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UFL/COEL-TR/102
THREE-DIMENSIONAL HYDRODYNAMICS AND
MORPHOLOGY ASSOCIATED WITH RIP CURRENTS
by
Tae-Myoung Oh
Dissertation
1994
THREE-DIMENSIONAL HYDRODYNAMICS AND MORPHOLOGY
ASSOCIATED WITH RIP CURRENTS
By
TAE-MYOUNG OH
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1994
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ACKNOWLEDGMENTS
My first sincere appreciation and gratitude should belong to God. Throughout
my study here, God has always kept His eyes on me, protected my spirits, and guided
and encouraged to do what I should do. Without His Hands, this study would not
have been possible.
I also would like to express my sincere appreciation and gratitude to my adviser,
Professor Robert G. Dean. He has provided me a goal which I am trying and of
course will try to follow. His encouragement, enthusiasm, unflagging inspiration,
timely advice have taught me not only how to finish the work but also how to access
and love nature. I wish to extend my thanks to my committee members, Dr. Hsiang
Wang, Dr. Daniel M. Hanes, Dr. Ashish J. Mehta, Dr. Robert J. Thieke and Dr.
Ulrich H. Kurzweg, for their precious teaching and assistance. I am also grateful
to all staff, especially Becky Hudson for providing me her generous hospitality and
Helen Twedell for helping me in exploring the Coastal Archives. Special thanks are
due to Subarna Malaker for computer help, Jim Joyner for his help with laboratory
work, and Sidney Schofield for his overall help at the laboratory. I also owe a lot
my fellow students for their hospitality and friendship, especially Taerim Kim and
Taewhan Lee.
Special thanks go to my church members, especially the members of the edito-
rial board for the Church News Letter (Dongil Kim, Dr. Jaekyung Baek, Joon Lee,
Yongjin Lim and Yoonwha Chung), and also the member of the Continuing Witness-
ing Training Team (Dongil Kim, Jongmin Lee, Insoo Cho and Ms. Inhee Chung).
Their encouragement and prayer will always be remembered.
Finally, I could not finish my acknowledgements without appreciating endless
support and prayer from my family, parents and parents in-law. My wife, Hyunsoon,
and my two daughters, Youngeun and Seieun, are always with me during good times
as well as the bad. My parents and parents in-law are specially thanked for their
support and continuous encouragement and prayer during the stay.
L
TABLE OF CONTENTS
ACKNOWLEDGMENTS ................... .......... iii
LIST OF FIGURES .................... ............ vi
LIST OF TABLES ................. ................ xvi
ABSTRACT ................... ............... xviii
CHAPTERS
1 INTRODUCTION ..................... .......... 1
1.1 Three-Dimensional Features in Nature ................ 2
1.2 Three-Dimensional Features in a Narrow Wave Tank ...... .. 6
1.3 Objectives ................. ............... 8
1.4 Scope of Study .............................. 10
2 LITERATURE REVIEW ........................... 11
2.1 Introduction ................. .............. 11
2.2 Profile Evolution and Equilibrium Profile ................ 11
2.3 Three-Dimensionality in a Narrow Wave Tank ............. 17
2.4 Rip Currents in Nature ................... ...... 21
2.5 Rip Current Models ............................ 26
2.6 Rip Current Generation Models ..................... 27
2.6.1 Introduction .......................... 27
2.6.2 Prismatic Topography Models .................. 30
2.6.3 Non-Prismatic Topography Models . . .... 35
3 LABORATORY STUDIES ON THREE-DIMENSIONAL MORPHOLOGY 41
3.1 Introduction .............. .. .............. 41
3.2 Laboratory Facilities and Experimental Conditions .......... 42
3.3 Description of Movable Bed Experiments . . . .... 45
3.4 Results and Discussions ................ ......... 47
3.4.1 Reference Test (Experiment MT01) . . . 47
3.4.2 Experimental Repeatability . . . .... 51
3.4.3 Perturbation Tests ........................ 54
3.5 Summary and Conclusions ........................ 78
3.5.1 Summary ................... .......... 78
3.5.2 Conclusions ............................ 82
4 THREE-DIMENSIONAL HYDRODYNAMICS ON A PRISMATIC BEACH 85
4.1 Introduction ................... ............ 85
4.2 Wave-Induced Three-Dimensional Hydrodynamics . . ... 86
4.2.1 Governing Equations ....................... 86
4.2.2 Force Balance Inside the Surf Zone . . . .
4.2.3 Wave-Induced Horizontal Torque Induced on the Surf Zone .
4.3 Description of Fixed Bed Experiments . . . . .
4.3.1 Laboratory Facilities and Experimental Conditions . .
4.3.2 Experimental Wave Conditions . . . . .
4.3.3 Experimental Devices . . . ..... ...
4.4 Vorticity Measurement Tests .......................
4.5 Circulation Tests with Jet Discharge . . . . .
4.5.1 Introduction ............................
4.5.2 Data Measurements ........................
4.5.3 Data Analyses and Discussions . . . . .
4.6 Edge W ave Tests ............................
4.7 Summary and Conclusions . . . . . .
4.7.1 Summ ary .............................
4.7.2 Conclusions ............................
5 THREE-DIMENSIONAL HYDRODYNAMICS ON A BARRED BEACH
5.1 Introduction .................
5.2 Circulation Tests on a Barred Beach .
5.2.1 Description of Circulation Tests .
5.2.2 Wave Heights Measurements .
5.2.3 Data Analyses and Discussions .
5.2.4 Summary ..............
5.3 Longshore Currents on Barred Coastlines .
5.3.1 Governing Equations and Boundary
5.3.2 Perturbation Methods . .
5.3.3 Perturbation Solutions . .
5.3.4 Numerical Example . . .
5.4 Conclusions .................
.......
Conditions
......0.
6 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS .......
6.1 Sum m ary ................................ .
6.2 Conclusions ................................
6.3 Recommendations for Further Study . . . . .
APPENDICES
A DISTRIBUTIONS OF TORQUE COMPONENTS (H, = 3cm) ......
B DISTRIBUTIONS OF TORQUE COMPONENTS (Ho = 9cm) ......
C FILTERING OF MARGINAL DISTRIBUTIONS . . . .
BIBLIOGRAPHY .................................
BIOGRAPHICAL SKETCH ...........................
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LIST OF FIGURES
1.1 Schematic Diagram of Nearshore Circulation System with Rip
Currents (from Shepard and Inman, 1950b) . . . 2
1.2 Bar Morphologies Created Digitally by Averaging Ten-Minute Time
Exposure Images. White bands indicate the shore break and re-
gions of concentrated wave breaking on distinct bar morphologies
at the Army Corps of Engineers Field Research Facility at Duck,
North Carolina. (from Holman and Lippmann, 1987) . 4
1.3 An Example of Three-Dimensional Morphology in Two-Dimensional
Narrow Wave Tank Tests (from Beach Erosion Board, 1947). Note
the temporal reduction of three-dimensionality after 10 hours wave
operation . . . . . . . .. 7
2.1 Bar Formation with Increasing Wave Height (drawn based on
Dean, 1985). Here, T is the wave period, and tF is the fall time of
the sediment, defined as the ratio of the suspended height to the
fall velocity of the sediment. . . . ..... 15
2.2 Cross-Tank Variations of the Profiles after 1 hour Wave Run (mod-
ified from Hughes and Fowler, 1990). Note 3-D morphology inside
the surf zone and near the bar area. .. . . . ... 20
2.3 Constitutive Elements of a Rip Current System: the feeder currents,
flows of water parallel to the shore; the neck, the main seaward-
flowing current extending beyond the breaker zone; and the rip head,
in which the rip current lose its identity as the water spreads out.
(from Shepard, Emery and La Fond, 1941) . . ...... 21
2.4 Measured Wave Height and Rip Current Intensity (from Shepard
et al., 1941). These data were obtained daily. Note the direct
relationship between these two variables. Note the definition of
zero intensity of rip current when wave height was 0.6 m (2 ft);
which would suggest that wave height must be larger than 0.6 m
for rip currents to form. ....................... 23
3.1 Schematic Diagram of the Initial Profile and Other Experimental
D details . . . . . . . .. 43
3.2 Three Profiles B1, B2 and B3 across the Tank . . ... 44
3.3 Mean Profile Evolution during Early Stages of Experiment MT01.
Elapsed Times = 0 (Initial Profile), 23, 69, 161 and 242 min. Note
that the profile approached an equilibrium and a level of profile
stability had occurred at 242 min. . . . .. 48
3.4 Comparison of the Mean Profile at 242 min during Experiment
MT01 with the Equilibrium Profile Proposed by Dean (1977).
Note the similarity between two profiles inside the surf zone. 49
3.5 Mean Profile Evolution after the Profile Approached an Equilib-
rium during Experiment MT01. Elapsed Times = 0 (Initial Pro-
file), 242, 297, 352, 407 and 476 min. Note the substantial erosion
of the area seaward of the bar and the deposition of the area imme-
diately landward of the bar trough. Note also landward movement
of the bar. . . . .. . . ...... 50
3.6 Initial Profile and Three Profiles B1, B2 and B3 and Mean Profile
at 476 min during Experiment MT01. Note three-dimensionality
inside the surf zone and near the bar area, which is 3-D morphology
associated with rip currents, and also that the centerline profile
B2 is almost the same as the mean profile. . . .... 51
3.7 Mean Profile Evolution after the Profile Approached an Equilib-
rium during Experiment MT02. Elapsed Times = 0 (Initial Pro-
file), 207, 297, 352, and 407 min. Note the substantial erosion
of the area seaward of the bar and the deposition of the area
immediately landward of the bar trough, which occurred during
Experiment MTO1 (see Figure 3.5), thus providing experimental
repeatability. ............. ............ 52
3.8 Comparison of the Mean Profile at 207 min during Experiment
MT03 with the Mean Profiles at the Same Time during Experi-
ments MT01 and MT02. Note the similarity between those three
m ean profiles ................... ......... 53
3.9 Mean Profile Evolution during Experiment MT03. Elapsed Times
= 0 (Initial Profile), 207, 352, 476 and 545 min. Note rapid change
during 476 to 545 min with landward movement of the bar and
another peak of the berm ...................... 55
3.10 Initial Profile and Three Profiles (Bl, B2 and B3) and the Profile
along Rip Channel inside the Surf Zone at 545 min during Ex-
periment MT03. Note overall 2-D conditions inside the surf zone
except a very narrow deep channel near glass side wall. . 56
3.11 Induced Perturbation to Deepen the Bar Trough at 69 min during
Experiment MT04: (a) Comparison of the Modified Mean Profile
with Pre-Modified Mean Profile, (b) Three Profiles before Modifi-
cation, and (c) Three Profiles after Modification . ... 57
I
3.12 Comparison of Pre-Modified Profile at 69 min with Pre-Modified
Profile at 138 min during Experiment MT04: (a) Comparison of
Mean Profiles, (b) Three Profiles before Modification at 69 min,
and (c) Three Profiles before Modification at 138 min. . 58
3.13 Induced Perturbation to Provide Asymmetric Area Seaward of
the Bar at 138 min during Experiment MT04: (a) Comparison
of the Modified Mean Profile with Pre-Modified Mean Profile, (b)
Three Profiles before Modification, and (c) Three Profiles after
Modification. Note the dominantly 2-D profile before modification
and the asymmetric area seaward of the bar after modification. .59
3.14 Profile Evolution from 138 min to 207 min during Experiment
MT04: (a) Comparison of the Modified Mean Profile at 138 min
with Mean Profile at 207 min, (b) Three Profiles after Modifica-
tion at 138 min, and (c) Three Profiles at 207 min. . ... 61
3.15 Profile Evolution from 297 min to 352 min during Experiment
MT04: (a) Comparison of the Mean Profile at 297 min with Mean
Profile at 352 min, (b) Three Profiles at 297 min, and (c) Three
Profiles at 352 min. Note the erosion of the area seaward of the bar
and deposition of the area immediately landward of the bar trough.
Note also the differences in the three-dimensionalities between two
elapsed tim es .............................. 63
3.16 Induced Perturbation to Remove Half Part of the Bar Crest at
352 min Wave Run during Experiment MT04: (a) Comparison
of the Modified Mean Profile with Pre-Modified Mean Profile, (b)
Three Profiles before Modification, and (c) Three Profiles after
Modification.................... .......... 64
3.17 Profile Evolution from 352 min to 407 min during Experiment
MT04: (a) Comparison of the Pre-Modified Mean Profile at 352
min with Mean Profile at 407 min, (b) Three Profiles before Mod-
ification at 352 min, and (c) Three Profiles at 407 min. Note the
complete recovery of the bar crest with a smaller rotational angle
(less than 10 degrees) ......................... 65
3.18 Isolines of Profile Elevations at 545 min during Experiment MT04.
These photographs showed: (1) the weak 3-D berm, (2) deep and
narrow channel near steel side wall, (3) depositional area occupy-
ing large portion of the surf zone, and (4) clockwise rotation of the
bar. Elevation contours were established by placing black yarn at
waterline during lowering of water level. . . ... 67
3.19 Initial Profile with 3-D Berm Area for Experiment MT05 . 69
3.20 Initial Profile and Three Profile B1, B2 and B3 at 476 min during
Experiment MT05. Note dominantly 2-D morphology. . 70
3.21 Mean Profile Evolution during Experiment MT05. Elapsed Times
= 0 (Initial Profile), 476, 545 and 614 min. Note rapid change
during 545 to 614 min with landward movement of the bar and
another peak of the berm. However, the overall shape of the profile
was surprisingly unchanged . . . . ... 71
3.22 Mean Profile Evolution during Experiment MT05. Elapsed Times
= 0 (Initial Profile), 890, 959 and 1028 min. Note the back-and-
forth movement of the bar, otherwise the profiles approached an
equilibrium ......................... .. .. 72
3.23 Mean Profile Evolution during Experiment MT06. Elapsed Times
= 0 (Initial Profile), 242, 352 and 476 min. Note the deep channel
across the tank at about the mean water line at 476 min 74
3.24 Comparison of the Mean Profile at 614 min during Experiment
MT05 with that during Experiment MT06. Note good agreement
of the overall shape. ....................... ... 74
3.25 Mean Profile Evolution during Experiment MT06. Elapsed Times
= 0 (Initial Profile), 614 and 683 min. Note the large berm with
peak moving seaward at 683 min ................... 76
3.26 Mean Profile Evolution during Experiment MT06. Elapsed Times
= 0 (Initial Profile), 683, 752, 890 and 1166 min. Note that the
profile approached an equilibrium with reduced overall slope. 77
4.1 Cartesian Coordinate System. An angle of incidence 0 is defined
as the angle made between x-axis and the wave direction . 86
4.2 Schematic Diagram of Fixed Bed Wave Tank Experiments. Based
on right-handed coordinate system, counter-clockwise rotation is
defined as positive ........................... 94
4.3 Vorticity Meter and its Supporting Rod . . ... 97
4.4 Water Jet with Four Nozzles for the Case of Ho = 9cm . 98
4.5 Schematic Diagram of the Vorticity Measurement Tests with the
Definition of Each Test. Each test is defined as a different combi-
nation of wave and/or jet operation time. Note the breakerline as
affected by jet-induced opposing currents to the incoming waves,
and also counter-clockwise circulation outside the surf zone. 100
4.6 Cumulative Rotation at a Point within the Surf Zone as Influenced
by Jet and Waves during the Vorticity Measurement Tests. Pos-
itive revolution means counter-clockwise rotation of the vorticity
m eter. . . . . . . . . 102
4.7 Wave Height Distributions without the Presence of Jet-Induced
Circulation when Ho = 3cm (Test F211). Wave heights are given
in cm ..... .. .... .. ...... ... .. 105
4.8 Wave Height Distributions in the Presence of Jet-Induced Circu-
lation when Ho = 3cm (Test F212). Wave heights are given in
cm. Note higher wave heights in the vicinity of the jet return flow
due to both the counter current and refraction around the return
current. ................... ........... 106
4.9 Wave Height Distributions without the Presence of Jet-Induced
Circulation when Ho = 9cm (Test F221). Wave heights are given
in cm ........... ..... .............. 107
4.10 Wave Height Distributions in the Presence of Jet-Induced Circu-
lation when Ho = 9cm (Test F222). Wave heights are given in
cm. Note higher wave heights in the vicinity of the jet return flow
due to both the counter current and refraction around the return
current. ................... ........... 108
4.11 Wave Crest Lines in the Presence of Jet-Induced Circulation: (a)
Ho = 3cm (Test F212) and (b) Ho = 9cm (Test F222). Dotted
lines represent the breakerline. This figure is drawn to the scale.
Note when Ho = 9cm that waves displaced due to the opposing
current appeared to overtake the waves riding jet-induced current
near x = lm, where the vorticity measurements were performed. 109
4.12 Distributions of the Total Torque when Ho = 3cm (Test F212).
The values varied from -5.77 to 4.82 in 1/sec2. Solid lines represent
positive isolines while dotted lines represent negative values. Note
the generally positive values within the surf zone, which indicates
positive circulation in opposition to the jet-induced circulation.
Near the breakerline, negative circulation occurs due to continuity. 112
4.13 Distributions of the Total Torque when Ho = 9cm (Test F222).
The values varied from -5.12 to 5.85 in 1/sec2. Solid lines represent
positive isolines while dotted lines represent negative values. Note
the generally positive values within the surf zone, which indicates
positive circulation in opposition to the jet-induced circulation.
