UFL/COEL89/011
FLOWFINE SEDIMENT HYSTERESIS IN SEDIMENT
STRATIFIED COASTAL WATERS
by
Rui C.F. Gameiro da Costa
Sponsor:
U.S. Army Engineer
Waterways Experiment Station
P.O. Box 631
Vicksburg, MS 39180
August, 1989
REPORT DOCUMENTATION PAGE
1. Report No. 2. 3. Recipient Accesloo o.
4. Title and Subtitle 5. Report Date
FLOWFINE SEDIMENT HYSTERESIS IN SEDIMENT August, 1989
STRATIFIED COASTAL WATERS 6.
7. Author(s) 8. Performing Organization Report No.
Rui C.F. Gameiro da Costa UFL/COEL89/011
9. Perforauti Organizatioa ame ad Address 10. Project/Task/uork Unit No.
Coastal and Oceanographic Engineering Dept.
University of Florida 11. contract or crant No.
336 Weil Hall DACW 3987K0023
Gainesville, FL 32611 _1. Typ otf Ro..r
12. Sponsoring Organizatioo eame and Address
U.S. Army Engineer Final
Waterways Experiment Station
P.O. Box 631
Vicksburg, MS 39180 14.
15. Supplementary Notes
16. Abstract
An examination of the causes for generation and dynamics of turbidity maxima
in estuaries reveals the critical role of sediment tidal pumping phenomenon and, to a
lesser extent, of the wellknown effect of residual gravitational circulation due to salt
water penetration. Both phenomena depend on the vertical sediment concentration
profile and, consequently, on the magnitude of the vertical mass transport fluxes.
Where high concentration suspensions occur regularly, the erosion/deposition fluxes
can be drastically modified by sediment stratification, consequently influencing sus
pended sediment response to currents and wave action. This influence is inherent in
flowsediment hysteresis, which therefore reflects the role of vertical mass transport
in the estuarine and coastal suspended fine sediment regime.
A vertical transport numerical model was used to investigate the influence of
several key parameters describing sediment settling, bed properties and stabilized
diffusion on the concentration profile. The model was also applied to simulate
the influence of the same parameters on the timelagged sediment response to flow
variations, reflected in the characteristics of flowsediment hysteresis loops.
Field data obtained in Hangzhou Bay (People's Republic of China), a high
concentration environment, showed typical features of flowsediment hysteresis and
confirmed the importance of the vertical mass fluxes in contributing to sediment
transport in the bay._ A qualitative simulation provided by the numerical model,
Continued 
17. Originator's Key Words 18. Availability Statement
Coastal bays
Coastal sediments
Finegrained sediment
Mud erosion
Sediment resuspension
19. U. S. Security Clasetf. of the Report 20. U. S. Security Claslf. of This Page 21. No. of Pages 22. Price
Unclassified Unclassified 172
using settling parameters corresponding to local sediment, while confirming the
importance of the hysteresis phenomenon, also revealed the critical need to use
algorithms describing adequately stabilized diffusion and bed fluxes.
Additional evidence of hysteresis was obtained through analysis of microscale
variables, such as the Reynolds stresses and the variances of the velocity components
resulting from combined effects of wave action and turbulence. Spectral analysis of
the measured random variations did not support the commonly accepted hypothe
sis of similarity between the responses to turbulent flow of sediment concentration
and temperature. The normalized turbulent intensities for all the measured veloc
ity components showed their highest values during the period of lowest sediment
concentration; this result is consistent with the hypothesis of turbulent intensity
damping by suspended sediment.
UFL/COEL89/011
FLOWFINE SEDIMENT HYSTERESIS IN SEDIMENT STRATIFIED
COASTAL WATERS
by
Rui C.F. Gameiro da Costa
Coastal and Oceanographic Engineering Department
University of Florida
336 Weil Hall
Gainesville, FL 32611
Sponsor:
U.S. Army Engineer
Waterways Experiment Station
P.O. Box 631
Vicksburg, MS 39180
August, 1989
ACKNOWLEDGEMENTS
The author would like to express his sincere gratitude to Dr. Ashish J. Mehta,
Professor of Coastal Engineering, for his guidance, advice and support during the
author's study period at the University of Florida. The author's appreciation is
extended to Dr. Hsiang Wang, Dr. D. M. Sheppard, and and Dr. Robert G. Dean
for their helpful comments.
The author is indebted to Dr. Mark A. Ross, who developed the numerical
transport model used in the present study, for his assistance in using the model
and for his helpful advice. The author would also like to thank Dr. Yixin Yan and
Sidney Schofield who participated in the field experiment, Subarna Malakar for his
assistance in some aspects of data processing, Jiang Feng who did the laboratory
tests at the Hohai University and Dr. Robert Kirby who calibrated the turbidity
sensors and provided most of the material included in Appendix A. Special thanks
are due to Helen Twedell of the Coastal Engineering Archives, Shannon Smyth and
Barry Underwood of the Engineering Publications Services.
Support for this study was made possible by contract No DACW 3987K0023,
"Investigation of Cohesive Bed and Fluid Mud Response to Current and Waves,"
from the U. S. Army Engineer Waterways Experiment Station (WES), Vicksburg,
MS. Thanks are due to Ms. Tamsen Dozier who was the project manager. The field
study in Hangzhou Bay, People's Republic of China, was conducted in cooperation
with Dr. Hsiang Wang, as a part of his WES supported study (contract No DACW
3986K009) "A Joint Research with People's Republic of China on Muddy Coast
Dynamics."
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ............................
LIST OF FIGURES ................................
LIST OF TABLES ................................
LIST OF SYMBOLS ...............................
CHAPTERS
1 INTRODUCTION ...............................
1.1 Problem Significance ...........................
1.2 Objective and Methodology ........................
1.3 Outline of Upcoming Chapters .....................
2 TRANSPORT PROCESSES IN ESTUARIES AND COASTAL BAYS .
2.1 The Turbidity Maximum .........................
2.1.1 General Aspects .........................
2.1.2 The Variability of the Turbidity Maximum ..........
2.1.3 Transport Mechanisms in Estuaries ..............
2.2 Salt and Sediment Fluxes and Mass Transports in Estuaries and
Coastal Bays ................. ..............
2.2.1 General Aspects .........................
2.2.2 Bowden (1963) ..........................
2.2.3
2.2.4
2.2.5
Hansen (1965) ....
Fischer (1972) ...
Dyer (1973) ......
1
1
3
3
5
5
5
6
8
13
13
13
2.2.6 Dyer (1974) ................... ......... 25
2.2.7 Murray and Siripong (1978) ................... 27
2.2.8 Dyer (1978) ................... ......... 30
2.2.9 Rattray and Dworsky (1980) . . . ... 32
2.2.10 Uncles, Elliot and Weston (1984) . . . .... 36
2.2.11 Uncles, Elliot and Weston (1985a) . . .... 39
2.2.12 Uncles, Elliot and Weston (1985b) . . . .. 41
2.2.13 Dyer (1989) . . . ..... .. ....... .. 43
2.2.14 Summary ................. ........... 44
3 SOME ASPECTS OF FINE SEDIMENT DYNAMICS .......... 48
3.1 The Transport Equation ......... ............. .. 48
3.2 Settling ................................... ... 51
3.2.1 General Aspects ........................ 51
3.2.2 Free Settling .......................... 52
3.2.3 Flocculation Settling ....................... 53
3.2.4 Hindered Settling ......................... 53
3.2.5 Settling Flux ................... ........ 55
3.3 Diffusion ................................. 55
3.3.1 General Aspects ......................... 55
3.3.2 Stabilized Diffusion ........................ 58
3.3.3 Diffusion Flux ................... ....... 60
3.4 Fluxes at the Bed ................... .......... 64
3.4.1 General Aspects ......................... 64
3.4.2 Deposition Flux ........................... 64
3.4.3 Erosion Flux ................... ........ 67
3.5 The Numerical Model ............ ............. ..68
3.5.1 General Aspects ......................... 68
3.5.2 Numerical Procedure ...... .... ........... 69
3.5.3 Discussion ............................. 72
4 FIELD AND LABORATORY EXPERIMENTS ........ ...... 74
4.1 General Aspects ............................. 74
4.2 Laboratory Tests ............ ... .............. 75
4.2.1 Grain Size Test ...... .... ................ 75
4.2.2 Settling Velocity Tests ............ .. ....... 77
4.2.3 Erosion Tests ........................... 78
4.3 The Field Experiment .......................... 81
4.3.1 The Measurement Site . . . . ... 81
4.3.2 Field Experimental Procedures . . . ... 84
4.3.3 Data Preprocessing ................ ...... 88
5 RESULTS AND DISCUSSION ....................... 94
5.1 General Aspects ............ ....... ........ 94
5.2 Sensitivity Analysis ............................ 94
5.2.1 General Aspects .............. ......... .. 94
5.2.2 Settling Velocity .............. ....... 96
5.2.3 Erosion Flux ................ ........ 97
5.2.4 Diffusion ............................ 102
5.3 Modeling of FlowSediment Hysteresis . . . ... 105
5.3.1 General Aspects ........................ .. 105
5.3.2 Modeling Results ................. .......... 109
5.4 Field Data Analysis ......... ............... 117
5.4.1 General Aspects ........................ .. 117
5.4.2 Stationarity Analysis .............. ... ...... .. 118
5.4.3 Spectral Analysis ....................... 119
5.5 FlowSediment Hysteresis in Hangzhou Bay . . .... 122
6 SUMMARY AND CONCLUSIONS .......
APPENDIX
CALIBRATION OF THE ELECTROOPTICAL
A.1 General Aspects ..............
A.2 Calibration Media .............
A.3 Siltmeter Calibration Procedure .......
A.3.1 S1000 Calibration ..........
A.3.2 SDM16 Calibration ..........
A.4 Quality Control and Assessment ......
BIBLIOGRAPHY ...................
. . .. . 135
TURBIDITY METERS
140
140
141
142
143
144
145
151
LIST OF FIGURES
2.1 Velocities with which different water masses move with the tides,
illustrating the effects of settling lag and scour lag. From Postma,
1967 .. . . . . . . . . 10
2.2 Decomposition of twodimensional profiles into components: a)
velocity; b) concentration. Adapted from Fischer, 1972 . 21
2.3 Decomposition of velocity components: a) Ud into steady and
fluctuating parts; b) Ud into transverse and vertical profiles.
