UFL/COEL98/011
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 Para, 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. Gusmao from AQUAMODELO, Consultoria e
Engenharia Ltda., in Rio de Janeiro.
This work was made possible by financial support from CAPES, Brazil.
TABLE OF CONTENTS
ACKNOWLEDGMENTS...................................................................... iii
LIST OF FIGURES........................................................................ viii
LIST OF TABLES......................................................................... xv
ABSTRACT........... ... ................................ ..................... ............ xvi
CHAPTERS
1 INTRODUCTION..................................................................................................... 1
1.1 Study Area and Problem Statement........................................................................ 1
1.2 O 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 W ind and W aves............................................................................................. 10
2.5 Amazon Plume and Salinity Distribution ............................................... .......... ... 11
2 .6 D ata Set................................................................................................................ 14
3 FORMULATION OF 1D 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 V iscosity ................................................................................................ 32
3.2.7 Shear strength ......................................................................................... 35
3.2.8 Permeability and effective stress .............................................. .......... 38
3.3 Vertical Flow Structure................................................................................... 42
3.3.1 G general ................................................................................................... 42
3.3.2 Tidal boundary layer approach ............................................... ........... .... 43
3.3.3 Fluid cinematic and eddy viscosities ........................................ ........... .. 45
3.3.4 A model for laminar boundary layer....................................... ........... .... 51
3.3.5 Shear stress from measured velocity profiles............................................. 54
3.3.6 M ass diffusivity.......................................................... ................................ 59
3.4 Bottom Boundary Condition......................................................................... 67
3.4.1 Bed level definition............................................................................... 67
3.4.2 Erosion mechanisms and erosion rate....................................... ........... .. 69
3.4.3 Bed shear strength............................................. ............... .... .. 80
3.4.4 R ate constant................................ ........... ............................................... 81
3.4.5 Deposition and consolidation .............................................................. 83
4 MUD DYNAMICS IN AMAZON ESTUARY.............................................................. 87
4.1 Introduction .................................................................................................... 87
4.2 Long Term Processes: Accumulation Mechanisms......................... ...... 89
4.2.1 General observations.................................................................................. 89
4.2.2 Salt wedge trapping mechanism ............................................. ........... .... 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 dynamics.............................................................. 97
4.3.3 Wave effects ............................................... 101
4.3.4 Time scale of sedimentary processes......................................................... 107
4.4 Validity of 1D Approach........................................ 109
4.4.1 Introduction......................................................................................... 109
4.4.2 A dvective transport.................................................................................. 110
4.4.3 Turbidity current.................................................... 114
5 TRANSPORT MODEL......................................................................................... 117
5.1 P ream ble............................................................................................................. 117
5.2 Main characteristics of the numerical model ..................................................... 117
5.3 Numerical experiments ....................................................................................... 120
5.4 Modeling 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 Mass diffusion coefficient......................................................................... 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
REFERENCES.................................................................. ......... 143
BIOGRAPHICAL 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 OS2 (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 Crossshelf salinity transects at the river mouth for (a) spring and (b) neap tides
(from K ineke, 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 offlocs determined by image analysis of plankton camera
photographs at 35 m depth and high river discharge (from Kineke, 1993)............. 26
9 A schematic representation offlocculation, sedimentation and consolidation
(from Im ai, 1981).................................................................................................... 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
B ingham 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 OS2 (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(30)] velocity profiles over
two tidal cycles recorded at anchor station OS1 (low river flowas4434)................. 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 OS1 and CN. ................................................................... 56
27 Observed shear stress [dotted line Equation (35)] and calculated shear stress
[solid line Equation (31)] over two tidal cycles for anchor stations OS1 and
CN ............................................................................. .................................... 57
28 Depth variation of the tidal eddy viscosity at anchor station CN (2428) (dots) and
eddy diffusivity 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 O S2(2418)...................................................................................... 64
31 Tidal median values of the stratification damping function [Equation (41)] versus
depth for anchor stations CN (2428) and OS2 (2418) ............................................ 65
32 Stratification damping function versus depth over the tidal cycle for anchor
station C N (2428).................................................................................................... 66
33 Timevariation 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=0.25m) and current velocity for anchor station OS1 (1154).
