and resistance of sand waves
in the vicinity of the Tone River mouth
S. Okamura

IDEA Consultants, Inc., Tokyo, Japan

S. Fukuoka

Chuo University, Tokyo, Japan

ABSTRACT: The long-term growth and decay processes and resistance of sand waves of the Tone
River mouth from 1961 to 2007 are clarified in this paper. At first, we examine characteristics of thearrangement of dunes which are seen in the downstream of the Choshi Ohashi Bridge. Next, the longterm growth and decay processes of dunes are indicated by the variation of cross-sectional bed forms
surveyed from 1961 to 2007. The variation of sand waves resistances and bed elevations during the floods
in different years is evaluated by a numerical unsteady flood flow and bed variation analysis based on theobserved temporal changes in water surface profiles. At last, characteristics of the sand wave resistancesin the Tone River mouth are clarified by comparing with the previous studies on sand wave resistances inalluvial rivers.
1

INTRODUCTION

1.1 Flood flow and sediment movement analysis
systemFor river management, it is important to estimate
the bed variations during floods and the resistancesin flood flows including sand waves resistances.
To figure out flood flows and bed variations
during floods, Fukuoka (2011) proposed the flood
flow and sediment movement analysis system
based on the observation of temporal changes
in water surface profiles shown in Figure 1. Thissystem is based on the idea that the influences ofchannel shape, resistance, bed variation and so on
are reflected in temporal changes in water surface
profiles of flood flows.
In the system, temporal changes in water surface profiles and discharge hydrographs during
the period of floods and bed elevations before andafter floods are observed at the sections in question. Then, we can figure out flood phenomena bycomputing the unsteady flood flow and bed variation so as to explain these observed data.
In this paper, we apply the system to the vicinityof the Tone River mouth in order to estimate thebed variations and the resistances of sand waves
during floods.
1.2 Problem of the Tone River mouth

Figure 1. Flood flow and sediment movement analysis
system based on the observation of temporal changes inwater surface profiles.

The rise of water level during floods in the Tone
River mouth is remarkable and brings a problemof flood inundation near the river. But the reason
why the water rise and the steep water surface gradient near the river mouth was generated has not
yet been explained.
In this paper, we indicate that the sand waves
are formed in the vicinity of the Tone River mouthand cause the rise of water level during floods.
Then, the long-term growth and decay processes ofsand waves and the variations of sand waves resistances are clarified by the relation with bed variations of the Tone River mouth from 1961 to 2007.

239

The ­variations of sand waves resistances and bed
elevations during floods in different years are evaluated by a numerical unsteady flood flow and bed
variation analysis based on the observed temporalchanges in water surface profiles. At last, the characteristics of the sand wave resistances in the Tone
River mouth are clarified by comparing with theprevious studies on sand wave resistances in alluvial rivers.
2 the bed form in the tone river
mouth
2.1

profiles of average grain size surveyed at left andright side of -0.5 km cross-section. The comparison of cross-sectional bed form before and after
2007 flood is also shown in Figure 3. At the left
side, although the sand exists in surface layer, thegravel exists in lower layer. At the right side, thegravel exists from river bed to the depth of about
5 m from river bed. The river bed elevation at theright side of -0.5 km cross-section is maintained
due to the gravel bed materials even though there
are located at the outer side of the narrow curved
channel.
2.2 Distribution of sand waves

Bed material grain size

Figure 2 shows the planer distribution of grain
size of bed material in the surface layer. Whereas
the bed material is composed of silt or sand in
most areas around the river mouth, the gravelcomponent dominates from −1.0 km to 0.5 km
section, where the channel width is decreasedby the training dikes. Figure 3 shows the verticalFigure 4 shows the plan form and bed elevationsof the Tone River mouth. Bed elevations are surveyed at 50 m intervals in 2003 and 20 m intervals
in 2009 longitudinally. Figure 5 shows the longitudinal bed forms along the channel center line
shown in Figure 4. While the undulation of bed
form is mild from 6 km to 10 km section, the bed
form in the downstream of 6 km section is rugged.
The sand waves are considered as dunes for thefollowing reason. The water depth of the section
is from 5 m to 10 m during floods. Generally, thewave length and wave height of the dune estimatedabout ten times longer than the water depth andfrom one-fifth to one-tenth of the water depth,
respectively.
2.3 Sand waves in the downstream of the Choshi
Ohashi Bridge

Figure 2.