Adapted from Fischer, 1972. .................... 22
2.4 Crosssectional area decompositions: a) Design i); b) Design ii);
c) Design iii). From Rattray and Dworsky, 1980. . ... 33
3.1 A general description of settling velocity and settling flux vari
ation with suspension concentration of fine grained sediment
(ni = 1.33, n2 = 5.0) ...................... 54
3.2 Diffusion flux as a function of aC/az for 6 = 4.17 and a = 2.0.
Adapted from Ross, 1988......................... 62
3.3 Concentration profile definitions. . . . .... 63
3.4 Typical time concentration relationship during deposition. From
Mehta and Lott, 1987 .......... ............ 66
3.5 Dependence of Cf on rb for uniform and non uniform sediments.
From Mehta and Lott, 1987. ..................... 66
4.1 Grain size distribution. ...... ................ 76
4.2 Settling velocity as a function of concentration for Hangzhou
Bay sediment.............................. 79
4.3 Erosion rate versus applied bottom shear stress for Hangzhou
Bay sediment ........................... .... .. 81
4.4 Location of the measurement site in Hangzhou Bay. ...... .82
4.5 Sediment transport patterns in Hangzhou Bay. Courtesy of Dr.
Hsiang Hwang. .. ........... ..... ...,........... 85
4.6 Measurement tower and positions of the equipment used in Hang
zhou Bay. ..................... ......... 87
4.7 Trend removal from the measured records (deployment C2, data
block 2). a) Measured c2 (upper level); b) c2 after trend removal;
c) measured u2 (upper level); c) u2 after trend removal .. 90
4.8 Examples of the filtering procedure applied to the measured u2,
velocity data at the upper level (Deployment C3, block 7). a)
measured u2; b) pressure record; c) i2; d) u . . .. 93
5.1 Initial concentration profile . . . .... 99
5.2 Simulated profiles with different settling velocities.. . .. 100
5.3 Simulated profiles with different erosion conditions. .. . 101
5.4 Simulated profiles with different values of diffusion parameter a. 103
5.5 Simulated profiles with different values of diffusion parameter f. 104
5.6 Simulated profiles with different values of 6 with high erosional
fluxes at the bottom ................. .........106
5.7 Flowsediment hysteresis. Resuspension, solid lines; deposition/settling,
dashed lines;indeterminate, dotted lines. From Nichols (1986). 110
5.8 Flowsediment hysteresis measured in the Humber River estuary
(courtesy of Prof. Keith Dyer, Plymouth Polythecnic, Plymouth
U.K.) ............ .... ... ......... ...... 110
5.9 Qualitative hysteresis loops at different elevations in cases of
lutocline formation . . . . . 111
5.10 C vs. U11 for case A ........................... 112
5.11 C vs. time for case A ......................... 115
5.12 Vertical gradient of the net flux vs. time for case A. .. . 115
5.13 C vs. fVV for case B .................. ...... 116
5.14 C vs. li l for case C ................. .......... 116
5.15 C vs. ilI for case D ......................... 123
5.16 Examples of computed spectra for variables of the el type. a)
C2; b) C1; c) ul; d) vi; e) w2; f) U2 g) C~w' h) uw. ........124
5.17 Examples of computed spectra for variables of the e' type. a)
C2; b) C1; c) ul; d) vi; e) w2; f) U2 g) C^w' h) uw . .... 126
5.18 Hysteresis loops simulated using Hangzhou Bay sediment set
tling properties........................... 128
5.19 Measured hysteresis loops: a) Deployment C3; b) Deployment
C2.. . . . . . . . . .129
5.20 Hysteresis in Reynolds stresses. . . . .. 130
5.21 Hysteresis in u variance . . . . . 131
5.22 Hysteresis in v variance . . . . . 131
5.23 Hysteresis in w variance. . . . . . 134
A.1 Calibration curves for the S1000 transducer and console 19264
using fresh water.......................... 147
A.2 Calibration curves for the SDM16 transducer and console 19265
using saline water. .......................... 148
A.3 Calibration curve for the SDM16 transducer and FSD1 channel
of console 19265 using fresh water. . . . .. 149
A.4 Calibration curves for the SDM16 transducer and channels FSD2
and FSD3 of console 19265 using fresh water. . . ... 150
LIST OF TABLES
4.1 Summary of wave data during the measurement period ..... 88
5.1 Measured mass diffusivities . . . . ... 133
5.2 Measured momentum diffusivities, Schmidt and flux Richardson
numbers. .................... ........... 133
5.3 Mass diffusivities computed by the model. . . ... 134
LIST OF SYMBOLS
a Crosssectional area; erosion parameter
A Tidal variation of the crosssectional area
(the same convention applies to other variables)
Ai. Partial area in row i of the crosssection
A.j Partial area in column j of the cross section
A Tidal mean of the crosssectional area
(an overbar denotes time averaging, except when otherwise noted)
A' Turbulent fluctuation of the crosssectional area
(primed variables have a similar meaning, except when otherwise noted)
AAij Partial area corresponding to position ij
c Concentration (of generic constituent or of suspended sediment)
Cdt Transverse deviation of the concentration
(similar subscripts apply to other variables)
Cdu Vertical deviation of the concentration
(similar subscripts apply to other variables)
c, Vertical deviation of concentration
(a similar subscript applies to other variables)
CA Crosssectional average of the concentration
(a similar subscript applies to other variables)
CT Celerity of the tidal wave
Cab Complex crossspectrum of variables a and b
Cf Residual concentration
CD Drag coefficient
CD Integral of the vertical concentration profile from the surface to a level D
Co Upper limit of the flocculation settling range; initial concentration
C, C Time average concentration of sediment
(the overbar may be omitted for convenience)
d Average depth of the pressure gage during a measurement period
D Molecular diffusion coefficient; dispersion coefficient;
sediment grain diameter
Db Mass of sediment accumulated at the bottom per unit area
e Generic measured variable
et Tidal trend in a generic measured variable
el Generic variable including wave induced and turbulent effects
e Timeaverage part of a generic measured variable
,n Average of n points of a record of a generic measured variable
e Wave coherent part of a generic measured variable
e' Turbulent part of a generic measured variable
E Momentum eddy diffusivity
Eb Mass of sediment eroded per unit bed area
En Momentum eddy diffusivity for non stratified conditions
(the same subscript applies to turbulent mass diffusivity)
E, Momentum eddy diffusivity for stratified conditions
(the same subscript applies to turbulent mass diffusivity)
f Sediment flux; frequency
/ Cutoff frequency
F Sediment flux
Fd Diffusive flux of sediment
Fe Erosional flux of sediment
Fp Depositional flux of sediment
F, Settling flux of sediment
g Acceleration of gravity
h Water depth; mean water depth
Hp Frequency response function of a filter
H. Significant wave height
k Inverse of the volume of a given suspension sample; wavenumber
ki, k2 Settling velocity parameters
k Vertical unit vector
K Turbulent mass diffusivity
K, Longitudinal mass dispersion coefficient
I Prandtl's mixing length
L MoninObukov length; complex transfer function
Lo Wavelength
M Mass transport; erosion rate constant
MT Total mass of a suspension sample
Mw Mass of a suspension sample volume of salt water
n Manning resistance coefficient; number of runs in stationarity test
nl, n2 Settling velocity parameters
N Net sediment flux
p Probability of sediment deposition; pressure
Q Transport of water; salt flux
q Fluid velocity vector
R River discharge; hydraulic radius
Re Reynolds number
Ri Richardson number
s Salinity
sd Deviation of salinity
SD Depth average of s
(the same subscript applies to other variables)
sij Measured salinity at position ij
(the same subscripts apply to other variables)
s'. Deviation of the lateral mean of the salinity
(the same subscripts apply to other variables)
s'. Deviation of the vertical mean of the salinity
(the same subscripts apply to other variables)
So Time averaged salinity
S Water surface slope
Sa Power spectrum of variable a
Sij Interaction constant the salinity at position ij
(the same notation applies to other variables)
St Schmidt number
s.. Cross sectional average salinity
(the same subscripts apply to other variables)
t Time
T Tidal period; finite time interval
Tm Modal period
u Velocity in the x direction
u, Friction velocity
UD River crosssectional mean velocity
v Velocity in the y direction
V, Erosion velocity
Vt Transport velocity
w Velocity in the z direction
W, Settling velocity of the sediment particles
W.o, W,0 Peak settling velocity, modified peak settling velocity
x Longitudinal cartesian coordinate; generic random process
Y Filtered time series of a generic random process
X Fourier transform of a generic random process x
y Transverse cartesian coordinate
z Vertical cartesian coordinate; distance from a solid boundary
zo Roughness length
f Mean height of sediment suspension
a, a' Stabilized diffusion empirical constants
a, Erosion resistance defining parameter
" Stratification parameter
/3, / Stabilized diffusion empirical constants
6 Boundary layer thickness
e Erosion rate; spectrum bandwidth parameter
t7 Tidal elevation relative to a given datum
Kr Von Karman universal constant
A Wavelength
p Dynamic viscosity
v Kinematic viscosity
p Density of the fluid
p, Grain density
Pw Water density
r Shear stress
Tr Bottom shear stress
Tcd, T* Critical shear stresses for deposition
To Shear stress at the wall
7, Bed shear strength
xvii
CHAPTER 1
INTRODUCTION
1.1 Problem Significance
Estuaries and coastal bays have traditionally offered multiple advantages for
the development of urban and industrial centers on their banks due to, among
other reasons, the existence of sheltered harbors and waste disposal sites and to the
possibility of inland navigation. The rapid development of many of those centers
has led to intensive use of estuarine and coastal waters and, as a consequence of
competing demands and negative environmental effects, to technical and ecologic
problems.
Of the problems resulting from human impact in estuarine and coastal areas,
some of the most important ones, both in economical and practical terms, are
directly related to sediment dynamics. Such aspects include:
1. Dredging of navigation channels and deepening of natural waterways;
2. Changes in natural topography and land reclamation;
3. Water quality problems due to transport of sorbed nutrients and contaminants
by fine sediment particles or to turbidity increases;
4. Shoaling or scouring of natural bottoms due to hydrodynamic changes;
5. Changes in the position of the zones of maximum turbidity and, in general,
in the patterns of sediment circulation within the estuary or bay.
This last aspect is particularly relevant, since one of the most generalized features
in estuarine environments is the existence of a turbidity maximum, a zone of high
2
suspended sediment concentration through which sediment is continuously circu
lated.
The understanding of the physical mechanisms contributing to sediment trans
port in estuaries is, consequently, fundamental in predicting any effects of anthro
pogenic activities. The nature and relative importance of estuarine transport pro
cesses has been investigated by several researchers through the analysis of velocity,
salinity and suspended sediment data. Although several procedures have been ap
plied to a variety of estuaries showing different geometries and stratification con
ditions, two transport mechanisms, tidal pumping and vertical shear, have been
generally found to be dominant.
Tidal pumping results from phase differences between crosssectional area vari
ations and average crosssectional velocities and concentrations of salt or sediment.
Transport by vertical shear effect is caused by the residual gravitational circula
tion due to salt water penetration. Both transport mechanisms depend strongly
on the vertical concentration profile and, in the case of sediment, are related to
erosion/deposition phenomena at the bed and to settling and diffusion in the water
column or, in more general terms, to the vertical sediment fluxes. Such fluxes reflect
the time lagged response of sediment to flow variations which is globally expressed
by the well known flowsediment hysteresis phenomenon. Moreover, the differences
between salt and suspended sediment behavior (resulting from negative buoyancy
and erosion/deposition) suggest the importance of the study of sedimentstratified
flows and of their differences relative to saltstratified flows; this approach, which
is supported by recent field results contradicts the assumption, implicit in some
early studies, that the dominant physical mechanisms transporting salt and sed
iment landward in an estuary or coastal bay should be the same. A description
of sediment dynamics in the vertical direction as a function of sediment stabilized
turbulent flow characteristics and of bed and sediment settling properties is, conse
quently, of fundamental importance.
1.2 Objective and Methodology
The main purpose of the present investigation was to study the effect of dif
ferent physical processes in the evolution of the vertical concentration profile in a
sedimentstratified coastal environment. In particular, the influence of the sedi
ment settling properties, stabilized diffusion parameters and bed properties on the
general features of the profile and their effects on the corresponding lag phenom
ena contributing to flowsediment hysteresis were investigated. A vertical transport
numerical model was used to generate concentration profiles and to study the ef
fect caused by the variation of settling velocities, and of stabilized diffusion and
erosion/deposition parameters. Measured field data of pressure, velocities and sus
pended sediment concentrations were obtained in a highconcentration coastal en
vironment (Hangzhou Bay, People's Republic of China). Laboratory tests of local
sediment allowed the evaluation of the pertinent physical parameters. The field data
were used to test the importance of fine sediment lagged response to flow changes
in a sediment stratified environment and compared with the model's results.
1.3 Outline of Upcoming Chapters
A summary of the different transport processes acting in estuaries and coastal
bays is presented in Chapter 2. An overview and comparison of the methods and re
sults obtained by several researchers, leading to the identification of the dominant
transport mechanisms acting to transport salt and sediment landward is also in
cluded in the chapter. In Chapter 3 a review of the physical processes described by
the vertical transport model is presented, together with the general mathematical
and numerical formulation of the problem. Some possible limitations of the model's
approach when dealing with high concentration environments are also discussed.
4
The field experiment carried out in Hangzhou Bay and the laboratory experiments
leading to the definition of the local sediment's settling and erosional properties,
the experiments' methodologies and data preprocessing methods are described in
Chapter 4. The results and discussion of model simulations and field data interpre
tation are presented in Chapter 5. Finally, Chapter 6 includes a summary and the
conclusions derived from the study.
CHAPTER 2
TRANSPORT PROCESSES IN ESTUARIES AND COASTAL BAYS
2.1 The Turbidity Maximum
2.1.1 General Aspects
One of the most generalized features in estuarine environments is the turbidity
maximum; in this zone, usually located at the limit of the salt intrusion and, gener
ally, containing more sediment than the annual estuarine supply, the concentrations
of suspended sediment are 10 to 100 times higher than landward (fluvial zone) or
seaward. Under the turbidity maximum, but more limited in size, another feature,
the mud reach, a place of continuous shoaling of fine particles in the channel, can
occur. Wellershaus (1981) presents a summary of the characteristics of 20 estuaries,
in different areas of the world, which shows the existence of a turbidity maximum
near the tip of the salt wedge in, at least, 15 of them, with a significant group also
showing a mud shoal.
The turbidity maximum can be understood as an indication of fine sediment
transport potential in an estuary; despite the opposing effects of flushing river cur
rents, mixing and dilution phenomena it contains a high percentage of the available
mobile fine sediment, corresponding to a narrow band of diameters with low settling
velocities. Moreover, its importance is fundamental in the circulation of sediment
within the estuary (since sediment is continuously circulated, in successive cycles of
deposition and reentrainment, through it) and in controlling its flow from the river
to the sea.
In a very general way, the existence of a turbidity maximum can be predicted
6
as a consequence of both net estuarine hydrodynamics and the properties of fine
sediment suspensions. Officer (1981) divides fine sediment transport in estuaries
into two forms: suspended sediment transport (with higher velocities and lower
concentrations) and near bed sediment transport (with higher concentrations but
lower velocities); fluid mud, a high concentration suspension, commonly found close
to the bed in many estuaries is included in the latter.
From the net hydrodynamics viewpoint, an estuarine gravitational circulation
pattern is commonly observed, with net flow seaward in the upper layers and land
ward in the lower layers; it follows that sediment, suspended due to mixing, and
flowing seaward, in the upper layer, in the middle to lower reaches of an estuary will,
in more quiescent areas, settle towards the bottom and be carried back landward,
to form a zone of maximum concentration. This mechanism can also explain the
narrow band of sediment diameters found in the turbidity maximum: the coarser
particles will deposit in the lower reaches of the estuary and will move only as bed
load; among the finer ones, some hardly deposit and are carried to the sea by the
mean flow, while the remaining settle down and are carried landward to form the
turbidity maximum.
The flow patterns close to the bottom can, additionally, contribute to the for
mation of this feature, since the net flow in the upper (fluvial) reaches is seaward,
while in the lower areas (due to gravitational circulation) is landward. As a conse
quence, the near bed sediment will be transported to a null point close to the tip
of the saline intrusion.
2.1.2 The Variability of the Turbidity Maximum
The previously mentioned processes may largely explain the existence of a tur
bidity maximum in microtidal (tidal range < 2 m) and mesotidal (tidal range be
tween 2 and 4 m) estuaries. Some of its features, especially in macrotidal estuaries
7
(tidal range > 4 m), cannot, however, be explained exclusively in terms of the net
transport. The peak concentration, for example, can vary by as much as one order
of magnitude, typical values being 100200 mg/l in low tidal range estuaries and
100010000 mg/l in high tidal range estuaries (Dyer, 1989). This variability can be
measured during the tidal cycle, in springneap cycles and due to seasonal varia
tions in river flow and is, consequently, associated with the erosion and deposition
of sediment.
Studies in the Gironde estuary (Allen et al., 1980, cited by Nichols and Biggs,
1985) show that the core of the turbidity maximum moves during the tidal cycle
10 to 30 km, while concentrations grow and decrease during the same period. As
expected, the turbidity maximum occupies its most landward position at high water
and its most seaward position at low water. At slack water the low flow velocities
allow deposition to occur and the dimensions of the turbidity maximum decay.
Fully developed ebb and flow currents cause erosion and transport and the growth
of the maximum; however, due to the asymmetry of the tidal currents (higher
flood than ebb currents), the net effect, after a number of tidal cycles, will be
landward transport of sediment, consequently supporting the existence of a turbidity
maximum.
The fortnightly transition from spring to neap tides also affects the turbidity
maximum, since decreasing peak currents and increasing slack durations allow in
creasing sedimentation and the decrease of the turbidity maximum; during spring
tides the turbidity maximum will be at its most landward position. The transi
tion from neap to spring tide will cause the opposite effect. However, some of the
sediment settled during the neap tides will have consolidated and will not be resus
pended, thus causing net shoaling (Allen et al., 1977 and Allen et al., 1980, cited
by Nichols and Biggs, 1985).
8
The effect of variations in river flow is apparent in the position of the turbidity
maximum, higher flows pushing it towards the mouth of the estuary. Low flow
situations allow the salt intrusion limit to migrate landward; the turbidity maxi
mum corresponding to this case will be located upstream of its normal position.
2.1.3 Transport Mechanisms in Estuaries
Several physical processes of conflicting effects can be identified in an estuary,
contributing to salt and sediment transport. Among such mechanisms can be in
cluded, according to Officer (1981), the following:
1. Gravitational, nontidal (net) circulation, and near bottom residual circulation,
together with the tidal average sediment concentration.
