Citation |

- Permanent Link:
- http://ufdc.ufl.edu/UF00091088/00001
## Material Information- Title:
- A preliminary examination of amazon shelf sediment dynamics
- Series Title:
- UFLCOEL-98011
- Creator:
- Vinzon, Susana Beatriz, 1960-
University of Florida -- Coastal and Oceanographic Engineering Dept - Place of Publication:
- Gainesville Fla
- Publisher:
- Coastal & Oceanographic Engineering Dept., University of Florida
- Publication Date:
- 1998
- Language:
- English
- Physical Description:
- xvii, 155 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Estuarine sediments -- Brazil -- Amazon River Estuary ( lcsh )
Sediment transport -- Brazil -- Amazon River Estuary ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Engineer)--University of Florida, 1998.
- Bibliography:
- Includes bibliographical references (leaves 143-154).
- Statement of Responsibility:
- by Susana Beatriz Vinzon.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 41567289 ( OCLC )
## UFDC Membership |

Full Text |

UFL/COEL-98/011
A PRELIMINARY EXAMINATION OF AMAZON SHELF SEDIMENT DYNAMICS by Susana Beatriz Vinzon Thesis 1998 A PRELIMINARY EXAMINATION OF AMAZON SHELF SEDIMENT DYNAMICS By SUSANA BEATRIZ VINZON A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF ENGINEER UNIVERSITY OF FLORIDA 1998 To Mauricio and my son, Lucas. ACKNOWLEDGMENTS The author would like to express her gratitude to Dr. Ashish J. Mehta, Professor of Coastal and Oceanographic Engineering, University of Florida, for his advice and support during the development of this work. The author's appreciation is also extended to Professor Cliudio F. Neves and Professor Paulo C. C. Rosman, both in the Ocean Engineering Program, Federal University of Rio de Janeiro, Brazil, for their assistance, friendship and support during this study. Acknowledgment is also due to Dr. R. G. Dean and Dr. Y. P. Sheng for their participation as supervisory committee members and their helpful assistance. The author would like sincerely to thank Professor Gail Kineke of the University of South Carolina, who kindly provided the database analyzed in this work, to Professor Alberto Figueiredo, of the Federal Fluminense University, Brazil, who supplied sediment samples from the Amazon Shelf, and to Professor Richard Faas of Lafayette College, Easton, Pennsylvania, who provided data on sediment rheology. The author would also like to thank Prof. Dimitrie Nechet of the Federal University of Park, Brazil, and to Dr. Robert Kayen of the U.S. Geological Survey, Menlo Park, California, for providing supplementary information. Special thanks are due to the staff members of the Coastal and Oceanographic Engineering Department. This work could not have been completed without the support of Jolo C. M. Cassar and Luis A. B. Gusmlo from AQUAMODELO, Consultoria e Engenharia Ltda., in Rio de Janeiro. This work was made possible by financial support from CAPES, Brazil. TABLE OF CONTENTS ACKNOWLEDGM1ENTS ...........................................................i LIST OF FIGURES ............................................................ viii LIST OF TABLES................................................................ XV ABSTRACT ..................................................................... xvi CHAPTERS 1 INTRODUCTION ............................................................................1. 1. 1 Study Area and Problem Statement...................................................... 1 1.2 Outline of Chapters....................................................................... 6 2 DESCRIPTION OF THE AMAZON SHELF ENVIRONMENT .......................... 8 2.1 Water and Sediment Discharge.......................................................... 8 2.2 Tidal Flow............................................................................... 9 2.3 Subtidal Flow ............................................................................ 10 2.4 Wind and Waves......................................................................... 10 2.5 Amazon Plume and Salinity Distribution.............................................. 11 2.6 Data Set ................................................................................ 14 3 FORMULATION OF 1-D VERTICAL MODULE FOR FINE SEDIMENT TRANSPORT MODELING ................................................................. 19 3.1 Governing Equation..................................................................... 19 3.2 Sediment Characteristics................................................................ 20 3.2.1 Introduction ........................................................................ 20 3.2.2 Sediment composition and cation exchange capacity .......................... 21 3.2.3 Particle size ........................................................................ 23 3.2.4 Flocculation and settling velocity ................................................ 25 3.2.5 Settling experiments............................................................... 27 3.2.6 Viscosity ........................................................................... 32 3.2.7 Shear strength ............................................................................................... 35 3.2.8 Permeability and effective stress ................................................................... 38 3.3 Vertical Flow Structure ......................................................................................... 42 3.3.1 General ......................................................................................................... 42 3.3.2 Tidal boundary layer approach ...................................................................... 43 3.3.3 Fluid kinem atic and eddy viscosities ............................................................. 45 3.3.4 A model for lam inar boundary layer .............................................................. 51 3.3.5 Shear stress from m easured velocity profiles ................................................. 54 3.3.6 M ass diflusivity ............................................................................................. 59 3.4 Bottom Boundary Condition ................................................................................. 67 3.4.1 Bed level definition ....................................................................................... 67 3.4.2 Erosion m echanism s and erosion rate ............................................................ 69 3.4.3 Bed shear strength ......................................................................................... 80 3.4.4 Rate constant ................................................................................................. 81 3.4.5 D eposition and consolidation ........................................................................ 83 4 M UD DYNAM ICS IN AM AZON ESTUARY .............................................................. 87 4. 1 Introduction .......................................................................................................... 87 4.2 Long Term Processes: Accumulation M echanism s ................................................ 89 4.2.1 General observations ..................................................................................... 89 4.2.2 Salt wedge trapping m echanism .................................................................... 89 4.2.3 Tidal trapping ................................................................................................ 90 4.3 Short Term Processes: Local Balance in the Water Column .................................. 93 4.3.1 Some dynamical features of the suspension over the water column ................ 93 4.3.2 Lutocline dynam ics ....................................................................................... 97 4.3.3 W ave effects ............................................................................................... 101 4.3.4 Tim e scale of sedim entary processes ........................................................... 107 4.4 Validity of I -D Approach ................................................................................... 109 4.4. 1 Introduction ................................................................................................. 109 4.4.2 Advective transport ..................................................................................... 110 4.4.3 Turbidity current ......................................................................................... 114 5 TR AN SPORT M ODEL ............................................................................................... 117 5.1 Preamble ............................................................................................................. 117 5.2 M ain characteristics of the num erical model ....................................................... 117 5.3 Numerical experiments ....................................................................................... 120 5.4 M odeling Quiescent Settling ............................................................................... 124 5.5 Amazon Shelf Concentration Profiles ................................................................. 129 5.5.1 Objective ..................................................................................................... 129 5.5.2 Bed level and bottom boundary condition .................................................... 129 5.5.3 Settling velocity .......................................................................................... 131 5.5.4 M ass diffusion coeffi cient ........................................................................... 132 5.5.5 Vertical component of the flow velocity ...................................................... 132 5.5.6 Sediment concentration time series .............................................................. 133 6 GENERAL OBSERVATIONS, SUMMARY AND CONCLUSIONS ......................... 138 6.1 General Observations .......................................................................................... 138 6.2 Summary and Conclusions .................................................................................. 140 REFEPENCES ................................................................................. 143 BIOGRAPFUCAL SKETCH ............................................................... 155 LIST OF FIGURES Figure Page 1 Velocity, suspended sediment concentration, and sediment flux profiles in the Amazon Shelf during high discharge at station 0S2 (from Kineke, 1993) ............ 4 2 Amazon water discharge from daily estimates at Obidos.................................. 8 3 Seasonal sea surface salinity (psu, i.e., practical salinity units) in the Amazon Shelf region (from Kineke, 1993)........................................................ 13 4 Bottom salinity (psu) in the Amazon Shelf region........................................ 14 5 Cross-shelf salinity transects at the river mouth for (a) spring and (b) neap tides (from Kineke, 1993) ...................................................................... 15 6 Topographic map of the Amazon continental margin with locations of AMASSEDS anchor stations (from Kineke, 1993) ..................................... 17 7 Size distribution of the Amazon Shelf bottom sediment for samples 3227 and 3230 ..................................................................................... 24 8 Size distribution of flocs determined by image analysis of plankton camera photographs at 35 mn depth and high river discharge (from Kineke, 1993)........... 26 9 A schematic representation of flocculation, sedimentation and consolidation (from Imai, 198 1) ......................................................................... 28 10 An example of temporal evolution of concentration profile in the settling column..31 11 Settling velocity versus concentration obtained from settling experiments through Equation (7) (small symbols) and from free settling stage (larger black dots) ....... 32 12 Flow behavior of Amazon sediment suspension at different densities (from Faas, 1986) ..................................................................................... 34 13 Apparent viscosity vs. shear rate for a core sample from the Amazon Shelf, as a function of density (from Faas, 1985) ................................................... 34 14 Schematic of shear stress versus shear rate flow curve, and definition of the upper Bingham yield stress ...................................................................... 36 15 Yield shear stress versus concentration for Amazon Shelf mud (from Dade, 1992, Table 6.7 and Faas, 1985, Figure 3)...................................................... 37 16 Shear strength versus concentration for the Amazon Shelf bed sediment (1) for incipient motion and (2) for bed destruction............................................. 38 17 Effective stress as a function of local density (from Toorman and Huysentruyt, 1994) ..................................................................................... 40 18 Effective stress versus void ratio data taken from experiments reported in the literature (Alexis et al., 1992) ............................................................ 41 19 Permeability versus void ratio data taken from experiments reported in the literature (Alexis et al., 1992) ............................................................ 41 20 v (circles) and vf (asterisks) median values for the tidal cycle, from Equation (25) and (11), respectively ..................................................................... 46 21 Velocity profiles recorded at anchor station 0S2 (3419)................................. 47 22 Boundary layer velocity profiles for increasing and decreasing viscosity with depth ..................................................................................... 49 23 Tidal Reynolds number [Equation (29)] vs. depth ........................................ 50 24 Observed (discrete points) and calculated [Equation(3 0)] velocity profiles over two tidal cycles recorded at anchor station 0S51 (low river flow-as443 4)............ 52 25 Absolute velocity errors arising from differences between measurements and Equation (30) (asterisks) and measurements and logarithmic fit (circles) ........... 53 26 Observed velocity (dotted line) and shear stress (solid line) over two tidal cycles for anchor stations OS 1 and CN ........................................................ 