Planer distribution of bed material.

Figure 3. Vertical profiles of average grain size of −0.5 k
cross-section.

Figure 6 shows planer distribution of bed elevations of downstream of the Choshi Ohashi Bridge(located in the reach 2.25 km upstream from theriver mouth) where the large dunes can be seen.
The dashed lines pass through the wave tops ofdunes in the figure. The arrangements of dunes are
fish-scale pattern.
Kinoshita (1984) found that longitudinal vortices are often seen and the sand waves often form
fish-scale pattern at downstream of bridge piers.
He indicated that stable longitudinal vortices
emerge when the cross-sectional intervals of dunes
are four times wider than the water depth by thechannel experiment.
Figure 7 shows the cross-sectional bed forms
at 1.5 km upstream from the river mouth andthe Choshi Ohashi Bridge (2.25 km). The­cross-sectional intervals of dunes (about 20 m)
are four times wider than the water depths during floods (about 5 m). The arrangement of dunes
which are seen in the downstream of the Choshi
Ohashi Bridge verified it to be scale-like dunes as­Kinoshita found.

240

Figure 4.

Plan form and surveyed bed elevation of the Tone River mouth.

Figure 7(b).

Figure 5.

Longitudinal bed forms along the center line.

3

Figure 6. Scale-like dunes in the downstream of the
Choshi Ohashi Bridge.

Figure 7(a). Cross-sectional bed form at Choshi Ohashi
Bridge.

Cross-sectional bed form at 1.5 km.

GROWTH AND DECAY PROCESSESOF SAND WAVES FROM 1961 TO 2007

In this section, we examine the growth and decay
processes of the sand waves with the cross-sectional
surveying data which were started in 1961.
Figure 8 shows the peak discharge of floods over
3,000 m3/s and the years when the cross-sectional
bed forms were surveyed.
The floods over 6,000 m3/s didn’t occur in the
1960’s and 1970’s. Floods came in groups in thethree years from 1981 to 1983. There are many
floods over 6,000 m3/s after 1998.
Figure 9(a)∼9(d) show the changes of crosssectional bed forms in the reach downstream of
6.5 km from 1961 to 2007. In these Figures, small
bumps existing on the bed surfaces are dunes. Fromthese Figures, the growth and decay processes ofthe sand waves are clarified as follows.
The sand waves whose heights are from 1 m
to 2 m had existed on many cross-sections before
1965 as shown in Figure 9(a). But most of themdisappeared with sediment deposition in the rivermouth area from 1965 to 1970 and the cross­sectional bed forms became smooth. Dredges
were concentrated in this age in the section from
20 km to 40 km (Moro et al., 2011). It is thoughtthat the fine sediments yielded with dredging in

241

Figure 8. Peak discharge of floods and the years when
the cross-sectional bed forms were surveyed.

Figure 9(b). Variation of cross-sectional bed forms
from 1970 to 1980.

Figure 9(a). Variation of cross-sectional bed forms
from 1961 to 1970.

the upstream ­section deposited in the reach downstream of 6.0 km from river mouth.
The bed elevations are stable and the sand waves
cannot be seen from 1970 to 1980 as shown in­Figure 9(b). In this period, large flood didn’t occur
except 1947.
After that, the sand waves grew again from 1980
to 1983 as shown in Figure 9(c). In this period, anumber of large floods occurred and the bed elevations degraded about 0.5 m∼1.0 m in the entire section of river mouth.
After 1983, the height and the distribution areaof the sand waves have grown gradually up to 2007
as shown in Figure 9(d).

Figure 9(c). Variation of cross-sectional bed forms
from 1980 to 1983.

