Difference between static and dynamic angle of

1 

Difference between static and dynamic angle of repose of uniform sediment grains

2  3  4  5 

ABSTRACT

6 

In the investigation of sediment transport, it is necessary to differentiate various definitions of

7 

angle of repose (AoR) available in the literature. The static AoR, composed of upper and

8 

lower angle of slope, forms just before and after slope instability, while the dynamic AoR can

9 

be observed when sediment grains are moving continuously down an inclined plane. In the

10 

present study, a series of laboratory experiments was conducted to measure static and

11 

dynamic angle of repose for uniform natural sediments with median diameter of 0.28-4.38

12 

mm. The results show that the different slope angles have different characteristics. The upper

13 

and dynamic AoR increase slightly with increasing grain diameter, while the lower AoR is

14 

not sensitive to changes in sediment size and may assume a constant value. The average of the

15 

upper and lower AoR is equivalent to the dynamic AoR, and the difference between them

16 

increases with increasing grain diameter. The present study suggests that the different angles

17 

of repose should be treated with caution when applying in investigations of bedload transport,

18 

dune migration and local scour development.

19 

Keywords: angle of repose, avalanche, sediment transport

20  21 

1   

22 

1. Introduction

23  24 

The concept of the angle of repose (AoR) of granular materials has been applied in various

25 

areas of science and engineering, such as sediment transport, geomorphology and chemical

26 

engineering.

27 

descriptions of initial sediment motion, dune formation, bedload transport process and scour-

28 

hole geometry, and in the investigations of riverbank stability, riprap protection and reservoir

29 

sediment removal, as demonstrated in ASCE Manual 110 (Garcia 2008), in which the term of

30 

“angle of repose” appears in 35 instances.

In the hydraulics of sediment transport, the AoR has been applied in the

31 

The AoR has been defined or measured differently, but with results open to

32 

interpretation (Carrigy 1970; Francis 1986). For example, the AoR can be obtained simply by

33 

pouring sand grains to form a conical pile. However, two different slopes can be differentiated

34 

during the pile formation. As grains are gradually added to a heap, they can pile up to an

35 

upper AoR (αU). Once masses slump, a new surface will form at a lower AoR (αL). As a

36 

result, the AoR varies repeatedly during the growth of the pile. The upper AoR may be also

37 

measured as a critical angle of a tilting box at which some grains start to roll down along the

38 

inclined surface. In comparison, the lower AoR may be achieved at the end of an avalanche,

39 

which can be generated by the removal of support from loose material. The AoR can also be

40 

measured by draining grains through a bottom opening of a container, by building up a cone

41 

over a fixed base, or by rotating a drum filled partially with grains (Eisen et al. 1998).

42 

In principle, the AoR can be considered to be the maximum angle at which grains can

43 

stand without becoming unstable. When conducting physical measurements, like in the pile

44 

formation, two relatively constant angles of repose could be obtained. The upper angle is

45 

associated with the onset of slope instability, and the lower angle is associated with the

46 

cessation of slope instability. Unfortunately, confusions often exist in the differentiation

2   

47 

between the two angles and thus the use of the term of AoR in the literature.

48 

For example, Simons and Senturk (1992) stated that the AoR is the angle of slope

49 

formed by particulate material under the critical equilibrium condition of incipient sliding.

50 

Soulsby (1997) applied the angle of final repose for the angle of lee slopes of dunes and the

51 

angle of slope of the conical scour around a circular vertical pile, which is observed at the end

52 

of avalanching. Garcia (2008) considered the AoR as a slope angle beyond which

53 

spontaneous failure of the slope occurs. An early differentiation between the upper and lower

54 

angles of slope was made by Bagnold (1966a), who called the upper angle the apparent

55 

limiting static friction angle of initial yield and the lower angle the residual angle. Allen (1969)

56 

described the upper angle as the angle of initial yield and the lower angle as the residual angle

57 

after shearing. Carrigy (1970) noted that there is no agreement reached as to which angle

58 

should be measured. Francis (1986) indicated that some confusion exists in the precise

59 

meaning of the term ‘angle of repose’, and a single angle of repose is inadequate to explain all

60 

observable characteristics of many scree slopes.

