UFLICOEL97/004
A MECHANISM FOR NONBREAKING WAVEINDUCED
TRANSPORT OF FLUID MUD AT OPEN COASTS
by
Hugo N. Rodriguez
Thesis
1997
A MECHANISM FOR NONBREAKING WAVEINDUCED TRANSPORT OF FLUID
MUD AT OPEN COASTS
By
HUGO N. RODRIGUEZ
A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
1997
ACKNOWLEDGMENT
First, I wish to express my deepest gratitude to my
advisor, Dr. A.J. Mehta, for giving me the opportunity To
study under his skillful guidance. His direction and valuable
advice contributed greatly to the final form of this work.
I would also like to thank Dr. R.G. Dean and Dr. R.J.
Thieke for their participation as supervisory committee
members, and for the knowledge given through their interesting
lectures and discussions.
Thanks and appreciation are due to the staff and fellow
students of the Coastal and Oceanographic Engineering
Department, with special acknowledgement to the laboratory
staff for helpful assistance during the experimental phase of
this study.
Finally, I wish to thank very specially Wally Li and
Ismael Piedra Cueva for their invaluable help in reviewing the
derivation of the equations and constructive suggestions.
TABLE OF CONTENTS
ACKNOWLEDGMENT . . . . . . . . . . .ii
LIST OF FIGURES . . . . . . . . ... . vi
LIST OF TABLES . . . . . . . . . . viii
LIST OF SYMBOLS . . . . . . . . ... .ix
ABSTRACT . . . . . . . . . . . xiv
CHAPTERS
1 INTRODUCTION . . . . . . . . ... . 1
1.1 Problem Statement . . . . . . . 1
1.2 Objective and Scope . . . . . . . 9
1.3 Thesis Outline . . . . . . . . 10
2 A MECHANISM FOR MUD TRANSPORT . . . . . 12
2.1 Introduction . . . . . . . . 12
2.2 Wave Mass Transport over a Rigid Bottom . . 12
2.2.1 Mass Transport under Wave Flow in
Constant Depth . . . . . . 13
2.2.2 Streaming . . . . .. . . 17
2.2.3 Nearshore Current . . .. . . 23
2.3 Wave Mass Transport over a Fluid Mud Bottom 29
2.3.1 Interface and Surface Setups . . . 30
2.3.2 Pressure at a Generic Position z
(Water Layer) . . . . .. .. . 33
2.3.3 Pressure at the Interface . . .. .35
iii
2.3.4 Pressure at a Generic Position z
(Mud Layer) . . . . .. . . 36
2.3.5 Vertical Integration of Pressure
(Water Layer) . . . .. . . 36
2.3.6 Vertical Integration of Pressure
(Mud Layer) . . . .. . . . 38
2.3.7 Free Surface Setup . . . . . 39
2.3.8 Interface Setup . . . . . . 41
2.3.9 Mean Current in the Wave Direction . .43
2.3.10 CrossShore and Alongshore Mean
Currents . . . . . . . . 46
3 LABORATORY EXPERIMENTS . . . . . . . 49
3.1 Introduction . . . . . . . . 49
3.2 Experimental Equipment . . . ... .. 49
3.2.1 Wave Basin . . . . . . . 49
3.2.2 Wave Gage . . . .. . . . 50
3.2.3 Velocity Measurement . . . . . 50
3.2.4 Measuring Carriage . . . . . 53
3.2.5 Other Apparatuses . . . . . 54
3.3 Test Preparation . . . . .. . . 54
3.3.1 Profile Sediment . . . . . 54
3.3.2 Profile Preparation ... . . . 55
3.3.3 Profile Measurement . .. . . 56
3.3.4 Fluid Mud Preparation . . . . 57
3.4 Experimental Procedure and Data . . . . 58
3.4.1 Experiments without Fluid Mud . . 59
3.4.2 Experiments with Fluid Mud . . . 61
3.5 Data Preparation and Corrections . . . 61
4 FIELD DATA ON MUD TRANSPORT . . . . . . 63
4.1 Introduction . . . . . . . . 63
4.2 Northeastern Coast of South America . . . 63
4.3 Louisiana Coast . . . . . .. .. . 68
4.4 Southwest Coast of India . . .. . . 70
5 DATA ANALYSIS . . . . . . . . . 75
5.1 Introduction . . . . . . . . 75
5.2 Dimensionless Forms of Mass Transport
Equations . . . . . . . . . 75
5.3 Simplifications of Mass Transport Equations 80
5.4 Laboratory Data Analysis . . . . . 82
5.4.1 Correction for Shoaling . . . . 83
5.4.2 Friction Coefficient Determination . .85
5.4.3 Determination of Fluid Mud Viscosity .88
5.4.4 Measured Alongshore Velocity . . . 89
5.4.5 Computed Alongshore Velocity . . . 92
5.4.6 Data Comparison . . . . . . 93
5.5 Field Data Application . . . . . . 94
5.5.1 Input Data Evaluation . . . . 95
5.5.2 Data Comparison . . ... . . 97
6 SUMMARY AND CONCLUSIONS . . . . . .. 101
6.1 Summary . . . . . . . . . 101
6.2 Conclusions . . . . . . . . 102
6.3 Recommendations for Further Studies . .. 104
LIST OF REFERENCES . . . . . . . . . 106
BIOGRAPHICAL SKETCH . . . . . . . . .. 112
LIST OF FIGURES
Figure page
1.1 Idealized depiction of seasonally varying,
normally incident waves in the absence of
alongshore sediment supply causing cyclic
fluctuation of shoreline (A), or recession
of "initial" shoreline (B), depending on
the distance over which eroded sediment
travels in relation to the width of the
littoral zone . . . . . . . . 4
1.2 Idealized depiction of obliquely incident
waves in the presence of steady alongshore
sediment supply, which can either lead to
a stable or prograding shoreline orienta
tion (A), or a receding shoreline (B) . . 5
1.3 Undulant low water shoreline associated
with obliquely incident waves and a signi
ficantly pulsating sediment load . . . 6
1.4 Undulant low water shoreline associated
with obliquely incident, seasonally vary
ing waves and a practically steady (in the
mean) sediment load . . . . . . . 7
1.5 Normally incident waves with offshore
sediment supply leading to a crenulate
low water shoreline due to wave
refraction effects . . . . . . . 8
1.6 Global map showing the location of the muddy
coasts of SurinamGuyana, Louisiana and
Kerala, India (adapted from A.N. Strahler,
1969) . . . . . . . . . . 9
2.1 Schematic sketch for an obliquely
incident wave . . . . . . . . 24
2.2 Schematic sketch of a twolayered
fluid flow . . . . . . . ... .30
2.3 Forces in sdirection . . . . . . 43
2.4 Schematic sketch for Um determination . . 45
3.1 Plan view of the experimental basin setup 51
3.2 Elevation view of the experimental basin
setup . . . . . . . . . . 51
3.3 Capacitance wave gage . . . . . . 52
3.4 Strip chart recorder . . . . . . 52
3.5 Metric scale ruler for alongshore velocity
calibration . . . . . . . . 53
3.6 Measuring carriage with video camera and
wave gage . . . . . . . . . 55
3.7 Representative bottom profiles . . . . 57
4.1 Northeastern coast of South America
(after Wells, 1983) . . . .. . . 64
4.2 Louisiana chenier plain (after Wells, 1983) .69
4.3 Southwest coast of India (after Mathew
et al., 1995) . . . . . . . . 72
4.4 Schematic mudbank along the southwest coast
of India (after Mehta et al., 1996) .. . . 73
5.1 Comparison between measured and shoaled
wave heights at position B without fluid
mud, using the stream function wave theory .85
5.2 Computed and measured alongshore velocities
comparison . . . . . . . . . 94
5.3 Wave damping over a Surinam mudbank (after
Wells, 1983) . . . .... . . . . 96
vii
LIST OF TABLES
Table page
3.1 Measured data without fluid mud . . . .. 60
3.2 Measured data with fluid mud . . . . 62
4.1 Characteristic data for the muddy coast
environment of the Guyanas . . . . . 68
4.2 Characteristic data for the muddy coast
environment of Louisiana . . . . . 71
4.3 Characteristic data for the muddy coast
environment of southwest India . . . . 74
5.1 Characteristic reference parameters . . . 76
5.2 Dimensionless variables . . . . . . 77
5.3 Dimensionless parameters . . . . . 79
5.4 Data for friction coefficient determination .88
5.5 Data for dynamic viscosity determination .. .90
5.6 Measured alongshore velocity data . . . 91
5.7 Computed alongshore velocity data . . . 92
5.8 Selected values of relevant parameters for