Top: timeseries of sediment flux and velocity. Bottom: sediment flux vs.
velocity after phase adjustment.......................................... ............................ 74
36 Sediment flux (z=0.25m) and current velocity for anchor station OS2 (2418).
Top: timeseries of sediment flux and velocity. Bottom: sediment flux vs.
velocity after adjusting for the phase. ................................................................. 75
37 Sediment flux (z=0.25m) and current velocity for anchor station CN (2428). Top:
timeseries 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:
timeseries 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:
timeseries of sediment flux and velocity. Bottom: sediment flux vs. velocity
after adjusting for the phase................................................. .......................... 78
40 Salinity and sediment concentration timeseries for anchor station RMo (2405)....... 79
41 Timeseries of sediment flux and current velocity for anchor station RMo (2405)...... 79
42 Timeseries of sediment flux and current velocity for anchor station OS1 (4434)....... 80
43 Observed (dotted line) and calculated (solid line) sediment fluxes at an elevation
of 0.25m for anchor stations OS2 (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)
OS1 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
environments (after Mehta, 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 ofkaolinite beds in flumes
(from M ehta, 1989). .............................................................................................. 102
53 Timeseries 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 Macaph, Amap................................................... 106
56 Current meter, optical backscatter output (OBS) and nearbed 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 tideinduced resuspension with a uniform settling
velocity (W,=lmm/s) and a depthindependent, timedependent diffusivity.
Hourly concentration profiles over a tidal cycle................................................... 121
63 Simulations similar to Figure 62, but with a smaller settling velocity
(W,=0.5mm/s). Hourly concentration profiles over a tidal cycle.......................... 122
64 Comparison of results for different particle masses. Solid lines: Cm=0.005g/l,
asterisk and circles: C jn=0.02g/l.. ..................................................................... 122
65 Same simulation as in Figure 63 with a constant settling velocity (W,=0.Smm/s),
but a with a nonuniform diffusivity (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 nonuniform 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 (51)]............................................. 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/100g .................. 22
3 Exchangeable cations and cation exchange capacities for Amazon samples, in
m eq/100g ...................................................................................................... . .......... 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 flowsediment 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 nearbed measurements of velocity and sediment concentration. Salt stratification
damping of turbulence is also included, and the shear strengthbed density relationship
required for the calculation of the erosion flux is derived from laboratory analysis of
Amazon sediment samples.
Measured timeseries of suspended sediment concentration are compared with
modelsimulations in order to better understand the sedimentary processes occurring in
the water column. It is found that flowsediment 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 turbulencedamping role of the high density fluid
mudlike suspension that dominates the lower water column. For the remainder of the
water column, stratified turbulent flow conditions can be assumed.
Longterm accumulation mechanisms as well as shortterm 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 densitydriven estuarine circulation, saltinduced stratification, and
flocculation enhance the trapping of sediments in fluid mud over the long term. On the
other hand, shortterm, 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 1
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 claysized 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 macrotidal 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 13 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 near
bed 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 al.
(1989b). With the continuous reduction of computational costs, threedimensional
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 nearbed layers in high concentration zones, and
advection of density gradients are some of the aspects in modeling which require three
dimensionality. 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.
14 14 14
12   12 ,,  I 12 ( _,  
S10   10   10   
4 .  8   
4 4
2  2 2 
o 0 0
0.8 1 1.8 0.01 1 100 0.01 1 100
velocity (Mn) concentraton (gil) edimert flux (km)
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 densitydriven 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 threedimensional manner. It is
hypothesized here, that densitydriven circulation is a mechanism which acts over long
term scales, of several months, and may play an important role in the turbidity maximum
formation at the Amazon Shelf. Salinityinduced 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 onedimensional 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
watersediment 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 shortterm 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 19891991 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).
E 1.8
% 1.5
e
g 1.2
*1
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 1113x108 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 85
95% of the sediment carried out to the mouth of the Amazon River was silt and clay
sized.
In addition to the large input of fresh water, strong tidal currents and winddriven
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 NorteCabo 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 Flow
Subtidal currents are vertically sheared, moving alongshelf toward the northwest
(Limeburner and Beardsley, 1995). With nearsurface currents that reach speeds of 1.5
m/s and near bottom velocities less than 0.2 m/s (Geyer et al., 1991), these currents show
a high degree of coherence with the alongshelf 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 monsoonlike
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
JanuaryMarch while blowing onshore from the northeast, and weaken and blow along
shelf 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 12 m less than 4m 99% of the year, and dominant periods of 67
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) swell
type 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 extra
tropical 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 100200 km
seaward of the mouth (Geyer, 1995). This shelf is oriented in NWSE direction and
extends from shore to the 100m isobath, between the latitudes of about 0 and 5N.