As mentioned before, the characteristic estuarine type of circu
lation consists, within the area of saline penetration, of a seaward
net flow in the upper layers of the water column and a net landward
flow in the lower layers. In the fluvial zone, landward of the saline
intrusion, the net transport is seaward in the whole water column.
If this pattern is combined with a tidal average sediment concentra
tion a net seaward sediment transport can be expected upstream of
the limit of the saline propagation; downstream, the transport can
be expected to be seaward in the upper layers and landward in the
lower layers. This combination of residual circulation and average
sediment concentration can, consequently, explain the dependence
of the turbidity maximum on the limit of the saline penetration,
this limit being related to variations in tidal range and runoff.
2. Flood and ebb tide variations in vertical concentrations of sediment in con
junction with a symmetric tidal current.
9
The variations in vertical sediment concentrations can be ex
plained by the directions of the vertical velocities associated with a
standing tidal wave (the dominating tidal effect in many estuaries)
which are upwards during the flood and downwards during the ebb
period. The resulting net effect of this superposition should cause
landward transport.
3. Lag effects between the vertical sediment distribution and the governing tidal
cycle in conjunction with an asymmetric tidal current.
This effect was described by Postma (1967) and can be ex
plained, with some initial simplifications by considering the follow
ing assumptions:
(a) Uniform velocities in the cross sections of the tidal channel.
(b) Symmetry of the tidal curve (sine curve) at all points.
(c) Simultaneous high and low tide along the tidal channel (stand
ing wave).
(d) Linear decrease, from the sea to the head, of the average tidal
current velocity.
(e) Constant tidal range.
If a plot of velocity versus position is made for various water masses
(fig. 2.1) it is apparent that, although the tide is symmetrical, the
curves are asymmetrical. The tangent P to the curves represents
the maximum current velocity at each point and corresponds, in
each curve, to a point attained by the water mass at half tide. Two
lag phenomena are related to these curves:
Shore
Figure 2.1: Velocities with which different water masses move with the tides, illus
trating the effects of settling lag and scour lag. From Postma, 1967.
(a) The settling lag, corresponding to the delay between the time
when the particle starts to settle and that when it reaches the
bed; during that time the particle is carried along some dis
tance.
(b) The scour lag, which is the time delay between the occurrence
of the transportation velocity Vt and the erosion velocity Ve, the
latter being higher.
If only the settling lag is considered, and using figure 2.1, it can be
seen that a particle at rest in point 1 and requiring a velocity V1 to
be suspended will be entrained in the flood flow by the water mass
coming from A; at point 3 the current has decreased below the lowest
transportation velocity and the particle begins to settle, while still
being carried landward. The particle will reach the bottom at point
5 and the water mass will come to rest at A' before returning in the
following ebb tide. The water mass coming from A' reaches point 5
11
with a velocity lower than Vi and cannot, consequently, resuspend
the particle which is only removed by water mass BB'. Following
the particle during the ebb tide, it follows that it will begin to settle
at point 7 and will reach the bottom at point 9. Consequently, a
net landward movement of the particle from points 1 to 9 happens
during a tidal cycle.
Alternatively, if only the scour lag is considered, again from
figure 2.1 (Ve > Vt), and considering Vi = V,, the particle eroded
in 1 will deposit at point 4 (Vt) and will settle to 5 instantaneously
(no settling lag). Again, only water mass BB' will be able to erode
the particle which will be carried and will begin to settle at point 8,
reaching the bottom at 9. As a consequence, a landward transport
from point 1 to point 9 will have occurred.
If both the settling lag and the scour lag act simultaneously, it is
obvious that the landward movement of a particle during the tidal
cycle will be greater. Furthermore, the effect of the timevelocity
asymmetry has to be considered, since for equal flood and ebb dis
charges the velocities have to be higher around low water, when
the cross sectional area is smaller. The slack water period is, then,
longer around high tide than around low tide, consequently allow
ing deposition of some of the sediment carried landward during the
flood; it should also be mentioned that, in general, the estuarine
cross sections change, from wide shapes close to high tide to con
fined channels close to low tide: the average depth is, accordingly,
lower at high tide and more suspended sediment will deposit. The
cumulative effect is, then, a landward shift of particles during the
tidal cycle.
12
4. Different ebb and flood durations and consequent net transport effects related
to the differing flood and ebb current magnitudes.
Another cause of asymmetry in velocitytime plots is the asym
metry of the tidal (heighttime) curve itself. In estuaries, where the
water depth is of comparable magnitude to the tidal amplitude (and
much lower than the wavelength), the tidal wave propagates as a
shallow water wave with celerity
CT = g(h +7) (2.1)
where h is the mean water depth, Y/ the local tidal elevation rela
tive to the mean water level and g the acceleration of gravity. As
a consequence the tidal crest (high water) travels faster than the
the trough (low water) and the tidal wave deforms while propagat
ing landward, the flood period decreasing while the ebb duration
increases. Flood velocities are, then, higher than ebb velocities and
produce greater erosion and higher sediment concentrations around
high water. This phenomenon, combined with the previously men
tioned longer duration of the slack period around high water, will
cause a landward net transport.
5. Hydrodynamic tidal mixing exchanges in the direction of the longitudinal
concentration gradient.
This Fickian transport process can be described by the negative
of the product of a longitudinal dispersion (tidal mixing) coefficient
by the longitudinal average concentration gradient, in the form
KI (2.2)
13
and can include, besides some of the above mentioned effects, tur
bulent transport contributions and those related to topographical
irregularities, such as tidal trapping.
The most evident transport mechanism in estuaries is, however, that associated
with seaward advection caused by the nontidal drift UA (mean flow); this flow can
be explained by the superposition of the river flow and of flows that compensate
the landward transport associated with the partial progressive wave nature of the
tide in estuaries.
2.2 Salt and Sediment Fluxes and Mass Transports in Estuaries and Coastal Bays
2.2.1 General Aspects
The variability of the turbidity maximum, as noted before, can be related to
several aspects of the estuarine hydrodynamics and sediment properties. Those
physical processes and their effects can be better understood by considering the
decomposition of the cross sectional, tidally averaged, fluxes (or mass transports)
of sediment and salt, occurring in different cross sections of an estuary. Different
types of flux and mass transport decompositions have been used by several authors,
initially with the purpose of determining salt fluxes (and dispersion coefficients)
and, later, for the calculation of suspended sediment fluxes. In the following sub
sections a brief summary of the different methods will be presented.
2.2.2 Bowden (1963)
Bowden was the first author to decompose the mass transport into different con
tributions; he examined the effect of vertical shear by assuming an estuary without
lateral variations. The instantaneous mass transport of salt through a unit width
area, perpendicular to the flow is
M= usdz = h(us)D (2.3)
where h is the depth and the subscript D indicates depth averaging. Considering
that, at any depth,
u = uo+u' (2.4)
8 = so + (2.5)
(where uo (so) are values averaged over a time interval of several minutes and u' (s')
are turbulent fluctuations) and further decomposing uo and so into depth average
values UD, SD and vertical deviations udv, sd,:
U = UD+Udv+U' (2.6)
S = SD + Sd + S (2.7)
Furthermore, UD and oD can be decomposed into tidal averages UD, SD and tidal
deviations U, S; consequently
u = UD+U+Ud,+U' (2.8)
s = SD+S+Sdv+S' (2.9)
Averaging the salt mass transport over a tidal cycle and over the depth
M 1T
M = h(us) dt =
= hUDOD + hUS + h(UdvSdv)D + T(U'SD = (2.10)
= MI+M+M3 + M4
where terms TShS, hUSD have been omitted by the author and the overbar denotes
tidal averaging. The partial mass transports have the following meaning:
15
M1 Contribution of the mean values of depth, velocity and salinity (or,
in Bowden's definition, the contribution due to advection by the mean
flow corresponding to the river discharge);
M2 Third order correlation of tidal deviations in depth and mean
velocity and salinity;
Ms Contribution of the correlation of depth deviations of velocity and
salinity (this term is associated with the vertical gravitational circula
tion);
M4 Turbulent mass transport of salt.
Bowden calculated the values of M1, MH and Ms for the Mersey Narrows on
four occasions; M4 was not computed due to the nature of the available data but
it was assumed to be small. From the computed mass transport terms, M1 was
downstream and compensated by the upstream effect of M2 and Ms, the former
being dominant. The computation of (M1 + M2 + Ms)/h showed a net upstream
salt transport which could be explained by:
1. Short period horizontal diffusion due to turbulence (term M4).
2. Non stationarity of the mean salinity.
3. Lateral effects in the transport.
Bowden and Sharaf el Din (1966) cited by Dyer (1973) confirmed, by analyzing three
stations in a cross section, the existence of lateral variations in (M1 + MZ + Ms)/h;
the net transport was, then, seaward. An additional difficulty was related to the
fact that Mi was computed using Ui = R/A where R is the river discharge and i
the mean fluvial cross sectional area; the value found by the authors for zD over the
estuarine cross section was one order of magnitude higher than the value calculated
using the river discharge, due, possibly to undetected variations in the velocity or
measurement imprecisions or, more likely, to compensation flows due to the partial
progressive nature of the tidal wave.
2.2.3 Hansen (1965)
Hansen considered the instantaneous velocity at any point of the cross section
(u) as composed by a cross sectional average (UA) and a deviation (ud):
u = UA + Ud (2.11)
A similar decomposition was applied to the salinity:
s = SA + Sd (2.12)
The instantaneous mass transport of salt through the cross section is, consequently:
M(t) = a usda = a[uASA + (UdSd)AI (2.13)
In this equation UASA corresponds to the salt flux due to the sectional means of
both the current and the salinity and has the direction of the mean current; the
second term is known as "shear effect" and accounts for the correlation between
deviations of velocity and salinity from their sectional mean values. This "shear
effect" is a consequence of density currents and shear induced currents, combined
with vertical and lateral salinity deviations. The mean longitudinal salt transport
over a tidal cycle is then
M = auAsA + a(udSd)A (2.14)
(the overbar denoting tidal cycle averaging).
Hansen further divided the average components of the velocity, UA, and of the
salinity, SA, into a tidal mean (denoted with an overbar), a tidal oscillation and a
turbulent fluctuation (denoted with a prime):
UA = UA+UA+UA (2.15)
SA = SA+SA+SA' (2.16)
17
and applied a similar subdivision to the cross sectional area:
a = + A + A' (2.17)
and to the deviation term of the instantaneous salt flux:
(UdSd)A = (UdSd)A + (U dSd)A + ( ) (2.18)
Expanding M :
( = ( + A + A')( A + UA + u)(A +SA+ s) +
+ (A + A + A')[(udSd)A + (UdSd)A + (UdSd)'A] (2.19)
and eliminating terms with uncorrelated or weakly correlated variables, the follow
ing result is obtained:
M = AUASA+AUASA+A'U'ASA+AUASA+
+At uAs + UAASA + AUASA + UAA'S'A + (2.20)
+A'usS'A + A(U1dS)A + A(UdSd)A + A'(udSd)'A
where it should be noted that terms A(dS)A and A(udSd)'A were not retained. If,
furthermore, the first three terms are grouped, by noticing that Mw = A~IA+AUA+
A'u'A is the mean transport of water through the section during the tidal cycle, and
the remaining terms corresponding to turbulent fluctuations are also included in a
term M':
M = MWwA+AUASA +i AASA+AUASA+
+;! (Us) A + A(UdSd)A + M# = (2.21)
= M1 +...+M7
The previous terms correspond to the mass transports associated with
M1 The mean river flow and salinity;
M2 Correlation of tidal period variations of the sectional mean salinity
and current;
Ms Correlation of tidal period variations of cross section and mean
salinity;
M4 Third order correlation of tidal period variations in cross section
and mean salinity and velocity;
Ms Mean shear effect;
Me Covariance of shear effect and cross section;
M7 Correlation between shorter period fluctuations of the various
quantities.
Hansen applied his decomposition scheme to data from a cross section of the Colum
bia River Estuary, assuming small changes in salt storage upestuary of the section,
and concluded that:
1. Only 70% of the measured nontidal drift VA corresponded to freshwater dis
charge; the remaining 30% were a compensation current for the inward Stokes
transport by the tidal wave AUA, the turbulent flux being assumed negligi
ble (this confirms the difference between Bowden's derived and computed M1
term);
2. Of the salt advected seaward with the mean river discharge, about 40% was
balanced by covariance between fluctuations of tidal periodicity (term 2), and
about 45% was balanced by shear effects (term 5). The remaining 15% were
attributed to short period fluctuations (term 7);
3. Terms 3,4 and 6 were small and of the order of magnitude of the uncertainty
in the evaluation of the major transport terms;
4. The importance of term 5 may be explained by the large vertical salinity
gradient coupled with systematic, even if small, density induced variations in
the velocity profile.
19
Dyer (1973) also points out that the large values obtained for AUA and A UASA
show that UA and SA are not 900 out of phase. However, since AUASA is small, the
progressive component of the tidal wave (reflected in the cross sectional variation)
cancels out the effect produced by UA and SA. It should also be noted that, since
the deviation terms were not decomposed into vertical and lateral contributions,
the relative importance of these contributions cannot be evaluated.
2.2.4 Fischer (1972)
Fischer studied the Mersey estuary, in an attempt to clarify the relative impor
tance of the mechanisms transporting dissolved constituents in partially mixed and
vertically homogeneous estuaries of the coastal plain type. The author decomposed
the observed velocity into an average value and a deviation, as before,
U = UA + Ud (2.22)
and considered also a separation into a tidal mean and the corresponding deviation:
UA = UA + UA (2.23)
Ud = Ud + Ud (2.24)
By considering a threedimensional profile, both Ui and Ud were explicitly separated
into variations in the vertical and transverse directions:
Ud = Udt + Udo (2.25)
Ud = Udt + Ud (2.26)
Udt and Udt are transverse velocity profiles, respectively the depth means of Ud and
Ud; Udv and Ud, are vertical variations from the local vertical mean.
Then
u = UA + UA + idt + Udt + dv + Ud,,
(2.27)
and, similarly, for the concentration
c = CA + CA + dt +Cdt + d + Cdv (2.28)
It should be noted that the turbulent fluctuations were neglected. A graphical
representation of the velocity decomposition is presented in figures 2.2 and 2.3.