calculation of field mud streaming velocity 97
5.9 Values obtained from Jiang's model (1993) .. 98
5.10 Calculated streaming velocities and measured
characteristic velocity ranges . . .. 100
viii
LIST OF SYMBOLS
a6 = amplitude of nearbed wave orbital motion
b = interfacial oscillatory amplitude
c = Wave celerity
cf = Bed friction coefficient
c, = Wave group celerity
D = Littoral zone width
ds = Increment in the sdirection
E = Wave energy density
g = Acceleration due to gravity
h = Water depth, or total depth(h1+h2)
hb = Water depth at breaking
h, = Water depth
h* = Dimensionless water depth
h2 = Mud layer thickness
h* = Dimensionless mud layer thickness
H = Wave height
HA = Measured wave height at position A
HE = Measured wave height at position B
Hb = Wave height at breaking
Ho = Characteristic wave height outside the mudbank
k = wave number
ki = damping coefficient
kN = bed roughness
m = Profile slope
p, = Pressure in the water layer
p2 = Pressure in the mud layer
P = Total pressure
q = Water discharge per unit width
Re. = Boundary layer Reynolds number
Res = Streaming Reynolds number
s = Horizontal coordinate in the direction of wave
propagation
s* = Dimensionless horizontal coordinate in the
direction of wave propagation
S.. = Mean momentum fluxes
S.. = Radiation stresses
S = Integrated Reynolds stresses
t Time in an Eulerian coordinate system
t = Time in a Lagrangian coordinate system
to = Time in a Lagrangian coordinate system
T = Wave period
u = Velocity component in the sdirection
= Eulerian velocity vector
u, = Irrotational part of u
U = Lagrangian velocity vector
U = Timeaverage Lagrangian component in the s
direction
um = Maximum orbital velocity u
uR = Rotational part of u
ux = Velocity component in the xdirection
fx = Oscillatory component of ux
ux' = Turbulent component of ux
uy = Velocity component in the ydirection
Uy = Oscillatory component of uy
uy' = Turbulent component of uy
ui = Velocity in sdirection in the water layer
u* = Dimensionless velocity in sdirection in the
water layer
2 = Velocity in the sdirection in the fluid mud
layer
u2 = Dimensionless velocity in the sdirection in
the fluid mud layer
u = Orbital velocity at the edge of the boundary
layer
U = Second order mean velocity in the fluid mud
layer in the sdirection
Ur = Depthaverage value of U
Un = Depthaverage value of U in the crossshore
xdirection
U = Depthaverage value of U in the alongshore
ydirection
U = Second order mean velocity in the water layer
in the sdirection
Ux = Depthaverage velocity in the xdirection
Uy = Depthaverage velocity in the ydirection
UM* = Dimensionless depthaverage value of U
Um* = Dimensionless depthaverage value of U in the
crossshore direction
U* = Dimensionless depthaverage value of U in the
alongshore direction
U = Mean second order velocity inside the boundary
layer in the sdirection
U. = Streaming velocity in the sdirection
U, = Total average mass transport velocity in the
sdirection
V = Measured alongshore velocity without fluid mud
Vbreak = Field computed alongshore velocity due to
breaking
Vfm = Measured alongshore velocity with fluid mud
Vmeas = Measured alongshore velocity due to viscous
streaming
= Timeaverage Lagrangian component in the z0
direction
w = Velocity component in the zdirection
wI = Irrotational part of w
WR = Rotational part of w
w, = Velocity component in the water layer in the
zdirection
w = Dimensionless velocity component in the water
layer in the zdirection
w2 = Velocity component in the fluid mud layer in
the zdirection
w = Dimensionless velocity component in the fluid
mud layer in the zdirection
x = Horizontal axis in the crossshore direction
= Space vector in an Eulerian coordinate system
0 = Space vector in a Lagrangian coordinate system
y = Horizontal axis in the alongshore direction
z = Vertical coordinate
zo = Lagrangian vertical coordinate
zi = Interface (total) displacement
z* = Dimensionless vertical coordinate
a = Angle between the s and x axes
3 = Modulating coefficient of nonlinear wave
energy with respect to linear wave energy
5 = Boundary layer thickness
S = Free surface displacement
ri = Free surface oscillatoryy) displacement
xii
[ = Free surface (total) displacement
r,ri = Free surface setups
nr* = Dimensionless free surface oscillatoryy)
displacement
ri* = Dimensionless free surface setup
r2 = Interface oscillatoryy) displacement
r = Interface (total) displacement
r2 = Interface setup
n2* = Dimensionless interface oscillatoryy)
displacement
12* = Dimensionless interface setup
K = Wave breaking index
12 = Dynamic viscosity of fluid mud
v,v, = Kinematic viscosity of water
\2 = Kinematic viscosity of fluid mud
p,p1 = Water density
p2 = Fluid mud density
2* = Dimensionless fluid mud density
o = Angular wave frequency
oii = Viscous normal stresses in the fluid mud layer
Tb = Mean bottom stress
Is = Mean surface stress
Tj = Viscous shear stresses in the fluid mud layer
Ib* = Dimensionless mean bottom stress
xiii
Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science
A MECHANISM FOR NONBREAKING WAVEINDUCED TRANSPORT OF FLUID
MUD AT OPEN COASTS
By
Hugo N. Rodriguez
May 1997
Chairperson: Dr. Ashish J. Mehta
Major Department: Coastal and Oceanographic Engineering
A preliminary study was carried out to examine wave
induced transport of fluid mud at open muddy coasts. Given the
large wave damping potential of bottom mud, an analytic
formula for nonbreaking waveinduced streaming of fluid mud
underflow was developed by considering mud to be a highly
viscous continuum.
Laboratory basin tests were carried out to examine the
validity of the formula for alongshore streaming. The formula
showed an order of magnitude agreement with measured
alongshore velocities generated by obliquely incident waves
over a muddy bottom profile.
The same formula was then applied to microtidal muddy
coasts in Surinam, Guyana, Louisiana and India to show that
xiv
alongshore and crossshore mud streaming can be an important
mechanism by which sediment is naturally supplied to sustain
coastal mudbanks.
CHAPTER 1
INTRODUCTION
1.1 Problem Statement
Along muddy coasts, large amounts of cohesive sediments
are present, and these sediments have important interactions
with the hydrodynamics of the littoral zone. However, even
though, for example, nearly onequarter of the total coastline
of North and South America is muddy (Wells and Coleman, 1977),
and although long muddy shores are present elsewhere in the
world, a clear understanding of the sedimentary processes that
sustain mudbank dominated coasts is lacking.
The effect of hydrodynamics on coastal mudbanks have been
documented, and it is found that in the microtidal (range <
2m) environment, the dominant forcing is the action of wind
waves. Free surface wind waves cause liquefaction of the upper
stratum of the bottom mud and generate fluid mud. In turn, the
interface between the water layer and the fluid mud layer thus
formed is forced to oscillate with the same period as that of
the surface wave. Wavemud interaction is further complicated
because the generated fluid mud in turn affects the wave by
2
absorbing and dissipating part of its energy. In fact, in many
situations the high viscous damping potential of fluid mud
causes it to absorb wave energy to such an extent that waves
tend to disappear entirely before they reach the shoreline.