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 lowsalinity 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
southeastwardd 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".
Springneap 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 alltidal cycle. Figure 5 shows two cross
shelf 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).
354 
25 8
20
5 
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
multicomponent 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 ConductivityTemperatureDepth (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 crossshore 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.
13 16
17 Is
0 1
30
45
so 75 100 125 13
Pirran ~I I hn
50 73 100 125 150
(b) Dizmm. Offshom =m)
Figure 5: Crossshelf 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.,
ConductivityTemperatureDepth) 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.' with a sampling rate
of 0.85 Hz, thus yielding a scan every 1518cm. The upanddown casts were bin
averaged in 0.3m bins, and the data were converted to height above the bottom in even
spaced 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 binaveraging, for the velocity
measurements a 17term 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 suspendedsediment
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 lowermost elevation, at z = 0.25m, considering z = 0 to be
the resting elevation of the profiler legs. Isoline elevations indicate the maximum
elevation reached by concentrations of 0.lg/1 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).
ell4 4 1~O 4 0 r
00 Oo.rR0 W0 CD ) W C>cQ )0
"i 0 0 66
~~ o o~ ~V)
~0 0 6 6 4 ,j C4,t Q
~e
t tn "t C4 0C4
00
4)~~~~c. rfNQ ( k o 0 k 0 t qC
NO 0% 0 f 00 N. en IfR \R \R W t eq CI '.0
C4e~ 00 'fNoo e c O C 00 0 Q 00 e t O
W) en I C7 en t ON '~
CHAPTER 3
FORMULATION OF 1D 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,
8C \(1)
= C(W w) + e,
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)
z ah (2)
w h
h t
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 bedsuspension 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, t,, a linear rate of erosion is
prescribed according to
(3)
z=0, To>r, E=M\ 1(
where M is an empirical erosion rate constant.
When no erosion is occurring, a deposition rate is prescribed according to
z=0, To 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 physicochemical 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 Xray diffraction (XRD) was conducted using a computercontrolled
diffractometer. Clay minerals identified from XRD patterns (Whittig and Allardice,
1986) were preponderantly consisted of kaolinite and smectite. Nonclay 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 Acid
Dichromate Digestion, with FeS04 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 (D297487 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+,
IH, 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 80150
Table 3: Exchangeable cations and cation exchange capacities for Amazon samples, in
meq/100g
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, d5o, is 2.5 pm and 3.7 pm, for samples 3227 and 3230,
respectively. Furthermore, the size is less than 10pm 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 pm, which indicates a slightly smaller particle size than for the
samples analyzed here.
Sediment Size Distribution Amazon Sample 3227
100
80 
60
9o ,
40  
20 
A I IL
diameter (pim)
Sediment Size Distribution Amazon Sample 3230
100
80  
S80
60 
40
40  
20 
n
diameter (prm)
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 2pm (clay) are platelike 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, shearinduced, 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 finegrained 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 50100l m, and a maximum of 200nm.
During AMASSEDS, photographs of the in situ suspended materials were obtained
showing that these materials were dominantly present as flocs, with modal size in the
range of 200500gm (Kineke, 1993). At the Open Shelf Transect, see Figure 6, flocs
with modal sizes of 4006001im were observed (Figure 8).
I\
I;
100 
1 'I
I ____...I._____
10 100 1000
Diameter wn)
Figure 8: Size distribution of flocs 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, floe density is inversely
related to floe diameter, decreasing with increasing order of aggregation. While Krones'
floc densities ranged between 1.07 to 1.27 g/cm3, Gibbs (1985) reported floe densities of
suspended sediments from the Chesapeake Bay ranging from 1.004 to 1.032 g/cm3, for
floe sizes of 50pm to 500pm. Kineke (1993) reported an estimated value of 1.08 g/cm3
for floe density during AMASSEDS measurements near the Open Shelf Transect.
Kranck and Milligan (1992) found floe densities in San Francisco Bay ranging from 1.04
to 1.48 g/cm3 for floe sizes of 500pm and 50pm. In recent in situ measurements of floe
size and settling velocity in the Tamar River estuary, England, Fennesy et al. (1994)
obtained floe densities ranging from 1.008 g/cm3 to 1.12 g/cm3 for 600gm to 50pgm
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 600pm, 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 (Imai, 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 sedimentwater 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 selfweight 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.