61 

In recent decades, sediment transport and morphodynamic processes at various stages

62 

have been widely simulated with numerical models, which often require a selection of a

63 

proper value of AoR. However, relevant studies in the literature show that the problem has

64 

only been approached with significant simplifications. For example, in the simulation of bank

65 

failure, Zech et al. (2008) applied two critical angles (for submerged and emerged conditions)

66 

above which a block failure occurs, and also two residual angles at which the failed materials

67 

depose. A reasonable evaluation of all such angles for particular cases could only resort to

68 

laboratory tests (Roulund et al. 2005; Zech et al. 2008; Evangelista et al. 2014; Evangelista et

69 

al. 2015).

70 

The present study aims to quantify differences among the different angles of repose. A

71 

series of experiments was conducted with a rotating drum to measure the AoR of uniform

3   

72 

sediments under static and dynamic conditions. Both terms of static AoR and dynamic AoR

73 

are used, with the static AoR consisting of lower (minimum) and upper (maximum) AoR and

74 

the dynamic AoR describing the inclination of a surface layer of continuous sediment motion.

75  76  77 

2. Experiments

78  79 

In the present study, measurements of AoR were conducted using a transparent, motor-driven

80 

rotating drum, of which the inner diameter was 28.90 cm and the depth was 11.50 cm (see Fig.

81 

1). Altogether nine uniform natural sediments, with median diameter D ranging from 0.28

82 

mm to 4.38 mm, were tested. In each test, the drum was half-filled with selected sediment

83 

grains. All slope measurements were conducted under both dry (with sediment grains exposed

84 

in air) and submerged condition (with the drum fully filled with water).

85 

When the drum rotated, the following changes were generally observed in the slope of

86 

the free surface of grains for each size of sediment. Initially at a very low rotating speed,

87 

sediment grains moved together with the drum, demonstrating a rigid body motion until the

88 

slope reached its upper angle, αU [see Fig. 2(a)]. Then, a further increase in the slope angle

89 

triggered an avalanche, transporting grains down the slope. At the end of the avalanche, a new

90 

slope formed at the lower angle, αL [see Fig. 2(b)]. Characterised by the repeated change in

91 

the slope, the motion of sediment grains is referred to as slumping (Henein et al. 1983). If the

92 

rotating speed was further increased, both αU and αL would disappear and the slope angle

93 

approached a constant while sediment grains kept rolling down the slope. This indicated the

94 

beginning of the rolling stage. The corresponding slope angle is referred to as the dynamic

95 

AoR (αD). At this stage, sediment grains move continuously from the upper to lower end of

96 

the slope, yielding a surface shear layer of grains that flow down the plane inclined at a fixed 4   

97 

angle. To achieve the rolling stage, a slow variation in the rotating speed was needed. From

98 

the slumping to rolling stage, the free surface at any instance remained planar and thus can be

99 

described with a single slope angle. However, after the rolling stage, an increase in the

100 

rotating speed resulted in cascading or cataracting without forming any planar free surface

101 

(Henein et al. 1983), which is beyond the subject of the present study.

102 

To avoid undesired sliding along the inner wall during rotation, a layer of sediment

103 

grains was glued to the cylindrical wall using silicon. To record the motion of sediment grains

104 

in the drum, a camera was positioned horizontally facing the frontal side of the drum, with the

105 

center of view being aligned with the drum axis. The back side of the drum was covered with

106 

black paper to provide dark background. A video was taken at a rate of 25 frames per second.

107 

The duration of each video was set so that at least 20 avalanches were captured in the

108 

slumping regime, and at least one-minute recording was taken in the rolling regime.

109 

With the colour difference between the sediment grains and the dark background, each

110 

frame was first converted to a black and white picture and a straight line composed of more

111 

than 400 pixels was then identified between the slope surface and the background. Finally, the

112 

angle between the straight line and the horizontal was calculated. The above processing was

113 

completed with the aid of Matlab (MathWorks Inc. 2007).