Near the shoreline and the breakerline, negative circulations occur
due to continuity............................ 113
4.14 Marginal Distributions of Individual Components along x-axis in
cm/sec2 when Ho = 3cm (Test F212): (a) Components I and
II and (b) Components III, IV and V. Note that all components
converge to zero in the area far offshore (x = 200 cm) and demon-
strate generally positive values within the surf zone (except Com-
ponent IV), which tend to induce counter-clockwise circulation in
opposition to the jet-induced flow. . . . .. 115
4.15 Marginal Distributions of Individual Components along x-axis in
cm/sec2 when Ho = 9cm (Test F222): (a) Components I and
II and (b) Components III, IV and V. Note that all components
converge to zero at near x = 400 cm. . . . .. 116
4.16 Marginal Distributions of Total Torque and the Contributions
of Radiation Stresses Components along x-axis in cm/sec2 when
Ho = 3cm (Test F212). Note that torque by S.. is dominant
over the torque contributed by Sx, and Sy. Note also that the
contribution by Sy tends to approximately cancel out the con-
tribution by Sx, within the surf zone. The total torque changes
from positive values near the shoreline and inside the surf zone (to
x = 60cm) to negative values near and beyond the breakerline,
thus indicating positive circulation over the surf zone, resulting in
opposing effects of waves on the jet-induced circulation. . 118
4.17 Marginal Distributions of Total Torque with the Contributions
of Radiation Stresses Components along x-axis in cm/sec2 when
Ho = 9cm (Test F222). Note that torque by Sx is dominant over
the torque contributed by Sxy and S,,. Note also the distribution
of total torque changing from negative values near the shoreline to
positive ones inside and outside the surf zone, thus indicating pos-
itive circulation within the surf zone, resulting in opposing effects
of waves on the jet-induced circulation. . . ... 120
4.18 Marginal Distributions of Total Torque and the Contributions of
Radiation Stresses Components based on Momentum Equations
along x-axis in N m/m when Ho = 3cm (Test F212). Note the
distribution of total torque changing from positive values near the
shoreline and inside the surf zone (to x = 0.7m) to negative values
near and beyond the breakerline (to x = 1.4m), thus indicating
positive circulation within the surf zone, resulting in opposing ef-
fects of waves on the jet-induced circulation. . . ... 122
4.19 Marginal Distributions of Total Torque and the Contributions of
Radiation Stresses Components based on Momentum Equations
along x-axis in N m/m when Ho = 9cm (Test F222). Note that
the contribution by Sy, is dominant within the surf zone, while the
torque by S., is dominant near and beyond the breakerline. Posi-
tive circulation occurs within the surf zone, resulting in opposing
effects of waves on the jet-induced circulation . . ... 124
5.1 Installation of Three-Dimensional Bar when Ho = 3cm. 3-D bar
extends from one wall (y = -0.3m) to two-thirds of the tank width
(y = 0.lm), while 2-D bar extends to the whole width of the tank. 134
5.2 Wave Height Distributions in the Presence of the Two-Dimensional
Bar when Ho = 3cm (Test F311). Wave heights are given in cm.
A dark area represents the area covered by the 2-D bar. Note the
uniformity of the wave crest lines including the breakerline in the
longshore direction.............. ............ 136
5.3 Wave Height Distributions in the Presence of Three-Dimensional
Bar when Ho = 3cm (Test F312). Wave heights are given in cm.
A dark area represents the area covered by the 3-D bar. Note that
wave fields are not greatly influenced by the presence of return flow
due to the 3-D bar, resulting in a more or less uniform breakerline. 137
5.4 Wave Height Distributions in the Presence of Two-Dimensional
Bar when Ho = 9cm (Test F321). Wave heights are given in
cm. A dark area represents the area covered by the 2-D bar.
Note the nonuniformity of the wave crest lines inside the surf zone
(especially near x = 120cm)....................... 138
5.5 Wave Height Distributions in the Presence of Three-Dimensional
Bar when Ho = 9cm (Test F322). Wave heights are given in cm.
A dark area represents the area covered by the 3-D bar. Note that
wave fields are substantially influenced by the return flow induced
due to the presence of the 3-D bar. Note also almost uniform
breakerline due to deeper water depth at the area of return flow
than the area over the bar ...................... .139
5.6 Marginal Distributions of Individual Components along x-axis in
cm/sec2 when Ho = 3cm: (a) Components I, II and VI in the
presence of the 2-D Bar (Test F311) and (b) Components I, II and
VI in the presence of the 3-D Bar (Test F312) . . .... 143
5.7 Marginal Distributions of Individual Components along x-axis in
cm/sec2 when Ho = 9cm: (a) Components I, II and VI in the
presence of the 2-D Bar (Test F321) and (b) Components I, II and
VI in the presence of the 3-D Bar (Test F322) . . ... 144
5.8 Marginal Distributions of Total Torque and the Contributions
of Radiation Stresses Components along x-axis in cm/sec2 when
Ho = 3cm in the Presence of the 3-D Bar (Test F312). Note that
torque by S., is dominant over the torque contributed by Sy.
Note also the positive values within the surf zone and in the area
seaward of the bar ........................... 145
5.9 Marginal Distributions of Total Torque and the Contributions
of Radiation Stresses Components along x-axis in cm/sec2 when
Ho = 9cm in the Presence of the 3-D Bar (Test F322). Note that
torque by S.x is dominant over the torque contributed by Sy.
Note also the positive values within the surf zone and in the area
seaward of the bar. Hence, positive circulation occurs within the
surf zone, which tends to suppress the bar-induced circulation. 146
5.10 Marginal Distributions of Total Torque and the Contributions of
Radiation Stresses Components based on Momentum Equations
along x-axis in N m/m when Ho = 3cm in the Presence of
the 3-D Bar (Test F312). Note the distribution of total torque
changing from positive values near the shoreline and inside the
surf zone (to x = 0.75m) to negative values over the bar region
and again to positive, thus indicating positive circulation within
the surf zone, resulting in opposing effects of waves on the 3-D
bar-induced circulation. ....................... 149
5.11 Marginal Distributions of Total Torque and the Contributions of
Radiation Stresses Components based on Momentum Equations
along x-axis in N m/m when Ho = 9cm with the 3-D Bar
(Test F322). Note that the contribution by Sy is dominant term
over the whole area of interest except over the bar area, where
the contribution by Sx, is dominant. Positive circulation occurs
within the surf zone, resulting in opposing effects of waves on the
3-D bar-induced circulation. . . . ..... 150
5.12 Definition Sketch for Mean Longshore Currents on Barred Coastlinesl53
5.13 Momentum Theory Applied to Net Flow over a Bar ...... ..156
5.14 Non-Dimensional Longshore Velocity. Here, L represents the half
length of the bar, and y does the longshore distance from the
center of the bar. Note the linear pattern of the velocity near
y = 0, while it demonstrates nonlinear pattern near y = L. . 163
5.15 Maximum Longshore Velocity. Note that the maximum velocity
approaches a limit value as the half length of the bar increases. 164
5.16 Non-Dimensional Set-Up. Here, L represents the half length of
the bar, and y does the longshore distance from the center of the
bar. ..................... ............. 165
A.1 Distributions of the Component I when Ho = 3cm (Test F212).
The values varied from -0.07 to 0.97 in 1/sec2. Solid lines represent
positive isolines while dotted lines represent negative values. Note
that Component I is generally positive and confined mainly near
the shoreline, where water depth is shallow. . . ... 174
A.2 Distributions of the Component II when Ho = 3cm (Test F212).
The values varied from -1.39 to 1.76 in 1/sec2. Solid lines represent
positive isolines while dotted lines represent negative values. Note
that Component II is generally positive within the surf zone and
negative near the breakerline. . . . . . 175
A.3 Distributions of the Component III when Ho = 3cm (Test F212).
The values varied from -1.30 to 0.56 in 1/sec2. Solid lines represent
positive isolines while dotted lines represent negative values. Note
that Component III is generally small except the area near the
breakerline. ................... .......... 176
A.4 Distributions of the Component IV when Ho = 3cm (Test F212).
The values varied from -5.80 to 2.44 in 1/sec2. Solid lines represent
positive isolines while dotted lines represent negative values. Note
large negative values in the vicinity of return currents. . 177
A.5 Distributions of the Component V when Ho = 3cm (Test F212).
The values varied from -0.87 to 1.96 in 1/sec2. Solid lines represent
positive isolines while dotted lines represent negative values. Note
that Component V shows mainly positive values near the area of
higher waves due to opposing currents. . . ... 178
A.6 Distributions of (Tv)1 when Ho = 3cm (Test F212). The values
varied from -4.18 to 5.30 in 1/sec2. Solid lines represent positive
isolines while dotted lines represent negative values. Note that
(Tv)1 is generally positive within the surf zone and negative near
the breakerline, which is essentially the same as the Component II. 179
A.7 Distributions of (Tv)2 when Ho = 3cm (Test F212). The values
varied from -1.16 to 1.77 in 1/sec2. Solid lines represent positive
isolines while dotted lines represent negative values. Note that
(Tv)2 is generally negative within the surf zone and positive near
the breakerline ............................. 180
A.8 Distributions of (Tv)3 when Ho = 3cm (Test F212). The values
varied from -5.23 to 2.73 in 1/sec2. Solid lines represent positive
isolines while dotted lines represent negative values. Note the very
complicated variations of (Tv)3. . . . . .. 181
B.1 Distributions of the Component I when Ho = 9cm (Test F222).
The values varied from -0.58 to 0.13 in 1/sec2. Solid lines represent
positive isolines while dotted lines represent negative values. Note
that Component I is generally negative and confined mainly near
the shoreline, where water depth is shallow. . . ... 183
B.2 Distributions of the Component II when Ho = 9cm (Test F222).
The values varied from -0.89 to 1.58 in 1/sec2. Solid lines represent
positive isolines while dotted lines represent negative values. Note
that Component II is generally positive within the surf zone and
negative near the breakerline. . . . ...... ...... 184
B.3 Distributions of the Component III when Ho = 9cm (Test F222).
The values varied from -2.66 to 1.46 in 1/sec2. Solid lines represent
positive isolines while dotted lines represent negative values. Note
that Component III is generally small except large negative values
near the breakerline. ............ .......... .. ..185
B.4 Distributions of the Component IV when Ho = 9cm (Test F222).
The values varied from -5.84 to 5.27 in 1/sec2. Solid lines represent
positive isolines while dotted lines represent negative values. Note
the very complicated variations of the Component IV. . 186
B.5 Distributions of the Component V when Ho = 9cm (Test F222).
The values varied from -2.35 to 2.35 in 1/sec2. Solid lines represent
positive isolines while dotted lines represent negative values. Note
that Component V shows mainly positive values near the area of
higher waves due to opposing currents . . . .. 187
B.6 Distributions of (Tv)1 when Ho = 9cm (Test F222). The values
varied from -2.68 to 4.73 in 1/sec2. Solid lines represent positive
isolines while dotted lines represent negative values. Note that
(Tv)i is generally positive within the surf zone and negative near
the breakerline, which is essentially the same as the Component II. 188
B.7 Distributions of (Tv)2 when Ho = 9cm (Test F222). The values
varied from -4.62 to 5.32 in 1/sec2. Solid lines represent positive
isolines while dotted lines represent negative values. Note that
(Tv)2 is generally negative within the surf zone and positive near
the breakerline ............................. 189
B.8 Distributions of (Tv)3 when Ho = 9cm (Test F222). The values
varied from -4.62 to 5.32 in 1/sec2. Solid lines represent positive
isolines while dotted lines represent negative values. Note that
(Tv)3 is generally positive near the breakerline. . ... 190
C.1 Filter Response Function versus Wave Number. Note the low-pass
character. ................ ................. 192
C.2 Filtered, Unfiltered and Removed High-Frequency Components of
the Marginal Distributions of the Total Torque presented in Figure
4.19. Note that the filter worked well except both end regions of
the data, in which the filtered data are affected by the finite length
of the filter ... .. .. .. .. ... .. .. .. ... ... 193
C.3 Wave Number Spectrum of the Filtered, Unfiltered and Removed
High-Frequency Components of the Marginal Distributions of the
Total Torque presented in Figure 4.19. . . .... 194
LIST OF TABLES
2.1 Profile Classification Criteria ................. .... 13
2.2 Relationship of Rip Currents Intensity to Various Factors (from
Shapard et al. (1941) ......................... 22
2.3 Factors Affecting the Occurrence of Rip Currents (from McKenzie,
1958) . . . . . . . . 24
2.4 Rip Current Generation Mechanisms. . . . 29
3.1 Description of Movable Bed Experiments. . . .. 45
3.2 Maximum Cross-Tank Differences of the Profile Elevations at 352
min during Experiment MT06 ................ .. 75
3.3 Periods Associated with Edge Waves. . . . 84
4.1 Wave Conditions for Fixed Bed Experiments . . ... 94
4.2 Jet Discharge for Fixed Bed Experiments . . ... 99
4.3 Description of Circulation Tests with and without the Presence of
the Jet . . . . . . . .. 104
4.4 Summary of Minimum and Maximum Values of the Components
Involved in Vorticity Driving Torque. Values are given in 1/sec2. 111
4.5 Summary of Total Vorticity Driving Torque based on the Vorticity
Equation. Values are given in cm2 /sec2 . . . 121
4.6 Comparison of Total Vorticity Driving Torque based on the Mo-
mentum Equation with Torque Induced by the Jet. . ... 125
4.7 Wave Period Conditions for Edge Wave Tests . . ... 126
5.1 Numerical Values of the Bar Dimensions with Their Definitions 133
5.2 Description of Circulation Tests on a Barred Beach . ... 135
5.3 Summary of Minimum and Maximum Values of the Components
Involved in Vorticity Driving Torque. Values are given in 1/sec2. 141
5.4 Summary of Total Vorticity Driving Torque based on the Vorticity
Equation. Values are given in cm2 sec2 . . ... 147
5.5 Summary of Total Vorticity Driving Torque based on the Momen-
tum Equation. ............................ 151
xvii
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
THREE-DIMENSIONAL HYDRODYNAMICS AND MORPHOLOGY
ASSOCIATED WITH RIP CURRENTS
By
TAE-MYOUNG OH
August 1994
Chairman: Dr. Robert G. Dean
Major Department: Coastal and Oceanographic Engineering
This study was conducted to develop an improved understanding and predictive
capability of the 3-D hydrodynamics and morphology associated with rip currents.
Six movable bed tests were carried out with a fairly fine sand to (1) monitor
profile evolution and the resulting equilibrium, and (2) investigate various possible
interrelationships and causes of the three-dimensionality, which included the effects
of (a) induced perturbations in the bar topography, (b) elevated water table in the
berm, and (c) an initially 3-D berm. These movable bed studies suggested that the
profiles appeared to be stable against the perturbations and there was a relatively
slow feedback between the hydrodynamics and the morphology that led to initiation
and growth of 3-D features.
Three series of fixed bed tests were performed to investigate whether or not an
existing rip current was self-reinforcing or stable on a 2-D prismatic topography,
including (1) vorticity measurements at a point within the surf zone, (2) circulation
tests with and without the presence of a jet discharge, and (3) edge wave tests by
varying wave heights and periods. It was found that rip currents were neither self-
reinforcing nor stable on a 2-D prismatic topography. Waves in the area with opposing
xviii
currents were higher due to both the refraction and the counter current. It was
these increased momentum fluxes that tended to stabilize the jet-induced cellular
circulations on a 2-D beach morphology.
Since rip currents are neither self-reinforcing nor stable on a 2-D prismatic beach,
a series of fixed bed tests was conducted to examine the effect of a 3-D bar morphology
on the tendency for cellular circulation. The results showed that rip currents were
stable on 3-D barred beach, although increased momentum fluxes of waves in the area
with opposing currents induced by the bar tended to exert a counter force against
them. It appeared that the water mass transported over the bar tended to return to
sea through deeper area in the bar morphology. This cause is believed to be due to
the greater hydraulic efficiency of flows through deeper portions of the profile, which
leads to a hydraulic/sedimentary instability which causes development of a vestigial
channel.
Finally, a simple model for rip currents on barred beaches was developed consid-
ering mass transport over the bar and the bar morphology.
CHAPTER 1
INTRODUCTION
A beach, the boundary between sea and land, is one of the most interesting places
in the world. The beach is quite complex in structure being composed of air, water
and sediment, resulting in dynamics characterized by nonlinear and nonequilibrium
processes; hence it presents an eternal challenge to the coastal engineer.
The challenge is due, in part, to so many processes, of which the underlying
physics are still obscure, occurring simultaneously on various temporal and spatial
scales with a large number of factors. One of these challenges is understanding and
predicting the three-dimensional (3-D) hydrodynamics and sedimentary features as-
sociated with rip currents.
Rip currents are seaward-flowing jet-like concentrated currents extending beyond
the breakerline, representing a major agent of surf zone water drainage, and carrying
large quantities of sediment offshore. They can occur along a long straight beach
periodically, near longshore barriers such as jetties or groins, and at relatively narrow
and deep channels in sand bars.
Rip currents constitute an integral part of the nearshore circulation pattern which
shows more or less a closed-form with a series of discrete cells along the beach, as
shown in Figure 1.1. Water carried landward through a broad portion of the breaker-
line runs along the beach and finally returns to the offshore through relatively narrow
and deep channels. The offshore velocity in a rip current can be large enough to
modify the wave field including refraction such that at the rips the wave crest lines
are displaced seaward and wave-current interaction such that the waves are higher at
the rip locations.
Figure 1.1: Schematic Diagram of Nearshore Circulation System with Rip Currents
(from Shepard and Inman, 1950b)
Even before Shepard (1936) first proposed the name 'rip current' to describe the
seaward-flowing jet-like concentrated currents, rip currents were well-known to life
guards and to experienced swimmers, as these currents were considered responsible
for pushing swimmers outside the breaker zone at irresistible speeds; but seemed to
have largely escaped the notice of scientists at that time. Since Shepard, however, rip
currents have attracted the interest of coastal engineers because (1) they can modify
the wave field by refraction and other interaction mechanism, (2) they can change the
coastal configuration by removing and transporting significant quantities of sediment
offshore, (3) they are a potential danger especially to unwary swimmers, and (4) they
can refresh the surf zone water, thus affecting water quality in the nearshore region.