Time
Figure 9: A schematic representation offlocculation, sedimentation and consolidation
(from Imai, 1981).
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 fluid
supported. 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 nonquiescent conditions should be
conducted to investigate the effects of shear stress or turbulent kinetic energy on the
growth and breakup 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 multidepth method (McLaughlin, 1958) must be used instead. Accordingly, for a
quiescent medium, the sediment mass conservation equation is reduced to
C a(wc)_ (5)
Integrating this equation with respect to vertical coordinate z gives
a (6)
(W.C) = f Cdz
t0
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
S1=P Cdz d (7)
C t,+1 000
where
,+! C'i+ +C' (8)
C 2 z D
.D 2
yields the pair, W,C, for each time tj+12. 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, Co, ranging between 2.1 to 37 g/1. Conditions of each test are given in
Table 4.
Table 4: Settling column test conditions
Test No. Mud Sample Suspension Height (cm) Co (g/l)
1 3230 166 151 21
2 3230 175.3163 37
3 3230 179167.5 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 Co = 21 g/1. 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 watermud 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 Co= 21 g/l
1 8 0 0..
160
140 15 min
120 30 min
100
E 56 min
80
40 180min
20
0 20 40 60 80 100 120 140
concentration (g/1)
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 mm/s, are in
agreement with the settling experiments. As could be expected, the settling velocities
determined in quiescent conditions may underestimate the fieldestimated 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 0.01C).6 mm/s
W, = 0.11C'6 mm/s
for C > 1.7 g/1
for C < 1.7 g/l
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
loo + a
a ; + A+ + +
100 + + .
0 1
SCo= 2.1 g/
103 + Co= 8.310.7 g/I
x C 21 g/I
o C0= 37 g/
101 100 101 102
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 netlike structure. This
structure causes the suspension to exhibit nonNewtonian 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 shear
thinning (pseudoplastic) behavior. At densities > 1200 kg/m3, the behavior changes from
shearthinning to shearthickening dilatantt). 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 shearthinning at low densities and shearthickening, 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 shearthickening or dilatant.
01
*O (1.1 1)
,l "(1.0
I.
M (1.052)
.01 
1 10 100
SHEAR RATE (se"')
Figure 12: Flow behavior of Amazon sediment suspension at different densities (from
Faas, 1986).
10 
.
II
A 10
10 60 100
SHEAR RATE (use1)
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 powerlaw model (Sisko, 1958)
v,(C) = Cexp(0.7810.24) for < 3.9 sec'
az za
v,(C) = Cexp(0.017 12.95) for a 3.9sec_1
az 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, T, can be determined from its
correlation with the yield stress measured from Amazon sediment samples. The upper
Bingham yield stress, Ty,, is defined from the stressversusshear 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:
az
Figure 14: Schematic of shear stress versus shear rate flow curve, and definition of the
upper Bingham yield stress.
For consolidated mud
T, = 0.25tPa ty >1.6 Pa (12)
and for weakly cohesive mud
= Oly Pa Ty <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 t. and yield stress ry for the
first group is
T,1 = 0.27%6Pa (14)
,2 = 0.79%tPa (15)
depending on the threshold states in mud transport considered; these being, according to
Otsubo, sT for incipient mud particle movement and T2 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 JuneJuly 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.
102
o o
10 o (o o0
S10 0
Figure 15: Yield shear stress versus concentration for Amazon Shelf mud (from Dade,
1992, Table 6.7 and Faas, 1985, Figure 3).
10o
Concentration (g 1)
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 l 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
..Otsubo(2)
10" .. Otsubo l
110
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 finegrained 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, o', 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, o', is defined as
o' = ot w, (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
C = p (18)
e+1
Thus, for a void ratio of 5 a sediment concentration of 441 g/1 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 selfweight
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..
1200 ,.
1800 ,.
S100
200
1 1.05 1.1 15 11 115 1.25 1.3 1.535 4 1.45 1.5
DENST (*1000 kg/m3)
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
*
e
15
.. :*.
O :
L 10 '. 1*
52I
5 9 */ ... .