114  115  116 

3. Results

117  118 

Fig. 3 shows typical time series of the slope angle measured under the dry condition. The time

119 

interval was 0.04 s as the video was taken at 25 frames per second. To calculate the slope

120 

angle for each frame, the interface identified between the slope surface and the dark

121 

background was fitted to a straight line. The correlation coefficients (R2) calculated for such

5   

122 

straight lines were all greater than 0.98. With the data plotted in Fig. 3, the calculated average

123 

values of αU, αL and αD are 40.6o, 35.3o and 37.7o, respectively. Significant difference exists

124 

in the measured angles from the slumping to rolling stage. At the slumping stage, the

125 

variation of the slope angle is relatively large, ranging from 35.2o to 40.8o. In comparison, at

126 

the rolling stage, the variation becomes much smaller, only from 37.0o to 38.1o, implying that

127 

the slope angle can be approximated as a constant. Furthermore, it seems reasonable to take

128 

the dynamic AoR as the average of the upper and lower AoR, i.e. αD ≈ 0.5(αU + αL). This

129 

approximation has also been found acceptable by Liu et al. (2005) for other shapes of grains.

130 

In the following, only the average values of αU, αL and αD, calculated at individual

131 

rotating speeds, are used for analysis. Fig. 4 shows variations of αU, αL and αD with the

132 

rotating speed under the dry condition. At the slumping stage, different variations can be

133 

observed in the lower AoR (αL) and the upper AoR (αU). For example, for each sediment in

134 

the range of D ≥ 0.73 mm, αL is not much affected by the rotating speed and can be assumed

135 

to be a constant. However, the corresponding αU increases slightly with increasing rotating

136 

speed. This implies that αU varies depending on the sediment supply at the upper end of the

137 

slope. By gradually increasing the rotating speed up to a critical value, both αU and αL merged

138 

into almost a single value (around the middle of αU and αL), indicating the inception of the

139 

rolling stage. However, it should be mentioned that the critical rotating speed was not realized

140 

for the two fine sediments of D = 0.28 mm and 0.46 mm. For these two cases, the difference

141 

between αU and αL was small, and in particular the surface of the slope did not appear to be

142 

planar and was subject to some local undulations, the latter disappearing for the cases of

143 

coarser grains. Fig. 4 shows that the dynamic AoR (αD) measured at the rolling stage

144 

increases with the sediment size, but it seems to be independent of the rotating speed.

145 

With the data plotted in Fig. 4, the dependence of the three different kinds of AoR on

146 

the sediment size for the dry condition is further examined in Fig. 5. To minimize effect of the 6   

147 

rotating speed, both the lower and upper AoR are taken as those measured at the lowest

148 

rotating speed. It can be seen that for D ≥ 0.73 mm, the lower AoR does not change much

149 

with the median diameter D (with an average of 35.4o), while both the upper and dynamic

150 

AoR increases with increasing D. In addition, Fig. 5 also shows that the average angle of the

151 

lower and upper AoR is close to the dynamic AoR.

152 

Fig. 6 shows variations of the average αU, αL and αD with the rotating speed under the

153 

submerged condition. The fashion of the variations is similar to that given in Fig. 4 under the

154 

dry condition. However, the measurements obtained under the submerged condition are

155 

subject to high uncertainties in comparison to those under the dry condition.

156 

Fig. 7 provides the average values of αU, αL and αD varying with the sediment size for

157 

the submerged condition. Both αU and αL were taken as those measured at the lowest rotating

158 

speed. By making a point-by-point comparison between all the individual data points plotted

159 

in Fig. 7 and the corresponding ones in Fig. 5, it can be found that the AoR measured under

160 

the submerged condition generally decreases. Among the 23 cases compared, there is only

161 

one case that shows the AoR under the submerged condition is larger than that under the dry

162 

condition. The decreased angle varies from 0.0o to 2.5o for the upper AoR, from − 0.5o to 3.9o

163 

for the lower AoR, and from 0.9o to 2.4o for the dynamic AoR.

164 

The above changes are associated with a few factors. Under the dry condition,

165 

sediment grains interact by direct contact, but they interact through a thin water layer when

166 

submerged in water (Jain et al. 2004). In other words, given the irregular surface of the natural

167 

sand and gravel, the inter-grain friction would be lower in water than in air. As a result, the

168 

so-induced hydrodynamic lubrication between grains may become significant, which could

169 

yield a decrease in the AoR. However, on the other hand, the inter-grain shear and thus the

170 

AoR could be increased by the viscous effect of the water. In addition, the reduced weight of

171 

each grain and thus its inertia may also alter the flow characteristics of the grains and thus the 7   

172 

AoR value. The observed decrease in the AoR under the submerged condition implies that the

173 

hydrodynamic lubrication may be dominant in comparison to other factors.