The tidal cycle average of the longitudinal mass transport is
M = ucdadt (2.29)
Fischer assumed that, in the area decomposition a = A + A (tidal average and
deviation), A is small and, consequently, a A. Accordingly, using Hansen's result
and neglecting the turbulent term:
M = MWCA + A UACA + AACA + AUACA +a(UdCd)A = (2.30)
= MIt + ... + Mb
In this equation the last term corresponds to terms 5 and 6 of Hansen's result; if, as
noted before, a A and terms 3 and 4 are neglected due to Hansen's calculations
then
M = MWCA + A UACA + A(dCd)A (2.31)
Using the previous decomposition of Ud and cd:
M = MWCA+AUACA+
+A[(dtCdt)A + (dt'Cdv)A + (UdtCdt)A + (UdvCde)A = (2.32)
= M' +... + M'6
The transport terms represent the following processes :
M'I Mean flow of the river discharge;
M'2 Correlations of tidal variations of sectional mean velocity and
X,U
Figure 2.2: Decomposition of twodimensional profiles into components: a) velocity;
b) concentration. Adapted from Fischer, 1972.
XU
(a)
Udt,
Transverse Velocity
Profile z
Udv
x,u
Profile
(b)
Figure 2.3: Decomposition of velocity components: a) ud into steady and fluctuating
parts; b) ~d into transverse and vertical profiles. Adapted from Fischer, 1972.
concentration;
M'3 Net transverse circulation;
M'4 Net vertical circulation;
M'5 Transverse oscillatory shear;
M'6 Vertical oscillatory shear.
Fischer considered salinity equilibrium (M = 0) and neglected the second term
(tidal pumping); through theoretical and empirical results applied to data from
the Mersey estuary he evaluated the contribution of the last four terms ( i.e. the
bracketed expression in 2.32) to a dispersion coefficient defined as
Udd (2.33)
dz
by dividing them by dc/dx.
1
D = d1[/dtCdt)A + (UdVCdv)A + (UdtCdt)A + (UdCd)A] = (2.34)
dc/dx
= DI+D2+D3+D4
These partial dispersion coefficients had the values:
Dr = 430 m2/sec (transverse (net) gravitational circulation, probably
overestimated, according to the author).
Ds = 6 m2/sec (transverse oscillatory shear).
D2 = 32 m2/sec (vertical (net) gravitational circulation).
D4 = 23 m2/sec (vertical oscillatory shear).
Fischer concluded from these values (although resulting from certain hypotheses)
that the vertical gravitational circulation (D2) is not, necessarily, the most impor
tant transport mechanism in estuaries. The author mentioned that the exclusion of
the transverse gravitational circulation from previous analyses might, indeed, mean
that the most important part of the estuarine circulation was omitted. The trans
verse gravitational circulation was, actually, in this case, one order of magnitude
24
higher than the combined vertical effects.
2.2.5 Dyer (1973)
Dyer extended the previous analyses by including a tidally fluctuating cross
section in the calculation of the mass transport and by considering a turbulent
fluctuation in the several variables:
a = A+A+A' (2.35)
UA = UA+UA+U' (2.36)
SA = SA+SA+SA (2.37)
Ud = d + Ud + U (2.38)
Sd = Sd+Sd+S 8' (2.39)
Ud = Ldt + Ud (2.40)
Sd = Sdt+Sdv (2.41)
Ud = Udt +Ud (2.42)
Sd = Sdt+Sda (2.43)
Ud = Udt + U (2.44)
S' = Sdt + s' (2.45)
where the symbols have the previous meanings but the deviations are more intu
itively defined as
idt : mean deviation of the depth mean at any position from the cross
sectional mean.
udvu : mean deviation of the mean value at any depth from the depth
mean value.
Udt : tidal fluctuation of the depth mean deviation at any position.
Udv : tidal fluctuation of the vertical deviation at any depth.
25
u' : turbulent transverse deviation.
u'd : turbulent vertical deviation.
The mass transport becomes, by neglecting some terms:
M = a(us)A =
= AUASA + AUA A + ASAUA + A UASA +
+AUASA + A(udtSdt)A + A(udOdu)A + (UdtSdt)A + (2.46)
+ i(USd)A + A(UdSdA + A(U + Sdv)A + A(7uSt)A + A(tiuS )A
= MI+...+Mls
Dyer considers
M' = A(Udtdt)A + A(u'dSd,)A = M12 + M13 (2.47)
and neglects such terms as A (UAA )A, UA(A'SA)A, WA(A'u4)A, A'(UAS'A)A, A'(u'dtSt)A
and A'(u7ds )A.
Also neglected were terms of similar nature and probably small such as
(Vdt(2Adt)}A, (A, (AS))A
(^dt(A))A,((d.A
(a(t (A'))A, (du A))A
(Vdt ( A'))A, (do(7d A))A
This decomposition scheme was not applied to any specific case, due, probably, to
practical difficulties that the evaluation of some terms would present.
2.2.6 Dyer (1974)
Dyer used a simplified version of his previous mass transport decomposition by
considering :
a = A+A (2.48)
26
u = uA+U A +U+Udt+Udt d+Ud (2.49)
s = SA+SA+Sdt+Sdt+ Sdv +Sdv (2.50)
The mean mass transport during a tidal cycle is then
M = a(us)A =
= UAASA + AUASA + ASAUA + A UASA +
+AUASA + A(dtdt)A + A(Vdvodv)A +
+A (UdtSdt)A + A (Ud SdA + A(UdtSdt)A + A(UdvSd)A = (2.51)
= MI + ...+ M1
In this expression terms (idv(ASdv))A, (ddt(ASdt))A, ((UdvA)Sdv)A, ((UdtA)dt)A are
neglected. Again, terms 1 and 2 correspond to the effects of the mean flow and
salinity, and terms 3 to 5 are correlations of tidal period fluctuations. Terms 6 to 9
correspond to the last four terms in Fischer's expression 2.32 or to terms 5 and 6
in Hansen's analysis.
Dyer applied several decomposition schemes to compute the salt transport in
the Vellar estuary (salt wedge/partially mixed), in Southampton Water (partially
mixed) and in the Mersey (partially mixed with lateral variations of salinity and
velocity).
A first computation was made with Southampton Water data, using Bowden's
decomposition; if a mean velocity V = R/A (where R is the river discharge) is con
sidered, the results show large downstream salt fluxes which are difficult to explain
without a decreasing salt content in the upper part of the estuary. If, however, the
measured VD values are used, term M1 (in Bowden's decomposition) becomes much
larger than the remaining ones but shows considerable lateral variation in the cross
section. This confirmed the need for a complete cross sectional analysis.
The application of Hansen's scheme to the three estuaries was considered from
an order of magnitude viewpoint due to errors in the evaluation of uA. In most
27
cases the salt balance was maintained by terms (in Hansen's notation) 2 and 5,
while term 3 was small; terms 3 and 4 were within the probable error range, term 6
and the additional A(UdSd)A were assumed negligible and term 7 was not considered
(assumed small).
Dyer's scheme was applied to the additional terms (6 to 9) resulting from the
decomposition of terms 5 and 6 in Hansen's equation, with good agreement between
the results of the different procedures. In the Vellar the largest term was term 7 with
a smaller contribution from term 9, the lateral terms being smaller. In Southampton
Water terms 6 and 7 were of the same order of magnitude, with terms 8 and 9 one
order of magnitude less. In the Mersey, all terms, 6 to 9 were of the same order of
magnitude, terms 8 and 9 being one order of magnitude higher than in the previous
case; these results are, as noted by Dyer, of different importance to that anticipated
by Fischer (1972), since, even in the case of the two partially mixed estuaries, lateral
shear terms never dominated.
Dyer concluded that the proportion of the salt balance effected by the lateral
circulation is greater in partially mixed than in salt wedge estuaries; furthermore,
with decreasing stratification and the development of a vertically homogeneous es
tuary, lateral effects should predominate.
2.2.7 Murray and Siripong (1978)
Murray and Siripong examined the salt flux in a shallow estuary under intense
tidal mixing in conditions of low runoff, with the purpose of showing that, in a
fairly straight channel, the lateral effect is larger than the gravitational vertical
component. Their decomposition scheme was applied to data from the Rio Guayas
in Ecuador, a shallow well mixed estuary.
The diurnal inequalities of the several measured parameters (tidal height, salin
ity, speed) were negligible and enabled the use of data collected in three successive
28
tidal cycles; a quasi steadystate salinity distribution was also assumed. Murray
and Siripong called their measured flux "advective salt flux" and the remaining, non
measurable part, the dispersivee salt flux" (turbulent and high frequency fluxes).
In their decomposition scheme vertical (column) and lateral (row) effects are
characterized by the deviations of the vertical and lateral means from the cross
sectional mean; those are defined as
1 m
ui. = u (2.52)
m
(lateral means, m columns)
n
U.i = U i (2.53)
n
(vertical means, n rows)
1 1' 1 .in m
uA = ;E i. = .i = Z ui (2.54)
n m mn
(cross sectional mean).
The deviations of each mean from the cross sectional mean are denoted by a
prime:
u'.. = Ui. UA (2.55)
(deviation of each lateral mean)
u.3 = u. UA (2.56)
(deviation of each vertical mean).
The sums of both sets of deviations are zero. The observed velocities, uj, are
written as the sum of the cross sectional mean, lateral and vertical deviations and
an interaction constant Uii.
uij =UA+ u. + u'j + Uij
(2.57)
29
the interaction constant being defined as
U,. = uy (UA + u. + u,) (2.58)
and having nonzero values only when a nonlinear trend exists in either the vertical
or lateral values of uij. The sum of the interaction constants is zero along lateral and
vertical lines and in the cross section. The salinity values are decomposed similarly:
sij = SA + s'. + s'. + Sij (2.59)
Expanding uijsij, averaging over time, and over the cross section, excluding
trivial terms, the advective salt flux becomes
Qa = UASA+ (.' )A + (u.s j)A + (u'.S1 )A +
+( s:j. )A + (U )A + (UjSij)A +
+( Ui,,s. )A + ( Ui, )A + (Uiij )A = (2.60)
= Q I+...+Q1o
where the cross sectional average, denoted by the subscript A, corresponds to
1
IA = E ,iAAi (2.61)
The authors expected only terms 1, 2 and 6 to be important, corresponding, re
spectively, to salt flux by mean current, to gravitational circulation and to flux by
lateral variations in salt and velocity.
The computed values for the time averaged mass transports, i.e. .i 4~iAAi j
showed that:
1. The terms corresponding to downstream transport were 1,3,7,8 and 10; term
1 accounted for 93% of the transport, the remaining terms being negligible;
2. Terms 2,4,5,6 and 9 produced upstream transport; from those, term 2 (gravita
tional circulation) accounted for 35.5%, term 6 (lateral variations) accounted
30
for 53.1% and term 5 for 10.1%; the authors considered this last term to,
eventually, result from spurious correlations;
3. All terms containing interaction constants were negligible.
The authors support the idea that their scheme is better than Fischer's since it
evaluates both vertical and lateral effects from a single vertical and lateral profile;
nevertheless they claim that for the data used both methods produced almost the
same results.
From the above results it can be seen that the lateral gradient transport was
1.5 times higher than the one produced by gravitational circulation, confirming Fis
cher's hypothesis. However, there was also a lack of balance between downstream
and upstream fluxes, the ratio being of 4 to 1; if the steady state hypothesis is ac
cepted this would imply that about 75% of the upstream transport was carried out
by the nonmeasured dispersivee component". Consequently, the most important
upstream transport process would be the one associated with turbulence and high
frequency advective fluxes rather than the ones corresponding to lateral variations
or gravitational circulation.
2.2.8 Dyer (1978)
With the purpose of studying salt and sediment mass transports, Dyer applied
a simplified version of his decomposition scheme to data from the Gironde (well
mixed) and Thames (partiallymixed) estuaries; both estuaries show a turbidity
maximum. The following equation was used:
M = AUACA + AUACA+ ACAUA +AUACA+
+AUACA + +dtdt)A+
+a(A,cd)A + i( UdtCdt)A + A( UdvCd )A = (2.62)
= + ...+ M
31
where the overbars denote tidal cycle averaging and the subscript A denotes av
eraging over the cross section; c corresponds to a generic concentration of salt or
sediment.
Terms 1 and 2 are related to the river flow R, since
R = A~iA + AUA (2.63)
where UA is the non tidal drift and AUA (negative) corresponds to the landward
transport due to the partially progressive nature of the tidal wave (Stokes drift).
Terms 3,4 and 5 are correlations between tidal oscillations of concentration, cross
sectional area and velocity, commonly called the tidal pumping terms. Terms 6
and 7 are due to the net transverse and vertical (gravitational) circulations, while
terms 8 and 9 represent the corresponding tidal effects. By defining a dispersion
coefficient as
D A (2.64)
dt
it can be seen that terms 1 and 2 are advective terms, while terms 3 to 9 correspond
to dispersive processes. Dyer's main purpose was to extend the previous applications
of the method to the case of well mixed estuaries. In these, a well developed turbidity
maximum exists near the head of the salt intrusion, suggesting that the mechanisms
that drive salt upstream should also be effective in transporting suspended sediment.
The calculations made for three sections of the Gironde estuary showed in all
of them a total landward dispersion of salt; the total dispersion of sediment was
seaward in the two upper sections and landward but small in the third. For the
Thames a total landward dispersion was found both of salt and sediment.
In the Gironde term 4 was consistently the largest, being partly compensated
by opposite contributions from terms 3 and 5. In the case of salt transport, term 7
was the largest of the shear terms at the upper section but term 8 became dominant
32
towards the mouth; for sediment transport, terms 7 and 9 were the largest at the
upper sections but terms 6 and 8 increased in importance towards the mouth.
In the case of the Thames the largest shear contribution to salt transport was
due to the vertical mean circulation (term 7) although the transverse effects were
large as happens with other partially mixed estuaries. The tidal terms (tidal pump
ing) were of comparable magnitude to the total shear transport. The suspended
sediment transport showed a large seaward contribution from term 9, with an op
posite contribution of term 4 of, approximately, twice its magnitude.
Dyer, assuming that the results of the Gironde are typical of well mixed estuaries
in low river discharge, steady state conditions, concluded that the normal situation
would correspond to landward dispersion of suspended sediment; at the mouth the
dominant mechanism should be tidal dispersion, while upstream the importance of
the shear effects should increase.
Summarizing, and relative to partially mixed estuaries, Dyer concluded that, by
analogy with the salt transport mechanisms, the shear terms would be dominant,
with the turbidity maximum maintained by the gravitational circulation. However,
and due to lags in sediment movement, the sediment distribution may respond more
slowly than the salt distribution estuarine dynamics.
2.2.9 Rattray and Dworsky (1980)
In an attempt to assess the validity of previous results, Rattray and Dworsky
examined the effects of three sampling procedures (crosssection decomposition into
subareas) for determining flux terms. In their analysis only the transport mecha
nisms associated with timemeans of velocity and salinity were considered.
The three sampling designs are shown in figure 2.4 and have the following char
acteristics:
Design i) : uniform vertical spacing, proportional transverse spacing;
I'?