56 27 Observed shear stress [dotted line Equation (35)] and calculated shear stress [solid line Equation (3 1)] over two tidal cycles for anchor stations OS 1 and CN ...................................................................................... 57 28 Depth variation of the tidal eddy viscosity at anchor station CN (2428) (dots) and eddy diffuasivity calculated according to Equation (37) ................................ 60 29 Tidal median values of the gradient Richardon number considering density affected by salinity and temperature (solid line), and salinity, temperature and sediment concentration (dotted line).................................................... 61 30 Gradient Richardson number as a function of depth for anchor stations CN (2428) and 0S2(2418) .................................................................. 64 31 Tidal median values of the stratification damping function [Equation (41)] versus depth for anchor stations CN (2428) and 0S2 (2418).................................. 65 32 Stratification damping function versus depth over the tidal cycle for anchor station CN (2428) ....................................................................... 66 33 Time-variation of water depth at anchor station CN (2428)............................. 67 34 Tidal median values of shear stress versus depth for anchor station RMi............. 72 35 Sediment flux (at z=O.25m) and current velocity for anchor station OSI (1154). Top: time-series of sediment flux and velocity. Bottom: sediment flux vs. velocity after phase adjustment........................................................... 74 36 Sediment flux (z=O0.25m) and current velocity for anchor station 052 (2418). Top: time-series of sediment flux and velocity. Bottom: sediment flux vs. velocity after adjusting for the phase..................................................... 75 37 Sediment flux (z=O0.25m) and current velocity for anchor station CN (2428). Top: time-series of sediment flux and velocity. Bottom: sediment flux vs. velocity after adjusting for the phase .............................................................. 76 38 Sediment flux (z--lm) and current velocity for anchor station CN (2428). Top: time-series of sediment flux and velocity. Bottom: sediment flux vs. velocity after adjusting for the phase .............................................................. 77 39 Sediment flux (z--lm) and current velocity for anchor station RMo (3455). Top: time-series of sediment flux and velocity. Bottom: sediment flux vs. velocity after adjusting for the phase .............................................................. 78 40 Salinity and sediment concentration time-series for anchor station RMo (2405)...79 41 Time-series of sediment flux and current velocity for anchor station RMo (2405) ...79 42 Time-series of sediment flux and current velocity for anchor station OSi1 (4434)...80 43 Observed (dotted line) and calculated (solid line) sediment fluxes at an elevation of 0.25m for anchor stations 0S2 (2418) and CN (2428) ............................. 82 44 Turbidity maximum caused by residual circulation (from DYER, 1986) .............. 90 45 Sediment trapping due to tidal asymmetry (from ALLEN et al., 1980) ................ 91 46 Examples of water levels and current velocities in the Amazon Shelf area. (a) OS 1 anchor station at the Open Shelf Transect and, (b) RMi anchor station at the River Mouth Transect (see Figure 6 for locations) ........................................ 94 47 Distribution of sediment accumulation rate (from KUEHL et al. 1986). Contours are in cm /yr ............................................................................................................. 95 48 Vertical concentration and velocity profiles in high concentration estuarine environm ents (after M ehta, 1989) ........................................................................ 97 49 Settling velocity and settling flux as functions of concentration (from Ross and M ehta, 1989) ........................................................................................................... 98 50 Suspended sediment concentration profiles (solid line), and water density gradients including salinity and temperature effects (circles), for anchor stations in the Amazon Shelf for cases of strong salinity stratification ................................ 100 51 Suspended sediment concentration profiles (solid line), and water density gradients including salinity and temperature effects (circles), for anchor stations at the Amazon Shelf for cases of weak salinity stratification .................................. 101 52 Influence of waves on shear resistance to erosion of kaolinite beds in flumes (from M ehta, 1989) ............................................................................................... 102 53 Time-series of a) pressure and b) velocity measured at OS3 anchor station while the profiler was positioned on the seabed. The sediment concentration at this level was about 300 g/l (from Kineke, 1993) ......................................................... 105 54 Depth, hourly current speed and sediment concentrations at the GEOPROBE deployment (from Cacchione et al., 1995) ............................................................. 106 55 Hourly wind velocity recorded at Macapik Amapi ................................................... 106 56 Current meter, optical backscatter output (OBS) and near-bed orbital velocity at St. Bees deployment, Irish sea, UK (from Aldridge and Rees, 1995) ..................... 108 57 Maximum isoline elevation reached during the tidal cycle versus tidal range (both made dimensionless by dividing by the local depth) for the 18 anchor stations ...... 109 58 Comparison between measured (dashed lines) and simulated (solid line) suspended sediment concentrations in the Weser estuary within the zone of turbidity maximum (from Lang et al., 1989). Only advection was simulated ......... 111 59 Current speed (dashed line) and optical backscatter output (solid line) measured outside of Morecambe Bay, Irish Sea, UK (from Aldridge and Rees, 1995) ........... 112 60 Concentration profiles (left) and time series of salinity and concentration (right) at anchor station RM i (2444) ................................................................................. 114 61 Flood and ebb currents in the Amazon Shelf and residual sediment density induced currents at anchor stations CN (2428) and OS1 (4434) ............................. 116 62 Numerical simulation of tide-induced resuspension with a uniform settling velocity (W=lmIm/s) and a depth-independent, time-dependent diffusivity. Hourly concentration profiles over a tidal cycle ..................................................... 121 63 Simulations similar to Figure 62, but with a smaller settling velocity (W5=0.Smm/s). Hourly concentration profiles over a tidal cycle ............................ 122 64 Comparison of results for different particle masses. Solid lines: Cmm=0.005g/l, asterisk and circles: C min=0.02g/l ........................................................................... 122 65 Same simulation as in Figure 63 with a constant settling velocity (W,=0.Smm/s), but a with a non-uniform difflsivity (set equal to zero from 3.5m level up to surface). Hourly concentration profiles over a tidal cycle ...................................... 123 66 Same simulation as in Figures 62 and 63, but with a non-uniform settling velocity. Hourly concentration profiles over a tidal cycle ...................................... 124 67 Observed (dotted line) and modeled (solid line) sediment concentration profiles obtained in settling experiments in quiescent conditions ........................................ 126 68 Observed (dotted line) and modeled (solid line) sediment concentration profiles with the inclusion of effective stress [Equation (5 1)] ............................................ 128 69 Bottom sediment flux based on measurements at anchor station CN and calculated by the model, according Equation (38) .................................................. 130 70 Dotted lines define the range of the settling velocity considered in the model ......... 131 71 Mass diffusion coefficient profile included in the model ......................................... 133 72 Sediment concentration profiles at anchor station CN (rising river flow- as2428), measured (solid line) and simulated (circles) ........................................................ 134 73 Hysteresis in sediment concentration at 3.25m level at anchor station CN (2428), measured (top) and simulated (bottom) ................................................................ 136 LIST OF TABLES Table Page 1 Summary of Amazon Shelf data........................................................... 18 2 Cation exchange capacities of three common clay minerals, in meq/lO0g ............ 22 3 Exchangeable cations and cation exchange capacities for Amazon samples, in meq/loog ................................................................................. 23 4 Settling column test conditions ............................................................ 30 5 Tidal mean and maximum shear stresses and shear velocities at the anchor stations ... 58 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Engineer A PRELIMINARY EXAMINATION OF AMAZON SHELF SEDIMENT DYNAMICS By Susana Beatriz Vinzon August 1998 Chairman: Ashish J. Mehta Major Department: Coastal and Oceanographical Engineering Experimental data on flow and sediment dynamics on the Amazon Shelf, previously obtained under AMASSEDS (A Multidisciplinary Amazon Shelf Sediment Study), have been interpreted with the help of a modeling approach to examine the vertical structure of flow-sediment interaction. The model solves the sediment transport equation following the particle tracking method. In this model, sediment settling velocity is considered to be concentration dependent, and erosion flux function is fitted using in situ near-bed measurements of velocity and sediment concentration. Salt stratification damping of turbulence is also included, and the shear strength-bed density relationship required for the calculation of the erosion flux is derived from laboratory analysis of Amazon sediment samples. Measured time-series of suspended sediment concentration are compared with model-simulations in order to better understand the sedimentary processes occurring in the water column. It is found that flow-sediment interaction in the bottom one meter of the water column can be described using an oscillatory viscous flow model. The reason for this behavior is the significant turbulence-damping role of the high density fluid mud-like suspension that dominates the lower water column. For the remainder of the water column, stratified turbulent flow conditions can be assumed. Long-term accumulation mechanisms as well as short-term processes related to the tides are examined in the context of the vertical transport processes. As observed in the AMASSEDS study, the findings of the present investigation corroborate the view that a combination of density-driven estuarine circulation, salt-induced stratification, and flocculation enhance the trapping of sediments in fluid mud over the long term. On the other hand, short-term, namely tidal, signatures indicate that sediment dynamics over this time scale is strongly influenced by resuspension events governed by tidal forcing. Sediment spreading over the shelf appears to be dominated by transport processes which occur within the bottom 2m of the water column. Further measurements are required to assess fully the role of fluid mud advection within this layer as the key component of shelf sediment transport in this area. It is also likely that swell activity contributes to the behavior of the oscillatory boundary layer and therefore to fluid mud dynamics over the shelf. xvii CHAPTER I INTRODUCTION 1. 1 Study Area and Problem Statement There are wide ranging engineering problems associated with the transport of fine sediments. Ship channel filling, and sedimentation problems along quay walls, under pontoons or in shadow zones in harbors, are well known problems for many port authorities. Water quality related to the transport of fine sediments is another engineering problem of increasing interest. The properties of clay-sized sediments, including large specific area of the particle, net negative electrical charge on their surfaces, and cation exchange capacity, facilitate the sorption of contaminants (Hayter and Mehta, 1983). Thus, the movement of fine sediments must be understood in order to account for contaminant transport and fate. Amazon River, having by far the largest river discharge on the planet, carries up to its mouth a great amount of sediment, which is mainly composed of fine sediments. Fresh and seawater mixing occurs on the Amazon Shelf, far from the river mouth in a very dynamic macro-tidal environment. High concentration of fine sediment can be found over an extensive area on the shelf, and a complex interaction between sediment and hydrodynamics occurs. Detailed measurements of sediment concentration profiles made recently, and available for further studies, offer the opportunity to take a close look at the mechanisms governing sediment transport in high concentration environments. Improving our understanding of the dynamics of these environments is a first step toward improving our ability to model the relevant processes, the ultimate objective of this work. A comprehensive research program of sediment transport on the Amazon shelf occurred from 1989 to 1991 as part of the AMASSEDS project. AMASSEDS (A Multidisciplinary Amazon Shelf Sediment Study) was an interdisciplinary research group, which developed geological, chemical, physical and biological studies undertaken by oceanographers from Brazil and the United States (AMASSEDS, 1990). As part of the AMASSEDS project, a suspended sediment transport study was designed and carried out to measure the flow and suspended sediment characteristics throughout the water column over several time scales: semidiurnal, fortnightly and seasonal (Kineke, 1993). Four cruises were undertaken to cover seasonal variability in the river discharge, wind stress, wave climate and transport of the North Brazil Current. Profiling from an anchored ship was done to account for the semidiurnal time scales. Profiling was repeated during both spring and neap tides. This data set was available for the analysis made in the present work. Also, a coastal profiling study was carried out (in shallow waters from 2 to 10m), and an instrumented bottom tripod (GEOPROBE) was