AN EXAMINATION OF ESTUARINE LUTOCLINE DYNAMICS
By
JIANHUA JIANG
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
1999
ACKNOWLEDGMENT
I would like to express my most profound gratitude to my advisor and the chairman
of my supervisory committee, Dr. Ashish J. Mehta, professor of Coastal and Oceanographic
Engineering, for his guidance and support throughout my study at the University of Florida.
It has been a challenging and a rewarding experience for me.
1 wish to express my deep appreciation to the members of my supervisory committee.
Graduate Research Professor Robert G. Dean, Associate Professor Kirk Hatfield, Professor
D. Max Sheppard and Assistant Professor Robert J. Thieke for their helpful advice,
comments and patience in reviewing this dissertation. Thanks are also due to all other
teaching faculty members in the Coastal & Oceanographic Engineering Department, as well
as those whose courses I attended in Aerospace Engineering, Mechanics & Engineering
Science and Environmental Engineering Sciences. They supplied me with knowledge
essential for the pursuit of this study through their creative teaching efforts. Special thanks
are due to Jim Joiner and Sydney Schofield of the Coastal Engineering Laboratory, and John
Davis and Twedell Helen of the Coastal Engineering Archives.
My thanks also extend to Dr. Eric Wolanski of the Australian Institute of Marine
Science, for his valuable suggestions and discussions. Our joint effort on the investigation
of vertical mixing due to breaking of internal waves at the lutocline in the Jiaojiang estuary,
China, gave me the initial direction of this study.
11
Many fellow student colleagues, including A1 Browder, Matt Henderson, Hugo
Rodriguez, Bill McAnally, Chenxia Qiu, Detong Sun, and others gave me support in various
ways.
This study has been made possible in part due to data gathered in the Jiaojiang. These
campaigns were successfully carried out due to the sincere efforts of Professors Xie Qinchun,
Li Yan, Li Bogen, Xia Xiaomin, Feng Yinjun, and others, all of the Second Institute of
Oceanography, Hangzhou.
Most important of all, I am deeply indebted to my wife, Tian Fang, and my daughter,
Jiang Ruiwei. Although we were separated from each other during my education at UF, they
always provided me their strong support, persuasion and love from the other side of the
Pacific Ocean. My family has been the source of my perseverance with this work.
Ill
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS ii
LIST OF FIGURES viii
LIST OF TABLES xviii
LIST OF SYMBOLS xix
ABSTRACT xl
CHAPTERS
1 INTRODUCTION 1
1 . 1 Problem Statement I
1.2 Objectives 3
1.3 Tasks 4
1 .4 Thesis Outline 5
2 LUTOCLINES IN ESTUARIES 7
2.1 Vertical Structure of Fine Sediment Suspension and the Lutocline 7
2.2 Causes of Fluid Mud and Lutocline Generation 9
2.2.1 Settling Velocity 11
2.2.2 Formation of Fluid Mud 12
2.2.3 Formation of Lutocline 12
2.3 Influence of Lutocline on Turbidity Transport 14
2.4 Dynamics of Turbid Estuaries 16
2.4.1 Amazon Shelf 17
2.4.2 Ariake Bay 18
2.4.3 Gironde River 19
2.4.4 Hangzhou Bay 20
2.4.5 James River 21
2.4.6 Orinoco River 22
IV
2.4.7 Rhine-Meuse River 23
2.4.8 Severn River 24
2.4.9 South Alligator River 24
2.4.10 Thames River 25
2.4.11 Yellow River 26
3 FLOW AND SEDIMENT TRANSPORT 28
3.1 Introduction 28
3.2 Hydrodynamics 29
3.2.1 Governing Equations 29
3.2.2 Boundary Conditions 31
3.3 Sediment Transport 35
3.3.1 Sediment Conservation Equation in the Water Column 35
3.3.2 Boundary Conditions 35
3.3.3 Fine Sediment Transport Processes 37
3.4 Flow-Sediment Coupling 43
3.4.1 Baroclinic Effects 43
3.4.2 Vertical Momentum and Mass Diffusion Coefficients 43
3.5 Solution Techniques 47
3.5.1 Discretization of Hydrodynamic Equations 48
3.5.2 Discretization of Sediment Transport Equation 50
3.5.3 Discretization of Consolidation Equation 51
3.5.4 Properties of the Finite-Differential Equations 52
3.6 Basic Simulations 54
3.6.1 Hydrodynamics 54
3.6.2 Sediment Transport 60
3.6.3 Consolidation 68
3.6.4 Interfacial Entrainment 73
4 FIELD INVESTIGATION AND DATA ANALYSIS 75
4. 1 Study Area Description 75
4.2 Experimental Plan, Methods and Instruments 77
4.2.1 Fluid Mud Observations 77
4.2.2 Lutocline Observations 79
4.2.3 Observations of SSC, Currents, Salinity and Temperature 80
4.2.4 Tidal Elevations 80
4.3 Experimental Data 80
4.3.1 Sediment Size 80
4.3.2 Tides 81
4.3.3 Profiles of SSC 83
4.3.4 ASSMData 83
V
4.3.5 Floe Settling Velocity 102
4.3.6 Erosion Rate Constant 105
4.4 Properties of Internal Waves 107
4.4.1 Effect of Ri^ on // 107
4.4.2 Effect of Ri^ on 108
4.4.3 Celerity and Wave Length Ill
5 TURBULENCE DAMPING IN FLUID MUD 114
5.1 Introduction 114
5.2 Turbulence Damping and its Effect on Lutocline Formation 115
5.3 Mixing Length in the Jiaojiang 125
5.4 Modified Vertical Momentum and Mass Diffusion Coefficients 127
6 LUTOCLINE DYNAMICS IN THE JIAOJIANG 128
6.1 Introduction 128
6.2 Parameters for Flow and Sedimentary Processes 128
6.3 Model Application 132
6.3.1 Modeled Domain, Initial and boundary Conditions 132
6.3.2 Sediment Deposition, Erosion, Consolidation
and Entrainment 134
6.4 Flow and Sediment Dynamics 136
6.4.1 FlowField 136
6.4.2 Tidal Variation of Velocity 137
6.4.3 Tidal Variation of SSC 142
6.4.4 Vertical Profiles of Velocity 146
6.4.5 Vertical Profiles of SSC 155
6.4.6 Lutocline Layer 163
6.4.7 Flow-SSC Hysteresis 168
6.4.8 Effect of Turbulence Damping on SSC and
Lutocline Formation 174
7 SUMMARY AND CONCLUSIONS 176
7.1 Summary 176
7.2 Conclusions 177
7.3 Recommendations for Future Studies 179
VI
APPENDICES
A DERIVATIONS OF THE GOVERNING EQUATIONS 181
A. 1 V ertical V elocities, w and o), and Continuity Equation (3.1) 181
A.2 Momentum Equations (3.2) and (3.3) 183
A. 3 Sediment Conservation Equation (3.15) 185
B NUMERICAL TECHNIQUES 187
B. l Back-Tracing Approach 187
B.2 Pre-conditioned Conjugate Gradient Method 188
C EFFECT OF TEMPERATURE ON SETTLING VELOCITY 191
D AN APPLICATION OF COHYD-UF:CONTRACTION SCOUR IN A
RIVER 197
D.l Scour Problem 197
D.2 Scour Simulation 200
D. 3 Results 203
E SIMULATION OF SEDIMENT DEPOSITION IN A FLUME 206
E. l Introduction 206
E.2 Flume Test 206
E.3 Settling Velocity in Moving Water 207
E.4 Deposition Simulation 210
BIBLIOGRAPHY 214
BIOGRAPHICAL SKETCH 226
Vll
LIST OF FIGURES
Figure page
2.1 Vertical dry SSC profile classification and associated velocity profile. Also
shown are unit transport processes which govern concentration profile
dynamics (after Mehta, 1989; 1991) 8
2.2 A representative description of settling velocity and flux variation with SSC
(after Mehta and Li, 1 997) 10
2.3 Mixing of a two-layered stratified fluid, with fluid mud beneath clear water ... 16
3.1 Schematic diagram showing o-transform 29
3 .2 Dimensionless median settling velocity as a ftinction of temperature, where (Ojq
is the median settling velocity at 15 °C 44
3.3 Schematic diagram of computational mesh and notation 48
3.4 Schematization of the simulated consolidation process, where (a) is the
original consolidating layer, (b) is the case of net deposition and (c) is the
case of net erosion 52
3 . 5 Modeling 1 D linear hydrodynamic equation for tidal flow in an open channel.
Lines are simulations and open circles represent analytical solutions 57
3 .6 Modeling 1 D linear hydrodynamic equation for tidal flow in an open channel.
Lines are simulations and open circles represent analytical solutions 57
3.7 Modeling ID non-linear hydrodynamic equation for tidal flow in an open
channel. Lines are simulations and open circles represent analytical solutions
59
3.8 Modeling ID non-linear hydrodynamic equation for tidal flow in an open
channel. Lines are simulations and open circles represent analytical solutions
viii
59
3.9 Modeling ID convection-diffusion equation. Line is analytical solution and
dots represent model simulations 62
3.10 Modeling 2D Laplace equation. Lines are analytical solution and open circles
represent model simulations 62
3.11 Modeling ID transient heat conduction. Lines are analytical solutions and
dots represent model simulations 64
3.12 Modeling heat conduction with radiation. Lines are analytical solutions and
dots represent model simulations 64
3.13 Modeling ID transient convection-diffusion equation. Lines are analytical
solutions and dots represent model simulations 66
3.14 Modeling 3D Laplace equation. Lines are analytical solutions, dots represent
model simulations, £ind plus signs are associated with contour number 67
3.15 Modeling SSC (unit: kg m '^). Solid lines are simulations and dashed lines
represent field data observed from 1800, 9/24/68 to 0400, 9/25/68 in
Savannah River estuary (after Ariathurai, et al., 1977). Contours from the
surface to the bottom are 0.1, 0.25, 0.5, 1, 1.5 and 2, respectively 70
3.16 Consolidation rate, , as a function of dry density for Doel Dock mud with
Cy^=80 kg m (after Toorman and Berlamont, 1993) 70
3.17 Modeling laboratory data of T oorman and Berlamont ( 1 993 ) on consolidation
without deposition at the bed-fluid interface. Lines are model simulations and
points represent data 71
3.18 Modeled consolidation curve (solid line) compared with the laboratory data
(open circles) of Toorman and Berlamont (1993) 71
3.19 Consolidation rate, o)^^, as a function of dry density for the laboratory tests
of Burt and Parker (1984) with c^=26.3 kg m 72
3.20 Modeling laboratory data of Burt and Parker (1984) on consolidation with
deposition at the bed-fluid interface. Lines are model simulations and points
represent data 73
3.21 Modeling laboratory data on entrainment by Mehta and Srinivas (1993).
Lines are model simulations and points represent data 74
IX
4. 1 Location map of Jiaojiang estuary, China. Depths are in meters below lowest
astronomical tide. Ml and M2 are mooring sites (Table 4.1); Cl, C2, C3 and
C4 are velocity measurement and SSC profile sampling stations (Table 4.1);
C6 is the site of ASSM (Table 4.1) and T1-T6 are tide stations. The region
between double dotted lines is the modeled domain 76
4.2 A representative frequency distribution of suspended sediment (dispersed)
size in the Jiaojiang estuary 81
4.3 Time series of tidal elevation at sites T1 and T5 during a spring tide from
0000 hr on 1 1/05/94 to 0100 hr on 1 1/06/94 82
4.4 Time series of velocity at site C4 during a spring tide. Observations began at
1700 hr on November 5, 1994. Positive numbers signify flood and negative
are for ebb 84
4.5 Time series of velocity at site C4 during a neap tide. Observations began at
0900 hr on November 10, 1994. Positive numbers signify flood and negative
are for ebb 84
4.6 Time series of velocity at site C6 during a neap tide. Observations began at
0630 hr on November 15, 1995. Positive numbers signify flood and negative
are for ebb 85
4.7 Time series of SSC at site C4 during a spring tide. Observations began at
1700 hr on November 5, 1994. Shaded area includes SSC greater than 20
kg m 85
4.8 Time series of SSC at site C4 during a neap tide. Observations began at 0900
hr on November 10, 1994. Shaded area includes SSC greater than 20 kg m
86
4.9 Time series of SSC at site C6 during a neap tide. Observations began at 0600
hr on November 15, 1995. Shaded area includes SSC greater than kg m ... 86
4. 1 0 Typical raw ASSM records during the neap tide on November 15,1 995, with
a horizontal time scale of 1 min and a vertical distance scale of 1 .25 m. (A)
was observed during a flood with a value of Richardson number Ri^ of about
2, and (b) during an ebb with Ri^ of about 150 87
4.11 Relationship between lutocline elevations above bottom detected by the
turbidimeter and by the ASSM. Data were collected during 0600-1600 hr on
November 15, 1995 88
X
4. 12 Time series of lutocline elevation at site C6 during a neap tide using ASSM
on November 15, 1995. (a) and (b) were sampled during flood with a value
of RIq of about 1, and (c) during ebb with Ri^ of about 150. Solid lines are
instantaneous elevations and dashed lines are mean trends 89
4.13 Time series of lutocline elevation after trend removal at site C6 during a neap
tide, where (a), (b) and ( c ) correspond to Figure 4.12 91
4.14 Typical profiles of internal waves exhibiting sharp crests and flat troughs.
Data were taken from example (a) in Figure 4.13. Wave heights range from
0.07 m to 0.23 m 92
4.15 Auto-correlation function against time interval, where (a), (b) and (c)
correspond to Figure 4.13 94
4.16 Internal wave spectrum corresponding to example (a) in Figure 4.13 94
4. 1 7 Internal wave spectrum corresponding to example (b) in Figure 4.13 95
4. 1 8 Internal wave spectrum corresponding to example (c ) in Figure 4.13 95
4.19 rsm of high-frequency internal wave height as a fimction of global
Richardson number 97
4.20 Modal frequency of high-frequency internal waves as a fimction of global
Richardson number 97
4.21 rms of high-frequency internal wave height during ebb and flood 98
4.22 Modal frequency of high-frequency internal waves during ebb and flood 98
4.23 rsm of low-frequency internal wave height as a function of global Richardson
number 99
4.24 Modal frequency of low-frequency internal waves as a function of global
Richardson number 99
4.25 rms of low-frequency internal wave height during ebb and flood 100
4.26 Modal frequency of low-frequency internal waves during ebb and flood 100
XI
4.27 Settling velocity as a function of SSC during a neap tide from 0900 hr on
1 1/10/94 to 1000 hr on 1 1/1 1/94. Solid line is the best-fit of the calculated
data points using Eq. (4.12) 104
4.28 Settling velocity as a function of SSC during a spring tide from 1700 hr on
1 1/05/94 to 1800 hr on 1 1/06/94. Solid line is the best-fit of the calculated
data points using Eq. (4.12) 104
4.29 Erosion rate as a function of excess bottom shear stress 107
4.30 Definition sketch of two-layered flow system 110
5.1 Definition of linear sediment concentration, c^, and its relationship with
sediment concentration, c. dj is the floe diameter 117
5.2 Relative momentum mixing length calculated from Eq. (5.11) (solid lines).
field data (data points) and settling flux (dashed line) as functions of SSC ... 123
5.3 Lutocline strength index as a function of turbulence energy production based
on measured profiles of SSC and velocity. The equation represents the best-
fit line 125
6. 1 Bathymetry in the modeled domain of the Jiaojiang (a), where the datum is
mean water level and the regions enclosed within dotted lines are mudflats,
and the numerical mesh in the horizontal plane (b) 133
6.2 Simulated peak flood flow during a spring tide at 2000 hr, 1 1/05/94 138
6.3 Simulated high water slack during a spring tide at 2245 hr, 1 1/05/94 138
6.4 Simulated peak ebb flow during a spring tide at 0100 hr, 1 1/06/94 139
6.5 Simulated low water slack during a spring tide at 061 5 hr, 1 1/06/94 139
6.6 Tidal velocity at QAH (a), and SSC at 0.25// (b) and at 0.75// (c) at C2
during a spring tide. Solid lines are simulations and dashed lines represent
field data collected during 1800 hr, 1 1/05/94 to 1700 hr, 1 1/06/94 140
6.7 Tidal velocity at 0.4// (a), and SSC at 0.25// (b) and at 0.75// (c) at C2
during a neap tide. Solid lines are simulations and dashed lines represent field
data collected during 1000 hr, 1 1/10/94 to 0900 hr, 11/1 1/94 140
XU
6.8 Tidal velocity at 0.4// (a), and SSC at 0.25// (b) and at 0.75// (c) at C4
during a spring tide. Solid lines are simulations and dashed lines represent
field data collected during 1800 hr, 1 1/05/94 to 1700 hr, 1 1/06/94 141
6.9 Tidal velocity at 0.4// (a), and SSC at 0.25// (b) and at 0.75// (c) at C4
during a neap tide. Solid lines are simulations and dashed lines represent field
data collected during 1000 hr, 1 1/10/94 to 0900 hr, 11/1 1/94 141
6.10 Time series of vertically-averaged SSC during a spring tide (a) and a neap
tide (b). During spring tide, the data at sites Cl and C3 began at 1700 hr,
1 1/04/94 and at sites C2 and C4 at 1800 hr, 1 1/05/94. During neap tide, the
data at sites Cl and C3 began at 1000 hr, 1 1/12/94 and at sites C2 and C4 at
2300 hr, 11/10/94 145
6.1 1 Velocity profiles at site C2 during a spring tide. Solid lines are simulations
and open circles represent field data obtained during 1900 hr, 1 1/05/94 to
0600 hr, 1 1/06/94. Positive values signify flood and negative denote ebb .... 147
6.12 Velocity profiles at site C2 during a neap tide. Solid lines are simulations and
open circles represent data obtained during 1100 hr to 2100 hr, 11/10/94.
Positive values signify flood and negative denote ebb 147
6.13 Velocity profiles at site C4 during a spring tide. Solid lines are simulations
and open circles represent data obtained during 1900 hr, 1 1/05/94 to 0500 hr,
1 1/06/94. Positive values signify flood and negative denote ebb 148
6. 14 Velocity profiles at site C4 during a neap tide. Solid lines are simulations and
open circles represent data obtained during 1 100 hr to 2100 hr, 1 1/10/94.
Positive values signify flood and negative denote ebb 148
6. 1 5 Distribution of transverse (a) and longitudinal (b) currents across the channel
due to the Coriolis effect, viewed in the direction of tidal wave propagation.
Solid lines are currents in Taizhou Bay and dashed lines in the Jiaojiang .... 151
6.16 Profiles of gravitational circulation without wind stress (after Hansen and
Rattray, 1965) 154
6.17 Tidal currents, vorticities and residual circulation in the neighborhood of a
headland. Currents are signified by solid arrows for flood and dashed arrows
for ebb. Currents are largest near the headland and decrease towards the
shoreline. Vorticity therefore has a maximum near the headland. Vorticities
generated by side-wall friction are shown by solid circles for flood and
dashed circles for ebb. Vorticities have highest strength near the headland and
diminish away from it (after Zimmerman, 1981) 155
Xlll
6.18 SSC profiles at site C2 during a spring tide. Solid lines are simulations and
open circles represent field data obtained during 1900 hr, 1 1/05/94 to 0600
hr, 11/06/94 156
6.19 SSC profiles at site C2 during a neap tide. Solid lines are simulations and
open circles represent field data obtained during 1 100 hr to 2100 hr, 1 1/10/94
156
6.20 SSC profiles at site C4 during a spring tide. Solid lines are simulations and
open circles represent field data obtained during 1900 hr, 1 1/05/94 to 0400
hr, 11/06/94 157
6.21 SSC profiles at site C4 during a neap tide. Solid lines are simulations and
open circles represent field data obtained during 1 100 hr to 2100 hr, 1 1/10/94
157
6.22 Time series of vertically-averaged SSC. Dark circles are from site C2, open
circles represent site C4. Solid lines signify simulations at C2, dashed lines
are simulations at the center of the flow section containing C2 and C4 and
dotted lines denote simulations at C4 160
6.23 Comparison of simulated vertically-averaged SSC using uniform and non-
uniform boundary conditions for SSC during a spring tide. Solid lines signify
simulations at site C2, dashed lines are that at site C4, and dark and open
circles represent that using uniform conditions of SSC 161
6.24 Vertical gradient of SSC as a function of elevation during a neap tide. Solid
lines are simulations and open circles represent field data obtained during
1100 hr to 2100 hr, 11/10/94 164
6.25 Simulated (solid lines) and measured (dashed lines) tidal variation of the
lutocline layer at site C2 during a spring tide from 1800 hr, 1 1/05/94 to 1700
hr, 11/06/94 165
6.26 Simulated (solid lines) and measured (dashed lines) tidal variation of the
lutocline layer at site C2 during a neap tide from 1000 hr, 1 1/10/94 to 0900
hr, 11/11/94 165
6.27 Simulated (solid lines) and measured (dashed lines) tidal variation of the
lutocline layer at site C4 during a spring tide from 1800 hr, 1 1/05/94 to 1700
hr, 11/06/94 166
XIV
6.28 Simulated (solid lines) and measured (dashed lines) tidal variation of the
lutocline layer at site C4 during a neap tide from 1000 hr, 1 1/10/94 to 0900
hr, 11/11/94 166
6.29 Lutocline layer thickness: simulation (6^^) and measurement (6^^) 167
6.30 Lutocline layer upper elevation: simulation (Z^^) and measurement (Z^^) ... 167
6.31 Simulated (solid line) and measured (dashed line) hysteresis loops at 1 m
above the bottom at C2 during a spring tide from 1800 hr, 1 1/05/94 to 0500
hr, 11/06/94 169
6.32 Simulated (solid line) and measured (dashed line) hysteresis loops at 1 m
below the surface at C2 during a spring tide from 1 800 hr, 1 1/05/94 to 0500
hr, 11/06/94 169
6.33 Simulated (solid line) and measured (dashed line) hysteresis loops at 1 m
above the bottom at C2 during a neap tide from 1000 hr to 2100 hr, 1 1/10/94
170
6.34 Simulated (solid line) and measured (dashed line) hysteresis loops at 1 m
below the surface at C2 during a neap tide from 1000 hr to 2100 hr, 1 1/10/94
170
6.35 Simulated (solid line) and measured (dashed line) hysteresis loops at 1 m
above the bottom at C4 during a spring tide from 1 800 hr, 1 1/05/94 to 0500
hr, 11/06/94 171
6.36 Simulated (solid line) and measured (dashed line) hysteresis loops at 1 m
below the surface at C4 during a spring tide from 1 800 hr, 1 1/05/94 to 0500
hr, 11/06/94 171
6.37 Simulated (solid line) and measured (dashed line) hysteresis loops at 1 m
above the bottom at C4 during a neap tide from 1000 hr to 2100 hr, 1 1/10/94
172
6.38 Simulated (solid line) and measured (dashed line) hysteresis loops at 1 m
below the surface at C4 during a neap tide from 1000 hr to 2100 hr, 1 1/10/94
172
XV
6.39 Modeling SSC profiles at site C4 during a neap tide. Open circles represent
field data from 1 100 hr to 2100 hr, 1 1/10/94, solid lines are simulations with
t/2=0.75, dashed lines signify simulations with d^^Q.lS and Rig=Ri^=0, and
dotted lines represent simulations with d2=0 175
B . 1 Schematic diagram of back-tracing approach, where dotted line is the pathline
of water particle, o is the position of water particle at current time step n+l
and p is the position of water particle at previous time step n 188
C. l Frequency distribution, (J)^, of the settling velocity of kaolinite 194
C.2 Time-concentration relationship during deposition at 26 °C. Open circles are
the experimental data of Lau ( 1 994) 194
C.3 Time-concentration relationship during deposition at 20 °C. Open circles are
the experimental data of Lau ( 1 994) 195
C.4 Time-concentration relationship during deposition at 10 °C. Open circles are
the experimental data of Lau ( 1 994) 195
C.5 Time-concentration relationship during deposition at 5 °C. Open circles are
the experimental data of Lau ( 1 994) 196
C. 6 Cumulative distribution of settling velocity of kaolinite at different
temperatures, 0 196
D. 1 Schematic diagram showing the Haldia oil pier and depth contours (m) in the
vicinity. Water depth are below mean low water 198
D.2 Location map of Haldia oil pier, India 199
D.3 Measured scour depths in fi'ont of the Haldia oil pier. The pier is shown as an
idealized rectangular protrusion. Unit: m 201
D.4 Bottom topography of the modeled segment of the river in the vicinity of the
Haldia pier. Water depths (unit: m) are below mean low water 201
D.5 Simulated flow field around pier at 0.057/ below the surface 204
D.6 Simulated flow field around pier at 0.05// above the bottom 204
XVI
D.7 Comparison of scour depths simulated (solid lines) and measured (dashed
lines) in front of the Haldia oil pier. Unit: m 205
D. 8 Comparison between simulated and measured areas at 2-3 m (•), 3-4 m (¥),
4-5 m (o) and >5 m (+) scour depths 205
E. 1 Settling velocity as a function of SSC in moving water. Data are from Mehta
(1973) 210
E.2 Simulated flow field around the barrier in the flume, (a) near the surface and
(b) near the bottom 212
E.3 Distribution of simulated (solid lines) and observed (numbers in circles)
deposition (thickness) at the down side of the barrier. Data are from
Ariathurai (1974) 213
xvii
LIST OF TABLES
Table page
3.1 Parameters for momentum and mass diffusion coefficients in a stratified flow . 47
4. 1 Summary of Jiaojiang field campaigns 78
4.2 rms height and angular frequency of internal waves as functions of global
Richardson number and tidal range 101
4.3 Celerity and length of internal waves 113
6.1 Flow and sedimentary process formulations and parameters 129
6.2 Vertically-averaged maximum velocities at sites C2 and C4 (Unit: ms'') . . 142
6.3 Vertically-averaged SSC during minor and significant non-uniformity of SSC
across the flow cross-section (Unit: kg m'^) 159
6.4 Average thickness and upper elevation of lutocline layer at different times
(Unit: m) 168
E. 1 Basic parameters in deposition experiments using the bay mud 208
xviii
LIST OF SYMBOLS
a Sediment-dependent empirical coefficient in Eq. (3.30)
a, Empirical constant in Eq. (3.33)
Empirical constant in Eq. (3.34)
^i±\i2j Coefficients depending on the time step in Eq. (B.4)
Aq Amplitude of the forcing tide at the open boundary (m)
Coefficient in Eq. (4. 1 9)
A^ Sediment-dependent constant in Eq. (3.26)
A ^ Measuredscourarea(m^)
A^ Horizontal turbulent momentum diffusion coefficient in the direction normal to the
shore boundary (m^ s'*)
A^ Simulated scour area ( m ^ )
A^ V ertical turbulent momentum diffusion coefficient ( m ^ s'*)
A^g Vertical momentum diffusion coefficient in homogenous flow (m ^ s '*)
A^^ Background value of the turbulent diffusion coefficient of momentum (m ^ s'*)
XIX
Horizontal turbulent momentum diffusion coefficient in the x-direction ( m ^ s’)
Ay Horizontal turbulent momentum diffusion coefficient in the >>-direction (m ^ s')
b Sediment-dependent empirical coefficient in Eq. (3.30)
Empirical constant in Eq. (3.33)
Empirical constant in Eq. (3.34)
b.j Coefficients depending on the time step in Eq. (B.4)
^ij±m Coefficients depending on the time step in Eq. (B.4)
B Sum of the discretized Coriolis, baroclinic, horizontal diffusion and Bingham yield
strength terms
Coefficient in Eq. (4. 1 9)
B^ Coefficient in Eq. (4.20)
B Mean width of the estuary (m)
c Suspended sediment concentration or SSC (kg m '^)
Cq Prescribed boundary condition
Cj Maximum SSC for free settling (kg m ‘^)
C2 Maximum SSC for flocculation settling (kg m “^)
C3 Maximum SSC for hindered settling (kg m '^)
XX
Near-bed SSC (kg m
Cg Prescribed SSC at open boundary (kg m )
Cj Initial concentration of deposited sediment (kg m
c. SSC from outside the modeled domain (kg m
Maximum sediment concentration corresponding to grain-grain contact
SSC of water particle at position p {kg m )
c^j Concentration corresponding to the maximum settling flux (kg m
c^2 Saturation concentration or the maximum compaction concentration (kg m “^ )
c, Transition concentration (kg m’^)
^vmcxx Maximum volumetric floe content corresponding to floc-floc contact
Linear sediment concentration
c ' Turbulent fluctuation of SSC ( kg m )
C Vertical mean SSC (kg m
Cq Initial vertical mean SSC (kg m “^)
C, Depth-mean sediment concentration in the upper, mixed, layer of height
(kg m“^)
XXI
Cj Depth-mean sediment eoneentration in the lower, fluid mud, layer of height
(kg m'^)
Wave speed ( m s'*)
Bottom drag coefficient
Vertically averaged steady state SSC (kg m '^)
Coefficient dependent on the granular density in Eqs. (3.26) and (3.32)
Vertical mean SSC of n‘^ sediment class (kg m '^)
Lateral friction coefficient
Water surface drag coefficient
C Bed average dry density or concentration (kg m '^ )
C * Fraction of depositable concentration
d Grain or floe size of sediment (m)
Sediment-dependent coefficient in Eq. (5.24)
d^ Sediment-dependent coefficient in Eq. (5.25)
d^ Grain size of n sediment class (m)
dx Total back-tracing distance in the x-direction (m)
dx"‘ Back-tracing distance in x direction at time step m (m)
xxii
dy Total back-tracing distance in the jj^-direction (m)
dy"' Back-tracing distance in the >^-direction at time step m (m)
dz Height between the in situ measured layers of SSC (m)
do Total back-tracing distance in the o-direction
do"" Back-tracing distance in the o-direction at time step m
D Sum of the discretized vertical settling and horizontal diffusion terms
Sediment-dependent constant in Eq. (3.26)
D/Dt Total derivative with respect to time
erfc Complementary error function
f Coriolis parameter (s’')
F Temperature function of flocculation
ASSM signal reading
F^ Flocculation factor
F^ Settling flux (kg m s'*)
F^^ Maximum settling flux (kg m s'*)
F^ Turbulence energy production (m ^ s '^ )
Fj Transition function
g Gravitational acceleration (m s '^)
XXlll
Phase lag of the n tidal harmonic component
g' Reduced Gravity (m s
G Any physical property
Any physical property of water particle at position p
h Undisturbed water depth (m)
Hq Maximum water depth at the channel center (m)
Sediment-dependent coefficient in Eq. (4.1) (m)
Bed thickness (m)
Fluid mud layer thickness (m)
^mix Mixed layer thickness (m)
h' Depth below the acoustic probe (m)
h Mean water depth of the flow cross-section (m)
H Total water depth (m)
Hq Initial thickness of the consolidating layer (m)
//, Water depth of the upper layer in the two-layer flow system (m)
H2 Water depth of the lower layer in the two-layer flow system (m)
Internal wave profile (m)
^an ” wave height (m)
XXIV
Effective water depth defined as the thickness affected by internal waves (m)
Amplitude of the n harmonic component (m)
rms wave height (m)
H' Thickness of the consolidating layer (m)
I Number index of the mesh cell centers in the x-direction
j Number index of the mesh cell centers in the y-direction
k Number index of the mesh cell centers in the o or o 'direction (subscript), and wave
number
Deposition rate constant in Eq. (E.l)
Sediment-dependent coefficient in Eq. (4.1)
k^ Proportional coefficient between fluid density and salinity
Kq Constant mass diffusion coefficient (m ^ s'')
Vertical turbulent mass diffusion coefficient (m ^ s'')
Vertical mass diffusion coefficient in homogenous flow (m^ s'')
Background value of the turbulent mass diffusion coefficient (m ^ s'')
Horizontal turbulent mass diffusion coefficient in the x-direction ( m ^ s'')
Ky Horizontal turbulent mass diffusion coefficient in the y-direction ( m ^ s'')
XXV
I Length of a basin (m)
Mass mixing length (m)
Mass mixing length in a homogeneous, non-cohesive flow (m)
Momentum mixing length (m)
Momentum mixing length in a homogeneous, non-cohesive flow (m)
Length of a rectangular domain (m)
ly Width of a rectangular domain (m)
Height of a cubic domain (m)
Sublayers of the water column in the hydrodynamic and sediment transport model
Sublayers of the consolidating bottom layer in the consolidation model
Lutocline strength index
Monin-Obukhov length scale (m)
m Modeling step in the consolidation model and back-tracing calculation (superscript)
and summation variable
Rate of SSC deposition (kg m s '*)
Rate of bottom sediment erosion (kg m s')
m Rate of interfacial entrainment (m s')
en ^ '
XXVI
Sediment-dependent constant in Eq. (3.23)
M Number of mesh grids in the >^-direction
Vertical flux of buoyancy (kg m s’')
Classes of sediment
Maximum erosion rate constant at (kg N ’’ s'*)
Mp Working vector at the step of water surface elevation iteration according to Eq.
(B.5)
Vertical mass flux (kg m s'*)
n Modeling step in the hydrodynamic and sediment transport model (superscript), and
summation variable
Manning’s bed resistance coefficient
Direction normal to the lateral solid boundary
Sediment-dependent constant in Eq. (3.23)
N Number of mesh grids in the x-direction
Total number of waves over a period
Consolidation step
Subdivided back-tracing step
Np^ Peclet number
XXVll
N, Total number of tidal harmonic components considered
o Position of water particle at the current time step n+\ (subscript)
p Position of water particle at the previous time step n (subscript), and water pressure
(Pa)
Pj Probability of sediment deposition
Working vector at the k‘^ step of water surface elevation iteration according to Eq.
(B.5)
p^ Pore water pressure (Pa)
q Deposition flux minus erosion flux at the bed-fluid interface (kg m s'*)
q Constant source-sink term
Working vector at the k‘^ step of water surface elevation iteration according to Eq.