1'4 ___ r. _
14 i / C) ,
0 02 04 04 0 I2 14 1 16 18 2O20 2 02 4 06 04 8 1O 0 02 04 d II0 20 2 0Z 04 0 0 *8 I0 12 6>4 6 IP. 20 2 2
NESW Setlon (mn)
Figure 2.4: Crosssectional area decompositions: a) Design i); b) Design ii); c)
Design iii). From Rattray and Dworsky, 1980.
Design ii) : proportional vertical spacing, uniform horizontal spacing;
Design iii) : uniform vertical spacing, uniform horizontal spacing.
Denoting the average values of salinity and velocity in each sub area, AAyi as sii
and uij (i denoting the row and j the column), the averages of the rows and columns
are defined as
s : sijAAii (2.65)
A.
(average salinity in the column)
si. = 1 siiAAij (2.66)
(average salinity in the row)
1 1 1
s.. = SA SAA. = .A s.iA., (2.67)
t
(average salinity in the area) where
A.j = EAAq, (2.68)
A.. = EAAi (2.69)
A = Z AAi, = = A. (2.70)
ii i I
Several deviations are also defined:
Sdvi = Si. s.. (2.71)
(primary vertical deviation)
8dtj = s., s.. (2.72)
(primary transverse deviation)
Sdvij = Sij Sdtj S.. = Sij S. (2.73)
(secondary vertical deviation)
Sdtij = 8si Sdti S.. = Sij Si. (2.74)
(secondary transverse deviation)
Sij = SiiSi. S.j + S..=
= Sil Sdvi Sdtj 8.. (2.75)
(verticaltransverse interaction).
The authors used twoway analysis of variance techniques to test three hypothe
ses regarding the dominant pattern of variation:
a) The primary variation is transverse;
Sij = S.. + Sdti + Sdvij
(2.76)
35
b) Both variations (vertical and transverse) are of comparable importance and
their effects are essentially additive;
Sij = S.. + Sdvi + Sdtj + Sij (2.77)
c) The primary variation is vertical.
Sij = s.. + Sdui + Sdtij (2.78)
Neglecting Design iii), hypothesis b) due to limited utility, the contributions to
the net advective salt flux due to spatial correlations over the cross section be
tween velocity and salinity components can be identified for each hypothesis as
Hypothesis a)
1 1
(us).. = u..s.. + UdtjSdtA.j + UdijSdvijAAi (2.79)
A A j
Hypothesis b)
1 1 1
(us).. = u..s.. + 1 udvisdviAi. + UdtjSdtjA. + 1 UijSijAAii (2.80)
A A A i
Hypothesis c)
1 1
(us).. = u..s.. + A UdviSv,dAA. + A E Udti,SdtijAAi, (2.81)
I A
Data from a cross section in Southampton Water was used to test the three
designs which, combined with the three hypotheses a), b), c) produced nine (eight,
by neglecting iiib)) distinct determinations of advective salt flux components. This
estuary was chosen due to the existence of marked lateral variation in its depth,
suitable for generating a transverse circulation as discussed by Fischer (1972); pre
vious computations by Dyer (1974, 1978) had reported comparable contributions of
the vertical and transverse circulations to the mean salt flux at the cross section.
For design i) all three hypotheses led to the conclusion that the advective flux
was mainly due to the vertical contribution. In the case of design ii) all the hypothe
ses showed comparable contributions from the vertical and transverse components.
36
Design iii) showed comparable vertical and transverse contributions under hypoth
esis a) (predominance of transverse effect), consequently denying the assumption,
but predominance of the vertical circulation contribution with hypothesis c), which
is consistent with the basic assumption.
The authors concluded that different methods of data treatment produce dif
ferent interpretations regarding the dominating transport processes, requiring ad
ditional criteria to be applied.
By comparing the assumed dominant patterns with the obtained results, espe
cially in case iii) in which the transverse component never dominates, even in case
a), Rattray and Dworsky concluded that the most important contribution arises
from the vertical effects; this interpretation is reinforced by physical reasoning in
dicating that salinity and velocity (due to longitudinal salinity gradients) should
be vertically stratified. Accordingly, a strong dependence of both variables upon
vertical position should lead to a dominant contribution from gravitational effects;
this contribution is implicit in designs i) and iii) but partly neglected in design ii) by
attributing part of the vertical effects to a transverse contribution (the horizontal
subareas are not at the same levels). Another reason for the small contribution of
the transverse effects to the advective flux is the low correlation between transverse
fluctuations of u and s, while their vertical deviations are physically related.
2.2.10 Uncles, Elliot and Weston (1984)
Uncles et al. (1984) investigated the transverse variations in water, salt and
sediment transports in the upper reaches of the Tamar estuary which is a partially
mixed estuary; data were collected for three cross sections, each during spring and
neap tidal cycles.
The transport of water per unit width is
QD = huD
(2.82)
37
where h is the tidal depth and UD the velocity (the subscript D denotes depth
averaging). The residual water transport is
QD = hD = QE+ QS= QL (2.83)
in which the overbar denotes a tidal average and where:
QE = hUD (2.84)
(Eulerian residual transport per unit width)
Qs = HU (2.85)
(residual transport associated with the tidal wave, equivalent to the Stokes transport
for 1D flows), with U = UD UD and H = h h.
QL is the residual transport of water per unit width.
Considering
u = UD + U + u, (2.86)
8 = SD + S + 8 (2.87)
(where u, and ,, are the vertical deviations from the depth averages), the residual,
depth averaged rate of transport of salt per unit width is
M = h(us)D =
= hD D + T(US) + D (HS) + SD(HU) +
+HUS + h(u)D + H(u,,,)D = (2.88)
= QEBD + QSD + QS + h(u,S)D + H(Us)D =
= ML + MTP + M
the three contributions corresponding, respectively, to the residual flow of water, to
tidal pumping and to vertical shear. Similarly, defining the instantaneous suspended
sediment concentration c = CD + C + c,, the depth averaged sediment transport is
M' = h(uc)D = M'L + M'TP + M'v (2.89)
where
M'L = QLCD (2.90)
M'TP = QC (2.91)
M', = h(uC) + H(uIc)D (2.92)
The measured data generally confirmed some well known features of the estuarine
hydrodynamics, such as higher currents during spring tides relative to neaps and
tidal asymmetry (higher flood currents over shorter duration). During spring tides
this last feature produced more local resuspension and higher sediment concentra
tions during the flood stages than during the ebb stages, in the two upper sections
under study.
The obtained results showed higher values of the Stokes water transport in
spring tides than in neap tides; this transport was directed upestuary in the deeper
parts of the cross sections and was small and downestuary in the intertidal ar
eas. The Eulerian transport was always directed downestuary; the total residual
transport tended to be downestuary in the shallower parts of the cross sections and
smaller (up or down estuary) in the deeper areas.
Data for salt transport showed vertical shear transport to be always up estuary
and larger in the deeper parts of the cross sections, where gravitational circulation
is more developed. Tidal pumping transport was directed upestuary in the deeper
parts of the cross sections and, in some cases, reversed direction in the shallower
areas; the advective salt transport followed the pattern of the residual flow of water.
The total residual transport of salt was upestuary in the deeper parts of the cross
sections and downestuary in the shallower.
The sediment transport due to vertical shear was negligible while the advective
transport of sediment due to the residual flow of water followed the latter. Transport
during spring tides was between one and two orders of magnitude larger, relative
to neap tides, due to strong local resuspension in higher currents. Tidal pumping
39
of sediment, due to increased sediment resuspension and upestuary transport by
flood currents was, as expected, larger during spring tides and was the dominant
pattern in the two upper sections. However some downestuary pumping occurred
in the deepest part of the middle section, where sediment was, probably, more dif
ficult to erode: the authors concluded that, in the absence of local resuspension,
tidal pumping of sediment acts in the opposite way to the tidal pumping of salt.
At the most downstream section, the slight asymmetry of tidal currents allowed
downestuary sediment pumping.
2.2.11 Uncles, Elliot and Weston (1985a)
Uncles, Elliot and Weston extended their previous analysis to seven stations
in the Tamar estuary, including the central stations of the three cross sections
previously studied by the authors (Uncles et al., 1984), which were located in the
upper part of the estuary.
The decomposition scheme used by the authors was similar to the one used
in their 1984 study, the variables being normalized by i, the tidal average depth.
Consequently:
QD = hUD = h(UDE + UDS) = hUDL (2.93)
where
UDE = UD (2.94)
UDS = HU/h (2.95)
lUDL = QDI (2.96)
A freshwater induced residual current UDp was also defined by considering A, the
tidally averaged area of a cross section, and QDP the tidally averaged rate of input
of freshwater volume up estuary of the section
UDF = QDF1/A (2.97)
40
The residual transport of salt was similarly defined as
M = ML + MTP + M. (2.98)
where
M = (h(us)D)/i (2.99)
ML = UDLSD (2.100)
MTP = QS/h (2.101)
M, = h(uJ)DI/h (2.102)
The residual transport of sediment per unit width of the column is also:
M' = M'L + M'Tp + M'T (2.103)
with corresponding definitions.
Uncles, Elliot and Weston completed the results obtained for the previously
studied three sections (the first two of which, stations 1 and 2, were located up
estuary) with a study of four new stations, one located between stations 2 and 3
and the remaining located downestuary of station 3.
The authors were able to confirm the springneap variability of suspended sed
iment concentration and local resuspension of sediment in the upper reaches of the
estuary. A turbidity maximum was detected in the region of the lower salinities.
In the lower reaches of the estuary bed sediment resuspension did not occur and
suspended sediment concentrations were low.
The residual flux of salt per unit width during spring tides was directed up
estuary, with tidal pumping generally dominating. In the absence of cross sectional
data, enabling the evaluation of transverse shear effect, equilibrium conditions could
only be assumed. The data for neap tide conditions showed a less systematic be
havior; tidal pumping was still upestuary but negligible in the deeper down estuary
sections where vertical shear dominated.
41
The residual flux of suspended sediment during spring tides was dominated by
tidal pumping, directed up estuary in the two upper sections (due to tidal asym
metry) and down estuary elsewhere. This down estuary nature of the tidal pump
ing was partly due to high freshwater inputs which caused higher ebb than flood
currents. In the upper sections, where resuspension from the bed occurred and
concentrations increased from bed to surface, the vertical shear flux was up estuary
while the opposite happened in the down estuary stations where resuspension did
not occur.
During neap tides the observed fluxes of sediment were either negligible or di
rected down estuary.
2.2.12 Uncles, Elliot and Weston (1985b)
The authors completed their previous study (Uncles et al., 1984) of three cross
sections of the Tamar estuary by further investigating the transverse contributions
to salt and sediment transport. Murray and Siripong's 1978 decomposition scheme
was used, in which the instantaneous rate of transport of salt through a section is
given by:
M = a(uisij)A (2.104)
Expanding:
M = auASA + a(u.s'.)A + a(u'S')A + M* (2.105)
where
= a[(.s,)A + (.)A + ( .)A + u'Si)A +
+(UiS'.)A + (Uij')A + (UijSij)A (2.106)
Decomposing the variables into a tidal mean and a tidal oscillation
(2.107)
42
s = +S (2.108)
and averaging over a tidal cycle
M = ML + MTP + Msv + Mt + M* (2.109)
where
ML = qsA (2.110)
MTP = QS (2.111)
M,, = a(u;,.s.)A (2.112)
Mot = a(u'.jS')A (2.113)
In these, ML is the rate of transport due to the residual flow of water over the
section, MTP is the rate of transport due to tidal pumping, M,, and M,t are due,
respectively, to vertical and transverse shear dispersion and M*, corresponds to
interactions between vertical and transverse deviations.
Similar expressions are found for sediment, by substituting M' for M and c for
M' = M'L + M'TP + M',, + M',t + Ma*, (2.114)
For the computation of the residual transport of salt the authors considered a sum
of the deviation terms, corresponding to the shear processes over the cross section:
M, = M,, + Mt + M*, (2.115)
which was found to be comparable or much larger than the tidal pumping term.
The dominant component in M, was M,,; from the remaining two, only M,t had
some importance in the wider section. The authors concluded that transverse shear
dispersion becomes more important with increased width.
Similarly, in the computation of the residual transport of suspended sediment
a global shear term M, was defined. The residual transport during spring tides
43
was between one and two orders of magnitude higher than during neaps, due to
sediment resuspension. The residual rate of sediment transport due to the resid
ual discharge of water was always directed down estuary. Tidal pumping was the
dominant mechanism at spring tides. M', (and its terms) were very small in all
cases.
The author's conclusions stress the importance of the vertical gravitational cir
culation in the transport of salt which was, commonly, dominant relative to trans
verse shear and allows bidimensional (depth, axis) models to be used in partly mixed
estuaries; those models, due to the importance of the tidal pumping terms cannot
have steady state characteristics.
2.2.13 Dyer (1989)
Dyer used a condensed version of his 1978 decomposition, in the form:
M =(U a)
UASAA + AUASA + ASAUA + A UASA +
+AUASA + (udSd)A + (UdSd)A (2.116)
Term 1 is the Eulerian flow and term 2 corresponds to inward transport due to the
Stokes effect. Terms 3 to 5 are tidal pumping terms, while terms 6 and 7 correspond
to vertical and transverse oscillatory shears.
Dyer complemented his 1978 data relative to the Gironde and Thames estuaries
with Schubel's data from the Susquehana estuary, corresponding to two stations
occupied in different conditions of river flow and stratification; one of the stations
was landward of the turbidity maximum while the second was seaward. Despite
the differences in stratification conditions, the flux terms corresponding to terms 6
(d d)A and 7 (UdSA were comparable and small, while tidal pumping (UASA)
had higher values.
44
Dyer summarizes his results by considering that tidal pumping is the dominant
process supporting the turbidity maximum in several estuaries. This mechanism
balances the net outflow of sediment due to the combination of river flow and com
pensation flows for the Stokes drift.
2.2.14 Summary
In estuarine environments a long term quasiequilibrium between seaward and
landward mass transport of dissolved and suspended constituents is generally found.
Tidallyaveraged salinity and suspended sediment concentration at a given point,
for example, usually show slow variations over time scales of days or even weeks.
This quasistationarity over time scales higher than a tidal period and the existence
of turbidity maxima have led to the investigation of the physical mechanisms that,
within the tidal cycle, transport salt and sediment landward, balancing the flushing
effect of the mean flow.
