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 (c0.54 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 of3010, 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, 0^0. 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 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> < 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 ■< ^ 6 G o C/a 3 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 _ , 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 c u 3 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 cd > >, .o 00 (N (N o „ Q CO 00 «-H a 'o rH o o 1—1 (U O" 0) * E N— ✓ o d d W) Q *CO 00 VO 3h 3 ^ (N o a T-H (N ^ s 00 1 (N m >> O o VO o . a CO VO m c C in . »o o 3 T3 fN (N CN 2 d d d O VO m o -- o? 6 in (N (N fN No. Cd .O o v> 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 129 O O u S O 00 'O Urn cd C B c/3 c o o o (U "O a < r- E c ^cd ^ ^ ^ oo c o > S3 u X5 — ^ 0> C -o 3 o S wS © ® © @ c o c .a ° c/3 *- 3 ^ £ -o T3 < C ”S Lh x> cd T3 *:3 3 Cd 0 3^ [/3 T3 s C ^ o o CO O 2 0 © @ @ s _o Urn rg "cd o "TD O C Id o ^ :2 cd ^ CJ “ ^ ^ 0) — c/2 -S 0> (1) o> O I^S s ^ s © © cd -j-j a — CA ^ 3 ^ CQ C "cd o a. E tQ S + II -5- s3 ■*-* CO C O U « ■o o E c 3 (U C !S u \S E u o o E _ c 3 O .y 'S? S| > T3 c/3 C/3 c .E o Cd E (U o Cd a o 3 t .2 a c/3 > E 3 3 ^ .2 « jq CO I S O rt3 o £ "O 'q c/2 ^ 22 ^ 3 w E o *- o u C o Cd a 3 C o N crt c/2 O cd 0> J= c/2 E o c O a ’■<-J O a cd o s ’■+-» cd -4-) C/5 C4-N o o o c o cd Urn cd Table 6.1— continued 130 Table 6.1 --continued 131 132 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 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 St uuuv nmu>- *UiVUi' vuuu* u\ui* vuu* mui- UUi4‘ UiUi‘ ‘J4UU‘ > ^ B ed OJ Im C/) Cl. D B o ti o •o > o cd ca .2 > iO Os o < cd 'imun: hAii'i lu'iu^'h o ta (U 'S M s 13 U 3 B CO fo H 3 CUQ £ X' X 3 u at tN i,ilil\Ul> lUiUlli \UVUVi uuiu uuu iuur um;> JliUP 'HiVUi 'jmull! uiDiviM U'mv'k u»»uu-l cd > "S os o < cd l! a o tJ o > w<;a;vL u • 'U4U^*- 'UiUlV' o -O ' JUiU'* o .£> ' »iUUV‘ uuitut -I G Mumu Cd n G ‘U jiUVi a o o c •3 O o c T3 U "3 S CO (N vd 2^ 3 bJO £ luring a spring tide at 2000 hr, 1 1/05/94. during a spring tide at 2245 hr, 1 1/05/94. 139 o o -f2 u ,—4 D c3 U( -C vn T3 o U c3 a (D 3 B ’•w W) .s lo *c C14 X) c/3 o c3 3 W) W) c E ‘C D T3 f I I I 1 \Vl Tiftmtl f tmiut tmiiitt iitinui. Mutt ^uttt ttttt^ tf tttt t tttf tt» ttffttt, tttitU \ St CS E o o JO > o -O ed a •4— » a > •52 u o> o Ugtt\\^ 11 » f f 1 t t| 1:3 |u » 1 1 1 1 1 »| ^ 5: o ^ X) zi X) X ex O o cS u. T3 U "a wi < (,.s UI) XjioojaA ui 331) DSS Q.ui 831) oss 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. 141 o ~a ^ § 5 Vi O - C ^ . o 52 ^ in 3 d.§ ^ 13 T3 (D C c6 60 G ■c G T3 u c3 O o\ \D G a; 60 in £ o •a u J3 ■It H; (u = o o cd cd •O 2 13 <4-l c o c/i H a, ‘o _o 13 > •o l-l CD Vi G SP c T}- pi G T3 u t3 u m •S 5 T3 £ x: uT "O T3 § o s 2 (,.« in) iC}|Ooi0A (^.IH Sji) OSS (e.™ 3^1) OSS 00 d a 3 60 E a; m r~ d T3 § 60 _C 'C 3 XJ 142 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 uiouoq SAoqc uoiibasis SAiicja'a ;o "o c/2 o o 00 ai T3 cb T3 ^ 2 § (U ^ bO ' _c 'C a- (A eb ba B s S) u ^ 0) o c/3 c/3 gs in p td 0/3 05 2 cx U 00 00 d cs d 2 3 ba lx "O uT c cd c/3 8 c os _o *4-^ i2 ba _E 3 'C E 3 'c/3 2^ T3 -a u c3 _E 0/3 'rt 05 C 4— t X) o mOUOq 3AOqB U0I}BA3I3 SAia^is-^ 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