(B.5)
R Auto-correlation function for internal waves
Rq Rossby number
Rj. River inflow rate ( m ^ s'*)
Ra Estuarine Rayleigh number
Ri Richardson number
RIq Global Richardson number
Rig Ratio of the Bingham yield strength to the Reynolds stress
xxviii
Ri^ Stream Richardson number
Ri^ Ratio of the viscous force due to interactions between floes in the fluid mud layer
to the Reynolds stress
Ri^ Ratio of potential energy of sediment settling flux to the production of turbulent
energy
5 Salinity (%o)
S Spectral density of internal waves (m^ s)
Sq Salinity at the estuarine mouth (%o)
t Time (s)
/jQ Time corresponding to c * =50% (s)
t' New time after o-transformation (s)
t * Non-dimensional time
T Period of tide (hr)
Duration of each segment of ASSM data (s)
Period over which is calculated (s)
Elapsed time
T Integration time-limit (s)
u Horizontal instantaneous velocity in the x-direction (m s “* )
XXIX
Gravitational circulation (m s ’* )
Maximum velocity in the x-direction within the modeled domain (m s'')
Up Velocity u of water particle at position p (m s'*)
u ' Turbulent fluctuation of horizontal velocity in the x-direction (m s'’)
M. Bottom frictional velocity (m s '')
Bottom frictional velocity in homogenous, non-cohesive flow (m s')
U Vertical mean velocity in the x-direction (m s *' )
f/j Depth-mean flow velocity in the upper, mixed, layer of height (m s')
U2 Depth-mean flow velocity in the lower, fluid mud, layer of height (m s')
Uj- Vertical mean inflow velocity (m s’')
Slip velocity near the bottom (m s"')
V Horizontal instantaneous velocity in the y-direction (m s')
Maximum velocity in the y-direction within the modeled domain (m s’')
Vp Velocity v of water particle at position p{m s'')
F, Velocity vector of the modeled layer closest to the bottom (m s'')
V Vector of the normal velocity at the lateral solid boundary (m s'')
XXX
Tangential velocity at the modeled grid closest to the shore boimdary (ms * )
w Vertical instantaneous velocity in the z-coordinate (m s'*)
w' Turbulent fluctuation of vertical velocity in the z-coordinate (m s'*)
W Wind velocity vector at a reference elevation (10 m above the water surface in the
prototype case) (m s'*)
X Longitudinal Cartesian coordinate located at the mean sea level (m)
Coordinate normal to the shore boundary (m)
x' New longitudinal Cartesian coordinate after o transformation (m)
y * Frictional Reynolds number
y Transverse Cartesian coordinate located at the mean sea level (m)
y' New transverse Cartesian coordinate after o-transformation (m)
z Vertical Cartesian coordinate originating from the mean sea level and positive
upward (m)
Zq Effective roughness of the bed (m)
Zj Elevation above the bottom of the modeled layer closest to the bottom (m)
z^ Elevation above bottom (m)
Zj Lower elevation of the lutocline layer (m)
z^ Upper elevation of the lutocline layer (m)
XXXI
Measured upper elevation of the lutocline layer (m)
Simulated upper elevation of the lutocline layer (m)
z' Vertical coordinate of the consolidating layer originating from the bottom and
positive upward (m)
a Sediment-dependent empirical coefficient in Eq. (3.30)
a, Sediment-dependent coefficient in Eq. (3.25)
«2 Bed-dependent coefficient in Eq. (3.24)
Sediment-dependent coefficient in Eq. (4. 1 )
ttg Sediment-dependent coefficient in Eq. (5.13)
Constant in the relationship of salinity distribution (6.5)
Coefficient for the k'^ step of water surface elevation iteration according to Eq. (B.5)
The dimensionless variable dependent on SSC in Eq. (5.1 1)
Empirical coefficient in Eq. (3.38)
Linear heat transfer coefficient
Empirical constant in Eq. (3 . 1 0)
A Sediment and elevation dependent constant in Eq. (5.20)
P Sediment-dependent empirical coefficient in Eq. (3.30)
Pi Sediment-dependent coefficient in Eq. (3.25)
xxxii
Bed-dependent coefficient in Eq. (3.24)
P2
P.
Ps
P.
pw
P™
P„
P.
Yi
Y2
6
/
6
Im
6
Is
6
V
aH
aH
max
aH'
Sediment-dependent coefficient in Eq. (4.1)
Sediment-dependent coefficient in Eq. (5.13)
Sediment-dependent constant in Eq. (E.3)
Coefficient at the k‘^ step of water surface elevation iteration according to Eq. (B.5)
Turbulent Schmidt number
Positive roots of Pcotp+a^/=0
Empirical coefficient in Eq. (3.38)
Empirical constant in Eq. (3.33)
Empirical constant in Eq. (3.34)
Lutocline layer thickness (m)
Measured lutocline layer thickness (m)
Simulated lutocline layer thickness (m)
Thickness of current shear layer (m)
Thickness of sedimentation (m)
Maximum scour depth (m)
Thickness of deposited sediment over time step At (m)
xxxiii
At Time step (s)
At' Consolidation time step (s)
At" Back-tracing time interval (s)
aT Time period for the scour hole to be stable (s)
AX Horizontal step length in the ^-direction (m)
Ay Horizontal step length in the >’-direction (m)
Az Incremental depth downward from the bed surface (m)
AO Vertical step length in the o-coordinate
AO' Vertical step length in the o '-coordinate
A Sediment-dependent coefficient in Eq. (3.29)
e Roughness parameter
C Instantaneous water surface elevation (m)
Co Mean water level (m)
C^ Lutocline elevation above the bottom (m)
Vector of water surface elevation at the k‘^ step of iteration according to Eq. (B.5)
C, Lutocline elevation above the bottom detected by the turbidimeter (m)
Z Vertical distribution function of salinity
0 Temperature (°C)
0 Absolute temperature (°K)
K von Karman constant
X Sediment-dependent coefficient in Eq. (3.29)
XXXIV
Wave length (m)
A Sediment-dependent coeffieient in Eq. (3.29)
p Dynamic viscosity of the fluid mud (m ^ s'')
V Fluid kinematic viscosity (m^ s'')
^ Integration variable in Eq. (3.74) (s)
p Fluid density (kg m '^ )
Pq Water density (kg m'^)
p j Fluid density of the upper layer in the two-layer flow system (kg m '^ )
P2 Fluid density of the lower layer in the two-layer flow system (kg m '^)
Air density (kg m'^)
Dry density of the bottom sediment (kg m '^)
Py Bulk density of floes (kg m '^ )
p^ Sediment granular density (kg m '^)
p^ Fluid density at the water surface (kg m '^ )
p' Turbulent fluctuation of density (kg m'^)
p Vertical mean fluid density (kg m '^ )
o Normalized vertical coordinate in the water column
XXXV
Standard deviation of SSC
o' Normalized vertical coordinate of the consolidating layer
X Shifting time and integration variable in Eq. (4.2) (s)
x^ Bottom shear stress (Pa)
Vector of bottom shear stress (Pa)
xl Bottom shear stress in the x-direction (Pa)
xl Bottom shear stress in the y-direction (Pa)
Bottom shear stress at the point where the maximum scour appeared (Pa)
Xg Bingham yield strength (Pa)
Critical shear stress for deposition (Pa)
x^^ Minimum critical shear stress for deposition of the sediment classes (Pa)
Xj^ Maximum critical shear stress for deposition of the sediment classes (Pa)
Reynolds stress (Pa)
Bed shear strength for erosion (Pa)
Shear strength of newly deposited bottom sediment (Pa)
T^' Critical shear stress at the point of maximum scour (Pa)
x^ Total (normal) stress (Pa)
XXXVl
Vector of wind-induced water surface stress (Pa)
Wind-induced water surface stress in the x-direction (Pa)
Wind-induced water surface stress in the >^-direction (Pa)
Shear stress near the bottom (Pa)
Shear stress due to the interactions between floes (Pa)
Shear stress due to cohesion and interactions between floes (Pa)
Effective (normal) stress (Pa)
Dimensionless wind stress
Solid weight fraction
Critical solid weight fraction
Dynamic angle of repose (°)
Frequency distribution of sediment
Latitude (°)
Sediment-dependent coefficient in Eq. (3.29)
Dimensionless horizontal coordinate
Stratification function
Vertical instantaneous velocity in the o-coordinate (s ’' )
Angular frequency of the forcing tide at the open boundary (rad s'*)
xxxvii
(Oq^ Settling velocity of the n sediment class (m s'*)
0)^ Angular frequency of the internal waves (rad s'*)
(0^ Angular frequency of the n"' tidal harmonic component (rad s’*)
Sediment settling velocity (m s’*)
Flocculation settling velocity at 15 °C
o)^j Minimum flocculation settling velocity of the sediment classes (m s’*)
0)^^ Near-bed settling velocity (m s’*)
0)^^ Rate of consolidation (m s’*)
0)^^ Free settling velocity (m s’*)
Rate of consolidation for the first mode (m s’*)
o)^^2 of consolidation for the second mode (m s’*)
^smax Maximum flocculation settling velocity (m s ’*)
Maximum flocculation settling velocity of the sediment classes (m s’*)
0)^^ Flocculation settling velocity of the fr* sediment class (m s’*)
0)^ Brunt-Vaisala frequency (rad s’*)
0)' Angular frequency (rad s’*)
XXXVlll
Q Angular frequency of earth’s rotation (rad s ’* )
e Allowed error in the iteration of water surface elevation
xxxix
Abstract of Dissertation Present to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
AN EXAMINATION OF ESTUARINE LUTOCLINE DYNAMICS
By
JIANHUA JIANG
August, 1999
Chairman: Dr. Ashish J. Mehta
Major Department: Coastal and Oceanographic Engineering
Three dynamical features associated with estuarine lutoclines are examined with
special reference to data from the Jiaojiang estuary in China. These features include
turbulence damping induced by high suspended sediment concentration, internal wave
behavior at the lutocline, and the response of the lutocline to tidal forcing. It is shown that
turbulence damping is governed by the settling flux, cohesion, interaction between floes and
sediment-induced stratification. Maximum turbulence damping in the water column occurs
at the lutocline, a finding which supports previous, qualitative observations of a similar
nature. Expressions for the vertical momentum and mass diffusion coefficients incorporating
these effects have been developed.
Observations from the Jiaojiang are examined for the height, angular frequeney,
celerity and length of internal waves at the lutocline. Both deep water high and shallow water
xl
low frequency waves are identified. The low frequency, at 0.09 rad s , is shown to be close
to the local Brunt-Vaisala frequency. The high frequency wave at 1.33 rad s is possibly
induced by interfacial shear, and is characterized by sharp crests and flat troughs. The height
and the angular frequency of both wave types are shown to decrease with increasing
Richardson number.
Lutocline variation with tide in the Jiaojiang is examined by applications of
three-dimensional, finite-difference codes developed for flow and sediment transport. It is
shown that lutocline responses reflect the cumulative effects of sediment settling and
entrainment, turbulence damping and tidal asymmetry.
xli
CHAPTER 1
INTRODUCTION
1.1 Problem Statement
A lutocline is a step structure in the vertical profile of fine-grained suspended
sediment concentration (SSC), and is a critically important pycnocline in addition to salinity
and temperature gradients governing estuarine dynamics (Parker and Kirby, 1979). From
experiments in the field and the laboratory in recent years, the lutocline’s significance in
governing the vertical mixing of suspended sediment and sedimentation patterns has been
assessed (e.g., Kirby, 1986; Wolanski, et al., 1989; Mehta and Srinivas, 1993; Winterwerp
and Kranenburg, 1997). As a result, it is now recognized that lutocline formation and
strength are controlled by numerous factors including tidal currents, interfacial waves formed
at the lutocline, turbulence damping in the fluid mud layer beneath the lutocline, floe settling,
suspended sediment deposition, bottom erosion, and consolidation of deposits.
Unfortunately, the inter-linkage among these unit processes has yet to be quantified on
theoretical grounds for an accurate prediction of lutocline dynamics. There is a critical need
for such a quantification, due to the importance of predicting fine-grained sediment transport
in estuarine engineering applications.
Turbid water is a characteristic feature of estuarine zones with high-load fine
sediments, especially around turbidity-maxima, sedimentary fronts, and dredging disposal
1
2
sites. In such cases, the spatial distribution of SSC and the horizontal transport of sediment
are significantly associated with the high concentration distribution near the bottom, i.e., the
lutocline and fluid mud. Lutoclines frequently can cause significant problems due to
associated sedimentation in engineering projects, especially where the natural
environment is dramatically altered by dredging operations or erection of structures.
Prediction of lutocline dynamics is presently hindered by a lack of adequate
understanding of lutocline-associated processes, namely the influence of SSC on turbulence
damping, hence diffusion, and the response of the lutocline interface to shear flow that leads
to interfacial waves and vertical mixing.
A high degree of turbulence damping is known to occur in the fluid mud layer
(Wolanski, et, al., 1992; Kranenburg and Winterwerp, 1997). Hence, vertical diffusion and
mixing at the lutocline are influenced not only by the sediment-induced buoyancy force, but
also by turbulence damping. It has also been shown that the characteristic, upward-
asymmetric, sediment entrainment caused by the instability and breaking of internal waves
at the lutocline occurs concurrently with high turbulence damping (Jiang and Wolanski,
1 998). However, there remains a lack of detailed theoretical analyses and direct evidence of
turbulence damping because of the difficulty in observing turbulent damping both in the field
and in the laboratory.
Substantial efforts have been devoted to understanding lutocline response to tidal
forcing (e.g., Wolanski, et al., 1988; Smith and Kirby, 1989; Costa and Mehta, 1990; Dong,
et al., 1997). It is found that the elevation and strength (i.e., the steepness and size of the SSC
3
step structure) of the lutocline vary with the tidal cycle under the combined effects of the
strongly current-dependent vertical and horizontal sediment transport processes. However,
a comprehensive quantification of lutocline response to tide is still difficult because of
limited field data and process-based formulations for the interactions between SSC and flow
as noted. This difficulty in quantification and the consequent need to examine SSC-flow
interaction was the main motivation behind the present study.
1.2 Objectives
In accordance with the above discussion, the objectives of the investigation were set
as follows:
1. To examine the effects of high SSC on the turbulent mixing length over the
estuarine water column, and develop an expression for the vertical diffusion coefficient
accounting for these effects.
2. To investigate the behavior of the interfacial waves at the lutocline, as a basis for
an improved understanding of the vertical mixing processes at the interface.
3. To develop a numerical code for estuarine flow and fine sediment transport,
incorporating the latest unit proeess models for sediment transport including erosion/
entrainment, diffusion, settling, deposition and consolidation.
4. To test the model against laboratory data and analytical solutions, and apply the
model to extensive field data obtained at Jiaojiang estuary in China, as a means to comment
on the utility of the model.
4
1.3 Tasks
With respect to the above objectives, the specific tasks established to conduct the
study consist of the following:
1. Review of Previous Studies: Previous studies on sedimentary process in fine
sediment dominated estuarine environments are reviewed, thereby providing the groundwork
for the subsequent tasks.
2. Theoretical Analyses: A analytical model for vertical diffusion in the water column
is developed and incorporated in the numerical model for the flow and sediment transport
developed to examine tide-induced lutocline dynamics (see below).
3. Flow and Sediment Transport Model: A three-dimensional, finite-difference
scheme numerical model code is developed. This code consists of two major parts, namely,
a flow model (called Coastal and Estuarine Hydrodynamic model - University of Florida or
COHYD-UF) and a fine-grained sediment transport model (called Coastal and Estuarine
Cohesive Sediment model - University of Florida or COSED-UF). The model system focuses
on fine sediment transport processes and flow-sediment coupling due to sediment-induced
stratification and turbulence damping in the fluid mud layer.
4. Model Parameters: All parameters defining the flow and sediment transport
processes required for COHYD-UF and COSED-UF are determined from works of previous
researchers, model calibrations as well as prototype data analysis.
5. Modeling Tests: For validation, COHYD-UF and COSED-UF are tested against
some special forms of the governing equations having exact or approximate solutions, as
• well as laboratory and field data. The sub-models for consolidation of deposits and interfacial
5
entrainment are tested against experimental data of previous researchers. Further tested are
the ability of the developed model in predicting local scour using data at a pier in a river
(Appendix D), and sedimentation behind a piled structure using flume test data (Appendix
E).
6. Field Data Analysis: Through analysis of field data from the Jiaojiang estuary, the
following tasks are carried out: (1) Examination of the behavior of internal waves at the
lutocline in conjunction with previous analyses; and (2) Examination of lutocline and fluid
mud response to tidal forcing.
1.4 Thesis Outline
This study is presented in the following order. Chapter 2 introduces the vertical
structure of fine sediment suspension, definitions of lutocline and fluid mud layers, causes
of fluid mud and lutocline generation, influence of lutocline on turbidity transport, and a
review of turbidity dynamics in several estuaries.
Chapter 3 describes model formulations including the governing equations and the
boundary conditions for flow and sediment transport, reviews of previous studies on flow-
sediment transport processes, and the respective mathematical formulas. Lastly, the results
of model tests are reported for hydrodynamics and sediment transport.
Analysis of field data and the results for the behavior of internal waves, tides, SSC,
settling velocity and erosion rate in the Jiaojiang are covered in Chapter 4. Also introduced
in this chapter are previous models for the behavior of internal wave height, angular
frequency, celerity and wave length at haloclines and thermoclines. These models are used
to explain the behavior of internal waves detected in the Jiaojiang.
6
Chapter 5 presents an analytieal model for the turbulent mixing length and the
associated diffusion coefficient. Data from the Jiaojiang are used to examine the validity of
the model.
Chapter 6 summarizes the hydrodynamic and sedimentary formulations and
parameters for numerical model application to the Jiaojiang. Simulations of lutocline and
fluid mud dynamics in the Jiaojiang and their comparisons with data are then presented.
Based on above results, major conclusions derived from this study are summarized
in Chapter 7.
Appendix A provides derivations of the governing equations for hydrodynamics and
sediment transport with respect to o-transformation.
Appendix B describes the numerical technique, including the back-tracing approach
used in the discretization of the inertial terms in hydrodynamic and sediment transport
equations, and the pre-conditioned conjugate gradient method used in solving the finite
differential equation for the water surface elevation.
Appendix C presents the analyzed results for the effects of temperature on cohesive
sediment settling using the experimental data of previous researchers.
Appendix D presents the simulated results of contraction scour at the Haldia Pier, in
the Hooghly River in India using COHYD-UF, and a comparison with observations in the
field.
Appendix E presents the simulated results of sediment deposition pattern behind a
piled barrier carried out in a flume test and comparison with experimental observations.
CHAPTER 2
LUTOCLINES IN ESTUARIES
2.1 Vertical Structure of Fine Sediment Suspension and the Lutocline
A typical instantaneous sediment concentration profile and associated velocity profile
is schematically shown in Figure 2.1, as might be observed in a macro- or meso-tidal
estuarine environment with high fine-grained sediment loads (Mehta, 1989; 1991a). A
noteworthy characteristic of the vertical profile of the suspended sediment concentration
(SSC) is the multiple step structures identified as secondary lutoclines and the primary
lutocline. A secondary lutocline (Mehta and Li, 1997) has a relatively low SSC, generally
~1.0 kg m'^, and is induced by the coupling between concentration dependent settling
velocity (i.e., settling velocity increasing with SSC as the floes become stronger, larger and
denser in the flocculation settling mode; see Figure 2.2), and the SSC gradient dependent
diffusion (i.e., the upward mass diffusion retarded by this gradient due to negative
buoyancy). The primary lutocline is a significant pycnocline which occurs near the bottom.
It is characterized by a high value of SSC, generally > -5-10 kg m and a significant
vertical gradient of SSC. It is formed due to a high sediment settling flux above it and high
turbulence damping as well as hindered settling; see Figure 2.2 below (Ross and Mehta,
1989; Mehta and Li, 1997).
7
8
Figure 2.1. Vertical SSC profile classification and associated
velocity profile. Also shown are unit transport processes which
govern concentration profile dynamics (after Mehta, 1989; 1991a).
9
As shown in Figure 2.1, the primary lutocline tends to separate the water column into
two reasonably well-defined zones. The near-bottom, non-Newtonian flow zone is
characterized by high SSC, generally > ~5-10 kg m'^, hindered settling, turbulence
damping, turbidity underflow, a sediment-modified velocity profile, etc. On the other hand,
the overlying zone, generally having SSC < 1.0 kg m and exceeding 2-3 kg m during
extreme energy events, is marked by free or flocculation settling (Figure 2.2), a higher level
of turbulence and near-Newtonian flow properties.
The typical SSC profile can be sub-divided into a layer of mobile suspension, the
lutocline layer, mobile fluid mud, stationary fluid mud, a consolidating mud layer and the
fully consolidated bed. The lutocline layer is characterized by a thickness 6^, between the
upper elevation, Z^, and lower elevation, Z^, of the vertical gradient of SSC (Figure 2.1).
The characteristic elevation of the lutocline above the bottom, C , is within the lutocline
layer. Approximately at elevation Z^, the lutocline layer transitions to layers consisting of
horizontally mobile and horizontally stationary fluid muds (Figure 2.1). Within the stationary
fluid mud layer, the sediment is maintained in suspension by turbulent diffusion. Below this
is the consolidating mud layer.
2.2 Causes of Fluid Mud and Lutocline Generation
As noted, the primary lutocline is characterized by a steep vertical gradient of SSC
and occurs within a thin transition zone (or the lutocline layer) from the upper mobile
suspension to the lower fluid mud (Ross, 1988). Thus, the lutocline occurs concurrently with
10
the fluid mud. The formation of fluid mud and lutocline are dependent on complex erosion,
deposition and mixing-settling processes, high turbulence damping due to settling flux
(Einstein and Chien, 1955), cohesion, interactions between floes (Bagnold, 1954 and 1956),
and the buoyancy effect due to sediment-induced stratification (Ross and Mehta, 1989). A
general description of the settling behavior as well as causes of fluid mud and lutocline
generation are given in the following sections.
Figure 2.2. A representative description of settling velocity
and flux variation with SSC (after Mehta and Li, 1997).
11
2.2.1 Settling Velocity
In water with salinity >0.1-0.5%o, the settling velocity, of cohesive sediment is
strongly dependent on SSC (=c), and can be divided into three sub-ranges in terms of
concentration. These ranges are free settling (c<c, ), flocculation settling (c,<c<C2) and
hindered-settling (OCj) with the maximum SSC for free settling c, -0.1 -0.3 kg m the
maximum SSC for flocculation settling Cj^S-lO kg m (Mehta and Li, 1997) and the
maximum SSC for hindered settling Cj =~80 kg m (Odd and Rodger, 1986). Free settling
occurs at low SSC when the settling velocity is independent of concentration. In the
flocculation range, the settling velocity increases with concentration due to the formation of
stronger, denser and larger floes with increasing concentration. In the hindered range, the
occurrence of an aggregated particulate network inhibits the upward escape of the interstitial
water present within the deposit. As a result, the settling velocity decreases with increasing
SSC. Figure 2.2 gives a representative description of the settling velocity and the settling flux
(=o)^c), variation with SSC, where a, b, a and P are sediment-dependent empirical
coefficients in the combined relationship between the settling velocity and SSC for the
flocculation and the hindered settling regions, is the free settling velocity, is the
maximum flocculation settling velocity, and is the maximum settling flux.
12
2.2.2 Formation of Fluid Mud
Fluid mud can form during rapid erosion or deposition. During erosion, if the erosion
rate greatly exceeds the net upward flux due to sediment diffusion and settling, i.e., the rate
at which sediment is mixed by turbulence into the upper column mobile suspension layer,
the SSC near the bottom continues to increase and tends to lead to the formation of fluid mud
(Maa and Mehta, 1987). During deposition, if the sediment deposition flux exceeds the rate
of upward transport of the pore fluid (i.e., the dewatering rate of the suspension), a dense
near-bed suspension is formed, grows upward and consolidates slowly (Ross, 1988). Once
a fluid mud layer forms, it is stabilized by buoyancy and may be enhanced by high turbulence
damping and hindered settling as well as further erosion or deposition (Maa and Mehta,
1987; Ross and Mehta, 1989).
2.2.3 Formation of Lutocline
Based solely on the concept of density stratification, the lutocline behaves, at least
qualitatively, like a halocline or a thermocline. Both the lutocline and other pycnoclines are
characterized by the coincidence of a vertical maximum in the concentration gradient and
minimum vertical mixing (Ross and Mehta, 1989). The local minimum in mixing due to the
buoyancy effect is associated with the concentration gradient. However, there are two
significant differences between a lutocline and other pycnoclines as follows:
(1) Sediment settling occurs with respect to the fluid. This has the effect of further
enhancing the stability of the lutocline for a given set of flow conditions (Ross and Mehta,
1989). Based on the non-linear variation of the settling velocity with concentration (Figure
2.2), at the maximum settling flux the gradient ofdF ^dc must be equal to zero, i.e..
13
dc dc/dz
(2.1)
where z is the vertical coordinate. Note that the vertical gradient of settling flux is not equal
to zero, i.e., dF^dz*0, due to the high gradient of SSC at the lutocline. Thus, only as
dc/dz^°°, Eq. (2.1) is satisfied. This analysis supports the observation that discontinuities
in the concentration profile are generated at elevations where the maximum settling flux
occurs.
(2) High turbulence damping occurs in the fluid mud layer due to the settling flux,
cohesion and interactions between floes as well as the sediment-stratified buoyancy effect.
Turbulence damping decreases vertical mixing and consequently enhances the lutocline
(discussed in details in Chapter 5).
From the above it can be expected that the lutocline would be much more persistent
in the macro-tidal environments than the halocline and the thermocline (Ross and Mehta,
1989). This inference has been confirmed from field data taken in the Severn estuary, U.K.
(Kirby, 1986), with a spring tide range of ~15 m, and in the Jiaojiang (Dong, et al., 1993,
Guan, et al., 1998 and Jiang and Wolanski, 1998), with a spring tide range of 6.3 m. At these
sites, lutoclines have been observed to persist through most of the tidal cycle, even during
spring tides.
14
2.3 Influence of Lutocline on Turbidity Transport
In a stratified flow, vertical mixing is damped due to the buoyancy force because
work must be done to raise heavy fluid parcels and lower lighter parcels. As noted, in the
case of the lutocline the mixing process is further complicated by sediment settling, cohesion
(Hopfinger and Linden, 1982; Ross and Mehta, 1989) and interactions between floes
(Bagnold, 1954; 1956). Thus, additional work is required to keep the sediment in suspension
and overcome cohesion and interactions between floes.
Without explicitly considering settling, cohesion and interactions between floes, the
mixing process over the lutocline can be simply demonstrated with reference to a sediment-
induced two-layer flow with initial densities Pj and Pj, and identical depths //, =H^=H/2\
this system representing a fluid mud layer beneath clear water moving parallel to the jc-
direction with velocities t/, and U^, respectively (Figure 2.3). After some time complete
mixing is assumed to take place, and the system now consists of single layer of mean density,
p =(p, +P2)/2 , flowing with mean velocity, U. Because initially the heavier fluid (of density
Pj) lies below the lighter fluid (of density Pj ), the center of gravity of the system is below
mid-depth level, whereas in the final state it is exactly at mid-depth. Thus, the center of
gravity is raised in the mixing process, for which a potential energy PE must be provided
to the system according to
0
0
(2.2)
15
where g is the gravitational acceleration. The source of potential energy increase, PE, must
be from the externally imposed kinetic energy so long as the initial velocity distribution is
uneven. Conservation of linear momentum in the absence of the effects of settling, cohesion
and interactions between floes implies that the final, uniform velocity is the mean of the
initial velocities of each layer, i.e., U=(U^ leading to a kinetic energy loss, KE,
given by
KE={
flp,uidz*fjp,ul
dz
H,
H
I Ip U^dz~j-p(_U,-U,fH (2.3)
8
Here, the Boussinesq approximation, p,=p2=p, has been invoked (Lesieur, 1997).
Complete vertical mixing is possible as long as the kinetic energy loss exceeds the
potential energy gain, i.e.,
(P2-Pi)g//^^
p{U,-U,f
(2.4)
Therefore, the initial density difference must be sufficiently weak in order not to present an
insurmountable gravitational barrier, or alternatively, the initial velocity-shear induced at the
interface must be sufficiently large to supply the necessary amount of energy. When criterion
(2.4) is not met, mixing occurs only in the vicinity of the initial interface and cannot extend
over the entire system. This situation can result in interfacial instabilities (Delisi and Corcos,
1973). This phenomenon will be further discussed in Section 4.4.2.
16
Vertical mixing is also highly dependent on the nature and intensity of turbulence in
both layers. As stated in Section 1.1, an upward-asymmetric mixing caused by the instability
and breaking of internal waves at the lutocline occurs concurrently with high turbulence
damping in the fluid mud layer.
Figure 2.3. Mixing of a two-layer stratified
Final state (after complete mixing)
id, with fluid mud beneath clear water.
2.4 Dynamics of Turbid Estuaries
Over the past decades there has been an increasing number of studies on the dynamics
of lutocline and fluid mud through combined physical and mathematical approaches and in
situ measurements of sediment movement. This has been made possible by improved
instrumentation such as the optical backscatter sensors, e.g., turbidimeters and nephelometers
(Wolanski, et al., 1988), and acoustic backscatter sensors, e.g., the recently developed
Acoustic Suspended Sediment Monitors or ASSM (Shi, 1997). Improvements have also
occurred in designing advanced methods of recording large quantities of data and computer
17
analysis techniques. The following sections hriefly describe the works of previous
researchers in some of the turbid estuaries of the world.
2.4.1 Amazon Shelf
The Amazon continental shelf in Brazil is the offshore region extending from the
mouth of the Amazon River. This river is one of the world’s largest with the highest fresh
water outflow; the annual mean rate of about 210,000 m ^ s'* is 1 8% of the global riverine
discharge. The Amazon also has one of the largest fluvial sediment discharge of about
1.2x10^ tons yr '* , or 10% of the global sediment output to the sea (Kineke, 1993). The
shelf region can be considered as an estuary because the fresh water discharge is so large that
low salinities (e.g., 20%o) occur 100 km from the river mouth and 100 km offshore.
Measurements of SSC and sediment transport were made over the middle and inner
Amazon shelf region during four oceanographic cruises between August 1 989 and November
1990 using OBS sensors (range 0-36 kg m CTD probes, current meters and a data logger
mounted on a tripod frame (AmasSeds Research Group, 1990). Silt- and clay-size sediments
were found to dominate the bottom. A major characteristic of the suspended sediment was
the vast extent of high-concentration near-bed suspension (or fluid mud), which accounted
for most of the suspended sediment inventory on the shelf. Fluid muds generally were
observed in the region of the turbidity maximum. Fluid mud of up to 7 m in thickness on the
inner and middle shelf was found to cover an extensive region, ranging from 5,700 to 10,000
km^, in water depths as great as 40 m. Observed SSC near the surface of the shelf ranged
from 0.01 to >0.54 kg m SSC in fluid mud layer ranged from 10 to 100 kg m with a
18
sharply decrease of SSC above, thus marking a distinct lutocline. Much of the inner shelf
mud deposit was found to accumulate sediment at a rate exceeding 2cm yr .
2.4.2 Ariake Bay
Ariake Bay in southern Japan is dominated by semi-diurnal tides with a spring range
of 3.86 m and a neap range of 1 .54 m. The maximum tidal velocity is in the range of 0.3-0. 5
m s ■* . A great amount of fine sediment is transported from the mountain area to the bay
through two main rivers, which lead to a wide tidal flat near Kumamoto City. The bed
material offshore consists of fine clay and silt, and fine sand and silt occur nearshore.
Extensive field measurements have been carried out at Kumamoto Port in order to obtain
information on the sedimentation mechanism in the port area (Tsuruya, et al., 1990a).
In field surveys, an echo-sounding measuring pole (measuring sedimentation
volume), electro-magnetic current meters and turbidity meters were employed. A noteworthy
engineering project was the selection of a submerged dike for reducing deposition within the
navigation channel. A multi-layered numerical model was used to take into account the
effects of the dike and to reproduce the vertical distribution of suspended mud particles in
the port area (Tsuruya et al., 1990b). It was found that there was a strong correlation between
the wave-induced oscillatory current and turbidity concentration. High concentrations
occurred at low tide when wave heights were large. Maximum concentrations were about 1 .0
kg m . SSC greater than 10.0 kg m was observed near the bottom with a steep vertical
gradient of SSC (or lutocline). It was also found that net bottom erosion took place when the
wave height was large and tidal level low. It was concluded that the bottom sediments around
19
Kumamoto Port were eroded mainly by shear stress induced by wind waves (Tsuruya et al.,
1990a).
2.4.3 Gironde River
The Gironde River estuary in France is one of the largest estuaries of the European
Atlantic coast. It has an annual mean inflow rate on the order of 1 ,000 m ^ s', and the tidal
range varies from 2 to 6 m at the mouth. The range increases towards the upper estuary due
to the convergence effect of the estuarine shape, and tides propagate up to 1 70 km upstream
during low river inflow periods. Turbidity maximum is an important feature of this estuary.
Numerous studies on the dynamics of this turbidity maximum have been made through field
investigations (e.g., Allen, et al., 1976; Castaing and Allen, 1981), as well as depth-averaged
and three-dimensional numerical modeling (Sottolichio, et al., 1999).
In order to examine the resuspension and dispersion of fluid mud in zone of the
turbidity maximum, a tracer study was carried out using sediments labeled with radioactive
Scandium 46. Five kilos of labeled, naturally occurring, fluid mud were injected into a fluid
mud pool in a channel during a neap tide in May, 1974. It was shown that the high turbidity
zone was characterized by SSC ranging between 1 to 10 kg m (Allen et al., 1976). During
slack water periods, and especially at neap tides, suspended sediment settling was enhanced,
and thick patches of fluid mud appeared on the channel bottom with concentrations up to 300
kg m . The estimated total sediment mass contained in the turbidity maximum-fluid mud
system was found to reach up to 5x10^ tons (Castaing and Allen, 1981). It was also
determined that in this estuary, tidal pumping is the major mechanism of the formation of the
20
turbidity maximum (Sottolichio, et al., 1999). Large-scale lateral transport of suspended
sediment occurs within the turbidity maximum, resulting in a general northward drift of the
sediment. This lateral transport occurs by advection, while the rates of lateral sediment
diffusion appear to be low (Allen, et al., 1976).
2.4.4 Hangzhou Bav
Hangzhou Bay on the coast of East China Sea is the outer region of the Qiantang
River estuary with an average inflow of 4.2x10*° m^ yr and an average suspended
sediment load of 7.9x 10° ton yr '* . This bay is a typical funnel-shaped estuary with a length
of about 100 km and an average depth of 10 m. Its width decreases from 90 km at the mouth
to 20 km at its western end. The tidal range is about 3 m at the mouth and increases rapidly
landward. A tidal bore develops about 10 km further upstream of the bay (Su, et al., 1992).
Sediment in the bay is predominantly fine and medium silt with the median size of the
suspended load ranging from 1 0 to 1 3 pm and that of the bed about 1 6 pm (Costa and Mehta,
1990).