0 rl rm irlriln q rT hlnqrri 1111irrunqlr'mnn*
10 *' 10' I0"' I 10 10' (0'
Effective stress (kPo)
Figure 18: Effective stress versus void ratio data taken from experiments reported in the
literature (Alexis et al., 1992).
52
507
48
46
44
42'
40
38
36
30 ;
a 28 
276
922
O 20 P.9
>18 1."
16 : /
12 :
10 g .B U
8 *
2
0'
0 us n" rq rn..... .qr.. .. ....q ....q, ... .
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
3.31 General
Having in mind the development of a vertical transport model for finegrained
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 wellunderstood 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 noncohesive 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 nearbed 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. Currentinduced 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
Bu 24A Du
layer, the relevant advective acceleration term can be written as u. 2 =,
ax L at
where u. is the flow velocity outside the boundary layer, A is the water particle semi
excursion length and L is the tidal wavelength. This term can be neglected depending on
the ratio 2xA/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
a u (19)
P (u u,) = 
at az
where u(z,t) is the velocity at elevation z above the bed level, u. is the velocity outside
the boundary layer and T is the shear stress at level z. Equation (19) can also be written
as
O(u u) ( )(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: quasisteady 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, quasisteady models can be
considered as a subclass of unsteady models, by solving the nonsteady oscillatory
boundary layer problem with a suitable viscosity function.
Expressing the tidal velocity profile as u(z,t) = u(z)e', where a is the tidal
frequency, Equation (20) becomes
d ( du (21)
v(z) = ia(u u( )(
dz\ dz)
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 K, is the
von Karman constant. Substitution of this relation in (21) yields a homogeneous, Bessel
type differential equation of zero order, known as the Kelvin's differential equation:
d2u 1 du (22)
+ + u= 0
Ss ds
where X=2  is imaginary and s = z
KU.
With the boundary conditions u = 0 at z = 0 and u = u at z oo, the general
solution of (22) has the form
u = u 1+aker 2 (23
where
1 (24)
kerf2f
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 p (u u,)dz
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.
107
106
105 104 103
Kinematic Viscosity (m's)
Figure 20: v (circles) and vf (asterisks) median values for the tidal cycle, from Equation
(25) and (11), respectively.
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 Im above the bottom, high
mud concentrations were observed and, accordingly, a change in the behavioral trend of
momentum diffusion is noticeable in Figure 20.
mw mWam a M aHBi
NeK W xW K xW iBWaB
3 UN "m M WK ( 00100
3 N NUUU NENE(E 1)mmm
3ME WKE XN N E XW W
mm mmm m XI @ sEoD
W2 eK NE i alQ
W NE NKE W on BB)
m am w x a asgi
NE 3I KN m @NH B
N1 E mm *Bag
ofok@fy @OE agWJ
rili93_. Q@WH wasgiHQ
~~ ~~~~~~ ~ CeW.. .... .... .. ...... ....... ....
A noteworthy feature of the velocity profiles near the bottom of the Amazon Shelf
32u
is the concavity ( > 0), a feature uncommon to estuaries in which logarithmic
az
profiles are observed. Figure 21 shows details of the velocity profiles, enhancing the
portion near the bottom.
OS2 (high river flow as3419)
0.5 1 1.5 2
Velocity (m/s)
Figure 21: Velocity profiles recorded at anchor station OS2 (3419).
Nearbed 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 nearbottom
horizontal velocity of the Avon River as an unsteady Couettetype 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, cz, where c, and c2 are constants, Equation (21) can be rewritten
as
d =du u ,. (26)
(ci c z) = i(uum)
dz dz)
Through a change of variable, s = c cz, a differential equation is obtained with the
parameter = 2 / The solution, compatible with the boundary conditions u=0 at
VC2
z=0 and u = u. at z > oo, is
u= u 1+b bei(2 (c 1 c2z) (27)
where
1 (28)
bei(22C
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.
7 T=12 hs
z u= 1 m/s
(m) 6 k=0.4 ,
u 2 cm/s (z)=0.001 0.0001 z /
5
4
3 /
2 /
./. izV)=u4k z
0 ...................
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
nonNewtonian fluid following the Bingham equation. They defined a characteristic
Reynolds number as
4puh (29)
1 r+ zh
where p is the fluid density, rT is the rigidity coefficient defined from the shear stress
versus shear rate curve (see Figure 14), , 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 0 0 0
00 00 C 0 0
S
a oW M
00 000oc
00 0 m 000
ao a 0m Go
0 0 MO 0
0 00 000 00
O 0cm 000 0
S 0 0000 0
0 0 0Oa 0
a O 0 000 a
0 0 0
10 104 106 108
Reynodls Number
Figure 23: Tidal Reynolds number [Equation (29)] vs. depth.