As above, in the reach downstream of 6.0 km
from the Tone River mouth, although the sandwaves had existed in the 1960’s, most of them disappeared with sediment deposition in the river mouth
area in the 1970’s. After that, the sand waves grew242

the general coordinate system. The bed load is calculated by the formula of Fukuoka (2010).
In the quasi-3D unsteady flow analysis, the horizontal velocity ui at ζ = ( zs − z ) h is assumed asshown in Eq. (1). Where i,j = x,y, δ ui = the difference between depth averaged velocity U i and bottom velocity ubi . δ ui is calculated by Eq. (2) using
depth integrated vorticity Ωi .

(

δ ui
1 − 3ζ 2
2

)

(1)

2
∂Wh 2
δ ui = U i − ubi = ε ij 3Ω j h −≈ ε ij 3Ω j h3
∂xi  3(2)

ui − U i =where spatial change of depth averaged verticalvelocity W is assumed to be negligible in shallow
flows. Depth averaged horizontal velocity U is calculated by shallow water equations employing thegeneral coordinate system (ξ ,η ).
Ωi is calculated by horizontal vorticity equation
as shown in Eq. (3).

(Figure 9(d). Variation of cross-sectional bed forms
from 1983 to 2007.

again with the degradation of bed elevation caused
by a series of floods from 1981 to 1983. Then theheight and the distribution area of the sand waves
have grown gradually up to the present date.
4

THE VARIATION OF SAND WAVERESISTANCES AND BED ELEVATIONSDURING THE FLOODS

We apply the flood flow and sediment movement analysis system based on the observation of
temporal changes in water surface profiles to thefloods in different years of the Tone River in orderto evaluate the variation of sand wave resistancesand bed elevations during the floods with growthand decay processes of sand waves.
4.1

)

(

∂h∆ξ Fiη + Diη∂JhΩi ∂h∆η Fiξ + Diξ
+
+
∂t
∂ξ
∂η

)

= J ( ERzi + Pω i )(3)

where Fiξ , Fiη = flux of Ωi with expansion, contraction and rotation, Diξ , Diη = flux of Ωi with diffusion, ERzi = rotation term, Pω i = production term.
The concentration of suspended sediment iscalculated by 3D advection-diffusion equations­(Okamura and Fukuoka, 2012) as shown in Eq. (4).
Where, the velocities are calculated by quasi-3D
unsteady flow analysis. This analysis can calculatevertical distribution of suspended ­sediment concentration due to secondary flow.
∂ck 1 ∂∆η ⋅ uξ ck ∂∆ξ ⋅ uη ck
+
+ 
∂η 
∂ξ
J
∂t
+

Computational method and conditions

We develop unsteady numerical analyses by solving the flows and bed variations during the floods
in different years so as to coincide with observed
temporal changes in water surface profiles. The
discharge hydrograph and bed forms after theflood are compared with observed data.
The analysis consists of quasi-3D unsteady flow
analysis (Uchida and Fukuoka, 2009) as shownin Eqs. (1)∼(3) and 2D bed variation analysis forgraded sediment (Fukuoka et al., 1998), ­employing

243

=

(

)

∂ck uσ − σ t − w0 k∂σ∂  ∂ck ∂  ∂ckε s ⋅   + ∂η ε s ⋅ ∂η 
∂ξ 
∂ξ 
 ∂  ∂c  ∂  ∂ck+ cos θ ηξ   ε s ⋅ k +ε s ⋅  ∂η  ∂η ∂ξ 
∂ξ 
+ε s ∂ck ∂θ ξ ∂ck ∂θ η  ∂  ∂ck+
 + ε s ⋅∂η ∂ξ  ∂z ∂z J  ∂ξ ∂η+ qsuk(4)

Figure 10.
simulation.