Available in situ data are from the following three surveys: (1) observations in the
bay carried out from 13'* to 21'* of December, 1987 and from 22'* of July to 2'* of August,
1988 using Partech 7000-3RP turbidimeter and ENDECO 174 current meter, aimed at
understanding of the plume front and its role in suspended sediment transport (Su, et al.,
1992); (2) Measurements of lutocline and flow-fine sediment hysteresis near the south shore
were conducted between 14'* and 16'* of May, 1988, deploying a pressure gage, two
turbidimeters (Partech SDMI 6) and two electromagnetic current meters mounted on a tower
21
(Costa and Mehta 1990); and (3) Fluid mud and interfacial waves in a proposed navigational
channel observed at a neap tide (October 23, 1993) using a ship-borne ASSM, soon after a
dredging operation (Shi, 1998). Horizontal, two-dimensional, numerical modeling of the
depositional patterns in Hangzhou Bay was carried out by Su and Xu (1984). It was found
that the tidal bore there traps the fine fluvial sediments. A persistent, year-around NE-SW
suspended sediment concentration fi-ont was found to exist inside the bay. This fi-ont acts to
concentrate suspended sediment and transport it southwestward into the bay (Su, et al.,
1992). ASSM data showed that high concentration suspensions (SSC > 10 kg m “^)
appeared close to the mud bed, occupying 30% of the water column. The ASSM also
detected a relatively well defined wave train with a wave period of about 144 s superimposed
by higher frequency oscillations with periods of a few seconds (Shi, 1998). Due to the
presence of the lutocline, reversals of flow-fine sediment hysteresis was observed during the
transition from accelerating to decelerating flows (Costa and Mehta, 1990).
2.4.5 James River
The James River estuary in Virginia discharges into the south end of the Chesapeake
Bay. Extensive surveys of the vertical profiles of SSC, tidal currents and sediment-water
interface were conducted using an optical turbidimeter, an electromagnetic current meter and
a nuclear transmission density probe, all mounted on a tripod frame (Nichols, 1984-1985).
It was shown that the flood peak at 6 cm above the bed reached 0.24 m s , whereas the ebb
reached 0.30 m s Fluid mud accumulated mainly as shallow pools and blanket deposits
greater than 0.2 m in thickness in the channel depressions in the middle reach of the estuary.
22
This occurred mainly in the turbidity maximum zone, a site of high near-bed concentration
(0. 5-2.0 kg m intensive resuspension and rapid sedimentation (10-80 kg m yr
Measurements of mud density and thickness from 85 continuous densitometer profiles
revealed two basic types of profiles: (1) those with an abrupt increase in density with depth
to more than 1,200 kg m in the upper 1-2 cm, i.e., mainly settled mud, and (2) those with
a moderate inerease in density with depth in the density range -1,003-1,200 kg m in the
upper 2-30 cm. It was found that accumulation of fluid mud was promoted by stratification
of the interfacial fluid and pore water, by the pseudoplastic behavior of the mud with
relatively high viscosity at low shear rates, by the high suspended sediment coneentrations,
and by the resultant rapid-settling flux in the hindered state relative to the consolidation rate.
2.4.6 Orinoco River
The Orinoeo River delta is located in a coastal plain extending from the Guyana
shield in the south to the Venezuelian corrdillera in the north. The river discharges through
the delta with an annual mean outflow of about 17,000 m ^ s and total suspended load on
the order of 10* tons yr “‘ (Eisma, et ah, 1978). Bottom deposits sampled in the delta
(Eisma, et al., 1978) were essentially mixtures of silt and elay-size materials, with generally
low sand content (less than 1 0%).
Turbidity was measured using the radiometric method (Goldberg and Bruland, 1974).
A layer of hyper-concentrated suspended material (i.e., fluid mud layer) with concentrations
reaching 500 kg m was located at the bottom of the navigation channel over a distanee
23
of up to 50 km and a thickness on the order of 6 m. This layer was confined to the navigation
channel, while turbidity exceeding 0.7kg m was not detected elsewhere on the deltaic
platform. The upper layer of lower turbidity was separated from the fluid mud by a sharp
discontinuity (lutocline), which caused strong reflections on the echograms, thus resulting
in a so-called “double-bottom” pattern. Within the fluid mud layer there was a significant
vertical gradient of turbidity, which in the deepest parts of the channel tended to reverse close
to the bottom in the lowest 0.5 m of the water column.
2.4.7 Rhine-Meuse River
The Rhine-Meuse River estuary (or Rotterdam Waterway) in the Netherlands is a
partially mixed estuary, through which the Rhine and the Meuse flow into the North Sea with
an annual mean discharge of 1,500 m^ s . It has a deepened navigation channel with a
depth of 25 m. In field investigations of the movement of fluid mud, echo-sounders and a
Partech turbidity meter with a range of 0-12 kg m were employed (Kirby and Parker,
1977). As described by van Leussen and van Velzen (1989), fluid mud layers with
thicknesses of 0.5-4.0 m and SSC < 15 kg m usually appear in this area. A distinct lutocline
was observed around high water slack. Fluid mud layers play a dominant role in sedimentation
in this estuary. They are responsible for high deposition rates in short times under rough
weather conditions. At times sediment deposition of about 2x10^ m ^ has been measured in
one week (Wiersma, 1984).
24
2.4.8 Severn River
The Severn River estuary in the UK is dominated by semi-diurnal macro-tides with
a spring range of 13 m and a neap range of 5 m. Extensive field studies in this estuary were
carried out in the early 1970's (Kirby and Parker, 1982 and Kirby, 1986). In these studies,
optical turbidity meters were used to obtain continuous horizontal and vertical traverses of
SSC in comparatively low concentration (0.1-20.0 kg m'^) areas, and gamma-ray
densimeters for high-resolution vertical profiles of dense stationary suspensions. The
movement of fluid mud was simulated by Odd and Cooper (1989) using a horizontal, two-
dimensional, numerical model. It was found that during spring tides the strong tidal currents,
with a flood peak on the order of 2.3 m s'* and associated turbulence, were sufficiently high
that entrained sediment reached the water surface at peak ebb and peak flood, leading to
“well-mixed”, homogeneous vertical SSC profiles. As the velocity decreased towards slack
water, sediment began to settle whilst the velocity was still high (>1.5 m s'*). A
discontinuity in concentration (i.e., a lutocline) caused by settling formed in the water
column, which then subsided towards the bed. By the time of slack water the layer settled to
a level only 2 or 3 meters above the bed to leave a residual low concentration upper zone and
a high concentration (~ >20.0 kg m '^) lower layer. At slack water the layer stagnated for
a short period before being re-entrained by the subsequent reversal of the tidal current.
2.4.9 South Alligator River
The South Alligator River estuary in the Northern Territory of Australia is a macro-
tidal and shallow estuary with the mean water depth in the thalweg on the order of only a few
25
meters, a spring tidal range up to 6 m and maximum spring tidal currents of up to 2 m s .
The tidal currents exhibit a strong asymmetry with the flood currents being much stronger
than ebb. The bed sediment is a mixture of sand and mud. The bulk of the suspended
sediment is composed of particulates 1-4 pm in diameter. Vertieal profiles of SSC were
obtained using an Analite optical fiber nephelometer, which recorded data at 3 Hz when
lowered at a speed of about 0.2-0. 3 ms’* through the water column. A 210-KHz narrow-
beam Deso-10 acoustie sounder was used to obtain a visual record of density stratification
induced by suspended sediment (Wolanski, et al., 1988). The estuary is very turbid, with
typical SSC values on order 1-6 kg m . A maximum SSC of 10 kg m was measured.
For most of the ebb duration a lutoeline separated a clear upper layer from an extremely
turbid bottom layer; both layers being of comparable thickness, whereas the vertical gradients
in SSC were small at flood tides (Wolanski, et al., 1988). A vertical, one-dimensional,
numerical model of SSC was used by Wolanski, et al. (1988). Simulations of the tidal
evolution of SSC were shown to be consistent with observations.
2.4.10 Thames River
The Thames River estuary in southern England is characterized by relatively uniform
depths, about 7.6 m at mean tide level, and an exponential variation of the cross sectional
area and charmel width along its length. From its seaward limit to the tidal limit it is about
100 km long; the widths at these limits being 7000 m and 85 m, respectively. The mean tide
range at the mouth is 4.3 m and increases up to 5.6 m at 63 km. With one major exception,
the estuary has a hard bed made up of gravel, clay and chalk. In the area known as the Mud
26
Reaches, 45-53 km upstream of the mouth, there are extensive deposits of silt, and a turbidity
maximum with SSC ranging between 0. 1-5.5 kg m ^^ (Owen and Odd, 1970). Field
investigations (Inglis and Allen, 1957) and two-layer numerical modeling of Odd and Owen
(1972) found that the null point is usually located in this area, and the turbidity maximum
and sedimentation there are due mainly to silt transported in the lower layers jfrom both the
upstream and downstream directions. A local pocket of silt also occurs at about 24 km
upstream of the mouth.
2.4.11 Yellow River
The Yellow River estuary in northern China discharges into the Bohai Sea with an
annual mean inflow rate of about 1,550 m^ s"' and a sediment load of about
1.2x10^ tons yr It carries the largest sediment load of any river in the world and is
dominated by loess silt and fine sand. On the order of 64% of the sediment is deposited on
the river delta and mudflats, while the remaining is transported deeper into the Bohai Sea.
As a result, a fan-shaped delta has been formed since 1855, which covers a distance of 160
km along shore and 20-28 km seaward (Wang, 1988). An extensive survey over the active
delta front was carried out in September-October 1987 and July- August 1988, aimed at
documenting the areal extent, vertical thickness, bulk densities, downslope velocities and
velocity gradients of the sediment underflows and their relationship to tidal and storm
forcing. It was reported that cross-isobath sediment dispersal into the shallow Bohai Sea is
dominated by the formation of a hyperpycnal plume (SSC>2 kg m ‘^) and gravity -driven
underflows (Wright, 1988). The strong tidal currents often had speeds of over 1.5 m s
27
near the surface and 0.8 ms'* at 1 m above the bed. The observed cross-isobath
components of underflows had downslope speeds of 0.05-0.30 ms'*. These underflows
descended the rapidly prograding delta front as hyperpycnal plumes of 1-4 m thickness. SSC
in the lower 2 m of the water column in the shallow parts (<5 m) normally exceeded 1
kg m and attained maxima of over 10 kg m '^ . SSC near the surface varied from <0.1
kg m to > 1 .0 kg m '^ . The highest turbidity values occurred landward of the front. In the
deeper part (~10 m), SSC near the surface was consistently less than 0.1 kg m and about
1 kg m near the bottom. Observations during a storm and immediately after the storm
revealed near-bottom layers of fluid mud with a thickness of 1-2 m and average SSC at -0.8
m above the bed of about 252 kg m '^ . Another process observed at the top of the
underflows was the activity of low frequency (2.5-5.0x10'^ Hz) internal waves of relatively
high amplitude (1.0-2. 5 m), which were believed having contribution to plume deceleration
(Wright, et al., 1988).
CHAPTER 3
FLOW AND SEDIMENT TRANSPORT
3.1 Introduction
In order to examine lutocline dynamics in estuaries through numerical modeling, it
is essential to carry out simulation for the following: (1) ambient flow field, (2) flow-
sediment coupling due to the stratification effect and turbulence damping in the fluid mud
layer, (3) erosion, entrainment, settling and deposition, (4) formation of fluid mud, (5)
interfacial mixing, (6) suspended sediment advection and diffusion, (7) consolidation of
bottom sediment, etc. Accordingly, a three-dimensional numerical model code is developed.
This code consists of two parts, namely, a hydrodynamic model (called Coastal and Estuarine
Hydrodynamic model - University of Florida or COHYD-UF) and a fine-grained sediment
transport model (called Coastal and Estuarine Cohesive Sediment model - University of
Florida or COSED-UF).
This chapter begins with descriptions of the governing equations of hydrodynamics
and sediment transport, relevant boundary conditions, fine sediment transport processes,
flow-sediment coupling and finite-difference schemes for solving these equations. Results
of tests related to model validation are then presented.
28
29
3.2 Hydrodynamics
3.2.1 Goyeming Equations
In the deriyations of the goyeming hydrodynamic equations, the following treatments
are considered: (1) o-transform is introduced (Figure 3.1) by transforming the temporal and
Cartesian coordinate system (t, x,y, z) to a new system (/', x',y', o) according to o^{z-(,)IH
(Stansby and Lloyd, 1995), (2) the flow continuity equation takes the yertically integrated
form, (3) the yertical distribution of pressure is assumed to be hydrostatic, and (4) higher
order terms related to diffusion inyolying o-coordinate are neglected. Thus, the continuity
and momentum equations in the new time and coordinate system, where the superscript
(prime) has been eliminated for conyenience, respectiyely are (see Appendix A for
deriyations):
a
A
x'
9 M M M M 3 9 M 3 M M M ,
Figure 3.1. Schematic diagram showing o-transform.
30
Continuity equation:
dt
dHu dHv], „
+ \da=0
dx dy )
(3.1)
Momentum equation in the ^-direction:
Du r du du du du r d( gH^rdp,
— -jv- — +u — +v — +0) — — / —da
Dt dt dx dy da dx p J dx
a
gm
p dx
op+J pda
dx‘
dy'
d
du
^ Hda
1 dXg
pH da ^
2+y2
(3.2)
Momentum equation in the y-direetion:
Dv . dv 6v dv dv . dC sH r 5p ,
— +/m= — +w — +v — +0) — +fu=-g—-^ / -^da
Dt dt dx dy da dy p J dy
gdH
P dy
op+J pda
*-.2 >'-,,.2
V
dx'
dy'
+1A
( A
dv
^ Hda
,11^ J
dXr
(3.3)
where D/Dt denotes the total derivative with respect to time, u, v and o) are the flow
velocity components in the x, y and o directions, respectively, t is time, g is the gravitational
acceleration, C is the instantaneous water surface elevation, H is the total water depth,
+h,h\s the imdisturbed water depth, /is the Coriolis parameter, A and A are the
X y
31
horizontal turbulent momentum diffusion coefficients in the x and>^ directions, respectively,
is the vertical turbulent momentum diffusion coefficient, p is the fluid density, and is the
Bingham yield strength (Odd and Cooper, 1989). The vertical velocity in the o coordinate,
(0, is determined as according to
dx dy J H
dHu^dHv\^^
dx dy j
(3.4)
and the vertical velocity in the z coordinate, w, is obtained from (see Appendix A)
W-H(j3+U
o-
.m
dx
.K
dx ,
dy dy^
dt dt
/
(3.5)
3.2.2 Boundary Conditions
The boundary conditions for solving the above equations are prescribed as follows:
1 . Water surface: At the water surface (o=0), the x and y components of the wind-
induced stresses, and , are respectively specified as
X PA,du V P^vdv
where the resultant vector, f ^ , is obtained from
(3.6)
withC,=0.00l(l+0.07H) (3.7)
Here p is the air density, C is the surface drag coefficient and W is the wind velocity
vector at a reference elevation (10 m above the water surface in the prototype case).
32
2. Bottom: The boundary condition at the bottom (o=-l) can be specified by either
no-slip or a shear stress condition. At the bottom, the water particles are attached to
(stationary) the solid surface, hence the velocity is zero, i.e., . Within the fluid, velocity
increases rapidly and reaches a value of (called slip velocity) over a small distance
(Woodruff, 1973). Thus, there is a very steep velocity gradient near the bottom. If one were
to specify the no-slip condition right at the bottom, a high-resolution numerical grid near the
bottom would become essential. In turn this would lead to overly extensive computation
time. In addition, viscous effects are important in the sublayer near the bottom, so that high
Reynolds number turbulence modeling is not applicable there. Therefore, the shear stress
condition is most commonly used. The bottom shear stress is expressed in terms of the
velocity components taken from the modeled layer closest to the bottom. Accordingly, the
corresponding stress components, and x^, can be related to the velocity gradient
according to
_ P^v du ^_P'^vdv
where the resultant vector, x^, is obtained from
V., with Cn=
K
U\ 1
1’ U
ln(z/zo)+il;(z,/Z;^)
(3.9)
for turbulent flow. Here Zj and Fj , respectively, are the elevation above the bottom and the
velocity vector of the modeled layer closest to the bottom, is the bottom drag coefficient.
33
K is the von Karman constant, Zg is related to the effective roughness of the bed and tj; is a
stratification function which takes the form (Monin and Obukhov, 1953):
'_z^
V
= 1 +a,
(3.10)
In Eq. (3.10) is an empirical constant in the range of 4.7-5.S, and L^is the Monin-
Obukhov length scale defined as
Kgw'p'/pg K^gz[(p^-p)/p] dc/dz
(3.11)
where D , is the bottom friction velocity defined as w*=^|?j|/p, c is the suspended sediment
concentration (SSC), Pg is the density of water, p^ is the sediment granular density, w' is
the turbulent fluctuation of the vertical velocity in the z coordinate and p' is the turbulent
fluctuation of density.
Since the shear stress condition is employed at the bottom boundary, the flow within
low turbulence region close to the wall can only be described by a semi-empirical wall
function which bridges the viscous sublayer by relating the values at the first numerical grid
point placed outside the viscous sublayer to conditions at the wall. Launder and Spalding
(1972) have proposed a wall function which is described by a logarithmic velocity profile
applicable to the solid wall and the first grid point adjacent to it. The standard formulation
of the wall function is
34
U 1
— =-ln(y^€)
K
(3.12)
where y*=u^z^/\ is a dimensionless distance or frictional Reynolds number, v is the
kinematic viscosity and e is a roughness parameter (e=9 for a hydraulically smooth wall and
0.05 for a hydraulically rough wall). As suggested by Rodi (1980), the wall function should
apply to a point whose y * value is in the range of30<y "^<100. Eq. (3.12) is then sufficiently
accurate for most of the situations.
4. Shore Boundaries: At the shore boundaries, the following impermeable condition
is prescribed:
(3.13a)
where the subscript n, denotes the direction normal to the solid boundaries.
Similar to the bottom, a no-slip condition at the shore boundary would require a high-
resolution grid near boundary. Thus, the lateral shear stress condition is specified at the shore
boundary according to
dv,
(3.13b)
where is the horizontal turbulent momentum diffusion coefficient in the direction normal
to the solid boundary, is the coordinate normal to the solid boundary, is the tangential
velocity at the modeled grid closest to the solid boundary, and C, is the lateral friction
coefficient.
35
5. Open Boundaries: At the open boundaries, the elevation of the water surface is
prescribed:
Ce=Co+E^„cos(o)„r-g„) (3.14)
n = \
where Cq is the mean water level, TV, is the total number of tidal harmonic components
considered, (o^ is the angular frequency of the «'* harmonic component, and and are
the amplitude and phase lag of the harmonic component, respectively.
3.3 Sediment Transport
3.3.1 Sediment Conservation Equation in the Water Column
The sediment conservation equation in the and o coordinate system can be stated
as (see Appendix A for derivation of the equation in the o coordinate):
Dc 1 dc dc dc dc 1
— = — +u — +v — +0) — ^
Dt H do dt dx dy do H do
{ d^c d^c]
K +K
1 ^ ^
^ ^ dx^ ^ dy^ ^
^ Hdo
[ //aoj
where O) is the sediment settling velocity, and AT,, are the horizontal turbulent mass
j X y
diffusion coefficients in the x and y directions, respectively, and is the vertical turbulent
mass diffusion coefficient.
36
3.3.2 Boundary Conditions
The boundary conditions for solving Eq. (3.15) are prescribed as follows:
1. Water Surface: No sediment crosses the water surface (o=0), i.e.,
1 c 1 1
H da Hda
H da
=0
(3.16)
2. Bottom: Sediment vertical flux at the bed-fluid interface (o=-l) is equal to the net
flux due to erosion and deposition, i.e.,
1 1 1
H da Hda
H da
(3.17)
where w^is the flux or rate of bottom sediment erosion (mass per unit area and time) and
is the flux or rate of SSC deposition.
3. Shore Boundaries: No sediment crosses the solid shore bovmdaries, i.e..
dc
dn.
=0
(3.18)
4. Open Boundaries: The condition at the open boundaries are dependent on the flow,
i.e., inflow or outflow. If fluid flows into the modeled region, the SSC of the water from the
outside region is prescribed, i.e.,
c,=cfx^,a,t) (3.19)
where is the prescribed SSC at the open boundary and c. is the SSC from outside the
modeled domain. In the case of outflow, the diffusion terms in Eq. (3.15) are neglected for
numerical purposes, so that the mass conservative condition becomes:
37
(3.20)
dt dx dy da H do
3.3.3 Fine Sediment Transport Processes
To use Eq. (3.15), the fine sediment transport processes must be expressed in terms
of mathematical formulas. The major relevant processes are as follows.
1 . Consolidation: In cohesive sediment transport models there is a need to take into
account consolidation of newly deposited sediment, since the shear strength and, thus, the
erosion resistance of the deposit increases with consolidation time (Verreet and Berlamont,
1 989). The erodibility of a mud bed is determined by the strength of its particle-to-particle
structure. This in turn is influenced by the state or degree of consolidation of the bed, which
is time-dependent. Thus, in a consolidating bed the resistance to erosion is a time-dependent
function of consolidation duration. Consequently, in a model of fine sediment transport
behavior, predicting bed consolidation occupies a crucial role.
Consolidation of the deposit is caused by the self-weight of sediment particles, and
is a process of expulsion of pore water from the bed (Parker and Lee, 1979). The settling
cohesive sediment begins to behave as a soil and consolidate when the resulting stationary
suspension develops particle-to-particle contacts and an effective stress appears (Hayter,
1983). At this stage, the total stress within the soil matrix is expressed as
where is the total (normal) stress, r'is the effective (normal) stress and is the pore
(3.21)
water pressure.
38
The aim of a consolidation model is to predict the vertical distribution of the time-
dependent dry density or concentration, c, which in turn is related to the soil shear strength.
There are two typical theoretical models describing the consolidation process, those based
on force balance (Gibson, et al., 1967) and those based on solid mass conservation (Kynch,
1952). Here the theoretical framework first developed by Kynch (1952) is applied. It
describes consolidation by a vertical transport equation. The consolidation rate, instead of
permeability, is used as the model parameter, which is a function of dry density (Toorman
and Berlamont, 1993). In the present work, the difference from the pure deposition model
of Kynch (1952) is that the deposition flux minus erosion flux at the bed-fluid interface, i.e.,
q=m^-m^, is incorporated in the model, and a normalized coordinate o' {=z'/H') is used.
Thus, the relevant transport equation is:
dH'c
= +q
dt do'
(3.22)
where z' is the vertical coordinate of the consolidating layer originating from the bottom and
positive upward, i/'is the thickness of the consolidating layer and is the rate of
consolidation. As for this rate, experimental evidence of Toorman and Berlamont (1993)
suggests that two distinct modes of consolidation can be recognized from the plot of
consolidation rate against concentration, i.e., loose soil consolidation and compacted soil
consolidation. For these two modes, they developed a combined relationship expressed as
( \
( \
ntt
( \
w,iexp
c
F,+(o ,
t sc2
1-^
(l-F,), F,=exp
-
c
(3.23)
39
where is a characteristic mode transition (from loose soil to compacted soil) function with
« >10, m^ is a sediment-dependent constant is the transition concentration, and 0)^^2
are the rates of consolidation for the first and the second modes, respectively, c^j is the
concentration corresponding to the maximum settling flux and the saturation
concentration, i.e., the maximum compaction concentration.
2. Fully Consolidated Bed: The deposited sediment is said to be fully consolidated
when c^c^2- The vertical profile of the dry density or concentration of a fully consolidated
bed can be expressed as (Mehta, et al., 1982):
c=Ca.
f U )P2
h.-LZ
b )
(3.24)
where h, is the bed thickness, C is the bed average concentration, az is the incremental
depth downward from the bed surface, and and P2 are bed-dependent coefficients.
3. Shear Strength: The bed shear strength with respect to erosion is the primary
measure of bed scour. As stated above, the shear strength increases with the consolidation
time, or the dry density. Hence, usually, the shear strength is related to the bottom sediment
concentration. In accordance with the description of Mehta (1991b), the bed shear strength,
T , can be considered to have the form
S ’
(3.25)
40
where is the shear strength of newly deposited bottom sediment, and Pj are sediment-
dependent coefficients, (j) is the solids weight fraction, (|)=c/p^, and (j)^is the critical solids
weight fraction below which mud has a fluid-like consistency. James, et al. (1 988) show that (j)^
is typically on the order of 0.03 to 0.05.
4. Interfacial Entrainment: Since diffusion-induced mixing over the lutocline is
damped due to strong stratification and turbulence damping within fluid mud, internal wave
breaking becomes a major mechanism contributing to vertical entrainment over the lutocline
(Scarlatos and Mehta, 1993). There are two primary modes of instability of the interface
depending on the relative thicknesses and positions of the current shear layer of thickness 6^
(Figure 2.1) and the density interfacial layer of thickness 6^ (Mehta and Srinivas, 1993). In
case the mid-axes of density and velocity gradients coincide and 6^ is approximately equal
to 6^, the primary mode of instability is of the Kelvin-Helmholtz type, and is characterized
by a roll-up and pairing of the interfacial vortices (Delisi and Corcos, 1973). When 6^ is
smaller than 6^ due to stratification, Holmboe type of instability results, as recognized by
sharp-crested interfacial cusps which protrude alternatively into both fluids (Browand and
Wang, 1972). Results from laboratory and field observations further show that turbulence
damping in the fluid mud layer introduces an upward-asymmetric mixing over the lutocline,
i.e., there is a net upward flux of mass over the lutocline (Wolanski, et al., 1989; 1992;
Mehta and Srinivas, 1993; Kranenburg and Winterwerp, 1997; Winterwerp and Kranenburg,
1997; Jiang and Wolanski, 1998).
41
Mehta and Srinivas (1993) established a semi-empirical formula for the rate of
interfacial entrainment, which accounts for the cumulative effects of settling, cohesion
and viscosity difference (between fluid mud and water) on mud entrainment
^en with Ri^ - -
(3.26)
where is defined as dh^dt, is the fluid mud layer thickness, and are
sediment-dependent constants, Ri^ is the global Richardson number, C, is the depth-mean
sediment concentration in the upper, mixed, layer of height is the corresponding
value for the lower layer (being entrained), (/, and (/, are the respective depth-mean flow
velocities, and is a coefficient dependent on the granular density as
m
P.-Po
(3.27)
5. Deposition: From flume experiments Krone (1962) concluded that the rate of
deposition is equal to the product of the near-bed settling velocity, SSC and the probability
that a settling floe becomes attached to the bed:
1--^
''dj
(3.28)
where w . and c, are the near-bed settling velocity and SSC, respectively, and t . is the
critical shear stress for deposition.
42
6. Bottom Erosion: Erosion of cohesive sediment, which is dependent on the
composition and structure of bottom material that characterizes bottom resistance, and on the
nature of the eroding force, can occur in two typical ways in estuaries (Mehta, 1991b). The
first mode is floc-by-floc surface erosion in which the floe at the bed-water interface, initially
attached to their neighbors by inter-particle electro-chemical bonds, breaks up and is
entrained as a result of hydrodynamic lift and drag. The second mode is referred to as mass
erosion, wherein the bed fails at a deeply embedded plane such that all the material above
that plane is rapidly brought into suspension.
Surface erosion under current-induced bottom stress has been studied extensively
(Parchure and Mehta, 1985). This process was subsequently examined ftirther by Lee and
Mehta (1994) and Mehta and Parchure (1999). From these studies, the effects of shear
strength and temperature on the erosion rate are incorporated as follows:
[0exp(A-A/0)] (3.29)
in which is the maximum erosion rate constant at tj=2T^, %, X, A and A are sediment-
dependent coefficients and 0 is the absolute temperature.
7. Settling Velocity: As stated in Section 2.2.1, in water with the salinity
.s>0.1 -0.5%, the settling velocity of cohesive sediment is strongly dependent on SSC and
can be divided into three sub-ranges in terms of concentration. These sub-ranges include free
settling, flocculation settling and hindered-settling (Figure 2.2). Hwang (1989) developed a
combined relationship between the settling velocity and SSC for flocculation and hindered
43
settling regions. Here, a modified relationship of Hwang, which includes the effects of
flocculation, salinity and temperature on settling velocity, is used:
where a, b, a and P are sediment-dependent empirical coefficients, 0 is the temperature, v(6)
is the temperature-dependent fluid kinematic viscosity, p(0,5,c) is the temperature, salinity
and SSC dependent fluid density, and F(d) is a temperature fimction which reflects the effect
of temperature on flocculation defined as (o^ , where o)^ is the flocculation settling
velocity at 15 °C. By reprocessing the experimental data of Lau (1994) at different
temperatures (see Appendix C), an empirical relationship for F(d) is obtained as (Figure 3.2):
3.4.1 Baroclinic Effects
Sediment-induced stratification is considered in the hydrodynamic equations, i.e., the
second and third terms on the right hand sides of Eqs. (3.2) and (3.3), where the bulk density
(p) of water/ sediment mixture are related to SSC (=c) by
(3.30)
F(0)=1.776-O.O5180, for 0-O-3O°C
(3.31)
3.4 Flow-Sediment Coupling
(3.32)
3.4.2 Vertical Momentiun and Mass Diffusion Coefficients
When the flow is stratified, the buoyancy effect tends to restore vertically moved
fluid lumps back to their original positions, and thereby causes a reduction of the turbulent
transfer of momentum and mass. From the theoretical relationship derived by Rossby and
44
Montgomery (1935), Munk and Anderson (1948) proposed following generalized semi-
empirical formulas;
Figure 3.2. Dimensionless median settling velocity as a function
of temperature, where o)^ is the median settling velocity at 15 °C.
Momentum diffusion coefficient:
Mass diffusion coefficient:
(3.34)
where a^, b^, Yp ^2 Y2 empirical constants, and and respectively are
the vertical momentum and mass diffusion coefficients in homogenous flow. In terms of the
45
well-known mixing length concept of Prandtl (1925), and have simple forms for
neutral turbulent diffusion:
Momentum diffusion coefficient:
A -I
^vO ‘mO
du
dz
=KW^//(-0)(l +o)
(3.35)
Mass diffusion coefficient:
^vO ^cO^mO
du
dz
=KM,/f(-0)(l +o)
(3.36)
Here a={z-(,)/H, 1^^ and are, respectively, the momentum and mass mixing lengths in
a homogeneous, non-cohesive flow, and Ri is the Richardson number defined as
Ri^
_ g dp/dz
P (du/dzf- +{dv/dzy
(3.37)
Jobson and Sayre (1970) noted that vertical mixing of suspended sediment in open
channel flow occurs as a result of at least two semi-independent processes which are shown
to be additive. These processes are: (1) Diffusion due to tangential components of turbulent
velocity fluctuations, which is the predominant turbulent mixing process for fine sediment
particles in general, and for all sediment particles in flows without strong vortex activity; and
(2) diffusion due to centrifugal force arising from the curvature of fluid particle path lines,
which is significant for coarse sediment in flows with strong vortex activity. They derived
theoretically the following expressions for the total turbulent transfer coefficient of sediment:
46
(3.38)
in which the first term represents turbulent transfer of sediment due to rectilinear velocity
fluctuations, and the second term represents turbulent transfer of sediment due to curvature
of fluid particle pathlines. Coefficients and are assumed to be functions of the particle
characteristics. Using the method of average curve fitting technique, Jobson and Sayre (1970)
reported that p, was 0.98 and 0.49 and a was 0.038 and 0.1 for fine and coarse sediments,
respeetively.
A drawback of Eqs. (3.33) and (3.34) is that the turbulent diffusion coefficients will
become zero if and where Richardson number tends to infinity, which is the case when the
vertical gradient of velocity is zero. However, this is not realistic, since diffusion is
practically never equal to zero. Hence, to overcome this disadvantage, an additional term
interpreted a “background” value of the diffusion coefficient can be introduced in the
formulas of Munk and Anderson (1948), i.e..
Momentum diffusion coefficient:
(3.39)
Mass diffusion coefficient:
(3.40)
47
where and are the background values of the turbulent diffusion coefficients of
momentum and mass, respectively. Representative values of a^, b^, y,, a^, and are
given in Table 3.1.
Table 3.1. Parameters for momentum and mass diffusion coefficients in a stratified flow
«1
Y,
*2
Y2
Source
1
60-160
-0.5
—
—
—
Rossby and Montgomery (1935)
1
10
-0.5
1
3.33
-1.5
Munk and Anderson (1948)
1
10-15
-1
—
—
—
Kent and Pritchard (1959)
3.5 Solution Techniques
To solve hydrodynamic Eqs. (3.1), (3.2) and (3.3) and sediment transport Eq. (3.15),
a semi-implicit finite difference method is applied, which discretizes the convective and
diffusive terms by an Eulerian-Lagrangian scheme (Casulli and Cheng, 1992). This solution
method has the advantage of a minimum degree of implicitness, good stability and
consistency, and high computational efficiency at a low computational cost.
As shown in Figure 3.3, a spatial mesh consisting of rectangular cells, totally
MxN>^Lj- and each of length ax and Ay and height ao, is introduced. Each cell is numbered
at its center with indices i J and k. The discrete w- velocity component is then defined at half-
integer i and integers j and k. Similarly, the v-velocity component is defined at integers i and
k and half-integer j. The vertical velocities, to and w, are defined at integers i and j and half-
48
integer k. Water surface elevation C is defined at integers / and j. The undisturbed water
depth h{x^) and the total water depth are specified at the both m and v points. Finally,
SSC, denoted by c, and fluid density p are defined at integers i ,j and k.
i-m i 1+1/2
;+l/2
j-\ 12-
ax
^:v, H,h
•: C, w, c, p
(a) Horizontal mesh
A. ♦ 1/2 A ^
1 k
K * 1/2 ’
•: M, V, c, p
Cl), w
(b) Vertical mesh
Figure 3.3. Schematic diagram of computational mesh and notation.
3.5.1 Discreti2ation of Hydrodynamic Equations
The general, semi-implicit, discretization of the continuity Eq. (3.1) and momentum
Eqs. (3.2) and (3.3) takes the following forms:
49
Differential continuity equation:
/
AO
n + l
^i*M2j,k
n*\
Ay
V + l/2.
V' 1/”^* ”
2Z^
k=\ k=\
+1
ij-m,k
(3.41)
Differential momentum equation in the x-direction:
/j+i
At
Cn*\ _rn*\
i*\j ^ij
AX
+B
n
i^Xnj^k
• *2,/, J
f n+\ n+1 \
^i*\/2j,k ^i*mj,k-lj
1
I^AO^
(3.42)
Differential momentum equation in the y-direction:
«+i
^ij^\n,k
At
rn*\ _r« + l
^i*\j ^ij
AX
+B
n
V + i/2,)t
^\j*\l2,k*\l}
f « + l /i+l ]
^ij*l/2,k*\ ^ij*y2,kj
1 « + l /j + 1 \
^ij*\/2,k ^ij^l/2,k-lj
1
PAO^
(3.43)
Here the subscripts denote spatial positions, superscripts denote time steps, At, B is the sum
of the Coriolis, baroclinic, horizontal diffusion and Bingham yield strength terms discretized
by the hydrodynamic values at time step n, and v" are respective values of w and v at
time step «, and subscript p denotes water particle position that is currently located at u or
V point. To obtain position p, it is assumed that the velocity field (u, v) remains unchanged
from the previous time step n to the present time step n+\.