The generation and transport of fluid mud is a unique and
critically important feature that can play a pivotal role in
governing bottom profile dynamics. Unfortunately the precise
manner in which fluid mud is transported within the littoral
zone between the shoreline and the depth of closure of
sediment transport is not well understood. Yet, field evidence
indicates that, for example, rapid recovery of a storm eroded
mud bottom profile can only be explained by taking fluid mud
transport into account (Mehta et al., 1996; Lee and Mehta,
1997). Furthermore, the observation that fluid mud transport
also occurs under nonbreaking wave conditions suggests that
formulations for littoral sand transport based on wave
shoaling and breaking over a practically rigid bottom (Bruun,
1983; Kriebel and Dean, 1993) are unlikely to be suitable for
explaining muddy coast evolution.
The response of muddy coasts to wave forcing can be
described within two main categories. In general, to describe
shoreline response in the plan form, the low water line, which
is always in contact with water, can be conveniently chosen.
Now if, under seasonally variable but normally incident waves,
3
fluid mud is transported toandfro in the crossshore
direction within the littoral zone, the "initial" shoreline
will remain stable without any net gain or loss of sediment
(case A in Figure 1.1). This type of (first category) response
occurs along the southwest coast of India in Kerala, where
fluid mud is transported shoreward during the high monsoonal
wave climate in the Indian Ocean. This transport results in
shorefast mudbanks, which dissipate at the end of the monsoon
as wave activity wanes and a gravity slide returns the mud to
the deeper offshore region within the littoral zone ( Mathew,
1992; Mathew et al., 1995; Mathew and Baba, 1995). This type
of response may also explain the stability of some muddy
coasts in Indonesia (Tarigan et al., 1996).
When the sediment transport pathway and travel distance
are such as to cause the eroded material to go out of bounds
of the littoral zone, the shoreline will recede (case B in
Figure 1.1). In some cases, this (second category) response
may be actuated at a stable shoreline by drastic changes in
the water level, as along the coasts in Lake Ontario, Canada
(Coakley et al., 1988).
Dredging can also lead to significant land loss. A
dramatic example is that of shoreline recession in excess of
1,200 m between 1972 and 1991 in the Ondo State in Nigeria,
possibly due to the dredging of a navigation passage across a
4
barrier island, which caused strongly scouring tidal currents
on both sides of the new entrance (Eedy et al., 1994; Ibe et
al., 1989).
  Receding LW Shoreline
I Mud Mud
Initial LW Shoreline
\ Mud B
Littoral
Zone
  Depth of Closure
T T [ Mud
Normally Incident Seasonal Waves
Figure 1.1 Idealized depiction of seasonally
varying, normally incident waves in the absence
of alongshore sediment supply causing cyclic
fluctuation of shoreline (A), or recession of
"initial" shoreline (B), depending on the
distance over which eroded sediment travels in
relation to the width of the littoral zone.
If waves approach the shoreline obliquely, the alongshore
mud stream they generate can lead to a stable or accreting
shoreline provided there is a source of sediment, as depicted
in Figure 1.2 (case A) (first category response), or lead to
recession if the profile cannot be replenished at the rate at
which it is eroded (case B)(second category response).
5
    Receding LW Shoreline: B
[ Mud I IM d I
Stable or Prograding LW
Steady Shoreline: A
Sediment A
Supply
      Depth of Closure
I Mud I
Obliquely Incident Waves
Figure 1.2 Idealized depiction of obliquely
incident waves in the presence of steady alongshore
sediment supply, which can either lead to a stable
or prograding shoreline orientation (A) or a
receding shoreline (B).
The stability of some of the muddy coasts that are
subject to high tidal variations and wave action in South
Korea has been attributed to adequate supply of sediment from
nearby rivers (Wells, 1983). Along the Selangor coast of
peninsular Malaysia, reduced sediment supply to the coast due
to fresh water diversion has led to sea encroachment into
cultivated lands (Midun and Lee, 1989). In Jiansu Province in
China, the long term effects of the shifting of the mouth of
the Yellow River has led to prograding beaches where sediment
supply from the river is ample, and significantly eroding
beaches where the supply is inadequate (Ren, 1992).
Pulsating Amplitude
Sediment
Load Undulant
LW Shoreline
  Depth of Closure
Obliquely Incident Waves
Figure 1.3 Undulant low water shoreline
associated with obliquely incident waves and a
significant pulsating sediment load.
Muddy coasts exhibit short and long term modulations that
are qualitatively akin to those at sandy coasts; yet the
mechanisms that cause these features in these two environments
are not always analogous. In the case of muddy coasts,
variations in alongshore supply and wave climate can cause the
shoreline to develop a spatially rhythmic or undulating
configuration, as in Surinam, Guyana and Louisiana. The
simplest way to explain shoreline response in the contiguous
coasts of Surinam and Guyana (Figure 1.3) is in terms of a
trade wind induced steady wave action from the northeast in
combination with a temporally pulsating sediment supply from
the east derived from the Amazon River. The wave damping
7
potential of fluid mud in this region is very high. It is
reported that as much as 96% of offshore wave energy is
dissipated by viscous damping (Wells, 1983).
Undulant
Constant LW Shoreline
Sediment
Load
  Depth of Closure
Obliquely Incident Seasonally VaryingWaves
Figure 1.4 Undulant low water shoreline
associated with obliquely incident, seasonally
varying waves and a practically steady (in the
mean) sediment load.
An undulant shoreline may also result if the alongshore
supply of sediment is steady, say, on an annual basis, but the
wave intensity and direction change seasonally between the
winter and summer months (Figure 1.4). The wavedominated
coastal mudbanks bordering the chenier plain in Louisiana west
of the mouth of the Atchafalaya River show features that
correspond to this description in an approximate way. Mud is
supplied by this river, which receives its sediment load from
the Mississippi River. Wave energy losses as high as 48% have
8
occurred due to viscous damping within fluid mud (Tubman and
Suhayda, 1976).
WS9w 9 Crenulate
LW Shoreline
Wave
Refraction Undulating
Offshore
I  I  Bathymetry
Normally Incident
OffshoreWaves
Figure 1.5 Normally incident waves with
offshore sediment supply leading to a
crenulate low water shoreline due to wave
refraction effects.
The monsoonal mudbanks of Kerala in India tend to be
disklike in plan form, and occur intermittently along a
nearly straight sandy shoreline, giving it a crenulate
appearance (Figure 1.5). As Mathew and Baba (1995) have shown,
the dominant crossshore mechanism for mud streaming there is
influenced by wave refraction due to offshore bathymetry, and
this leads to mudbank formation at known locations where wave
energy is concentrated. As a result, monsoonal wave breaking
activity shows a distinct variability ranging from practically
nil in the mudbank areas to plunging breakers along the
9
intervening sandy beaches. Mathew et al. (1995) have noted
that during monsoon, as much as 90% of the wave energy is
damped out over the mudbanks.
Three regions of the world (SurinamGuyana, Louisiana and
Kerala, India) that are further described in this study are
shown in Figure 1.6.
90 Vu
Figure 1.6 Global map showing the location of the muddy coasts
of SurinamGuyana, Louisiana and Kerala, India (adapted from
A.N. Strahler, 1969).
1.2 Objective and Scope
In this study, a mechanism for fluid mud mass transport
at muddy open coasts where nonbreaking waveinduced forcing
may be the dominant cause of transport has been explored. The
action of progressive waves in a twolayered (waterfluid mud)
10
system is studied by means of the vertically integrated
momentum equations. The upper layer is considered to be an
inviscid fluid and the bottom layer a Newtonian viscous fluid.