Initial grain size distributions in thewhere ck = concentration of suspended sediment ofdk (arbitrary grain diameter), w0 k = falling velocity
of grain, ε = coefficient of diffusion. Pick up rate ofsuspended sediment from bed surface (qsuk ) is calculated by the formula of Itakura and Kishi (1980).
The boundary conditions of upstream anddownstream ends are defined as observed waterlevels around 30 km point and astronomical tidelevels of the Choshi tidal observatory at sea of 4 km
from the river mouth, respectively. F­ igure 10 shows
initial grain size distributions.
We evaluate the resistance of flood flows using
the Manning’s roughness coefficient including
sand wave resistances. The section in question isdivided in three subsections. The bed material iscomposed of gravel in the subsection from -1.0 km
to 0.5 km. The sand waves exist in the subsection
from 0.5 km to 6.0 km. Sand waves don’t exist andthe bed material is composed of sand in the subsection upstream of 6.0 km. The Manning’s roughness coefficients are determined so as to minimize
the differences between the observed and the calculated water surface profiles in each subsection,
respectively.
4.2

Computation of the September 2007 flood

Figure 11 shows locations of observation stationsof water levels and discharge to 30 km from the rivermouth. Temporal data of water levels were measured at many observation stations. ­Figure 12 showsobserved water level hydrographs of the September
2007 flood. The flood peak occurred in the stageof low tide.
The observed temporal changes in water surface profiles are accurately reproduced by deciding the Manning’s roughness coefficient n = 0.028
from −1.0 km to 6.0 km section where the sandwaves exist and n = 0.014 upstream from 6.0 km,
respectively as shown in Figure 13.

Figure 11.

Observation stations during the 2007 flood.

Figure 12. Observed water level hydrographs of theSeptember 2007 flood.

Figure 13. Comparison between observed and calculated water surface profiles during the September 2007
flood.

Figure 14 shows the comparison of observedand calculated bed elevation before and after theflood. The calculated bed variation is similar toobserved bed variation. The bed elevation is maintained in most parts of the section in question anddecreases about 0.5 m at 0.0 km point.
The calculated discharge hydrograph corresponds with the observed discharge hydrograph asshown in Figure 15.
As above, high resistances due to sand waves
appeared to act on the flood flow in the section
downstream of 6.0 km from river mouth.

244Figure 14. Comparison between observed and calculated bed elevation before and after the September 2007
flood.

Figure 15. Comparison between observed and calculated discharge hydrographs.

4.3

Figure 16. Observed water level hydrographs of thefloods from 1981 to 1983.

Figure 17. Comparison between temporal changes inobserved and calculated water surface profiles during the
September 1982 flood.

Computation of the 1981∼1983 floods

The sand waves grew again with the bed degradation caused by a series of floods from 1981 to 1983.
Figure 16 shows observed water level hydrographsof the floods from 1981 to 1983.
Figure 17 shows the comparison between temporal changes in observed and calculated water
surface profiles during the September 1982 flood.
Although the numbers of the water level measurement points are not so many, the results of
the analysis correspond with temporal changes
in observed water surface profiles during theflood. Where the Manning’s roughness coefficient n = 0.022 from -1.0 km to 0.5 km section,n = 0.016 from 0.5 km to 6.0 km section and
n = 0.012 upstream from 6.0 km. The Manning’s
roughness coefficient is smaller than that of the2007 flood in the section downstream of 6.0 km.
Temporal changes in observed water surface profiles of other floods occurred in the period from1981 to 1983 are re-created by the same conditionsof Manning’s roughness coefficients as shown in
Figure 18.
Figure 19 shows the changes in averaged bed
elevations and deepest bed elevations from 1980
to 1983. The solid lines indicate the calculated
bed elevations after the each flood. Bed elevations degraded every flood. The calculated bed

Figure 18. Comparison between observed and calculated peak water surface profiles during the floods from1981 to 1983.

e­ levations after the September 1983 flood explain
well the observed bed elevations.
Figure 20 shows the observed and the calculated
contours of bed variations from 1980 to 1983. Calculation results explain the bed degradation caused
by a series of floods from 1981 to 1983.
The calculated discharge hydrographs correspond with the observed discharge hydrographs asshown in Figure 21.
4.4 Computation of the September 1972 flood
The September 1972 flood occurred after most ofsand waves disappeared with sediment depositionin the river mouth area from 1965 to 1970.

245

Figure 19. Comparison between observed and calculated bed elevation before and after the floods from 1981 to1983.

Figure 20. Observed and the calculated contour of bed variations from 1980 to 1983.