The basic tool used by several authors to investigate the relative importance
of tidally averaged longitudinal mass transport phenomena in estuaries has been
the decomposition of the relevant variables (velocity, concentration, crosssectional
area) into different components, either in terms of average values or values related
to spatial or time variations. Through tidal cycle and crosssectional averaging pro
cedures and elimination of uncorrelated terms, the different processes contributing
to transport salt or sediment are identified and evaluated through field measure
ments. Moreover, the hypothesis that the physical processes transporting salt and
sediment are of similar importance has often been implicitly assumed.
A source of error in early studies proved to be the difference between the cross
sectional and tidally averaged velocity and the velocity that results from dividing
the fresh water flow discharge by the crosssectional area. The former velocity, in
fact, accounts not only for the latter but also for seaward flows compensating the
45
landward Stokes effect (Hansen, 1965). A second source of uncertainty derives from
the fact that the turbulent components of the mass transports were never evaluated;
this, for example, makes an assessment of the results obtained by Bowden (1963)
and Murray and Siripong (1978) difficult. A third source of uncertainty results from
the fact that most authors do not explicitly describe their sampling procedure in
the crosssection, which, if not correctly done, as shown by Rattray and Dworsky
(1980), can lead to the overestimation of certain components. Finally, the fact that
some terms are omitted in some of the mass transport expansions, without explicit
evaluation of their magnitudes, provides an additional source of uncertainty.
The early results by Bowden (1963) showed the importance of the vertical grav
itational circulation and tidal pumping terms, while confirming the existence of
crosssectional variations.
Hansen's study (Hansen, 1965) confirmed the dominance of tidal pumping and
of the mean shear effect, although the latter was not decomposed into its lateral
and vertical contributions.
Fischer (1972) used Hansen's decomposition scheme and through the use of the
oretical and empirical results previously obtained by other authors concluded that
the net transverse circulation might be the most important transport mechanism in
estuaries. It should be noted, however, that the tidal pumping term was neglected
in this analysis.
Dyer (1974) applied several of the previous decomposition schemesto three
estuaries, confirming the dominance of tidal pumping and shear effect terms. De
composition of the shear terms into vertical and transversal components showed
vertical shear to prevail in the case of a salt wedge/partially mixed estuary and
similar magnitudes for transverse and vertical shear in the case of partially mixed
estuaries. Contrary to what had been anticipated by Fischer, lateral shear terms
did not dominate. It should also be noted that Dyer's decomposition scheme cor
46
responds to the hypothesis that the primary deviation is transverse (Rattray and
Dworsky, 1980); information on the crosssectional area decomposition should, con
sequently, allow a better assessment of the results.
Murray and Siripong (1978) applied a different decomposition scheme (assuming
vertical and transverse variations of similar magnitude) to a shallow, well mixed
estuary. This scheme did not allow for an explicit evaluation of tidal pumping.
The obtained results showed the transverse shear and the vertical gravitational
circulation to be of the same order of magnitude, the former being higher. The
lack of balance between the measured landward and seaward transports, however,
makes the significance of these results difficult to assess.
Dyer (1978) confirmed the dominance of tidal pumping and vertical shear terms
in salt and sediment transport, although complete equivalence between the relative
magnitudes of the transport processes in the cases of salt and sediment was not
observed. A similar analysis by Dyer (1989) showed dominance of tidal pumping
over shear terms in the case of sediment transport.
Rattray and Dworsky (1980) evaluated the effects produced by different meth
ods of data treatment if either the transverse or the vertical deviations of the ve
locity and concentration are assumed to dominate and in case both variations are
of comparable importance. The investigators concluded that the way in which the
crosssection is divided may lead to overestimation of certain physical processes;
in fact, a part of the transverse shear terms evaluated in previous studies might
correspond to vertical effects.
Uncles et al. (1984, 1985a, 1985b) confirmed the importance of tidal pumping
and vertical shear in the transport of salt and sediment. The dominant processes
were not, however, entirely coincident; tidal pumping and vertical shear were dom
inant in the cases of sediment and salt, respectively. The difference between the
relative importance of the main transport mechanisms in the cases of salt and sedi
47
ment contradicts the assumption by previous investigators (Dyer, 1978, for example)
of complete analogy between the causes for landward transport of both constituents.
A further result obtained by Uncles et al. (1984, 1985a) relates the magnitude and
direction of the sediment tidal pumping term to the erodability of the bed at the
crosssections.
The main transport processes acting to transport sediment landward seem, con
sequently, to be related to tidal pumping effects and, probably to a lesser degree,
to the vertical gravitational circulation. Tidal pumping is a consequence of phase
differences between tidal variations of the crosssectional area and average cross
sectional velocity and sediment concentration. Such phase differences between the
flow and the sediment concentrations are due to lags in sediment response to hydro
dynamic changes and can be related to erosionresuspension and settlingdeposition
phenomena during the tidal cycle. Sediment transport due to the vertical grav
itational circulation is also strongly dependent on settling, diffusion and bottom
conditions since these define the vertical concentration profile. A description of the
physical phenomena contributing to sediment dynamics in the vertical direction is
presented in Chapter 3.
CHAPTER 3
SOME ASPECTS OF FINE SEDIMENT DYNAMICS
3.1 The Transport Equation
Suspended sediment dynamics in a water body are described by the advection
diffusion equation which is a particular case of a mass conservation equation. The
equation can be derived, in cartesian coordinates (x longitudinal, y lateral and
z vertical, positive upwards from the water surface) by considering a differential
control volume and equating the time rate of sediment accumulation inside the
volume to the net flux of sediment through its boundaries. This continuity principle
is a consequence of the assumption that, despite the continuous process of sediment
floc formation and destruction within the control volume, the overall sediment size
distribution remains constant and there is no net generation or destruction of the
flocs of any particular size (Mehta, 1973). As a result, no production or decay terms
need to be added to the equation and suspended sediment can be assumed to be
conservative. The equation is then
ac
S= V N (3.1)
where C is the instantaneous sediment concentration (mass of sediment/volume of
suspension) and Ni is the net sediment flux vector. This vector can be decomposed
into an advective component (NA), a molecular diffusion component (Ni), and a
settling component (Ns), since the sediment is not neutrally buoyant
NA = QC (3.2)
ND = DmVC (3.3)
Ns = WCk (3.4)
49
where q is the fluid velocity vector, Dm the Fickian molecular diffusion coefficient
(assumed isotropic), W, the terminal settling velocity of the sediment particles or
flocs and k the vertical unit vector (since several sediment sizes are present, the
settling flux should be interpreted as the sum of the partial fluxes corresponding to
the different sizes). This leads to
aC
at = V (qC DmVC W,Ck) (3.5)
Due to the turbulent characteristics of natural flows and the impossibility of track
ing individual particle movements, flow velocity components and concentrations are
usually decomposed into time averaged parts (over a longer period than the tur
bulent time scales involved), denoted by an overbar, and fluctuating components,
denoted by a prime; for example
u = u + u' (3.6)
C = + C' (3.7)
where u is the z component of the velocity vector in the cartesian coordinate system.
Inserting these terms into equation 3.5, expanding, averaging over time (using the
same time scale as before) and using the fluid continuity equation, the following
equation is obtained:
t + q VC = Dm C + V (W,k) + V (C') (3.8)
which can also be written as
aC aC auC aC a2 a2 aOC a
a +  + Dm( + + ) +(W ) 
a (U z (* r (3.9)
ax ay az
The last three terms in equation 3.9 correspond to gradients of turbulent diffusion
fluxes and are commonly modeled, by analogy with the molecular diffusion case as
a 'x (3a10)
ax aTC = (K ) (3.10)
50
= 4(K^~) (3.11)
ay '(7 ay)
a = K, aC (3.12)
where K,, K, and K, are the turbulent mass diffusivities in the x, y, z directions,
respectively. Turbulent diffusion coefficients are, however, much larger than the
molecular diffusivities and the terms corresponding to the latter phenomenon can
be neglected in equation 3.9 (Harleman, 1988). In order to investigate the vertical
structure of sediment concentration a scaling analysis of the remaining terms in
equation 3.9 is required. Ross (1988) shows that for estuarine flows, in which the
advective vertical velocity w is negligibly small and the advective travel time trough
the estuary is greater than the characteristic time for sediment settling, equation
3.9 reduces to
ac a ac
S (WC + K, ) (3.13)
at az az
where the overbars (denoting time average values) have been omitted. The scaling
analysis has, consequently, allowed horizontal and vertical advective fluxes to be
neglected relative to vertical settling and diffusive fluxes. Equation 3.13 is valid in
the water column (between the bed and the water surface) and requires appropriate
boundary conditions. These are (Ross, 1988):
1. Bed Boundary Condition
At the bed, z = Zb, a bed flux term, Fb, (mass of sediment per unit bed
area per unit time) must be defined, corresponding to a source or sink for
the suspended sediment in conditions, respectively, of erosion or deposition.
Consequently, in the z direction, and at the bed:
N(Zb,t) = Fb = Fe Fp (3.14)
where F, and Fp are the erosion and deposition fluxes, respectively.
2. Surface boundary condition
At the water surface, z = 0, a no flux condition is necessary, correspond
ing to a zero total flux, the diffusion flux always balancing the settling flux.
Consequently:
N(O,t) = W.Clo + K, a o = 0 (3.15)
In the water column, away from the boundaries, equation 3.13 applies; Ross (1988),
solved this equation numerically, with the above boundary conditions, using a finite
difference scheme. In the following paragraphs some of the physical phenomena
described by the model will be presented.
3.2 Settling
3.2.1 General Aspects
The settling flux of cohesive sediment in turbulent flows is strongly dependent
on the sediment concentration; this is due to the fact that the settling velocity itself
depends on the concentration for a wide range of values. Moreover, the settling
velocity of cohesive materials is a function of the suspension and not exclusively
of the sediment (Mehta, 1988). This aspect can be understood if the causes for
aggregation of cohesive particles are considered.
Aggregation occurs as a consequence of net attractive forces between parti
cles, brought close enough by Brownian motion, differential settling and current
shear. Although the relative importance of collision frequency due to the above
mechanisms depends on the particle diameter, current shear seems to be the most
important factor contributing to aggregation, with the exception of slack water peri
ods when differential settling becomes dominant (Mehta, 1988). Aggregates or flocs
are formed of individual particles and can, themselves, form aggregates of higher
orders. They differ from primary particles in four main aspects:
52
1. their size is larger than that of individual particles;
2. their density is less than that of the particles due to interstitial water;
3. their shape is more spherical than the platelike shape of the primary particles,
which corresponds to reduced drag;
4. they are extremely weak, tending to break up.
From the above factors, the increase in fall diameter and the reduction of the drag
are more significant than the decrease in density and the settling velocities of the
flocs are higher than those of individual particles. The magnitude of the aggregate
diameter and settling velocity are, moreover, only slightly dependent on the primary
particle diameter. The dependence of the settling velocity of cohesive sediment par
ticles (primary particles or flocs) on the concentration, neglecting secondary effects
such as those of temperature and salinity falls within three cases (figure 3.1).
3.2.2 Free Settling
Free settling occurs for low concentrations, typically lower than 100 to 300 mg/i
(Mehta, 1988). In this range the particles settle freely, without mutual interference;
their terminal settling velocity is a result of the force balance between drag and net
negative buoyancy. In the viscous range (Re, < 1) the drag coefficient is
24
CD =  (3.16)
Re,
(where Re, = WsD/v) and the terminal settling velocity is (Vanoni, 1975)
W. D( ))g (3.17)
18 g
where D is the grain diameter, p, and pw are the grain and fluid densities, g is the
acceleration of gravity and p the fluid dynamic viscosity. Fine estuarial sediment
in dispersed or quiescent conditions typically falls within these conditions although
53
the shape of the particles requires the use of an effective particle diameter (Ross,
1988).
3.2.3 Flocculation Settling
When the suspension concentration becomes higher than the free settling limit,
increased concentration and interparticle collision frequency cause an increase in
aggregation and higher settling velocities. The general expression for the settling
velocities in the flocculation range is
W, = kiC"' (3.18)
The coefficients in equation 3.18 may be determined in laboratory settling columns
or in field tests; values determined by the latter procedure have been found to
be higher by as much as an order of magnitude than those corresponding to the
former, using the same sediment (Owen, 1971). This is due to the effect of local
flow shearing rate on kl.
3.2.4 Hindered Settling
For concentrations higher than a value Co of about 2 to 5 g/l the settling
velocity decreases with the concentration. This is the result of hindered settling, a
phenomenon in which aggregates become so closely packed that the fluid is forced
to flow between them, through increasingly smaller pores. The general expression
for the settling velocity in the hindered settling range is
W. = W,0[1 (C Co)]"
SWo(1 k2C)" (3.19)
where Wo is the settling velocity that corresponds to Co, i.e., the maximum velocity
of the flocculation range.
C1 C2 C3 C4;
LOG (CONCENTRATION)
I
Negligible
Settling
Figure 3.1: A general description of settling velocity and settling flux variation with
suspension concentration of fine grained sediment (ni = 1.33, n2 = 5.0).
3.2.5 Settling Flux
The behavior of the settling flux, itself, can be seen in figure 3.1 .The settling
flux grows with the concentration within the flocculation settling range and for the
lower concentrations of the hindered settling range. However, within the hindered
settling zone the settling flux reaches a maximum, and decreases for higher concen
trations. This is known as hindered flux and requires special consideration in the
definition of numerical modeling schemes. The settling flux shows, consequently, a
nonlinear dependence on the concentration.
3.3 Diffusion
3.3.1 General Aspects
The vertical turbulent flux, expressed in equation 3.12 by the product of the
vertical velocity and concentration deviations is the counterbalancing effect in the
water column of the gravitational settling flux. It can be expressed by means of the
Fickian analogy, as in equation 3.13 or by the use of more complex closure models.
Similarly, turbulent fluxes of momentum can be defined, using the Fickian analogy
by
i = E.j (3.20)
where Eij are the components of an eddy viscosity tensor; if turbulence is assumed to
be isotropic Ej = E. The ratio of the turbulent diffusion coefficients for momentum
and mass (in the same direction) is called the Schmidt number (Harleman, 1988)
E
St = (3.21)
and can be taken as unity for particles in the Stokes range of settling, corresponding
to sediment smaller than about 100 microns (Teeter, 1986), which includes typical
cohesive sediment sizes. This fact allows the use of results, obtained theoretically
for the vertical distribution of momentum diffusivity in the determination of mass
56
diffusivities in the water column, under conditions of nonstratified flow. For this
purpose a description of the velocity profile in natural flows is necessary.