1.1 Three-Dimensional Features in Nature
If uniform waves in the longshore direction approach shore over straight and
parallel bottom contours, under some conditions beach morphology is known to be
characterized by a well-developed two-dimensional (2-D) longshore bar. In nature,
SCHEMATIC OIAGRAM OF SURFACE
I LOW IN COASTAL ANO NEAN3HOXE
CURRENT SYSTEMS
N -
~* L[NGtTh OlARROW INOICATIS (LArlyVl
U CUR r NT MA*NITUO( of THE CURRiiT
4,1 -"*, -, -,
ARENT
O "- "-.-- *. --. -. -
/ .. ..... '
" o -/ : .. ."
- MASS TRASP*iloRT ri WAVES
.. ...O. ^ t CU tR ^ -
3
however, waves are not 2-D and the beach morphology would be characterized by a
range of states including some 3-D forms that are fairly regular, and which appear
to be associated with rip currents. Figure 1.2 (from Holman and Lippmann, 1987),
which was created digitally by averaging ten-minute time exposure images, shows a
typical well-developed 2-D longshore bar and well-formed 3-D crescentic bar.
Since Shepard, Emery and La Fond (1941), who first carried out detailed field
studies, field observations were emphasized to obtain some basic idea of the depen-
dence of nearshore circulation and beach morphology on the prevailing wave condi-
tions. After obtaining a detailed description of the main features of the rip currents,
laboratory and simple analytical (including numerical) models were developed to ex-
plain those features and to determine the strength of and the spacing between the rip
currents.
Based on previous studies, it is generally accepted that nearshore cellular circu-
lation at natural beaches could be produced and maintained by the driving forces
induced by longshore variability in the breaking wave heights; however, it is not yet
entirely clear which mechanism can provide those variations in the longshore direction,
which are sufficient to drive rip currents. To quote from Sonu (1972), "Undulation
of the surf zone bed was an essential condition to the formation of the two major
patterns of wave-induced nearshore currents: circulation and meander" (p. 3245).
However, Dalrymple and Losano (1978) showed through analytical and numerical
analyses that wave-current interaction could generate stable rip current circulation
cells on a prismatic beach. Hence, a relevant question is: Can rip currents occur on
a prismatic beach?
Furthermore, there is a related question about the direction of rip current circu-
lation. It is known that circulation is shorewards near the regions of higher breaking
waves and seawards near low wave zones; but some observations showed the opposite
case (e.g., McKenzie, 1958), where the currents flowed shorewards from low wave area
(a) Well-Developed Two-Dimensional Longshore Bar System
(b) Well-Formed Three-Dimensional Crescentic Bar System
Figure 1.2: Bar Morphologies Created Digitally by Averaging Ten-Minute Time Ex-
posure Images. White bands indicate the shore break and regions of concentrated
wave breaking on distinct bar morphologies at the Army Corps of Engineers Field
Research Facility at )uck, North Carolina. (from Holnman and Lippmann, 1987)
5
(shoals) and seawards through higher wave area (embayments). Hence, this argument
raises the question: "Which path do rip currents voluntarily follow in Nature? Where
are they generated? From the shoals, or from the embayments?"
Another important characteristics of rip currents is the modification of wave field
through refraction. As discussed earlier, it is known that wave crests generally show a
lag in the neck of the rips due to refraction and wave heights increase due to shoaling.
But sometimes a gap in advancing breakers exists where rips move seaward (Shepard,
Emery and La Fond, 1941), which means low waves at the location of rips. This
simply raises the question: "What is the usual wave-current interaction mechanism?"
The last point of disagreement is the causes) of the three-dimensionality in na-
ture. Several mechanisms appear to dominate this interaction between the hydrody-
namics and morphology. Candidate explanations include edge waves and a hydrody-
namic/sediment instability that reinforces the nearshore system and 3-D morphology.
This argument raises the question: "Which mechanisms) can initiate and form a bar
morphology with rip currents present?"
In the present study, it was decided to conduct laboratory experiments in a nar-
row wave tank in attempts to investigate the tendency for cellular circulation in the
planform and to investigate various possible interrelationships and causes of the three-
dimensionality. Laboratory studies can isolate the selected governing parameters of
the processes and can control their effects more easily than field studies. Furthermore,
a narrow tank can remove any 3-D effects inherent to a wide tank, e.g., nonuniform
wave crests in the longshore direction or cross waves. It should be noted that a rel-
atively wide tank condition can be satisfied by reducing generated wave heights and
periods, which makes the tests for edge waves mechanism possible in a narrow wave
tank.
6
1.2 Three-Dimensional Features in a Narrow Wave Tank
When conducting movable bed tests in a narrow wave tank, it is usually assumed
that the hydrodynamics and morphology are 2-D. Both water and sand particles are
considered to remain within the plane of motion and to move horizontally back-and-
forth only. Hence, any 3-D patterns such as horizontal circulation of the water and/or
cross-shore variations of the profiles are considered as 'errors' or extraneous features in
the experiments and subsequently have not been documented extensively. However,
investigators have found evidence of three-dimensionality in narrow wave tank tests.
Bagnold (1940) first observed noticeable 3-D circular water motions and resulting
profile changes over the beach face during 2-D laboratory experiments. The beach
face was always 2-D with coarser materials. With sands, however, 3-D beach face
and resulting circular sweeping motion of the water were observed with sands always
deposited on one side or the other side of the tank over the beach face. Bagnold found
that the slope of sand deposition was 14 degrees and the other side of the tank, where
there was no deposition, retained 5 degrees slope. It should be noted that the tank
used in Bagnold's experiment was only 0.53 m wide.
Three-Dimensional morphology in 2-D wave tank tests were also observed in ear-
lier tests by the Beach Erosion Board (BEB, 1947). The experiments were conducted
in a tank approximately 26 m long, 4.3 m wide and 1.2 m deep, and with a deep water
wave height of 11.6 cm; hence, the experiments could be considered as 3-D since the
ratio of the tank width to the wave height would suggest that the experiments were
carried out in a relatively wide tank. Even though BEB considered the experiments
as 2-D, BEB measured five profiles across the tank to account for observable three-
dimensionality during the experiments and averaged to represent the mean profiles at
the measuring time. Although BEB did not document those profiles, they presented
photographs as shown in Figure 1.3, which clearly exhibited 3-D morphology during
the tests.
(a) Bed after 1 hour Wave Operation
--
* '; U.-
(b) Bed after 10 hours Wave Operation
Figure 1.3: An Example of Three-Dimensional Morphology in Two-Dimensional Nar-
row Wave Tank Tests (from Beach Erosion Board, 1947). Note the temporal reduction
of three-dimensionality after 10 hours wave operation.
During the investigation of vertical velocity profiles, Russel and Osorio (1958)
stated; "It is probable however that the velocities would be disordered by circula-
tions in a horizontal plane, if the waves were not confined to a narrow channel" (p.
183). Bagnold (1963) has observed when the channel width in Russel and Osorio's
experiments exceeded a certain multiple of the water depth, that the wave drift along
the bed, which was otherwise uniform across the channel section, showed signs of
instability, becoming greater on one side than on the other. Bagnold suggested that
in wider channels, horizontal circulation would appear and a random scattering of
sediment would tend to be superimposed upon a smaller forward drift, which might
result in 3-D morphology.
3-D morphology over the entire surf zone in relatively narrow wave tanks have
been noted by Kriebel et al. (1986a) and documented by Hughes and Fowler (1990).
There are several possible causes of 3-D flows and morphology in a wave tank,
some of which could also be representative of those occurring in nature: (1) instabil-
ity of 2-D flows on 2-D morphology, (2) organized longshore wave motions that could
induce edge waves in nature or in a tank or cross-tank waves under laboratory condi-
tions, and (3) a feedback between the hydrodynamics and morphology that reinforces
3-D morphological features under certain stages of profile development.
1.3 Objectives
This study was conducted to understand and predict 3-D hydrodynamics and
morphology associated with rip currents. As previous studies were not yet entirely
clear to explain rip current generation, the following questions were addressed as the
first step:
(a) Can rip currents occur on a prismatic beach?
(b) Which path do rip currents voluntarily follow in Nature? Where are they gen-
erated? From the shoals, or from the embayments?
(c) What is the usual wave-current interaction mechanism?
(d) Which mechanisms) can initiate and form a bar morphology with rip currents
present?
For these questions, it was decided to review the previous studies extensively to
address the questions in a more orderly fashion to determine the scope of present
study. And then, as explained earlier, both fixed and movable bed tests were con-
ducted in two narrow wave tanks of the Coastal and Oceanographic Engineering
Laboratory of the University of Florida.
At first, movable bed experiments were designed (1) to monitor profile evolution
and the resulting equilibrium, and (2) to investigate various possible interrelation-
ships and causes of the three-dimensionality. For 3-D morphology, the following were
examined:
(a) Effects of an induced perturbation in the bar topography,
(b) Effects of an elevated water table in the berm, and
(c) Effects of an initially 3-D berm.
Upon knowing the main features of profile evolution with rip currents present, the
next objective was to investigate the tendency for cellular circulation on a prismatic
beach and to measure following:
(a) Cumulative rotation at a point within the surf zone as influenced by waves and
a landward directed jet, and
(b) Wave height distributions in the presence of a jet-induced circulation
The final objectives were to (1) investigate the tendency for cellular circulation
on a barred beach through wave tank experiments, and (2) develop a simple model
for rip currents on barred beaches.
10
1.4 Scope of Study
Chapter 2 provides a comprehensive review of previous studies on 3-D cellular
circulation and morphology both in nature and in narrow wave tanks. Also included
are previous studies on profile evolution and the resulting equilibrium.
Chapter 3 summarizes the experimental facilities, conditions and experimental
cases for movable bed laboratory studies, and presents the results of the generation
and development of 3-D morphology and the resulting equilibrium of the profile in a
narrow wave tank.
Through a series of fixed bed wave tank experiments, Chapter 4 discusses the
generation and existence of 3-D flows on a prismatic beach with a fixed bed. The
method to analyze the test results are presented and followed by the conclusions.
In Chapter 5, 3-D flows on a barred beach are studied through wave tank exper-
iments, and a simple model for rip currents on barred beaches is developed.
Finally, conclusions from this study and recommendations for future efforts are
presented in Chapter 6.
CHAPTER 2
LITERATURE REVIEW
2.1 Introduction
Considerable research has been carried out during the past 50 years, and has
provided a wealth of knowledge in understanding and predicting quite complex beach
processes associated with rip currents. This chapter is devoted to exploring and
acknowledging some of those very major contributions.
This chapter starts with reviews focused mainly on laboratory studies of beach
profile evolution and the resulting equilibrium profile, which seems to be logical in
the present study since laboratory experiments were selected to investigate the three-
dimensionality associated with rip currents under 2-D controlled wave conditions. As
appropriate, previous studies addressing the physical mechanism will be reviewed.
Next, a brief review is made of the literature describing 3-D flows and morphology
in a narrow wave tank. These are followed by a comprehensive review describing field
studies on rip currents which have established the terminology and main features of
rip current systems. Finally, an extensive review on analytical and numerical models
to investigate various characteristics and causes of the three-dimensionality associated
with rip currents are presented.
2.2 Profile Evolution and Equilibrium Profile
A beach profile is the result of the natural forces (both "destructive" and "con-
structive") acting on the sand composing the beach. The destructive forces including
gravity act to flatten the profile by transporting sediments seaward; while the con-
structive forces act to move sand onto the beach, resulting in a steep beach. Hence
12
for sediment particles of a given size and for the given wave conditions, there will be
an 'equilibrium' profile as a balance of destructive versus constructive forces. If any
of these forces are altered, the beach responds and evolves toward a new equilibrium.
The equilibrium beach profile is an idealized profile, but is very useful in predicting
qualitatively the evolution of a particular beach profile for changes in waves and
sediment conditions, or for given initial conditions. In the field, the equilibrium profile
is considered to be 'dynamic' as the tide and incident wave field change continuously
in nature and therefore the profile changes shape as well. In the laboratory, however,
it is relatively easy to establish an equilibrium profile, by running a steady wave train
onto a beach for a long time. After the remolding of the initial profile, a 'final' profile
results, which changes little with time. This is the equilibrium profile for that beach
material and wave conditions in the laboratory. Hence, as a beach profile approaches
an equilibrium, the incident wave energy is dissipated without any significant profile
changes and the time-averaged sediment transport rate converges to zero at all points
along the profile.
Considerable laboratory research in a 2-D narrow wave tank has been directed
toward determining the relationship between dominant forces and complex processes
of beach profile evolution and resulting equilibrium profile (e.g., Rector, 1954; Saville,
1957; Kriebel et al., 1986a and 1986b; Larson and Kraus, 1989, etc.). Based on those
studies, it was generally accepted that storm wave conditions cause offshore transport
of sand resulting in an eroded beach face and an offshore breakpoint bar, while normal
conditions cause onshore transport resulting in a developed berm and no offshore bar,
and coarser sands result in steeper profiles while finer sands result in milder ones.
Due to its uniqueness and importance on the cross-shore hydrodynamic and sed-
iment transport processes, the longshore bar has been the focus of many previous
studies. However, an understanding of the profile evolution, the rates at which the
profile changes occur, becomes equally important recently since the beach recovery
13
processes have been paid attentions to achieve the overall stability of the beaches
following severe erosion events, the storm.
Based on the bar formation, profile classification criteria in terms of wave steep-
ness, sediment size, sediment fall velocity and beach slopes, etc. were proposed to
predict the behavior of the profile. Some of these criteria are summarized in Table 2.1;
while more detail review is given in Kriebel et al. (1986a).
Table 2.1: Profile Classification Criteria
Researchers) Bar Formation Criteria Scales
Johnson (1949) Ho/Lo > 0.03 small
Rector (1954) Ho/Lo > 29.4(Dso%/Lo)0- small
Saville (1957) Ho/Lo > 0.025 small
Ho/Lo > 0.0064 large
Dean (1973) Ho/wT > 0.85 small
Sunamura and Horikawa (1974) Ho/Lo > C(tan/ )-0.27(D5o%/Lo)067 small
Kriebel et al. (1986a) Ho/Lo > Arw/gT small, large
Larson and Kraus (1989) Ho/Lo < 0.0007(Ho/wT)3 large
Dalrymple (1992) gH2/w3T > 9000 10400 large
where, in this table
Ho = Deep water wave height
Lo = Deep water wave length (=gT2/2r)
T = Wave period
g = Gravitational acceleration
tan = Initial beach slope
D50% = Median diameter of sand
w = Sediment fall velocity
A, C = Constants
As presented in Table 2.1, the empirical constants determined based on the large
scale experiments are always much smaller than those based on the small scale ex-
periments. It appeared that the constants in the small scale experiments contained
scale effects due to the use of relatively larger sand sizes (Kriebel et al., 1986a).
Based on small-scale model tests, Rector (1954) described the behavior of the
bar as follows. The bar generated rapidly on an initially planar beach and moved
14
onshore during early times and remained stationary at later times. It was then that
the equilibrium shape of the profile started to take form. Bottom ripples formed
rapidly at the offshore area of the wave breaking zone and progressed steadily onshore;
eventually they lost their identities at the bar. Rector noted that waves sorted the
sands continuously during the evolution process to the equilibrium profile; hence the
median diameter of the sand in the bar area was always larger than that of the original
sand.
Dean (1973) developed a heuristic model based on the suspension of sediments
induced by wave breaking and possible types of sediment fall trajectories. Assuming
instant suspension of the sand particles under the passing of wave crests and set-
tlement during one wave period, Dean suggested that the direction of net sediment
transport would depend on the ratio of the fall time of a sand particle to the wave
period. Net offshore transport would occur if the suspended particle required a time
longer than one half the wave period to fall back to the bottom, and vice versa. As-
suming that the suspension height is proportional to the wave breaking height, Dean
finally expressed the onshore/offshore transport criteria as
Ho
S= 0.85 (2.1)
wT
Although Dean presented the criteria for the direction of sediment transport only,
his model can be easily extended to predict the sediment transport rate for determin-
ing the evolution process of the beach profiles (e.g., Dally and Dean, 1984). It is
worth noting that the role of the sediment size and wave period become clear in the
formation of the different profile shapes.
Dean's heuristic model also can be extended to explain the bar formation by
increasing wave heights with a constant period, as shown in Figure 2.1 (Dean, 1985).
When small waves act on a beach profile, they break in shallow water resulting in
deposition of the suspended sediment landward of its suspension point as the fall time
4 NO BAR p BAR
., 4 V: "3 3
(a)t
SF< T4 < tF
Figure 2.1: Bar Formation with Increasing Wave Height (drawn based on Dean, 1985).
Here, T is the wave period, and tF is the fall time of the sediment, defined as the
ratio of the suspended height to the fall velocity of the sediment.
is still less than one-quarter of the wave period (T/4). With increasing wave heights,
the waves break in deeper water, the sediment is suspended to a higher elevation
above the bed and thus the fall time increases. If the fall time is longer than T/4 but
still less than T/2, the sediment still experiences a net onshore movement but it is
influenced by offshore velocities under the wave trough. If the fall time is equal to be
T/2 at some wave height, the sediment will fall back to the same location where it was
suspended. For larger waves, the sediment will experience a net offshore displacement
as the fall time increases further, thus causing a bar to form.