deployed in 63m depth and measured boundary layer parameters continuously over 19 days. Many noteworthy works have reported the main results and findings of the AMASSEDS project (Kineke, 1993, Beardsley et al., 1995, Cacchione et al., 1995, Kineke and Sternberg, 1995, Kuehl et al., 1995, Geyer, 1995, Lentz, 1995, Nittrouer et al., 1995, Trowbridge and Kineke, 1994, among others). Most of these studies are reviewed in Nittrouer and DeMaster (1986). One of the more important findings during the AMASSEDS project was the record of thick layers of mud, which can reach up to 7m in the inner and middle shelf, covering an extensive region (Kineke, 1993). Careful profiling of current velocity, salinity, temperature and sediment concentration was carried out over the shelf. The lowermost measurements were 0.25m above the level where the profiler rested. An interesting plot of data related to sediment transport is shown in Figure 1, which gives instantaneous vertical profiles of velocity, concentration and flux (Kineke, 1993). The maximum flux occurs in the lower 2m, while the suspended sediment flux in the main water column and surface plume is 1-3 orders of magnitude less. This demonstrates how critical it is to measure the sediment transport rate as a function of elevation and to model velocity and sediment concentration distributions near the bottom. In this study it is intended to formulate a model which accounts for, in the simplest way, the characteristics and processes that occur in the water column including the nearbed layers responsible for the larger part of sediment transport. Also, it is sought to bring further insight into the understanding of the processes responsible for the genesis and dynamics of sediment concentration profiles in the Amazon Shelf area. Numerical modeling is one of the most useful prediction techniques meant to solve hydrodynamic and transport problems in estuaries. Fine sediment transport models solve the mass transport equation for suspended sediment, and different approaches can be found in the literature, according to the problem of interest. A review of numerical models on fine sediment transport can be found in Mehta et al. (1989a) and Mehta et at. (1989b). With the continuous reduction of computational costs, three-dimensional models are more and more commonly used (Sheng, 1986; Nicholson and O'Connor, 1986; Lang et al., 1989). Density stratification effects on turbulence and density driven currents, rheologically controlled near-bed layers in high concentration zones, and advection of density gradients are some of the aspects in modeling which require threedimensionality. However, the use of a simple suitable model to represent the processes occurring in the water column, in order to reach a better understanding of the physics of sediment motion in the Amazon Shelf, is the "track" chosen in this work. It v-i1; 1< . ... ... S0 0. 0.5 1 1.5 0.01 1 100 0.01 1 100 veloc-ty (M/s) ce=ntn (g/i) medit flux (k/r ) Figure 1: Velocity, suspended sediment concentration, and sediment flux profiles in the Amazon Shelf during high discharge at station OS2 (from Kineke, 1993). Fine sediment transport models are very demanding of numerical methods. The quantities usually involved, like settling velocity or diffusion, vary by several orders of magnitude over the physical domain. In the vertical dimension, strong sediment concentration gradients can be observed, and these raise severe problems in solving the transport equation. An alternative numerical method based on what is called the particle tracking technique is introduced in the present work in order to attempt to overcome common problems found with the usual methods. Particle tracking methods are being used increasingly in transport problems, and their introduction into sediment transport problems may be'very useful. The main characteristics of the method are described in the present work, and its advantages and disadvantages are discussed in order to look for future improvements. The choice of the numerical method represents an important first step in the modeling process, since it must ensure that the numerical results correspond to the equivalent mathematical model. It should be noted, however, that mathematical models are always restricted. They are mere representations of physics which require a knowledge of the underlying processes and of the scales associated with them. Thus, before the presentation of the proposed model, a comprehensive examination of the relevant mechanisms which occur in high concentration fine sediment environments is presented. It should be noted that one of the conclusions arrived at by Kineke (1993) and Kineke and Sternberg (1995), concerning the mechanism that explains the presence of thick layers of fluid mud in the Amazon Shelf, is The combination of density-driven estuarine circulation, salt induced stratification, and flocculation enhance the trapping of sediments in fluid muds. Although tidal currents are very strong on the Amazon shelf, and surface waves probably are important in maintaining fluid muds as a suspension, fluid muds mainly are not the result of resuspension or erosion from the seabed, but the result of enhanced settling flux. (Kineke and Sternberg 1995, p.227) If correct this assertion precludes the possibility of there existing a local sedimentary flux balance which could explain the observed sediment concentration profiles. According to this conclusion, modeling for representing the processes in the water column must be thought of only in a two- or three-dimensional manner. It is hypothesized here, that density-driven circulation is a mechanism which acts over longterm scales, of several months, and may play an important role in the turbidity maximum formation at the Amazon Shelf. Salinity-induced stratification may exert influence over the tidal time scale due to the damping of the turbulence. However, the observed suspended sediment concentration profiles, and particularly the quantity of sediment found in suspension, can be the result of local sediment flux balances, which can therefore be explained in a simple way by using a one-dimensional vertical model. 1.2 Outline of Chapters A general description of the physical environment and a brief description of the data set are provided in Chapter 2. In Chapter 3 the mathematical model for sediment transport in the water column is described. A characterization of the native sediments was necessary for comparative purposes, and also to obtain the sediment dependent functions to be included in the model, i.e., for settling velocity, for the viscosity of the water-sediment mixture, for bottom shear strength and for consolidation. This aspect is presented in Section 3.2. Flow velocity and shear characteristics are also needed for modeling purposes. Thus, a preliminary analysis of the measured velocity profiles is provided in Section 3.3, which highlights the main features of the flow. In Section 3.4.2 an estimate of the erosion rate, also a model parameter, is provided. A discussion of the several long and short-term mechanisms underlying the transport of fine sediments related to the Amazon Shelf is presented in Chapter 4. The importance of short period wave and tidal action mechanisms is addressed in Section 4.3.3. At the end of this Chapter, in Section 4.3. 1, some of the main features related to sediment vertical profiles are noted. In Chapter 5 the numerical model is described briefly. After the presentation of a few examples showing the response of the model to some of the important modeled processes, settling experiments in quiescent conditions are reproduced with the model. The most important results relative to the modeling of a time series of sediment concentration on the Amazon Shelf are presented and discussed in section 5.4. Finally, conclusions are provided in chapter 6. CHAPTER 2 DESCRIPTION OF THE AMAZON SHELF ENVIRONMENT 2.1 Water and Sediment Discharge Draining an area of 6.9 million km2, the Amazon River is the largest riverine freshwater source to the oceans (Kineke 1993, Gibbs, 1970). With an average discharge of 1.8x105 m3/s, it exhibits strong seasonal variations. Figure 2 shows daily estimates of Amazon River discharge for 1989-1991 based on stage data from Obidos, with an observed maximum discharge in May of about 2.4x105 m3/s, and a minimum discharge in November of 0.8x105 m3/s (Figueiredo et al., 1991). O E 1.8 W0 1.5 F * .2 0.9 0.6 1989 1990 1991 Figure 2: Amazon water discharge from daily estimates at Obidos, from Figueiredo et al. (1991). Relative to the sediment discharge of the Amazon River, Meade et al. (1985) reported a mean value of 11-13x108 tons/year. Richey et al. (1986) measured sediment discharge at several locations along the Amazon River. At Obidos, the sediment flow varied from 2.5 to 19.2x108 tons/year, with minimum and maximum recorded concentrations of 0.09 and 0.35 g/l, respectively. Most of the suspended material transported was reported to be fine (<0.063 mm). Gibbs (1967) also reported that 8595% of the sediment carried out to the mouth of the Amazon River was silt- and claysized. In addition to the large input of fresh water, strong tidal currents and wind-driven currents contribute to one of the most energetic continental shelf environments in the world. 2.2 Tidal Flow The tidal forcing is strong, with a range of up to about 5m (Gibbs, 1970), which generates strong currents with a large fortnightly variability. The dominant tidal constituent, the M2 (12.42 h), plus other principal semidiurnal constituents, the S2 (12.00 h) and the N2 (12.66 h), account for about 85% of the total tidal elevation variance, and are the most energetic tides in this region. Beardsley et al. (1995) observed that M2 tidal currents are oriented primarily across the local isobaths over the shelf Over the adjacent shelf north of Cabo Norte, the M2 tide approaches a partial standing wave, with large amplitudes near the coast due to resonance. One consequence of the enhanced tide within the Cabo Norte-Cabo Cassipore embayment is the formation of large tidal bores, locally known as 'pororocas'. Near the Amazon River mouth, the M2 tide propagates as a progressive wave, with decreasing amplitude in relation to the vertical structure of the M2 tidal current. Beardsley et al. (1995) observed a small ellipticity with a clockwise rotation. The M2 phase increased with height above bottom, with flood at the bottom leading that at the surface by about 1 hour. 2.3 Subtidal Flo Subtidal currents are vertically sheared, moving along-shelf toward the northwest (Limeburner and Beardsley, 1995). With near-surface currents that reach speeds of 1.5 m/s and near bottom velocities less than 0. 2 rn/s (Geyer et al., 199 1), these currents show a high degree of coherence with the along-shelf wind stress (Lentz, 1995). The influence of the North Brazilian Current is not completely determined, but there is evidence that it could help explain the drift of the Amazon plume in the opposite direction to most buoyant plumes, with the coast on its left, in the Northern Hemisphere. 2.4 Wind and Waves The trade winds blowing over the Amazon shelf do not show monsoon-like reversals of the prevailing wind direction, or hurricanes and tropical storms, as at other similar latitudes. According to Nittrouer et al. (1995), "winds generally strengthen in January-March while blowing onshore from the northeast, and weaken and blow alongshelf from the southeast during June to November". This wind climate causes locally generated waves that are dominated by short periods coming from the eastern quadrant (significant height of 1-2 m less than 4m 99% of the year, and dominant periods of 6-7 sec less than 10sec 95% of year, from Kineke, 1993). Nevertheless, observations carried out in the area along the coast of Ceari (40 S) indicated long period (up to 20 s) swelltype waves coming from a more northerly direction from December to March. Melo et al. (1995) asserted that these waves might have as a distal source, namely the extratropical storms in the North Atlantic. 2.5 Amazon Plume and Salinity Distribution The Amazon River discharge is so large that seawater never enters the river mouth. The riverine water meets the seawater on the shallow continental shelf 100-200 km seaward of the mouth (Geyer, 1995). This shelf is oriented in NW-SE direction and extends from shore to the 100m isobath, between the latitudes of about 00 and 5*N. The Amazon River plume follows the coast northwestward from the river mouth. Sea surface salinity maps (Figure 3) show the plume spreading in the vicinity of the river mouth and extending northwestward along the shelf. The lowest salinity in the vicinity of the river mouth is found near the North Channel, suggesting that this is the strongest source of fresh water. Lentz and Limeburner (1995) describe the behavior of the riverine plume as follows. "The Amazon River discharge forms a plume of low-salinity water that spreads from the river mouth at the equator, 5m thick layer of relatively fresh water. The plume intersects the bottom sea water between the 10m and 20m isobaths forming a bottom front that appears to be locked to the local bathymetry" (Figure 4). Gibbs (1970) suggested that the North Brazilian Current (referred to as Guiana Current in his work) carries the plume out into the ocean, while the prevailing trade winds and waves result in a steady longshore current flowing northwestwardly along the shore. Lentz (1995) pointed out that the general tendency of the Amazon plume to flow northwestward is probably related to the North Brazilian Current, but he also observed large variations in plume width, salinity and currents of time scales of days to weeks due to the local wind stress. Lentz and Limeburner noted the importance of local winds, asserting that "southeastward winds may impede or even block the normal northwestward transport of fresh water resulting in a pool of fresh water. This process, more frequent in spring, when the river discharge is large and southeastward winds are most likely, may have important implications in sediment supply since it results in both longer residence times and broader areal extent in the vicinity of the river mouth". Spring-neap variations in vertical density structure on the Amazon shelf have been observed (Geyer, 1995). During neap tides the water column is strongly stratified, resembling an estuarine salt wedge. During spring tides, however, stratification is greatly reduced resembling a partially mixed estuary. In all cases, surface and bottom waters flow in the same direction during the nearly all-tidal cycle. Figure 5 shows two crossshelf salinity transects for spring and neap tides observed at the river mouth. Different salinity structures in different tidal regimes are apparent. Figure 3: Seasonal sea surface salinity (psu, i.e., practical salinity units) in the Amazon Shelf region (from Kineke, 1993). 