It is known that close to a wall or flow bed the flow velocity is reduced due
to friction between the flowing water and the solid boundary. The layer in which
velocity reduction, relative to the free stream velocity occurs is called the bound
ary layer, and, in the case of shallow water flows, can fill the whole water depth
(Dyer,1986). Within the boundary layer, of thickness 6 two main zones can be
defined (Dyer, 1986):
1. An inner zone, close to the wall, with thickness 0.1 to 0.26 in which the shear
stress can be assumed to be constant and the flow is not affected by external
conditions; this zone can extend to the bed, if a rough bed exists, but can also
be limited below by a buffer layer that separates it from a viscous sublayer
(contiguous to the solid boundary) in the case of a smooth bed.
2. An outer layer which includes the remaining 80 to 90 per cent of the boundary
layer and in which the flow is affected by external conditions, particularly by
the free stream velocity.
Furthermore, in order to define velocity profiles which are adequate to describe
turbulent flows, several differences have to be considered relative to viscous flows.
Contrary to what happens in the case of molecular diffusion, the turbulent diffusiv
ities are a function of the flow characteristics rather than of the fluid. Defining the
turbulent shear stress in an unidirectional flow in the x direction as
du
7 = pE (3.22)
and noting that viscous forces in turbulent flow are approximately proportional to
the square of the mean velocity rather than to its first power, as happens in the
case of laminar flow (Schlichtling, 1979), it follows that E should be proportional
to the mean velocity. Prandtl introduced the concept of a mixing length which is
57
a local function of the flow, measuring the length scale of turbulence. This length
can be used as an indicator of the eddy size or of the average distance traveled by
fluid parcels in their random movements, a similar concept to that of the "mean
free path" in the kinetic theory of gases (Henderson, 1966). The eddy diffusivity is
then defined through theoretical reasoning and experimental evidence as
E = 12 1 (3.23)
(since the sign of r must change with du/dz) and the shear stress as
r =p I2Izz (3.24)
In the vicinity of a solid boundary it was assumed by Prandtl that the mixing length
is directly proportional to the distance from the boundary, z
I = kz (3.25)
where the constant of proportionality k is, in fact, the Von Karman universal con
stant K equal to 0.4 for homogeneous clear fluids (rc has been reported to decrease
with increasing suspended solids concentration (Vanoni, 1975)); if, furthermore, it
is assumed that in the same neighborhood the shear stress is constant (equal to the
shear stress at the wall o0)
To = pKz2(Z)2 (3.26)
and a friction velocity is defined as
u, = (3.27)
it is found that
d u.
dz = z (3.28)
dz =z
Integration of equation 3.28 produces a logarithmic velocity distribution law, of the
form
u .9)
u =I z + const (3.29)
/C
58
This velocity law, applied to the inner zone of the boundary layer is known as the
Prandtl law (Dyer, 1986)
n() (3.30)
U* C zo
where zo is the roughness length, a distance from the bottom at which the mean
velocity becomes zero. In the outer layer equation 3.29 can tentatively be used,
despite the fact that it was derived for a narrow zone, close to the solid boundary,
of constant shear stress. Given that, at the surface z = H and u = ax, integration
produces an universal velocitydefect law (Schlichtling, 1979)
U max 1 Z
S In() (3.31)
U* /K 1
which describes the velocity distribution very accurately, even at substantial dis
tances from the boundary where r is different from ro (Henderson, 1966). Since
in both cases equation 3.28 applies, as noted before, assuming a linear shear stress
variation in the water column, in the form
Hz
r = ro( H (3.32)
and using equation 3.26, an expression for the momentum diffusivity is obtained as
E = cu,(H z) (3.33)
H
This expression can also be used for the mass diffusivity, since St = 1 was assumed.
3.3.2 Stabilized Diffusion
In the presence of density stratification which can be caused by salinity, tem
perature or suspended sediment, the vertical diffusion of mass and momentum is
affected, since a stable density gradient will tend to damp turbulence, strongly in
hibiting mixing; in the limit, a large density gradient could lead to the formation of
a stable interface with little turbulent exchange taking place across it. Furthermore,
59
mass and momentum diffusivities are not affected in the same manner, the latter
having larger values (Oduyemi, 1986). A measure of the relative importance of the
stabilizing gravitational forces to the destabilizing shear induced turbulence forces
is the gradient Richardson number
Ri = pdz (3.34)
\dz)
which takes positive values in the case of stable stratification and is negative in un
stable stratification cases; neutral conditions have been found to occur for 0 < Ri <
0.03. If the stratification becomes significant (Ri > 0.25) turbulence is damped out
and the flow becomes essentially laminar (Dyer, 1986).
Density gradients also affect the velocity profile. Stratification changes the
turbulent mixing length which becomes dependent not only on the distance from
the wall but also on the length scale associated with stratification, L, known as the
MoninObukov length. The gradient of the average velocity becomes
dt = I (3.35)
dz vnz
where D is a function of z/L and
L = us (3.36)
Kgw'p'
(p is the depth averaged density, w' and p' are the fluctuating parts of the vertical
velocity and of the density, respectively, and w'p' is the timeaverage value of their
product). The ratio z/L depends on Ri, being zero for Ri = 0 and increasing
rapidly as Ri approaches 0.25. The function 4 is defined as (Dyer, 1986)
az
S= (1 + ) (3.37)
which, substituting in equation 3.35 and for small zo leads to
iiz z
S (In z + ) (3.38)
u. zo L
60
where a has a value in the range from 4.7 to 5.2 (Dyer, 1986). Equation 3.38 is
valid over small values of Ri and is similar to the Prandtl equation for the neutral
case with an added correction term. Since z/L depends on Ri, 4 is often written
as
= (1 + #'Ri)a' (3.39)
where a' f' are empirical positive constants and, consequently
= (1 + f'Ri)" (3.40)
dz x z
If equations 3.26 and 3.32 are again used, an expression for the momentum
diffusivity in stratified conditions is obtained as
u(H z) (3.41)
E, = (3.41)
(1 + I'Ri)a'
or
E= (1 + Ri)"' (3.42)
En
where the subscripts s and n indicate, respectively, a stratified and a neutral sit
uation. Since the Schmidt number is no longer unity (Oduyemi, 1986) a similar
relationship for the mass diffusivity will require different empirical constants a and
Kn
The previous equations follow the general Munk and Anderson (1948) expression,
relating diffusivities in stratified and neutral conditions; the empirical coefficients
in equations 3.42 and 3.43, however, show a certain degree of variation, as reported
in different studies (Ross, 1988).
3.3.3 Diffusion Flux
Another important effect of gravitational stabilization is the nonlinearity be
tween the diffusive flux, Fd, and the vertical concentration gradient ac/az. This
61
is a result of the inverse dependence of the diffusion coefficient on a power a of the
Richardson number, itself dependent on ap/az and consequently on aC/az, while
the diffusive flux is also directly dependent on aC/az A plot of Fd vs. ac/az
(Ross, 88) is presented in figure 3.2 It can be seen that IFdI reaches a maximum
for a given value of laC/azl, say C, For low gravitational stability laC/azl < C,,
perturbations of the concentration profile are smoothed by diffusion, since the dif
fusive flux increases with laC/az ; the opposite effect occurs when laC/az > Z
(high gravitational stability), since IFdI decreases with increasing laC/azi, mass
accumulates and perturbations increase. However, for very high values of laC/azl,
the gradient of IFdI with it tends to zero and perturbations will stabilize, forming
steplike structures known as lutoclines. Consequently, an effect of the nonlinearity
between Fd and aC/az is the promotion of growth and stability of lutoclines, layers
in which (analogously to what happens in the case of haloclines and thermoclines)
steep concentration gradients and local minima in mixing and vertical diffusion oc
cur. Further properties of such pycnocline layers include significant shear production
and interfacial instability.
The above analysis, based on the hypothesis of similarity between lutocline
phenomena and other pycnoclines should, however, be reviewed in the light of a
fundamental difference: sediment is negatively buoyant, relatively to the surround
ing fluid and the effects of settling should be added to the purely diffusive type of
analysis. In this case, consequently, a settling flux counteracting the effect of dif
fusion in the water column, will contribute to enhance lutocline stability relatively
to other types of pycnoclines; the net vertical flux (positive upwards) will still show
minimum values at the levels of the lutoclines.
Examples of lutocline features (several of which may happen in a water column)
are shown in figure 3.3 which shows a typical concentration profile, as observed in
high sediment load environments. This figure shows the suspension layers which
Figure 3.2: Diffusion flux as a function of aC/az for #f = 4.17 and a = 2.0. Adapted
from Ross, 1988.
CONCENTRATION
Suspension
(No Effective Stress)
Bed
(Measurable
Effective Stress)
Newtonian
(Free Settling)
Increasingly
NonNewtonial
with Increasing
Concentration
(Flocculation Settling)
Highly NonNewtonian
(Hindered Settling)
Two Phased
Skeletal Framework
(Consolidation)
Figure 3.3: Concentration profile definitions.
n
64
correspond to the settling ranges described in section 3.2 and the bed layers (with
concentrations of 200 g/l or higher) in which the particle framework supports a
measurable effective stress. In particular, the hindered settling layer includes a pri
mary lutocline shear layer, a mobile hyperpycnal layer (with both horizontal and
vertical particle movement and low rate of dewatering) and the stationary mud layer
in which horizontal particle movement no longer occurs. The two bed layers can be
distinguished by their degree of consolidation and deformation: while the deform
ing cohesive bed is only partially consolidated and deforms viscoelastically under
oscillatory forcing by waves, the stationary bed is well consolidated and shows little
deformation.
3.4 Fluxes at the Bed
3.4.1 General Aspects
The bed fluxes are the deposition flux, F,, and the erosion flux, F,, which sat
isfy the bed boundary condition. In the case of cohesive sediment, deposition and
erosion of sediment at the bed can be treated as nonsimultaneous phenomena, oc
curring over different ranges of bed shear stress values. A short description of the
physical phenomena associated with the bed fluxes is given below.
3.4.2 Deposition Flux
The time rate of sediment deposition per unit bed area (or deposition flux, F,)
can be defined as
dm
Fp = = pWC (3.44)
where p is the probability of sediment deposition, W, the settling velocity and C
the nearbed sediment concentration. The probability of deposition, due to Krone,
is defined as
= (1 ) (3.45)
Ted
where rb and rcd are the bottom shear stress and a critical shear stress for deposition,
respectively. This concept reflects the fact that the deposition of flocs is controlled
by near bed turbulence or, more specifically, by the rate of shearing au/az at z = Zb.
For a floc to stick to the bed it must be strong enough to withstand the near bed
shear stress; weaker flocs are disrupted and resuspended. The deposition process
is, then, also an effective sorting mechanism, controlling the size distribution of the
suspended flocs.
For nonuniform sediment, experimental data (Mehta and Lott, 1987) show that
complete deposition will occur if the bed shear stress, rb, drops below a critical value
Tei (see figure 3.4). For increasing bed shear stress a residual concentration Cf (less
than the initial concentration Co) will remain, as long as rT is less than another
critical value rTM (for Trc < Tb < TcM the ratio C1/Co is a function of rb only and
increases with it). For shear stresses higher than TeM no deposition takes place
and the initial concentration remains; TcM is, consequently, an upper value of shear
stress for deposition. The solution of 3.44 corresponding to the case of nonuniform
sediment is then (Mehta and Lott, 1987)
C ,n,() Trb W mi" W:
= W exp{[1 ( )"] t} (3.46)
CO i=l
where
In(iM)
Tm = In (3.47)
and h is the flow depth, 4(Wi) is the frequency distribution of W} (settling velocity
class), W,'" and Wa" being the extreme values that define the range of the settling
velocities; in the case of uniform sediment, for which r1 = TcM = Ted
C Trb W,
= exp[(1 )tj (3.48)
Co Ted h
Co
Tb CM; Cf = CO
TIME
Figure 3.4: Typical time concentration relationship during deposition. From Mehta
and Lott, 1987.
Uniform
Sediment
NonUniform
Sediment
TCM
BED SHEAR STRESS,Tb
Figure 3.5: Dependence of Cf
Mehta and Lott, 1987.
on rb for uniform and non uniform sediments. From
Co
67
Furthermore the probability of deposition will, for the case of nonuniform sediment,
be expressed as
Pi = 1 (3.49)
rd
which corresponds to a "settling by class" concept: for a given fraction i of the
suspended sediment, if r7 is greater than re (but less than TcM ) no sediment will
deposit while, eventually, another class, j, of coarser sediment may have rj >> r
and virtually no sediment will remain in suspension. The size distribution of the
suspended sediment is, consequently, not only a function of the bottom shear stress
but also of the initial size distribution and settling properties of each sediment class.
3.4.3 Erosion Flux
Erosion of cohesive sediment has generally been observed to occur in one of
two modes: particle by particle and mass erosion. The former mode corresponds
to the case in which particles separate from the bed in an individual basis, as a
result of hydrodynamic erosional forces exceeding cohesive bonding, frictional and
gravitational forces; in the latter case portions of the bed become unstable and
large masses of sediment are resuspended, bed failure occurring below the surface.
Particle by particle erosion is, however, the most common erosional mechanism
in estuaries; under the action of bottom shear stresses higher than the bed shear
strength, removal of particles and decrease in bed elevation (scour) will proceed
until a bed layer of higher strength, equal to the applied stress, is found. This
increase in bed shear strength with depth is due to changes in the floc structure after
deposition, during consolidation and gelling. In general and for uniform sediment,
the bed shear stresses which are necessary to keep sediment in suspension are much
lower than those necessary to erode it; consequently the critical shear stress for
erosion is higher than the previously defined rca.
The time rate of increase of suspended sediment mass per unit area of the bed
(rate of erosion) can be described as
dm
Fe =f( T, 7 r, 7 al2 ... an) (3.50)
where rb is the time mean bottom shear stress, r, the bed shear strength and the 0a
are other erosion resistance defining parameters. The difference Tb r, the excess
shear stress, is one of the common features of some of the existing formulas for the
erosion flux, such as the proposed by Kandiah (1974) for uniform beds
F, = a,( ') (3.51)
where al is an empirical erosion parameter defined as
a1 = Felr=2r. (3.52)
For nonuniform beds, Parchure and Mehta (1985) proposed
F, = a, exp{a3[7 r(z)1 } (3.53)
where a2 is known as the floc erosion rate
a2 = FIjr=r, (3.54)
and as is a factor which is inversely proportional to the absolute temperature. It
should be noted that rb is defined as a mean value and, consequently, some sediment
particles will still be eroded when it equals the bed shear strength. This is taken
into account by equation 3.53 but not by equation 3.51.
3.5 The Numerical Model
3.5.1 General Aspects
The vertical transport model developed by Ross (Ross, 1988) solves equa
tion 3.13 through a finite difference scheme, using boundary conditions 3.14 and 3.15.