50
Substituting Eqs. (3.42) and (3.43) into Eq. (3.41), a linear, five-diagonal system of
equations for the water surface elevation, C, is obtained. This system is symmetric and strictly
diagonally dominant with positive elements on the main diagonal and negative ones
elsewhere. Thus it is positive-definite and has an unique solution. A substantial part of the
computational time is utilized in solving this linear system (Casulli and Cheng, 1992). In
practice, this system can be solved efficiently by the pre-conditioned conjugate gradient
method (see Appendix B for details), which is fast and requires a minimum amount of
computer memory (Bertolazzi, 1990). Then the velocity components, u and v are obtained
from Eqs. (3.42) and (3.43).
3.5.2 Discretization of Sediment Transport Equation
Similar to the discretization of the hydrodynamic equations, the semi-implicit
discretization of Eq. (3.15) is given by
where the notations are the same as in Eqs. (3.42) and (3.43), D is the sum of the discretized
vertical settling and horizontal diffusion terms using the flow conditions and SSC at the
previous time step n.
3.5.3 Discretization of Consolidation Equation
The consolidating bottom layer of thickness H' is divided into sublayers, with
each sublayer defined by a concentration c and thickness H'/L^ . At each time step At, the
sediment transport model (COSED-UF) provides the net sedimentation (mass per unit area).
51
qAt, at each grid point. Also, at each time step, qAt is introduced onto the consolidating layer
in following way (Figure 3.4); (1) When net deposition takes place, i.e., ^^0, the initial
concentration of the deposited sediment is prescribed as Cj- (typically Cy.=80 kg m ; Odd
and Cooper, 1989), and the corresponding thickness aH' (=qAt/Cj) is added to the top of the
consolidating layer. Then the consolidating layer is redivided into sublayers, and new
initial concentrations at each computational element are obtained by cubic spline
interpolation. (2) In the case of net erosion an erosion depth, aH' , is subtracted from the top
of the consolidating layer. Then, through redivision and cubic spline interpolation, new initial
concentrations at each element are obtained as before. Following this procedure, the
consolidation process is modeled using the discretized form of Eq. (3.22). To increase the
modeling accuracy, a higher time resolution is applied, i.e., the consolidation time step
At' = At /N^, taking
An explicit scheme is used for descretizating Eq. (3.22):
Here the subscripts denote spatial positions, superscripts denote time steps and ao ' is the
vertical step length. Then, at each time step the total thickness of the consolidating layer is
calculated from mass conservation according to:
(3.45)
0
(3.46)
52
Original
Deposition
Erosion
Figure 3.4. Schematization of the simulated consolidation process,
where (a) is the original consolidating layer, (b) is the case
of net deposition and (c) is the case of net erosion.
3.5.4 Properties of the Finite-Differential Equations
The accuracy, stability, numerical diftusion and spurious oscillations of the finite-
differential equations (3.41), (3.42), (3.43) and (3.44) depend on the discretization scheme
of the convective terms, when an Eulerian-Lagrangian approximation is adapted. Actually,
as expressed in Eqs (3.2), (3.3) and (3.15), the convective terms can be rewritten more
compactly as Lagrangian derivatives according to
DG dG dG dG dG
= +u +V +(0
Dt dt dx dy da
(3.47)
53
where G denotes any physical quantity, e.g., the velocity or SSC. Then the Eulerian-
Lagrangian scheme discretizes the convective terms as
At
where subscript o denotes the current position of a water particle.
To obtain values of Gp , or v^” and Cp in Eqs. (3.42), (3.43) and (3.44), the
Eulerian-Lagrangian scheme uses the back-tracing approach incorporated by a suitable
interpolation method using three or more mesh points (Casulli and Cheng, 1992), in which,
as stated before, it is assumed that the velocity field («, v) remains unchanged from the
previous time step n to present time step n+\. Here, these values are approximated by a
bilinear interpolation over the eight surrounding mesh points (see Appendix B for details).
The back-tracing time interval At"=AtlNp, where Np is the back-tracing step at each modeling
time step, is usually selected to be ^5. In this event, the Eulerian-Lagrangian scheme
becomes free of spurious oscillations. Moreover, numerical diffusion, which can be regarded
as the interpolation error, is reduced when compared with numerical diffusion induced by
the “up-wind” method. Further reduction in artificial diffusion can be obtained by decreasing
the spatial steps ax. Ay and ao and increasing back-tracing step, Np . Complete elimination
of numerical diffusion can be achieved by using a higher-order interpolation formula, but the
resulting method may introduce some spurious oscillations. Applications of this scheme to
problems with large vertical diffusion, or K^, or small vertical spacings, ao, have
54
suggested the use of an implicit discretization only for the vertical diffusion terms. In fact,
it can be shown that the stability condition for the above scheme is simply given by
(Greenspan and Casulli, 1988)
A^, K^, Ky
-1
(3.49)
Evidently, when A^=Ay=K^-Ky=Q ,ibis scheme becomes unconditionally stable. However,
if one restricts the back-tracing process to within one cell, i.e., the back-traced water particle
is not allowed to emerge out of the cell where it would start from the beginning, the stability
condition becomes
where and respectively are the maximum velocities in the x and>^ directions within
the modeled domain.
3.6.1 Hydrodynamics
In this section, results of the COHYD-UF model will be compared with two
analytical solutions. The first comparison will primarily investigate the model’s capability
to simulate the time propagation step. The second comparison will attempt to validate the
model’s ability to compute nonlinear effects.
(3.50)
3.6 Basic Simulations
1 . Comparison with a Linear Analytical Solution: Consider tidal flow through an
open channel connected to the sea at the mouth (x=0) and closed at the uphead end (x=/).
55
Neglecting the advective terms, bottom friction and wind surface stress, the one-dimensional
hydrodynamic equations for flow through the channel are (Ippen 1966)
Momentum:
du ac n
— +g— =0
dt dx
(3.51)
Continuity:
dt dx
(3.52)
where U is the vertical mean velocity in the x direction and depth h is assumed to be
constant. The selected boundary conditions associated with Eqs. (3.51) and (3.52) are:
At the uphead end of the basin:
U{l,t)=Q (3.53)
At the mouth of the basin:
C(0,0=v4oSino)or (3.54)
where and 0)q respectively, are the amplitude and angular frequency of the forcing tide
at the open boundary. Selecting a uniform rectangular cross-section of the channel and only
considering the first mode of oscillation, the solutions of Eqs. (3.51) and (3.52) are:
Water surface elevation:
C(V)=
AqCosHI-x)
coskl
sincOpt
(3.55)
Velocity:
56
Af.C sink(/-x)
cosWq/ (3.56)
hcoskl
where is the wave speed ( -y/^) and k is the wave number ( =g)q/C^).
To test the hydrodynamic model results against the above solutions, a rectangular
basin with a constant water depth of 20 m and basin length of 59 km is considered. Assume
that a periodic tide with an amplitude of 1 m and a period of 7= 12 hr is forced at the mouth
of the basin. The numerical solution is obtained by discretizing the basin into 30 grids with
ax=2 km and time step At=20 min.
Figure 3.5 shows the water surface elevation near the mouth (x=10 km), at the mid-
point of the basin (x=30 km) and near the closed boundary (x=58 km). Likewise, Figure 3.6
presents the velocity near the mouth (x=l 1 km), at the middle point (x=3 1 km) and near the
closed boundary (x=51 km). Both Figures 3.5 and 3.6 demonstrate that there is reasonably
good agreement between theoretical and numerical solutions, even though a slight phase shift
of velocity between the theoretical and numerical results occurs because terms higher than
zero order are neglected in the analytical solution.
2. Comparison with a Non-linear Analytical Solution: When nonlinear terms are
included in the one-dimensional shallow water equations, it is not possible to obtain an exact
solution. However, one can use harmonic analysis to develop an analytical solution, which
is still cumbersome because the high order terms are difficult to solve for. Accordingly, here
only the zeroth and first order harmonic solutions are considered (Liu, 1988).
57
(ui) UOI)«A3]a
58
Without the inclusion of Coriolis and bottom friction forces, the governing equations
for one-dimensional nonlinear tidal motion can be written as
dU j.dU dC „
— +U — +^— =^0
dt dx dx
(3.57)
Equation (3.52) remains unchanged. The boundary conditions and the geometry of the basin
considered are the same as before. Neglecting solution terms higher than first order, the
solutions are (Liu, 1988):
Water surface elevation:
Af.cosk(l-x)
C(x,/)=-5 ^
coski
smwr
A^k
ihcos^kl
Velocity:
xsin2^/ -x) + — - — (sin2A:(/ +x) -tai)2klcos2k{l -x))
cos4kl
rrr a
U{x,t)= ^ COSO)t+-
(3.58)
cos2o)/
hcoskl
%h ^co^kl
xcoslkQ -x)+—s\tQ.kil -x) - — ^ — [coslkil +x) -XzrQ.klsirQ.Jdl -x))
2k cos4kl
(3.59)
sin2o)/
The same forcing condition at the basin mouth as before is used to compare the
solutions with model results. Figure 3.7 shows the comparison of water surface elevation,
while Figure 3.8 presents the velocity. It can be see that the model results are reasonably
close to the analytical solutions. As before, the slight phase shift of velocity between the
theoretical and numerical results occurs because terms higher than first order have been
neglected in the analytical solution.
59
(,.S Ul) X1130I3A
(ui) UOpBASia
Time (hr) Time (hr)
Figure 3.7. Modeling ID non-linear hydrodynamic equation Figure 3.8. Modeling ID non-linear hydrodynamic equation
for tidal flow in an open channel. Lines are simulations and for tidal flow in an open channel. Lines are simulations and
open circles represent analytical solutions. open circles represent analytical solutions.
60
3.6.2 Sediment Transport
To check the applicability of the COSED-UF numerical scheme, the following
modeling tests against some special forms of the governing equations having exact solutions
as well as field data in a river are carried out. Plots of numerical and analytical solutions are
accordingly presented.
1 . Steady State One-dimensional Convection-diffusion: The basic equation is
at
dx
(3.60)
where is the constant source-sink term and / is the length of the system.
The boundary conditions are prescribed as
ax
=0
lx=/
(3.61)
The analytical solution is
u
where Np^ is the Peclet number, ul/K^, which is the ratio of convective transport to
diffusive transport.
A rectangular grid with equal lengths of the spatial step was used in the numerical
solution. Values of the parameters used were: q^=5, l=\, m=1, Cq=1 and A!^q=1. Figure 3.9
shows the comparison between the modeled results and the analytical solution.
61
2. Laplace Equation: The Laplace equation was solved for a rectangular domain,
0<x<l^, 0<y^ly, with a parabolic boundary condition for a quantity such as temperature,
specified on>^0.
The basic equation is
K^+K—^0
(3.63)
The boundary conditions are given by
K.
c{x,0)-—x{l -x), c-Q on other faces
By taking K^=K -K^, the exact solution takes the form
/ ^ 4ii:o(l -cos«7r)
c{x^,t)=2_^ — smi
/ N
mix
n=i s\r\h{rml /l )n^TZ^
y X'
sinh
V * /
rm{ly-y)
(3.64)
(3.65)
The values 1^-3, 1^=4 and ^"^=40 were used in the test problem. Figure 3.10 shows
the comparison between the modeled results and the analytical solution.
3. One-dimensional Transient Heat Conduction: The basic equation is
5c 5^c
at 0<x</
(3.66)
The initial condition is
c=0, 0<x^/, /=0
(3.67)
62
Figure 3.9. Modeling ID convection-diffusion equation.
Line is analytical solution and dots represent model simulations.
0 0.5 1 1.5 2 2.5 3
Distance x
Figure 3.10. Modeling 2D Laplace equation. Lines are analytical
solutions and open circles represent model simulations.
I
63
The boundary conditions are
c(l,t)=c., ^
dx
=0
x=0
(3.68)
The analytical solution is (Carslaw and Jaeger, 1959)
^ = \-ly illZe -(2»-i)VV4^Q.(2n+l):tx
^ 21
Cq 7I„=o 2n+l
(3.69)
where is the elapsed time with T^i=Kj/l^. The values Cq=1, 1=2 and A:^=0.5 were used
in the test problem. Comparison between the modeled results and the analytical solution is
shown in Figure 3.11.
4. Heat Conduction with Radiation: This problem was chosen to check the flux
boundary condition formulation necessary in the model due to the resuspension term at the
bed. The governing equation and initial condition are the same as Eqs. (3.66) and (3.67),
respectively with the boundary conditions
c(/,0=0.
dc
— +a,c
dx ‘
=0
x=0
(3.70)
The exact solution is (Carslaw and Jaeger, 1959)
/ \
1 +C£^
» 2(p
sin
i x]
1--
c _
-V
~ n
1 1)
^0
1 +al
\ ‘ y
n = l
P
„(a/+a?/2+p^)
(3.71)
where is the linear heat transfer coefficient, is the linear heat transfer coefficient,
is the positive roots of Pcotp+a^^O and is the elapsed time as defined before. The
64
0 0.5 1 1.5 2
Distance x
Figure 3.1 1. Modeling ID transient heat conduction. Lines
are analytical solutions and dots represent model simulations.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Distance x
Figure 3.12. Modeling heat conduction with radiation. Lines
are analytical solutions and dots represent model simulations.
65
values CL -0.7, c^-l and /=1 were used. Comparison between the modeled results and the
analytical solution is shown in Figure 3.12.
5. Transient Convection-diffusion: The governing equation is
dc dc „ d^c „ A ^
— +u — -K^ =0, at
dt dx
(3.72)
The boundary conditions are
c(0,/)=Co , c(~,/)=0 (3,73)
By taking the initial condition as c(x,0)=0, the analytical solution takes the form
(Ogata and Banks, 1961)
c{x,t)
=exp
ux u t
/erfc
4K.
-exp
u\t-0
4K
-t-erfc
X
(3.74)
Here ^ is the integration variable and erfc is the complementary error function. The values
u=\ and .^^=0.5 were used in the modeling test. Comparison between the modeled result and
the analytical solution is shown in Figure 3.13.
6. Three-dimensional Laplace Equation: The three-dimensional equation was solved
for a cubic domain, 0<x</^, 0iy<l^, 0<z</^ with a power boundary condition for a quantity
such as temperature, specified on z=0.
The governing equation is
d^c
K —+K —+K —=0
^-.2
dx"
dy"
(3.75)
66
Figure 3.13. Modeling ID transient conveetion-diffusion equation.
Lines are analytical solutions and dots represent model simulations.
The boundary conditions are given by
I I
•'ll
c=0 on other faces
By taking K^=K^=K^=Kq, the exact solution has the form
EE
leArji-c-irHi-c-i)")
I n-i m 3/7
. rrnzx . mzy . , ,
sm sm — ^smh (
( ^2
miz
\ * /
/ \i
mr
V
(3.76)
(3.77)
D {stance y
67
The values ly-26, /^=14 and /Tq=160 were used in the modeling test.
Comparison between the modeled results and the analytical solution is shown in Figure 3.14.
Figure 3.14. Modeling 3D Laplace equation. Lines are
analytical solutions and dots represent model simulations.
7. Modeling Test Against Field Observations: To further test the sediment transport
model, observed data of SSC and velocity profiles in the Savannah River estuary by
z//,=2/14 z//,= 3/14 r//„=5/14
68
Ariathurai, et al. (1977) were used. Neglecting lateral variations of SSC and velocity due to
the fact that it is a narrow estuary, the model is used in the vertical 2D code. In the modeled
domain only three grid points - located upstream, mid-point and downstream of the 6.7 km
long estuarine segment considered - were set. The velocity profiles measured at these three
points were inputted for using in the sediment transport model. The SSC profiles measured
at the two end sites of the segment were prescribed as boundary conditions and that at the
middle site was used for verification. In the simulation, the following effects were included:
longitudinal and vertical turbulent diffusion, longitudinal advection, sediment settling,
suspended sediment deposition, and bottom erosion. The results are shown in Figure 3.15.
It is seen that the simulated results compare approximately with observations. It is likely that
the degree of comparison may improve provided: (1) modeling parameters are further fine-
tuned, and (2) the full 3D code is used along with more accurate boundary conditions for
SSC.
3.6.3 Consolidation
For testing the consolidation model two sets of experimental data on consolidation,
without and with the net deposition flux at the bed-fluid interface {q), are considered as
follows.
1 • Consolidation without Deposition: Consolidation without deposition at the bed-
fluid interface was examined by Toorman and Berlamont (1993) using an estuarine mud
from Doel Dock in Belgium. In their experiments, which were conducted in a settling
column described by Van den Bosch et al. (1988; 1989; 1990), both the consolidation rate
and the density profiles of the consolidating layer were measured. The rate of consolidation
69
takes the form of Eq. (3.23), for which Toorman and Berlamont (1993) selected
m s-‘, m s-^ c^j=20 kg m-^ c^2=205 kg m'^, c=160
kg m'^, m=3and n=\3 (Figure 3.16), along with the initial concentration of deposited
sediment Cy=80 kg m . The results are shown in Figure 3.17 for the vertical profiles of
sediment concentration within the consolidating layer, and in Figure 3.18 for the time
variation of surface elevation of the consolidating layer, where is the initial thickness of
the layer. Both Figures 3.17 and 3.18 show that the modeled results compare favorably with
the experimental data.
2. Consolidation with Deposition: Experiments on consolidation with deposition at
the bed-fluid interface were carried out by Burt and Parker (1984). In their tests, fixed masses
of an estuarine mud at a fixed concentration were added at 24 hour intervals to a settling
column of 10 m height and 0.092 m diameter. A total of 7 beds were added; density profiles
were measured 24 hours after each bed was added and immediately prior to the next addition.
Density was also measured 15 days after the last bed was added. The material added for each
layer comprised of 4 liters of suspension at 26.3 kg m solids content giving a dryweight
mass of 105.2 g solids.
Elevation above bottom (m)
70
Figure 3.15. Modeling SSC (unit: kg m Solid lines are simulations and
dashed lines represent field data observed from 1800, 9/24/68 to 0400, 9/25/68
in Savannah River estuary (after Ariathurai, et al., 1977). Contours from the
surface to the bottom are 0.1, 0.25, 0.5, 1, 1.5 and 2, respectively.
-5
Figure 3.16. Consolidation rate, o)^^, as a function of dry density for Doel
Dock mud with c^=80 kg m'^ (after Toorman and Berlamont, 1993).
71
Figure 3.17. Modeling laboratory data of Toorman and Berlamont (1993)
on consolidation. Lines are model simulations and points represent data.
Figure 3.18. Modeled consolidation curve (solid line) compared with
the laboratory data (open circles) of Toorman and Berlamont (1993).
72
Through calibration, the coefficients in Eq. (3.23) are taken as = 1 x 1 0'^ m s ,
^ ^ ^ ^^8 m'^,c^2=680 kg m'^,c,=15 kg m and n,=15
(Figure 3.19). is taken as 26.3 kg m In the modeling test, the deposited materials were
introduced into the consolidating layer in the same way as the experiments. The results of the
vertical profiles of sediment concentration within the consolidating layer are shown in Figure
3.20. It is seen that the modeled results are in reasonably good agreement with the
experimental data.
Dry density, c (kg m'^)
Figure 3.19. Consolidation rate, as a function of dry density for
the laboratory tests of Burt and Parker (1984) with c/=26.3 kg m'\
Hlevation above bottom (m)
73
0.6
0.4
0.2
0
0 400 800 0 400 800
0.6
0.4
0.2
0
Dry density, c (kg m ’)
Figure 3.20. Modeling laboratory data of Burt and Parker (1984)
on consolidation with deposition at the bed-fluid interface.
Lines are model simulations and points represent data.
3.6.4 Interfacial Entrainment
Modeled sediment entrainment over the lutocline was tested against experimental
data of Mehta and Srinivas (1993) using Eq. (3.26). In these experiments, values of
^jj=3xl0'^ and D^=1.6xl0'^ were reported. In their race-track flume experiments, the
74
initially introduced kaolinite mud layer, with a clear upper layer of water, was entrained as
the upper layer velocity was increased gradually. In the modeling test, the measured velocity
profiles were inputted. The same initial mud layer as in the experiment was also inputted.
Then the mud layer was allowed to entrain with time using Eq. (3.26). The modeled and
observed results of the vertical profiles of SSC at different times are shown in Figure 3.21.
The modeled results are seen to be in reasonable agreement with the experimental data.
Figure 3.21. Modeling laboratory data on entrainment by Mehta and
Srinivas (1993). Lines are model simulations and points represent data.
CHAPTER 4
FIELD INVESTIGATION AND DATA ANALYSIS
4.1 Study Area Description
Jiaojiang estuary is located on the east coast of China, about 200 km south of the
Yangtze River (Figure 4.1). Jiaojiang is 200 km long and drains a basin of 6,500 kml The
estuarine segment is only about 35 km in length. The mean water depth below mean sea level
(MSL) in this segment is 4-7 m, and the mean width is about 1 .2 km with a maximum of 1 .8
km. Seaward of its mouth the width increases rapidly, forming Taizhou Bay in shallow
coastal waters. The annual inflow rate is about 6.66x10^ m^, with a mean discharge of
210 m^ s . Semi-diurnal macro-tides prevail there. At the mouth the mean tidal range is
about 4 m with a spring range of 6.3 m. The depth-mean peak tidal current can be up to 2.0
ms"'. The tidal wave is strongly distorted in the estuary within a short distance from the
mouth. Within the estuary itself, the duration of ebb exceeds flood by 1-2 hr. As a result the
tidal currents exhibit a asymmetry; e.g., at Haimen the measured maximum flood and ebb
currents are 2.1 and 1.8 m s , respectively (Zhou, 1986; Dong, et al., 1997; Guan, et al.,
1998).
75
76
Figure 4. 1 . Location map of Jiaojiang estuary, China. Depths are in meters
below lowest astronomical tide. Ml and M2 are mooring sites (Table 4.1);
Cl, C2, C3 and C4 are velocity measurement and SSC profile sampling
stations (Table 4.1); C6 is the site of ASSM (Table 4.1) and T1-T6 are tide
stations. The region between the double dotted lines is the modeled domain.
Jiaojiang is highly turbid with near-bottom SSC often exceeding 10 kg m and
depth-mean SSC greater than 5 kg m’^ (Li, et al., 1993; Dong, et al., 1997). Lutoclines
along with fluid mud of 1-3 m thickness regularly appear except around flood slack. The
river sediment is mostly in the clay and fine silt size range. The suspended sediment is
dominated by clayey-silt with a dispersed mean particle size of 4-6 pm. The fluvial sediment
is estimated to be 1.23x10^ tons yr , with a mean inflow SSC of 0.18 kg m i.e., about
4% the SSC in the estuary. Despite this high load, sediment discharge into Taizhou Bay is
77
only about 1 % that of the Y angtze river (1.2x10* tons yr ' ' ) (Yu, 1987). The sedimentation
rate in this estuary is up to 0.2 m yr‘‘, requiring regular maintenance dredging for
navigation from the sea to the port of Haimen (Li, et al., 1992). Haimen is located near the
mouth of the estuary and is an economic center of this region. Thus, it is most important to
maintain the depth of the navigation channel. For that reason, extensive engineering and
scientific investigations have been carried out in this estuary. These studies have focused on
tidal hydrodynamics, sediment transport and sedimentation by means of field investigations
and horizontal and vertical 2D numerical modeling (e.g., Bi and Sun, 1984; Fu and Bi, 1989;
Li, et al., 1993; Dong, et al., 1997; Guan, et al., 1998).
4.2 Experimental Plan. Methods and Instruments
In order to examine flow and sediment dynamics in the Jiaojiang estuary, three field
campaigns were conducted. Table 4.1 provides the dates, measurements, locations, elevations
at which measurements were made and apparatuses used. Sampling sites are shown in Figure
4.1. In terms of the objectives of these investigations, the observations can be classified into
four types, namely: (1) fluid mud observations at sites Ml and M2, (2) lutocline observations
at site C6, (3) vertical profiles of SSC, currents, temperature and salinity at sites Cl , C2, C3
and C4, and (4) tidal elevations at sites T1-T6.
4.2.1 Fluid Mud Observations
Fluid mud related observations were carried out at site Ml during April 13-23, 1991,
and at M2 during November 9-25, 1995. In both investigations, the monitoring system
consisted of a 2.7 m high metal frame containing an Inter-Ocean (model S4) vector-
averaging electromagnetic current meter and six self-logging optical fiber analite
Table 4.1. Summary of Jiaojiang field campaigns
78
5 cd
c
c
.s
3
(J
'g, C
s
s
J=
G
G
O
Vi
U
X
B c
c/:
V)
o
o
VO
d
O en
O t*a
ro
a >
o
'r-
00 •
00 •
O
00 <
0>
hi) • —
1
fN E
rn ^
^ s
rn G
ts G
(A E
5 c
1
O 60
P 00
^ GO
M
o ea
1
1
1
s ^
o ^
OC f/i
O
O
® M
CJ
c
-2
ts
u.
B
"S
o
x>
o
Xi
kM
O
"S
E
c
4>
Apparatus
•4^
c
u
s
-2
"53
,'-S
Q ."3
c
rT ^
U t/i
t3
6
B
N
X
s
0>
q3
g
t
3
U
o
O
Urn
a.
c
GO
C
'5b
GO
o
H Z
« *' c
^ CO
1 u
O O
U o
T i
Fi <-
U i>
i-j t>
on g
via
Q
x
CL
c
GO
■ S
GO
GO
o
t:
3
O
§
0>
u
O
tL
V)
^ t
on g>
< <D
3
a*
Mutilayer
Niskin bottl
«a
Q
13
:g
•e
H
U
it3
13
c
k«
Uh
H
U
C/D
C/D
3
H
C/D
3*
C/D
I- -a
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4-. cs
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s m
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a:
(/)
c
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CQ
>
CQ
<
s
o
va 0^
go
> o
p
00
f/i O
fe VO
d
i/a
C
"cO
>
^ CQ
cv C/D
00 <-
-- B
a:t
(S ^
CQ c
S^vS
E 'S
S
u
3
O
*4-*
’o
s) m
«>a on
■<
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G
o
C/a
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to
>
13
0)
ed ^
fli nI
X
S'
o
0>
u
r-
> B
o u
>< O
on o
fc
yi m
b
lO
— Q
Ui
o in
4- O
E
d
c "2
i/1 (s'
o
<N
d
■e
<
-0'S
o
■£
<
o i
o
d
o
d
>
_ , Tt
. ^
m
Site
Ml
Ml
(N rj
u ^
(N rj
u ^
u
vy
M2
M2
C6
C6
C6
CJ ^
^ u
^ u
CN
u
Measure.
Current
Fluid mud
Velocity and
SSC profiles
Velocity &
SSC profile
STD'&
Turbidity
Fluid mud
Current
Lutocline
SSC for
Calibration of
ASSM signal
STD&
Turbidity
tie day at
ach site
a.
S B
s-
S 2
Date
12-22
13-23
cy
GO cO
C 0>
C *5o
GO cO
c o>
9-25
9-25
o
o
1
o
Same
Same
O “
™ 3
CL
1- 3
CL
o
o
C/D
C/D
o
-C .
Mont
Year
Apr.
1991
Nov.
1994
Nov.
1995
Superscripts: a: Above sea bed; b: Fractions of instantaneous water depth, //; c: Digital current meter; d: Conductivity-Temperature-Depth sensor;
e: Salinity, temperature and depth; f: Acoustic Suspended Sediment Monitor.
79
nephelometers. The nephelometers were fixed to the frame at six elevations near the bottom
(Table 4.1). However, in the 1991 campaign, useful data were recorded at only three
elevations. These transducers recorded data at intervals of 5 min in 1991 and 10 min in 1995.
The data were bin-averaged over interval of 1 min with sampling at 1 s interval.
The Inter-Ocean current meter was mounted on the frame at 1.75 m in 1991 and 1.5
m in 1995, respectively, above the bottom. The meter logged velocity data sampled at 0.5 s
interval and bin-averaged over 1 min. The logged interval was 5 min in 1991 and 10 min in
1995, so that the meter operated for only 1 min every 5 or 10 min. Practically calm water
surface prevailed throughout the experiments with negligible waves.
4.2.2 Lutocline Observations
Lutocline observations were conducted at site C6 during 0600-1600 on November
15, 1995. A ship-borne Acoustic Suspended Sediment Monitor (ASSM) (made by the
Shanghai Acoustics Laboratory, Academia Sinica) was used to detect the lutocline. The
ASSM consisted of a 0.5 MHZ acoustic transducer/receiver. The acoustic probe was
deployed 1-2 m below the water surface. The entire system was under control of a PC for
synchronization of sampling, preliminary data reduction and storage. The device had a pulse
length of about 40 ps, and measured the vertical profiles of sound scattered from suspended
sediments in the range bins at 0.6 sec interval with a vertical resolution of 5 cm. The data
were sampled at a rate of approximately 75 kHz for 9 min bursts. Each data burst consisted
of 900 profiles of backscattered acoustic energy from suspended sediment particles between
the bed and the acoustic probe.
80
4.2.3 Observations of SSC. Currents. Salinity and Temperature
Vertical profiles of SSC, tidal currents, temperature and salinity were obtained at sites
Cl, C2, C3 and C4 (Figure 4.1), respectively using Niskin bottles (each 60 cm long and 10
cm in diameter), SLC9-1 digital current meters (made by the Institute of Marine Instrument,
Qingdao), CTD probes, turbidimeters and a ‘mud probe’ which consisted of a CTD
transducer equipped with an Analite, infra-red, backscattering nephelometer (Wolanski, et
al., 1988). This profiler was able to measure SSC from 0.03 to 80 kg m The time series
of such profiles were collected over 1 or 2 tidal cycles. Water samples were collected in the
Niskin bottles at six elevations between the water surface to the bottom. Additionally, water
samples were collected at 0.3, 0.6 and 1 m elevations above the bed using three horizontally
deployed Niskin bottles mounted on a solid frame. These bottles were raised on board within
about half a minute of sampling underwater, and water samples were then drawn
immediately for analysis of SSC, salinity and sediment size (Li, et al., 1993; Dong et al.,
1997).
4.2.4 Tidal Elevations
Tides were obtained at six sites from T1 to T6 (Figure 4.1) for a one-month period.
Each time-series was processed by harmonic analysis (Dong, et al., 1997; Guan, et al., 1998).
4.3 Experimental Data
4.3.1 Sediment Size
The river sediment is mostly in the clay and fine silt size range. The suspended
sediment is dominated by clayey-silt with a dispersed mean particle size of 4-6 pm (Figure
4.2). About 50% by volume of the particles constitute very fine silt (size range between 4-16
81
urn). The particle fraction with size less than 4 pm accounts for about 40% of the material
by volume (Li, et al., 1993; Li, et al., 1999).
<u
B
s
o
>
>>
u
c
u
3
<T
U
b
Diameter (pm)
Figure 4.2. A representative frequency distribution of
suspended sediment (dispersed) size in the Jiaojiang estuary.
4.3.2 Tides
1. Elevation: Harmonic analysis showed that the Mj and S2 constituents dominate
this estuary. Figure 4.3 shows the time series of tidal elevations at sites T1 and T5 during a
spring tide. It is seen that the tide was distorted as it propagated upstream from the mouth.
For example, at T1 (outside the mouth), the flood lasted 5.78 hr and the ebb 6.64 hr. In
82
contrast, at T5 (inside the estuary) the flood was 5.29 hr and ebb was 7. 13 hr. In other words,
within a short distance of about 1 8 km from the mouth to upstream, the duration of flood
decreased by about one-half hour.
Figure 4.3. Time series of tidal elevation at sites T1 and T5 during
a spring tide from 0000 hr on 1 1/05/94 to 0100 hr on 1 1/06/94.
2. Currents: As a result of tidal distortion, the tidal currents also exhibited an
asymmetry as shown in Figures 4.4-4.6. It is seen that the flood current duration was shorter,
and its strength was greater, than the corresponding ebb current values. Thus, for example,
at C4 the ratio of depth-mean maximum flood current to ebb current was about 1 .4 during
the spring tide and about 1 .25 during the neap tide.
83
4.3.3 Profiles ofSSC
Figures 4.7 and 4.8 show typical time series of SSC profiles at C4 during a spring and
a neap tide, respectively, and Figure 4.9 for C6 during a neap tide. It is seen that during low
water slack and neap tide, the lutocline was distinct and well defined. In contrast, during
spring tide and peak flows, the lutocline became indistinct even though the fluid mud layer
(as defined in Section 2.1) became thicker. This observation supports the conclusion stated
in later Section 5.2 regarding the relationship between the robustness of the lutocline and the
flow conditions. Also at these sites, the vertical mean SSC was always greater than 5
kg m and often exceeded 10 kg m .
4.3.4 ASSM Data
Figures 4.10 (a) and (b) show two typical ASSM outputs, each over a 60 s time
segment. Observe that most of the time the ASSM signal had a distinct step structure. As
demonstrated in Figure 4.1 1, this step structure matches the real lutocline position detected
by the turbidimeter. Hence it is reasonable to treat the step structure in the ASSM signal as
the location of lutocline. Also observed are lutocline undulations, suggesting noteworthy
internal wave generation and likely sediment entrainment activity. A smoother interface was
observed during ebb [Figure 4.10(b)] than during flood [Figure 4.10(a)].
The SSC(=c) can be calculated from the ASSM reading, F^, from (Thome et al.,
1994)
(4.1)
where h' is the depth below the acoustic probe and k^, h^, and are sediment-
84
Figure 4.4. Time series of velocity at site C4 during a spring tide. Observations began at
1700 hr on November 5, 1994. Positive numbers signify flood and negative are for ebb.
Figure 4.5. Time series of velocity at site C4 during a neap tide. Observations began at
0900 hr on Novermber 10, 1994. Positive numbers signify flood and negative are for ebb.
85
Figure 4.6. Time series of velocity at site C6 during a neap tide. Observations began at
0630 hr on November 15, 1995. Positive numbers signifie flood and negative are for ebb.
Figure 4.7. Time series of SSC at site C4 during a spring tide. Observations began
at 1700 hr on November 5, 1994. Shaded area includes SSC greater than 20 kg m'^.
86
Time (hr)
Figure 4.8. Time series of SSC at site C4 during a neap tide. Observations began
at 0900 hr on November 10, 1994. Shaded area includes SSC greater than 20 kg m ^
8 10 12 14 16 18
Time (hr)
Figure 4.9. Time series of SSC at site C6 during a neap tide. Observations began
at 0600 hr on November 15, 1995. Shaded area includes SSC greater than 20 kg m 'I
87
(b)
Figure 4.10. Typical raw ASSM records during the neap tide on Nov. 15,
1995, with a horizontal time scale of 1 min and a vertical distance scale
of 1.25 m. (a) was observed during a flood with a value of Richardson
number Ri^ of about 2, and (b) during an ebb with Ri^ of about 150.
88
dependent constants. As mentioned above, the location of the largest vertical gradient of
can be taken as the lutocline elevation, . above the bottom. From the time series of C^(0 ,
a spectral analysis of internal waves can be carried out, and related to local hydrodynamics.