It should be noted that the threshold of Re=10,000 for the turbulent regime was
established for laboratory conditions. The Amazon Shelf data, however, do provide
evidence of values for R& below 10,000 within a welldefined, 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 nearbed zone due to
the enhanced viscosity of the sediment suspension.
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'it lexp([1+ i] (30)
from which the bottom shear stress is obtained as
,(0, t)= Aoe'" (1+i) V (31)
and the corresponding shear velocity as
ru = 12 (32)
[Ae l 2v/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 semidiurnal tidal
frequency, and Aa = 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) = li(z) u(z) (33)
Here, ii 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., z (34)
u = U In (34)
K z
where u* was calculated from the velocity profiles as described in section 3.3.5. The
values of the roughness height, Zo, 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.
OS1 (low river flow)
0 0 0 >
0 0 0 0 0 0 0
0 0 0 0
\U 1 U I. U I U0 I
1. O 1. O 1. O 1. O 1. O 1.
U 1. U 1. U 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 OS1 (low river flowas4434).
1. U 1. U i. U 1.
OS1 (lowriver flow a4434)
It
3.5 M
(K
3 K
Of
2.5
S2 o
0
1.5 K
oo
0.5X 0
0
0.5 X 0
0 0.2 0.4 0.6 0.8 1
absolute 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
lm level. It should be noted, however, that the model was overextended 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 nearbed
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/1 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 Im to the total height of the boundary layer).
3.3.5 Shear Stress from Measured Velocity Profiles
Integrating Equation (19) between z and oo, and recalling that T, = 0, the shear
stress can be calculated from
S8 <(35)
W(z) = pf(u u.)dz
and the shear velocity, u., can be then expressed as
,_ . 1 (36)
u. = = p (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, 6, must be determined
carefully. A crude estimate of 6 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 (31), the free
stream flow velocity was taken from the measured velocity at 4m level, and the nearbed
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 underestimation of the shear stress, and, also, a discrepancy in phase. Note
that Equation (31) 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 nearbed 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, viscosityconcentration 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 7" 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.
OS1 (4434)
4 3
S / 
31
0 5 10 15 20 25
time (h)
CN (2428)
8  10 15 20 25
Figure 26: Observed velocity (dotted line) and shear stress (solid line) over two tidal
cycles for anchor stations OS1 and CN.
OS1 (4434)
n;
e 0
0.1
.n4 I
0 5 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 OS1 and CN.
Table 5: Tidal mean and maximum shear stresses and shear velocities at the anchor
stations
Stn. 8 Mean Max. Mean Max. Mean Mean Mean calc. u.
obs. T obs. rT calc.T. obs. u. obs. u. calc. u. Cb=320 g/l
(m) (Pa) (Pa) (Pa) (m/s) (m/s) (m/s) (m/s)
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) = KU.Z 1 (37)
which applies above the viscous layer. Figure 28 shows an example of the momentum
diffusivity 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 shearinduced turbulence can be
measured by the gradient Richardson number:
CN (2428)
:10
01 i 1 i l i i IIl i
104 103 102
diffusivity (rf/s)
Figure 28: Depth variation of the tidal eddy viscosity at anchor station CN (2428) (dots)
and eddy diffusivity calculated according to Equation (37).
g p (38)
where g is the acceleration due to gravity, p is the mixture density and u is the local
horizontal flow velocity. High Ri indicates high stratification, while low values indicate
wellmixed 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:
Al = 5890+38 T0.375 T2 +3S
A2 = 1779.5 +11.25 T 0.0745 T2 (3.8+0.01 T)S (39)
1+A1
p(S,T) = 1000
A2 + 0.698 Al
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) = CI + p(S, r)
Pa (40)
OS1 (4434)
10
8
6
4
2
103 102 101 100 101 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 salinityinduced 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 0.017 m/s, with a mean value of zo =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/1.
Mass diffusivity is related to the momentum diffusivity by the turbulent Schmidt
number
v (41)
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)
6, = (Z)
s,
oeh
where 4 accounts for stratification effects over the water column. A simple relationship
for 4 based on the gradient Richardson number is the well known function proposed by
Munk and Anderson (1948)
= (1+ aRib (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 OS2 (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 31.