Bruun (1954) first showed and Dean (1977), after analyzing more than 500 profiles,
presented equilibrium beach profile by the form
h(x) = Ax 213 (2.2)
where h is the water depth at a distance z which is oriented in an offshore direction
with the origin at the mean water line and A is a profile scale factor which depends on
sediment size. Dean demonstrated that this form of the profile could be interpreted in
sediment size. Dean demonstrated that this form of the profile could be interpreted in
16
terms of uniform wave energy dissipation per unit water volume. Later Dean (1987)
presented the relationship between the parameter A and the sediment fall velocity,
W, as
A = 0.067w0.44 (2.3)
where A is given in m4 while w in cm/sec. This relationship is surprisingly linear on
a log-log plot.
Kriebel et al. (1986a, 1986b) performed comprehensive small-scale laboratory
experiments to: (1) test the validity of the model scaling law proposed by Dean
(1985) using an undistorted Froude similarity with sediment scaled according to fall
velocity, (2) investigate the effects of initial profile shape (not slope) on the equilibrium
profile configuration, (3) examine the onshore/offshore sediment criterion given in
Equation (2.1) (Dean, 1973), and finally (4) investigate the beach recovery process
following severe erosion events.
Based on comparisons with large scale results of Saville (1957), the model scaling
law provided good agreement for erosive wave conditions, but moderate agreement
for acceretive waves mainly due to wave reflections. It appeared that the initial
profile shape was very important in determining the profile evolution and resulting
equilibrium as the concave profile was found to have more realistic wave shoaling
and breaking characteristics. Results for the sediment transport criterion indicated
that Dean's original criterion included scale effects and suggested a higher value than
that given by the right hand side of Equation (2.1). For the post-storm recovery
experiments, the storm-generated bar did not migrate landward easily under normal
waves of constant height, while berm formed rapidly. However, the bar migration
occurred in a related experiment as wave conditions were finely tuned such that the
waves always broke just landward of the migrating bar.
Kraus and Larson (1988) documented all the experimental data obtained at the
Large Wave Tank of the Coastal Engineering Research Center (CERC, successor orga-
nization to the BEB). Based on these data, Larson and Kraus (1989) described general
characteristics of cross-shore transport and derived empirically-based transport rate
equations to develop a numerical model, called SBEACH.
2.3 Three-Dimensionality in a Narrow Wave Tank
It will be most appropriate to start this section with a contribution by Bagnold
(1940). Through a series of wave tank experiments and with very careful observations,
Bagnold presented a pioneering description of noticeable 3-D circular motions over
the beach face during 2-D experiments using both rounded pebbles with a median
diameter of 7 mm and sands of diameter ranging 0.5 to 3 mm under regular waves.
The width of the tank was only 0.53 m, as discussed earlier.
With coarser material like shingle, Bagnold observed that a 2-D beach was always
formed with a straight shoreline parallel to the wave crests. When more material was
added on one side of the tank, this soon became distributed evenly over the width of
the mobile portion of the beach. However, the beach changed to 3-D forms with fine
materials such as sands and sands always deposited on one side or the other side of
the tank over the beach face, resulting in a circular sweeping motion across the tank
instead of moving directly up and down. As presented earlier, Bagnold found that
the slope of sand deposition was 14 degrees and the other side of the tank, where
there was no deposition, retained 5 degrees slope.
Rector (1954) performed laboratory studies in the same tank used by BEB (1947),
of which size is 25 m long, 4.3 m wide and 1.2 m deep, but with four different sizes of
sands. The median diameters of the sands were 0.22, 0.46, 1.20 and 3.44 mm, in which
the sand of 1.20 mm was a mixture of 0.46 and 3.44 mm sands. Considering 2-D tests
with width being of little importance, Rector tested those sands simultaneously by
dividing the tank longitudinally into four equal widths. Rector also utilized full and
18
half widths of the tank. Although Rector did not note 3-D topography, some of his
photographs clearly showed the cross-tank variations of the beach profiles. Rector's
experiments demonstrated that the three-dimensionality inside a wave tank strongly
depended on the width of utilized tank as well as on the size of the sand.
Maruyama and Horikawa (1977) performed laboratory experiments in a wave
basin (6 m long, 1.2 m wide) in order to investigate the generation mechanism of rip
current using regular waves with a period of 0.96 seconds and heights of 0.8 and 1.5
cm. They measured wave heights across the tank, which were smaller at the center
of the tank where the rip currents flowed offshore than those near both side walls,
resulting in a circulation from high to low waves. This variation of the wave height
remained unchanged even with strong rip currents with the maximum velocity of 6.8
cm/sec under about 1 cm wave height experiments. As pointed out by Maruyama and
Horikawa, this tendency is contrary to the usual wave-current interaction mechanism.
But the measured wave height distributions across the tank agreed well with the
calculated wave heights by superimposing the cross waves into the incident waves.
Maruyama and Horikawa concluded that rip currents would occur due to the interac-
tion between incident waves and the synchronous progressive cross waves having crests
at certain angles to the wave maker and therefore cross waves should be suppressed
in a laboratory experiment to simulate a natural coast.
However, cross waves are subharmonic standing waves with crests at right angles
to the wave maker, of which energy are supplied by the wave maker through depth
independent second-order pressures (Garrett, 1970; Mahony, 1972). Maruyama and
Horikawa also pointed out that synchronous cross waves observed in their experiments
were physically differerit from subharmonic cross waves predicted theoretically by
Garrett and Mahony and that the mechanism was not yet definite how the wave
maker supplied the energy to cross waves in their experiments.
19
At this stage, it is worth noting that cross waves are usually observed in wave
tanks where the wave maker is substantially wider than the length of the waves
generated; which is another good reason to select a narrow wave tank in investigating
three-dimensionality.
During the small-scale laboratory experiments, as presented earlier, Kriebel et al.
(1986a) first noted 3-D cross-tank variations of the profiles over the entire surf zone,
noticeable in the seeming lack of volume closure in several of the profiles. Hence,
Kriebel et al. took several profiles across the tank and averaged to represent mean
profiles. The tank used was approximately 30 m long, 1.3 m deep, and 0.9 m wide.
Barnett and Wang (1988) conducted 2-D laboratory tests in the same tank used
by Kriebel et al. (1986a) using regular waves and a fine sand with a median diameter
of 0.15 mm. Barnett and Wang took two profiles at approximately one-third the cross-
tank distance from each of the flume walls to represent mean profiles. They noted
that these variations were observed periodically and exhibited a temporal reduction
such that, by the end of most of the tests, the cross-tank variations appeared to be
negligible. They considered that possible causes are wall boundary layer producing
wave refraction, uneven bed compaction, and reflected wave interaction with incident
waves.
From a series of 2-D laboratory tests in a wave tank of 1.83 m width, Hughes and
Fowler (1990) observed that noticeable cross-tank variations occurred after the profile
reached a quasi-equilibrium condition and documented these variations by measuring
three profiles across the tank, as shown in Figure 2.2 which were measured after 1650
waves (equivalent to 1 hour wave run) in their experiment T03.
It was thought that these cross-tank variations have been caused by a small
misalignment of the revetment in the flume which in turn caused reflection of waves
from the exposed concrete revetment since similar variations were not present in the
prototype scale tests of Dette and Uliczka (1986). After another test with increased
1.5
-- -- CENTER PROFILE
1.2 -----GLASS SIDEWALL PROFILE
O -- CONCRETE SIDEWALL PROFILE
--- ---- INITIAL PROFILE
Z
S 0.9
a 0.6
0
0.3
0.0
-2.0 -1.0 -0.0 1.0 2.0 3.0 4.0 5.0 6.0
DISTANCE (m)
Figure 2.2: Cross-Tank Variations of the Profiles after 1 hour Wave Run (modified
from Hughes and Fowler, 1990). Note 3-D morphology inside the surf zone and near
the bar area.
wave height by approximately 10 percent (their experiment number T04), Hughes and
Fowler noted that increased wave heights test exhibited a similar cross-tank profile
variations as the test approached. equilibrium, but apparently made the variations a
little more severe than observed in the previous test (Test T03).
Hughes and Fowler described that these variations did not materialize until after
the profile was close to an equilibrium, which might indicate that the profile was
more susceptible to cross-tank perturbations when the profile had reached a quasi-
equilibrium state. If the profile was not close to equilibrium, the onshore/offshore
transport of sand seemed to overwhelm any cross-tank-induced sediment transport.
Hughes and Fowler then related this trend to the field data of Howd and Birkemeier
(1987), and stated "A prestorm breakpoint bar that exhibited nonuniform along-
shore variation became quite linear and moved offshore during the storm. Near the
end of the storm, when presumably a near-equilibrium had been reached, alongshore
variation in the bar began to reappear" (p. 37).
** *
-iJH
S"O*r UNE
S ZO C 0.*' "'* *
i I OC
from the land. Shepard et al. described the general characteristics of rip currents,
s.oar UNC
Figure 2.3: Constitutive Elements of a Rip Current System: the feeder currents flows
of water parallel to the shore; the neck, the main seaward-flowing current extending
beyond the breaker zone; and the rip head, in which the rip current lose its identity
as the water spreads out. (from Shepard, Emery and La Fond, 1941)
2.4 Rip Currents in Nature
Shepard et al. (1941) reported the first scientific observations of rip currents, and
recognized them as a main feature of the nearshore circulation system, which returned
the water piled onto the beach by the waves and carried seaward fine sediments derived
from the land. Shepard et al. described the general characteristics of rip currents,
their relation to coastal and shoreline configuration, and the constitutive elements of
a rip current system, as shown in Fig 2.3.
By measuring waves (height, period and direction), winds, longshore currents,
and tides, Shepard et al. related rip current intensity (its definition was not clear in
their paper, but it appeared to be represented by the flow velocity in the rip neck) to
those factors, and the results are summarized in Table 2.2.
It will be interesting to note that Shepard et al. defined zero intensity of rip
current when wave height was 0.6 m (2 ft), as shown in Figure 2.4; which would
Table 2.2: Relationship of Rip Currents Intensity to Various Factors (from Shapard
et al. (1941)
Factors Observed Relations
Wave Height Largest rip current intensities corresponded
with the largest waves (see Figure 2.4).
Wave Period No pronounced relation was observed.
and Steepness
Longshore Rip current appeared to vary with the direction
Currents and intensity of the currents, but completely
satisfactory relation was not well defined.
Rip currents subsided with large waves, and strong
Winds rips did not occur on the windy days when waves
were small. Rip currents also appeared to be small
when wind was blowing along the coast.
Rip currents were more pronounced at low tide,
Tides but no evident relation of the rips to the spring
and neap tides was observed.
suggest that wave height must be larger than 0.6 m for rip currents to form. If waves
were smaller than 0.6 m, then rip currents would disappear.
Another important result was the existence of the channels in the path of a rip
current, which were largely confined to the surf zone and deeper than their surround-
ings. The floor of the channels was found to be decidedly irregular mainly due to the
strong rip current. Shepard et al. noted that the position of the channels varied rel-
atively rapidly as well as the positions of rip currents, as stated "three days after the
survey, the inner channel had completely disappeared and no appreciable rip could
be observed in the locality" (p. 355).
Shepard and Inman (1950a) investigated the nearshore circulation system near
areas where diversified submarine topography occurs off relatively straight shorelines
and found that the nearshore circulation system was definitely influenced by the
wave divergence at the heads of submarine valleys and by the wave convergence over
submarine ridges. Longshore currents adjacent to the shore diverged from areas of
ca
C,
'1
'U
C
I-'
C!)
fC& MARCH APPl. MAY 41:C
S(-- WAVS & roDC (--) RIPCUARCtHr3
Figure 2.4: Measured Wave Height and Rip Current Intensity (from Shepard et al.,
1941). These data were obtained daily. Note the direct relationship between these
two variables. Note the definition of zero intensity of rip current when wave height
was 0.6 m (2 ft); which would suggest that wave height must be larger than 0.6 m
for rip currents to form.
wave convergence and flowed seaward as rip currents at areas of wave divergence.
This work implied the importance of the longshore wave height variations caused by
wave refraction due to irregular offshore bottom topography.
Shepard and Inman (1950b) performed a comprehensive series of field measure-
ments, and described the general circulation system on most beaches including two
straight beaches with parallel bottom contours. It was found that the direction of the
longshore currents was primary dependent not only on the angle of wave incidence to
the shoreline, but also on the longshore distribution of the wave set-up, greater in the
zones of higher breakers along the beach. The longshore currents commonly flowed
away from the zones of highest breakers toward the rip current. Although Shepard
and Inman did not note why there were those variations of the set-up in the longshore
direction on uniform beaches, they concluded that cellular circulation systems could
occur and be maintained even under normal wave incidence on straight beaches with
parallel contours.
N 1 A -2
a_, ___ ', ,, \4
_____"ijirI!V f V
24
McKenzie (1958) observed rip current systems on beaches with smooth offshore
topography but undulatory surf zone topography of alternate shoals and channels,
and categorized main factors affecting the occurrence of rip currents as summarized
in Table 2.3.
Table 2.3: Factors Affecting the Occurrence of Rip Currents (from McKenzie, 1958)
Factors Importance Observation(s)
Size and Regularity Determine the strength Small and numerous rips appeared
of waves of rip currents under moderate waves; while larger
but fewer rips were developed under
large waves.
Tides Affect the position Falling tide caused the change of
of rip currents channel angle with the beach, or
gradually moved channel into
new position with the same angle.
Wave Control wave angle Rips tended to turn into the waves
Direction and determine the rip within the surf zone, and tended to
currents direction turn away from the waves
outside the surf zone.
Coastal Control wave angle No direct relationship was observed
Configuration and wave energy between the rips and the distribution
distribution by wave of wave energy.
refraction
As listed in Table 2.3, McKenzie could not find any direct relationship between the
rip currents and longshore wave energy distributions, i.e., wave height distributions;
instead, he observed that rip currents "do not, as might be expected, seek that part of
the beach with least energy concentration but tend to move seaward in the vicinity of
greatest wave activity" (p. 107). This argument seemed to contradict the circulation
system flowing from higher energy zones to lower ones, as observed by Shepard and
Inman (1950a). However, this could be explained by the difference in the nearshore
zone topography, i.e., difference between undulatory offshore but planar surf zone
as in the observation by Shepard and Inman and smooth offshore but undulatory
25
surf zone topography as in McKenzie's observation, and also by considering both the
correlation between the circulation and the surf zone topography and wave-current
interaction at the location of rip currents, as will be discussed later.
Bowen and Inman (1969) performed field studies on the beach having smooth
offshore topography and planar surf zone bed in order to confirm their laboratory
results, which showed that the rip currents occurred at alternate antinodes of standing
edge waves of the same frequency as the incident waves. Bowen and Inman measured
incoming wave heights, the breaking wave heights and water depth, the width of the
surf zone and also the spacing of the rip currents to confirm whether or not these were
equal to the longshore wavelength of the edge waves, and found that rather regular
spacing was in good agreement with the calculated longshore wave length of an edge
wave of a particular mode. Bowen and Inman suggested that stationary interaction
between incident waves and synchronous edge waves resulted in periodic longshore
variations of breaking wave heights, which could drive such regular circulations on
plane beaches.
Sonu (1972) observed wave-induced nearshore circulation and meandering cur-
rents on a beach with smooth offshore topography and surf zone undulations under
essentially uniform breaking wave heights. By measuring the spatial distribution of
the horizontal velocities, the current patterns, the wave set-up in the surf zone, and
the time series of velocities inside the rip channels, he observed that the currents near
the shoreline moved from shoal areas of lower waves to rip channel areas of higher
waves, of which pattern seemed to contradict Shepard and Inman (1950a) but to
agree with McKenzie (1958).
Sonu then found that the current patterns followed precisely the same spacings
as the undulation wavelength; he concluded that for uniform waves the surf zone
undulation was an essential factor to the cellular circulation. By observing that
floating balls followed the directions of the measured gradient of water surface, Sonu
26
demonstrated that the currents were not driven by the gradient of the wave heights,
but were driven by the gradient of the mean water surface, which were caused by
radiation stresses.
2.5 Rip Current Models
In this section, a brief description will be given some rip current models which
explained various characteristic features of rip currents as observed in the field. Those
features include the width change of the rip neck, the rate of velocity decrease in the
current, the formation of rip head, and the turning of rip currents into the direction
of wave approach. These models consider that rip currents were already generated
by the merging of feeder currents.
Arthur (1962) first theoretically examined the dynamics of rip currents using the
steady-state shallow water continuity and momentum equations, in which the pressure
gradient term was the only remained driving force term after neglecting friction and
the Coriolis force terms. Arthur then eliminated the forcing term by employing the
vorticity equation and explained a relatively narrow, concentrated pattern of the rip
current by the conservation of vorticity along a stream line extending from shallow
to deeper water. Arthur noted that the friction would tend to broaden streamlines
of rip current. He also implied the importance of refraction effects of rip currents
on the incident wave field and noted that this interaction might be important in the
dynamics of rip currents.
Tam (1973) proposed a simple mathematical model which reduced a rip current
to an identical 2-D incompressible jet after assuming steady-state conditions, shallow
water approximations, energy dissipation mainly by horizontal mixing, and boundary
layer approximation for a narrow rip current. Tam also considered that the gradients
of the radiation stresses were balanced by the slopes of the mean sea level within the
narrow rip current; hence he assumed that the radiation stresses could be negligible
as long as the rip current was already formed and driven by the feeder currents.
27
For a plane beach, Tam found the linearly proportional width (this contradicts
Arthur, 1962) and inversely proportional maximum velocity to the distance from the
shoreline, resulting in constant flow rate, and described the rip head formation by
imposing a sudden increase in the plane beach bottom slope in deeper water depth,
which resulted in dramatic broadening of the width and reducing of the maximum
velocity, thus forming a rip head.