Figure 4.1 1. Relationship between lutocline elevations above bottom detected by the
turbidimeter and by the ASSM. Data were collected during 0600-1600, Nov. 15, 1995.
Figure 4.12 shows three typical time-series, i.e., examples (a), (b) and (c), of
lutocline elevation, C,Jj), traced from 9 min long ASSM records. It is evident that there are
two types of internal waves riding on the lutocline, namely, low frequency internal waves
with a period on the order of 1 min and high frequency internal waves with a period of about
5 s.
89
C3
O
(N (N (N
VO (N (N
ri (N (N
VO Tt (N (N
ri (N (N
(ui) uioijoq 9AoqB uoijba9|9 9uijoojnq
Figure 4.12. Time series of lutocline elevation at site C6 during a neap tide using ASSM on November
15, 1995. (a) and (b) were sampled during flood with a value of Ri^ of about 1, and (c) during ebb
with Ri^of about 150. Solid lines are instantanious elevations and dashed lines are mean trends.
90
Two distinct types of internal waves on the lutocline have also been reported by other
researchers. From the observation of internal waves on the lutocline in a race-track flume
experiment, Mehta and Srinivas (1993) described two types of internal waves; One was
characterized by high frequency and breaking. While the other included large-amplitude
solitary waves which interspersed with the breaking waves and seemed to decay without
breaking. Shi (1998) also observed two types of internal waves on the interface between the
mobile and stationary fluid mud in Hangzhou Bay in China using an ASSM meter. These
included low frequency waves with a period on the order of minutes and high frequency
waves with a period of seconds. Wright, et al. (1988) observed low frequency (-6x10"'*
rad s‘*) and relatively large amplitude internal waves on the upper interface of the
underflows over the active delta front of the Yellow River in China. They found that these
internal waves had frequencies near the local Brunt-Vaisala frequency (~ 5 .3 x 1 0 rad s ‘ ‘ ).
For the purpose of spectral analysis of the high frequency internal waves, the
lutocline elevation time-series processed for trend removal using an approach used by Costa
(1989), i.e., filtering out the low frequency internal waves. The results after trend removal
are shown in Figure 4.13. Figure 4.14 shows an enlarged portion of the time-series
accentuating the internal waves. It is seen that the internal waves usually have sharp crests
and flat troughs, which is most similar to the Holmboe-type interfacial instability (Broward
and Wang 1972). In experiments of Srinivas (1989), in general both Kelvin-Helmoltz and
Holmboe modes of interfacial instabilities appeared. However, here the Kelvin-Helmoltz
instability is difficult to identify. Due to the interfacial instability, a net upward-asymmetric
91
during a neap tide, where (a), (b) and (c) correspond to Figure 4.12.
92
mass mixing can be expected in this case. Mixing can be further enhanced if the internal
waves begin to break.
Figure 4.14. Typical profiles of internal waves exhibiting sharp
crests and flat troughs. Data were taken from example (a) in
Figure 4.13. Wave heights range from 0.07 m to 0.23 m.
Based on the Wiener-Khintchine theorem, the internal waves spectrum, S'Cto '), can
be calculated using (Bendat and Piersol, 1971)
5((0^)=— J R(x)cos((O^T)dv
^ 0
with the auto-correlation fimction, /?(t) , defined as
(4.2)
1 ^
R(x)=lim^ f HJ,t)HJJ^x)dt
/ J
(4.3)
-T
where is the internal wave profile, x is the time interval or shifting time, T'is the
integration time-limit and o)^ is the angular frequency (Ochi, 1990). The auto-correlation
93
functions for examples (a), (b) and (c) are shown in Figure 4.15. The resulting respective
spectral densities are plotted in Figures 4.16-4.18. It is seen that the maximum spectral
density is located around the modal frequency o)^=T3 rad s '' during flood, with Ri^ of
about 1 (Figures 4.16 and 4.17), and around w^=l.l rad s '' during ebb, with Ri^ of about
150 (Figure 4.18).
A further examination of internal waves was made by relating the root mean square
(rms) of the height of the internal wave as well as its modal frequency (both for high and low
frequency waves) to the global Richardson number and the tidal range. The rms of the high
frequency wave height and the corresponding angular frequency were taken over 1 min
segments of the ASSM output. One segment was selected from each 9 min ASSM burst
(totally 21 bursts) for this purpose. The rms of the low frequency wave height and the
corresponding angular frequency were taken over each 9 min segment of ASSM burst. The
global Richardson number, Ri^, was calculated from Eq. (3.26) using the current velocity
and SSC profiles. The lutocline was considered to be at the near bottom elevation having the
maximum vertical gradient of SSC. The average value of Ri^ over 1 min was taken for the
high frequency waves and 9 min for the low frequency waves. The rms wave height, is
calculated from
\
1
2
an
(4.4)
where H is the n wave height and N is the total number of waves over the selected data
Spectral density, S (m s) Autocorelation function, /?(x) (m
94
X lO '^
Figure 4.15. Auto-correlation function against time interval,
where (a), (b) and (c) correspond to Figure 4.12.
X 10
Angular frequency, co^ (rad s'')
Figure 4.16. Internal wave spectrum corresponding to example (a) in Figure 4.13.
95
-4
X 10
Angular frequency, o)„ (rad s"')
Figure 4.18. Internal wave spectrum corresponding to example (c) in Figure 4.13.
96
segment, i.e., 1 min or 9 min. The modal frequeney is caleulated as
^a=^r- (4.5)
where is the duration of each segment.
The processed results are shown in Figures 4.19-4.26 and summarized in Table 4.2.
It is seen that // and both for high and low frequencies, have reasonably good
relationships Avith the global Richardson number. In general, the lower the global Richardson
number, the higher the and values. This behavior is examined in Section 4.4.
There appears to be no identifiable relationships between // , (o^ and the tidal range.
However, the Richardson number was typically much higher during ebb than during flood,
because the ebb exhibited comparatively more uniform velocity profiles (Figure 4.6). As
observed in Figures 4.21, 4.22, 4.25 and 4.26, internal waves had lower heights and angular
frequencies during ebb than during flood. During ebb the majority of the data points are
located below the mean trend-line. In contrast, the majority of the points are located above
this line during flood.
Following Wright, et al. (1988), a possible relationship between the internal wave
frequency and the Brunt-Vaisala frequency is examined here. The Brunt-Vaisala frequency,
0)^,, is defined as
p dz
(4.6)
Modal frequency, co„ (rad s ‘) nns internal wave height, (m)
97
Global Richardson number, Ri^
Figure 4.19. rms of high-frequency internal wave height
as a function of global Richardson number.
Global Richardson number, Ri^
Figure 4.20. Modal frequency of high-frequency internal
waves as a function of global Rechardson number.
Modal frequency, co, (rad s'*) ‘"Vernal wave height, H, (m)
98
Time (hr)
Figure 4.21. rms of high-frequency internal wave height during ebb and flood.
1.6
1.5
1.4
1.3
1.2
1.1
1
6 8 10 12 14 16 18
Time (hr)
Figure 4.22. Modal frequency of high-frequency internal waves during ebb and flood.
Ebb
O
Flood
Ebb
OD 000
Mean o) =1.326 s'
00
GOOD
Modal frequency (rad s ‘) ^ rms internal wave height, H, (m)
99
Global Richardson number, Ri^
Figure 4.23. rms of low-frequency internal wave height
as a function of global Richardson number.
Global Richardson number,
Figure 4.24. Modal frequency of low-frequency internal
waves as a function of global Richardson number.
Modal frequency, co, (rad s *) rms internal wave height, (m)
100
0.55
0.5
0.45
0.4
0.35
0.3
0.25
6 8 10 12 14 16 18
Time (hr)
Figure 4.25. rsm of low-frequency internal wave height during ebb and flood.
Ebb
Flood
Ebb
(DO
CP
^
Mean =0.382 m
C?
cP
0.12
0.11
0.1
0.09
0.08
0.07
0.06
0.05
6 8 10 12 14 16 18
Time (hr)
Figure 4.26. Modal frequency of low-frequency internal waves during ebb and flood.
OOO
Ebb
cio
ODOOO
Mean « =0.089 s'
O OO
Ebb
Flood
OO
101
Table 4.2. rsm height and angular frequency of internal waves
as functions of global Richardson number and tidal range.
Type of
internal
wave
Parameter
Relationships
H^{m)
M s'*)
/?/q
i/^=0.25 -0.041og(R/Q)
a)^=1.4-0.11og(R/o)
High
frequency
Tidal
range
Flood: 9 out of 10 points are
above the mean value of
// =0.205 m
Ebb: 5 out of 6 points are
below the mean value of
//j =0.205 m
Flood: 6 out of 9 points are
above the mean value of
o)^=1.326 rad s '*
Ebb: 5 out of 6 points are
below the mean value of
(0^=1.326 rad s '*
R/q
7/^=0.4375 -0.05251og(/?/o)
co^=0.1025-0.01251og(R/o)
Low
frequency
Tidal
range
Flood: 1 1 out of 12 points are
above the mean value of
//^ =0.382 m
Ebb: 7 out of 8 points are
below the mean value of
// =0.382 m
Flood: 1 lout of 12 points
are above the mean value
of 0)^ =0.089 rad s
Ebb: 6 out of 8 points are
below the mean value of
(0^=0.089 rad s “*
Similar to the global Richardson number formula (3.26), one can approximate O) as
0) =
V
SCJC,-C,)
N \ Kt
g'
(4.7)
where g' is the reduced gravity. To estimate using Eq. (4.7), the SSC profiles (Figure 4.9)
observed at the same time as the ASSM data were employed. The results indicate that o) has
a range of 0.12-0.26 rad s' '. As shown in Figures 4.24, 4.26 and Table 4.2, the low
frequency waves had frequencies ranging from 0.06 to 0.12 rad s'* and a mean value of
102
about 0.09 rad s * . Thus, the low frequency waves had modal frequencies near the local
Brunt- Vaisala frequency. This type of internal wave at haloclines or thermoclines has been
extensively studied by previous researchers (e.g., Lamb, 1945; Neumann and Pierson, 1966;
Phillips, 1977). Theoretically, in a stratified fluid a vertically-moved fluid particle due to any
disturbance will be restored to its original level under buoyancy force and overshift inertially.
Consequently, it will oscillate about the original level at the Brunt-Vaisala frequency. Thus,
it is believed that the low frequency wave detected in the Jiaojiang exhibited a natural
oscillation about the lutocline.
4.3.5 Floe Settling Velocity
In order to examine the characteristic dependence of the settling velocity as a function
of SSC in this estuary, the SSC profile data shown in Figures 4. 7-4. 9 were used. One may
conveniently select the SSC profiles during slack water for this purpose, since the quiescent
water assumption is approximately satisfied then. From the sediment conservation Eq. (3.1 5),
the simplified vertical equation for SSC transport is
dc 1 d(o,c
dt H da
(4.8)
For solving Eq. (4.8) to obtain the SSC profile c(o,f), the initial and boundary conditions are
specified as follows:
Initial condition:
C(O,0)=Cg(O)
(4.9)
Zero settling flux boundary condition at the water surface (o=0):
103
(4.10)
Zero settling flux boundary condition at the bottom (o=-l):
(4.11)
Based on the above boundary value formulation, one can approximately calculate the
settling velocity, o)^, as a function of SSC. This analysis has been programmed by Mehta and
Li (1997). Accordingly, the constants a, 6, a and P in the settling velocity formula (3.30) can
be determined by best-fitting Eq. (3.30) against the data points (for settling velocity against
SSC). To determine the coefficients in settling velocity formula (3.30), the effects of
temperature and salinity on flocculation are neglected and the settling velocity formula is
simplified as
CO.
ac
{c^+b^f
(4.12)
Through best-fitting of the calculated data points, the values of the coefficients in Eq.
(4.12) are obtained as: a=0.045, 6=6, a=1.5 and P=1.51 during neap tide (Figure 4.27), and
0=0.23, 6=10, a=1.5 and P=1.8 during spring tide (Figure 4.28). These values are consistent
with c (=SSC) measured in kg m and co^ measured in m s . It is observed that the
maximum SSC for flocculation settling, (approximately equal to 6), increases during
spring tide compared with neap tide, with C2=6.0 kg m during neap and C2=8.4 kg m
during spring. In contrast, the settling velocity during spring tide is less than during neap tide
(Figures 4.27 and 4.28). Also shown in Figures 4.27 and 4.28 is the data point of Li, et al.
104
-1 0 1 •>
10 10 10 10
SSC (kg m'^)
Figure 4.27. Settling velocity as a function of SSC during a neap
tide from 0900 hr on 1 1/10/94 to 1000 hr on 1 1/1 1/94. Solid line
is the best-fit of the calculated data points using Eq. (4.12).
Figure 4.28. Settling velocity as a function of SSC during a spring
tide from 1700 hr on 1 1/05 /94 to 1800 hr on 1 1/06/94. Solid line
is best-fit of the calculated data points using Eq. (4.12).
105
(1993) determined using the Postma’s ‘pipette’ method (McCave, 1979) during the 1991
campaign in the Jiaojiang (Table 4.1). Finally note that in both plots is the free settling
velocity.
As described by Burt (1984), the tidal range, or associated turbulence has two effects
on flocculation and, consequently, the settling velocity. Increasing turbulence may enhance
flocculation and at the same time limit the size of floes that can be sustained. In other words,
depending on its cumulative effects in enhancing flocculation and limiting floe size,
increasing turbulence may either increase or decrease the settling velocity. In the Jiaojiang,
limiting floe size apparently dominated during spring tide, while enhanced flocculation
occurred during neap.
4.3.6 Erosion Rate Constant
The erosion rate constant, M^, in Eq. (3.29) can be determined from the vertical
profiles of current velocity and SSC shown in Figures 4.4-4.9. The vertical profiles chosen
for this purpose are those corresponding to periods when the suspended sediment mass in the
entire water column increased gradually, usually 3-4 hr during each period of measurement.
Also, the following assumptions are made:
(1) SSC is approximately uniform along the estuary, so that the longitudinal
advection term in the sediment conservation equation (3.15) can be omitted, i.e., udc/dx~0.
(2) Within a short erosion period of 3-4 hr, the bottom shear strength can be
approximately taken as constant.
106
(3) The effect of temperature on erosion rate is negligible.
From the data two parameters are obtained, i.e., the terms in Eq. (3.29) including the
excess bottom shear stress, exp|-XT^) and the erosion rate, m^. The bottom shear
stress is calculated from the vertical mean velocity according to
_ 9grif t/
Z/l/3
(4.13)
where rij. is the Manning’s bed resistance coefficient taken as 0.015 (Dong et al., 1997), and
U is the vertical mean velocity. The bottom shear strength is taken as the bottom shear stress
at the beginning of each selected time period. Also, according to Lee and Mehta (1994), the
values of x=8 and A=0.5 are applicable. Finally, the erosion rate is calculated from the SSC
profiles according to
m
e
At
(4.14)
where C is the vertical mean SSC, superscript n denotes time and At is the time interval
between two profiles.
Through best-fitting of the data points (Figure 4.29), the erosion rate constant,
0.29 kg N s ■' is obtained. Vinzon (1998) obtained erosion rate constants by analyzing
prototype data collected by Kineke (1993) in the same way. Her data yielded the range of
M to be 0.25-0.34 kg N-‘ s-‘.
max ®
107
4.4 Properties of Internal Waves
The results in Section 4.3.4 show that both the rms height, // , and the modal
frequency, (o^, of internal waves decrease with increasing Ri^. An attempt is made here to
explain this behavior by referring to works of previous researchers. Also examined in this
section are the celerity and length of internal waves.
4.4. 1 Effect of RIq on
Since high Ri^ implies high buoyancy-induced stabilization of the lutocline,
increasing Ri^ should correlate with decreasing wave height. As described in Section 1 .2,
this phenomenon has been observed by previous investigators in laboratory experiments on
108
lutoclines (e.g., Mehta and Srinivas, 1993) and on other pycnocline (e.g., Chou, 1975;
Narimousa and Fernando 1987). By assuming that the interfacial undulations are due to the
energy-containing, mixed-layer eddies impinging on the density interface, Narimousa and
Fernando (1987) established an empirical relationship between the wave height, // , and
as follows
-1/2
Y~ (4.15)
'%ix
In Eq. (4.15), is the mixed-layer thickness, i.e., the upper layer depth. According to this
expression, decreases with increasing Ri^ with a slope of 0.5 on a log-log plot. As noted
in Section 4.3.4, the observations in Jiaojiang do show that decreases with increasing
RIq, although slopes much smaller than 0.5 were found (Figures 4.19 and 4.23, and Table
4.2). This difference in slopes between the Jiaojiang and the laboratory results of Narimousa
and Fernando (1987) is believed to be mainly due to different physical scales and associated
hydrodynamic effects including the degree of turbulence and eddy lengths.
4.4.2 Effect of Rif. on (o
In order to examine the influence of the Richardson number on the modal frequency
of internal waves, the work of Lamb (1945) is introduced here. Lamb analytically examined
internal waves at the interface of two inviscid fluids of densities p, and beneath the
other, and moving parallel to the x-axis with velocities f/, and , respectively (Figure
109
4.30). By assuming both fluids to be of unlimited depth and taking the wave profile as
i(u)J-hc)
(4.16)
Lamb derived the following expression of the dispersion relationship for the waves:
P|^+P2^
P1+P2
±
g(p2~Pl)
^Pl+P2)
P1P2
(Pl+P2)^
1
2
(4.17)
where k is the wave number. The first term on the right-hand side of Eq. (4.17) is referred
to as the vertically-averaged velocity, U, of the two layers. It is seen that the values of
given by (4. 1 7) are imaginary if
g(p2~Pi) ^ P2 „1
P,+P2~2
and it is also recognized that for two fluids of nearly equal densities, such as water and fluid
mud, p2/(p,+p2)==0.5.
It is evident that under the condition imposed by (4. 1 8), two possible cases can arise
with respect to the sign of the second term of Eq. (4.17). Considering Eqs. (4.16) and (4.17),
it is seen that taking the plus sign the wave height will dissipate with time. This inherently
implies that the interface will be stable. On the other hand, if the minus sign is taken, the
wave height will grow with time. In other words, the interface will be unstable. For the
present analysis, only the unstable mode is of interest, i.e., with the minus sign relative to the
second term in Eq. (4.15). Thus, as soon as (4.18) is satisfied the interface will become
110
unstable. This is known as the Kelvin-Helmholtz instability (Delisi and Corcos, 1973). If
now one considers the internal waves of all likely wavelengths in the estuarine setting such
as in the Jiaojiang, it can be concluded that sufficiently short waves will be present to cause
interfacial instability. Therefore, a two-layered estuarine shear flow characteristically
unstable.
Figure 4.30. Definition sketch of two-layered flow system.
Based on the above, holding p, and P2 constant, Eq. (4.17) can be restated as
(0
‘u,-vr
a a eft
V )
(4.19)
where /l^-g(p2-p,)/^(p,+P2), 5^=p,P2/(Pi+p2)^ and //^ is the effective water depth
defined as the thickness affected by internal waves. Further holding the wave number k and
the mean velocity U constant, one may quantitatively evaluate the influence of the velocity
Ill
gradient, \U^-U^\/H^j^, on o)^. Thus, Eq. (4.19) can be expressed in terms of a stream
Richardson number, Ri , as
. a eff
Ri.
i_
2
where 5^'=5^(p2-p,)/ Pj and Ri^ can be conveniently defined as
(4.20)
P,- _^P2-Pi)^.,r
(4.21)
Thus Ri^ is conceptually analogous to Ri^ [Eq. (3.26)]. From Eq. (4.20), it is seen that
decreases as Ri^ increases, an observation that is consistent with the data in Figures 4.20 and
4.24.
4.4.3 Celerity and Wave Length
In order to further understand the properties of internal waves at the lutocline in the
Jiaojiang, the celerity and wave length are calculated here. For simplification, the flow is
treated as two-layered system with fluid densities of p, and P2 in upper and lower layers,
respectively. Once the reduced gravity g' [ =g(p2-p,)/p] and the lutocline elevation above
the bottom are known, the celerity and the wave length can be calculated according
to (Lamb, 1945)
^tanh(A:CJ ,
to:
(4.22)
where k is the wave number ( =2-k/XJ. From the measured SSC profiles (Figure 4.9),
112
lutocline elevation (Figure 4.12) and the modal frequency of internal waves (Section 4.3.4),
the calculated results for examples (a), (b) and (c) in Figure 4.12 are presented in Table 4.3.
For these calculations, the mean lutocline elevation was taken for each ASSM segment, and
an iteration method was used in solving Eq. (4.22).
From Eq. (4.18), one can calculate the critical wave length, below which the
interface will become unstable, i.e., the Kelvin-Helmholtz (Delisi and Corcos, 1973) or
Holmboe (Browand and Wang, 1972) instabilities due to interfacial shear. By equating the
two sides of Eq. (4.18) one obtains
X =
g(pl~p])
RL
(4.23)
Values of calculated from Eq. (4.23) are given in Table 4.3. Also listed in Table 4.3 are
the Brunt- Vaisala frequencies calculated from Eq. (4.7).
From Table 4.3 it is seen that the high frequency waves were characteristically in
deep water, with the ratio on the order of 5 (»0.5), whereas the low frequency waves
were close to the shallow water regime, with on the order of 0.07 (=0.05). It is also
observed that the celerity and wave length increased with increasing Richardson number for
both high and low frequency waves. Observe that X^^ decreased with increasing Richardson
number (ranging from 2.71 m to 0.06 m). Thus, the wave lengths for high frequency internal
waves (ranging from 0.39 m to 0.65 m) are between the maximum and minimum critical
wave lengths for stability. This suggests that the high frequency waves are generated by
forcing due to interfacial shear.
113
cd
B
(U
.s
o
C3
<U
*a
u
cn
jj
IS
cd
00
VO
c^
00
O
(N
d
o^
iO
S
VO
o'
d
d
>
cd
>
>,
.o
00
(N
(N
o
„ Q CO
00
«-H
a
'o rH
o
o
1—1
(U
O"
0)
*
E
N— ✓
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d
d
W)
Q *CO
00
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3h
3 ^
(N
o
a
T-H
(N
^ s
00
1
(N
m
>>
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o
VO
o
. a CO
VO
m
c
C
in
<U
;3
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0)
4-1
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d
d
%
T
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hJ
a
o
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3 "S
I—*
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o
Cd
t-i
d
d
d
>.
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3 T3
fN
(N
CN
2
d
d
d
O
VO
m
o
--
<N
in
<N
VO
ON
o^
o^
1-H
tX) -
O
O
d
d
d
a o
00
o
VO
5 S
00
-s: w
'-'
(N
o
m
1
>o? 6
in
(N
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CHAPTER 5
TURBULENCE DAMPING IN FLUID MUD
5.1 Introduction
As stated in Chapters 1 and 2, the vertical mixing pattern of suspended sediment over
the lutocline is highly dependent on the nature and intensity of turbulence in both layers.
Laboratory experiment results (e.g., Wolanski, et al., 1989; Mehta and Srinivas 1993;
Winterwerp and Kranenburg, 1997) and field investigations (e.g., Jiang and Wolanski, 1998)
have revealed that upward mixing caused by the instability and breaking of internal waves
at the lutocline occurs concurrently Avith turbulence damping within the fluid mud layer
below.
Turbulence damping by suspended sediment was early examined by Einstein and
Chien (1955). They argued that since part of the turbulent energy is used to maintain the
sediment particles in suspension, turbulence is damped by the suspended sediment. Since
then, turbulence damping in the fluid mud layer and consequent (vertically) asymmetric
mixing over the lutocline have been commonly reported (e.g., Wolanski, et al., 1992;
Scarlatos and Mehta, 1993; Kranenburg and Winterwerp, 1997; Jiang and Wolanski, 1998).
However, there remains a lack of a theoretical basis as well as any direct evidence of this
feature of mixing because of the difficulty in observing it in the field.
114
115
Here, turbulence damping in the fluid mud layer is examined on a phenomenological
basis. Its effect on lutocline formation in the Jiaojiang is then explained.
5.2 Turbulence Damning and its Effect on Lutocline Formation
For a simplified treatment, we will consider a steady uniform flow, treat fluid mud
as a Bingham plastic and omit advective effects.
The rate of production of turbulent energy associated with the Reynolds stress (per
unit volume) can be expressed as x^^du/dz (Rossby and Montgomery, 1935). Based on the
Prandtl mixing theory, Einstein and Chien (1955) proposed following formula for the
Reynolds stress, t^, in a uniform flow
— m'(P,-c)w-m'c(w-o) )
P.Po ,
Po^-
du
dz ,
(5.1)
where u' is the turbulent fluctuation of the horizontal velocity in the x direction, / is the
momentum mixing length, and the overbars denote time-averaging. Hence the rate of
production of turbulent energy becomes
du
' dz
P,Po
^41)'
(5.2)
The rate of work done against buoyancy due to stratification is (Odd and Rodger,
1978)
dz
du
du
dz
fn fn
dz
(5.3)
116
where is the vertical flux of buoyancy and is the ratio of the mass mixing length, /^,
to the momentum mixing length, /^, i.e., the turbulent Schmidt number.
The rate of work done against gravity is (Hunt, 1954)
(5.4)
where A/ is the vertical mass flux and c'is the turbulent fluctuation of SSC. For a steady
flow, from the sediment and water continuity considerations and the Prandtl mixing theory,
Einstein and Chien (1955) obtained
w'c'=-
p -c
-CO).
(5.5)
Substituting Eqs. (3.27) and (5.5) into Eq. (5.4), rate of work done against gravity becomes
g(p,-Po)(p,-c)
gM ^ : cw.
2
p.
(5.6)
The rate of work done against cohesion and interactions between the floes in the fluid
mud is
^ du _/ \du
dz
(5.7)
where is the total shear stress due to cohesion and interactions between the floes and
is the shear stress due to the interactions between floes. Bagnold (1954; 1956) examined the
normal and tangential stresses in granular flows and suggested that they may be expressed
117
as functions of the shear rate, du/dz, and a “linear sediment concentration”,
characterizes the relative surface proximity between sediment particles (Figure
related to the sediment concentration, c, by
1
where is the maximum concentration corresponding to grain-grain contact.
c^, which
5.1). is
(5.8)
< >
Figure 5.1. Definition of linear sediment concentration, c^, and its
relationship with sediment concentration, c. the floe diameter.
Observe that increases drastically as c approaches (Figure 5.1). For small.
light grains in a very viscous fluid, Bagnold (1954; 1956) found that the behavior of the
118
mixture of fluid and cohesionless grains is dominated by viscosity due to interactions
between particles, and termed it the macro-viscous regime. In this regime the shear stress has
the form
tancj)
V = 1.3(l+c,)
1+-A
/
du
(5.9)
where (})^ is the dynamic angle of repose, which was found to be (J)^=37° (Bagnold, 1954;
1956), and p is the dynamic viscosity of the fluid. To extend the applicability of this formula
to fluid mud, it is reasonable to assume that the behavior of a mixture of fluid and cohesive
sediment is phenomenologically analogous to the behavior of a mixture of fluid and
cohesionless grains. Accordingly, Eq. (5.9) can be rendered applicable to the fluid mud
merely by taking the dynamic viscosity p to be that of the fluid mud, and treating floes as
basic particles.
Following the original arguments of Rossby and Montgomery (1935), under an
assumed equilibrium condition the sum of the kinetic, potential and dissipated energy in a
stratified and cohesive flow per unit mass can be considered to be the same as that in a
homogeneous, non-cohesive flow at identical shear rates. Thus, combining Eqs. (5.2), (5.3),
(5.5), (5.7) and (5.9) one obtains
2
dz I
P.Po
Po^i
li dwV
ATz
Sr mm ^ ^
dz dz
. ^(P.-PoXP.-^^) , X ^
+ C0)^+I.3tan4)/1 +q)
P.
f
1+^
1 2j
du
^ dz
(5.10)
119
Solving Eq. (5.10) for /pleads to
(5.11)
with
P.Po
; _ (P.~P0)(P,-g)^
pjpo
i?/^ = 1.3tanc|)j(l +q)(l +-^) ti
(5.12)
Ri^
9(floidu/dzf
Here is a dimensionless variable dependent on SSC, Ri^ is the ratio of the potential
energy of the sediment settling flux to the production of turbulent energy, ll^{du/dzf ,
(denoted as F^), Ri^ is the ratio of the viscous force due to the interactions between floes in
the fluid mud to the Reynolds stress, and Rig is the ratio of Bingham yield stress to the
Reynolds stress.
Equation (5.1 1) shows that at equilibrium the higher the potential settling flux of
suspended sediment (as would occur in quiescent water), the smaller the turbulent mixing
length. In other words, the turbulent kinetic energy is damped because part of it is used to
maintain floes in suspension. Due to hindered settling, the sediment settling flux will
120
decrease with increasing SSC (Ross and Mehta 1989). Thereafter, cohesion and the
interactions between floes can be expected to play an increasingly important role in
turbulence damping. To demonstrate this behavior, one may introduce the following
formulas and parameters:
(1) Settling Velocity: Here the floe settling velocity formula (3.30) is applied, with
a=0.085, 6=10 kg m'^ a=1.5, P=1.6, and without considering the effects of salinity and
temperature on flocculation.
(2) Bingham Yield Strength: The Bingham yield strength is expressed as a function
of SSC as follows
(5.13)
Here and are sediment-dependent coefficients. Following Owen (1970) and Odd and
Rodger (1986), ag=7.36xl0“^ and P^=2.33 are chosen.
(3) Momentum Mixing Length: In the near-bottom layer of a homogeneous, non-
cohesive flow, the following Prandtl (1925) approximation for the momentum mixing length
can be applied
(5.14)
where is the elevation above the bottom, and von Karman constant k is normally taken
as 0.4.
(4) Velocity Profile: In the near-bottom layer, the velocity distribution will be
considered to be a locally logarithmic, i.e..
121
“.n
w(z.)=— In—
K
(5.15)
Where is the bottom frictional velocity in homogenous flow.
(5) Viscosity of Fluid Mud: Following Odd and Rodger (1986), the viscosity of fluid
mud, |i, with SSC<80 kg m ^ , which is the critical SSC for soil formation, will be taken as
0.01 Pa s.
(6) Water and Granular Densities: Values of water and granular densities taken here
are Pq=1,000 kg m'^ and =2,650 kg m■^
(7) Maximum Concentration: As mentioned above, for interactions between particles
in fluid mud the floes are treated as basic particle units. Hence the maximum concentration,
^max’ should Correspond to floc-floc contact. Floes tend to be very loosely bound and are
light in water, with a typical bulk density, p^, of about 1,080-1,150 kg m which will
contain nearly 95% by volume of water locked within the interstitial particulate fabric (Mehta
and Li, 1997). Accordingly, the maximum concentration corresponding to the floc-floc
contact can be expressed as
=
(Pp-P/K
C„
(5.16)
Where is the maximum volumetric floe content corresponding to floc-floc contact. In
analogy with the water-sand mixture (Bagnold, 1956), c^^=0.65 will be chosen.
122
Substituting the above parameters and expressions into Eq. (5.1 1), the ratio of /
is calculated and plotted in Figure 5.2 as a function of SSC and the production of turbulent
energy, F^, below SSC<80kg m ^ without considering the effects of buoyancy. Also shown
is the sediment settling flux as a function of SSC. It is observed that within the flocculation
settling range, maximum turbulence damping is consistent with the maximum settling flux.
Thus, a lutocline can be expected at the elevation of the maximum settling flux in the water
column. As soon as the lutocline is formed, the buoyancy effect due to sediment-induced
stratification will enhance turbulence damping as indicated in Eq. (5.1 1). This in turn will
accentuate the lutocline. Figure 5.2 also shows that in the range of hindered settling,
turbulence damping decreases due to decreasing sediment settling flux for F <0.0001
m^ s However, above SSC>40-50 kg m damping increases drastically as the effects
of cohesion and interactions between floes become significant. Turbulence collapse
subsequently occurs. Thus, turbulence damping is governed by the potential settling flux in
the flocculation settling range, and by cohesion and interactions between floes in the
hindered settling range with SS040-50 kg m Overall, flocculation settling, turbulence
damping due to settling flux, cohesion and interactions between floes in fluid mud as well
as the buoyancy effect due to sediment-induced stratification contribute to the formation of
a lutocline. This analysis thus replicates the conclusion arrived at by Ross and Mehta (1989)
based solely on the concept of density stratification. However, the explanation provided here
introduces new parameters, namely Ri^, Ri^ and Ri^.
123
Settling flux, Cl»,cx10^ (kg s ')
0.002 0.004 0.006 0.008 0.01
Figure 5.2. Relative momentum mixing length calculated from theoretical formula (solid
lines) and field data (data points) and settling flux (dashed line) as fimctions of SSC.
Also observed in Figure 5.2 is that turbulence damping is dependent not only on SSC,
but also on flow. Thus, beyond F^>~0.0002 m ^ s the ratio / approaches unity, i.e.,
124
there is no significant damping and, as a result, mixing will occur in two ways, i.e., both
upward and downward. The lutocline will be ruptured and become less distinct. This
conclusion is supported by the observations of Jiang and Wolanski (1998) in the Jiaojiang,
where it was foimd that a distinct lutocline only appears during slack water at neap tide when
the production of turbulent energy, F^, is usually less than 0.0002 m^ s .
To examine the relationship between lutocline formation and the flow condition, an
analysis of vertical profiles of SSC and velocity in the 1991 and 1994 field campaigns in the
Jiaojiang (Table 4.1) is considered here. The result is shown in Figure 5.3 in terms of a
lutocline strength index, as a function of the production of turbulent energy. Here, is
defined as
[dc/ dz)
r _ ' ' 'mean
where subscripts “max" and “mean" denote the maximum and the mean values of the SSC
gradient, dc/ dz , over the water column, respectively. = 1 indicates that there is a uniform
vertical distribution of the SSC and no lutocline. The vertical gradient of SSC is calculated
from
dc _^k*\
dz dz
(5.18)
where subscript “Z” denotes the measurement elevation and dz is the height between k and
Z+l. Figure 5.3 shows the results and the mean trend line. It is evident that the lutocline
becomes less distinct as the turbulent energy production F^>~ 0.0003 m^ s Relatively
125
high value of F^, at which the lutocline became less distinct, are found because the buoyancy
effect due to sediment-induced stratification was not considered in Figure 5.2.
Production of turbulent energy, F, (m^ s'^)
Figure 5.3. Lutocline strength index as a fimction of turbulence
energy production based on measured profiles of SSC and
velocity. The equation represents the best-fit line.