The free surface and interfacial setups generated as well as
the second order depthaverage drift are calculated. To
examine the validity of the expression for alongshore drift,
laboratory basin tests are carried out. Field data from three
different muddy coastal areas (Figure 1.6), where good quality
data on alongshore and crossshore drift are available, are
analyzed in conjunction with theory.
1.3 Thesis Outline
Chapter 2 presents the derivations of setups and mass
transport velocity equations. First, a review of mass
transport mechanisms over a rigid bottom, i.e., a onelayer
model, is presented. Then, the momentum equations for a two
layered model are integrated in the vertical to obtain the
free surface and interfacial setup as well as the crossshore
and alongshore mass transport velocities.
Chapter 3 deals with laboratory basin experiments. In
this chapter, laboratory test preparation and performance are
mentioned. Wave height measurements at two locations and
alongshore currents over a muddy profile built in a basin are
described.
11
Information on the hydrodynamic and sedimentary
environments of the northern coast of South America, Louisiana
and the southwest coast of India is reported in Chapter 4.
Chapter 5 gives a comparison and discussion of the
theoretical velocity values with the field and laboratory
measured values. For theoretical analysis, a non
dimensionalization of the mass transport is carried out.
Finally, in Chapter 6 concluding remarks and
recommendations for future studies are made.
CHAPTER 2
A MECHANISM FOR MUD TRANSPORT
2.1 Introduction
At an infinitely long shoreline in the presence of fluid
mudbanks, different hydrodynamic mechanisms may cause cross
shore and alongshore movements of these banks. In this
chapter, an analytical description of such movements caused by
waves is presented. First the classical case of a rigid bottom
is stated. Then the case of a twolayered fluid, corresponding
to the presence of a fluid mud bottom, is analyzed.
2.2 Wave Mass Transport over a Rigid Bottom
The first order solution for the propagation of waves in
water of uniform depth h, over a rigid bottom, is given by the
linear wave theory. For simplicity a monochromatic sinusoidal
perturbation of the water surface is considered. This
assumption does not imply a loss of generality, because a
perturbation of the water surface can be reduced to a
summation of sinusoidal functions by means of Fourier
13
transform. As the first order linear theory is being
considered, the different sinusoidal constituents propagate
independently, and their linear superposition is valid.
From the linear wave theory, different nonlinear
properties that are correct to second order can be derived.
These properties are the second order momentum flux, the setup
and mass transport. They are obtained averaging over the wave
period and integrating over depth.
2.2.1 Mass Transport under Wave Flow in Constant Depth
Linear wave theory predicts the trajectories of the water
particles to be closed ellipses. However, if the particle
displacement under a wave is measured, a net movement in the
direction of wave propagation can be observed after a wave
length has passed. This movement increases from the bottom
towards the free surface. If the Lagrangian description of
water motion is used, this second order mass transport, or
drift velocity, can readily be derived.
In an Eulerian description of water motion under a wave,
the physical properties are given as functions of fixed
positions in space 5 and time t. Therefore, the velocity field
is represented as q=q(k,t). The Lagrangian description follows
the movement of a particle that, at a given time to, was at a
position x = (x, z,) so that the independent variables are
the initial coordinates x0 and the elapsed time tt0. The
14
relation between these two descriptions of the same physical
phenomenon is
S= K(x0, tt0) ; x(x, 0) = x0 (2.1)
As the velocity of a fluid element is the timederivative of
its position,
x (x,, tto)
1(Xo,tto) = 0 t (2.2)
one obtains
t
S= x0 + f u (o, tto) dt (2.3)
to
where = (u ,w,) is the Lagrangian velocity, which can be
equated with the Eulerian velocity,
t
u1(Xo,t) = (i(0+f 1 ( 0o, tt0)dt, t) (2.4)
to
Using the Taylor series expansion to the second order, the
final expression for the Lagrangian velocity as a function of
the Eulerian velocity is obtained as
t
+ (0,t) = ((50,tU) + I(0,tto) dt .Vx (5 0,t) (2.5)
to
This expression shows that, to the first order, both the
Lagrangian and the Eulerian velocities are the same and that
their mean values over a wave period are zero. On the other
hand, the second order term of the Lagrangian velocity has a
nonzero mean value. If the linear wave theory is used, the
second order mean value components of the Lagrangian velocity
ul, are
okH2cosh2k(zo+h)
1 8sinh2kh (2.6)
w, =0
with o and k being the wave frequency and the wave number,
respectively, H the wave height, u the velocity component in
the direction of wave propagation s, and w the velocity
component in the vertical direction z. The coordinate axis, s,
is horizontal at the still water level and positive in the
direction of wave propagation, and z is positive upwards from
the still water level. The total water discharge per unit
width over the water column associated with this mass
16
transport velocity is obtained by integrating over the water
depth as
nH2 1
q (2.7)
4T tanhkh
where T is the wave period. Due to the linear approximation
the Eulerian velocity is periodic under the wave trough and
thus its wavemean value is zero in that region. However, in
the region between the trough and the crest, the wavemean
velocity must be obtained by means of Taylor expansion and is
found to have a nonzero value in this region. The total water
discharge per unit width can therefore be calculated at a
given position s, and the same result as equation (2.7) using
the Lagrangian approach is obtained. The Eulerian analysis
thus yields the same total discharge associated with the net
drift, but instead of being distributed over the entire water
column, it is confined to the region between the trough and
the crest of the wave. Both analyses are different approaches
to account for the same phenomenon.
The above results are derived from the linear wave theory
considering inviscid irrotational motion, which yields a
finite value of the horizontal velocity at the bottom. In a
real fluid the noslip condition at the bottom must be
considered and, therefore, the viscous stress term in the
17
NavierStokes equation must be taken into account in the near
bed region.
2.2.2 Streamina
As noted, the potential theory of irrotational flow
predicts a finite value of the horizontal velocity u at the
bed. In a real fluid this velocity must be zero and
accordingly leads to the development of a boundary layer in
which the viscous terms in the momentum equation must be taken
into account because they are comparable to the other terms.
A local coordinate system is adopted in the boundary
layer with s positive in the direction of wave propagation at
the horizontal bottom boundary and z in the upward vertical
direction from the bottom.
The continuity equation for an incompressible flow is
au aw
+ = 0 (2.8)
as az
with u and w the horizontal and vertical velocity components,
respectively. As the boundary layer thickness 5 is much
smaller than the wave length L, the spatial derivatives in the
sdirection can be neglected with respect to derivatives in
the vertical zdirection, so that the NavierStokes equation
in the s direction can be approximated as,
au auw 1 ap, au(
p+ = +a (2.9)
at az p as z z2
where P is the total pressure and p and v the density and the
kinematic viscosity of the fluid, respectively. As w is much
smaller than u the NavierStokes equation in the zdirection
reduces to
1 aP
0 1 g (2.10)
p az
with g the acceleration of gravity. Noting that p = P pgz is
constant in the zdirection according to equation (2.10), the
Euler equation, i.e. equation of motion excluding the viscous
contribution, applied to the inviscid flow region can be used
to define the pressure gradient in the boundary layer.
To the first order, the horizontal velocity component u
in the boundary layer can be considered to be the
superposition of an irrotational part uz, given by the linear
wave theory, plus a rotational part uR, since to the first
order, the equations are linear.
The irrotational part satisfies the linearized Euler
equation
aui 1 ap (2.11)
at p as
19
and, after substitution of equation (2.11) into equation
(2.9), the resulting equation is
auR R (2.12)
V (2.12)
at az2
with the boundary condition at the bottom
u = u,+uR = 0; i.e. uR = uI (2.13)
The solution of equation (2.12) with the boundary condition
(2.13) is
z
H e ((2.14)
u os ksot+ 214)
2 sinhkh 6)
in which
5 = [ (2.15)
is the boundary layer thickness.