Figure 21. Comparison between observed and calculated
discharge hydrographs of the floods from 1981 to 1983.

Figure 22 show the comparison between temporal changes in observed and calculated water
surface profiles during the September 1972 flood.
Temporal changes in calculated water surface profiles correspond with observed ones, where theManning’s roughness coefficient is n = 0.012 in theentire section in question. This value of Manning’s
roughness coefficient is almost equal to the resistance on flat bed.
Figure 23 show the comparison between
observed and calculated discharge hydrographs.
The differences of bed elevations between before
and after the flood are smaller than 0.5 m both incalculation and observation as shown in Figure 24.
4.5

Figure 22. Comparison between observed and calculated
water surface profiles during the September 1972 flood.

Computation of the August 1959 flood

The sand waves had existed on many cross-sections
before 1965.
Figure 25 shows the comparison of temporal changes in observed and calculated water­surface profiles during the August 1959 flood. The
Manning’s roughness coefficient was n = 0.020
from -1.0 km to 6.0 km section and n = 0.013
upstream from 6.0 km. Temporal changes in­calculated water surface profiles correspond with246

Figure 23. Comparison between observed and calculated discharge hydrographs of the September 1972
flood.

Figure 26. Comparison between observed and calculated discharge hydrographs of the September 1959
flood.

Figure 24. Comparison between observed and calculated bed elevation before and after the September 1972
flood.

Figure 27. Long-term variations of the Manning’s
roughness coefficients in each section.

It is presumed that the Manning’s roughness
coefficient would have been n = 0.020 in the reachdownstream of 6.0 km except the flood peak.
4.6 Long-term variation of sand wave resistances

Figure 25. Comparison between observed and calculated water surface profiles during the September 1959
flood.

observed ones in flood rising period (before 8/14
18:00) and flood falling period (after 8/17 12:00).
But calculated water surface profiles are higher
than observed ones in flood peak (8/15 6:00∼8/16
6:00). The bed resistance could have changed during the flood, but it’s not sure whether is insufficient of observation data of water level. The bed
variations during the flood are not clear, because
the cross-sectional bed forms had not been surveyed till before 1961.
Figure 26 shows the comparison between
observed and calculated discharge hydrographs.

Figure 27 shows long-term variations of the­Manning’s roughness coefficients in each section which were evaluated by the unsteady flood
flow and bed variation analysis based on theobserved temporal changes in water surface profiles from 1959 to 2007. In the section upstreamof 6.0 km where the sand waves have not been
existed, the Manning’s roughness coefficients aren = 0.012∼0.014 in every flood. They are usual
value of Manning’s roughness coefficients in thevicinity of river mouth. In the section from -1.0 km
to 0.5 km where the bed material is composed ofgravel, the Manning’s coefficients are high valuen = 0.020∼0.028 in every flood.
On the other hand, in the section from 0.5 km to
6.0 km, the Manning’s roughness coefficients havechanged with the long-term growing and decaying
processes of sand waves. The Manning’s roughness
coefficient was n = 0.020 when the sand waves had
existed except the peak of the August 1959 flood.
The roughness coefficient became small (n = 0.012)
in the September 1972 flood which occurred after

247most of sand waves disappeared with sediment deposition in the river mouth. After that, the roughness coefficient increased a little bit (n = 0.016) ina series of floods from 1981 to 1983 when the sandwaves grew again. Then the roughness coefficientbecame large (n = 0.028) in the September 2007
flood the height and the distribution area of thesand waves have grown up.
5

RESISTANCE OF SAND WAVESIN THE VICINITY OF THE TONE
RIVER MOUTH

 gs 
D* = d50  2 ν There have been a lot of researches on the resistance of sand waves introducing the “effective bed
shear stress” in order to discuss the resistance ofsand waves since Einstein and Barbarossa (1952).
In this section, we attempt to describe the resistances in the Tone River mouth calculated by thequasi-three dimensional unsteady flow and bed
variation analysis by one dimensional quantities.
Then the resistances in the Tone River mouth arecompared with the results of previous researcheson the resistance of sand waves studied in one
dimensional analysis.
5.1