Wind and Vreugdenhil (1986) presented a numerical model to generate a rip
current due to the interaction of longshore current with a longshore barrier such as
groin and then compared the numerical results with the results of experiments in a
closed basin with fixed bed. Their numerical model included all the terms in the
depth-integrated momentum equations and the continuity equation, and investigated
the relative importance of convection, diffusion and bottom friction in the flow over
a sloping bottom.
They demonstrated that the combined effects of the bottom topography and con-
vective terms caused the convergence of rip current streamlines over a seaward sloping
bottom, which agreed with Arthur (1962). When convective terms were excluded from
the momentum equations, the rip current disappeared. They also found that the ef-
fect of bottom friction was to decelerate the rip current with diverging streamlines,
as discussed by Arthur (1962). For a given forcing, the total circulating flow rate was
regulated by the bottom friction rather than the magnitude of the viscosity for the
lateral shear stresses or convective term, which would be expected from the angular
momentum balance of the circulating water mass. The lateral shear stresses were
found to be responsible for closed streamlines outside the surf zone.
2.6 Rip Current Generation Models
2.6.1 Introduction
Since field studies have recognized a longshore variation in the radiation stresses
field and the wave-induced set-up as the main driving forces in the formation of rip
28
currents, various analytical and numerical models were developed based on several
mechanisms to explain the longshore wave height variability resulting in the variation
of the radiation stresses field. The two factors are usually considered: (1) longshore
variation in the breaking wave height, and (2) longshore variation in the bottom
topography.
In the present analysis, the various mechanisms are divided into two categories:
(1) prismatic and (2) non-prismatic topography models. The distinction between the
two is that prismatic models can explain the occurrence of a rip current on a planar
beach, while non-prismatic model needs undulatory topography inside the surf zone
to generate and maintain a rip current. It should be noted in present classification
that the mechanisms by the presence of coastal structures were not included, since the
special interest here was to investigate whether or not rip currents could be caused
by interaction of the beach topography with the incident waves. Similarly, Dalrymple
(1978) classified the models into (1) wave interaction and (2) structural interaction
according to the same criteria as the present study, but included the interaction with
coastal structures.
Prismatic topography models require hydrodynamic longshore perturbations on a
prismatic beach to provide the longshore variability of wave heights, which are enough
to drive the rip currents. For the case of normally incident waves on a straight
beach with fixed bed, no horizontal circulation is expected with uniform set-up in
the longshore direction. However, the prismatic model considers that the wedge-
shaped 3-D beach is very sensitive to instabilities in the longshore direction; hence the
beach could not maintain the uniform state against longshore perturbations, which are
considered mainly due to the wave-wave interactions (Bowen, 1969; Bowen and Inman,
1969, Sasaki and Horikawa, 1975; Dalrymple, 1975) or wave-currents interactions
(LeBlond and Tang, 1974; Dalrymple and Losano, 1978) or some instability inherent
to the nearshore hydrodynamics (Hino, 1974).
Table 2.4: Rip Current Generation Mechanisms
(a) Prismatic Topography Models
Mechanisms Representative Researcher(s)
Wave wave interaction
Synchronous edge wave Bowen (1969), Bowen and Inman (1969)
Infra-gravity wave Sasaki and Horikawa (1975)
Intersecting wave trains Dalrymple (1975)
Wave current interaction LeBlond and Tang (1974)
Dalrymple and Losano (1978)
2-D Instability with movable bed Hino (1974)
(b) Non-Prismatic Topography Models
Mechanisms Representative Researcher(s)
Undulatory bottom topography Bowen (1969), Noda (1974)
Mei and Liu (1977), Schmidt (1986)
Bar morphology Dalrymple (1978)
without sedimentary feedback Deigaard (1986)
Bar morphology Dalrymple, Dean and Stern (1976)
with sedimentary feedback Deigaard (1990)
Non-prismatic topography models need undulatory bottom topography (Bowen,
1969; Noda, 1974; Mei and Liu, 1977; Schmidt, 1986) or 3-D bar morphology with rip
channels (Dalrymple et al., 1976; Dalrymple, 1978; Deigaard, 1986, 1990) to provide
the driving forces which generate rip currents. Table 2.4 lists a number of the various
models according to the present classification.
Before presenting details, it would be better to start by reviewing Bowen's work
(1969) since his model first theoretically approached the rip current problem and
furthermore could be classified as either prismatic or non-prismatic model since it
was not clear in his analytical and numerical examples whether or not a longshore
variation of the radiation stresses field was induced by the bottom topography's effects
on the wave field.
30
Bowen first presented a theoretical model to generate rip currents on a planar
beach under normally incident waves having a longshore variation in wave height.
Considering two possible mechanisms for such longshore perturbations: (1) a long-
shore undulation of the surf zone bed, and (2) the interaction of synchronous edge
waves with the incident wave field, Bowen demonstrated that cellular circulations
were driven by a longshore variation in the radiation stresses field in the surf zone,
resulting in the currents flowing from higher to lower waves.
Considering that Bowen assumed a linear relationship of local wave heights to
the total water depth even at the location of the rips, the circulation would flow from
the embayments of higher waves to the shoal areas of low waves, which seemed to be
contradictory to most field studies carried out on undulatory beaches.
2.6.2 Prismatic Topography Models
Bowen and Inman (1969) found that progressive or standing edge waves with
the same frequency as the incident wave (synchronous edge waves) could generate a
nearshore circulation and presented the rip current spacing as,
L, = L, = L, sin[(2n + 1)03] (2.4)
where, Lr is the rip current spacing, Le the edge wave length, Lo = gT2/27, the deep
water wave length, g the gravitational acceleration, / the planar beach slope and n
is the mode of the edge wave, which is equal to the number of zero crossings of the
water surface elevations in the offshore direction. The spacing of rip currents is not
dependent on the wave heights but strongly dependent on the incident wave period,
which was in good agreement with their field observations but appeared to contradict
most of field studies (e.g., Shepard et al., 1941), and has a maximum value of deep
water wave length for the case of very steep beaches and/or high mode edge waves.
31
As given by Equation (2.4), it is always possible to select a combination of wave
period, edge wave mode numbers and beach slope, which can nearly match the ob-
served spacings; this would be a reason why the edge wave model is so attractive.
However, Guza and Davis (1974) have shown that only subharmonic edge waves could
be excited on a plane beach through a nonlinear resonant mechanism. These subhar-
monic edge waves were different from those suggested by Bowen and Inman (1969)
and could not produce rip currents. Guza and Davis also have shown that surging
conditions might be required for an edge wave model to be effective in causing rip
currents.
Hino (1974) proposed a rip current generation model based on stability analysis
of the steady-state uniform beach system on an initially plane beach but allowing a
feedback between the movable bed and the flow field, and found that the system was
hydrodynamically unstable for infinitesimal longshore perturbations, resulting in the
most preferred spacing of rip currents of about four times the surf zone width,
L, 4 Xb (2.5)
where Xb is the surf zone width, the distance from the shoreline to the breakers. Hino
showed that these spacings agreed well with the observed data. Although this model
allowed a sedimentary feedback for the growing and maintenance of rip currents, it
could generate rip currents on an initially planar beach by hydrodynamic instability;
hence, Hino's model was classified as prismatic topography model in the present study.
LeBlond and Tang (1974) questioned Bowen's (1969) assumption that rip currents
were sufficiently small so that their interaction with the field was negligible, and first
applied energy equation including advection, wave-current interaction and dissipation
terms. Together with the shallow water continuity and momentum equations and
the fixed bed consideration, LeBlond and Tang posed an eigenvalue problem for the
circulation cells inside the surf zone and performed a perturbation analysis. However,
32
in solving the posed eigenvalue problem it was necessary to assume that rip currents
would be most likely to occur where the relative rate of energy dissipation is least,
i.e., LeBlond and Tang have looked for the longshore wave number that minimized
the ratio of energy dissipation rate to the total kinetic energy present in a rip current
system.
The resulting circulation pattern was found to be essentially the same as in the
uncoupled case where wave-current interaction was neglected, with a somewhat mod-
ified wave field such that the energy coupling with the currents attenuated the waves
in that area (this contradicts the usual wave-current interaction considerations, espe-
cially on a prismatic beach where there is no undulatory bottom topography, i.e., no
differences in water depth along the longshore direction.), but the currents predicted
by the coupled case were weaker due to hydrodynamic feedback, as excepted. How-
ever, their computed values of the longshore wave number were too small and did
not compare well with available data. Furthermore, Dalrymple and Losano (1978)
later found that LeBlond and Tang's work contained a significant numerical error,
and concluded that their results were invalid.
Sasaki and Horikawa (1975) analyzed rip current spacings given by Bowen and
Inman (1969) and by Hino (1974) according to the deep water surf similarity param-
eter, which is defined as the ratio of beach slope to the square root of deep water
wave steepness, and found that these two models predicted the spacings which were
always smaller than those observed in the field for the very mild beaches and could be
applied only on beaches with steep and medium slopes; hence Sasaki and Horikawa
proposed an infra-gravity wave model for the gentle beaches with spacings given by
tanf tan #
L, = 157 ( ta- )2 Xb (0.22 > tn > 0.08) (2.6)
VHo//Lo .XHo/Lo
in which Ho/Lo is the deep water wave steepness. However, the mechanism to generate
rip currents was actually the same as that of edge wave except with the forcing given
by infra-gravity wave.
Dalrymple (1975) has shown on an open coast that intersecting wave trains of
the same period could cause rip currents. If two wave trains of the same period
intersect, they superimpose and cancel each other spatially, resulting in periodically
spaced nodal lines along the shoreline; at these lines, rip currents occur with a spacing
determined by the deep water wave length and directions of the waves, as follows:
L, = o (2.7)
sin 01 sin 02
where 01 and 02 are the deep water wave angle of the two wave trains. It was noted
that this model had no theoretical maximum spacing but a minimum of one half the
deep water wave length. Dalrymple also carried out laboratory experiments to verify
this model and the observed results agreed well with the predicted spacings. The
spacings of rip current were strongly dependent on the incident wave period but had
no relationship with the wave heights or the surf zone width, which was similar to
the relationship proposed by edge wave mechanism.
Assuming an existing rip current, Dalrymple and Losano (1978) developed two
analytical models to provide steady rip current system on a prismatic beach based
on a hydrodynamic feedback through wave-current interaction. It was noted in their
models that the energy equation was considered indirectly by a linear relationship
between the wave height H and the total water depth (h + 77) as
H = n (h + ) (2.8)
where h is the still water depth, r is the set-up, and n is a breaking index of the order
of unity (about 0.8).
34
The first model extended previous studies by LeBlond and Tang (1974) to include
only the changes of the local wave length in the presence of rip currents, yet no rip
currents occurred. Dalrymple and Losano then included the refraction of the waves
on the currents in the second model, and found that this refraction caused the waves
to impinge on the beach obliquely by forcing the incident waves to slow over the
rip, thus generating longshore currents flowing from regions of high wave energy and
set-up towards regions of low energy, i.e., the base of the rip, as suggested by Bowen
(1969).
They presented the non-dimensional rip current spacing as a function of one
parameter, defined as the ratio of the bottom slope to the friction, and later Dalrymple
(1978) proposed an approximate equation
1
A X +2.8 (2.9)
AD
where AXb 2rXb/L, is the non-dimensional rip current spacing, AD = nKr tan Pl[f(8+
3,c2)], is the breaking index and f is a Darcy-Weisbach (constant) friction coeffi-
cient. Equation (2.9) predicts that the rip current spacing increases with increasing
wave height as observed in the field. The spacing given in Equation (2.9) also increase
with more smooth bottom, which is not clear in the field since it is difficult to define
the bed smoothness in the field.
It was noted that work done by the currents against the radiation stresses ap-
peared to reduce the wave energy at the location of rip currents. As discussed earlier,
however, this mechanism seems to contradict the usual wave-current interaction such
that the opposing current tends to increase the wave height due to the wave refraction
and interaction on a counter current.
2.6.3 Non-Prismatic Topography Models
Noda (1974) developed a numerical model to obtain a steady-state nearshore
circulation pattern considering the effects of bottom topography on an incident wave
field, and found that, for an undulatory bottom, wave-bottom topography interaction
changed the incident wave field according to bottom undulation, thereby causing
spatial variation of the radiation stresses field, and that this variation of the radiation
stresses field inside the surf zone ultimately derived the nearshore circulation flowing
from the shoals to the embayments.
Although Noda obtained unrealistically large value of the maximum current ve-
locity, numerical examples of his work to various bottom topography have verified
the driving mechanism for the nearshore circulation due to the bottom topography's
effects on the wave field. It was noted that a feedback between circulation currents
and movable bottom should be provided to develop an equilibrium bottom configu-
ration for a given wave forcing. Noda also pointed out that wave-current interaction
would tend to modify the incident wave field, resulting in a more uniform breakerline
as observed by Sonu (1972), thus reducing spatial variation of wave heights, hence
consequently reducing the magnitude of the circulation velocity.
Dalrymple et al. (1976) suggested a nearshore circulation model on a 3-D long-
shore bar crest-trough morphology, with the main driving forces given by the gradients
in the set-up values behind the bar. Dalrymple et al. considered that these gradients
in set-up could be induced both by wave reflection from the submerged sand bar and
mass transport over the bar, and presented the circulation flows to regions of lesser
set-up at rip channels. It was noted that by roughly considering sedimentary feedback,
they could obtain minimum rip current spacings on the barred coastlines. Although
no details were given, the basic idea seemed to be correct. The detail formulations of
the hydrodynamics are given in Dalrymple (1978).
36
Mei and Liu (1977) developed a linear analytical model for the nearshore circu-
lation driven by the effects of periodically varying topography confined within and
near the surf zone. Assuming a small depth deviation from a plane beach, Mei and
Liu found that the circulation pattern would be determined based on two effects:
(1) variations in the set-up and in the tangential and transverse components of the
radiation stresses to the wave direction which would tend to drive rip currents from
the shoals to the embayments, and (2) variations in an additional component of the
radiation stresses, representing the flux in the wave direction of the transverse com-
ponent of momentum, which would tend to drive rip currents from the embayments
to the shoals.
Relative magnitude of those two effects was controlled by both the bottom to-
pography and the ratio of the surf zone width to the longshore wave length of the
topography variation. However, Mei and Liu found that the circulation was always
shorewards near the shoals and seawards near the embayments if the bottom undu-
lations were entirely confined within the surf zone. They also obtained a counter-
rotating circulation in a small region near the shoreline.
Dalrymple (1978) presented a model to include the effect of wave reflection from
the bar as suggested by Dalrymple et al. (1976), and examined mean currents behind
a longshore bar by considering (1) the continuity equation, which states that the mass
transport over the bar crest should equal the increase in flow in the longshore trough
between the bar and the beach, (2) the equation of motion within the trough, which
is driven by the set-up differences, and finally (3) the momentum equation, which
includes the radiation stresses and the reverse effect of momentum flux due to wave
reflection by the bar.
By considering the differences between the set-up corresponding to uniform con-
ditions without net flow (designated as potential set-up) and the set-up with net flow
(designated as actual set-up), Dalrymple found that the mass transport over the bar
37
increased continuously toward the rip channels, resulting in the longshore current
velocity increasing from zero at the midpoint between two rip channels to the maxi-
mum value at the locations of the rip channels. In driving his equations, Dalrymple
imposed no mass transport condition (i.e., actual set-up = potential set-up) at the
center point of the bar, which seemed to be physically incorrect since mass transport
at that point might occur depending on the length of the bar. If the bar length is
short, then water would be transported over that point. The symmetric condition for
the longshore velocity (i.e., zero longshore velocity) would be enough at that point.
Expecting that the set-up would be zero at the rip channels, Dalrymple obtained
the minimum stable spacing of the channels as a function of the wave steepness and
sediment size through bottom friction effects as well as several geometric parameters
representing 3-D bar morphology. Dalrymple noted that, in addition to wave-current
interaction, the sedimentary feedback mechanism should be incorporated into this
hydrodynamic model. Even though some equations appears to be incorrect, the basic
idea to consider the momentum equation near the bar area seems to be correct to
describe the mean currents behind the longshore bar.
Deigaard (1986) presented an analytical model to calculate the longshore currents
behind the bar, which was similar to Dalrymple's model (1978) but the reflection from
the bar was not taken into account. Deigaard assumed that the flow rate over the
bar was simply determined by the energy loss caused by the differences between the
potential set-up and the actual set-up, and expressed the flow rate as;
q = h V2g(o 7) (2.10)
in which q is the flow rate, h the mean water depth over the bar crest, j0 the potential
set-up, and J the actual set-up. Equation (2.10) considers that the total head loss in
set-up values is fully contributed to the velocity head loss, thus neglecting unknown
loss of internal energy which is usually involved in wave breaking process. Hence,
38
Equation (2.10) tends to overestimate the flow rate, resulting in decreased longshore
velocity. In this aspect, the momentum equation would be better application to the
problem of determining the flow rate. This will be discussed in details later.
Deigaard also considered both the momentum equation within the trough ne-
glecting bed friction and the continuity equation. By allowing mass transport at the
center point of the bar, which depended on the length of the bar, he then obtained
the longshore current velocity as a function of potential set-up and geometric pa-
rameters such as the length of the bar and cross sectional area of the trough. His
results demonstrated the same trend as Dalrymple's model as the velocity increased
and approached the maximum magnitude toward the rip channels.