35- ~30j 25 E o 15 0. 0 20 40 60 80 100 water depth (m) Figure 4: Bottom salinity (psu) in the Amazon Shelf region (from Lentz and Limeburner, 1995). 2.6 Data Set The data set examined in this study is part of data collected during the AMASSEDS study (AMASSEDS, 1990). As noted in chapter 1, this program was a multi-component U.S./Brazil program meant to study physical, geological, and geochemical processes that control sediment transport and accumulation in the Amazon Shelf region. The physical oceanographic component included four regional shipboard Acoustic Doppler Current Profiler (ADCP) and Conductivity-Temperature-Depth (CTD) surveys timed to sample the shelf during four stages of the Amazon River discharge, an instrumented bottom tripod deployment (GEOPROBE), and a coastal study. Figure 6 shows the sampling locations. Measurements were taken seasonally, according to the river stages shown in Figure 2 along 6 cross-shore transects, at 19 anchor stations with hourly profiles during one or two tidal cycles, and during coastal profiling, totaling more than 600 profiles. Synchronous profiles of current, sediment concentration, salinity and temperature over the tidal cycle were available for the present study. lMtonm Offasbm (O ) 50 75 100 125 150 (b) Dssalnc offabm cko Figure 5: Cross-shelf salinity transects at the river mouth for (a) spring and (b) neap tides (from Kineke, 1993). A detailed description of the collected velocity profiles and data analysis has been given by Kineke (1993). For completeness of treatment, the main features of the measurement procedure are reproduced here. The measurements were performed with a profiling system which included an optical suspended sediment sensor, a water/suspended sediment pumping system to collect samples, a CTD (i.e., Conductivity-Temperature-Depth) sensor, an electromagnetic current meter and a compass. The profiles were continuous from the sea surface to -0.25m above the seabed. The profiler was lowered through the water column at -10m min."1 with a sampling rate of -0.85 Hz, thus yielding a scan every 15-18cm. The up-and-down casts were binaveraged in 0.3m bins, and the data were converted to height above the bottom in evenspaced intervals (0.25m). The bottom reading elevation (0.25m) was an averaged value over the time the tripod was on the seabed. In addition to bin-averaging, for the velocity measurements a 17-term cubic/quadratic smoothing function was used to filter out some of the high frequency oscillations. Measurements were referenced to the height above the bed where the bed was considered to be defined by the limit of downward movement of the suspended-sediment profiler (Kineke and Sternberg, 1995). Table 1 provides a summary of the main characteristics of each time series recorded at the 18 anchor stations, identified by a name with reference to its location (showed in Figure 6) and a number. Note that mean and minimum bottom velocities denote values recorded at the lower-most elevation, at z = 0.25n, considering z = 0 to be the resting elevation of the profiler legs. Isoline elevations indicate the maximum elevation reached by concentrations of 0. lg/l or lg/l over the tidal cycle. Total mass denotes the mean total mass in suspension per unit area over the tidal cycle. Figure 6: Topographic map of the Amazon continental margin with locations of AMASSEDS anchor stations (from Kineke, 1993). Table 1: Summary of Amazon Shelf data Anchor River Depth Tidal Max. Min. Max. Min. Mean Max. Max. Max. Total Station Stage Range Salinity Salinity Conc. Bottom Bottom Velocity 0.1g/l lg/l Mass Vel. Vel. Isoline Isoline Elev. Elev. (M) (M) (psu) (psu) (g/l) (m/s) (m/s) (m/s) (M) (m) (kg/n2) OS2 1154 falling 17.8 2.05 34.4 5.9 0.72 0.0039 0.299 1.51 10.0 0.0 1.3 OSi 4434 low 11.9 5.12 24.2 16.3 321.0 0.003 0.0165 1.49 8.0 3.5 203.4 OS2 4438 low 17.8 4.54 34.1 27.7 11.5 0.027 0.404 1.49 17.8 10.0 9.6 OS3 4441 low 37.4 2.22 36.3 26.7 1.28 0.006 0.232 1.20 27.0 0.0 2.7 RMc 4413 low 14.5 2.11 34.6 0.4 292.5 0.0008 0.0641 1.20 6.0 4.0 133.8 OSi 2420 rising 11.8 5.06 16.8 15.6 124.0 0.025 0.386 1.54 11.8 11.8 16.3 OS2 2418 rising 18.0 3.80 33.5 6.7 129.2 0.018 0.0864 2.01 18.0 8.5 120.2 OS3 2415 rising 39.3 3.23 35.6 25.6 289.3 0.028 0.0747 1.84 10.0 3.0 164.0 CN 2428 rising 17.3 3.13 34.1 9.3 233.6 0.015 0.0374 1.67 6.5 3.0 122.3 RMi 2444 rising 13.5 1.69 19.2 0.1 245.6 0.018 0.0542 1.45 4.0 2.5 87.5 RMo 2445 rising 10.9 1.39 26.0 1.3 266.6 0.0357 0.0429 1.35 3.5 2.8 118.5 RMo 2405 rising 11.1 3.25 21.0 3.1 86.5 0.034 0.237 1.74 11.1 11.1 110.4 OSI 3420 high 12.1 5.17 23.6 13.3 291.7 0.012 0.245 2.05 12.1 6.5 90.3 OS2 3419 high 18.3 3.41 33.6 15.3 285.2 0.0081 0.0181 1.97 14.0 8.0 203.8 OS3 3418 high 40.3 2.19 35.3 12.1 304.8 0.012 0.0412 1.97 10.0 1.0 128.8 CN 3442 high 16.4 3.91 34.3 24.3 287.2 0.019 0.279 1.83 10.0 7.0 92.9 RMo 3455 high 10.6 2.78 13.7 0.6 145.2 0.014 0.169 1.69 10.6 6.5 53.3 RMo 3405 high 10.8 2.26 27.0 2.1 290.2 0.011 0.222 1.81 6.0 4.0 63.9 CHAPTER 3 FORMULATION OF 1-D VERTICAL MODULE FOR FINE SEDIMENT TRANSPORT MODELING 3.1 Governing Equation The governing equation for the vertical suspended sediment transport is the conservation of mass, 6C =9 CO(w,-) + _.(1) where C is the suspended sediment concentration, W. is the settling velocity, e, is the mass diffusion coefficient, andw is the vertical component of the flow velocity. This last variable is usually neglected, but it may be important where noticeable bathymetry variations occur, or where tides have high ranges, which is the case in the Amazon Shelf environment. Vertical velocity values are low, but they can be of the order of the settling velocity of fine sediments. The vertical velocity was estimated as (Dean and Dalrymple, 1991) ozah (2) h at where h is the water column depth. Equation (1) requires two boundary conditions, one at the free surface and another at the bottom level. At the free surface, the boundary condition corresponds to no net sediment flux. As for the bottom condition, the conceptual model assumes that bed-suspension sediment exchange occurs only in one direction, i.e. erosion or deposition. In other words, erosion and deposition are not considered to occur simultaneously. For erosion, i.e., when the flow bed shear stress, to., is greater than the shear strength of the overlying bed layer, -c., a linear rate of erosion is prescribed according to z =0, -' > r, E=MCTL 1D where M is an empirical erosion rate constant. When no erosion is occurring, a deposition rate is prescribed according to z=0, Dr t D=C(W, -w) (4) To solve Equation (1) with the corresponding boundary conditions it is necessary to prescribe the settling velocity, mass diffusivity coefficient, flow shear stress, bed shear strength and the erosion rate constant. These parameters depend on the sediment and flow characteristics, which is the subject matter of this chapter. 3.2 Sediment Characteristics 3.2.1 Introduction Factors characterizing the physico-chemical properties of the Amazon Shelf sediment including mineralogy, dispersed particle size, cation exchange capacity and organic content, are important for comparative purposes. These properties were determined for two samples obtained from the shelf bottom, and were available for this work (samples numbers 3227 and 3230). Other properties directly involved in sediment transport modeling, such as settling velocity, viscosity and shear strength, were also determined in laboratory, or derived from previous studies of Amazon shelf mud (Faas, 1985, Dade, 1992). 3.2.2 Sediment Composition and Cation Exchange Capacity The geochemistry of the sediment discharged by the Amazon River is controlled by the weathering and erosion processes in the mountainous Andes regions of Bolivia and Peru (Gibbs 1967). Mineralogical composition, organic content and the cation exchange capacity (CEC) were determined for the two mentioned bottom sediment samples, in the Department of Soil and Water Sciences of the University of Florida. Analysis by X-ray diffraction (XRD) was conducted using a computer-controlled diffractometer. Clay minerals identified from XRD patterns (Whittig and Allardice, 1986) were preponderantly consisted of kaolinite and smectite. Non-clay minerals included quartz and mica. The mineralogy found in these samples is consistent with that reported by Barreto et al. (1975), cited in Dade (1992). The percentage of organic carbon in the samples was determined using AcidDichromate Digestion, with F.S04 tritation (Walkley and Black, 1934). The low carbonate percentages found (0.81 and 0.83 % for the samples 3227 and 3230, respectively) suggests poor biological activity, probably due to turbidity and anoxia, which are known to be unfavorable for the growth of benthic organisms. Organic matter, determined by loss of ignition (D2974-87 of ASTM, 1987) was 3.0 and 3.1 % for samples 3227 and 3230, respectively. Using the same technique, Dade (1992) and Faas (Dr. Richard Faas, Lafayette College, Easton, Pennsylvania, personal communication), reported organic matter ranging from 3.3 to 10.3 %. Clay minerals have the property of sorbing certain anions and cations and retaining them in an exchangeable state. The most common exchangeable cations are Ca++, Mg,+ I, K', NH4+ and Na+. Table 2 gives the CEC ranges for kaolinite, illite and smectite (Grim, 1968). Note that kaolinite and smectite are also present in the Amazon samples. Table 2: Cation exchange capacities of three common clay minerals, in meq/100g The CEC for the Amazon samples were determined using the technique of Extractable Bases (SOIL CONSERVATION SERVICE, 1992). The CEC values found in this way are given in the Table 3. Clay CEC Kaolinite 3 5 Illite 10 40 Smectite 80- 150 Table 3: Exchangeable cations and cation exchange capacities for Amazon samples, in mec/100 _Sample Ca Mg K CEC 3227 9 12 3 24 3230 12 13 2 27 The Mg ion could have come from the marine source of the samples. If Mg is reduced in the proportion of the Mg/Na ratio, typical of marine samples, then 5 meq/100g must be subtracted, so that the CEC would reduce to 19 and 22. Since CEC is a measure of the degree of cohesion, it could be said that Amazon Shelf sediments exhibit a medium or moderate degree of cohesion. 3.2.3 Particle Size The dispersed particle size of the Amazon sediment was determined in the Coastal Engineering Laboratory of the University of Florida using the bottom withdrawal tube method (for a detailed description of this method see Vanoni, 1975), recommended for fine sediments. The resulting size distribution curves for the two samples are shown in Figure 7. The median size, d50, is 2.5 pm and 3.7 gm, for samples 3227 and 3230, respectively. Furthermore, the size is less than 101m for 80% and 70% for the respective samples. Dade (1992) reported Micrometrics Sedigraph grain size analysis of a composite sample of dispersed Amazon mud, it being 90 percent less than 10 Pm with a median grain size 1.2 gm, which indicates a slightly smaller particle size than for the samples analyzed here. Sediment Size Distribution Amazon Sample 3227 0.1 1 10 diameter (g~m) Sediment Size Distribution Amazon Sample 3230 100 0 .0 40 20 -- diameter (gim) Figure 7: Size distribution of the Amazon Shelf bottom sediment for samples 3227 and 3230. 3.2.4 Flocculation and Settling Velocity Fine particles with diameters less than 2iim (clay) are plate-like having on their surfaces ionic charges which creates forces comparable to or exceeding the gravitation force, and cause the particles to interact electrostatically (Dyer, 1986). Consequently, they do not act as separate or individual particles, but as aggregates of particles called flocs. In fact, where in situ observations have been made, only rarely has unflocculated material been found in the clay size range. Salinity modifies the surface ionic charges by sorption of cations and the formation of an electric double layer. Interparticle collisions causing flocculation result from Brownian motion, flow shear, and differential settling velocities of particles (Einstein and Krone, 1962; Dyer, 1986). At low concentrations a small amount of shearing enhances the collision potential of the particles, but higher, shear-induced, collisions will tend to disrupt flocs rather than promote their growth (Van Leussen, 1988; Kranck, 1981). Suspended aggregates often include a certain amount of organic matter. Organic polymers released by algae and bacteria are sticky and therefore significantly affect the process of aggregation. Studying in situ aggregates, Brinke (1997) found that biological processes also affect the collision efficiency, aggregate strength and density, thus causing seasonal variability of flocs properties. Due to flocculation the dispersed particle size is not useful for characterizing the settling velocity of fine-grained sediment in the marine environment, and settling velocities of flocs on the Amazon Shelf are unknown. In general, settling velocities of the flocs can be computed from measured floc size and estimated floc density data, through the Stokes' particle settling equation. Observations have shown that flocs have an irregular shape, and are often joined biologically into 'stringers' (Fennessy et al., 1994). Since the Stokes' equation was formulated for spheres falling within the viscous regime, it should be used only as a rough approximation of the actual settling behavior in the field environment. Gibbs and Konwar (1986) measured floc sizes in the shelf region seaward of the Amazon River mouth obtaining mean values of 50-100 m, and a maximum of 200gim. During AMASSEDS, photographs of the in situ suspended materials were obtained showing that these materials were dominantly present as floes, with modal size in the range of 200-500g.tm (Kineke, 1993). At the Open Shelf Transect, see Figure 6, flocs with modal sizes of 400-6001m were observed (Figure 8). 