5.3 Mixing Length in the Jiaoiiang
From the Jiaojiang ASSM field data (e.g., Figure 4.10), it is possible to examine the
standard deviation of SSC, o^, which has the form
126
,'2
-
^4
'^d( \ 2
^-1
V c )
dt
(5.19)
where T^ is the period over which is calculated and the overbar expresses time averaging
over a period of T^. Here 7)^ will be taken as 2.5 min as a characteristic duration, c is
calculated from Eq. (4.1) by using ASSM records. Considering that Eq. (5.19) is applied to
a fixed elevation above bottom, i.e., /i'=constant, Eq. (4.1) becomes
c=AF„
(5.20)
where A is a sediment and elevation dependent constant. By assuming that is proportional
to the local Prandtl mixing length and the vertical gradient of SSC, one obtains
( -\2
dc
\ dzj
(5.21)
For a homogeneous flow, assuming that the shear stress near the bottom, x , is constant, the
local Prandtl mixing length is obtained as
Co
\ {du/dzf du/dz
Combining Eqs. (5.19), (5.20), (5.21) and (5.22), one obtains
(5.22)
du
C dz P'a
Co Pm". BFl/dz\
±r^-i
T J ^2
dt
(5.23)
127
UCS of ^ rrJ^ mO were evaluated by Eq. (5.23) using ASSM records and measured
velocity profiles (Figure 4.6). The results are plotted in Figure 5.2, where it is observed that
the trends in mixing length follow Eq. (5.1 1).
5.4 Modified Vertical Momentum and Mass Diffusion Coefficients
The above results suggest that to calculate the turbulent diffusion coefficients in a
cohesive sediment transport model, it is essential to consider the effects of potential settling
flux, cohesion and interactions between floes as well as sediment-induced stratification, as
embodied in Eq. (5.1 1). Accordingly, the turbulent diffusion coefficient formulas (3.39) and
(3.40) are modified following Munk and Anderson (1948) analog as:
Momentum diffusion coefficient:
^v=^v0
1
a.+b,Ri
fn \
+A
vb
(5.24)
Mass diffusion coefficient:
a+b.Ri
tn I
+K.
vb
(5.25)
where (<1) and (<1) are the sediment-dependent coefficients relating the effect of
suspended sediment on the mixing length.
CHAPTER 6
LUTOCLINE DYNAMICS IN THE JIAOJING
6.1 Introduction
COHYD-UF and COSED-UF are now applied to simulate lutocline and associated
fluid mud dynamics in the Jiaojiang. The modeled results include tidal variations of velocity,
SSC and the lutocline layer, lutocline layer thickness and hysteresis loops of SSC, and these
are compared with the data obtained in situ. Relevant formulations of the flow and
sedimentary processes modeled are summarized in Section 6.2. Applications of the numerical
models are described in Section 6.3. Finally, simulations along with the data are presented
in Section 6.4.
6.2 Parameters for Flow and Sedimentary Processes
Descriptions of the models and relevant flow and sedimentary formulations are given
in Chapters 3, 4 and 5. Table 6.1 summarizes these formulations and relevant parameters in
modeling sediment dynamics in the Jiaojiang. The most important parameters, including the
settling velocity, erosion rate constant, momentum and mass diffusion coefficients,
consolidation rate, and the effective roughness of the bed, were determined through analyses
of field data and/ or model calibration. Others were introduced from the works of previous
researchers.
128
Table 6.1. Flow and sedimentary process formulations and parameters
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Table 6.1 --continued
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132
6.3 Model Application
6.3.1 Modeled Domain. Initial and BoundatY Conditions
The modeled region includes the area from the mouth to the upstream tributary
(Figure 4.1), with a length of 13.4 km. The domain was discretized using spatial steps
m and ao=0.1, with the total number of rectangular cells
=67x13x10=8,710. Figure 6.1 shows the bathymetry and numerical mesh in the horizontal
plane within the modeled domain. A time step a/=30 s was adopted based on stability
constraints resulting from the numerical scheme involving the back-tracing approach
mentioned in Section 3.5.4 [Eq. (3.50)]. The lowest layer near the bottom was 0.05// above
the bed and the highest layer near the surface was 0.05// below the instantaneous water
surface.
The initial conditions for COHYD-UF runs were as follows: (a) All velocities were
set equal to zero, and (b) water elevation at each point was generated through linear
interpolation of the prescribed values at the open boundaries based on measurements. For
COSED-UF, the initial SSC was obtained by the same interpolation approach using the
values at the open boundaries from measurements. Once a stable flow field resulted from
COHYD-UF model, which took about 6 hr (one-half tidal cycle), COSED-UF run was
initiated.
Stations T1 and T5 (Figure 4.1) were selected to prescribe the open boundary
conditions for the water surface elevation. The following 1 1 tidal constituents were
considered at these boundaries: Q,, Oj, P,, Kj, N2, M2, S^, K^, M^, M^ and M . All
6 ^6
133
harmonic components were assumed to be constant in phase and amplitude over the cross-
section of each open boundary.
(a)
Upstream
dx=dy=2QQ m
— ►x
Mouth
(b)
Figure 6.1. Bathymetry in the modeled domain of the Jiaojiang (a), where
the datum is mean water level and the regions enclosed within dotted lines
are mudflats, and the numerical plane mesh in the horizontal plane (b).
Measured SSC values at sites Cl and C3 (Figure 4.1) were inputted as the open
boundary conditions for COSED-UF. Since there were only one measurement station over
the cross-section of each open boundary (located in the middle as shown in Figure 4.1), it
was assumed that the SSC was uniform over the entire cross-section of the open boundaries.
134
In reality there is always a certain amount of lateral variation of SSC, even in narrow
channels. Consequently, this uniformity approximation can lead to a noticeable error in
simulations results within the modeled domain. This aspect will be discussed further in
Section 6.4. Because SSC observations were made at variable elevations using a turbidimeter
lowered from the boat, the SSC values at fixed elevations had to be obtained through cubic
spline interpolations.
Currents and SSC at sites C2 and C4 (Figure 4.1) were used to verify the simulations.
Model tests were run over the same period as the field campaign, i.e., from November 4,
1994 (spring tide) to November 12, 1994 (neap tide). Simulated flow field and SSC profiles
at the same time as observations at C2 and C4 were outputted for verification and analysis.
6-3.2 Sediment Deposition. Erosion. Consolidation and Entrainment
1 . Deposition and Erosion: When sediment deposition occurs (Section 3.3.3
and Table 6.1). Within a time step, At, the rate of deposition, m^, is assumed to be constant
and calculated by the formula in Table 6.1. When where is a function of sediment
concentration, bottom erosion occurs (Section 3.3.3 and Table 6.1). First eroded is the
consolidating layer and then the fully consolidated bed. Within time step. At, erosion
continues until is encountered. When it is assumed that deposition and
erosion are in equilibrium, i.e., the sediment vertical flux at the bed-fluid interface m -m ,
is zero. Deposition and erosion are incorporated in COSED-UF through the bottom boundary
condition, i.e., Eq. (3.17).
135
2. Consolidation: The deposited material is combined with the consolidating sediment
layer having an initial sediment concentration of (Table 6.1), and goes through
consolidation (Sections 3.3.3 and 3.5.3). When sediment concentration near the bottom of
the consolidating layer is greater than the maximum compaction concentration, this
portion of the deposit is combined with the fully consolidated bed.
During deposition, bottom erosion and consolidation of freshly deposited sediment,
although the heights of the consolidating and the fully consolidated layers vary with time, the
bottom datum remains constant.
3. Entrainment: Interfacial entrainment discussed in Section 3.3.3 and turbulent
diffusion in Section 3.4.2 and Chapter 5 are facets of interfacial mixing in a stratified flow,
and are referred to as pure entrainment mixing and the pure turbulent-diffusion mixing.
Grubert (1990) noted that pure entrainment mixing takes place when the interfacial transition
layer (or lutocline layer) is in a subcritical state. In this condition, cusp motions generated by
interfacial instabilities, transfer volumes of fluid in either direction to a greater or lesser
extent, depending on turbulence in each layer.
Pure turbulent-diffusion mixing occurs when the transition layer is in a supercritical
state. In this condition, violent vortex motions exchange equal volumes of fluid between
layers. In reality, these two mixing processes are usually superimposed on each other, and
the critical condition is dependent on local site-specific processes. By reexamining the
experimental data of previous researchers including Ellison and Turner (1959), Lofquist
(1960), Kato and Phillips (1969), Moore and Long (1971), Wu (1973), Chu and Vanvari
136
(1976), Kantha, et al., (1977), Bo Pedersen (1980), Kit, et al., (1980), Buch (1981), and
Narimousa, et al., (1986), Christodoulou (1986) found that in the Richardson number range
0.01 </?/q:£ 1 the interface is turbulent and mixing is produced directly by eddies in the mixed
layer. In the range 0.1 10, mixing is less turbulent and is produced by Kelvin-
Helmholtz instabilities. In the range 1 100, where shear is weak, mixing is produced
by intermittent, cusp-generated entrainment. Finally, when /?/q^100, mixing is due to
molecular diffusion. Mehta and Srinivas (1993) conducted experiments of fluid mud
entrainment in a shear flow using a race-track flume. They observed that for Ri^<5,
interfacial entrainment appeared to be turbulence dominated. As Ri^ increased above ~5, the
interface become convoluted with large, irregular undulations. Entrainment appeared to be
dominated by interfacial wave breaking in which wisps of fluid were episodically ejected into
the mixed layer.
Thus, considering the fact that at very low Richardson number, i.e., Ri^->0, mixing
over the lutocline is dominated by turbulent diffusion, and in this situation the entrainment
formula (Table 6.1) becomes inapplicable due to The model only activates the
entrainment formula when Ri^^ 1 .
6.4 Flow and Sediment Dynamics
6.4.1 Flow Field
Spring tide peak flows and slack water within the modeled domain are shown in
Figures 6.2-6.5. The flow field shows the following noteworthy features:
137
(1) During peak flow the velocity decreases gradually from the mouth to upstream,
while during slack water it decreases from upstream to the mouth.
(2) Maximum velocities occur along the channel center, in the channel segment near
convex shorelines, i.e., shorelines protruding into the water area, and at other narrow
sections.
(3) Near concave shorelines the velocities are lower. There mudflats are usually found
[Figure 6.1(a)].
6.4.2 Tidal Variation of Velocity
Model simulated tidal velocities are shown in Figures 6.6(a)-6.9(a). The flood
current was stronger than the ebb current. For example, at 0.4// elevation the flood peak at
C2 during spring tide was around 1.70 m s , and the ebb peak around 1.08 m s'' [Figure
6.6(a)], with a ratio of 1.57. However, the peak ebb flow lasted considerably longer than the
peak flood flow. From Figures 6.6(a)-6.9(a) it is found that the duration of peak ebb was
around 5.34 hr, whereas that of the peak flood was around 2.05 hr, with a ratio of 2.60.
A noteworthy difference between simulations and observations is that the model
under-predicted the ebb peak at C4 [Figures 6.8(a) and 6.9(a)]. To highlight this difference
further, vertically-averaged maximum velocities at C2 and C4 are given in Table 6.2. It is
evident that the greatest difference between simulations and observations occurred at C4
during peak ebb, with a difference of about 0.37 m s . At C4 the measured peak ebb was
greater than peak flood in contrast to other sites where flood was dominant. Table 6.2 also
shows that the flood current at C2 was greater than at C4, whereas, ebb at C4 was greater
138
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Figure 6.6. Tidal velocity at 0.4// (a), and SSC at 0.25// (b) Figure 6.7. Tidal velocity at 0.4// (a), and SSC at 0.25// (b)
and at 0.75// (c) at C2 during a spring tide. Solid lines are and 0.75// (c) at C2 during a neap tide. Solid lines are
simulations and dashed lines represent field data collected simulations and dashed lines represent field data collected
during 1800 hr, 1 1/05/94 to 1700 hr, 1 1/06/94. during 1000 hr, 1 1/10/94 to 0900 hr, 1 1/1 1/94.
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than at C2. These observations colleetively suggest that there was a horizontal eirculation in
the clockwise sense wdthin the domain. This effect is discussed further in Section 6.4.4.
Table 6.2. Vertically-averaged maximum velocities at sites C2 and C4 (Unit: ms')
Site
Tide range
Observation
Simulation
Flood
Ebb
Flood
Ebb
r?
Spring tide
1.60
1.42
1.66
1.40
Neap tide
1.33
0.99
1.56
1.16
C4
Spring tide
1.24
1.38
1.26
1.03
Neap tide
1.16
1.21
1.36
0.82
6.4.3 Tidal Variation of SSC
The tidal variation of SSC near the bottom (specifically 0.25// above the bottom) and
near the surface (0.25// below the surface) are presented in Figures 6.6(b)-6.9(b) and 6.6(c)-
6.9(c), respectively. The following noteworthy differences in the magnitude of SSC between
simulations and observations occur:
(1) Near the bottom, the model mostly under-predicts SSC during flood and over-
predicts it during ebb [Figures 6.6(b)-6.9(b)].
(2) Near the surface the model mostly under-predicts SSC over the entire tidal cycle
[Figures 6.6(c)-6.9(c)].
Tidal variations of SSC show the following characteristics:
(1) Variations of SSC with Tidal Range: SSC exhibits quantitatively the same
behavioral pattern during spring and neap tides. However, near the bottom, SSC during
143
spring tide is less than at neap [Figures 6.6(b)-6.9(b)], being usually 10-20 kg m during
spring and 10-25 kg m during neap. In contrast, near the surface, SSC during spring is
greater than during neap [Figures 6.6(c)-6.9(c)], being usually 5-10 kg m during spring
and 2-5 kg m during neap.
(2) Vertical Variations of SSC: As might be expected, SSC near the bottom was
always greater than near the surface. Near the bottom it was seldom less than 10 kg m
[Figures 6.6(b)-6.9(b)], whereas, near the surface it was seldom greater than 10 kg m
[Figures 6.6(c)-6.9(c)].
(3) Variations of SSC during Flood and Ebh: During high water slack, “low” SSC,
which is defined here as the value at the “trough” of SSC tidal variation, occurred both near
the bottom and the surface, with a lag of 0.5-1 hr near the surface. During low water slack
peak SSC, which is defined in an analogous way as the value at the crest of SSC tidal
variation, occurred near the bottom whereas low SSC occurred near the surface. During peak
ebb, peak SSC occurred over the entire water column. During peak flood, peak SSC occurred
near the surface whereas low SSC occurred near the bottom.
The behavior of SSC reflects cumulative effects on SSC due to floe settling, tidal
current asymmetry (with a stronger flood peak and a weaker, longer duration of ebb peak),
entrainment of the lutocline, turbulence damping in the fluid mud layer and the large ratio
of tidal range to mean water depth.
144
During slack water, due to comparatively low vertical turbulent diffusion over the
water column, higher turbulence damping in the fluid mud layer and decreasing mixing over
the lutocline, sediment tends to settle and concentrate near the bottom where, as a result, high
SSC occurs in the hindered settling range. As a further result, peak SSC occurs near the
bottom and low SSC near the surface, along with a thick fluid mud layer and a stable
lutocline (Odd and Rodger, 1986; Wolanski, et al., 1988). This was the situation during slack
water in the Jiaojiang with the exception of the condition near the bottom during high water
slack. It is believed that the large ratio of the tidal range to the water depth causes this latter
phenomenon. As stated in Section 4.1, the mean tidal range in the studied domain was about
4 m and the mean water depth (below MSL) about 4-7 m with a ratio of about 0.8 between
tidal range and water depth. Thus, during flood the water depth increased dramatically. This
increase plus advection of lower SSC water from the region beyond the modeled domain
caused the SSC near the bottom to be diluted. As shown in Figure 6.10, the vertically-
averaged SSC at Cl during flood was always less than that at C2 and C4, with a difference
of about 4-13 kg m , which accounts for about 50%-85% of the vertically-averaged values
of SSC at C2 and C4. It is this difference plus the large ratio of tidal range to water depth that
causes dilution during flood. It also implies that sediment transport seems to be influenced
strongly by advection.
145
4>
(a)
(b)
Figure 6.10. Time series of vertically-averaged SSC during a spring tide (a) and a neap
tide (b). During spring tide, the data at sites Cl and C3 began at 1700 hr, 1 1/04/94
and at sites C2 and C4 at 1800 hr, 1 1/05/94. During neap tide, the data at sites Cl
and C3 began at 1000 hr, 1 1/12/94 and at sites C2 and C4 at 2300 hr, 1 1/10/94.
During peak current, due to significant vertical turbulent diffusion and strong upward
mixing over the lutocline (caused by internal wave breaking and high rate of erosion of
freshly deposited sediment during the previous slack), SSC over the water column usually
increases, the elevation of lutocline layer rises, and both fluid mud and lutocline layers
become comparatively thicker (Odd and Rodger, 1986; Wolanski, et al., 1988). In the
Jiaojiang, as noted, although the ebb peak is weaker than flood, it lasts longer than flood. As
146
a resiilt, during peak ebb, diffusion, vertieal mixing over the lutocline and bottom erosion are
also comparatively high. Consequently, both during peak ebb and peak flood, SSC exhibits
quantitatively the same behavior as noted, except near the bottom during peak flood due to
the cumulative effects of dilution, stronger entrainment over the lutocline and mixing within
the fluid mud layer.
6.4.4 Vertical Profiles of Velocity
Typical vertical profiles of velocity over a tidal cycle are shown in Figures 6.1 1-6.14.
Many of the simulated velocity profiles are in reasonable agreement with the observations,
although the simulations for spring tide show better agreement than at neap. It is also
observed that at peak flood, the velocity over the entire water column was greater than that
at peak ebb. Flood peak near the bottom ranges 0.8-1. 0 m s at spring tide and 0.5-0.8
m s at neap. Ebb peak near the bottom ranges 0.6-0.8 m s at spring tide and 0.4-0.65
m s'* at neap.
Noteworthy differences between simulations and observations are as follows; during
the flood at neap tide, the model over-predicted some profiles near the bottom, e.g., at 2 hr
and 3 hr in Figures 6.12 and 6.14, respectively. During ebb, the model over-predicted some
profiles near the surface at C2, e.g., at 8 hr and 9 hr in Figure 6.12 and under-predicted some
profiles near the surface at C4, e.g., at 9 hr and 10 hr in Figure 6.14. These differences
between simulations and observations are perhaps caused by following factors:
(i) Use of Approximate Stratification Function: The stratification function [Eq.
(3.10)] used in the model can be expected to affect the bottom drag coefficient, [Eq. (3.9)
147
krj o o'
o <=;
uiouoq SAoq-e uoi^basis
Figure 6.11. Velocity profiles at site C2 during a spring tide. Figure 6.12. Velocity profiles at site C2 during a neap tide.
Solid lines are simulations and open circles represent data Solid lines are simulations and open circles represent data
obtained during 1900 hr, 1 1/05/94 to 0600 hr, 1 1/06/94. obtained during 1 100 hr to 2100 hr, 1 1/10/94. Positive
Positive values signify flood and negative denote ebb. values signify flood and negative denote ebb.
148
Uioiaoq 3AOq^ U0I}^A3{3 3A11BI3-^
Solid lines are simulations and open circles represent data Solid lines are simulations and open circles represent data
obtained during 1900 hr, 1 1/05/94 to 0500 hr, 1 1/06/94. obtained during 1 100 hr to 2100 hr, 1 1/10/94. Positive
Positive values signify flood and negative denote ebb. values signify flood and negative denote ebb.
149
and Table 6.1], and consequently the near-bottom velocity near the bottom. As a result, an
over-prediction of the stratification effect can imply reduced and enhanced velocity near
the bottom.
(2) Coriolis Effect: The model omitted the Coriolis effect due to the high Rossby
number within the modeled domain, R^-U/f 5=12.0»1, where the Coriolis parameter
/=2Qsin(t),= 7.29x10'^ s the angular velocity of the earth’s rotation Q =7.29x10"^ s’,
the latitude (j), -30° , the characteristic mean velocity C/=1.0 m s and the mean width of
the estuary b =1.2 km (Section 4.1). Note that signifies the effect of flow inertia relative
to Coriolis acceleration. Thus, a high R^ implies comparatively low Coriolis effect. Note
however, that Taizhou Bay outside the estuary (Figure 4.1) is a much wider water body with
a mean width of about 6 km and R^ of about 2.0. Thus, one may expect that Coriolis
acceleration would have a greater effect on the tidal current in the bay, which in turn perhaps
has an effect on the tidal currents within the estuary. However, in that context the following
analysis should be noted.
To examine the Coriolis effect fiirther, the work of Proudman (1925) is introduced
here. Proudman calculated the vertically-averaged transverse and longitudinal tidal currents,
V and U, across a channel under the Coriolis effect for the special case of the angular
frequency of the n'^ tidal constituent, which corresponds to the diurnal tide, S2, at latitude
30°. A parabolic section is assumed for the channel as
150
( \
2'
1-
y_
l
.Bj
/
(6.1)
where h is the mean water depth below MSL, is the maximum water depth at the channel
center and>^ is the transverse Cartesian coordinate originating from the channel center. The
relevant formulas are as follows:
Non-dimensional transverse current:
O) pkB
JLhV= ^
SK Ik^'^
2(0^ . , -
sK
, (6.2)
sinhA(5 +y) +[A: -y ^) -k(B +;/)]coshA(5 +y)
Non-dimensional longitudinal current:
-^hU=- ^
SK 2k^'^
2o)^ , , -
l^—^+k\W-y^)-k(B-y)
sK
(6.3)
sinhA(5 +>^) -[A: -y +k{B +>;)]coshA(5 +y)
with the tidal wave number k=(jiy\[^, where h is the mean water depth of the parabolic
flow cross-section, i.e., h=2h^3 . Figure 6.15 shows the distributions of the transverse and
longitudinal currents across channel in the Jiaojiang and also Taizhou Bay. h^=5 m and
(0^=7.27x10'^ rad s'* (diurnal tide) were taken both for the estuary and the bay, and b^= 1.2
km and 6 km were taken in the estuary and the bay, respectively. It is observed that even in
151
the Taizhou Bay the Coriolis effect is almost negligible. The circulation in the bay due to this
effect will rotate in the counterclockwise sense. This will lead to a clockwise circulation
within the estuary due to the vorticity transfer by shear, consistent with the sense of rotation
observed (Section 6.4.2).
y/B
(a)
(b)
Figure 6.15. Distributions of transverse (a) and longitudinal (b) currents across the
channel due to the Coriolis effect, viewed in the direction of tidal wave propagation.
Solid lines are currents in Taizhou Bay and dashed lines in the Jiaojiang.
(3) Gravitational Circulation due to Salinity: Previous observations in the Jiaojiang
have showed that salinity is typically well-mixed vertically during the entire tidal cycle
(Zhou, 1986; Li, et al., 1993; Dong, et al., 1997). Hence, salinity-induced transport was not
152
considered in the present modeling effort. However, based on their simulations of salinity
distribution in the Jiaojiang with and without sediment-induced buoyancy effects, Guan, et
al. (1998) observed that suspended sediment could measurably modify the vertical
distribution of salinity due to the fact that suspended sediment causes the currents to be
strongly sheared along the vertical direction. They predicted that under sediment-induced
buoyancy effects, salinity stratification locally increased during flood tide, thus in turn
modifying the vertical profile of velocity.
Gravitational circulation due to salinity stratification can be approximately expressed
in a formulation including river inflow and surface wind stress as follows (Hansen and
Rattray, 1965):
^ ^ nn R.CI
^=_(l -o2)+l(l +40+30^)+^— (1 -9o2-8o3) (6.4)
where the first term on the right hand side is due to the effect of river inflow, the second term
is due to the effect of wind, the third term is due to salinity-induced density gradient, is
the gravitational circulation and Uj. is the vertical-mean inflow velocity. Further, T is the
dimensionless wind stress, equal to is the river inflow rate, Ra is the
estuarine Rayleigh number, defined as the ratio of free convection to diffusion (Hansen and
Rattray, 1965), i.e., equal to is a proportionality coefficient between
fluid density and salinity (=0.78x 10’^), is the salinity at the mouth of the estuary, and
is a constant in the relationship of salinity distribution introduced as follows
153
X+Z(o) ..
Sq ^ (6-5)
Here X is the dimensionless horizontal coordinate, i.e., X=R^/BHA^, and Z is the vertical
distribution flmction of salinity.
Figure 6.16 shows profiles of for different a^Ra without considering the effect
of surface wind stress. This type of gravitational circulation can cause ebb current to
increase near the surface and decrease near the bottom, and vice the versa during flood. The
observations at C4 (Figure 6.13 and 14) were characteristically consistent with this trend. It
should be noted that in light of the likely interaction between salinity and SSC in the
Jiaojiang, Hansen and Rattray’s analysis based on salinity effect alone must be interpreted
more broadly for this estuary with regard to the likely combined effects of the salinity and
SSC in inducing the gravitational circulation shown in Figure 6.16
(4) Geomorphologic Effects: Due to non-linear effects of bottom friction and of
advection, tidal residual currents usually exist in the vicinity of topographies such as basins,
headlands, sand ridges, etc. (Zimmerman, 1981). Noteworthy features in the Jiaojiang are the
headlands located at the mouth of Taizhou Bay (Figure 4.1). It is likely that tidal residual
currents induced by these headlands have effects on the residual currents within the estuary
itself As shown in Figure 6.17 (Zimmerman, 1981), along a coastal promontory tidal flow
tends to accelerate as well as decelerate, with a maximum near the headland. Concurrently,
a frictional boundary layer develops with diminishing current velocity towards the coast.
Thus, vorticity is generated along the coast with orientation alternating between the flood and
154
ebb. As the vorticity is largest near the headland, during flood there is a net flux of
counterclockwise vorticity into the left closed curve and out of the right one. During the ebb
the situation does not reverse because of a net flux of clockwise vorticity out of the left
quadrilateral and into the right quadrilateral, which is equivalent to the situation during the
flood. Hence there is a net flux of counterclockwise vorticity into the left quadrilateral and
of clockwise vorticity into the right quadrilateral, giving rise to a vortex pair of opposite
signs at either sides of the headland. In other words, residual circulation exists at both sides
of the headland.
-15 -10 -5 0 5 10 15 20
u,/U,
Figure 6.16. Profiles of gravitational circulation without
wind stress, after Hansen and Rattray (1965).
155
Thus, a clockwise circulation can be expected in the north region of Taizhou Bay and
a counterclockwise one in its southern region. These circulations possibly have extended
effects on residual currents within the Jiaojiang. Unfortunately, Taizhou Bay is beyond the
modeled domain, and there are no observations available to back-up this assertion.
Figure 6.17. Tidal currents, vorticities and residual circulation in the neighborhood of a
headland. Currents are signified by solid arrows for flood and dashed arrows for ebb.
Currents are largest near the headland and decrease towards the shoreline. Vorticity
therefore has a maximum near the headland. Vorticities generated by side-wall fnction
are shown by solid circles for flood and dashed circles for ebb. Vorticities have highest
strength near the headland and diminish away from it (after Zimmerman, 1981).
6.4.5 Vertical Profiles of SSC
Figures 6.18-6.21 show vertical profiles of SSC simulated and observed at sites C2
and C4 over one tidal cycle during spring and neap tides, respectively. The following features
of these SSC profiles are noteworthy:
(1) SSC changes in the following way: near the bottom during ebb, SSC continues
to increase until flow deceleration occurs, then decreases. During flood it either continues
to decrease, or increases somewhat during the accelerating phase of flow, then decreases.
156
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obtained during 1900 hr, 1 1/05/94 to 0600 hr, 1 1/06/94. obtained during 1 100 hr to 2100 hr, 1 1/10/94.
157
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158
Near the surface, it increases during accelerating flow and decreases during the decelerating
flow both for flood and ebb.
(2) During peak currents, the lutocline layer (defined in Section 2.1) rises and
becomes thicker, with gentler upper and lower gradients. During slack water it fall and
becomes thinner, with steeper upper and lower gradients.
(3) The lutocline layer is lower and thinner during neap tide than during spring.
(4) Fluid mud and lutocline layers occur over the entire tidal cycle except during the
period around high water slack.
This behavior of SSC reflects the cumulative effects of floe settling, tidal asymmetry,
lutocline entrainment, turbulence damping in fluid mud layer and the large ratio of tidal
range to water depth (see also Section 6.4.3).
Differences between the simulations and observations of SSC are as follows: during
ebb tide, the model over-predicts SSC for some profiles near the bottom, e.g., 5 hr, 7 hr and
1 1 hr in Figure 6. 1 9, and under-predicts near the surface, e.g., 1 0 hr and 1 1 hr in Figure 6.18
and 8 hr, 9 hr and 10 hr in Figure 6.20. During flood tide, the model under-predicts SSC in
some profiles over the entire water column, e.g., 2 hr and 3 hr in Figures 6.18 and 6.20,
respectively. These differences likely arise due to following modeling approximations as well
as observation errors:
(1) Approximate Open Boundary Conditions for SSC: SSC at the open boundaries
were measured 30 hr before and after those at sites C2 and C4 during spring and neap tides,
respectively, and were taken as uniform over the cross-sections of the open boundaries (see
also in Section 6.3.3).
159
In order to demonstrate the lateral non-uniformity of SSC in the Jiaojiang, model
simulations and observations of vertically-averaged SSC at the flow section including sites
C2 and C4 are plotted in Figure 6.22. From this figure together with the tidal variation of
velocity [Figures 6.6.6(a)-6.9(a)], it is found that during flood, SSC at C4 was greater than
at C2, and during ebb it gradually increased at C2 and ultimately became greater than at C4.
A difference (in the vertically-averaged SSC) of about 0-7 kg m occurred between C2 and
C4, which accounts for about 0%-70% of the vertically-averaged values of SSC at these sites.
Because uniform SSC over the cross-sections of the open boundaries were employed in
modeling, the model generated a more uniform SSC across the estuary within the modeled
domain than observations (Figure 6.22). Table 6.3 includes typical values from Figure 6.22
at 4 hr, where two exhibit significant SSC non-uniformity and the other two do not. It is
evident that better comparisons of vertically-averaged SSC between simulations and
observations resulted at times of comparatively minor non-uniformity than when significant
lateral non-uniformity occurred.
Table 6.3. Vertically-averaged SSC during minor and significant
non-uniformity of SSC across the flow cross-section (Unit: kg m '^)
Method
Site
Minor non-uniformity
Significant non-imiformity
Spring tide
Neap tide
Sprin
g tide
Neap tide
5hr
lOhr
5hr
8hr
2hr
15hr
2hr
lOhr
Observation
C2
5.50
12.50
4.00
6.00
11.50
6.75
6.75
13.20
C4
6.50
12.90
6.00
7.00
16.25
14.00
12.10
9.00
Simulation
C2
5.70
13.25
4.00
7.00
11.50
10.75
13.50
12.50
C4
5.70
15.00
3.60
7.50
9.50
11.75
14.00
15.50
160
Figure 6.22. Time series of vertically-averaged SSC. Dark circles are from
site C2, open circles represent site C4, solid lines signify simulations at C2,
dashed lines are simulations at the center of the flow section containing C2
and C4 and dotted lines denote simulations at C4.
In order to demonstrate the significance of SSC non-uniformity, a model test using
non-uniform boundary conditions for SSC was carried out for a spring tide. The simulated
results are shown in Figure 6.23. Also shown in this figure are results using uniform SSC
boundary conditions (Figure 6.22). Non-uniform SSC at the boundaries was generated using
a linear relationship as follows: (1) the mean value of SSC over each open boundary was
taken as the observation at site Cl or C3; (2) the slope of the lateral variation of SSC was
taken as the ratio of vertically-averaged SSC observed between sites C2 and C4 (Figure
161
6.22). From Figure 6.23, it is observed that considerable differences in the simulated SSC
were generated during peak ebb. This suggests, at least qualitatively, that non-uniformity of
SSC at the open boundaries may have had a noteworthy effect on simulations within the
estuary.
Time starting 2000 hr, 1 1/10/94.
Figure 6.23. Comparision of simulated vertically-averaged SSC using uniform
and nonuniform boundary conditions of SSC during a spring tide. Solid lines
signify simulations at site C2, dashed lines are that at site C4 and dark and
open circles represent that using uniform boundary conditions of SSC.
(2) Approximate Parameters: The selection of modeling parameters given in Table
6. 1 was a difficult task. Although most parameters were determined optimally from previous
works, data analysis and model calibration, parametric selection can in general cause
162
measurable differences between simulations and observations due to the approximate nature
of the parameters. Noted in the following are those parameters that are sediment-dependent
and highly influential in controlling the sensitivity of the simulations of SSC;
Parameters Determined bv Model Calibrations: Model calibrations were carried out
based on comparisons of SSC between simulations and observations. Because all parameters
affected each other on an inter-dependent way in the calibration tests, it was at times difficult
to identify the desired value of a certain parameter. In this regard, the most noteworthy ones
are; the efficiency coefficient of turbulence damping, d^, in the vertical mass diffusion
equation (3 .20) (see also in Table 6. 1 ), and in the consolidation
rate formula (2.23) (see also in Table 6.1) and C, Pj and in the vertical distribution
formula (2.24) of dry sediment concentration for a fully-consolidated bed (Table 6.1). There
were no direct experimental data available for determining these parameters for the Jiaojiang.
Bottom Erosion Rate Constant: The bottom erosion rate constant, was obtained
from analysis of field data (Section 4.3.6 and Table 6.1). Data analysis was carried out based
on three basic assumptions (Section 4.3.6). It is known that the second and third assumptions
seem reasonable but the first my not be so. As stated in Section 6.4.3, advection is a major
factor affecting sediment transport in the Jiaojiang, and consequently has a significant effect
on SSC. Thus it can not be ignored. In fact, the scatter of data points in Figure 4.29
demonstrates the effect of advection. Thus, the correct erosion rate constant can only be
determined through erosion experiments in laboratory using the same mud as in the
Jiaojiang.
163
(3) Observation Errors: As noted in Chapter 4, a ship-borne turbidimeter was used
for measuring the SSC. The ship was anchored by a mooring chain with a length = 10//=50
m. Thus, the ship could turn along with tidal currents and change its orientation within the
circle defined by the chain. Given the laterally non-uniform distribution of SSC in the
Jiaojiang as described above and in Section 6.4.3, changing ship position could have led to
a degree of error in the observed SSC.
6.4.6 Lutocline Laver
The lutocline layer was identified by its definition in Section 2. 1 , i.e., the near-bottom
layer between the upper elevation and lower elevation of the maximum vertical gradient of
SSC (Figure 2.1). Figure 6.24 shows typical vertical distributions of the vertical gradient of
SSC. The lutocline layer is identified by the zone over which the SSC gradient is
comparatively high.
Time variations of this layer, both simulated and observed at sites C2 and C4 during
spring and neap tides, are shown in Figures 6.24-6.27. It is seen that both the measured
pattern and elevation of the lutocline layer are reproduced approximately by the model.
Except for periods of about 1-2 hr around high water slack, a lutocline layer with a thickness
of about 1-3 m (Figure 6.29) and an upper elevation of 1-4 m (Figure 6.30) consistently
occurred, irrespective of whether the tide was spring or neap.
From Figures 6.25-6.28 and Table 6.4, combined with Figures 6.6(a)-6.9(a), the
following characteristics of the lutocline layer can be gleaned:
(1) The lutocline layer was at higher elevation during peak flows than during slack
water, with the highest elevations during peak floods.