Previous researches on the resistanceof sand waves

Einstein and Barbarossa (1952) tried to estimate
the resistance of sand waves dividing the total
resistance into the friction resistance and the form
resistance as shown below.
R = R ′ + R ″, u* = u* ′ + u* ″(5)

u* ′ = ( gR ′ I ) , u* ″ = ( gR ″ I )(6)u u* ′ = 5.75 log (12.2R ′ x/ks )

(7)

1/2

1/2

where, d = grain diameter.
Kishi and Kuroki (1973) thought that τ * was afunction of τ * ′ and R/d. They proposed the relational expressions among τ *∼τ * ′∼R/d in each bed
configurations by the data of the experimental
channels.
Wang and White (1993) proposed the relational
expressions among τ * ∼ τ * ′∼D* by the data of the
alluvial rivers.
1/ 3

(9)

where, s = specific gravity of sediment, ν = kinematic viscosity.
5.2

The resistance of sand waves in the Ishikari
River during August 1981 flood

The peak discharge of the August 1981 flood in the
Ishikari River exceeded the design discharge. During
the flood, the longitudinal bed forms were surveyed(Takagi et al., 1982). The bed configuration changed
from Transition to Dune in the flood falling period.
Figure 28 shows the comparison of the relations between τ * and τ * ′ of the equations of Kishi
and Kuroki (R/d = 25,000), the curve of Wang andWhite (D* = 4.75, d = 0.19 mm) with those in 2.5 km
upstream from the Ishikari River mouth (D* = 7.6,
d = 0.3 mm). Because a large bed scouring occurredin the Ishikari River mouth during the flood,the τ * and τ * ′ are obtained from the ­quasi-three
dimensional unsteady flood flow and bed variation analysis based on the temporal changes inobserved water surface profiles ­(Okamura andFukuoka, 2012).

where, R = hydraulic radius, R′ = hydraulic radiusrelated to the friction resistance, R″ = hydraulicradius related to the form resistance, u = velocity, u* = shear velocity, g = gravity acceleration,
I = energy gradient, x = coefficient, ks = equivalent roughness.
Engelund (1967) also investigate the resistancesin alluvial rivers and the experimental channels basing on the “effective bed shear stress”. ­Engelundindicated that the total bed shear stress τ * was afunction of the effective bed shear stress τ * ′ by thedata of experimental channels.

τ * = RI /sd, τ * ′ = R ′ I /sduR′= 6.0 + 5.75 log
2dgR ′ I(8)(9)

Figure 28. Relations between τ* and τ*′ in the Tone
River mouth.

248

The relations between τ * and τ * ′ in the flood
rising period correspond to the Flat bed. On theother hand, the degrees of τ * in the flood falling
period are relatively large than those in the flood
rising period. They are thought to be the resistance of the dunes surveyed in the flood falling
period.
The degrees of the resistance of dunes are smaller
than those of the equations of Kishi and Kuroki
basing on the data of the steady flow experiments.
The sand waves might not develop enough in the
Ishikari River because of time lag due to unsteadiness of the flood. On the other hand, the curve of
Wang and White basing on the data in the alluvial
rivers are corresponds with the τ * ∼ τ * ′ relation in
the Ishikari River.
5.3 The resistance of sand waves in the Tone
River mouth during the floods
Figure 28 shows the relations between τ * and τ * ′in 4.0 km upstream from the Tone River mouth(D* = 4.3, d = 0.17 mm) calculated by the unsteadyflood flow and bed variation analysis basing on thetemporal changes in observed water surface profiles comparing with the equations of Kishi andKuroki (R/d = 25,000) and the curve of Wang andWhite (D* = 4.75, d = 0.19 mm).
The resistance in the Tone River during theAugust 1959 flood is comparable to that in the
Ishikari River. But, the value of τ * is equal to τ * ′(it means Flat bed) in the September 1972 flood
which occurred after most of sand waves disappeared. After that, the resistance increases in aseries of floods from 1981 to 1983 when the sandwaves grew again. Then the resistance during the
September 2007 flood when the height and the
distribution area of the sand waves have grown
up became larger than the curve of Wang andWhite and smaller than the equations of Kishi andKuroki.
There are little changes in the ratios between τ *and τ * ′ during a flood in the Tone River mouth.
They change with the long-term growth and decayof sand waves.