The flow is consequently oscillatory, and as the
horizontal velocity given by equation (2.14) is a function of
s, it induces a vertical rotational flow velocity wR in the
boundary layer that can be found from the continuity equation
20
(2.8), along with the boundary condition wR = 0 at z = 0 (Dean
1996b)
H ok6
wR(z) 
2/2 sinhkh
sinhkh (2.16)
e cos(ksot+ ) cos (ksat
6 4 4
As observed from equation (2.16), wR does not go to zero
outside the boundary layer, that is, as z tends to infinity, wR
approaches a finite value. This can be described in physical
terms from the continuity equation applied to a control volume
including the entire boundary layer depth. As the horizontal
discharge per unit width, corresponding to uR, changes with
s along the boundary layer, to satisfy mass conservation a
vertical discharge associated with wR must exist.
Referring to w. as the vertical velocity just outside the
boundary layer, an additional shear stress uw is induced.
This stress is balanced by the viscous stress generated by a
mean flow in the boundary layer. The mean momentum equation
which states this balance is (Dean 1996b)
auw a2
= (2.17)
az az2
in which U is a mean second order velocity (overbar indicates
mean values over a wave period), or
(2.18)
au
uw uw = v 
8z
Also,
uw = (u+uR) (wI +wR)
and, correct to the second order,
H2 2k6
uw =
16 sinh2 kh
Sh e z 2)
2 ( e) ) W sin + 2e cos e 1
Therefore,
(2.19)
(2.20)
H2ok
U(z) = o
16 s inh2 kh
(z z 8z z 2( (2.21)
[32( z+2 e cos( ) 2( 1 eW sin( )+ + e
which gives a second order mean streaming velocity, U_,
outside the boundary layer in the direction of wave
propagation equal to
3 H2ok
U 16 sih2k(2.22)
16 sinh2kh
This mean velocity is independent of the viscosity v even
though viscosity must be nonzero for its existence (Dean,
1996b; Phillips, 1966).
The above analysis was carried out considering a laminar
boundary layer, for which a Reynolds number can be defined as
u,6
Re (2.23)
where u6 is the orbital velocity at the edge of the boundary
layer. For values of Re8 > 160 a turbulent boundary layer
develops (Phillips, 1966). In this case, an eddy viscosity Ve
must be introduced in equation (2.17) in place of the
kinematic viscosity v. However, in equations (2.9) and (2.12)
the fluid kinematic viscosity v should be retained. A
streaming velocity is also found in this case, but its value
is less than that given by equation (2.22). In this case,
streaming is no longer independent of viscosity but becomes a
a.
weak function of parameter ; a6 being the amplitude in the
kN
nearbed motion just outside the boundary layer and kN the bed
roughness (Fredsoe and Deigaard, 1992).
For laminar fluid mud flow relevant to this study, the
total mass transport velocity just outside the boundary layer,
UL, correct to the second order, is therefore the sum of the
irrotational expression given by (2.6) at z0 = h and U_; i.e.,
5 H2ok
U (2.24)
UL 16 sinh2kh
2.2.3 Nearshore Current
In the previous section the mass transport generated by
waves over a constant depth of water was analyzed. However,
coastal areas generally have a bottom slope that causes other
effects, such as breaking, that affect mass transport in other
ways.
The radiation stress is another mean wave quantity
introduced by LonguetHiggins (1953), which represents the sum
of the momentum flux and the mean pressure effects, and which
must be accounted for describing mass transport in the
littoral (surf) zone.
The equations of conservation of mass and momentum are
considered, but instead of using their differential forms,
they will be used after integration over the total
instantaneous depth h + ri (where r~ is the water free surface
displacement) and averaging in time over a wave period.
The mean waveinduced velocities in the horizontal
horizontal directions, x and y (Figure 2.1), can be defined as
(Thornton and Guza, 1989)
Ux = _1 luxdz
h+l h
1 I d
U = 1fu dz
h h
where r is the mean deviation of r from the still water level.
The corresponding velocity components are
x = U +Ux+Ux ;
U = U +i +u
y y y Y
in which denotes oscillatory component and the prime denotes
turbulent component.
Coast
x
S
Figure 2.1 Schematic sketch for an obliquely
incident wave.
(2.25)
(2.26)
25
After integrating the continuity equation over depth and
averaging the resulting expression in time, the following
equation is obtained:
aUx,(h +) aUy (h+) = (2.27)
ax ay
Integrating the momentum equation, the components of the
mean horizontal momentum per unit area in the x and y
directions are obtained as:
au aux au as
p(h+rl) X+U x+U + xx
at ax x ay ax
(2.28)
aS 8 s b
+ x pg(h+l) +Tx Tx
ay ax
[au au au_ as
p(h+)) i ++U +U + x
at ax ax y ax
(2.29)
as a s b
+ = pg(h+ ) +TyT
ay ay 
In equations (2.28) and (2.29), Ts and Tb are the mean
surface and bottom stresses, respectively, and the first term
on the right hand side is the horizontal force per unit area
caused by the slope of the mean water level. The mean momentum
flux Sij (with ij = xx, yy or xy) is defined as
Sij = ij + Sij
The contribution due to
integrated Reynolds stress
eddy viscosity. The wave
radiation stresses and are
turbulent motion, S'ij, is the
and can be parameterized using the
motion contributions are called
given by
T1 rl
S=p f dz+ Pdz Pgh2
h h
TYY =P Pfadz+fPdz 2 Pgh2
h h
XY=p j>fjddz
h
(2.31)
(2.32)
(2.33)
where P is the total pressure. If the linear wave theory is
used, the following expressions are obtained (Dean and
Dalrymple, 1991):
= E[2 c Cos2t+ 2Cg1]
X 2 c c
=s E 2 'sin2Ct + (2fc
yY 2 c
(2.34)
(2.35)
(2.30)
c
= Eg sinacosa (2.36)
xy c
where E is the energy density, c, the group velocity, c the
wave celerity and a the angle of wave incidence with the
crossshore direction. Equations (2.34) through (2.36) show
that the wave momentum fluxes (radiation stresses) are
proportional to the wave energy. Therefore any change in the
energy, such as by dissipation due to breaking, will cause the
radiation stresses to change. These changes must be balanced
by external forces so that the momentum equations (2.28) to
(2.29) are satisfied. That is, inside the surf zone forces are
induced that in turn account for water mass transport in this
zone.
In the crossshore direction, as there is a boundary at
the shore, the balancing force is a pressure gradient
(associated with the wave setup T). Given x the crossshore
direction, the final equation derived from (2.28) is
ap 1 as
ar  (2.37)
ax pg(h+r) ax
In the case of an open and unbounded shore in the alongshore
direction, there is no adverse pressure gradient capable of
balancing the radiation stress. As a result, a mean current is
28
produced that generates a bottom shear stress which balance
the radiation stress.
For a long and straight shoreline that can be considered
to be infinite in extent, all derivatives with respect to y
must be zero, so that the balance of forces in the alongshore
ydirection, when the coupling term is neglected, is
as
x (2.38)
ax
Given the turbulent nature of flow in the surf zone, the mean
bottom shear stress (due to turbulent flow) is usually
expressed by the quadratic law:
?b = pcf Tu (2.39)
where cf is the bed friction coefficient and q is the total
velocity. For the case of small angle of incidence of the
waves, LonguetHiggins (1970) simplified and linearized the
expression for the bed shear stress to
b Pcf
S= PCu Uy (2.40)
Y 411 m Y
where u is the maximum orbital velocity. Solving for U ,
LonguetHiggins gave the following expression:
U() = 5ngKm(h+fr) sina
U ( x) \
S K2) c (2.41)
2cf 1+3 
8
where K is the wave breaking index, m the bottom slope and c
the wave celerity. The index K is given by the breaker
criterion of McCowan(1894), which states that the breaker
height Hb = Khb with K = 0.78; the subscript b denoting the
values at breaking (Dean and Dalrymple, 1991). This criterion
will be used for accounting for the wave breaking effect in
laboratory measurements in Chapter 5.