However, his momentum equation, as presented in Equation (2.11), appeared to
be incorrect since he neglected the first order term, i.e., the bed shear stress, and
considered only the second order convective acceleration term. Furthermore, this
second order term was considered incorrectly. This will be discussed later.
d + ) = 0 (2.11)
Deigaard suggested the minimum stable spacings of rip channels by considering
that the longshore velocity in the trough approached the possible maximum longshore
velocity beyond certain distance from the center of the bar, which were of the order
of two or three times the ratio of cross sectional area of the trough to the mean water
depth over the bar crest.
Schmidt (1986) carried out experimental investigations on the hydrodynamics
of wave-induced circulation over bar-channel topography which simulated a periodic
form parallel to the coast superposed on a regular offshore sloping bottom. By measur-
ing the distribution of wave heights, mean currents and time-dependent wave-induced
vorticity, Schmidt concluded that this circulation associated with the bar morphol-
39
ogy might be interpreted as a self-maintenance mechanism. During the experiments,
Schmidt observed a shoreward-directed flow over the bar crest and a return flow from
the trough regions, and concluded that higher breaking waves over the bar crest in-
duced a larger set-up driving the flows to bar trough areas of lower waves and smaller
set-up. But he could not observe the counter-circulation in a small region near the
shoreline as obtained theoretically by Mei and Liu (1977). This effect was considered
to be due to the absence of lateral mixing in the theoretical considerations by Mei
and Liu.
Deigaard (1990) presented a model to explain the formation of rip channels on
a barred coastline by performing a linear stability analysis. The equilibrium state
was characterized by a uniform bar system under normally incident waves along the
beach with no net current. The flow was described using essentially the same method
as Deigaard (1986), However, in determining the flow rate over the bar crest, he
included the momentum exchange inside the surf zone, resulting in an increase in the
magnitude of the longshore velocity, thus reducing rip current spacings compared to
those suggested by Deigaard (1986).
Deigaard then provided a perturbation to the longshore bar, which was periodic
in the longshore direction, causing longshore variations in the set-up, finally resulting
in a net circulation with shoreward flows over the bar crest, longshore currents in the
trough and rip currents in the channels. By considering sediment transport due both
to bed slope of the perturbation and to linearly-varying cross-shore transport with
the water depth in the surf zone, Deigaard obtained the spacing of the rip channels, at
which the perturbation would grow or decay in time with the maximum rate. When
Deigaard considered a specific example, he could obtain the spacing of the order of
about twice the surf zone width, which was defined as the distance from the shoreline
to the bar crest.
40
It was noted that low period oscillations of the cross-shore discharge over the bar
crest was necessary to obtain a linear theory; however, it is very questionable how
he derived his cross-shore flow rate. Furthermore, his sediment transport equation,
which showed a linearly varying transport rate with water depth, also seems to be
incorrect.
CHAPTER 3
LABORATORY STUDIES ON THREE-DIMENSIONAL MORPHOLOGY
3.1 Introduction
As reviewed in Chapter 2, there are several possible causes of 3-D flows in a wave
tank, some of which could also be representative of those occurring in nature: (1)
Instability of 2-D flows on 2-D morphology, (2) Organized longshore wave motions that
could induce edge waves in nature or in a tank or cross-tank waves under laboratory
conditions, and (3) A feedback between the hydrodynamics and morphology that
reinforces 3-D morphological features, resulting in cross-tank profile variations under
certain stages, especially when the profile approached an equilibrium with overall
stability.
In the present study, as discussed earlier, movable bed experiments in a narrow
wave tank were carried out to (1) monitor profile evolution and the resulting equi-
librium, (2) provide experimental repeatability, and (3) investigate various possible
interrelationships and causes of the three-dimensionality, which included:
(a) Effects of induced perturbations in the bar topography, including deepening of
the bar trough, asymmetric area seaward of the bar, and removing half the bar
crest,
(b) Effects of controlled water table levels in the berm, which were maintained
higher than the mean sea level, and
(c) Effects of an initially 3-D berm.
42
It should be noted that present experiments were performed with the fine sands
which have been utilized during previous tests, thus have been subjected to continuous
submergence. This prolonged submergence appeared to allow microorganism active
although the experiments were carried out in winter season, resulting in unpleasant
odors and black-colored cohered-looking sands inside the beach, especially near the
tank bottom. Hence, one experiment was designed to investigate the effects of mi-
croorganism inside the beach on the profile evolution. Since the breaks between wave
runs to measure profiles appear to retard the development of steady-state for a given
beach system, an additional test was performed to examine the effects of continuous
wave run without intermissions on the occurrence of the three-dimensionality.
Based on the results of these experiments, the mechanisms) which can initiate
and form a bar morphology with the presence of rip currents will be discussed. Also,
by observing the path of rip currents during the experiments, it was examined whether
rip currents were generated from the shoals or from the embayments.
3.2 Laboratory Facilities and Experimental Conditions
A total of 6 experiments was carried out in the tilting flume of the Coastal and
Oceanographic Engineering Laboratory of the University of Florida. This tank is 15.5
m long, 0.9 m high and 0.6 m wide, and is equipped with a piston type wavemaker
with a mechanically controlled motion and with one glass wall panel and one steel
wall.
The planar beach of initial slope 1:18 was formed of well-sorted fine quartz sand
with a mean diameter of 0.21 mm (2.25 in < unit), a sorting coefficient of 0.58 and
fall velocity of approximately 2.3 cm/sec. The grain size distribution include sizes
ranging between 0.1 to 0.5 mm. The water depth in the horizontal portion of the
tank was 0.275 m. Regular waves with a period of 1.5 sec and wave height of 0.11 m
were utilized, and were measured by a capacitance-type wave gage. Figure 3.1 shows
a schematic diagram of the initial profile and other experimental details.
43
0
EE
C.)d
0
00
Ob
II C
C)
< A
E5
II
-o
a to CD
CD1 zQ -q q
J 0o
II I I-
(, rA 1c0
-o z
C)
cn )
ze
cj 5-
~Q:L NYI
I I I I
(ui EIAIVSN~ 3OVNLV3
44
The profile was measured manually by a point gage modified by replacing the
point with a small, rectangular foot (7.5 cm by 2.5 cm) which provided a flat surface
to rest on the sand. In order to avoid consolidation at the measuring point, the gage
was constructed of light aluminum. Three profiles, designated as B1, B2 and B3, were
measured over the entire length of the beach to document the three-dimensionality
at various times. As shown in Figure 3.2, profile B1 was 0.15 m from the glass side
wall while profile B3 was 0.15 m from the steel side wall. Profile B2 represented the
center line profile along the tank.
Wave
Steel Side Wall
Profile B3
0.6 m Profile B2 (Center Line)
Profile B1
Glass Side Wall
Figure 3.2: Three Profiles B1, B2 and B3 across the Tank
The profiles were documented at locations spaced 0.1 m along each of these lines,
and then these three profiles were averaged to represent the mean profile at the
measuring time. Since three profiles were not sufficient to document 3-D features
when a fairly deep and narrow channel appeared near one of the tank walls, the
profile elevations at both sides of the tank were measured additionally to document
the maximum cross-tank variations of the profile.
The desired water table level in the berm was maintained by excavating 2-D
depressions across the tank. These excavations were connected to the constant head
reservoir through plastic tubes so that water was siphoned out of or into the excavated
45
holes to maintain the desired water table level. During the experiments, this method
has worked very well. However, bubbles appeared sometimes inside the tubes due
to prolonged experimental duration; at which time, they were removed by allowing a
small amount of flow from the excavated holes to the reservoir, or vice versa.
3.3 Description of Movable Bed Experiments
Table 3.1 lists experiment identification number, total wave run duration, water
table level conditions, and brief descriptions of each experiment. Each experiment
was defined as a collection of sequential profiles for an initially planar beach subjected
to regular waves.
Table 3.1: Description of Movable Bed Experiments
Exp. Duration Water Table Note
No. (min) Level* (cm)
MTO1 0-476 0.0 Reference test
MT02 0-407 0.0 Biological effects (quick start)
MT03 0-545 0.0 Wave run duration effects
(run waves without intermission)
0-821 Perturbations in the bar topography
MT04 (0-69) (no change)
(69-138) 0.0 (bar trough is deepened)
(138-352) (asymmetric offshore scour area)
(352-821) (remove half of bar crest)
MT05 0-1166 +11.0 Initially 3-D berm
with elevated water table level
MT06 0-1166 +16.5 Highly elevated water table in the berm
Zero refers to the mean sea level (MSL). Hence, positive
value represents the water table level above MSL.
The six experiments were defined as: (1) Reference test (Experiment MT01),
(2) Repeat tests (Experiment MT02 and MT03), (3) Perturbation tests (Experiment
MT04 MT05 and MT06).
Reference Test
The first movable bed experiment (Experiment MT01) was designed to monitor
the profile evolution and to determine whether or not three-dimensionalities might
occur, hence providing a reference for future experiments.
The profiles were surveyed at intervals of 23 min during initial stages to document
rapid evolution of the profiles and were increased later up to 69 min depending on the
rate of profile change. The same intervals were used in the subsequent experiments
to compare the profiles at the same elapsed times.
Experimental Repeatability
Experiments MT02 and MT03 were carried out to (1) provide experimental re-
peatability of Experiment MTO1 and (2) investigate the possible effects of experi-
mental conditions on the three-dimensionality occurred at Experiment MT01, which
included the effects of the microorganism inside the beach and the effects of continu-
ous wave run.
Experiment MT02 was conducted to investigate the effects of microorganism, of
which presence could be perceived by unpleasant odors and black-colored cohered
sands located near the tank bottom, as discussed earlier. Since the biological film
inside sand was believed to be disturbed after long wave run, Experiment MT02
started with the remolding of the initial profile immediately after the last wave run
of Experiment MT01.
The purpose of Experiment MT03 was to investigate the effects of required
time for the beach system to reach steady-state on the occurrence of the three-
dimensionality. In this experiment, hence, the wave was run continuously without
intermissions to measure profiles.
Perturbation Tests
Experiments MT04, MT05 and MT06 represent the main attempts in the present
study to investigate the three-dimensionality.
47
Experiment MT04 was carried out to investigate the effects of induced pertur-
bations in the bar topography, which included (1) deepening of the bar trough, (2)
asymmetric area seaward of the bar, and (3) removing half the bar crest, and was
continued further to investigate resulting equilibrium.
Experiment MT05 started with an initially 3-D berm superimposed on a planar
initial profile to investigate whether or not resulting 3-D flows in the beach face would
facilitate the occurrence of three-dimensionality. This experiment also included the
effects of elevated water table in the berm.
Experiment MT06 was carried out to examine the effects of highly elevated water
table level in the berm on the profile evolution and resulting 3-D morphology.
3.4 Results and Discussions
3.4.1 Reference Test (Experiment MT01)
During the early stages (t < 242 min), as suggested by the criteria presented in
Table 2.1, the offshore bar formed quickly from a linear profile and remained station-
ary. However, the berm also started to accrete slowly at the initial times and more
rapidly at the later times. A small bar appeared between the main offshore bar and
the berm. This bar moved landward continuously and finally remained stationary
just landward of the berm. The profiles appeared to approach an intermediate equi-
librium with overall 2-D conditions. The mean profiles during early stages at elapsed
times 23, 69, 161, and 242 min with the initial profile are presented in Figure 3.3.
As shown in Figure 3.4, the mean profile inside the surf zone at 242 min was
in good agreement with the equilibrium profile, presented in Equation (2.2) using
the profile scale parameter presented in Equation (2.3); this indicated that, after 242
min, the profile really approached an equilibrium for the given forcing.
After the profile approached an equilibrium and a level of profile stability had
occurred at 242 min, a weak counter-clockwise 3-D circulation occurred after about
270 min, flowing from the steel side wall to the glass side wall inside the surf zone.
0.15
0.10
-0.05
S-0.
w
Z -0.05
S-0.10
0
Z -0.15
0
O 0
r -0.20
-0.25
S TANK BOTTOM
-0.30 I I '
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
DISTANCE FROM. TANK END (m)
Figure 3.3: Mean Profile Evolution during Early Stages of Experiment MT01. Elapsed
Times = 0 (Initial Profile), 23, 69, 161 and 242 min. Note that the profile approached
an equilibrium and a level of profile stability had occurred at 242 min.
49
0.15
S........ mean profile at 242 min.
S0.10 -
-- equilibrium profile. (y=Ax-2/3)
o 0.05 ........
.M.S.L.
> -0.05
0
S-0.10
O -0.15
[--
S-0.20 ...
-0.25 TANK BOTTOM "'"............
-0.30 i 1 I I I I
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
DISTANCE FROM TANK END (m)
Figure 3.4: Comparison of the Mean Profile at 242 min during Experiment MTO1
with the Equilibrium Profile Proposed by Dean (1977). Note the similarity between
two profiles inside the surf zone.
This flow initiated a new stage of profile evolution. However, only small changes
occurred between 242 and 297 min since the circulation was not sufficiently strong
to transport the sediments. Figure 3.5 shows the mean profiles at elapsed times 0
(initial profile), 242, 297, 352' 407 and 476 min.
After 297 min, the sedimentary feedback appeared to reinforce the circulation
sufficient to transport sands landward. As shown in Figure 3.5, the area seaward
of the bar eroded with a substantial deepening and the eroded sand was deposited
immediately landward of the bar trough, thus changing this area from mildly erosional
to strongly depositional.
Furthermore, this area changed from overall 2-D conditions to 3-D features with
higher parts near the steel side wall, as shown in Figure 3.6 which presents three
profiles Bl, B2 and B3 to document three-dimensionality occurred after 476 min.
Bar started to move landward with counter-clockwise rotation of approximately 20
degrees about the direction of wave propagation. The bar trough became shallower
0.15
476 min.
--- 407 min.
0.10 -- 352 min.
--- 297 min.
0.05 -- 242 min.
SIY \. --.....- initial profile
C;1 -0.00 -M.S.L.
0 .0 0 ... .............. ............ ..._....__...__..._........ ...
Z -0.05
'-0.10
z -0.15
0
-0.20
-0.25 \
TANK BOTTOM
-0.30 I I
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
DISTANCE FROM TANK END (m)
Figure 3.5: Mean Profile Evolution after the Profile Approached an Equilibrium dur-
ing Experiment MT01. Elapsed Times = 0 (Initial Profile), 242, 297, 352, 407 and 476
min. Note the substantial erosion of the area seaward of the bar and the deposition
of the area immediately landward of the bar trough. Note also landward movement
of the bar.
0.15
S mean profile profile B3
0.10 initial profile ----. profile B2
0.05 --- profile BI
1.4 M.S.L.
0.00 0 .0 -...... .... ...... ..........................................-,...... -, ............ ................ ...- -. .. .......
O -0.10
-0.05
0
S-0.20
S- 0.25
5 TAqNK BOTTOM
-0.30
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
DISTANCE FROM TANK END (m)
Figure 3.6: Initial Profile and Three Profiles B1, B2 and B3 and Mean Profile at 476
min during Experiment MT01. Note three-dimensionality inside the surf zone and
near the bar area, which is 3-D morphology associated with rip currents, and also
that the centerline profile B2 is almost the same as the mean profile.
at the later times as sand was deposited over the area landward of the bar. This
profile morphology was reminiscent of a bar morphology with a rip current present.
After 476 min, the maximum cross-tank difference of the profile elevation at the
depositional area inside the surf zone was estimated to be 6.0 cm.
3.4.2 Experimental Repeatability
Experiment MT02
As explained earlier, Experiment MT02 started with the remolding of the initial
profile right after the last wave run of Experiment MT01, at which time biological
film inside sand was believed to be disturbed.
The trend of the profile evolution was generally the same as Experiment MT01,
as shown in Figure 3.7, thus providing the experimental repeatability rather than
proving the effects of microorganism.
I
0.15
407 min.
----- 352 min.
0.10 -- 297 min.
207 min.
-.--*- initial profile
0.05
-0 M.S.L.
'l 0.00 .. ....... .... ................. ..................... ..
Z -0.05
S-0.10
Z -0.15 1 \
S-0.20
-0.30 I I
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
DISTANCE FROM TANK END (m)
Figure 3.7: Mean Profile Evolution after the Profile Approached an Equilibrium dur-
ing Experiment MT02. Elapsed Times = 0 (Initial Profile), 207, 297, 352, and 407
min. Note the substantial erosion of the area seaward of the bar and the deposition of
the area immediately landward of the bar trough, which occurred during Experiment
MT01 (see Figure 3.5), thus providing experimental repeatability.
0.15
Experiment MT03
0.10 -- Experiment MT02
0.05 ------ Experiment MT01
M.S.L.
0 .0 0 ...... ................ ... ........................................... ............. .......................... ._... : _._ ._s... .... .. ......
-o.o0 ....... .............
-0.05 -
0 -0.10
Z -0.15 .
O
E -0.20 "\^
-0.20
S-0.25 TANK BOTTOM
-0.30 i I I
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
DISTANCE FROM TANK END (m)
Figure 3.8: Comparison of the Mean Profile at 207 min during Experiment MT03
with the Mean Profiles at the Same Time during Experiments MTO1 and MT02. Note
the similarity between those three mean profiles.
Although the three-dimensionality occurred slightly earlier at about 240 min than
that during Experiment MT01, no clear conclusions could be drawn to the effects of
microorganism on the occurrence of the three-dimensionality, which might be due to
low temperature during this experiment, as explained earlier.
Experiment MT03
Experiment MT03 was performed to investigate the effects of required time for
the beach system to reach steady-state, as discussed earlier. Hence, the wave was
operated continuously without intermissions for the first 207 min. After 207 min,
however, the wavemaker malfunctioned, resulting in interruption of the continuous
wave run.