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 lowwater 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 downwater movement, increasing somewhat its effect
during slack water.
CN (2428)
S.... 
20
:o
b
nc
101 100
Richardson number
2 OS2 (2418)
.......0 .... . .... .. .. ..... .
0 C C
t 0
10    ...... ..
Si111
10
0
103 10(2 101 100 1
Richardson number
101 102
Figure 30: Gradient Richardson number as a function of depth for anchor stations CN
(2428) and OS2(2418).
CN (2428)
20
0 i ii 
103 2 1101 1 101
I Ii'S I 241I)
i1 ......  ...... ....
C
S10 .... ........... ...... .. .. ...
) C
5 1    1 4 
103 2 101 101
stratification function
08 d 2 (2418)
i11111 i I 111
i I I it ti i iii
i i i i II I i l
I io i i I
I I I i I
i i !,t" iii i
I 0 ,,," I' i I
iIo 0 i i
i102 101 100
stratification function
Figure 31: Tidal median values of the stratification damping function [Equation (41)]
versus depth for anchor stations CN (2428) and OS2 (2418).
102 100
stratification function
16 h
stratification function
21 h
104 102 100 104 102 100
stratification function stratification function
Figure 32: Stratification damping function versus depth over the tidal cycle for anchor
station CN (2428).
0I
104
'1oQ
I J
ias
17.
165
15.5
0 5 10 15 20
tirre(h)
Figure 33: Timevariation 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
timescale of interest. Naturally, this is a dynamic definition that will depend on the flow
condition, the sediment characteristics and the timescale of analysis, as it is locally
determined by the timevarying balance between the applied fluid shear stress and the
A
t I
5. . 1.....4..... *..... *..............
.. ....... ... ... . .
......... .......I .... ... ..t........ ...
L....4*.......t... l**.... ..................I ... ........................*.........* .* ...4*.t .* *.. .....
^
J
^
P
_ 1"
shear strength of the bed. Also, this level should preferably be timeinvariant in the
context of the time scale of analysis.
Another bed definition is based on the transition from where the sediment is fluid
supported to sediment supported by its own continuous framework, with a transition
zone which is partly fluidsupported, and partly frameworksupported (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 nonzero 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/1. 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 nearbed 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 particlebyparticle, as mass erosion, or by
liquefaction. Particlebyparticle 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, particlebyparticle, 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,
T, ;,, where to is the applied shear stress and r, is the shear strength, as discussed in
section_3.2.7. T, 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:
E=MI (44)
E=M
where M is a rate constant. A relationship between M and c, can be found in Lee and
Mehta (1994). According to Equation (44), for a given applied stress, C., 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
E( v\(45)
InE = a (z ()]'
where a (PaP) and y (=0.5) are sediment dependent constants, Ef is the floc erosion rate
(kg/m2s) and r,(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 (Mehta and Srinivas, 1993). When entrainment occurs from the top
of a quiescent lower layer, the upper layer deepens and resembles saltstratified 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
S du (46)
r(z)= vp
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 nearbed
layer, and decreases upward rather abruptly. Figure 34 shows an example of tidal median
values of the shear stress for anchor station OS1 (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, nearbed sediment concentrations in excess of 200g/l
along with appreciable horizontal motion were recorded.
OS1 (low water
40
0
0
3 0
I o
12
1
0 0 I
108 10,6 104 102 100
shear stress (Pa)
Figure 34: Tidal median values of shear stress versus depth for anchor station RMi.
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=0, 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 time
series as the timederivative 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 lowermost 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 nearbed zone can thus easily
lead to errors in measurements, especially when the profiles are recorded with nonfixed
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).
QA
> 0.:
x 104
OS2 (1154)
 .. .7 .7 .7
0 5 10 15 20 25
' 
0 5 10 15 20 25
time (h)
8 .......... .... .................................................................................................. ...................................
I. ... ............ ... ......... . .. ......
0 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
0
0 0
0
0
0 0
1.5 1 0.5 0 0.5 1 1.5
sediment flux (kg/m's) x 10"
Figure 35: Sediment flux (at z=0.25m) and current velocity for anchor station OS1
(1154). Top: timeseries of sediment flux and velocity. Bottom: sediment flux vs.
velocity after phase adjustment.