2.3 Wave Mass Transport over a Fluid Mud Bottom
A twolayered (waterfluid mud) model is now introduced
in order to calculate the momentum fluxes and mass transport
when a fluid mud bottom occurs under a water column. An
inviscid water column and a viscous fluid mud are assumed,
subject to a progressive, monochromatic, nonbreaking wave at
the free water surface (Figure 2.2). For simplicity, both
layers are considered to be of constant depth. First, the
derivation will be carried in the wave propagation direction
s, and the setups at the free surface and the interface will
be derived. Then the bottom stress and mean velocity in the
wave propagation direction in the absence of a setup will be
30
derived. Finally the crossshore and alongshore velocities
under an obliquely incident wave will be considered.
2.3.1 Interface and Surface Setups
In order to obtain mean quantities within the twolayered
domain, the NavierStokes momentum equations are used. With
reference to Figure 2.2, a 2D problem in a vertical plane is
considered. The sdirection is horizontal at the still free
surface along wave progression and z is the vertical
direction, positive upwards from the still free surface.
Figure 2.2 Schematic sketch of a twolayered
fluid flow.
31
Subscript 1 refers to the water region and 2 to the fluid
mud layer. The overbar denotes mean value in time due to
second order effects. Also, rl and '2 are the oscillatory
components of the free surface and interface displacements,
respectively, and rl and r2 are the total free surface and
interface displacements, respectively. The water layer depth h1
and the mud layer thickness h2 are considered constants. Water
is considered to be an inviscid fluid, thus neglecting its
viscosity v1 in the momentum equations. Fluid mud is
considered to be a Newtonian viscous continuum with kinematic
viscosity v.2
The horizontal and vertical components of the general
NavierStokes equations, after introducing the continuity
equation for an incompressible fluid, are
au+ + a +v 2 z (2.42)
8t as 9z p as [s2 2
aw auw aw2 1 ap a2W a2W
w+ + g + (2.43)
at as az p az as 2 az2
where u and w are the velocities in the s and zdirections,
respectively.
32
In order to integrate equations (2.42) and (2.43) in both
the water and fluid mud layers, the following kinematic and
dynamic boundary conditions are required:
At the water free surface the kinematic condition is
1 +u (s, tt) w1(s, rt,t) = 0 (2.44)
at 1s
The dynamic condition at the free surface states that the
pressure at the free surface is the atmospheric pressure po,
which, without any loss of generality, can be considered to be
equal to zero.
At the interface between the two layers the kinematic
condition states that
t +u2(s,zi,t) w2(s,zi,t) = 0 (2.45)
at as
where zi = r2h is the total displacement of the interface
for the coordinate system chosen (Figure 2.2). Also at the
bottom z = h = (hi+h2) both u2 and w2 are zero.
The dynamic condition at the interface asserts that the
stresses in both directions for both fluids must be the same,
as follows:
ufar1
P = ( ss +2) s + z
(2.46)
P1 = zzP2 sz as
in which oi. and .ij respectively are the normal and shear
stresses given by the viscous stress tensor:
au2 1 ( 2 u+ aU2)
sx azz ( ] 2 s(2.4
2 s 9z az
After having established the appropriate governing
equations and boundary/interfacial conditions, the necessary
integral values of the pressure gradients that are required to
solve for the free surface and interfacial setups will be now
determined. The pressure terms are derived first, followed by
setups.
2.3.2 Pressure at a Generic Position z (Water Layer)
In order to obtain the pressure at a generic point z
within the water layer, equation (2.43) is integrated over the
vertical between z and the free surface. For this layer,
34
neglecting the kinematic viscosity of water vi, this
integration yields
t t
0I awl i aUlW1 2 r
p dz+ p dz+pwll = pg9(1z) p1 (2.48)
z z
Then, making use of Leibnitz rule,
a b(x) b(x ) aF(xi,x )
f F(x,xj) dxj= f a dx.
Sa(xi) a(xi) (2.49)
(2.49)
ab(x.) Oa(x.)
+F[xi,b(xi) ] F[xi,a(xi) I
and taking into account the kinematic condition given by
equation (2.44), as well as noting that the pressure is zero
at the free surface, the pressure at a generic position z in
the water region is obtained as
t t
p1(z) pwdz+ puwdzp w +pg (r z) (2.50)
p,(z) =atJ fsfP 1
z z
To obtain the pressure at the interface it cannot be simply
stated that z = zi, because the generic position z is a fixed
coordinate point that does not vary with time or s, whereas
the interface is a function of time and the propagation
direction s, i.e., zi = z.(s,t). Thus, in order to obtain the
35
pressure at the interface, a different integration must be
carried out.
2.3.3 Pressure at the Interface
To obtain the pressure at the interface, equation (2.43)
is integrated between the interface zi = zi(s,t) and the free
surface rl:
f dz+ p dz +p = g(r1z 1 (2.51)
Zi zi
Then, making use of the Leibnitz rule given by equation
(2.49), introducing the dynamic condition p = 0 at the free
surface, and considering the kinematic condition (2.44) at the
free surface and the kinematic condition (2.45) at the
interface, the pressure at the interface is obtained as
t t
P (zi) a= aPw1ldz+sJ pul dz +plg (rzi) (2.52)
zi zi
As can be seen by comparing equations (2.52) and (2.50),
the introduction of the interfacial kinematic condition (2.45)
eliminates the term pw2 in the interfacial pressure. If the
pressure at the interface were computed by considering z = zi
in equation (2.50), that term would have remained because
36
interfacial variation with time and s, and the condition
(2.45) would not have been considered.
2.3.4 Pressure at a Generic Position z (Mud Layer)
For the fluid mud layer, equation (2.43) is integrated
between a generic position z and the interface zi:
2 d dz +pow 2 2 (Zi Z)
SP2t +fP2 dz+P2w2 = P2g
z z
(2.53)
z
P2 z f as z,
and again making use of Leibnitz rule (2.49) and the
kinematic condition at the interface (2.45), the expression
for pressure in the mud layer is obtained as
2
p2(z) = pl(zi) +p2g (ZiZ) P2W2
zi zi (2.54)
z SITdz +
+ p 2dz + p Uw2 dz  IS dz +ozz
at J 2 s 2 u22 9s Js zz
z z z
2.3.5 Vertical Integration of Pressure (Water Layer)
The pressure p, obtained in equation (2.50) must be
integrated over the vertical between the interface zi and the
t
free surface 'ri in order for it to be used in the integrated
momentum equation in the sdirection. This integration yields
t F t
rl, l a n
fP1(z) dz = f tf Pw,, dz dz
Zi Zi z
t t t
+ a Pjjjzd pw2d
f as f f 1,..4wld
Zil Z Zi
(2.55)
+2 t
t z z i
+pg 1z_ 2 
After taking the derivatives outside the integral and
differentiating with respect to s,
t
s fp,(z)dz =
zi
TSat f il"t
+ Oz p [ d
s i /
+az, a
as f J
t
at as
1 2 s
zi
p2gg ad( zi)2
p w, dz +1
1 2 as
Noting that the mean value of the oscillatory terms on the
left hand size of (2.56) over a wave period is zero, and
neglecting terms of order greater than two, becomes
(2.56)
l2 I ni
dS p1(z)dz : 8zi a + 12 1 dz
+ dz pudZ dz (2.57)
Z Z
as 2f P2 ( )2
S PPdw dz + P ad
s J s2 s
zi
__ plg a(qtzi)2
as plwldz 2 as
In this expression the overline stands for a temporal average
value.