164
E
o
o
•Q
O
•o
a
c:
o
Q
>
>
o
«>
oa
Vertical gradient ofSSC (kg in'*)
Vertical gradient ofSSC (kg in'*)
Figure 6.24. Vertical gradient of SSC as a function of elevation during
a neap tide. Solid lines are simulations and open circles represent
field data obtained during 1 100 hr to 2100 hr, 1 1/10/94.
(2) The lutocline layer was thinner during slack water than during peak flow, and also
thinner during neap tide than during spring.
(3) The lutocline layer rose during period of flow acceleration and fell during flow
decelerating periods.
From comparisons of the thickness of the lutocline layer between simulations and
measurements (Figure 6.29), it is observed that the predicted thickness of the lutocline layer
is in better agreement with measurement at neap tide than at spring (Figure 6.29). Also, there
is a degree of over-prediction of its upper limit at neap tide and under-prediction at spring
(Figure 6.30).
Elevation above bed (m) Elevation above bed (m)
165
Figure 6.24. Simulated (solid lines) and measured (dashed lines)
tidal variation of the lutocline layer at site C2 during a spring
tide from 1800 hr, 1 1/05/94 to 1700 hr, 1 1/06/94.
Figure 6.25. Simulated (solid lines) and measured (dashed lines)
tidal variation of the lutocline layer at site C2 during a neap
tide from 1000 hr, 1 1/10/94 to 0900 hr, 1 1/1 1/94.
Elevation above bed (m) Elevation above bed (m)
166
0 5 10 15 20 25
Time (hr)
Figure 6.26. Simulated (solid lines) and measured (dashed lines)
tidal variation of the lutocline layer at site C4 during a spring
tide from 1800 hr, 1 1/05/94 to 1700 hr, 1 1/06/94.
Figure 6.27. Simulated (solid lines) and measured (dashed lines)
tidal variation of the lutocline layer at site C4 during a neap
tide from 1000 hr, 1 1/10/94 to 0900 hr, 1 1/1 1/94.
167
lo” io‘
(m)
Figure 6.28. Lutocline layer thickness: simulation (6,J and measurement (6,J.
168
Table 6.4. Average thickness and upper elevation
__ofTutocline layer at different times (Unit: m)
Item
Tidal range
Observation
Simulation
Spring tide
Nea
3 tide
Spring tide
Neap tide
C2
C4
C2
C4
C2
C4
C2
C4
Lutocline
layer
thickness
Peak flood
1.83
1.61
1.17
1.02
1.98
1.83
1.46
1.42
Peak ebb
1.46
1.54
1.10
1.02
1.61
1.24
1.10
1.17
High slack
0.00
0.00
0.58
0.88
0.00
0.00
0.44
1.02
Low slack
1.32
1.17
1.02
0.80
1.17
1.02
1.02
0.88
Tidal mean
1.27
1.30
0.97
0.97
1.29
1.43
0.99
1.33
Upper
elevation
of lutocline
layer
Peak flood
3.07
3.80
2.63
3.07
3.29
3.59
3.15
3.50
Peak ebb
3.15
3.80
1.98
1.54
3.15
3.07
2.34
2.56
High slack
0.00
0.00
1.10
1.17
0.00
0.00
0.59
1.17
Low slack
1.90
2.20
1.61
1.61
2.05
1.76
1.61
2.05
Tidal mean
2.41
3.31
1.64
1.80
2.44
2.41
2.09
2.51
6.4.7 Flow-SSC Hysteresis
Due to time-lag effects associated with settling, diffusion, bed erosion, entrainment
and consolidation, SSC in estuaries is usually higher during decreasing currents than when
currents are increasing (Postma, 1967; Dyer and Evans, 1989). As a result, when the SSC at
a certain elevation is plotted against the bottom shear stress, this hysteresis becomes visually
evident (Costa and Mehta, 1990).
Simulated and measured hysteresis loops of SSC for sites C2 and C4 are shown in
Figures 6.31-6.38. These are at 1 m above the bottom and 1 m below the instantaneous
surface during spring and neap tides.
169
} ' 1 1 j
-6 -4 -2 0 2 4 6
(Pa)
Figure 6.30. Simulated (solid line) and measured (dashed line) hysteresis loops at 1 m
above bottom at C2 during a spring tide from 1800 hr, 1 1/05/94 to 0500 hr, 1 1/06/94.
(Pa)
Figure 6.31. Simulated (solid line) and measured (dashed line) hysteresis loops at 1 m
below surface at C2 during a spring tide from 1800 hr, 1 1/05/94 to 0500 hr, 1 1/06/94.
170
Figure 6.32. Simulated (solid line) and measured (dashed line) hysteresis loops at
1 m above bottom at C2 during a neap tide from 1000 hr to 2100 hr, 1 1/10/94.
(Pa)
Figure 6.33. Simulated (solid line) and measured (dashed line) hysteresis loops at
1 m below surface at C2 during a neap tide from 1000 hr to 2100 hr, 1 1/10/94.
171
3 ' 1- 1 1
-3-2-1 0 1 2 3 4
•>^4 (Pa)
Figure 6.34. Simulated (solid line) and measured (dashed line) hysteresis loops at 1 m
above bottom at C4 during a spring tide from 1800 hr, 1 1/05/94 to 0500 hr, 1 1/06/94.
•^4 (Pa)
Figure 6.35. Simulated (solid line) and measured (dashed line) hysteresis loops at 1 m
below surface at C4 during a spring tide from 1800 hr, 1 1/05/94 to 0500 hr, 1 1/06/94.
172
(Pa)
Figure 6.36. Simulated (solid line) and measured (dashed line) hysteresis loops at
1 m above bottom at C4 during a neap tide from 1000 hr to 2100 hr, 1 1/10/94.
X* (Pa)
Figure 6.37. Simulated (solid line) and measured (dashed line) hysteresis loops at
1 m below surface at C4 during a neap tide from 1000 hr to 2100 hr, 1 1/10/94.
173
During ebb (Figures 6.31-6.38), the simulated and measured loops near the bottom
and near the surface follow the same trend. During accelerating flow SSC increases gradually
(by resuspension). As the flow begins to decelerate, at first SSC increases dramatically until
it reaches a maximum, then starts to decrease (due to deposition). This behavior results in
the loops rotation in the clockwise sense.
During flood, the loops near the bottom and near the surface follow different patterns.
Near the bottom (Figures 6.31, 6.33, 6.35 and 6.37), SSC either decreases over all stages, or
increases somewhat during accelerating flow, then decreases. This causes the loops to rotate
clockwise. Near the surface (Figures 6.32, 6.34, 6.36 and 6.38), SSC increases during
accelerating flows due to vertical diffusion and mixing and reaches a maximum at the end
of the period of acceleration. Subsequently, during decelerating flows, SSC begins to
decrease due to settling and dilution. This causes the loops rotating in the counterclockwise
sense.
Thus, during flood, loop-reversals appeared near the bottom in the Jiaojiang. Costa
and Mehta (1990) observed such loop-reversals in Hangzhou Bay, China, during the
transition from accelerating to decelerating flow. They showed that reversal was induced by
the presence of the lutocline at that level. During accelerating flow the lutocline forms, and
entrainment over it occurs concurrently. Due to the entrainment lag, SSC near the bottom
increases and reaches a maximum during accelerating flow, then decreases. This leads to a
loop-reversal.
In the Jiaojiang, more significant loop-reversals occurred near the bottom during the
entire flood tide period (Figures 6.31, 6.33, 6.35 and 6.37) with the SSC either decreasing
174
over this period, or increasing somewhat during aecelerating flow, then decreasing. As noted
in section 6.4.3, this behavior is believed to be caused by the effects of dilution, vertical
diffusion as well as the presence of the lutocline.
6-4.8 Effect of Turbulence Damping on SSC and Lutocline Formation
With reference to Section 6.4.3, in order to further examine the effect of turbulence
damping on SSC and the lutocline formation, the following numerical tests were carried out:
(1) Test 1 : Considering turbulence damping and taking d^=Q.lS ;
(2) Test 2: Neglecting turbulence damping, i.e., taking
(3) Tgst 3: Neglecting the effects of cohesion and interactions between floes, i.e.,
taking Ri^^Ri^^O and d^^Q.lS.
In the above tests, all other parameters were taken to be the same as in previous
simulations. Figure 6.39 shows the results of these tests along with the observations at C4
during a neap tide. It is seen that without considering turbulence damping (Test 2), the model
resulted in relatively uniform profiles of SSC and consequently less distinct lutoclines (2-5
hrs and 9-1 1 hrs in Figure 6.39). If compared with the observations, it is seen that Test 2 had
the worst results. Test 1 had the best with Test 3 in-between. However, there were smaller
differences among these tests during low slack water (1 hr in Figure 6.39) and high slack
water (7 and 8 hrs in Figure 6.39). This implies that during high and low slack waters, the
lutocline is mainly governed by floe settling. It is also seen that considerable effect of
turbulence damping appeared near the bottom (2-5 hrs and 9-1 1 hrs in Figure 6.39), because
of damping induced by the relatively high SSC there. Near the surfaee, the effect of damping
175
during flood (2-5 hrs in Figure 6.39) is greater than during ebb (9-1 1 hrs in Figure 6.39), due
to the relatively higher SSC near the surface during flood. Without considering cohesion and
interactions between floes (Test 3), the simulated results are very close to Test 1 and the
differences are mainly near the bottom. This supports the conclusion in Chapter 5 that
cohesion and interactions between floes govern turbulence damping only at high SSC (>40-
50 kg m ^), in the hindered settling range.
E
o
o
ja
u
>
o
XI
(a
c
Q
R)
>■
u
.5
CK
40
40
SSC (kg m-')
20
40
40
SSC (kg m-^)
Figure 6.39. Modeling SSC profiles at site C4 during a neap tide. Open
circles represent field data from 1 100 hr to 2100 hr, 1 1/10/94, solid lines
are simulations with d2=0.75, dashed lines signify simulations with d2 =
0.75 and Rig=Ri^=0, and dotted lines represent simulations with d2=0.
CHAPTER 7
SUMMARY AND CONCLUSIONS
7.1 Summary
Lutoclines are common features in estuaries vvith high-load fine sediments. They are
significantly associated with interfacial waves and turbulenee damping in fluid mud, and tend
to diminish vertical mixing under tidal forcing. In order to better understand these dynamical
features, the following studies were carried out:
1 . Turbulenee Damping: Following Rossby and Montgomery (1935), assuming the
sum of the kinetic, potential and dissipated energy in a stratified, cohesive flow to be equal
to that in a homogeneous, non-cohesive flow at identical shear rates, turbulence damping in
fluid mud was phenomenologically examined. Accordingly, sediment settling flux, cohesion,
interaetions between floes and sediment-induced stratifieation were quantified by the
respective Richardson numbers Ri^, Ri^, Ri^ and Ri. The resulting formulations for the
turbulent mixing length were examined using observations in the Jiaojiang estuary. The
corresponding expressions for the vertical momentum and mass diffusion coefficients were
incorporated in the developed numerical model codes for flow and sediment transport.
2. Internal Waves: ASSM data from the Jiaojiang estuary were used to examine the
height, angular frequency, celerity and length of internal waves at the lutocline. Their
176
177
variation with the global Richardson number was examined in the light of results of previous
studies on the lutocline and other pycnoclines.
3- Lutocline Response to Tidal Forcing: Three-dimensional, finite-difference
hydrodynamic and sediment transport codes, COHYD-UF and COSED-UF, were developed,
incorporating the latest unit process models for fine sediment transport including
erosion/ entrainment, diffusion, settling, deposition and consolidation. Following model
testing against analytical solutions, laboratory data and field observations, tidal lutocline
dynamics was examined by applications of COHYD-UF and COSED-UF and comparisons
with field observations on flow and sediment transport from the Jiaojiang.
7.2 Conclusions
Important conclusions drawn from this study are as follows:
1 . It is shown that turbulence damping in the water column at high SSC is governed
by the settling flux in the flocculation settling range, and by cohesion and interaction between
floes in the hindered settling range. Maximum turbulence damping is shown to occur at the
lutocline, supporting a similar, but qualitative observation (e.g., Ross and Mehta, 1989). Data
derived from the Jiaojiang estuary are shown to support this observation.
2. Numerical model tests using SSC data from the Jiaojiang showed that without
considering turbulence damping, the simulated results of SSC become less tenable when
compared with the field observations, resulting in comparatively more uniform profiles of
SSC and less distinct lutoclines than observed.
178
3. Observations in the Jiaojiang showed that the lutocline strength was highly
associated with turbulence energy. The lutocline became comparatively less distinct as the
turbulent energy production F >~0.0003 m ^ s This supports the qualitative conclusion
from the derived expression of turbulence damping.
4. High and low frequency internal waves were detected at the lutocline in the
Jiaojiang. The shallow water low frequency wave had a representative rms height of 0.38 m
and a modal frequency of 0.09 rad s * , which was near the local Brunt-Vaisala frequency.
The deep water high frequency wave was characterized by sharp crests and flat troughs, with
a rms height of 0.21 m and a modal frequency of 1.33 rad s '* . This wave was possibly
induced by interfacial shear at the lutocline.
5. The height and the angular frequency of both high and low frequency waves
decreased with increasing Richardson number. The height and angular frequency versus
logi?/Q plots exhibited linear trends and, in turn, the celerity and wave length increased with
increasing Richardson number.
6. Numerical modeling tests showed that the developed codes, COHYD-UF and
COSED-UF, are able to adequately simulate transport processes including sediment
propagation, advection, diffusion, entrainment and consolidation. The simulated results
compare reasonably with laboratory and field observations, once the appropriate process-
related parameters are adopted.
7. From the simulations and observations in the Jiaojiang estuary, the lutocline was
found to behave as follows: (1) The lutocline elevation was higher during peak flows than
179
during slack water, with the highest elevations during peak floods; (2) the lutocline layer was
thinner during slack water than during peak flows, and also thinner during neap tide than
spring; (3) the lutocline layer rose during flow acceleration and fell during flow decelerating
periods; and (4) the lutocline was observed to persist through most of the tidal cycle, except
1-2 hr around high water. The overall behavior of lutocline reflects the cumulative effects
of tidal current asymmetry (with a stronger flood peak and a weaker, longer duration of ebb
peak), sediment settling and entrainment, turbulence damping and also the large ratio of tidal
range to mean water depth resulting in a dilution effect during flood.
7.3 Recommendations for Future Studies
The present study is based on the limited data from the Jiaojiang. There persists a
lack of quantitative information on internal wave generation, propagation, interfacial
instability and vertical mixing at the lutocline. The only observations used to verify the
derived analytical model of mixing length in the sediment-stratified water column were the
results obtained from the vertical excursion of SSC using ASSM signals.
In the sub-model for sediment erosion, it was assumed that the existence of fluid mud
has no tangible effect on bottom erosion. However, in general fluid mud appears as a
“protective” cover over the bottom and thereby bottom erosion.
Based on the conclusions of this study, the following recommendations are made for
future research:
1 . In order to fully understand the internal wave behavior, collecting long time series
of continuous in situ records of the interface are essential.
180
2. Direct measurement of turbulent mixing in the fluid mud is necessary to verify the
mixing length model.
3. Extensive experiments in flumes are required for understanding the way in which
the fluid mud layer affects bottom erosion.
APPENDIX A
DERIVATIONS OF THE GOVERNING EQUATIONS
A.l Vertical Velocities, w and (o. and Continuity Equation Q.D
The o-transform converts the time and Cartesian coordinate system (t, x, y, z) to the
new time and coordinate system (/', x',y', a) according to
t'=t
x'=x
y'=y
(A.1)
H
The vertical velocity in the o-coordinate, w, is defined as
(0= —
Dt
(A.2)
The vertical velocity in the z-coordinate, w, is
w= — =H — +o
Dt Dt
Do DH DC
— +o + — -
Dt Dt Dt
(A.3)
The continuity equation is
du 0v dw
— + — + —
(A.4)
dx dy dz
The transformation of each term in Eq. (A.4) is as follows
I8I
182
The first term:
— da _ du _du' a 8H ^ 1 5C '
dx dx' dx da dx dx' da ^ H dx' H dx\
(A.5)
The second term:
The third term:
dv _ dv dy' ^ dv do _ 5v dv
dy dy' dy da dy dy' da
a dH ^ 1 ac
Hdy' Hdy'^
(A.6)
dw _dw da _ 1 dw
dz da dz H da
(A.7)
Differentiating Eq. (A.3) vsdth respect to o, one obtains
aw jjdiji du
— -H — + —
da da da
dH ac
o — +—
dx' dx'
dH dv
+u +
dx' da
dH aci dH dH
a — + — — \ +v — +
, dy' dy' J dy' dt'
(A.8)
Substituting Eqs. (A.5)-(A.8) into Eq. ( A.4), simplifying and eliminating the superscript
(prime) for convenience, one obtains
ac duH dvH „ao) „
—5. + + +H — =0
dt dx dy da
(A.9)
Integrating Eq. (A.9) with o from -1 to 0 and considering the facts that -0, the
continuity equation (3.1) becomes
dHu dHv], „
+ k/o=0
a^: dy )
(A.10)
A.2 Momentum Equations (3.2) and OJ)
The derivation of the momentum Equation (3.2) in the x-direction is given here. The
derivation of equation (3.3) in the >^-direction is analogous. The momentum equation in the
x-direction is
du ^ du du du r 1 dp
— +u — +v — +w — -fv=—^ +
dt dx dy dz p
. . d^u
+A.
‘dx^ ^ dy^
dz
du
A —
"az
, 1 U
+ :(A.ll)
p * >/?
+v
Here the rheological effect (the last term on the right hand side) of fluid mud is simply
considered by assuming that its rheological behavior satisfies the Bingham model (Odd and
Cooper, 1989).
The o-transform of terms on the left hand side of Eq. (A.l 1) is
The first term:
du _ du dt' ^du da _ du du
[ o a//^ 1 ac)
dt dt' dt da dt dt' da
i Hdt' * Hdt' ^
(A.12)
The third term:
du _ du dy' ^du da _ du du( a dH 1 aC '
dy dy' dy da dy dy' da[Hdy' Hdy'j
(A. 13)
The fourth term:
^_du da _ 1 du
dz da dz H da (^-14)
Assuming pressure to be hydrostatic, the pressure term in right side of Eq. (A.l 1) becomes
184
where is the fluid density at the water surface. Thus the a-transform of the pressure term
is
1 dp_ 89^
p dx p
acax'^ac aoVgr
oJ
ax' dx do dx'
dx' dal Hdx'^Hdx',
Hda
p dx p J dx' pdx'Jda pdx'J da
(A.16)
-_g^C _gH fdp _gdH
ap.fpda
J dx D dx'
dx' p J dx' p dx'
The transformation of the turbulent diffusion term in the x-direction is
d^u
_ a
du
_ du(
ax 2
dx
dx'
da
//dx'^Ndx'l
=-^^+Higher Order Terms
ax'2 dx'^
Similar to Eq. (A. 17), the turbulent diffusion term in the >^-direction is
-^=-^^+Higher Order Terms=-^^
ay 2 a/2 ^,2
The turbulent diffusion term in the vertical direction has the form
_a
=1A
du
dz
i ’azj
Hda
^ H da J
(A. 17)
(A. 18)
(A. 19)
Substituting Eqs. (A.3), (A.5) and (A.12)-(A.19) into Eq. ( A.ll), simplifying and
eliminating the superscript (prime) for convenience, the momentum equation (3.2) in the x-
185
direction is obtained as
5m, du du du r dC sH^do ,
■ +M— +v — +(0 — -/v= -a_» - / ^da
da dx p J dx
dt dx dy
gm
p dx
op+J pda
. d^u . d^u
+A.
dx‘
' +1 ^
du
^ Hda
~H~da^
1 5t
B U
Pff3o ^
2 2
A.3 Sediment Conservation Equation (3. 151
The sediment conservation equation is
dc dc dc dc ^
-+M +V +W-
dt dx dy dz dz
dx^ ^dy^
dz
K — \
''dz
Similar to Eq. (A.20), the following forms are obtained:
The first term on the left hand side (LHS):
dc dc dt' dc da dc dc
dt dt' dt da dt dt' da
a dH I dC
Hdt' Hdt'
The second term on the LHS:
dc _
dc dx' ^ dc da
_ dc
dc(
dx
dx' dx da dx
dx'
da
the LHS:
dc _
dc dy' ^ dc da
_ dc
dc(
dy
dy' dy da dy
dy'
da ^
o dH^ 1 ac
Hdx'* Hdx'
a dH ^ 1 aC
\Hdy' Hdy',
(A.20)
(A.21)
(A.22)
(A.23)
(A.24)
The forth term on the LHS
186
dc _dc do _ 1 dc
dz do dz H do
(A.25)
The fifth term on the LHS:
3o) c 1 5(0 c
J _ 1 S
dz ~~H do
The first term on right hand side (RHS):
5^c 5^c
dx~^ dx'^
The second term on the RHS:
+Higher Order Terms=
d^c
dx'^
dy'^ dy'^
The third term on the RHS:
ci^C u n, A 'T
+Higher Order Terms
dy
a
d
_ 1 d
Xac'
dz
, ’'0zj Hdo
. H do^
(A.26)
(A.27)
(A.28)
(A.29)
Substituting Eqs. (A.22)-(A.29) into Eq. (A.21), simplifying and eliminating the superscript
(prime) for convenience, the sediment conservation equation (3.15) becomes
dc dc dc (
■+u — +v +(0-
dt dx dy do H do
1 d(^f
K +K
, 1 d
i^dc]
H do
[‘dx^
Hdo
[ Hdo)
(A.30)
APPENDIX B
NUMERICAL TECHNIQUES
B.I Back-Tracing Approach
In the Eulerian-Lagrangian differential scheme, the physical properties of water
particles at time step n are obtained by the back-tracing approach (Casulli and Cheng, 1992).
The back-tracing time interval At"=At/N^. At each back-tracing step m, the water particle
will lag by small distances
dx"' = -u" —
AX Ay
da'"=-w‘
n At"
AO
(B.I)
where dx dy ""and do"’ are the backward distances at time step m in the x, and o
directions, respectively, and u^, and (o^ are the local velocities of the water particle in
the X, y and a directions, respectively, that are approximated by bilinear interpolation over
the eight surrounding mesh points. Finally, the water particle is traced back to the position
p, using back-tracing distances dx, dy and do given by (Figure B.I)
dx = ^ dx dy='^ dy do = '^ do""
m = I
m = l
(B.2)
m = l
Then, the physical property at point p is obtained by bilinear interpolation as follows
187
188
Gp={\ -da){\ -dx){\ -dy)G^^^{\ -dx)dyG2 +dx{\ -dy)G"+dxdyG”
+i/o[(l -dx){\ -dy)Gs+(\ -dx)dyG^"+dx(\ -dy)G"+dxdyG”
(B.3)
Figure B.l. Schematic diagram of back-tracing approach, where dotted line is
the pathline of water particle, o is the position of water particle at current time
step n+1 and p is the position of water particle at the previous time step n.
B.2 Pre-conditioned Conjugate Gradient Method
The linear five-diagonal system of equations for the water surface elevation,
C, can be written in the following general form
r -
a.
^ij*\ ^ij
(B.4)
189
where a.^j/2 and b.. are coefficients that are dependent on the time step. Note
that for notational simplicity, superscript («+l) of water surface elevation, C, has been
omitted. Note also that the coefficients and cijj^y2 non-negative and their
sum is strictly less than unity. Thus the system formed by these equations is
normalized, symmetric and positive-definite (Casulli and Cheng, 1992).
The pre-conditioned conjugate gradient algorithm to solve the system of
equations (B.4) takes the following steps (Bertolazzi, 1990):.
(B.2.1) Guess c!?.
(B.2.2) Set
, -A
(B.2.3) Carry out following calculations for k=0, 1, 2, and until
(^(9, ^(*))
(B.5)
190
where e is the allowed error. At each iteration the essential calculations consist of a
matrix-vector multiplication Mp^ ^ as specified in Eq. (B.5), two scalar products
between vectors, namely and and three sums between vectors,
namely and
APPENDIX C
EFFECT OF TEMPERATURE ON SETTLING VELOCITY
The experimental results of Lau (1994) related to temperature effect on the
deposition of cohesive sediments in an annular flume indicated that temperature affects the
settling velocity not only through changes in the viscosity and density of water, but also
through floe aggregation. The floes become stronger and denser with decrease of
temperature, and consequently their settling velocity increases. To examine this effect, Lau’s
data are reprocessed here. His experiments were carried out at different temperatures and
under the same flow condition with bottom shear stress t^=0.2 Pa and initial vertical mean
SSC, Cq=9 kg m . The available data are the concentration-time curves for runs at various
temperatures. Commercial kaolinite was used in the experiments. Figure C.l shows the
frequency distribution, 4y, of the settling velocity based on the standard size distribution of
a similar kaolinite (Yeh, 1979), where the settling velocity is calculated from the Stokes law
0«
2(
18v
P.
— -1
iPo
(C.l)
In Eq. (C.l), and o)q^, respectively, are the grain size and settling velocity of the
kaolinite class, and =2,650 kg m^ po=l,000 kg m^and v = 10‘® m^ s"‘.
191
192
In accordance with the method of Mehta and Lott (1987), sorting by size
characteristically occurs for non-uniform fine sediment. Given C(t) as the instantaneous
depth-averaged concentration, under a given depositional flow condition C will decrease
with time and ultimately reach a steady state value, Cj. If and are respectively
defined as the minimum and maximum critical shear stress for deposition of the m ^ kaolinite
classes, then when no sediment can deposit, whereas when , all sediment
eventually deposits. When a certain fi-action of the initial sediment, represented
by C^, will ultimately remain in suspension, and the remainder, represented byCg-C^ will
settle out. The occurrence of Cj less than Cq but greater than zero is an indication of
sediment sorting. A deposition law for a non-uniform cohesive sediment was developed by
Mehta and Lott (1987) based on the consideration that the instantaneous concentration, C,
is obtained by summation of the corresponding concentrations, C^, obtained fi-om the
deposition relationship of Krone (1962) for each class. This leads to
C(0 _ 1
M,
M.
E C„(0=E 4>/wJexp^
^0 ^0
where is the settling velocity of the kaolinite class, which is simplified as
1— ^
( \
0)
sn ^
H
(C.2)
(C.3)
where and are respectively the minimum and maximum settling velocities of the
kaolinite classes and is a flocculation factor.
193
Using Eqs. (C.2) and (C.3), one can attempt to best-fit the experimental data of Lau
(1994) for different temperatures by changing the minimum settling velocity, , and the
flocculation factor, F^. The results are shown in Figures (C.2)-(C.5), where
taken. The plot of cumulative weight finer against the
settling velocity for different temperatures is shown in Figure (C.6). It is observed that the
settling velocity increases with decreasing of temperature. In other words, the floes become
stronger and denser with decreasing of temperature.
To express the temperature effect mathematically, the (median) settling velocity at
50% of cumulative weight (finer) is taken. The results are shown in Figure (2.3), where the
dimensionless median settling velocity is plotted against the temperature with as the
settling velocity at 15 °C. Finally, from Figure (2.3) the temperature function, F, i.e., Eq.
(2.28), is obtained as
1.776 -0.05 186, for 0=0-30 °C
(C.4)
Frequency distribution,
194
Figure C.l. Frequency distribution, (|y, of the settling velocity of kaolinite.
Figure C.l. Time-concentration relationship during deposition at
26 °C. Open circles are the experimental data of Lau (1994).
Normalized concentration C/C„ Normalized concentration C/C
195
Time Chr)
Figure C.3. Time-concentration relationship during deposition at
20 °C. Open circles are the experimental data of Lau (1994).
Figure C.4. Time-concentration relationship during deposition at
10 °C. Open circles are the experimental data of Lau (1994).
196
Figure C.5. Time-concentration relationship during deposition at
5 °C. Open circles are the experimental data of Lau (1994).
Figure C.6. Cumulative distribution of settling
velocity of kaolinite at different temperatures, 0.
APPENDIX D
AN APPLICATION OF COHYD-UF: CONTRACTION SCOUR IN A RIVER
D.l Scour Problem
An application of COHYD-UF is reported here for the simulation of a contraction
scour problem at the Haldia Jetty, or Pier (Figure D. 1 ), in the Hooghly River in India (Figure
D.2). The pier, designed in the 1960s, served as an oil unloading terminal that enabled
40,000 DWT oil tankers to transfer oil to storage tanks at the Haldia Port. The Hooghly
estuary transports high loads of fine-grained material. Since constmction of the pier, a scour
occurred in front of the pier. The presence of the hole near the tip of the pier was detrimental
to the pier piles and hence the superstructure (Engineers India Limited, 1980; Rao, et al.,
1980). The surficial shape of the scour hole was nearly elliptical, with the major axis and
minor axis approximately perpendicular and parallel, respectively, to the longitudinal axis
of the pier. The scour depth varied between 7 m and 13 m, and the hole migrated southward
over a 40-month period up to May, 1980. The contours in Figure D.3 show the hole as it
appeared in the May, 1980 survey. Note that the scour depths are taken as those below the
original bottom elevation before the construction of the pier.
During the 40-month period the average rate of migration was 1 .53 m/month along
the direction of the main flow and 0.56 m/ month perpendicular to the main flow and away
from the pier. This mode and rate of migration suggested a slight dominance of the ebb
197
198
current over flood in the bottom region immediately adjacent to the pier. The strength of ebb
current in this region was on the order of 2 m s ‘. The presence of such a strong current,
coupled with the fact that a 40,000 DWT tanker was often docked at the pier, exacerbated
the scour problem and necessitated action for filling up the hole in order to stabilize the pier
(Engineers India Limited, 1980).
Figure D. 1 . Schematic diagram showing the Haldia oil pier and depth
contours (m) in the vieinity. Water depth are below mean low water.
An evident conclusion that can be drawn is that the reduetion of the flow area due to
the pier resulted in an increase in the high ebb current and associated bed shear stress in front
of the pier. Consequently, there was an inerease in the erosive force in the effectively
199
contracted area in front of the pier, which is believed to be the main cause of the scour hole.
Hence, it can be called contraction scour (HEC-18, 1995).
Calcutta
Garden Reach
Study area
Beaumont %
Gut
Figure D.2. Location map of Haldia oil pier, India.
200
D.2 Scour Simulation
As described above, the disturbance of the pier on the local flow and the dominance
of ebb current were thought to be the main reasons for the development of the hole.
Accordingly, the scour depth can be calculated through the modeled ebb flow, which will be
assumed to be constant. Figure D.4 shows the bathymetry within the modeled segment of the
river. The bed erosion rate is taken to be proportional to the bottom shear stress t. as
follows
m=M ^
e max
(D.l)
\ ^ y
By assuming the bottom to be in sedimentary equilibrium before the construction of the pier,
one can take the critical shear stress for erosion, to be the bottom shear stress before pier
construction. The erosion rate constant, can be estimated using the maximum scour
depth, a//^, according to
aH
max
PqaT
(
T,'
-^-1
X '
V
\ -1
/
(D.2)
where x^' and x^' respectively are the bottom and critical shear stresses at the point where
the maximum scour occurred, aT is the time period over which a//^^ occurred, and is
the dry density of the bottom sediment. Accordingly, the scour depth at any position, aH, is
obtained from
201
Figure D.3. Measured scour depths in front of the Haldia oil pier.
The pier is shown as an idealized rectangular protrusion. Unit: m.
East Bank of River
Figure D.4. Bottom topography of the modeled segment of the river in the
vicinity of the Haldia pier. Water depths (unit: m) are below mean low water.
202
a//=
\
(D.3)
where the bottom shear stress, was calculated from Eq. (2.9) without considering
stratification effects. The values of Zq=0.1 mm [effective roughness of the bed in Eq. (2.9)]
^ taken in these calculations. From the modeling tests, it was found that
the ratio of t,' to x 'was about 1.51.
D S
The modeled region includes the area from 2.25 km upstream to 2.25 km downstream
of the pier; thus with a length of 4.5 km (Figure D.4). The domain was discretised with
spatial steps ax=50 m, a>'=25 m and ao=0.1, with the total number of rectangular cells
MxiVxZ^=90x 104x10 =93,600. A time step a/=5 s was adopted due to stability constraints
resulting from the numerical scheme involving in the back-tracing approach mentioned in
Chapter 2 [Eq. (2.50)].The lowest layer near the bottom was 0.05// above the bed, and the
highest layer near the surface was 0.05// below the local water surface.
The upstream and downstream open boundary conditions were prescribed by constant
water surface elevations at both ends. Elevations of 0.25 m above mean low water level at
the upstream boundary and 0.00 m at downstream open boundary were taken. Model run was
initiated at zero velocity and a constant water surface slope of 5.56x10'^ over the domain.
COHYD-UF was run until a stable ebb flow field resulted. The model was run for two cases:
before and after the construction of the pier, and both results were outputted for above
203
calculations. At each grid point, the bottom shear stress, in Eq. (D.3), was taken as the
simulated value after the construction of the pier, and the bottom shear strength, x^ in Eq.
(D.3), was taken as the x^ value before the construction of the pier.
D.3 Results
Figures D.5 and D.6 show simulated flow fields around the pier near the surface and
the bottom, respectively. It is observed that the current in the front of the pier became
stronger due to the disturbance of the pier on flow, with a maximum ratio between velocities
after and before the construction of the pier of about 1 .23.
The solid contours in Figure D.7 show the simulated scour depths in the front of the
pier. In Figure D.8, the simulated areas, A^, corresponding to 2-3 m, 3-4 m, 4-5 m and >5
m scour depths are plotted against the corresponding measured areas, A^ . This comparison
shows that the model reasonably reproduced the area of the scour hole, and that the deeper
the scour depth, the better the comparison of the simulated versus measured scour area.
Distance (m) Distance (m)
204
Distance (m)
Figure D.5. Simulated flow field around pier at 0.05// below the surface.
Distance (m)
Figure D.6. Simulated flow field around pier at 0.05// above the bottom.
Simulated scour area, (m^)
205
Figure D.7. Comparison of scour depths simulated (solid lines)
and measured (dashed lines) in front of the Haldia oil pier. Unit: m.
Figure D.8. Comparison between simulated and measured areas
at 2-3 m (•), 3-4 m (^), 4-5 m (O) and >5 m (+) scour depths.
APPENDIX E
SIMULATION OF SEDIMENT DEPOSITION IN A FLUME
E.l Introduction
In order to test the ability of COSED-UF in predicting sedimentation, a simulation
was conducted to compare modeled and observed shoaling patterns in the single flume test
of Ariathurai (1974). In the simulation, the settling velocity in moving water was determined
using experimental data of cohesive sediment deposition from Mehta (1973).