174  175 

The variation of the dynamic AoR with the median diameter of sediment grains for the submerged condition can be described using the following empirical function: tan(αD) = 0.74D0.05

(1)

176 

where D is the median diameter in mm. For comparison, also superimposed in Fig. 7 is

177 

another empirical formula:

αD = 36.45 + 4.294 log(D)

(2)

178 

with D given in mm, which was proposed by Xiong (1989) based on slope angles measured

179 

from underwater conical piles for sediment grains of 0.06-6 mm. The two empirical formulas

180 

agree well to each other, in particular for the range of D = 0.5-3 mm. This agreement suggests

181 

that the average AoR derived from conical piles can be considered equivalent to the dynamic

182 

AoR and also to the average of the upper and lower AoR.

183 

In comparison to the upper and dynamic AoR, both varying with the grain diameter,

184 

the lower AoR seems not to be sensitive to the change in the grain diameter for the submerged

185 

condition. It varies from 33.2o to 35.6o around an average of 34.4o. Such variations were also

186 

reported by Froehlich (2011), who conducted large-scale measurements of the lower AoR at

187 

74 stockpiles of dumped natural and crushed rock (D = 3.2-355 mm) for the dry condition.

188 

Froehlich (2011) classified the shape of rock as being round, subround, subangular or angular.

189 

It is interesting to note that his data of subround and subangular rock (in the similar shape of

190 

natural sand and gravels as used in the present study) show that the lower AoR does not

191 

depend on the grain size. It varied in the range of 34.4±1.2o, in spite of the wide change in the

192 

rock size.

193 

To quantify to what extend the AoR fluctuates between the upper to lower limit, the

194 

relative differences, (αU − αL)/αave where αave = 0.5(αU +αL), were calculated for both dry 8   

195 

and submerged conditions and the results are plotted against the grain diameter in Fig. 8. It

196 

shows that the difference (αU − αL) significantly increases with D, up to 25% of αave for D =

197 

3.68 mm. The data trend can be approximated as

αU − αL = 0.12 D αave 198 

(3)

where D is given in mm.

199  200  201 

4. Discussions

202  203 

In spite of the fact that the granular motion in a rotating drum is different from the sediment

204 

transport observed in an open channel flow, the rotating drum clearly demonstrates that the

205 

AoR varies repeatedly from the upper AoR to the lower AoR. Such information is useful by

206 

noting that the typical variations in the slope angle also take place in the lee side of a

207 

migrating dune (Allen 1985) or in the development of a scour hole (Roulund et al. 2005). The

208 

dune migration results from individual grains rolling down the slip face in the lee side of the

209 

dune. When the slope over-steepens, large numbers of grains move down the slope in an

210 

avalanche fashion. Therefore, the slope angle of the lee side cannot be simply taken as a

211 

constant, as it generally varies from the upper AoR at the beginning of the avalanche to the

212 

lower AoR at the end of the avalanche. In addition, the experimental results obtained in the

213 

present study can be used for the selection of the upper and lower AoR for a particular

214 

sediment in the implementation of a sand-slide model in the simulation of scour around a pier

215 

(Roulund et al. 2005).

216 

The rolling stage appearing in the rotating drum may resemble the sheet flow of the

217 

bedload at high transport rates, which occurs over a planar sediment bed. Therefore, the

218 

dynamic AoR measured at the rolling stage could be applied to estimate the friction

9   

219 

coefficient for the sheet flow. By noting that the dynamic AoR is close to the average of the

220 

upper and lower AoR, the angle of friction for sheet flows would be generally higher than the

221 

lower AoR measured at the end of an avalanche but smaller than the upper AoR. The above

222 

conjecture should be limited to intensive bedload transport with high sediment concentration.

223 

The dynamic AoR and thus the corresponding angle of friction would reduce for low

224 

sediment concentration (Allen 1985; Nino and Garcia 1998).

225 

In addition, to explore how the dynamic AoR varies with sediment transport rate, the

226 

following calculation can be performed for the case of the rotating drum. First, if ignoring the

227 

effect of sediment concentration, the sediment transport rate per unit width in the drum can be

228 

estimated as ωR2/2 (Cheng 2012), where R is the drum radius and ω is the rotating speed in

229 

rad/s. It is equal to the total rate of sediment supply in the drum, and also the maximum

230 

transport rate of the flowing layer of sediment grains at the center of the drum (Cheng et al.