2.3.6 Vertical Integration of the Pressure (Mud Layer)
As was done for p, in section 2.3.5, in this section the
pressure p2(z) given by equation (2.54) is integrated in the
mud layer from z = h = (h1+h2) to the interface z = zi. After
taking the derivatives outside the integrals, the following
expression is obtained:
i a
fp2(z)dz = c
h h
+ p2g
2
Zi Zi Zi
h z h
dzfp2u2w2dz fp2w2 dz
h h (2.58)
zi
+f odz+p (z) (z +h)
h
Next, as was done for p,(z), equation (2.58) is differentiated
with respect to s and the mean value over a wave period is
39
obtained. The final temporalmean value of the pressure term
in the mud layer thus is given by
a i a 2 z z P2U2W2 dz I a z
P2 (z) dz = dzf 2u2w2dz 2 w2dz
h h z h
2g a(zi+h) a2 i a (2.59)
2 as as2 z s zz
h z I
a(zi zi+dp (z)
+pl(z) as + (zi+h) as
as as
2.3.7 Free Surface Setup
In order to obtain the free surface setup, the equation
of momentum in the sdirection (2.42) is integrated in the
water layer from the interface zi to the free surface ri:
t t t
l au l Bu2 t 1 ap
Pf dz+ p dz +u w as dz (2.60)
at f as 1 as
zi zi zi
If the derivatives are taken outside the integrals and the
kinematic conditions at the free surface and interface are
introduced, it follows that
dt t t
a III a Ial 2 I~l 9Z
f P1dz + s plul2dz a spldz pl(zi) (2.61)
zi zi zi
After inserting the expression for p1(zi) from equation (2.52)
and averaging over time, the following expression is found:
a f pi u2dz p dz p1g (rt zj) 5i zi a d (2.62)
s 1 as s s  f262wi
i i
When equation (2.57) is introduced and the terms of order
greater than second are neglected, the final expression for
the free surface setup is obtained:
Sa a 2 2
g (h + ) a a 0 (U1 w )dz
as as J
h,
a2 0 0 a1
s2 fdz uwdz g(r 2) as (2.63)
h z
at 0 2s 0 a22 1as
h hi 1
2.3.8 Interface Setup
Next, in order to obtain the interfacial setup, equation
(2.42) have to be integrated in the mud region from the total
depth h to the interface zi. This time, the fluid is
considered to be a viscous fluid. Therefore, the viscous terms
must be taken into account. After taking the derivatives
outside the integrals and considering the kinematic and
dynamic conditions at the interface, the following equation
results:
P2 u2dz +P2 ~u2dz = P2dz
h h h
(2.64)
az. a zI
+p(z + ass dz
Z s ash
Recall a similar procedure as the one used in the water
region, that is, the methodology proceeded to insert p1(zi)
from equation (2.52) and then averaging in time over a wave
period, taking into consideration equation (2.59). The
equation for the interfacial setup is obtained as
au2
(p2P))g(h2+12) aS
h2 h1 h d
fP S (U2 wdz p2 S dz2 uw dz
P hs _h z Z
(2.65)
f dz dz (aOssO)dz
s2 i as _h
h zh
_i __ a9
Pg2as (P2 P 1) 2 g Pg(h2+r2) a
If the relations given in equation (2.47) for the viscous
stresses are considered, after keeping the second order terms,
the final expression for the interfacial setup is
_2
(P2 P)g(h2+2) as
h, 2 hi hi
as as
a h (u2w2)dz a2 / dz lu wdz
P2as f 2 2 P2 aS2 f Jf
h h z
(2.66)
S a + 2 I+22 a z2
r2 112. a + az 2I. a a az
_ _S 1 2 p1g (h2 2
Plgr12 (P2Pl)gO2sPlg(h2+r2) as
The two unknowns in equations (2.63) and (2.66) are the
water surface setup 1l and the mudwater interfacial setup r12
43
In the case of a wave train which approaches the coast at a
certain angle a (Figure 2.1), both crossshore and alongshore
components of waves forces will be present. At an unbounded
coast, in the alongshore direction, a setup cannot occur and
the mean forces must be balanced by a shear stress at the
bottom. This shear stress can only be generated by a mean
current.
2.3.9 Mean Current in the Wave Direction
When a setup cannot occur, the mean shear stress produced
is equivalent to the setup pressure gradient. With reference
to Figure 2.3, the following steady state force balance can be
stated:
2t
Figure 2.3 Forces in sdirection.
i r2
tods = h1+ 1++ ds r a2ds
+h2 2+ ds+2h +1+ 0ds h2n2 2ds (2.67)
2 2 s s 2 2 s 2
(h )2 [h 2 +(h1+ )](h+) 2
r21 2 [h2 + 2h' 2 +12 r+2 2
Neglecting the quadratic terms in ds, one obtains
To(s) = (h +rr2) p,+ (h2+r2) P2g a
(2.68)
+[(h2+nr+hl+n,) p2(hl+rin)p ]g2
The mean bottom stress To, which in effect is equivalent to
the setup pressure gradient, must be generated by a mean
velocity, as shown in Figure 2.4. If a steady state, fully
developed viscous flow is considered in the mud layer, the
differential equation to be satisfied by the mean velocity U
is
d 2U 1 dP
dz2 ds (2.69
This is a Poiseuille flow, in which the total pressure P is
the driving force and a function only of s. Since U does not
dP
vary with s, is a constant and can be expressed as a
ds
function of To.
F/////////////ck
Figure 2.4 Schematic sketch for U,
determination.
Also, the following boundary conditions must be satisfied:
U = 0
at z = (h +h2)
a 0 at z = h
dz 1
(2.70)
Given that
dU
To 92 dz z=(h +h2)
(2.71)
the average mean velocity in the mud layer can readily be
obtained as
T, (s)h2
Um(s) = (2.72)
3 P2
and the mean velocity in the water layer is
To(s)h2
Uw(s) = (2.73)
212
2.3.10 CrossShore and Alonashore Mean Currents
When a wave train approaches a muddy shoreline at an
angle a as shown in Figure 2.1, both a crossshore component Ux
and an alongshore component Uy are present. In the crossshore
direction, when the fluid mud is far from the coast, the wave
effect will move it towards the shoreline until the shore will
prevent further movement of the mud and a setup will build up
in that direction, thus leading to the formation of a convex,
shorefast mudbank (Jiang and Mehta, 1996). In the alongshore
direction the second order effects considered will produce an
alongshore current because in at open coast no setup is
possible. For small angles of incidence of the wave train and
considering a nonreflecting shore, the mean velocities in
both directions can be expressed as
U m(s) = Um(s) cosa;
U (s) = Um(s)sina
(2.74)
47
or, introducing the linearized equation (2.68) and
equation(2.72):
h 2
Um= 3 (hlpl+h2p2)gF+[(h2+hl)p2hpl]g cosa (2.75)
h =a+[(h+h 8)phpg1sin
Umy = 32 (hpl+h2 2)g +[(h2 +h) 2 hl sin (2.76)
It should be again emphasized that equations (2.75) and
(2.76) are valid only for small angles a. Higher values of the
angle of incidence introduce modifications due to refraction
that were neglected in the above derivations. Furthermore,
equations (2.75) and (2.76) were obtained based on the
assumption of a 2D model in the vertical. Thus, no variations
in wave energy were considered in the horizontal direction
perpendicular to s, i.e. along the wave crest. If a is not
small, this assumption is no longer valid because at a given
time t the wave front travels different distances over the
mudbank in the direction perpendicular to s and the wave
height is no longer constant in that direction. In this case,
momentum transfer along the wave crest must be taken into
account.
48
Equation (2.75) and (2.76) yield the desired crossshore
and alongshore mass transport velocities in a twolayer system
inclusive of the irrotational and rotational second order
effects.
CHAPTER 3
LABORATORY EXPERIMENTS
3.1 Introduction
To examine the validity of equation (2.76) derived in
Chapter 2, which gives the alongshore mass transport velocity
in the mud layer, laboratory experiments were conducted in
order to compare the theoretically predicted results with
experimental measurements. The tests were carried out in a
wave basin at the Coastal Engineering Laboratory of the
University of Florida. The experimental setup and conditions
together with the test data obtained are presented in this
chapter. Experimental results along with field data are
analyzed and discussed in Chapter 5.