00 O0
0
E
X
x 103
S2 (2418)
0 ..... . ............ ........ ....
5 . ..... ............ ......................... ... ........................ ................. ... ............ ..
I I I
0 i.V, ..  
5rs  i 
time (h)
[ !III!
i 2T 7Ti
S.. ......... ..........................  ', ..1............... 2"*"" ............ ............. ". '
5 10 15 20 25
time (h)
1.8 0
0
0 ,8
o0.8
13
0 0
0 o
o
S o
0 0o
0
O
0 0
10 8 6 4 2 0 2 4 6 8
sediment flux (kg/rrs) x 10
Figure 36: Sediment flux (z=0.25m) and current velocity for anchor station OS2 (2418).
Top: timeseries of sediment flux and velocity. Bottom: sediment flux vs. velocity after
adjusting for the phase.
1
a)
t .
i.
I
w
LIJ
CN rising river flow (Stn 2428)
x 103
E
0.
I 0.
0 5 10 15 20 25
time (h)
1
o0.8
0.6
0 2
sediment flux (kg/mrns)
6
x 10
Figure 37: Sediment flux (z=0.25m) and current velocity for anchor station CN (2428).
Top: timeseries of sediment flux and velocity. Bottom: sediment flux vs. velocity after
adjusting for the phase.
0 5 10 15 20 25
time (h)
O
11I I
R _
CN (2428)
0 5 10 15 20
time (h)
e
1171.
SI i I
time (h)
A
i1
S0.8
1,0.6
0.2 
4 3 2 1 0
sediment flux (kg/nms)
1 2 3
x 10
Figure 38: Sediment flux (z=lm) and current velocity for anchor station CN (2428). Top:
timeseries of sediment flux and velocity. Bottom: sediment flux vs. velocity after
adjusting for the phase.
Ch
/oo o
0 0o
1 .. .
III I 1
x 103
4
..... . ...... ... ....... . ....... . ..... .
4 I. ... i .
!~ \ /  \
2
I2
0 2 4 6
time (h)
8 10 12
0 2 4 6 8 10 12
time (h)
1.6 0 o
1.4 
1.2 O 0
0.8 O
0 0.8 0 0
0.6
0.4 0
0
0.2
4 3 2 1 0 1 2 3
sediment flux (kgfm2s) x 10
Figure 39: Sediment flux (z=lm) and current velocity for anchor station RMo (3455).
Top: timeseries of sediment flux and velocity. Bottom: sediment flux vs. velocity after
adjusting for the phase.
RMo (3455)
x 103
RMo (rising river flowas2405)
saliny (pau)
30
25
20
15
10
5
0
time (h)
Figure 40: Salinity and sediment concentration timeseries for anchor station RMo
(2405).
x 103
5
0
. 5
.s
RMo (2405)
0 2 4 6 8 10 12 14 16
time (h)
1.5.... ............
ii
0.5   6    4 
0 2 4 6 8 10 12 14 16
time (h)
Figure 41: Timeseries of sediment flux and current velocity for anchor station RMo
(2405).
80
x 10 OS1 (4434)
.5 . .. ...
5 .... .  .
1.
0.
0.
0.
0.
0 5 10 15 20 25
time (h)
2............... ..., ...
2. . .. ... .....
.I . .....1 .... ................ .. ............ .... ..... ..... ...
.
6
2 . ... .....   
0 5 10 15 20 25
time (h)
Figure 42: Timeseries of sediment flux and current velocity for anchor station OS1
(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/1
observed in the study (Table 1, anchor station OS1), one would obtain a shear strength of
0.91Pa according to, for example, Otsubo's criterion for incipient motion given in Figure
16. However, the estimated currentinduced 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
likely that in the prototype environment, wave action measurably lowers bed shear
strength, as well as enhances currentinduced bottom shear stress, as was seen in section
4.3.3. With the purpose of incorporating these effects a coefficient, 091, 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
S= 01.03xO4C1.57 (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. r,) 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 r kg m'n2sPa'1. Similarly, for anchor station OS2
(2818) M=0.027 c, kg m'2s'Pa' was obtained, using the same value of 9.
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.
x 10 OS2 (2418)
i \ / /
6
4
x 0
4 2
6 '
44
8
10
0 5 10 15 20 25
time (h)
x 1 0 CN (2428)
4
2
4 2
I a,
6
4
8
8
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).