231 

2011). Finally the dimensionless transport rate or Einstein number φ can be expressed as

φ=

ωR2

(4)

2 ΔgD 3

232 

where Δ = (ρs − ρ)/ρ with ρs and ρ denoting the sediment and water densities, respectively,

233 

and g is the gravitational acceleration. Fig. 9 shows that the dynamic AoR decreases with

234 

increasing φ for the submerged condition. This result may explain in part why the dynamic

235 

friction coefficient cannot be taken as a constant in the investigation of bedload transport

236 

(Bagnold 1973; Seminara et al. 2002). The variation shown in Fig. 9 agrees qualitatively with

237 

the result obtained by Bagnold (1966b), who stated that the dynamic bedload friction

238 

coefficient decreases with increasing the Shields number (and thus the bedload transport rate).

239 

All the experiments in the present study were conducted only with the fixed-size drum.

240 

By noting that a variation in the diameter of the drum will change the grain supply at the

241 

upper end of the slope, the drum size will affect flow characteristics, such as, the thickness of

10   

242 

the flowing layer (Felix et al. 2007) and the slope angles (Liu et al. 2005). Such scale effects

243 

remain a challenging task at present and need future research efforts.

244  245  246 

5. Summary

247  248 

With a rotating drum half-filled with uniform sediment grains, three different angles of repose

249 

are clearly identified. They are the upper AoR formed at the inception of an avalanche, the

250 

lower AoR at the end of an avalanche, and the dynamic AoR characterised with continuous

251 

sediment transport down the slope. The measurements show that the average of the upper and

252 

lower AoR is approximately equal to the dynamic AoR, and the difference between the upper

253 

and lower AoR increases with increasing sediment size, up to 25% of the dynamic AoR for D

254 

= 3.68 mm. Both the upper and dynamic AoR increase slightly with increasing grain diameter,

255 

while the lower AoR is not sensitive to changes in sediment size and may assume a constant.

256 

The dependence of the dynamic AoR on the grain diameter, which was derived from the

257 

present data, agrees well with the previous AoR measured from underwater conical piles.

258 

The results obtained in the present study could be useful for modelling sediment

259 

transport in open channel flows by noting that similar variations in the AoR also occur during

260 

dune migration and scour-hole development.

261  262  263 

Notations

264  265 

AoR

Angle of repose

266 

D

Median diameter of sediment grains

267 

g

Gravitational acceleration 11 

 

268 

R2

Correlation coefficient

269 

αave

0.5(αU +αL)

270 

αD

Dynamic AoR

271 

αL

Lower AoR

272 

αU

Upper AoR

273 

Δ

(ρs − ρ)/ρ

274 

ρ

Water density

275 

ρs

Sediment density

276 

φ

Dimensionless transport rate or Einstein number

277  278  279  280  281 

12   

282 

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283  284  285 

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  MathWorks Inc. (2007). Getting started with MATLAB 7, MathWorks, Nattick, Mass. 

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  Nino, Y., and Garcia, M. (1998). "Experiments on saltation of sand in water." Journal of Hydraulic  Engineering‐ASCE, 124(10), 1014‐1025. 

348  349  350 

  Roulund, A., Sumer, B. M., Fredsøe, J., and Michelsen, J. (2005). "Numerical and experimental  investigation of flow and scour around a circular pile." Journal of Fluid Mechanics, 534, 351‐401. 

351  352  353 

  Seminara, G., Solari, L., and Parker, G. (2002). "Bed load at low Shields stress on arbitrarily sloping  beds: Failure of the Bagnold hypothesis." Water Resources Research, 38(11), 10.1029/2001wr000681. 

354  355  356 

  Simons, D. B., and Senturk, F. (1992). Sediment transport technology: water and sediment dynamics,  Water Resources Publications, Littleton, Colo., USA. 

357  358 

  Soulsby, R. (1997). Dynamics of marine sands: a manual for practical applications, Telford, London. 

359  360  361 

  Xiong, S. L. (1989). "Study of scour‐hole geometry of cohensionless sediment grains." Journal of  Sediment Research(2), 76‐83. 