E.2. Flume Test
In the test of Anathurai (1974), a 20 m long, 61 cm wide tilting recireulating flume
was used. Approximately two-thirds of the way down the flume, a grating made of vertieal
steel rods, each of 6.35 mm diameter, was plaeed across one-half of the flume width so that
it partially blocked the flow. The barrier allowed eonsiderable flow through itself
Reeonstituted seawater with a salinity of about 3 1 %o was added to the flume to give a flow
depth of 10 cm. The flume pump was then set so as to generate an average veloeity of about
1 7 cm s . The slope of the flume bottom was adjusted simultaneously to produee a uniform
flow depth along the length.
Before starting the experiment, the barrier was removed and the flow velocity
increased. Sediment was then added to the flume gradually. San Francisco Bay sediment (bay
mud) obtained from deposits in a yaeht harbor in Mare Island was used for the test. Sea
206
207
shells, silt and other coarse materials present in the sample were removed by sedimentation
and the remaining cohesive sediment with some silt in it was mixed in sea water and poured
into running water in the flume slowly until an initial mean SSC of 0.2 kg m was reached.
After all of the sediment had been added the barrier was placed and the velocity
decreased to the same value as earlier (i.e., 17 cm s ’' ). Afterwards, the suspended sediment
settled gradually in the region in the lee of the barrier having relatively low velocities. The
test was earned out for a period of about 3.5 days. Water was then drained out slowly and
the deposition pattern around the barrier was recorded.
It was found that the SSC decreased with time according to (Ariathurai, 1974)
C-CqIO (E.l)
where C is the vertically-averaged SSC, Cg is the initial SSC (==0.2 kg m and is a
deposition rate constant, which was found to be 1.0x10"^ s by calibration (Krone, 1962;
Ariathurai, 1974).
E.3 Settling Velocity in Moving Water
As stated in Section 4.3.5, the settling velocity in moving water is characteristically
different from that in quiescent water, since increasing turbulence can enhance flocculation
and at the same time limit the size of floes that can be sustained. To determine the relevant
settling velocity, experimental data on bay mud deposition reported in Mehta (1973) were
used. These experiments were carried out in flumes under different bottom shear stress, t,.
208
and initial SSC, Cq . The data have been reported as the fraction of depositable concentration,
C * [=(Cq-C)/ (Cq-C^)] , as a function of non-dimensional time, t * where Cj is
the final (steady state) SSC and is the time corresponding to C ’=50%. Table E.l gives
the basic parameters in these experiments.
Table^BJ_^_Basic^p^ameter^^ experiments using the bay mud
No.
Co
(kg m'3)
H
(m)
(Pa)
(Pa)
9
(kg m-5)
^50
(hr)
Investigator
1
8.45
0.152
0.165
0.194
4.23
10
Mehta (1973)
2
0.50
0.305
0.067
0.070
0.05
63
Krone (1962)
3
0.92
0.305
0.049
0.065
0.00
23
Krone (1962)
4
1.92
0.305
0.037
0.065
0.00
7
Partheniades
(1962)
5
21.00
0.152
0.031
0.065
0.00
5.9
Krone ( 1 962)
Once C {t ), /jQ, Cq, Cj and water depth H are known, the settling velocity can be
calculated from
)
Ca(p,
(E.2)
where C” and C"’* are SSC at two consecutive measurements, a/ is the time interval
between these two measurements, is the probability of sediment deposition, which is
209
taken as 1 x^/x^ (Krone, 1 962), x^ is the critical shear stress for deposition, and the
instantaneous concentration C=(C ”+C"^‘)/2 . When applying Eq. (E.2), one should consider
the non-uniformity of the sediment. In this case each class of the sediment has a different
value of (Appendix C). By assuming that there is no interaction among different sediment
classes, Mehta and Lott (1987) related the critical stress, x^^, to the settling velocity,
for the n sediment class by
where [ -ln(T^J^^/T^,)/ln(a)^J^/(o^,)] is a sediment-dependent constant, e.g., p^=0.5 for
kaolinite (Mehta and Lott, 1987). The other symbols are the same as in Appendix C. For the
bay mud, t^j=^0.065 Pa, t^j^^=0.318 Pa, and 2.3x10'^ ms"* were chosen (Mehta,
1973). Considering that under a given flow only sediment classes having x, are
® dn b
depositable, one can take x^ in Eq. (E.2) as the minimum value of the critical stress of the
depositable sediment classes, i.e., t^=MIN(t^Jt^^^Tj). Table E.l gives for each test.
The calculated results using Eq. (E.2) are shown in Figure E.l, where the solid line
is the best- fit of the calculated data points using the settling velocity - SSC relation given by
Eq. (4. 12). It is seen that for SSC<~0.2 kg m the particles become practically free settling,
with a settling velocity of about 4.0 x 10"^ m s .
210
Figure E.l. Settling velocity as a function of SSC
in moving water. Data are from Mehta (1973).
E.4 Deposition Simulation
In the simulation, the flume was discretized with spatial steps ax=0.26 m,
Ay=0.02032 m and ao=0.1, with the total number of rectangular cells A/XiVxZ^=77x30xio
=23,100. A time step a/=0.02 s was adopted due to the same reason as stated in Appendix
D. The lowest layer near the bottom was 0.057/ above the bed and the highest layer near the
surface was 0.05// below the local water surface.
211
The upstream and downstream boundary conditions were prescribed by constant
water surface elevations at both ends. Elevations of 2.5 cm below the horizontal datum at the
downstream and 0.00 m at the upstream boundary were taken. This yielded a mean water
surface slope of 1.25x10 ^ . The flume bottom was tilted at the same slope as the water
surface. The water depth was 10 cm. The following values of the hydrodynamic parameters
were empirically selected: ^^=0.05 m^ s"’, ^^=0.005 m^ s'* and z^=0.5 mm. Under
these conditions, the model generated a cross-sectional mean velocity of 17 cm s before
the setting of the barrier.
In the model grid, the real barrier was approximated by alternatively placed solid
lines at the same location as in the flume test. The model was initiated at zero velocity and
a constant water surface slope of 1.25x10-3. COHYD-UF was then run until a stable flow
field resulted. The stable result was outputted for calculation of deposition. Figure E.2 shows
the simulated flow field. It is observed that the velocity is relatively weak in the regions
before and below the barrier, and also near the side walls, which are potential areas of
deposition (Ariathurai, 1974).
As described earlier, during the deposition process the SSC in the flume was found
to be a function of time, as defined by Eq. (E.l). Thus, the deposition depth at each grid point
can be simply evaluated from
^ Co) At
1— ^
(E.4)
212
where is the dry density of the newly deposited sediment (140 kg m'^). The
instantaneous concentration, C, was calculated by Eq. (E.l), and the bottom shear, from
’ o’
Eq. (3.9).
0 0.5 1 1.5 2 2.5 3 3.5
(a)
0 0.5 1 1.5 2 2.5 3 3.5
Distance (m)
Figure E.2. Simulated flow field around the barrier in
the flume, (a) near the surface and (b) near the bottom.
The calculated sediment deposition in the region downstream the barrier are shown
in Figure E.3. Also shown in this figure are the measured thickness of the deposit. It is seen
that the majority of deposition took place downstream the barrier due to a dramatic
decreasing in the velocity there (Figure E.2).
Distance (m)
213
0.6
0.4
0.2
0
Figure E.3. Distribution of simulated (solid lines) and observed (numbers in circles)
deposition (thickness) at the down side of the barrier. Data are from Ariathurai (1974).
O.S 1 1.5 2
Distance (m)
BIBLOGRAPHY
Allen, G. P., Sauzay, G., Castaing, P. and Jouanneau, J. M. (1976). Transport and deposition
of suspended sediment in the Gironde estuary, France. In: Estuarine Processes, Vol.
II, M. Wiley ed.. Academic Press, New York, 63-81.
AmasSeds Research Group, (1990). A multidisciplinaiy Amazon shelf sediment study. EOS.
Transactions, American Geophysical Union, 71(45), 1776-1777.
Anathurai, R. (1974). A finite element model for sediment transport in estuaries. Ph. D.
Thesis, University of California, Davis, California, 192p.
Ariathurai, R. and MacArthur, R. C. and Krone, R. B. (1977). Mathematical model of
estuarial sediment transport. Dredged Material Research Program, Technical Report
D-77-12. U. S. Army Engineering Waterways Station, Vicksburg, Mississippi, 77p.
Bagnold, R. A. (1954). Experiments on a gravity-free dispersion of large solid spheres in a
Newtonian fluid under shear. Proceedings of the Royal Society of London, A 225, 49-
Bagnold, R. A. (1956). The flow of cohesionless grains in fluids. Proceedings of the Royal
Society of London, Philosophical Transactions, B 249, 235-297.
Bendat, J. S. and Piersol, A. G. (1971). Random Data: Analysis and Measurement
Procedures, Wiley-Interscience, New York, 407p.
Bertolazzi, E. (1990). Metodo PCG ed applicazione ad un modello di acque basse. Thesis,
University of Trento, Trento, Italy, 135p.
Bi, A. and Sun, Z. (1984). A preliminary study on the estuarine process in Jiaojiang River,
China. Journal of Sediment Research, 3, 1 2-26 (in Chinese).
Bosworth, R. C. L. (1956). The kinetics of collective sedimentation. Journal of Colloidal
Science, 11, 496-500.
214
215
Broward, F. K. and Wang, Y. H. (1972). An experiment on the growth of small disturbances
at the interface between two streams of different densities and velocities. Proceedings
of the International Symposium on Stratified Flows, Novosibirsk, USSR, 491-498.
Buch, E. (1981). On entrainment and vertical mixing in stably stratified ^ords. Estuarine
Coastal and Shelf Science, 1 2(4), 46 1 -469.
Burt, T. N. (1984). Field settling velocities of estuary muds. In Lecture Notes on Coastal and
Estuarine Studies 14: Estuarine Cohesive Sediment Dynamics. A. J. Mehta ed
Springer-Verlag, Berlin, 126-150.
Burt, T. N. and Parker, W. R. (1984). Settlement and density in beds of natural mud during
successive sedimentation. Report IT 262, Hydraulics Research Limited, Wallingford
Oxfordshire, UK, 15p.
/
Businger, J. A., Wyngaard, J. C., Izumi, Y. and Bradley, E. F. (1971). Flux-profile
relationships in the atmospheric surface layer. Journal of Atmospheric Science, 28,
Carslaw, H. S. and Jaeger, J. C. (1959). Condition of Heat in Solids, Clarendon Press
Oxford, UK, 53p.
Castaing, P. and Allen, G. P. (1981). Mechanisms controlling seawards escape of suspended
sediment from the Gironde: a macrotidal estuary in France. Marine Geology, 40, 101-
Casulli, V. and Cheng, R. T. (1992). Semi-implicit finite-difference method for three-
dimensional shallow water flow. International Journal for Numerical Methods in
Fluids, 15, 629-648.
Chou, I. B. (1975). An experimental investigation of interfacial waves generated by low
frequency internal waves. M. S Thesis, University of Florida, Gainesville Florida
103p.
Christodoulou, G. C. (1986). Interfacial mixing in stratified flows. Journal of Hydraulic
Research. 24(2), 77-92.
Chu, V. H. and Vanvari, M. R. (1976). Experimental study of turbulent stratified shearing
flow. Journal of the Hydraulics Division, ASCE, 102(6), 691-706.
Costa, R. G. (1989). Flow-fine sediment hysteresis in sediment-stratified coastal waters, M
S. Thesis, University of Florida, Gainesville, Florida, 155p.
216
Costa, R. G. and Mehta, A. J. (1990). Flow-fine sediment hysteresis in sediment-stratified
coastal waters. Proceedings of the 20th Coastal Engineering Conference Vol 2
ASCE, New York, 2047-2060.
Delisi, D. and Corcos, G. M. (1973). A study of internal waves in a wind tunnel. Boundary
Layer Meteorology, 5, 121-137.
Dong, L. X., Wolanski, E. and Li, Y. (1997). Field and modeling studies of fine sediment
dynamics in extremely turbid Jiaojiang River estuary, China. Journal of Coastal
Research, 13(4), 995-1003.
Dyer, K. R. (1986). Coastal and Estuarine Sediment Dynamics, Wiley, New York, 342p.
Dyer, K. R. and Evans, E. M. (1989). Dynamics of turbidity maximum in a homogeneous
tidal channel. Journal of Coastal Research, SI 5, 23-30.
Einstein, H. A. and Chien N. (1955). Effects of heavy sediment concentration near the bed
on velocity and sediment distribution. Missouri River Division Series, No. 8,
University of California, Berkeley, California, 76p.
Eisma, D., van der Gaast, S. J., Martin, J. M. and Thomas, A. J. (1978). Suspended matter
and bottom deposition of the Orinoco delta; turbidity, mineralogy and elementary
composition. Netherlands Journal of Sea Research, 12(2), 224-251.
Ellison, T. H. and Turner, J. S. (1959). Turbulent entrainment in stratified flows. Journal of
Fluid Mechanics, 6, 423-448.
Engineers India Limited, (1980). Technical feasibility report - second oil unloading terminal
for Haldia, Vol. II, Submitted to the Ministry of Shipping and Transport, Government
of India, New Delhi, 53p.
Fu, N. and Bi, A. (1989). Discussion on the problems in sediment transport of Jiaojiang
river. Journal of Sedimentary Research, 3, 52-57 (in Chinese).
Gibson, R. E., Englund, G. L. and Hussey, M. J. L. (1967). The theory of one-dimensional
consolidation of saturated clays - 1. Geotechnique, 17, 261-273.
Goldberg, E. and Bruland, K. (1974). Radioactive chronologies. In: The Sea, Vol. 5, E. D.
Goldberg ed., John Wiley & Sons, New York, 93-103.
Greenspan, D. and Casulli, V. (1988). Numerical Analysis for Applied Mathematics. Science
and Engineering, Addison-Wesley, Reading, MA, 139p.
217
Grubert, J. P. (1990). Interfacial mixing in estuaries and ^ords. Journal of Hydraulic
Engineering, 1 1 6(2), 176-195.
Guan, W. B., Wolanski, E. and Dong, L. X. (1998). Cohesive sediment transport in the
Jiaojiang River estuary, China. Estuarine, Coastal and Shelf Science, 46(6), 861-871 .
Hansen, D. V. and Rattray, M. Jr. (1965). Gravitational circulation in straits and estuaries.
Journal of Marine Research, 23(2), 104-122.
Hayter, E. J. (1983). Prediction of cohesive sediment movement in estuarial waters, Ph. D.
Thesis, University of Florida, Gainesville, Florida, 348p.
Hopfmger, E. J. and Linden, P. F. (1982). Formation of thermoclines in zero-mean-shear
turbulence subjected to a stabilizing buoyancy flux. Journal of Fluid mechanics, 1 14
157-173.
Hunt, J. N. (1954). The turbulent transport of suspended sediment in open channel.
Proceedings of the Royal Society of London, A 224, 322-335.
Hydraulic Engineering Circular No. 18 (HEC-18), (1995). Evaluating Scour at Bridges
(Third Edition). U.S. Department of Transportation, Federal Highway Administration,
Washington DC, 225p.
Hwang, K. N. (1989). Erodibility of fine sediment in wave-dominated environments. M S.
Thesis, University of Florida, Gainesville, Florida, 158p.
Inglis, C. C. and Allen, F. H. (1957). The regimen of the Thames estuary as affected
currents, salinities and river ^\o'^ .Proceedings of the Institution of Civil Engineers
7, 827-868.
Ippen, A. T. and Harleman, D. R. F. (1966). Tidal dynamics in estuaries. In: Estuary and
Coastline Hydrodynamics, A. T. Ippen ed., McGraw-Hill, New York, 493-545.
James, A. E., Williams, D. J. A. and Williams, P. R. (1988). Small strain, low shear
rheometry of cohesive sediments. In: Physical Processes in Estuaries, J. Dronkers and
W. van Leussen eds.. Springer- Verlag, Berlin, 488-500.
Jiang, J. H. and Wolanski, E. (1998). Vertical mixing by internal wave breaking at the
lutocline, Jiaojiang River estuary, China. Journal of Coastal Research, 14(4), 1426-
1431.
Jobson, H. E. and Sayre, W. W. (1970). Vertical transfer in open channel flow. Journal of
the Hydraulics Division, ASCE, 96(3), 7148-7152.
218
Kantha, L., Phillips, O. and Azad, R. (1977). On turbulent entrainment at a stable density
interface. Journal of Fluid Mechanics, 79, 753-768.
Kato, H. and Phillips, O. M. (1969). On the penetration of a turbulent layer into stratified
fluid. Journal of Fluid Mechanics, 37, 643-655.
Kent, R. E. and Pritchard, D. W. (1959). A test of mixing length theories in a coastal plain
estuary. Journal of Marine Research, 1, 62-72.
Kineke, G. C. (1993). Fluid muds on the amazon continential shelf Ph. D. Thesis, University
of Washington, Seattle, Washington, 259p.
Kirby, R. (1986). Suspended fine cohesive sediment in Severn estuary and Inner Bristol
channel, U.K. Report ETSU-STP-4042, United Kingdom Atomic Energy Authority,
Harwell, UK, 243p.
Kirby, R. and Parker, W. R. (1977). The physical characteristics and environmental
significance of fine-sediment suspension in estuaroes. In: Estuaries, Geophysics and
the Environment, National Academy of Science, Washington DC, 1 10-120.
Kirby, R. and Parker, W. R. (1982). A suspended sediment in the Severn estuary. Nature
295:5848, 396-399.
Kit, E., Berent, E. and Vajda, A. (1980). Vertical mixing induced by wind and a rotating
screen in a stratified fluid in a channel. Journal of Hydraulic Research, 18(1), 35-58.
Kranenburg, C. and Winterwerp, J. C. (1997). Erosion of fluid mud layers. I: entrainment
model. Journal of Hydraulic Engineering, 123(6), 504-5 1 1 .
Krone, R. B. (1962). Flume studies of the transport of sediment in estuarial shoaling
processes. Final Report, Hydraulic Engineering Laboratory and Sanitary Engineering
Research Laboratory, University of California, Berkeley, California, 1 lOp.
Kynch, G. J. (1952). A theory of sedimentation. Transaction of the Faraday Society, 48
166-176.
Lamb, H. (1932). Hydrodynamics. Sixth edition, Dover Publications, New York, 738p.
Lau, Y. L. (1994). Temperature effect on settling velocity and deposition of cohesive
sediments. Journal of Hydraulic Research, 32(1), 41-51.
Launder, B. E. and Spalding, D. B. (1972). Lecture in Mathematical Models of Turbulence,
London, New York, Academic Press, 169p.
219
Lee, S. C. and Mehta, A. J. (1994). Cohesive Sediment Erosion. Dredging Research
Program, Contract Report DRP-94-6, U. S. Army Engineering Waterways Station,
Vicksburg, Mississippi, 4 Ip.
Lesieur, M. (1997). Turbulence in Fluids. Third Revised and Enlarged Edition, Kluwer
Academic Publishers, Dordrecht, The Netherlands, 515p.
Li, B. G., Xie, Q. C., Xia, X. M., Li, Y. and Eisma, D. (1999). Size distribution of suspended
sediment in maximum turbidity zone and its response to tidal dynamics in Jiaojiang
River estuary, China. Journal of Sediment Research, 1, 18-26 (in Chinese).
Li, Y., Pan, S., Shi, X. and Li, B. (1992). Recent sedimentation rates for the zone of the
turbidity maximum in the Jiaojiang estuary. Journal of Nanjing University, Natural
Science Edition, 28(4), 623-632 (in Chinese).
Li, Y., Wolanski, E. and Xie, Q. C. (1993). Coagulation and settling of suspended sediment
in the Jiaojiang River estuary, China. Journal of Coastal Research, 9(2), 390-402.
Liu, Y. M. (1988). A two-dimensional finite-difference model for moving boundary
hydrodynamic problems, M.S. Thesis, University of Florida, Gainesville, Florida,
134p.
Lofquist, L. (1960). Flow and stress near an interface between stratified liquids. The Physics
of Fluids, 3(2), 158-175.
Maa, P. Y. and Mehta, A. J. (1987). Mud erosion by waves: a laboratory study. Continental
Shelf Research, 7(11/12), 1269-1284.
McCave, I. N. (1979). Suspended sediment. In: Estuarine Hydrography and Sedimentation,
A Handbook, K. R. Dyer ed., Cambridge University Press, Cambridge, UK, 131-183.
McLaughlin, R. T. (1959). The settling properties of suspension. Journal of the Hydraulics
Division, ASCE, 85(12), 9-14.
Mehta, A. J. (1973). Depositional behavior of cohesive sediments. Ph. D. Thesis, University
of Florida, Gainesville, Florida, 275p.
Mehta A. J. (1989). Fine sediment stratification in coastal water. Proceedings of Third
National Conference on Dock & Harbour Engineering, K.R.E.C., Surathkal, India,
487-492.
Mehta, A. J. (1991a). Understanding fluid mud in a dynamic environment. Geo-Marine
Letters, 11, 113-118.
220
Mehta, A. J. (1991b). Characterization of Cohesive Soil Bed Surface Erosion, With Special
Reference to the Relationship between Erosion Shear strength and Bed Density.
Report No. UFL/COE-MP-91/4, University of Florida, Gainesville, Florida, 83p.
Mehta, A. J. and Li, Y. G. (1997). A PC-based short course on fine-grained sediment
transport engineering. Coastal and Oceanographic Engineering Department,
University of Florida, Gainesville, Florida, 9 Ip.
Mehta, A. J. and Lott, J. W. (1987). Sorting of fine sediment during deposition. Proceedings
of Coastal Sediment '87, ASCE, New York, 348-362.
Mehta A. J. and Parchure, T. M. (1999). Surface erosion of fine-grained sediment revisited.
In: Muddy Coasts: Processes and Products, B. W. Flemming, M. T. Delafontaine and
G. Liebezeit eds., Elsevier, Amsterdam (in press).
Mehta, A. J., Parchure, T. M., Dixit, J. G. and Anathurai, R. (1982). Resuspension potential
of deposited cohesive sediment beds. In: Estuarine Comparisons. V. S. Kennedy ed..
Academic Press, New York, 591-609.
Mehta, A. J. and Srinivas, R. (1993). Observations on the entrainment of fluid mud in shear
flow. In: Nearshore Estuarine Cohesive Sediment Transport, A. J. Mehta ed.,
American Geophysical Union, Washington, DC, 224-246.
Monin, A. S. and Obukhov, A. M. (1953). Dimensionless characteristics of turbulence in
atmospheric surface layer. Doklady Akad Nauk SSSR, 98, 223-226 (in Russian).
Moore, M. J. and Long, R. R. (1971). An experimental investigation of turbulent stratified
shearing flow. Journal of Fluid Mechanics, 49, 635-655.
Munk, W. H. and Anderson, E. A. (1948). Notes on a theory of the thermocline. Journal of
Marine Research, 1, 276-295.
Narimousa, S. and Fernando, H. J. S. (1987). On the sheared interface of an entraining
stratified UmA. Journal of Fluid Mechanics, 174, 1-22.
Narimousa, S., Long, R. R. and Kitaigorodskii, S. A. (1986). Entraimnent due to turbulent
shear flow at interface of a stably straitified fluid. Tellus, 38A(1), 76-87.
Neumann, G. and Pierson, W. J. (1966). Principles of Physical Oceanography. Prentice-
Hall, Englewood Cliffs, New Jersey, 545p.
Nichols, M. M. (1984-1985). Fluid mud accumulation processes in an estuary. Geo-Marine
Letters, 4, 171-176.
221
Nicholson, J. and O'Connor, B. A. (1986). Cohesive sediment transport model. Journal of
Hydraulic Engineering, 1 12, 621-640.
Ochi, M. K. (1990). Applied Probability & Stochastic Processes, John Wiley & Sons, New
York, 499p.
Odd, N. V. M. and Cooper, A. J. (1989). A two-dimensional model of the movement of fluid
mud in a high energy turbid estuary. Journal of Coastal Research, SI(5), 85-193.
Odd, N. V. M. and Owen, M. W. (1972). A two-layer model of mud transport in the Thames
estuary. Proceedings of the Institution of Civil Engineers, Supplement IX
Paper75I7S, 175-205.
Odd, N. V. M. and Rodger J. G. (1978). Vertical mixing in stratified tidal flows. Journal of
the Hydraulics Division, ASCE, 104(3), 337-351.
Odd, N. V. M. and Rodger, J. G. (1986). An analysis of the behaviour of fluid mud in
estuaries. Report No. SR 84, Hydraulics Research Limited, Wallingford, Oxfordshire
25p.
Ogata, A. and Banks, R. B. (1961). A solution of the differential equation of longitudinal
dispersion in porous media. Professional Paper 4II-A, U. S. Geological Survey Al-
A9.
Owen, M. A. (1970). Properties of a consolidating mud. Report No. INT 83, Hydraulics
Research Station, Willingford, UK, 35p.
Owen, M. A. and Odd, N. V. M. (1970). A mathematical model of the effect of a tidal barrier
on siltation in an estuary. Proceedings of an International Conference on the
Utilization of Tidal Power, Halifax, Nova Scotia, Canada, 457-484.
Parchure, T. M. and Mehta, A. J. (1985). Erosion of soft cohesive sediment deposits. Journal
of Hydraulic Engineering, 111(10), 1308-1326.
Parker, W. R. and Kirby, R. (1979). Fine sediment studies relevant to dredging practice and
control. Proceedings of the Second International Symposium on Dredging
Technology, BHRA, Paper B2, Texas A & M University, College Station, Texas, 13-
26.
Parker, W. R. and Lee, K. (1979). The behaviour of fine sediment relevant to the dispersal
of pollutants. ICES Workshop on Sediment and Pollutant Interchange in Shallow Sea,
Tecel, UK, 28-34.
222
Partheniades, E. (1962). A study of erosion and deposition of cohesive soil in salt water. Ph.
D. Thesis, University of California, Berkeley, 192p.
Pedersen, F. B. (1980). A monograph on turbulent entrainment and friction in two-layer
stratified flow. Series Paper No. 25, Technical University of Denmark, Lvnebv
Denmark, 397p.
Phillips, O. M. (1977). The Dynamics of the Upper Ocean. 2nd ed., Cambridge University
Press, London, 336p.
Postma, H. (1967). Sediment transport and sedimentation in the estuarine environment. In:
Estuaries, American Association for the Advancement of Science, Publication No.
83, Washington DC, 158-179.
Prandtl, Z. A. (1925). Bericht iiber untersuchugen zur ansgebildeten turbulenz. Zs Angew
Math, Mech., 5, 136-169.
Proudman, J. (1925). Tides in a channel. Philosophical Magazine and Journal of Science
16, 465-475.
Rao, P. V., Emerson, J. J., Emerson, J. A. and Mehta, A. J. (1980). A survey of small-craft
recreational marinas in florida. Technical Report, No. 151, Department of Statistics,
University of Florida, Gainesville, Florida, 4 1 p.
Rodi, W. (1980). Mathematical modeling of turbulence in estuaries. Proceedings of the
International Symposium on Mathematical Modeling of Estuarine Physics, J.
Sundermann and K. P. Holz ed., German Hydrographic Institute, Hamburg, 14-26.
Ross, M. A. (1988). Vertical Structure of Estuarine Fine Sediment Suspensions. Ph. D.
Thesis, University of Florida, Gainesville, Florida, 187p.
Ross, M. A. and Mehta, A. J. (1989). On the mechanics of lutoclines and fluid mud. Journal
of Coastal Research, SI 5, 51-61.
Rossby, C. G. and Montgomery, R. B. (1935). The layer of functional influence in wind and
ocean currents. Papers of Physical Oceanography, 3(3), I -101 .
Scarlatos, P. D. and Mehta, A. J. (1993). Instability and entrainment mechanisms at the
stratified fluid mud-water interface. In: Nearshore and Estuarine Cohesive Sediment
T ransport, A. J. Mehta ed., American Geophysical Union, Washington, DC, 205-223.
Shi, Z. (1998). Acoustic observations of fluid mud and interfacial waves, Hangzhou Bay,
China. Journal of Coastal Research, 14(4), 1348-1353.
223
Shi, Z., Ren, L. F., Zhang, S. Y. and Chen, J. Y. (1997). Acoustic imaging of cohesive
sediment resuspension and re-entrainment in the Changjiang Estuary, East China Sea.
Geo-Marine Letters, 17, 162-168.
Smith, T. J. and Kirby, R. (1989). Generation, stabilization and dissipation of layered fine
sediment suspensions. Journal of Coastal Research, SI 5, 63-73.
Sottolichio, A., Hir, P. L. and Castaing, P. (1999). Modeling mechanisms for the turbidity
maximum stability in the Gironde estuary. In: Coastal and Estuarine Fine Sediment
Processes, W. H. McAnally and A. J. Mehta, ed., Elsevier, Amsterdam (in Press).
Srinivas, R. (1989). Response of fine sediment-water interface to shear flow. M.S. Thesis,
University of Florida, Gainesville, Florida, 137p.
Stansby, P. K. and Lloyd. P. M. (1995). A semi-implicit lagrangian scheme for 3D shallow
water flow with a two-layer turbulence model. InternationalJournal of Numerical
methods, 20, 115-133.
Su, J. L., Wang, K. S. and Li, Y. (1992). Fronts and transport of suspended matter in the
Hangzhou Bay. Acta Oceanologica Sinica, 12(1), 1-15.
Su, J. L. and Xu, W. Y. (1984). Modeling of the depositional patterns in Hangzhou Bay,
Coastal Engineering, 8,2181-2191.
Toorman, E. A., and Berlamont, J. E. (1993). Mathematical modeling of cohesive sediment
settling and consolidation. In: Nearshore Estuarine Cohesive Sediment Transport, A.
J. Mehta ed., American Geophysical Union, Washington, DC, 148-184.
Tsuruya, H., Murakami, K. and Irie, I. (1990a). Mathematical modeling of mud transport in
ports with a multi-layered model: application to Kumamoto Port. Report of the Port
and Harbour Research Institute, 29(1), 5 Ip.
Tsuruya, H., Murakami, K. and Irie, I. (1990b). Numerical simulations of mud transport by
a multi-layered nested grid model. Proceedings of the 22th Coastal Engineering
Conference, Vol. 3, ASCE, New York, 2098-3012.
van den Bosch, L., Toorman, E. and Berlamont, J. (1988; 1989; 1990). Settling column
experiments and in situ measurements. Reports to IMDC, Hydraulics Laboratory,
Katholieke University Leuven, Belgium, variously paginated (in Dutch).
van Leussen, W. and van Velzen, E. (1989). High concentration suspensions: their origin and
importance in Dutch Estuaries and coastal waters. Journal of Coastal Research, SI(5),
224
Verreet, G. and Berlamont, J. (1989). Rheology and non-Newtonian behaviour of sea and
estuarine mud. Encyclopedia of Fluid Mechanics, Vol. VIh Rheology & Non-
Newtonian Flow, N. P. Cheremisinoff, Ed., Gulf Publishing Co., Houston, Texas
135-149.
Vinzon, S. B. (1998). A preliminary examination of amazon shelf sediment dynamics.
Engineer Degree Thesis, University of Florida, Gainesville, Florida, 154p.
Wiersma, J. (1984). Acoustisch onderzoek bodemslib in relatie tot sedimentatie in
toegangsgeulen en zeehavens. Report NZ-N-84.07, Rijkswaterstaat, North Sea
Directorate, 74p.
Winterwerp, J. C. and Kranenburg, C. (1997). Erosion of fluid mud layers. II: experiment
and model validation. Journal of Hydraulic Engineering, 123(6), 512-519.
Wolanski, E., Asaeda, T. and Imberger, J. (1989). Mixing across a lutocline. Limnology and
Oceanography, 34(5), 931-938.
Wolanski, E., Chappell, J., Ridd, P. and Vertessy, R. (1988). Fluidization of mud in
estuaries. Journal of Geophysical Research, 93(C3), 2351-2361.
Wolanski, E., Gibbs, R. J., Mazda, Y., Mehta, A. and King, B. (1992). The role of turbulence
in the settling of mud floes. Journal of Coastal Research, 8, 35-46.
Woodruff, D. P. (1973). The Solid-Liquid Interface. Cambridge University Press,
Cambridge, UK, 182p.
Wright, L. D., Wiseman, W. J., Bomhold, B. D., Prior, D. B., Suhayda, J. N., Keller, G. H.,
Yang, Z. S. and Fan, Y. B. (1988). Marine dispersal and deposition of Yellow River
silts by gravity-driven underflows. Nature, 332 (14), No. 6164, 629-632.
Wu, J. (1973). Wind-induced turbulent entrainment across a stable density interface. Journal
of Fluid Mechanics, 61, 275-287.
Yeh, H. Y. (1979). Resuspension of properties of flow deposited cohesive sediment beds.
M.S. Thesis, University of Florida, Gainesville, Florida, 1 18p.
^'5 Martin, J. M., Zhou, J., Windom, H. and Dawson, R. (1990). Biogeochemical study
of the Changjiang estuary, China. Ocean Press, London, 898p.
Zhou, Y. K. (1986). Some characteristics of stream-like macro-tidal estuary (Jiaojiang).
Geographical Study, 5(1) (in Chinese).
225
Zimmemian, J. T. F. (1981). Dynamics, diffusion and Geomorphological significance of
tidal residual eddies, Nature, 290(16), No. 5807, 549-555.
BIOGRAPHICAL SKETCH
Jianhua Jiang was bom on March 15, 1961 in the village of Hezhai, Zhejiang, China.
He received his Bachelor of Engineering degree in hydromechanics from the Hohai
University, Nanjing, in 1983, and then worked as an Assistant Engineer in the East China
Institute of Hydro-Electric Investigation and Design, Hangzhou, for three years. He obtained
his Master of Science degree in physical oceanography in 1989 from the Seeond Institute of
Oceanography, Hangzhou. Subsequently, he was hired as a coastal engineer by the same
institute and was engaged in investigations of estuarine and coastal hydrodynamics and
sediment transport for about six years. In 1996, he was accepted by the Coastal and
Oceanographic Engineering Department of the University of Florida as a doctoral student
and research assistant. After three years of study, he eventually earned the Doctor of
Philosophy degree, and looks forward to contributing his newly acquired knowledge towards
solving various coastal and estuarine engineering problems.
226
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
Ashish J. Mehta, Chairman
Professor of Coastal and Oceanographic
Engineering
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
Robert G. Dean
Graduate Research Professor of Coastal
and Oceanographic Engineering
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
Kirk Hatfield
Associate Professor of Civil Engineering
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
D. Max Sheppard !/ f ^
Professor of Coastal and Oceanographic
Engineering
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
Robert J. Thiekef j
Assistant ProfesW^of Coastal and
Oceanographic Engineering
This dissertation was submitted to the Graduate Faeulty of the College of Engineering
and the Graduate Sehool and was aeeepted as partial fulfillment of the requirements for the
degree of Doetor of Philosophy.
August, 1999
M. Jaek Ohanian
Dean, College of Engineering
Winfred M. Phillips
Dean, Graduate Sehool
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