10 II I I I 10 100 1000 Dizzter Oum) Figure 8: Size distribution of floes determined by image analysis of plankton camera photographs at 35 m depth and high river discharge (from Kineke, 1993). According to Krone (1986), there occurs "an order of aggregation" in formed flocs, starting with the most compact and strong flocs (having densities of 1.16 1.27 g/cm3). Increasing order of aggregation occurs by combination of flocs of the next lower order resulting in weaker aggregates of decreasing density. Thus, floc density is inversely related to floc diameter, decreasing with increasing order of aggregation. While Krones' floc densities ranged between 1.07 to 1.27 g/cm3, Gibbs (1985) reported floc densities of suspended sediments from the Chesapeake Bay ranging from 1.004 to 1.032 g/cm3, for floc sizes of 50gm to 500gtm. Kineke (1993) reported an estimated value of 1.08 g/cm3 for floc density during AMASSEDS measurements near the Open Shelf Transect. Kranck and Milligan (1992) found floc densities in San Francisco Bay ranging from 1.04 to 1.48 g/cm3 for floc sizes of 500pxm and 501gm. In recent in situ measurements of floc size and settling velocity in the Tamar River estuary, England, Fennesy et al. (1994) obtained floc densities ranging from 1.008 g/cm3 to 1.12 g/cm3 for 600gm to 50p.m flocs. The corresponding settling velocity ranged between 0.1 to 2 mm/s. Considering the size range for the Amazon Shelf sediment to be 50 to 600gxm, with corresponding densities of 1.3 and 1.01 g/cm3, the settling velocity according to Stokes' equation would range between 0.02 to 2.8 mm/s. 3.2.5 Settling Experiments A descriptive model of the processes occurring in a settling column is shown in Figure 9 (lmai, 1981). In the initial stage flocculation occurs and no measurable settling takes place. In the second stage, the flocs gradually settle and form a layer of sediment. At first, when the sediment-water mixture is truly a suspension, i.e., at relatively low concentrations, effective stresses are absent. As additional material settles on top, the interparticle spacing decreases with the expulsion of pore water from within and between the flocs. Consolidation occurs as a result of self-weight of the soil particles. A highly compressible soil framework develops with associated effective stresses. In the last stage, all of the sediment deposit undergoes consolidation and approaches an equilibrium state. 0 t I Time t2 Figure 9: A schematic representation of flocculation, sedimentation and consolidation (from Imai, 198 1). Settling velocity is a property of the suspension, i.e., it is the relative fall velocity of the particles or flocs occurring in the settling stage, where they are only fluidsupported. By carrying out laboratory tests with mud samples from the Amazon Shelf, the settling velocity was determined in quiescent conditions and considering its concentration dependence. Further research under non-quiescent conditions should be conducted to investigate the effects of shear stress or turbulent kinetic energy on the growth and break-up of flocs. The bottom withdrawal tube method, typically used for determining the dispersed particle size, assumes that there are no temporal variations in the settling velocity. Therefore, during the experiment if flocculation occurs this method is not suitable, and the multi-depth method (McLaughlin, 1958) must be used instead. Accordingly, for a quiescent medium, the sediment mass conservation equation is reduced to c a(wC) = 0 (5) & az Integrating this equation with respect to vertical coordinate z gives a D (6) (WC),= fCdz at0 where D is a given elevation in the water column. Thus, using the vertical distribution of sediment concentration tracked through time i.e., C(z,t), W. can be calculated. For each selected value of elevation D, the integral in Equation (6) can be evaluated for different times from the measured concentration profiles. Then, the spatially and temporally discretized equation + (7) I 2=Cdz Cdz ",,+ 0 C,.o(t,+1 i ) where i+! Ci+1 +c (8) C2 -=D .-D z=D 2 yields the pair, W3C, for each time tj+1,2. W, is thus obtained, knowing C. The settling velocity tests were carried out in the Coastal Engineering Laboratory of the University of Florida. Six settling column tests were performed, with initial concentrations, C., ranging between 2.1 to 37 g/l. Conditions of each test are given in Table 4. ________Table 4: Settling column test conditions Test No. Mud Sample Suspension Height (cm) C. (g/l) 1 3230 166 -151 21 2 3230 175.3-163 37 3 3230 179-167.5 8.3 4 3227 166.8 156.8 8.5 5 3227 173- 164 2.1 6 3227 170.8 161.7 10.7 Figure 10 shows an example of the concentration profiles obtained for an experiment with the sample 3230 and C. = 21 g/l. In every test, the mud particles were aggregated, and therefore an interface appeared between the upper water layer and the top of the suspension. This interface settled with time. From the settling experiment, the settling velocity can also be estimated as the fall velocity of the water-mud interface, which is constant during the settling stage (schematized in Figure 9). These values were also plotted in Figure 11 together with the results obtained using the Equation (7). Settling Test Amazon Sample 2730 C=21 g/l 180 160 140 15mmfi 120 30min 100 E 56min 9.80 i 60 120min 40 r 180min 20 0 20 40 60 80 100 120 140 concentration (g/l) Figure 10: An example of temporal evolution of concentration profile in the settling column The data points indicate an increasing velocity (with increasing concentration) region due to flocculation effects and a decreasing velocity region (with increasing concentration) due to hindered settling. The settling velocities derived from floc sizes measured in the Amazon Shelf and estimated floc densities, 0.02 to 2.8 min/s, are in agreement with the settling experiments. As could be expected, the settling velocities determined in quiescent conditions may underestimate the field-estimated values (Wolanski et al., 1992). In any event, from the above experiments the following empirical relationships between settling velocity and concentration are obtained: W =0.05 (1.35- O.O1C)s6 mm/s = 0. 11C16 mm/s forC> 1.7 g/l forC< 1.7 g/1 which are drawn in Figure 11. These equations are similar to those proposed by Ross (1988); however, the presently adopted coefficients better reproduced the laboratory experiments than those of Ross, when included in the numerical model. Details of numerical simulations are provided in section 5.4. 101 E +N 1001 =l E o o- + 0s + P = 1-2 '=2 -a a-+ + xC 21 g/I C =37 g/I .9 + 104+ 10-1 100 101 102 concentration (g/l) Figure 11: Settling velocity versus concentration obtained from settling experiments through Equation (7) (small symbols) and from free settling stage (larger black dots). 3.2.6 Viscosity A single viscosity characterizes Newtonian fluids, e.g., water. This value is obtained from the slope of a linear plot of shear stress vs. shear rate. Fine sediment suspensions at high concentrations behave differently from pure water. Clay particles are (10) mostly flat in shape and, if the concentration is high, build a net-like structure. This structure causes the suspension to exhibit non-Newtonian characteristics (Mignot, 1968, Wang and Wang, 1994). Thus, the viscosity is not constant, but is a function of the shear rate, shear stress and time. Thus, the term 'apparent' viscosity is used to refer to this property. Faas (1985, 1986) analyzed dense suspensions of Amazon sediment taken from tops of box cores. Figure 12 shows the flow diagram for one of the analyzed samples, at several densities. With the log of shear rate on the horizontal axis and the log of shear stress on the vertical axis, the diagram shows how the flow behavior changes between successive increments in shear rate. Low density mud (<1200 kg/m3) exhibits a shearthinning (pseudoplastic) behavior. At densities > 1200 kg/m3, the behavior changes from shear-thinning to shear-thickening (dilatant). This behavior can be also observed in Figure 13, which is a plot of the apparent viscosity vs. shear rate, showing the complex behavior of shear-thinning at low densities and shear-thickening, in certain ranges of shear rate, for denser suspensions. This change enhances the resistance of mud to resuspension of dense suspensions with increasing shear stress and, as was pointed out by Faas (1986), this effect in turn can control sediment resuspension, reducing or preventing further resuspension. It is interesting to note that in the AMASSEDS profiles, the maximum measured concentration was 321 g/l, which is just below the limit found by Faas above which the mud behaved as shear-thickening or dilatant. ILa ((1.0 5) gc .1 w x (1.052) U) I 0O1 10 100 SHEAR RATE (sec1) Figure 12: Flow behavior of Amazon sediment suspension at different densities (from Faas, 1986). lot 6 C 10 *(t .053)I S. ..................... ..................................... 10 50 100 SHEAR RATE (see1 Figure 13: Apparent viscosity vs. shear rate for a core sample from the Amazon Shelf, as a function of density (from Faas, 1985). In order to obtain analytical equations to fit the data in the range of concentrations found in movement at the Amazon Shelf (< 1200 kg/m3), the data set obtained by Faas were fitted to the power-law model (Sisko, 1958) vf(C) = Cexp(-0.78 & 10.24) for 3.9sec'(11) az vf(C) = Cexp(-0.017 & 12.95) for 3.9sec-1 az 3.2.7 Shear Strength Following Migniot (1968), Otsubo et al. (1986) and Dade (1992), among others, it is considered here that the yield stress represents a measure of the interparticle bond strength per unit area. Thus, the bed shear strength, 'r, can be determined from its correlation with the yield stress measured from Amazon sediment samples. The upper Bingham yield stress, ry,, is defined from the stress-versus-shear rate flow curve by extrapolation from the low values of shear rate, as indicated in the Figure 14. Migniot (1968) and Otsubo et al. (1986) performed extensive experiments to relate shear strength to the yield stress for sediments of different mineral compositions and water contents. Migniot suggested the following relations: Figure 14: Schematic of shear stress versus shear rate flow curve, and definition of the upper Bingham yield stress. For consolidated mud , = 0.25cyPa Ty >1.6 Pa (12) and for weakly cohesive mud c, = 0.1u, Pa -c, <1.6 Pa (13) In the experiments of Otsubo et al. (1986), the tested materials were grouped in two categories, depending on the type of exchangeable cation. The Amazon samples fell within the first group corresponding to natural clay mixtures. The functional relationships obtained by Otsubo between shear strength uc and yield stress -ry for the first group is -uI = 0.270.6Pa (14) 'T,2 = 0.79X'94Pa (15) depending on the threshold states in mud transport considered; these being, according to Otsubo, r,l for incipient mud particle movement and t82 for the onset of bed failure. Faas (1985) made yield stress measurements on superficial sediments of the Amazon Shelf bottom obtained from box cores during a June-July 1983 cruise (during a high discharge period in the river). Similarly, Dade (1992) studied two samples collected by a pump 20cm above the limit of downward instrument penetration in the bed. Using both sets of data, Ty as a function of sediment concentration, is presented in Figure 15. X- X 00 101 0 0"'e 0 o 0 0 00pd 0 i i i I i 2 Concentration (g/l) Figure 15: Yield shear stress versus concentration for Amazon Shelf mud (from Dade, 1992, Table 6.7 and Faas, 1985, Figure 3). The best fit line for the combined data is "ty= 2.02x10-6 C2.6233 (16) Thus Equations (12) through (13) and Equation (15), provide relationships between shear strength and concentration, which are plotted in Figure 16. The choice of the appropriate relationship for modeling purposes will be examined in connection with the current induced shear stress and associated erosion flux (see section 3.3.5). 101 10P Migniot or, .. Otsubo (1) Otsubo (2) 102 Concentration (g/I) Figure 16: Shear strength versus concentration for the Amazon Shelf bed sediment (1) for incipient motion and (2) for bed destruction. 3.2.8 Permeability and Effective Stress When fine-grained particles deposit on the bottom they form an open network structure. Under the weight of the accumulating particles above, this network slowly collapses during which pore water is expelled. Consolidation models solve the mass balance equation for the solid particles, and for that purpose additional information related to momentum exchange between the fluid and the solid phase is necessary (Toorman and Huysentryt, 1997). Thus, two parameters are introduced, namely the permeability, k, and the effective stress, a', for which empirical relationships must be found for each particular case. Through Darcy's law, permeability relates seepage velocity to the excess pressure head in the soil element, provided the interstitial flow remains laminar. The effective normal stress, a', is defined as a', = y P (17) where at is the total stress and w. is the pore water pressure. It represents the part of the total stress supported by grains. Effective stress is usually associated with sediment concentration (or void ratio in soil mechanics) but creep (i.e., time dependent change in concentration without any change in effective stress) can influence the results (Sills, 1997). Consolidation tests with sediment from the Amazon shelf have been carried out (Dr. Robert Kayen, U.S. Geological Survey, California, personal communication). However, in those tests, performed using the Constant Rate of Strain method (CRS), the initial void ratio was always below 5. The void ratio, e, is related to sediment concentration, C, through the granular density of the sediment, p., according to __=__ (18) e+I Thus, for a void ratio of 5 a sediment concentration of 441 g/l is obtained. Since it is of interest to analyze the set of data from at the Amazon Shelf where the maximum recorded sediment concentrations reached 321 g/l, the experiments of Kayen do not provide useful information for the purpose of this work. Effective stress is usually obtained (in settling columns through self-weight consolidation tests) as the difference between the total normal stress, computed from measured density profiles, and the pore pressure, measured with capillary tubes or pore pressure transducers (Berlamont et al., 1992, Been and Sills, 1981). Since the accuracy of the effective stress measurements depends on the accuracy of the measured pore pressures, in the beginning of consolidation, when the effective stress tends to be small, the error in the effective stress measurement can reach 100%. Figure 17 is an example of experimental data obtained by Toorman and Huysentruyt (1997) showing the spread of results at low densities. 2000. ,800 'r :",,' 1800- fr I 1600. . IL 1200 -O 01000.~ 800. I.