362  363  364  365 

  Zech, Y., Soares‐Frazão, S., Spinewine, B., and Le Grelle, N. (2008). "Dam‐break induced sediment  movement: experimental approaches and numerical modeling." Journal of Hydraulic Research, 46(2),  176–190.  14   

366 

 

Camera Motor 

11.50 cm Drum diameter = 28.90 cm 

367 

 

368 

 

369 

 

370 

 

371 

 

372 

 

373 

 

374 

Fig. 1. Schematic of experimental setup

375 

 

 

15   

ω

(a) αU

ω

(b)

αL

376 

 

377 

 

378 

Fig. 2. A Angle of sloope varies between uppper and lower AoR in the slumpingg regime

379 

 

380 

  16   

43

αU αL

42

Slumping regime Rolling regime

Angle (deg)

41 40 39 38 37 36 35 0

10

20

30

Time (s)

40  

381  382 

 

383 

 

384 

 

385 

 

386 

Fig. 3. Measurements of upper and lower AoR (slumping regime, ω = 0.035 rad/s), and

387 

dynamic AoR (rolling regime, ω = 0.131 rad/s) for D = 1.29 mm under dry condition

388 

 

 

17   

389 

  48

 D = 0.28 mm D          0.46         0.73         1.02         1.29         2.40         3.08         3.68         4.38

46

upper 

AoR (deg)

44 42 40 38

lower 

dynamic 

36 34 0

2

4

6

8

10

12

14

16

18

20

Rotating speed (deg/s)

390  391 

 

392 

 

393 

 

394 

 

395 

 

396 

 

397 

Fig. 4. Variations of upper, lower and dynamic AoR with rotating speed for dry condition for different diameters of sediment grains. For D ≥ 0.73 mm, the upper AoR increases slightly with increasing rotating speed while the lower AoR can be approximated as a constant. Both upper and lower AoR merge suddenly into the dynamic AoR at the end of slumping regime.

398  399  400 

18   

 

401 

  50  Lower

48

 Upper

46

 Average  Dynamic

AoR (deg)

44 42 40 38 36 34 32 0

1

2

3

4

5

D D (mm)

402  403 

 

404 

 

405 

 

406 

 

407 

 

408 

 

409 

 

410 

Fig. 5. Variations of upper, lower and dynamic AoR with grain diameter for dry condition

19   

 

411 

 

412 

  48

 D = 0.28 mm D          0.43         0.73         1.02         1.29         2.40         3.08         3.68         4.38

46

42

upper 

AoR (deg)

44

40 38

dynamic  lower 

36 34 32 0

2

4

6

8

10

12

Rotating speed (deg/s)

413  414 

 

415 

 

416 

 

417 

 

418 

 

419 

 

420  421  422  423  424 

Fig. 6. Variations of upper, lower and dynamic AoR with rotating speed for submerged condition for different diameters of sediment grains. For most of the data series, the upper AoR increases slightly with increasing rotating speed, while the lower AoR can be approximated as a constant. Both upper and lower AoR merge suddenly into the dynamic AoR at the end of slumping regime.

20   

14

 

425 

  50  Lower  Upper  Average  Dynamic  Eq. (1)  Xiong (1989)

48 46

AoR (deg)

44 42 40 38 36 34 32 0

1

2

3

4

D D (mm)

426  427 

 

428 

 

429 

 

430 

 

431 

 

432 

 

433  434 

Fig. 7. Variations of upper, lower and dynamic AoR with grain diameter for submerged condition.

435 

 

 

21   

5

 

436 

 

437 

 

Relative difference of AoR 

0.3 0.25 0.2 0.15 0.1 Dry Submerged  Eq. (3)

0.05 0 0

1

2

3

4

DD (mm)

438  439 

 

440 

 

441 

 

442 

 

443 

 

444 

 

445 

 

446 

 

447 

Fig. 8. Variation of relative difference, (αU - αL)/αave , with grain diameter.

448 

 

 

22   

5  

449 

 

450 

 

39.5

dynamic AoR (deg)

39.0

38.5

38.0

37.5

37.0

36.5 0

451 

0.5

1

1.5

2

2.5

φ 

3

452 

 

453 

 

454 

 

455 

 

456 

 

457 

Fig. 9. Dynamic AoR decreases with increasing dimensionless sediment transport rate

458  459 

23   

3.5