As shown in Figure 3.8, the mean profile at 207 min was amazingly the same as
those during Experiments MT01 and MT02. Also, three-dimensionality occurred at
almost the same time (240 min) as Experiment MT02. These facts implied that the
effect of continuous wave run was unexpectedly small.
54
Up to 476 min, the profiles evolved in the same manner as previous experiments.
Counter-clockwise circulation occurred and transported sands from the area offshore
of the bar to the depositional area immediately landward of the bar trough. As sand
was deposited, the circulation seemed to be reinforced again.
When the experiment was conducted continuously beyond 476 min, strong circu-
lation occurred, resulting in substantial onshore transport of the sands from the area
seaward of the bar, thus causing the landward movement of the bar and rapid build
up of the berm with another peak which moved seaward. Figure 3.9 shows the mean
profiles at elapsed times 0 (initial profile), 207, 352, 476 and 545 min.
This depositional area changed from overall 3-D feature to overall 2-D one, as
shown in Figure 3.10, except a very narrow deep channel near glass side wall of tank,
where the maximum depth was found to be about 5.0 cm below the mean profile.
The overall slope of the profile at 545 min was estimated to be 1:10, which was
defined as the slope between the seaward end of the profile and the peak of berm. The
slope of the beach face was estimated to be 1:7, which was approximately the same as
that of the area seaward of the bar. After 545 min, wavemaker again malfunctioned,
resulting in termination of this experiment.
3.4.3 Perturbation Tests
Experiment MT04
Experiment MT04 was carried out to investigate the effects of induced perturba-
tions in the bar topography on the occurrence of three-dimensionality inside the surf
zone, and continued further to investigate resulting equilibrium.
During the first 69 min of wave operation, the trend was the same as previous
experiments as the morphology was dominantly 2-D. As a first perturbation to the
bar topography, the bar trough was deepened by approximately 4 cm after 69 min of
wave run, as shown in Figure 3.11, to investigate whether or not the morphology is
stable against the perturbations during the initial phases of the profile evolution.
0.15
0.10
0.05
-0.05 -
-0.10
-0.15
-0.20
-0.25
TANK Bi
-0.30
0.0 0.5 1.0
1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
DISTANCE FROM TANK END (m)
Figure 3.9: Mean Profile Evolution during Experiment MT03. Elapsed Times = 0
(Initial Profile), 207, 352, 476 and 545 min. Note rapid change during 476 to 545
min with landward movement of the bar and another peak of the berm
0.15
-0- profile along rip channel profile B3
E 0'.-- initial profile ------ profile B2
S0.05 '_ --- profile Bl
^ O O ............. :,............ ,- .............................................................. .. ..... .................. .... ............................. ........... .... ... ... ........................
S............ .
-0.00 *-..- M.SL.-.
> -0.05 -.*...
0 ....... .. ..
-0.10 -
-0.15 -
E.-
-0.20 -
-0.25 ........
0.25 TANK BOTTOM
-0.30 I
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
DISTANCE FROM TANK END (m)
Figure 3.10: Initial Profile and Three Profiles (BI, B2 and B3) and the Profile along
Rip Channel inside the Surf Zone at 545 min during Experiment MT03. Note overall
2-D conditions inside the surf zone except a very narrow deep channel near glass side
wall.
During 69 to 138 min, the profile surprisingly returned back to the pre-modified
profile at 69 min, as shown in Figure 3.12. The deepened 2-D bar trough completely
filled in and the profile morphology was dominantly 2-D; which implied that the
profile morphology was stable against the perturbation given by deepening of the bar
trough.
After 138 min wave run, the area seaward of the bar, which was almost 2-D,
was modified by relocating sand from the steel side wall to the glass side wall, as
shown in Figure 3.13, resulting in asymmetric offshore area of the bar. However, the
mean profile remained approximately the same. The maximum cross-tank difference
of the profile elevation in this area at the two sides of the tank was about 6.0 cm.
This asymmetric offshore area was expected to change wave shoaling and breaking
characteristics over the bar region, hence resulting in longshore variations of the
breaking wave heights.
1
0.1 modified mean profile
........ pre-modified mean profile
-0.0 ............... M.S.L.
-0.1
-0.2
(a) Comparison of Mean Profiles
S-0.3 I i i
O 0. --- pre-modified profile B3
------ pre-modified profile B2
-0.1
.! t -0.2
W- _(b) Three Profiles before Modification-- ..
-0.3 -- I Ii I
0.1 modified profile B3
------ modified profile B2
-0. --- modified profile B1 M.S.L.
-0.0.
-0.1
-0.2
(c) Three Profiles after Modification -...
-0.3 i I i' I 'I
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
DISTANCE FROM TANK END (m)
Figure 3.11: Induced Perturbation to Deepen the Bar Trough at 69 min during
Experiment MT04: (a) Comparison of the Modified Mean Profile with Pre-Modified
Mean Profile, (b) Three Profiles before Modification, and (c) Three Profiles after
Modification
0.1
-0.0
-0.1
-0.2
-0.3
0.1
-0.0
-0.1
-0.2
-0.3
0.1
-0.0
-0.1
-0.2
-0 -3
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
DISTANCE FROM TANK END (m)
5.0 5.5 6.0
Figure 3.12: Comparison of Pre-Modified Profile at 69 min with Pre-Modified Profile
at 138 min during Experiment MT04: (a) Comparison of Mean Profiles, (b) Three
Profiles before Modification at 69 min, and (c) Three Profiles before Modification at
138 min.
--- pre-modified mean profile at 69 min.
........ pre-modified mean profile at 138 min.
M.S.L.
-------------- ---------------i----------------------
(a) Comparison of Mean Profiles
--- pre-modified profile B3
------ pre-modified profile B2
--- pre-modified profile B1 M.S.L.
-
(b) Three Profiles before modification at 69 min.
I I I I I I I
pre-modified profile B3
------ pre-modified profile B2
.'.'. --- pre-modified profile B1 M.S.L.
. _.....................t.......... .......................
-..
(c) Three Profiles before modification at 138 min. -'- .
f 1 I I I I I I I I I
v.*
0.1 -- modified mean profile
........ pre-modified mean profile
-0.0. .- M.S.L.
0 .0 .......o .. ..... ..................................... .... ... ....... .... ... ... ... ... .
-0.1
-0.2
(a) Comparison of Mean Profiles
-0.3 I i i
S .1- pre-modified profile B3
------ pre-modified profile B2
S--- --- pre-modified profile B1 M.S.L.
S-0.0
-0.1
oE
I -0.2
S(b) Three Profiles before Modification -" .-.
-0.3 I I
0.1 --- modidied profile B3
------ modified profile B2
S '* --- modified profile B1 M.S.L.
-0.0 ---...................---- --... .............
-0.2
(c) Three Profiles after Modification R'-"--. ..
-0.3 I 2 I
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
DISTANCE FROM TANK END (m)
Figure 3.13: Induced Perturbation to Provide Asymmetric Area Seaward of the Bar
at 138 min during Experiment MT04: (a) Comparison of the Modified Mean Profile
with Pre-Modified Mean Profile, (b) Three Profiles before Modification, and (c) Three
Profiles after Modification. Note the dominantly 2-D profile before modification and
the asymmetric area seaward of the bar after modification.
60
As shown in Figure 3.14, again the profile returned to pre-modified 2-D profile
morphology at 138 min within an hour after resumption of wave action. The mean
profile at 207 min agreed well with the pre-modified mean profile at 138 min, as
shown in Figure 3.14(a). And the perturbed area seaward of the bar was changed
to the pre-modified 2-D one. These again demonstrated the stability of the profile
morphology against the perturbations.
At this stage, it will be interesting to consider the direction of the sand transport
during 138 min to 207 min. During this time, sands could be transported either (1)
directly from the glass side wall to the steel side wall by gravity and/or 3-D circulation
confined in the area seaward of the bar, or (2) by overall 3-D circulation starting at
the higher perturbed area, passing over the bar, circulating inside the surf zone in
clockwise direction, and returning to the lower perturbed offshore area.
By considering the clockwise rotation of the bar by approximately 20 degrees, as
shown in Figure 3.14(c), the sands appeared to be transported by overall 3-D circula-
tion as described above; this was exactly what was observed during the experiment.
It appeared that modification of the area seaward of the bar caused different
characteristics of the wave breaking at two sides of the tank (wave breaking occurred
at first in the higher perturbed area), thus producing nonuniform distributions of the
set-up across the tank, resulting in the clockwise 3-D circulation. At the beginning,
this 3-D flow appeared to be weak to transport sands; however, it was sufficiently
strong to rotate the bar. As the bar was rotated, the 3-D flow was reinforced by the
sedimentary feedback. This 3-D flow then started to transport sands from the higher
perturbed area to the area immediately landward of the bar trough. Small portion of
those sands were transported landward continuously and built up the berm slightly,
while main portion of the transported sands were carried back to the lower perturbed
area seaward of the bar, resulting in the pre-modified 2-D offshore area.
61
0.1 ........ modified mean profile at 138 min.
-- mean profile at 207 min.
M.S.L.
-0.0 ...... ......................... ........................... .. ......
-0.1
-0.2
S (a) Comparison of Mean Profiles .
-0.3 I I
v --- modified profile B3
cc 0.1
------ modified profile B2
S --- modified profile B1 M.S.L.
v> 0 .0 ...................... ............................................................... ..... ....... . .. ...... .... ....
-0.0
O
E-
(b) Three Profiles after modification at 138 MIAl & -.
-0.3
0.- profile B3
----- profile B2
---- profile B1 M.S.L.
-0.1
S-0.2
(b) Three Profiles after modification at 138 min.
-0.3 i I---
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
DISTANCE FROM TANK END (m)
Figure 3.14: Profile Evolution from 138 in to 207 in during Experiment MT4:
--(a) Comparison of the Modified Mean Profile at 138 in with Mean Profile at 207
____ --- profile B1 M.S.L.
-0.1
-0.2
(c) Three Profiles at 207 min. -'
-0.3 I I I I
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
DISTANCE FROM TANK END (m)
Figure 3.14: Profile Evolution from 138 mm to 207 min during Experiment MT04:
(a) Comparison of the Modified Mean Profile at 138 min with Mean Profile at 207
min, (b) Three Profiles after Modification at 138 min, and (c) Three Profiles at 207
min.
62
During 207 to 297 min, the 3-D flows remained weak to cause strong onshore
transport of the sands from the 2-D area seaward of the bar and the profile was
dominantly 2-D except the bar area rotated clockwisely by about 20 degrees.
However, during 297 to 352 min, fairly strong 3-D circulations started to occur,
thus causing a rapid landward sand transport, increasing the three-dimensionality
inside the surf zone, as shown in Figure 3.15. In this figure, the mean profile at 352
min appeared to agree well with that at 297 min. However, the three-dimensional
features appeared inside the surf zone at 352 min, which were associated with a fairly
strong 3-D circulation and a rapid net landward sand transport. It is noted that the
rotation of the bar unchanged and the angle remained as about 20 degrees.
After 352 min of wave run, half of the bar crest was removed to fill the bar trough,
resulting in a linear profile without prominent bar near the steel side wall, as shown
in Figure 3.16. This was designed to impose an extreme three-dimensionality into
the bar crest, thus accelerating the development of the 3-D features already existed
inside the surf zone.
The bar crest again recovered its pre-modified shape from the imposed 3-D feature
over a testing time of approximately one hour, resulting in a smaller rotational angle
(less than 10 degrees), as shown in Figure 3.17. During the recovery, the other part
of the profile remained almost the same as the pre-modified one, and no distinct
circulation was observed.
After overall recovery occurred approximately at 400 min, the bar started to move
landward with increasing clockwise rotation due to reinforcing 3-D circulation and the
area seaward of the bar eroded substantially, resulting in the deposition of sand at the
area immediately landward of the bar trough and the increase in clockwise rotation
of the bar crest. This morphology and hydrodynamics are strongly reminiscent of the
morphology associated with rip currents, as shown in Figure 3.18.
63
0.1 -...... mean profile at 297 min.
mean profile at 352 min.
-0.1 ............
-0.2
(a) Comparison of Mean Profiles .
0.3 ,- ,1
S 0.1 --- profile B3
------- profile B2
--- profile Bl M.S.L.
-0.0
o
E- -
S-0.2 "-S,
W (b) Three Profiles at 297 min. '-..
-0.3 -- I
0.1 profile B3
------ profile B2
-0 .0 --- profile Bl M.S.L.
0.0 ........ ................ ... M L
0 1 % / ,
-0.2
(c) Three Profiles at 352 min.
-0.3 I I -
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
DISTANCE FROM TANK END (m)
Figure 3.15: Profile Evolution from 297 min to 352 min during Experiment MT04:
(a) Comparison of the Mean Profile at 297 min with Mean Profile at 352 min, (b)
Three Profiles at 297 min, and (c) Three Profiles at 352 min. Note the erosion of the
area seaward of the bar and deposition of the area immediately landward of the bar
trough. Note also the differences in the three-dimensionalities between two elapsed
times.
0.1
-0.0
-0.1
-0.2
-0.3
0.1
-0.0
-0.1
-0.2
-0.3
0.1
-0.0
-0.1
-0.2
-0.3
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
DISTANCE FROM TANK END (m)
5.5 6.0
Figure 3.16: Induced Perturbation to Remove Half Part of the Bar Crest at 352 min
Wave Run during Experiment MT04: (a) Comparison of the Modified Mean Profile
with Pre-Modified Mean Profile, (b) Three Profiles before Modification, and (c) Three
Profiles after Modification
modified mean profile
........ pre-modified mean profile
M.S.L.
................ ............. ................................................... ....... ..... ........................... ..... .... ....................................... ......... .. ..... M
(a) Comparison of Mean Profiles
Ipre-modified profile
--- pre-modified profile B3
------. ------ pre-modified profile B2
--- pre-modified profile BI M.S.L.
(b) Three Profiles before Modification
imodidied profile B3
-- modified profile B3
.------ modified profile B2
-\ --- modified profile BI M.S.L.
--~
(c) Three Profiles after Modification
i-I I I I I
0.1 ........ pre-modified mean profile at 352 min.
mean profile at 407 min.
M.S.L.
0.0 -- ....- ........- ..... ...... ................... ..................................... ------------ ------------- -- ------ ---------------- ----
-0.0 ..............l.......
-0.1
-0.2
(a) Comparison of Mean Profiles
S-0.3
C 0.1 --- pre-modified profile B3
------ pre-modified profile B2
-\ ---- pre-modified profile B1 M.S.L.
> 0. ..................0 ---- .............................. ................... ..... ..... ............... .
-0.1 zw
0
--o.1 ,,,
-0.2
(b) Three Profiles before Modification at 352 min-- s .
S -0.3 i I I I
0.1 profile B3
------ profile B2
--- profile Bl M.S.L.
-0.0 ....... --
0.1
-0.3 1 ----------file
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
DISTANCE FROM TANK END (m)
Figure 3.17: Profile Evolution from 352 min to 407 min during Experiment MT04:
-... -b-- -.,,
(a) Comparison of the Pre-Modified Mean Profile at 352 min with Mean Profile at
407 min, (b) Three Profiles before modification at 352 min, and (c) Three Profiles at
407 min. Note the complete recovery of the bar crest with a smaller rotational angle
(less than 10 degrees).
66
Figure 3.18 presents isolines of profile elevations at 545 min, which were taken
at intervals of 1 cm by placing blank yarn along the waterline as the water was
lowered. As shown in this figure, weakly 3-D berm formed responding to clockwise
3-D circulation. Also a deep and narrow channel appeared near steel side wall, which
had the same characteristics as described by Shepard et al. (1941) as the floor of the
channels was found to be decidedly irregular mainly due to the strong rip current.
Also the depositional area occupied a large portion of the surf zone, which also can be
observed in the field near shoal areas. It can be seen that the bar rotated clockwise
as the results of clockwise 3-D circulation flowing from the depositional area to the
channel. The maximum cross-tank difference of the profile elevations between the
two sides of tank was estimated to be 4.0 cm at 545 min, which was slightly smaller
than that during Experiment MT03.
The profile seemed to approach an overall 2-D equilibrium up to 614 min with a
clockwise rotated bar crest except slight deposition of sand at the area just seaward of
berm. The maximum cross-tank difference in the profile elevations remained almost
constant as 4.5 cm even though the channel, shown in Figure 3.18, became more
distinct and clear.
During 614 min to 821 min, the area seaward of the bar maintained its slope as
1:7 with actually no changes. The berm built up rapidly in the same manner as the
profile at 545 min during Experiment MT03, as shown in Figure 3.9. However, the
slope of the beach face remained almost constant as 1:7, which was approximately
the same as that of the area seaward of the bar. The overall slope of the profile after
821 min was about 1:10, which was actually the same as that of final profile during
Experiment MT03.
In summary, this experiment was carried out to investigate the effects of induced
perturbations in the bar topography on the profile evolution and resulting equilibrium.
When profile modifications such as deepening of the bar trough, asymmetric area
(a) Isolines at Shoreward Area
(b) Isolines near the Bar Region
Figure 3.18: Isolines of Profile Elevations at 545 min during Experiment MT04.
These photographs showed: (1) the weak 3-D bcrm, (2) deep and narrow channel
near steel side wall, (3) depositional area occupying large portion of the surf zone,
and (4) clockwise rotation of the bar. Elevation contours were established by placing
back yarn at waterline during lowering of water level.
68
seaward of the bar, and removing half the bar crest were imposed on the 2-D profile
sequentially at 69, 138 and 352 min, respectively, it was found that the profiles were
stable against those perturbations and returned to the previous 2-D profiles before
modification when waves were resumed.
The 3-D flows appeared after 160 min and started to rotate the bar by about 20
degrees. These 3-D flows were then reinforced by a sedimentary feedback, resulting in
a rapid net landward sand transport. This stage was reminiscent of the morphology
associated with rip currents (see Figure 3.18).