-v 2600 400- / Pr 200 ---0 1 1.05 1.1 1.15 1.2 1.2.5 1.3 1.35 1.4 1.45 1.5 DENSITY (01000 kg/rn3) Figure 17: Effective stress as a function of local density (from Toorman and Huysentruyt, 1997). From the literature, we noted that Alexis et al. (1992) gathered data on the effective stress and the permeability as functions of the void ratio, which are reproduced in Figure 18 and Figure 19, respectively. Both figures will be used to obtain approximate values of permeability and effective stress for the Amazon mud. 41 20 15 .. .* . O ** 0 .. * o :.: S .~..'..' 5 . literature (Alexis et al., 1992). 52 50-* 48 46. 44 42 40 38 36 ) 34" 32 30 - 28- , 24 9_22- ,= = 0 20 46 14 12 . 14 10 ....*..:._:,':* . 40 83 -- il 2 o76 ..n ,, .., ... ,,,. .* :.. . 10-' 10" 10- 10- 10 10- 10 10" 10" PERMEABILITY (cm/s) Figure 19: Permeability versus void ratio data taken from experiments reported in the literature (Alexis et al., 1992). 3.3 Vertical Flow Structure 3.3. 1 General Having in mind the development of a vertical transport model for fine-grained sediment applicable to the Amazon Shelf, it was necessary to examine the structure of the tidal boundary layer at the study site. High concentrations of fine suspended sediment induce two particular structural features that affect the flow, both with stabilizing effects: density gradients and increased fluid viscosity. The stability of flows in which both viscosity and buoyancy are important is not a well-understood matter; hence each issue needs to be addressed separately. Clay suspensions at very high concentrations can possess a yield stress and a high viscosity, and may exhibit a laminar behavior even at higher velocities, which is quite different from similar hydrodynamic conditions in clear water, or low concentration suspensions (Wang and Plate, 1996). Comparisons with non-cohesive suspensions have indicated that a change in viscosity plays an important role in governing the boundary layer structure. In laboratory experiments with a suspension of clay in seawater, Gust (1976) reported noticeable thickening of the viscous sublayer, and a reduction of the friction velocity. The laminar character of the near-bed flow in high concentration environments has been recognized by many investigators (Ross and Mehta, 1988, Kineke and Stenrberg, 1995). On the other hand, in turbulent flows, wherever there is a density gradient, turbulence is required to do more work to entrain the denser fluid upwards. Thus, the presence of the density gradient tends to damp turbulence and reduce the boundary shear stress (Sheng and Villaret, 1989). Recorded velocity profiles in the Amazon Shelf region are examined here, and the above two effects are evaluated. Current-induced bed shear stress and shear velocity as well as diffusivity through the water column are calculated for that purpose. 3.3.2 Tidal Boundary Layer Approach The tidal boundary layer is characteristically considered to be essentially horizontal (w w 0), and horizontally uniform [u=u(z,t)]. For such an oscillatory boundary 8uu0 2rA 0u layer, the relevant advective acceleration term can be written as u. & &x L 0 where u* is the flow velocity outside the boundary layer, A is the water particle semiexcursion length and L is the tidal wavelength. This term can be neglected depending on the ratio 27tA/L. Thus, for example, for tidal movement in 15m depth, this value is of order of 10-1. Under the assumption of a hydrostatic pressure distribution, and considering that the shear stresses vanish outside the boundary layer, the equation of motion, valid for the boundary layer, becomes P-a ( U) (19) &t az where u(zt) is the velocity at elevation z above the bed level, u. is the velocity outside the boundary layer and c is the shear stress at level z. Equation (19) can also be written as -U.) C (20) where v is a characteristic viscosity. If the flow regime is laminar, v is the fluid kinematic viscosity, an exclusive fluid property. However, if the flow is turbulent, v represents the eddy viscosity, which is dependent on the flow characteristics. In both cases it represents a momentum diffusion coefficient. Nielsen (1992) subdivided horizontally uniform models for oscillatory boundary layers in two categories: quasi-steady models, which assume that the velocity distribution is at all times logarithmic, and unsteady models, based on the above boundary layer approach. However, as it will be seen next, quasi-steady models can be considered as a sub-class of unsteady models, by solving the non-steady oscillatory boundary layer problem with a suitable viscosity function. Expressing the tidal velocity profile as u(z, t) = u(z)eit, where u is the tidal frequency, Equation (20) becomes (v(z)- = ic(u-u.) (21) A usual functional form found in the literature for the flow viscosity in turbulent boundary layers is the linear dependence with depth, i.e., v(z) = K u. z, where X, is the von Karman constant. Substitution of this relation in (21) yields a homogeneous, Besseltype differential equation of zero order, known as the Kelvin's differential equation: d2u ldU (22) ds+2%U= 0 s ds where X =2 1uis imaginary and s = With the boundary conditions u = 0 at z = 0 and u = u. at z -+ oo, the general solution of (22) has the form u = u1 +aker(2jI (23) where -1 (24) ker(2 and where ker is one of the Kelvin functions. For small arguments of ker, this velocity profile approaches the logarithmic form (Abramowitz and Stegun, 1972) as would be expected for a turbulent boundary layer. 3.3.3 Fluid Kinematic and Eddy Viscosities To evaluate the form of the viscosity function suitable for flow over the Amazon Shelf, it is necessary to emphasize the difference between the fluid kinematic viscosity and the eddy viscosity, especially in the case of a flow which contains a very high concentration of suspended sediment, and flow regime which is turbulent. Thus, the momentum diffusion coefficient, v, combines a fluid, whose viscous properties change gradually, with a flow regime which tends to vary from laminar to turbulent over the water column. For the present case, v was determined from the boundary layer approach, Equation (20), using measured velocity profiles, as follows: (25) - p (u u.)dz (Z p at v(z, t) =' az Median values of v over the tidal cycle, obtained according to the Equation (25) in the water column for each station, are plotted in Figure 20. Also, median values of the fluid kinematic viscosity, vf, obtained from concentration profiles and Equation (11), are plotted in Figure 20 as functions of depth. 10-7 10-6 10-5 104 10o-3 Kinematic Viscosity (m2Is) Figure 20: v (circles) and vf (asterisks) median values for the tidal cycle, (25) and (11), respectively. from Equation It is interesting to note that both sets of data converge around the same range of values in the lower part of the profile. In that zone, about lm above the bottom, high mud concentrations were observed and, accordingly, a change in the behavioral trend of momentum diffusion is noticeable in Figure 20. 4 m xxK *K WeXe NBIE~~ ))K)U U W 3 xx Aa QO D 0 UK NUK We W xli 3- -m a m agggD@ 2 mm as all @ N ma MW @ YR I@ W ~-m e liD 0 @eN g@ga 0@ @@0(n Ollame ., ,L .. .. ..i . . .. . .... ,,surn .....is . .... A noteworthy feature of the velocity profiles near the bottom of the Amazon Shelf is the concavity (a2u > 0), a feature uncommon to estuaries in which logarithmic az2 profiles are observed. Figure 21 shows details of the velocity profiles, enhancing the portion near the bottom. 0S2 (high fiver flow- as3419) 0.5 1 1.5 2 Velocity (m/s) Figure 21: Velocity profiles recorded at anchor station OS2 (3419). Near-bed layer velocity and concentration profiles measured in the Avon River (U.K.), a macrotidal and high concentration environment, also showed a similar trend in the velocity profile. Ross and Mehta (1988) numerically modeled the near-bottom horizontal velocity of the Avon River as an unsteady Couette-type flow driven predominantly by the shear stress imposed at the level of the lutocline. The result confirmed the observed concavity. They explained this trend to Raleigh flow effect arising from momentum diffusion into the fluid mud layer due to shear flow in the water column above the lutocline. Considering a viscosity which decreases linearly with elevation (z) above the bottom, v(z) = c c2z, where cl and c2 are constants, Equation (21) can be rewritten as d( duz I~i~~w (26) Through a change of variable, s = cl c2z, a differential equation is obtained with the parameter X = 2 -__. The solution, compatible with the boundary conditions u-0 at C2 z=O and u = uo at z -4 co, is -( + b b {2 a C1 2Z) (27) where -1 (28) be{ 2jC2 and where bei is one of the Kelvin functions. The resulting velocity profile is shown in Figure 22, and is compared with the result obtained from Equation (23). Note that the top of the figure at 8m does not represent the free surface, but is well below it. This simple analytic result implies that the adopted decreasing viscosity model generates concave velocity profiles, a characteristic found in the measured profiles, as was mentioned earlier. 8 7 T=12 hs z um= 1 m/s (M) 6 k=0.4 5 2 3 / 2/ / ,(z)-uC z 0 0.2 0.4 0.6 0.8 1 velocity (m/s) Figure 22: Boundary layer velocity profiles for increasing and decreasing viscosity with depth. Wang and Plate (1996) studied the turbulent structure of clay suspensions as a non-Newtonian fluid following the Bingham equation. They defined a characteristic Reynolds number as R 4puh (29) {+ ;nAJ where p is the fluid density, 'n is the rigidity coefficient defined from the shear stress versus shear rate curve (see Figure 14), rY is the yield stress, h is the flow depth and u is the average velocity in the water column. Following this definition, they found that turbulence developed in the entire channel only for Reynolds numbers above 10,000. Figure 23 shows the median values of the characteristic Reynolds number calculated over the tidal cycle, and as functions of depth, for the 18 anchor stations. 0 0 00 0 00 0 0 OO M O (X 0 a Som on 00 0 OOD 00 000000 00 a0 00000 0 0 0 CD0 0O 0 0 0 0000 0O a 0c 0 0 a 0 00 000 00 0 0 0 @=a 0 a 0D 000 000 0 0 0 00000 0 cc00 0 00 0 0 0 0 1O 104 106 108 ReynodIs Number Figure 23: Tidal Reynolds number [Equation (29)] vs. depth. It should be noted that the threshold of Rl=10,000 for the turbulent regime was established for laboratory conditions. The Amazon Shelf data, however, do provide evidence of values for &. below 10,000 within a well-defined, Im thick, layer near the bottom. This evidence, as well as the concave form of the velocity profile, supports the suggestion of the presence of a thick layer of viscous flow in the near-bed zone due to the enhanced viscosity of the sediment suspension. I I I 3.3.4 A Model for Laminar Boundary Layer Assuming the local validity of Equation (19) with uniform viscosity, an analytical solution for the velocity profile has been given by Nielsen (1985) as u(z, t) = Aae'0 [l exp(-[1 + i]__2- (30) from which the bottom shear stress is obtained as (O, t) = Aaei'a (1 + i) (31) and the corresponding shear velocity as U Aae t(1+i)v -1/2 (32) L 4f2v / a Here, Equation (30) is applied to tidal motion, and the kinematic viscosity depends on the local concentration according to Equation (11), a is the semi-diurnal tidal frequency, and Ao = u., is the velocity at the outer edge of the boundary layer. Following Geyer (1995), who considered the boundary layer in the Amazon Shelf region to be confined between 3 to 5 m for neap and spring tides respectively, the top of the boundary layer was considered here at z = 4 m for all measured profiles. Figure 24 compares Equation (3) with measured hourly profiles along two tidal cycles for the anchor station OS1 (4434). A measure of the agreement between the observed and the calculated velocity profiles is given in Figure 25, where the absolute error was computed according to er(z) = 1z(z)-u(z)3 (33) Here, ji is the calculated velocity according to Equation (30), and u is the observed velocity. For comparison purposes, the absolute error is also calculated for the logarithmic fit, i.e., u = U, In z(34) K zo where u- was calculated from the velocity profiles as described in section 3.3.5. The values of the roughness height, z., was obtained by matching the measured velocity at the top of the boundary layer, considered to be at z=4m. In both cases the errors are tidally averaged. OSI (low river flow) Ci a( 3. 0 0 o O o 0 , o 00 2 00 0 00 2.o 0 0 0 0 a 0 0 Do 0 1 0 .25a0 -1. 0 -1. 0 -1. 0 -1. 0 -1. 0 -1. 0 1.0 1.01. 0 1. 0 1. 1. Velocity (m/s) ,.0 0 0 0 Z2 0 0 0 1 0 00 0 0 o .2l -1. 0 -1. 0 -1. 0 -1. 0 -1. 0 -1. 0 1. 0 1. 0 1. 0 1.0 1.0 1. Velocity (m/s) flood -+ ebb Figure 24: Observed (discrete points) and calculated [Equation(30)] velocity profiles over two tidal cycles recorded at anchor station OS 1 (low river flow-as4434). OSI gow rivertflow -as4434) 3.5-M 2.5 U ~~W 0 0.5 0 0 0.2 0.4 0.6 0.8 1 absolut error (m/s) Figure 25: Absolute velocity errors arising from differences between measurements and Equation (30) (asterisks) and measurements and logarithmic fit (circles). For the logarithmic profile, a noticeable increase in error can be observed below 1m level. It should be noted, however, that the model was over-extended in the upper layer where turbulence was present merely to simplify the imposed boundary condition, namely the free stream velocity. Since the objective of these calculations was to estimate bed shear stress and the shear velocity from the velocity profile, the relevant results are those obtained in the neighborhood of the bottom. The high concentration near the bottom found in the Amazon Shelf, and the associated enhanced fluid viscosity, inhibit turbulence development in the near-bed layer. This effect markedly affects the boundary layer structure, including velocity profiles in a region of extreme importance for sediment transport evaluation. As was mentioned in the introductory chapter, most of the total sediment transport in the water column was concentrated in the first meters from the bottom, where the concentrations were around 10 g/l or higher (Kineke, 1993). Above this elevation there was an evident deviation from the laminar model, and, therefore, a turbulent profile must be adopted for the upper column (from about lm to the total height of the boundary layer). 