6

CONCLUSION

Main conclusions are drawn as follows.
1. It is found that the sand waves are dunes by
surveying in the Tone River mouth. We examined the arrangement of dunes which are seenin the downstream of the Choshi Ohashi Bridge
located in the reach 2.0 km upstream from theriver mouth and verified it to be scale-like dunes
as Kinoshita found out.

2. The long-term growing and decaying processesof dunes are indicated by the consideration ofvariation of cross-sectional bed forms surveyedfrom 1961 to 2007.
3. We apply the flood flow and sediment movement analysis system based on the observationof temporal changes in water surface profiles tomany floods of the Tone River. The variation ofsand wave resistances and bed elevations during the floods in different years is evaluated bya numerical analysis of quasi-three dimensional
unsteady flood flow and bed variation based on
the observed temporal changes in water surface
profiles. It is clarified that the degrees of sandwave resistances during floods have changed
with the long-term growing and decaying processes of sand waves.
4. The sand wave resistance in the Tone Rivermouth is featured by comparing with the previous studies on sand wave resistances in alluvial
rivers.
REFERENCES
Einstein, H.A. and Barbarossa, N.L. (1952) River channelroughness. Transactions, ASCE, Vol. 117, 1121–1146.
Engelund, F. (1967) Closure to “Hydraulic resistance of
alluvial streams”. Journal of the Hydraulics Division,
ASCE, Vol. 93, HY.4, 287–297.
Fukuoka, S. (2010) River engineering adaptations against
the global warming—Towards generalization of closeto-nature rivers. Proceedings of the Japan Society of
Civil Engineers, F, Vol. 66, No. 4, 471–489, JSCE.
Fukuoka, S. (2011) What is the fundamentals of river
design—Utilization of visible techniques of sedimentladen-flood flows. Advances in River Engineering,Vol. 17, 83–88, JSCE.
Fukuoka, S., Watanabe, A. and Okada, S. (1998) Bed
topography analysis in a compound meandering channel by using 3D numerical model with approximationof hydrostatics pressure. Annual Journal of HydraulicEngineering, Vol. 42, 1015–1020, JSCE.
Itakura, T. and Kishi, T. (1980) Open channel flow withsuspended sediments. Journal of Hydraulics Division,
Proceedings of ASCE, Vol. 106, HY.8, 1325–1343.
Kinoshita, R. (1984) Present status and future prospects of river flow analysis by the aerial photograph.
Proceedings of the Japan Society of Civil Engineers,Vol. 345/II-1, 1–19, JSCE.
Kishi, T. and Kuroki, M. (1972) Bed forms and resistanceto flow in erodible-bed channels (I), Bulletin of theFaculty of Engineering Hokkaido University, Vol. 67.
Moro, Y., Kazama, S. and Fukuoka, S. (2011) Change ofriver improvement works of the lower Tone River andeffectiveness of channel dredging. Advances in River
Engineering, Vol. 17, 101–106, JSCE.
Okamura, S. and Fukuoka, S. (2012) Numerical analysis of unsteady flow and bed variation using temporalchanges in water surface profiles during 1981 flood of
the Ishikari River mouth. Proceedings of 3rd ISSF,
Vol. 3.

249

Takagi, J., Makino, N., Takemoto, N. and Morita, Y.
(1982) Field survey of flood flow and bed evolution in
the lower Ishikari River. Annual Journal of HydraulicEngineering, Vol. 26, 57–62, JSCE.
Uchida, T. and Fukuoka, S. (2009) A depth integrated
model for 3D turbulence flows using shallow waterequations and horizontal vorticity equations. 33rd
IAHR Congress, Water Engineering for a Sustainable
Environment, 1428–1435.

Wang, S. and White, W.R. (1993) Alluvial resistance intransition regime, Journal of the Hydraulic Engineering, ASCE, Vol. 119, No. 6, 725–741.

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