If applied to the simulation of estuarine tidal conditions the following data are re
quired:
1. Tidal hydrodynamics data:
Tidal period, tidally averaged values of both the surface elevation and the
depth averaged flow velocity, tidal amplitudes and time lags (relative to the
time origin of computation) of both variables.
2. Sediment parameters:
These include the concentrations defining the limits of the free settling range,
parameters kI and nl in equation 3.18 and, for each sediment fraction, pa
rameters Wo, k2 and n2 (from equation 3.19), concentration Co defining the
lower limit of the hindered settling range and the percentage by weight of the
sediment fraction in the total sample.
3. Stabilized diffusion parameters:
Empirical parameters a and / (equation 3.43).
4. Bed characteristics:
These include the upper limit of the bed shear stress range for deposition
TcM = Ted (see equations 3.44, 3.45 and 3.48), the critical shear stress for
erosion r, and the erosion empirical constant a (see equation 3.66).
5. Initial concentration profile:
The values of sediment concentration at each grid point are required.
3.5.2 Numerical Procedure
For each time step a hydrodynamics routine computes the surface elevation H
and the depth averaged velocity i using the input tidal data; the bed average shear
stress rb is computed using the relation
rb = pgRS
(3.55)
70
where S is the water surface slope, and Manning's formula
U= R Si (3.56)
n
which, considering R H leads to
2
rb = pgrn 2 (3.57)
The main limitations of this approach are related to the fact that Manning's formula
was originally obtained from openchannel, hard bed, nonstratified flow data. It
should be noticed that the DarcyWeisbach friction factor (which can be related to
Manning's n) changes with concentration and that the shear stress increases in the
case of flows with suspended sediment (Ippen, 1971).
Within the water column, at the elevations corresponding to grid points, i, below
the water surface, the neutral mass diffusivities are calculated through equation
3.33. Mass diffusivities corresponding to a stratified case are then obtained through
equation 3.43. The diffusion fluxes are computed for each sediment class j through
a forward difference scheme ( i increasing downwards)
fd(i, j) = K(i) C(i + 1,j) C(i,j)(3.58)
Az
and the diffusion flux gradient is computed through backward differencing
dfd(i, j) fd(,j) fd U i ) (5
" ~ hz (3.59)
dz Az
The settling fluxes are computed at each grid point, i, by
fMi,j) = W,(ij)C(i,j) (3.60)
after W,(i,j) = W,(i,j)(C(i,j)) is computed, using formulas 3.18 and 3.19 for
flocculation and hindered settling, respectively, or a constant value, if the concen
tration falls within the free settling range. The settling flux gradient, in the range
71
of concentration for which the settling flux grows with C is, again, computed using
backward differences
df (i,j) f (i,j) f(i 1, )
~ (3.61)
dz Az
but a forward difference scheme is used in the hindered flux range
df,(i,j) f.(i + 1, j) f,(i, j)
" ~ z (3.62)
dz Az
The concentration at every grid point within the water column is, then, computed
as
Ct+At(ij) = Ct(ij) At(df(i, + dfd(ij)) (3.63)
dz dz
(where the negative sign is required for consistency with equation 3.13) and the
accumulated concentration as
ct+At(i,0) = C'+t(i,j) (3.64)
At the bed a flux is computed, corresponding to one of three cases as defined by
the value of the bed shear stress rb:
1. For rb < Td a depositional flux is defined for each class of sediment as
Fb(j) = F,(j) = W,(j)C(j)(1 T ) (3.65)
Ted
In this case, although different sediment fractions are considered for the set
tling velocities and concentrations a single value for red is used. The sediment
deposited during each calculation step of a depositional period is accumulated
into a variable Db which corresponds to the mass of sediment accumulated at
the bottom per unit area.
2. For Ted < 7b < r, an entrainment flux is defined, allowing the freshly deposited
sediment (during the period in which rb < ed) to be resuspended at a constant
rate during a given time T (specified from field data interpretation). If when
72
rb = r, some of the previously suspended sediment still remains at the bed, an
entrainment flux corresponding to the resuspension of that sediment quantity
is specified, consequently allowing mass conservation in the water column.
3. For rb > r, an erosional flux is defined as
Fb(j) = Fe(j) = a ( 1) (3.66)
which corresponds to a normalized excess shear stress concept. In this case
ai = a exp(2.33r,) as determined by Villaret and Paulic (1986).
The mass of sediment eroded per unit bed area is accumulated into a variable Eb,
allowing r, to be recalculated at each time step, in the form
Tr+At = r+ + kEb (3.67)
This increase in the bed shear strength reflects increasing bed resistance to erosion
with depth, due to consolidation and gelling. The new value of the concentration
at the bed, for each sediment fraction, is computed as
CA(b,j) = Ct(bj) At(( + df(b,j)+ dfd(b,j)) (3.68)
A Z Az Az
and the accumulated concentration as
Ct+A (b, O) = C t (b, j) (3.69)
3.5.3 Discussion
The description of the physical phenomena included in the model in the form
presented in the previous sections shows some limitations. These are essentially
related to the description of conditions close to the bed and to the effects of sediment
stratification on the flow.
73
The first aspect is related to the bed boundary condition. Bed definition has
been associated with the development of effective shear stress due to particle inter
action leading to the formation of a structure. Above the structured bed two layers,
the stationary fluid mud and the mobile fluid mud, exist (see figure 3.3). The near
bed phenomena include bed fluidization and fluid mud entrainment into the mobile
suspension layers and settling and bed formation as the opposite phenomena, to
gether with consolidation and gelling of the structured bed. These complex features
are, obviously, not completely described by a simple erosiondeposition model. In
particular, fluidized mud has no shear strength but wave generation and breaking,
under shear flow, at the fluid mudmobile suspension interface, easily cause fluid
mud entrainment; the rate of erosion expression, defined as a function of the excess
shear stress, although generally applicable to moderate concentration environments,
consequently gives a poor description of the behavior of high concentration fluid
muds.
The effects of sediment stratification on the flow have been discussed by several
authors. Although the Von Karman constant has been reported to decrease with C
(Vanoni, 1975, Ippen, 1971) this fact has recently been contested (Coleman, 1981).
Changes in the velocity profile and, specifically, in du/dz are, however, commonly
accepted, as shown in equation 3.35 by 4. This function has classically been ex
pressed in the form of equation 3.39, following Munk and Anderson (1948), in which
the coefficients a' and #' reflect globally, the effect of stratification. A more detailed
analysis was presented by Mc Cutcheon (1981) leading to a velocity profile that in
cludes the effect of the density gradient; other corrected forms of the logarithmic
profile for the case of flows with suspended sediment can be found in Ippen (1971)
and Coleman (1981). These analyses could allow the direct computation of E, and,
through the Schmidt number of K,.
CHAPTER 4
FIELD AND LABORATORY EXPERIMENTS
4.1 General Aspects
As described in Chapter 3, modeling of the processes involved in the simulation
of the vertical concentration profile requires the collection of data defining both
suspended sediment and bed properties. These data are essentially related to the
suspended sediment settling velocities and to the bed erosion parameters and can
be obtained through laboratory tests. Additionally, the definition of the flow's
stabilized diffusion parameters can be obtained through adequately designed field
experiments, enabling the computation of turbulent mass diffusion coefficients and
mass fluxes. Field data are also necessary to evaluate the overall accuracy of a
model's predictions.
The laboratory and field experiments carried out within the scope of the present
study are described in this chapter. The laboratory tests were done at the Hohai
University, Nanjing (People's Republic of China) using sediment collected at the
field measurement site located in Hangzhou Bay. A description of the laboratory
procedures and obtained results is presented in the section 4.2. The field environ
ment, the experiment's methodology and the field data preprocessing methods are
included in section 4.3.
75
4.2 Laboratory Tests
4.2.1 Grain Size Test
The grain size test was done according to standard hydrometer test proce
dures (ASTM, 1988) with slight modifications. These modifications are related to
the sample preparation, which generally followed Standard Practice ASTM D2217
(procedure B, applicable to samples at moisture content equal or higher than the
natural moisture content). The steps followed in the sample preparation were:
1. Collection of a moist sample containing at least 65 g of particles passing the
2.0 mm sieve.
2. Wet sieving of the sample through a 0.1 mm sieve which confirmed that the
sample only contained finer particles.
3. Removal of salt from the sample. For this purpose distilled water was added
and the suspension shaken, before being allowed to decant for 24 hours. After
this period the excess water was carefully removed. This step was repeated
once.
Hydrometer test procedures generally followed Standard Practice ASTM D422 with
the exception of sediment sample dispersion which was not done. The grain size
distribution is presented in figure 4.1 and shows the floc distribution existing in the
natural environment. Although the clay fraction cannot be determined it is assumed
to be small (Su et al., 1984). The median floc diameter is 23 Am, slightly higher
than the range of median diameters indicated by Su at al. (1984) for Hangzhou Bay
(10 to 13 pm for the suspended load and 16 Lm for the bed material); this might
be a result of the nondeflocculation of the sample.
SAND
SILT
CLAY
Coarse Medium Fine Coarse Medium Fine
   
 ..... .. ...
     
"Il ~  
_______   . 
     ______
::::::::: : :; 0:
:' :
=::::::=:;:^::::ff
0 0 0
* 00 0 0 0
0 0 *
00 0 O O0
GRAIN SIZE IN mm.
Figure 4.1: Grain size distribution.
00
o
00
0
0
o
100
90
80
70
60
50
40
30
20
10
0
00
90
80
70
60
50
40
30
20
10
0
Cm 
4.2.2 Settling Velocity Tests
A settling column is needed for this test. The column used at Hohai University
consisted of a 18.7 cm diameter tube, 1 m long, fitted with 5 mm inner diameter
taps at six elevations. Tap hoses were 5 cm long and were fitted with clamps. The
elevations of the taps from the bottom were (in cm): 15, 30, 45, 60, 75 and 90.
The experimental procedure used for the test was as follows:
1. High concentration sediment slurry was diluted with salt water to desired ini
tial concentration and required volume (27.45 liters to fill the settling column).
The suspension was thoroughly premixed in a large container.
2. After preparation the suspension was poured into the settling column and
completely mixed for at least 2 minutes. This was achieved through mechan
ical mixing.
3. Immediately after removal of mixing device the first set of samples (= 100 ml
per sample) was taken. Samples were collected in glass bottles which were
tightly capped, labeled and stored. Sampling was repeated at fixed times (5,
15, 30, 45, 60, 90, 120 and 180 minutes, for example) after the beginning of the
test. Water depths and temperatures were recorded at each sampling time.
The sampling hoses were always flushed (to remove the suspension left in the
previous sampling) before each withdrawal.
4. Gravimetric analysis was used to determine the concentration of each sample.
This was done by weighing a known volume of well mixed suspension (100 ml)
in a laboratory beaker of known weight. The sediment concentration in the
volume was then obtained through the use of the equation
C = k(MT Mw)
(4.1)
78
where k is the inverse of the volume of suspension in the beaker and MT, Mw
are the masses of the suspension and of the same volume of salt water.
The test was carried out fourteen times, using several combinations of initial
concentrations 2.0, 10.0, 20.0 and 30.0 g/l and salinities of 2.0, 10.0 and 30.0 ppt.
From the test results, values of the settling velocity, W,, corresponding to a given
concentration, C, can be determined (Vanoni, 1975, Ross, 1988) by considering that,
in a settling column (z = 0 at the surface, increasing downwards):
aC aW,C
a + a =0 (4.2)
at az
which is a continuity equation for sediment settling under quiescent conditions.
Integration of equation 4.2 with respect to z gives
a 8D a_
(WC) z=D = C dz= a (4.3)
5t l at
At t = 0 the concentration is uniform throughout the settling column; for t > 0,
integration of the concentration profile between the free surface and the D levels
produces CD values. The slope of a plot of CD versus t at each depth will produce
the righthand side of equation 4.2 and, since the C values are known, W, can be
computed. A plot of W, versus C values obtained in the settling tests done with
Hangzhou Bay sediment is presented in figure 4.2; also shown in the figure are the
curves of equations 3.18 and 3.19. Such curves were obtained by leastsquares
fitting, considering W,o = 1.094 mm/sec and Co = 4.0 g/l.
4.2.3 Erosion Tests
An annular flume was used for the erosion test. The flume that was used for this
test is similar to the one at the University of Florida, with a channel width of 20 cm,
depth of 46 cm and a mean radius of 76 cm. A plexiglass annular ring of width
slightly less than the channel width of 20 cm is suspended inside the channel in such
10I i 1 1i L
W=0.231 C 1122
W' o _=1.094 [10.00808 (C4.0)] 14.269
E 0
E, 0 a 0
S "o 0 \
I
0 10
I O
0 0
SCo
1 0 2 I I I I I ll I I i ,III I I I I IIIn
101' 1 10 102
C (g/1)
Figure 4.2: Settling velocity as a function of concentration for Hangzhou Bay sedi
ment.
80
a way that it is in complete contact with the water column. During operation the
ring and channel are rotated in opposite directions to minimize secondary currents.
Taps are located on the outside wall of the channel, allowing sampling from the
water column. The concentrations of the samples are determined by gravimetric
analysis, similar to that used in the settling column tests.
The steps followed in the erosion tests were:
1. A placed bed was is used to obtain a relationship between the bed shear
strength and the uniform density of the bed. A thick slurry (with selected
approximate density) of local bed sediment (sieved to remove shell and plants)
was mixed for one hour and placed into the annular flume to uniform depth.
Salt water was carefully added to the flume, until adequate depth was reached.
A similar slurry was placed in a bucket to allow bed density determination.
2. Four different shear stresses, rb, were selected and applied in a stepwise manner
during 90 min each. The first was 0.1 N/m2 and the remaining were obtained
by increments of 0.2 N/m2. A suspension sample was taken at the beginning
of the test. Suspension samples (= 100 ml) were also taken at times 2, 5, 10,
15, 20, 25, 30, 40, 50, 60, 75 and 90 min after the beginning of each period
of constant applied stress. Samples were taken, in each case, from taps at
the top and at the bottom of the water column and an average suspension
concentration was calculated and assumed representative of the entire water
column. Salt water was periodically added to maintain the initial water depth.
Each test was carried out four times for the first three values of rb, with bed
bulk densities of 1.2, 1.3, 1.4 and 1.7 g/cm3 and once for rb = 0.7 N/m2 with a
bed bulk density of 1.7 g/cm3. Plotting the rate of erosion (suspended sediment
mass eroded per unit bed surface area per unit time) versus the applied shear stress
(fig. 4.3) a critical shear strength value (r, = 0.05 N/m2) is obtained. From this