3.3.5 Shear Stress from Measured Velocity Profiles Integrating Equation (19) between z and oc, and recalling that -T. = 0, the shear stress can be calculated from tE(z) = p a(u- ,)dz(35) z a and the shear velocity, u., can be then expressed as (36) u. (0) = !p2 .a(u-u.)dz In this formulation, no assumption regarding the momentum diffusion coefficient is made. However, it should be noted that oo refers to the top of the boundary layer and this level, or the thickness of the corresponding boundary layer, 8, must be determined carefully. A crude estimate of 8 was obtained from the measured velocity profiles by determining the height at which the first local maximum in flow speed occurred, starting from the bottom. Table 5 gives the results of the bottom shear stress calculated from Equation (35) (setting z=0), and referred to as 'observed', while the bottom shear stress using Equation (31), is referred to as 'calculated'. Also included are observed, Equation (36), and calculated, Equation(32), shear velocities. Observed values of shear stress and shear velocity were obtained entirely from the measurements, so that the effects of viscosity, stratification etc. on the momentum diffusion coefficient are already included. The boundary layer thickness, used in the calculations of the observed shear stress and shear velocity at each anchor station, is also included in Table 5. In Equation (3 1), the free stream flow velocity was taken from the measured velocity at 4m level, and the near-bed concentration was the lowermost measured value. Even though the shear stress calculated from data and that obtained using the analytical expression appear to be in reasonable agreement in some cases, in general there is an under-estimation of the shear stress, and, also, a discrepancy in phase. Note that Equation (3 1) predicts that shear stress should lead velocity by about 45 degrees. While the observed shear stress lags velocity by about the same value for anchor station OS1, the lag is greater for anchor station CN, as is observed in Figure 26. Figure 27 shows comparisons between calculated and observed shear stresses, after adjusting for the phase lag. The discrepancies between predicted and observed shear stresses were found to be enhanced when the measured near-bed velocity flow was far from zero (stations 1154, 4438, 4441 or 2420), or when the assumed thickness of the boundary layer was greater than 4m (stations 2418, 2405 and 3442). It should be remarked, however, that there are many underlying uncertainties in the calculations [Equations (31), (32), (35), and (36)]. It should be noted, for example, that measurement technique used were not filtered for turbulence (for details of the measurement technique see section 2.6), the boundary layer thickness estimation is crude, viscosity-concentration relationship was taken from laboratory experiments, and the concentration at the bed level was taken as the lowermost measured concentration, which may not be truly representative of the near- bed zone. In fact, if the bed concentration were fixed at 320g/l, the maximum concentration at which mobile fluid mud occurs, the observed and calculated values of u. approach each other adequately, as can be seen from a comparison of shear velocities listed in Table 5, in the 7h and the last columns, respectively. This consideration would thus reduce the difference between the observed and the calculated shear stresses. Accordingly, the proposed model given by Equations (30) and (31) can be considered appropriate for describing the lower region of the Amazon velocity profiles. OSI (4434) 0 3~ 2 -. -..- --- -o 1 ..... ... a. 0 0 5 10 15 20 25 30 time (h) CN (2428) __3 2 ---------- a. 0~~ ---- -0510 15 20 25 time (h) Figure 26: Observed velocity (dotted line) and shear stress (solid line) over two tidal cycles for anchor stations OS 1 and CN. 0S1 (4434) n z; 0 10 15 time (h) CN (2428) 15 time (h) 20 25 30 Figure 27: Observed shear stress [dotted line Equation (35)] and calculated shear stress [solid line Equation (31)] over two tidal cycles for anchor stations OS 1 and CN. 0. 0.1 0 i-0.1 -0.2 -0.3 -n A/ I I iI Table 5: Tidal mean and maximum shear stresses and shear velocities at the anchor stations Stn. 6 Mean Max. Mean Max. Mean Mean Mean calc. u. obs. -r, obs. r. calc.'tr obs. u. obs. u. calc. u, Cb=320 g/l (m) (Pa) (Pa) (Pa) (mIs) (mIs) (mIs) (mIs) 1154 6.25 0.20 0.45 0.001 0.021 0.014 0.001 0.013 4434 2.50 0.21 0.42 0.15 0.020 0.014 0.016 0.017 4438 2.50 0.17 0.33 0.01 0.018 0.013 0.002 0.014 4441 2.50 0.12 0.22 0.002 0.015 0.011 0.001 0.011 4413 3.50 0.18 0.38 0.09 0.019 0.013 0.015 0.015 2420 2.25 0.09 0.19 0.01 0.014 0.010 0.003 0.014 2418 4.50 0.43 0.67 0.08 0.026 0.021 0.010 0.017 2415 2.50 0.14 0.27 0.06 0.016 0.012 0.010 0.012 2428 3.75 0.25 0.52 0.10 0.022 0.016 0.015 0.017 2444 2.75 0.12 0.22 0.06 0.014 0.011 0.012 0.015 2445 3.75 0.18 0.30 0.06 0.017 0.013 0.013 0.014 2405 4.50 0.63 1.07 0.07 0.032 0.025 0.010 0.019 3420 2.50 0.22 0.43 0.08 0.020 0.015 0.010 0.018 3419 3.75 0.39 0.87 0.17 0.028 0.019 0.018 0.019 3418 2.50 0.11 0.21 0.06 0.014 0.010 0.011 0.012 3442 5.00 0.52 1.04 0.07 0.032 0.023 0.010 0.020 3455 2.50 0.18 0.33 0.06 0.018 0.013 0.010 0.018 3405 2.50 0.25 0.62 0.08 0.025 0.016 0.011 0.019 3.3.6 Mass Diffusivity In the lower part of the suspension, where the flow behavior is viscous, molecular mass diffusivity is negligible, considering that molecular diffusion, usually small, decreases with increasing fluid viscosity. In this layer, however, shear caused by wave action can enhance mixing. Under quiescent conditions, or during tidal slack water, an effective stress developed by virtue of contact between particles can be of greater importance in preventing complete deposition than molecular diffusion. In contrast with the lower zone, in the upper suspension mass diffusivity is directly related to the momentum diffusivity, which can be large. There is a general agreement among researchers that the simple eddy viscosity model, following the parabolic distribution, is adequate for homogeneous fluids v(Z)= ru.z(1-hZ (37) which applies above the viscous layer. Figure 28 shows an example of the momentum diffiusivity taken from the velocity profile according to Equation (25), and calculated with Equation (37), above the 1 m level. Stable gradients of density caused by suspended sediment concentrations, salinity or temperature can affect flow dynamics, because work must be done on the fluid to mix it, and to raise its potential energy. Many researchers have considered turbulence damping due to density gradients caused by sediments, salt or temperature (Soulsby and Wainwright, 1987; Geyer, 1995; Trowbridge and Kineke, 1993) by modifying the universal logarithmic velocity law (Adams and Weatherly, 1981, Green and McCave, 1995; Glenn and Grant, 1987, Wolanski and Brush, 1975). The relative magnitudes of the stabilizing density forces and the destabilizing shear-induced turbulence can be measured by the gradient Richardson number: CN (2428) ~10 diffusivity (rrils) Figure 28: Depth variation of the tidal eddy viscosity at anchor station CN (2428) (dots) and eddy diffusivity calculated according to Equation (37). g 19P (38) where g is the acceleration due to gravity, p is the mixture density and u is the local horizontal flow velocity. High R, indicates high stratification, while low values indicate well-mixed conditions. Figure 29 shows an example of the tidal median value of the gradient Richardson number calculated according to Equation (38). Density was considered to be affected by salinity and temperature, and also by suspended sediment concentration. The adopted equation of state is given by the Eckart formula: A1 = 5890 + 38 T- 0.375 T2 +3S A2 = 1779.5 +11.25 T 0.0745 T2 (3.8 + 0.01 T) S (39) p(S, T) = 1000 1+AL A2 + 0.698 A1 where S is salinity in practical salinity units (psu), and T is temperature in degrees Celsius. Sediment concentration, C, effect on density is calculated according to p(S,T,C) = C [P'- p(ST)] + p(S,T) " (40) OS1 (4434) 8 E 0 lo-3 1o-2 10-1 100 101 10 103 Richardson number Figure 29: Tidal median values of the gradient Richardon number considering density affected by salinity and temperature (solid line), and salinity, temperature and sediment concentration (dotted line). It can be seen that the effect of suspended sediment is more important near the bed, while salinity stratification affects the upper suspension. A similar correlation is found for all the profiles in which high concentrations of sediment were measured. As stated before, it is not clear if laminar flow in the lower layer, where high concentration sediment exists, was due to the increased fluid viscosity, or if it was due to the density step. In any case, it can be considered that turbulence develops mainly in the upper layer, where the effect of salinity-induced stratification also plays an important role. Cacchione et al. (1995) adjusted the logarithmic velocity profile to measured profiles obtained from the GEOPROBE deployment at 63m depth. The results, after corrections due to accelerating and decelerating flows, showed an average shear velocity of 0.0 17 m/s, with a mean value of z,, =2.7 mm. The effect of tidally induced acceleration and deceleration resulted in changes in shear velocity of 10%. In Cacchione's data the stratification effect was comparatively less important. The considered velocity profiles were measured 15 cm from the bottom, where suspended sediment concentration did not exceed 2 g/l. Mass diffusivity is related to the momentum diffusivity by the turbulent Schmidt number Sc,h = v (41) e While Schmidt number for sediment mass diffusion varies from 0.5 to 1 for low concentration, homogeneous fluids, in stratified conditions, Teisson et al. (1991) found variations in the turbulent Schmidt number ranging from 0.7 to 10. Costa and Mehta (1990) reported values ranging from 0.94 to 2.4, from measurements in Hangzhou Bay, China, while Yamada (1975) reported values of 0.8 to 2. The mass diffusion coefficient in stratified fluids can be written as (Teisson, 1992) V (42) ac where accounts for stratification effects over the water column. A simple relationship for j) based on the gradient Richardson number is the well known function proposed by Munk and Anderson (1948) S= (I + aRi'f (43) with a=3.33 and b=-1.5, for mass diffusion. The gradient Richardson number variation over depth was calculated for anchor stations CN (2428) and 0S2 (2418), and the median values over the tidal cycle are plotted in Figure 30. Both effects, i.e. due to salinity and sediment stratification, are again evident (see also Figure 29). The stratification damping function 4 was calculated for both anchor stations according to Equation (43), and the median values over the tidal cycle are shown in Figure 3 1. In order to have an insight into the time variation of the stratification damping function over the tidal cycle, the damping function is plotted in Figure 32 for the high and low-water stages as a function of depth for anchor station CN (2428). Figure 33 shows the time series of depth at this site. An interesting observation is that the damping function follows the up- and down-water movement, increasing somewhat its effect during slack water. 64 CN (2428) 20 2CC wilt t1 Ii 5 1 0 ....... -L .... ..... J ........ ..... ..... .. . .......... .. 0 1 1 9 t I 10-3 10-2 10-1 100 101 102 Richardson number 20S2 (2418) I i fi 20 M . !t Ad I I ~ I I I'j 'iii if::' ,J ,I I, ; ,! 5s .: .......... ...... .. .. ............ ...... ," a-10 7 10 t1 i | I, o .... 1,31-210-1 100 101 Richardson number Figure 30: Gradient Richardson number as a function of depth for anchor stations CN (2428) and 0S2(24 18). 65 CN (2428) 20 i l i I I H I I I I ...il . 15 .- 5 t .... t 20 i Ii :11111 ~ I aiii C~Hil 1 5 ... .... f .... i l .. ... .. .1 -4 ........... ..... . .*. . 10-3 10-2 10-1 ~ 100 101 stratification function 20 _____ S2 (2418) I li a 15 ...Iiii i f iI ilii o 0 0a o I 11 0 o 0 1 1i0 i 1a t iiici ii Fiue31ia median vaue of th strtifcaiondamin ft...,..ion.[Equation(41) versu dp." t fr a r It C ( i 11.1 5 ................. ... . ... .I i l ii:" i ii. i i +- 0i S ,I l l I l] t | I i I i i i i lll l i !! i I l i li I (y-2 10 i 1 1 II~lI I t" a. F 3 1 T median values ll o r n ri [a ( 1)] 00 103 02 01 0 101 20! I I I I t!1 i 0 60 1 I Il !i IIi I t1111 i0i'I 1i-2 t0-1 10 strtiicaio function t Fiue31 ialmda vaue of th trtfiaiondmigfntoEuto4) vesu depth....................... fo anchor.... stations CN............ (2 28 and...... 0S (2418).+"" '"""- 10-2 100 stratification function 16 h stratification function 21 h 0.0 0 Ej VC - 00 ~ i 1 . 0 0 0 0 5 ............................... .-. .0 ....... .. .. 5 ........................ +_ ............+ 0 104 10-2 100 104 10-2 100 stratification function stratification function Figure 32: Stratification damping function versus depth over the tidal cycle for anchor station CN (2428). 01 10-4 0 510 15 bwe (h) Figure 33: Time-variation of water depth at anchor station CN (2428). 3.4 Bottom Boundary Condition 3.4.1 Bed Level Definition For modeling purposes, the bed level is considered following one of the earliest bed definitions (Krone, 1972) as the level where the current is not appreciable within the time-scale of interest. Naturally, this is a dynamic definition that will depend on the flow condition, the sediment characteristics and the time-scale of analysis, as it is locally determined by the time-varying balance between the applied fluid shear stress and the shear strength of the bed. Also, this level should preferably be time-invariant in the context of the time scale of analysis. Another bed definition is based on the transition from where the sediment is fluidsupported to sediment supported by its own continuous framework, with a transition zone which is partly fluid-supported, and partly framework-supported (Sills and Elder, 1986). Following this criterion, the bed level is taken where particles begin to develop an effective normal stress, a', defined by Equation (17). Fluid mud at sufficiently high sediment concentrations can develop a measurable effective stress, which is also a mechanism that prevents it from depositing. In the measurements of Toorman and Huysentruyt (1997), appreciable effective stresses were found for sediment concentrations as low as 80 g/l. Been and Sills (1981) measured pore pressures in laboratory consolidation experiments and concluded that there is no unique correlation between sediment concentration and the development a of non-zero effective stress. However, effective stress was always apparent in concentrations above approximately 220 g/l. In the Amazon data, mobile fluid mud was detected with concentrations up to 320 g/l. Thus, a bed definition based on classical soil mechanical definition of effective stress, as proposed by Been and Sills, may not be in agreement with the concepts mentioned here. Interesting discussions regarding the bed level definition can be found in Parker (1989) and Mehta et al. (1989). Finally, it should be noted that the bed definition adopted during the AMASSEDS survey followed a criterion dependent on the depth reached by the profiler (Kineke and Sternberg, 1995). 