The profiles seemed to approach a 2-D equilibrium up to 614 min with rotated
bar crest line except slight deposition of sand at the area landward of the bar trough.
After 614 min, the bar started to move seaward with fairly smaller rotation angle
(less than 10 degrees). The overall slope of the profile after 821 min was estimated
to be 1:10, which was actually the same as that of the final profile of Experiment
MT03. The slope of the beach face was estimated to be 1:7, which was approximately
the same as that of the area seaward of the bar.
Experiment MT05
Experiment MT05 was conducted with an initially planar profile perturbed by a
3-D berm to investigate whether or not resulting 3-D flows in the beach face would
facilitate the occurrence of the three-dimensionality, and was continued further to
investigate profile equilibrium. Figure 3.19 shows the initial profile with the 3-D
berm for Experiment MT05. The elevation difference at the two sides of the tank
was about 4.0 cm. This experiment also investigated the effects of an elevated water
table in the berm (+11.0 cm above Mean Sea Level).
During the first 23 min, the initially 3-D berm changed to fairly 2-D without
inducing any noticeable 3-D circulation inside the surf zone, which indicated that the
beach morphology was again stable against perturbations during the initial phases of
the experiment.
69
0.15
_-- mean profile
S 0--- profile B3
S 0.05 ------ profile B2
0.0 -- -- profile B1
-0.00 --- ............. M.S.L.
> -0.05
0
-0.10 -
O
S-0.15
E-
S-0.20
-0.25 TANK BOTTOM
-0.30 1 i I it
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
DISTANCE FROM TANK END (m)
Figure 3.19: Initial Profile with 3-D Berm Area for Experiment MT05
The profile seemed to approach an equilibrium at 207 min. The results were
almost the same as those of previous experiments, which implied that neither the
effect of elevated water table level in the berm nor that of an initially 3-D berm was
significant in this fine sand experiment.
3-D features appeared to start during 207 to 242 min, which was almost the same
as the previous experiments. After 297 min, the bar rotated in counter-clockwise
direction about 20 degrees and maintained its angle up to 407 min although 3-D
circulation appeared to be reinforced by sedimentary feedback continuously. The
three-dimensionality inside the surf zone continuously increased up to 407 min. The
maximum cross-tank differences in profile elevations inside the surf zone were esti-
mated to be 5 cm at 407 min. The deepest part was located along the glass side wall
of the tank, which could be expected from the counter-clockwise rotation of the bar
crest, and this deep channel resembled a rip channel morphology.
70
0.15
profile B3
S 0.10 .---- profile B2
S0.05 ,- --- profile BI
....-............- initial profile
S ........... .... .M.S.L.
> -0.00
S-0.05 .....
S-0.10 A
S-0.15 i
r -0.20
-0.25 X
-0.25 TANK BOTTOM -
-0.30 i 1 1 --
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
DISTANCE FROM TANK END (m)
Figure 3.20: Initial Profile and Three Profile Bi, B2 and B3 at 476 min during
Experiment MT05. Note dominantly 2-D morphology.
During 407 to 476 min, the profile changed to a fairly 2-D, as shown in Figure 3.20,
which appeared to be associated with a fairly strong 3-D circulation. This strong 3-D
circulation transported sands from the area seaward of the bar to the area immediately
landward of the bar trough and distributed sands evenly over that area across the
tank. However, the return channel near the glass side wall of the tank remained with
approximate dimensions of 6 cm depth below the mean profile and 8 cm width.
During 476 to 614 min, the 3-D circulation has been continuously reinforced
by the sedimentary feedback, resulting in very strong onshore transport. As strong
landward transport occurred continuously, the channel finally filled in while the profile
remained in a fairly 2-D. This strong landward transport resulted in the landward
movement of the bar and a rapid build up of the berm with another peak which
moved seaward, as shown in Figure 3.21.
The overall slope of the profile after 545 min was approximated as 1:10, which
resembled those of final profiles during Experiment MT03 and Experiment MT04.
0.15
S614 min.
S0.10 -- 545 min.
S0.05 ---- 476 min.
.. ............... initial profile
cii -M.S.L.
S -0.00 -- ........ ............. .. .................M .S.L............................. ...... ....... ......... .
> -0.05 *".
O
0 -0.10 \
S-0.15 -
E-\
-0.20 -
-0.25 TANK BOTTOM -- ----
-0.30 I I 1
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
DISTANCE FROM TANK END (m)
Figure 3.21: Mean Profile Evolution during Experiment MT05. Elapsed Times = 0
(Initial Profile), 476, 545 and 614 min. Note rapid change during 545 to 614 min
with landward movement of the bar and another peak of the berm. However, the
overall shape of the profile was surprisingly unchanged.
The slope of the berm was estimated to be 1:7, which was again approximately the
same as that of the area seaward of the bar. Even though the profile changed sub-
stantially during 545 to 614 min, the overall shape of the profile at 614 min remained
approximately the same as that at 545 min, as shown in Figure 3.21.
During 614 to 890 min, the new berm built up continuously, resulting in steeper
slope of about 1:5. The area seaward of the bar eroded substantially due to landward
transport with a slope of 1:5.5, which was again similar to that of the berm. The
bar moved offshore at the initial times and, at the later times, remained stationary
and changed to almost 2-D feature with rotation angle less than 10 degrees. As
time approached 890 min, profiles appeared to approach an another intermediate
equilibrium.
During 890 to 1028 min, the berm eroded slightly at the initial times and main-
tained its form with a slope of about 1:5, as shown in Figure 3.22. The bar moved
72
0.15
1028 min.
S010 --- 959 min.
0.05 /.--- 890 min.
...... ---- initial profile
c oo M.S.L.
0.00 .......-.........-.......> .................... . .. ........_ M ...
S-0.00 --r
> -0.05 -
0
< -0.10 o-
-0.15 .
> -0.20
-0.25 '*'
-0.25 TANK BOTTOM "
-0.30 1 1 1 1I
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
DISTANCE FROM TANK END (m)
Figure 3.22: Mean Profile Evolution during Experiment MT05. Elapsed Times = 0
(Initial Profile), 890, 959 and 1028 min. Note the back-and-forth movement of the
bar, otherwise the profiles approached an equilibrium.
back-and-forth with increased rotation angle of about 15 degrees. However, the pro-
files appeared overall 2-D. The slope of the area seaward of the bar changed from
1:5.5 to 1:4.5. The bar moved onshore if slope was near 1:4.5, while the bar moved
offshore if slope was near 1:5.5; these implied the equilibrium slope of 1:5.
After 1028 min, the profile approached an equilibrium with overall 2-D morphol-
ogy except the oscillation of the bar.
Experiment MT06
Experiment MT06 was carried out with a more highly elevated water table in the
berm (+16.5 cm) since moderately elevated water table during Experiment MT05
appeared to produce no effects on the profile evolution and 3-D morphology.
Up to 138 min, the trend was the same as the previous experiments as the profile
approached an overall 2-D equilibrium. Berm built up rapidly with onshore transport
at the initial times. Bar moved offshore at the initial times and moved onshore back
at the later times.
73
From 138 min to 207 min, the bar remained stationary. At the later stages (about
180 min), the bar rotated quickly and a 3-D feature appeared immediately landward
of the bar trough. At 207 min, the bar rotated approximately 25 degrees. The three-
dimensionality at this experiment occurred earlier (about 180 min) than previous
experiments, which might imply that the elevated high water table accelerated the
beach profile evolution as it caused more active transport by reducing the stability of
the bottom particles.
During 207 min to 242 min, the bar remained stationary. However, as the bar
trough became shallow and the area seaward of the bar eroded mildly, weak on-
shore transport continued and the overall three-dimensionality inside the bar region
increased continuously.
During 242 min to 476 min, as shown in Figure 3.23, the bar moved slightly
onshore at the initial times and stayed stationary at the later times. The bar trough
deepened considerably and rapidly during the first 30 min and maintained its depth
later. The area seaward of the bar eroded continuously and the sand eroded was
carried up and deposited on the depositional area immediately landward of the bar
trough. Up to 352 min, the overall three-dimensionality increased continuously. Ta-
ble 3.2 presents a summary of the maximum cross-tank differences of the profile
elevations at the two sides of the tank at 352 min. As strong onshore transport
continued during this time, the depositional area continued to grow up and to move
onshore, resulting in a deep channel across the tank at about the mean water line at
476 min.
During 476 min to 614 min, the depositional area moved continuously landward,
as observed in the previous experiments. However, it should be noted during this time
that the overall shapes of the profiles amazingly agreed well with those of Experiment
MT05, as shown in Figure 3.24 which presents a comparison of mean profiles at 614
min.
0.15
0.10
0.05
-0.00
-0.05
-0.10
-0.15
-0.20
-0.25
-0.30 1 I
0.0 0.5 1.0 1.5
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
DISTANCE FROM TANK END (m)
Figure 3.23: Mean Profile Evolution during Experiment MT06. Elapsed Times = 0
(Initial Profile), 242, 352 and 476 min. Note the deep channel across the tank at
about the mean water line at 476 min.
0.15
Experiment MT06
010 ----- Experiment MT05
S o.oo -................ ................. _
> -0.05 -
O
Z
o 0
S-0.15
-0.20
-0.25 -
-0TANK BOTTOM
-0.30 I I I I I I
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
DISTANCE FROM TANK END (m)
5.0 5.5 6.0
Figure 3.24: Comparison of the Mean Profile at 614 min during Experiment MT05
with that during Experiment MT06. Note good agreement of the overall shape.
Table 3.2: Maximum Cross-Tank Differences of the Profile Elevations at 352 min
during Experiment MT06
Region Location Maximum
(m) Difference (cm)
berm 0.5 2.0
shoreline 0.9 3.5
depositional area 1.8 4.5
bar trough 2.3 6.0
bar crest 2.5 3.5
During 614 to 683 min, the depositional area finally attached to the berm, result-
ing in a large increase in the size of the berm and another peak of the berm moving
seaward, as shown in Figure 3.25. Also the 3-D profile returned to a fairly 2-D one
except the bar rotation of about 15 degrees and a narrow deep channel on the glass
side wall of the tank. The water depth of the channel was estimated to be 7.0 cm,
which was about 3.5 cm below its surroundings. The slope of the area seaward of the
bar at 683 min was estimated to be 1:4.7, which was much steeper than that of the
beach face (about 1:7.3).
During 683 min to 890 min, the bar moved onshore very quickly at the initial
times since the slope of the area seaward of the bar was steeper than 1:5, as presented
in the results of Experiment MT05. Hence, the back-and-forth movement of the
bar around the slope of 1:5 was expected. However, the slope of the offshore area
continuously reduced to 1:6.5 at 752 min, and finally to about 1:11 after 890 min,
as shown in Figure 3.26. Also the slope of the beach face reduced from 1:4 at 752
min to 1:11 at 890 min continuously. Even the bar moved onshore, strong offshore
transport occurred at the initial times, which resulted in erosion of the berm area and
deposition of the offshore area of the bar. During this time, the bar moved onshore
slightly and profile remained fairly 2-D. After 890 min, the profile approached an
equilibrium with the overall slope of about 1:12.
0.15
683 min.
0.10 ----- 614 min.
S 0.05 ----- initial profile
M.S.L.
0 .0 0 .................................... ... .. .......... ... ... ...... ... ................ -- -..................
-y 0.0 0** ---- :*---- -- -- -- .---...--.---. ..-*...- --.--- ---. --. *..------.
> -0.05
-0.10 -
z
o
S-0.15 -
W 0.20
-0.25
-025 TANK BOTTOM -
-0.30 I I
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
DISTANCE FROM TANK END (m)
Figure 3.25: Mean Profile Evolution during Experiment MT06. Elapsed Times = 0
(Initial Profile), 614 and 683 min. Note the large berm with peak moving seaward at
683 min.
It appeared that the elevated high water table had accelerated the beach profile
evolution at the initial times as it facilitated sediment transport by destabilizing the
bottom particles, resulting in earlier occurrence of the three-dimensionality than the
previous experiments. Then the profile evolved as the same way as the previous
experiments without introducing any significant effects of the highly elevated water
table. However, when the profile became steeper, the elevated high water table level
seemed to affect the profile evolution, resulting in the milder equilibrium slope.
0.15
0.10
0.05
-0.00
-0.05
-0.10
-0.15
-0.20
-0.25 '
TANK BOTTOM
-0.30 I I i
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
DISTANCE FROM TANK END (m)
Figure 3.26: Mean Profile Evolution during Experiment MT06. Elapsed Times = 0
(Initial Profile), 683, 752, 890 and 1166 min. Note that the profile approached an
equilibrium with reduced overall slope.
78
3.5 Summary and Conclusions
3.5.1 Summary
A series of movable bed tests was carried out in a tank approximately 15.5 m
long, 0.9 m high and 0.6 m wide. The beach was formed of sand of median diameter
0.21 mm to an initial slope of 1:18. Regular waves of 0.11 m height and 1.5 sec period
were used in the experiments. Based on the profile classification criteria, presented
in Chapter 2, the bar would form under above experimental conditions. Profiles were
documented at various times using a modified point gage by measuring three profile
lines along the tank centerline and at quarter positions across the tank. The profiles
were documented at locations spaced at 0.1 m along each of these lines.
Six experiments were carried out to investigate various possible interrelationships
and causes of the three-dimensionality. The following were examined: (1) Effects of
an induced perturbation in the bar topography, (2) Elevated water table in the berm,
and (3) An initially 3-D berm.
Experiment MTO1 was conducted to investigate whether or not 3-D features
might occur inside the surf zone and to provide the reference for future tests.
At the initial stages, the bar formed quickly as expected, but also the berm built
up. Profiles approached an equilibrium at 240 min with overall 2-D conditions. After
about 270 min, 3-D circulation occurred inside the surf zone after a level of profile
stability had occurred at 242 min. This 3-D flows were generally weak to transport
the sands at the initial times. However, later they were reinforced by sedimentary
feedback so that they could transport the sediments onshore, thus initiating a new
stage of profile evolution, which was reminiscent of a bar morphology with a rip
current present.
Experiment MT02 was conducted to investigate the effects of microorganisms
inside the beach subjected to prolonged submergence. Hence, this experiment was
started immediately after the last wave run for Experiment MT01, at which time
79
the biological film inside sand was believed to be ineffective in clogging the sands.
Generally the trend of the profile evolution was similar to that during Experiment
MT01, which provided the experimental repeatability rather than proving the ef-
fects of microorganisms. Although three-dimensionality at this experiment occurred
slightly earlier at 240 min than that at Experiment MT01, no clear conclusions could
be drawn to the effects of microorganisms.
Experiment MT03 was designed to investigate the effect of required time for the
beach system to reach steady-state conditioned with continuous wave run. Hence,
the wave was run without intermission during 207 min. After 207 min, however,
the wavemaker malfunctioned, resulting in interruption of the experiment. Three-
dimensionality at Experiment MT03 occurred at the same time (240 min) as Exper-
iment MT02, which implied that the effect of continuous wave run was unexpectedly
small.
This experiment was continued beyond 407 min to investigate equilibrium of the
profile evolution. After 407 min, the area seaward of the bar eroded continuously
and the eroded sand was deposited immediately landward of the bar trough. This
depositional area changed from an overall 3-D to 2-D character except for a very
narrow deep channel near one side of tank, the maximum depth of which was found
to be about 5.0 cm below the mean profile.
Experiment MT04 was carried out to investigate the effect of induced perturba-
tions in the bar topography on the profile evolution and resulting equilibrium. Profile
modifications such as deepening of the bar trough, asymmetric area seaward of the
bar, and removing half the bar crest were imposed on the 2-D profile sequentially at
69, 138 and 352 min, respectively. Surprisingly, the profiles returned to the previous
2-D profiles when waves were resumed; hence, the profiles were stable against the
perturbations during the initial stages of the profile evolution.
80
The 3-D flows appeared after 160 min and started to rotate the bar by about
20 degrees. These 3-D flows were then reinforced by a sedimentary feedback, result-
ing in a rapid net landward sand transport. This stage was reminiscent of the bar
morphology associated with rip currents (see Figure 3.18).
The profiles seemed to approach a 2-D equilibrium up to 614 min with rotated
bar crest line except slight deposition of sand at the area landward of the bar trough.
After 614 min, the bar started to move seaward with a fairly small rotation angle less
than 10 degrees. The overall slope of the profile after 821 min was estimated to be
1:10, which was actually the same as that of the final profile of Experiment MT03.
The slope of the beach face was estimated to be 1:7, which was approximately the
same as that of the area seaward of the bar.
Experiment MT05 was carried out to investigate the effect of an initially 3-D
berm with moderately elevated high water table in the berm. During the first 23 min,
the 3-D berm changed to approximately 2-D without introducing any noticeable 3-
D circulation inside the surf zone; this again indicated that the morphology was
stable during mne initial phases of the experiment. The magnitude of the berm,
however, was somewhat larger than that of previous experiments due to initially
deposited sand. The profile seemed to approach an equilibrium up to 207 min and
the results were almost the same as those of previous experiments, which meant that
the effects of moderately elevated high water table level was negligible for these fine
sand experiments.
Three-dimensionality appeared to start during 207 to 242 min, which was almost
the same as the previous experiments. From 242 to 545 min, the bar rotated again 20
degrees and maintained its angle up to 407 min. The maximum cross-tank difference
of the 3-D features inside the surf zone was about 5.0 cm and occurred at 407 min.
The deepest profile was located at the narrow channel on one side of the tank. During
407 to 476 min, 3-D depositional area changed to a fairly 2-D one. The overall slope
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