3.4.2 Erosion Mechanisms and Erosion Rate Two distinct mechanisms for the entrainment of the near-bed sediment by the current can be recognized (Mehta et al., 1989). If the mud develops a measurable shear strength, erosion of the mud layer occurs. For a newly placed fluid mud subjected to shear flow, the interface between mud and water, with no measurable shear strength, can be warped by flow and entrained by mixing into the upper layer. Erosion of the bed can occur particle-by-particle, as mass erosion, or by liquefaction. Particle-by-particle and mass erosion occur when failure occurs where the shear strength is exceeded by the fluid stress; in the former case the characteristic shear strength is dependent on the interparticle cohesive bonds, and in the latter case the shear strength is a bulk property of the soil defined by Coulomb's law for failure. As noted, particle-by-particle, or surface erosion, is an individual floc breakup process that begins when the flow stress exceeds the critical shear strength of the bed surface. Mass erosion occurs by bed failure when the applied stress exceeds the bulk strength of the material, and the failed material is instantaneously suspended up to the depth where failure occurs, i.e. where the bed shear strength equals the applied shear stress (Mehta et al., 1989, Wright and Krone, 1989). Surface erosion occurs at low to moderate values of the excess shear stress, c. "c,, where ", is the applied shear stress and t, is the shear strength, as discussed in section_3.2.7. c, is also referred to as the critical shear stress for erosion, in analogy with cohesionless sediment transport. The erosion rate, E, i.e., the mass of sediment eroded per unit bed area per unit time, depends on the excess shear stress as well as on the erosion shear strength. For uniform, consolidated beds the erosion function suggested by Kandiah (1974) is often used: M -(44) where M is a rate constant. A relationship between M and x, can be found in Lee and Mehta (1994). According to Equation (44), for a given applied stress, %, the erosion rate remains constant because the shear strength is uniform within the bed. Mud layers formed by deposition, however, show a gradual decrease of the erosion rate with time due to an increasing strength with depth, caused by consolidation. Parchure and Mehta (1985) proposed an erosion rate function of the form I(E)= a[It. ;(z) (45) where a (Pa7O) and y (=0.5) are sediment dependent constants, Ef is the floc erosion rate (kg/m2s) and % (z) is the shear strength profile (Pa). When the excess shear stress becomes large mass erosion prevails. In such a case, erosion can be approximately described by an expression of the form of Equation (44), but with a constant M much greater than for surface erosion (Mehta et al., 1989). Entrainment is a basic concept in free turbulent flows and represents mixing of bottom fluid with turbulent flow. Fluid mud entrainment following slack water has been compared qualitatively with the entrainment of a stratified layer of salt water underneath flowing fresh water (Mebta and Srinivas, 1993). When entrainment occurs from the top of a quiescent lower layer, the upper layer deepens and resembles salt-stratified layer entrainment. If turbulence is produced at the base of the bottom layer the layer thickens with entrainment, as for example clear water entrainment by turbidity or density currents. In the work of Kranenburg (1994), the complete equations to solve both entrainment process, based on the turbulent kinetic energy balance, can be found. The entrainment process implies enhanced mixing driven by shear at the interface. The existence of highly sheared layers in the Amazon Shelf area has been pointed out by Geyer and Kineke (1995). The shear stress can be calculated according to -C(z) = fPdu (46) where vf is the kinematic viscosity of the mixture, calculated according to Equation (11), and p is the mixture density. The shear stress is maximum within the viscous near-bed layer, and decreases upward rather abruptly. Figure 34 shows an example of tidal median values of the shear stress for anchor station OS 1 (4413) plotted against depth. It is interesting to note that a maximum was reached at the 0.75m level, and then the stress decreased downward. At this site, near-bed sediment concentrations in excess of 200g/l along with appreciable horizontal motion were recorded. OSI (low water 5 Flri'eO 4-0 0 3- 0 0 12 0 10.8 10.6 104 10-2 100 shear stress (Pa) Figure 34: Tidal median values of shear stress versus depth for anchor station RM. From Figure 34 it can be seen that entrainment may occur above the primary lutocline, around the elevation of z--lm. At this elevation, due to the lower concentrations, it is likely that the mud does not develop an appreciable shear strength. However, a model with a boundary condition at the top of the fluid mud layer prescribing entrainment in the Amazon Shelf would not account for an important part of the transported sediment, namely fluid mud below the boundary. Given this, if the bed level were set at z=-O, i.e., at the bottom of the fluid mud layer, the entrainment process would have to be accounted for in the mass diffusion function established for the water column, and the prescribed flux boundary condition would correspond to a bed erosion rate, examined in the next section. The sediment flux over the tidal cycle was calculated for the available 18 timeseries as the time-derivative of the total mass in suspension, and was correlated with a characteristic "free stream" velocity as a measure of tidal action. Considering the flux at the lower-most measured elevation (z=0.25m) a degree of correlation was found, as can be seen in Figure 35, 36 and 37 for anchor stations 1154, 2418 and 2428, respectively. Poor correlation found in other cases (not shown) were due to imprecision in the measurements. In fact, the correlation improved when the flux was calculated considering the bottom level at higher elevations, as can be seen for anchor stations 2428 and 3455 in Figure 38 for the flux calculated at z=lm. At these stations the maximum resuspension flux occurred in association with the movements of the lower lutocline (below about the 2.5 m level for all the anchor stations). The high gradient of sediment concentration and the high rate of erosion occurring in this near-bed zone can thus easily lead to errors in measurements, especially when the profiles are recorded with non-fixed stations as in the AMASSEDS profiling surveys. The occurrence of a phase difference between the sediment flux and the velocity cannot be explained in a straightforward way. In some cases the velocity led the sediment flux (for example, station RMi 3455 in Figure 39), while in others velocity lagged sediment flux (as at station OS2 1154). As mentioned in section 3.3.5, the flow velocity at the top of the boundary layer does not always present a consistent phase relationship with the bed shear stress. However, the analysis is retained here in terms of the flow velocity because it is the model input parameter from which the shear stress is calculated. The above type of resuspension behavior was not found at the station RMo (2445), which showed the highest stratification level of all the measurements, with the lowest vertical sediment fluxes. At RMo (2405), an event that could be characterized as sediment advection from the shallower zone arriving at the anchor station at the end of 74 the ebb tide could have interfered with the cyclic behavior (compare Figures 40 and 41). Stations 4434 and 4441 also showed unusual responses with respect to resuspension (see for example Figure 42). x10' 0S2 (1154) 0 5 10 15 20 25 time (h) 0.8 0.6 0.4 > 0.2 SI I ... ... .. .......... ........ .......................... .................................... ... 5 10 15 20 25 time (h) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 n- 00 0 0 0 0 0 0 0 -1.5 -1 -0.5 0 0.5 1 1.5 sediment flux (kg/m2s) X 10, Figure 35: Sediment flux (at z=0.25m) and current velocity for anchor station OS 1 (1154). Top: time-series of sediment flux and velocity. Bottom: sediment flux vs. velocity after phase adjustment. x 10" 0S2 (2418) II I! II I I 0 ..... ....................... ? ................................. .................................. ".................. .............. !. .............................. ?.... 5 ........................ .. ..... ........... ........ ........ ....................... ................ ........ ... . ... ...... .... . . ...... ..... ........ ............................. .. . /iI i_____ Ii____ 15 time (h) X 1. 0 ? 0. 0 5 10 15 20 25 time (h) 1.8 0 1.60 1.4 0 0 0 1.2 0 0 0 0 S0.8 0 00 > 0 0 0.6 0.4 0 0 0.2 0 0 0 0 I IJI -10 -8 -6 -4 -2 0 2 4 6 8 sediment flux (kg/mns) X 10, Figure 36: Sediment flux (z=0.25m) and current velocity for anchor station OS2 (2418). Top: time-series of sediment flux and velocity. Bottom: sediment flux vs. velocity after adjusting for the phase. 5 1 5 CN rising river flow (Stn 2428) x 10 time (h) 0 5 10 15 20 25 time (h) P.. 0 2 sediment flux (kg/ns) 6 x 10- Figure 37: Sediment flux (z=0.25m) and current velocity for anchor station CN (2428). Top: time-series of sediment flux and velocity. Bottom: sediment flux vs. velocity after adjusting for the phase. E - -2 U) ..... ;.............................. ................. .. ...... .... ..... .. ... ............ .. .. .............................. ..... ..... ..... ... ... .... .... ........................................ s ...... ......... .......... ~~ .....I................ I (0. 0 X 10, ON (2428) I I I I . .............. .................. ..... .............. ..... ..... I ................ ......................I ............. ...... ............... ............ ..... 52 ............. ... .................................................... time (h) 0. >0.6 0.2 F- -4 -3 -2 -1 0 sediment flux (kglrt~s) 1 2 3 x 10, Figure 38: Sediment flux (z=-1m) and current velocity for anchor station CN (2428). Top: time-series of sediment flux and velocity. Bottom: sediment flux vs. velocity after adjusting for the phase. 000 0 *1... ...... ....... -2 .. ... ....... 4 ....................... 4 ......................... .......... .......... ... ... ............ ............... .. ... ...... ................ 0 2 4 6 time (h) 0 2 4 6 time (h) 0.41- -4 -3 -2 -1 0 sediment flux (kg/m2s) 8 10 12 8 10 12 1 2 3 x 10 Figure 39: Sediment flux (z=lm) and current velocity for anchor station RMo (3455). Top: time-series of sediment flux and velocity. Bottom: sediment flux vs. velocity after adjusting for the phase. II J I I I I I i RMo (3455) x 103 RMo (rising river flow-as2405) uinimy (psu) 30 25 20 15 10 5 0 time (h) Figure 40: Salinity and sediment concentration time-series for anchor station RMo (2405). x 103 RMo (2405) 0 2 4 6 8 time (h) 10 12 14 16 0 2 4 6 8 time (h) 10 12 14 16 Figure 41: Time-series of sediment flux and current velocity for anchor station RMo (2405). X 10, OSI (4434) I I i. 0 ...... ..... .. ..... 1....... .. . 5 ............. 1. ................ ................ ...................I*...................... 0 5 10 15 20 25 time (h) 2 ...... . ........................ .......... ................. ..... ........ ..1 .......... .. .... ..... ... ......... ..... ... ......... .. .... ... 6 ............. ............. ........... .. ... ... . ....... 4........ ... ............ .............. ......... .. ........ 2 I 1I 2 ................t..... ........... .......... .. ... ...... 0 5 10 15 20 25 time (h) Figure 42: Time-series of sediment flux and current velocity for anchor station 05 1 (4434). 3.4.3 Bed Shear Strength Bed shear strength can be calculated using the expressions derived in section 3.2.7 as a function of sediment concentration. For the maximum concentration of 321 g/l observed in the study (Table 1, anchor station OS 1), one would obtain a shear strength of 0.9lPa according to, for example, Otsubo's criterion for incipient motion given in Figure 16. However, the estimated current-induced bed shear stress reached a maximum value of 0.42 Pa at this site (see Table 5), which would not be enough to move a mud layer which, for instance, had a horizontal velocity of 7 mm/s. Beside the assumptions made in obtaining both the shear strength and the shear stress, other physical environmental factors may also change the properties determined in the laboratory. Thus, it is highly 0. 6. 0. 0. likely that in the prototype environment, wave action measurably lowers bed shear strength, as well as enhances current-induced bottom shear stress, as was seen in section 4.3.3. With the purpose of incorporating these effects a coefficient, 0<1, will be included and evaluated from the measurements. Considering the empirical equation given by Otsubo for incipient movement, the bed shear strength is obtained from Equations (14) and (16) as follows = 1.03x10-4 CI17 (47) 3.4.4 Rate Constant Selecting anchor stations OS2 (2418) and CN (2428) as examples, the erosion rate constant M in Equation (44) and the coefficient 0 were determined. The erosion rate is defined with respect to the positive sediment flux. Bottom shear stress was calculated according to Equation (31). The best fit between the excess shear stress (r, t,) and the erosion rate for anchor station CN (2428) gives a coefficient [in Equation (47)] 0 =0.4 and a erosion rate constant M=0.02 x. kg m-2s'Pa"1. Similarly, for anchor station OS2 (2818) M=0.027 ; kg m2s'Pa'1 was obtained, using the same value of 0. Thus, taking the calibrated M values, the observed and calculated erosion rates at z=0.25m for anchor stations 2428 and 2418 are as given in Figure 43. Note that negative flux values, i.e., when the bed shear stress is lower than the bed shear strength, should be treated as deposition, which will be considered later. x103 0S2 (2418) 6 A S2 ( 4 I 1 -to -6 -8 -10I I 0 5 10 15 20 25 time (h) x 10' CN (2428) 6 4 A , a I -2 a ,\ , a ,l 1 I J aI I -4 -6 -8 II 0 5 10 15 20 25 time (h) Figure 43: Observed (dotted line) and calculated (solid line) sediment fluxes at an elevation of 0.25m for anchor stations OS2 (2418) and CN (2428). |