3.2 Experimental Equipment
3.2.1 Wave Basin
The wave basin used for the experiment measured 3.1 m
wide, 0.9 m deep and had a total length of 33.5 m. A
monochromatic wave generator was located at one end of the
50
basin. It was a pistontype wavemaker consisting of a vertical
paddle driven by a 7 HP motor. The wave amplitude could be
adjusted by changing the piston displacement, and the wave
period by changing the angular frequency of the transmission
from the motor to the piston.
Only a portion of the basin was used for the experiment.
A muddy bottom profile, obliquely placed in relation to
incident waves, was built 16 m from the wavemaker (Figures 3.1
and 3.2).
3.2.2 Wave Gage
A capacitancetype wave gage was used to measure the wave
height (Figure 3.3). The data were recorded on a strip
analogue chart recorder (Figure 3.4). Wave height and period
were measured at two locations situated along the centerline
of the basin. The first, called A (Figure 3.2) had a still
water depth of 0.144 m and was located 12.0 m from the
wavemaker. The water depth at the second site, called B, was
0.114 m and was 1.2 m shoreward of location A.
3.2.3 Velocity Measurement
Alongshore waveinduced mean velocities were measured
using colored styrofoam tracers and recording their movements
with a videocamera. For the purpose of calibration of tracer
movement a linear scale ruler was fixed on the beach (Figure
51
3.5). The tracers used were flat cylinders with a diameter of
2.5 cm and 1 cm width. They were colored in order to
facilitate the measurements.
17.3 m
Z Sill 2m12.Om
4 I 1.2 m 12.0 m
Sill
Sil 16.0 m
Figure 3.1 Plan view of the experimental basin set
up.
Channel
Wavemaker
77 r;
Figure 3.2 Elevation view of the experimental basin
setup.
Figure
Capacitance wave
. .. .............
figure 3.4 Strip chart record
?li~i~'
::
''*' ;;:'':l~ii ~
I:
Figure 3.5 Metric scale ruler
alongshore velocity calibration.
for
3.2.4 Measurina Carriage
Both the videocamera and the wave gage were mounted on
a measuring carriage (Figure 3.6). This carriage was of steel
54
construction mounted on wheels that moved on iron rails placed
on both side walls of the basin.
3.2.5 Other Apparatuses
Other apparatuses to ease the preparation and running of
the experiment were used. Some examples are a water pump,
buckets, spades, rulers, etc. For setting the final profile of
the mud bottom(at a slope of 1:40), a carriagemounted point
gage with a flared tip was used.
3.3 Test Preparation
3.3.1 Profile Sediment
In order to run the experiments a mud profile was built
in the basin. The sediment used for the profile was a clayey
mixture of an attapulgite and a kaolinite in equal proportions
by weight. The mixture was made with commercially available
clays: attapulgite from Floridian Company in Quincy, FL and
kaolinite from Feldspar Corporation in Edgar, FL. Kaolinite is
a pulverized kaolin, which is a light beige powder, and
attapulgite is a greenishwhite powder. The median (dispersed)
size for both clays was approximately 1.5 pm.
Well water was used to mix with the powder mixture in
order to generate mud for the profile. This mud had a
55
relatively high density of 1,355 kg/m3 to ensure that it could
sustain itself in a sloping configuration.
Figure 3.6 Measuring carriage with video
camera and wave gage.
3.3.2 Profile Preparation
Mud profile preparation was carried out by first building
a wall with concrete blocks at the shoreward end of the slope
56
to be constructed (Figure 3.1). This wall was placed at an
angle of 70 with the sides of the basin so that the incident
waves would have an angle of 200 with the beach. This angle
was chosen so as to generate a measurable alongshore velocity
without introducing a high degree of distortion due to wave
refraction. Two sills 10 cm wide were placed on each side of
the back wall in order to allow flow recirculation in a back
return flow channel constructed for this purpose. The back
channel was made with another concrete brick wall 58.5 cm
apart from the previous one (Figure 3.1). In general, due to
the comparatively short length of the mud profile, the
alongshore current cannot reach its equilibrium unless certain
procedures are followed in order to avoid the boundary effects
generated by a finite length basin. Visser (1991) studied
different experimental setups previously used in longshore
currents studies, and found that layouts similar to the one
used in the present study usually work adequately provided the
return flow in the offshore region is properly reduced by the
recirculation procedure as devised.
3.3.3 Profile Measurement
Profile depths were measured in a rectangular grid
pattern with 20 cm spacing. Profile depth was measured at each
grid position by lowering the point gage until its flared tip
touched the muddy bottom. Using the moving carriage, depths
57
were determined throughout the gridded area. With these data
three representative profiles were obtained, one at the center
and the two others 50 cm on each side. In Figure 3.7 the three
profiles are shown. As can be seen in each case there is a
nearshore segment of steeper slope and a offshore segment with
a mean slope of 1/40. The steeper nearshore segment was built
in order to reduce any surf zone that would have occurred
invariably, and to have the desired water depth in the basin.
15
E Western profile
10 ..Center profile
Eastern profile
( 5
SStill water level
0
5
>10
15...
0 50 100 150 200 250 300
Crossshore distance (cm)
Figure 1.7 Representative bottom profiles.
3.3.4 Fluid Mud Preparation
Due to limitations in the basin size, wave
characteristics and test durations, fluid mud could not be
generated insitu by the action of waves on the mud bottom.
58
Therefore, fluid mud was prepared in large containers and then
poured over the profile layout. The final fluid mud layer was
2.5 cm thick in the mean with a density of 1,150 kg/m3.
Additional fluid mud forming a 12 cm thick layer over the 2.5
cm layer was placed along the updrift end to serve as a
sediment source (Figure 3.1).
3.4 Experimental Procedure and Data
Once the bottom profile was prepared, the basin was
filled with well water. Some preliminary tests without fluid
mud were carried out in order to check for the effectiveness
of model design and measuring procedures. Waves with periods
ranging from about 1 to 4 s and heights of 3 to 6 cm at a
water depth of 14.4 cm at the toe of the profile were used.
The flow recirculation setup proved to work generally without
the need of forcing the flow. However, it was observed that
generating a wave train over durations exceeding a few waves
led to return flows and basin circulation which interfered
with the alongshore flow. As a result, it became necessary to
restrict the duration of each test to a few waves only. The
procedure was to run five to six waves and to use the last
three for the measurements. The use of floating tracers for
the velocity measurement proved to work reasonably well. Data
59
scatter in these cases was not overly significant and with
different tests the results obtained were similar.
The procedure devised to obtain the alongshore velocity
was to carefully place the colored tracers at a distance of
approximately 40 cm from the shore. This distance guaranteed
that the complete littoral zone was covered, and that tracer
movements were within the range of the video camera. During
the processing of the data three tracers that moved within the
camera range were chosen, and a representative alongshore
water layer velocity V was obtained by averaging the
velocities of these three floats.
During the preliminary tests some wave reflection from
the basin walls as well as from the shore was observed.
Lateral reflection was produced along the walls mainly due to
a slight wave refraction caused by the oblique wave incidence.
The shore reflection was due to the steeper profile slope near
the shore. Horsehair was put against the basin walls along
the profile and at the shoreline to minimize wave reflection.
3.4.1 Experiments without Fluid Mud
The first set of tests were run without the addition of
fluid mud. The purpose was to determine the effect of any wave
breaking on the alongshore fluid mud current in the subsequent
tests with fluid mud. Twelve different tests were performed in
this situation following the procedure explained below.
60
Three different wavemaker displacements and, for each,
four different periods were run. For each test condition the
measuring carriage was positioned at the deepest wave
measuring point and the offshore wave height HA was measured
there. Then, the carriage was moved so that the wave gage was
positioned at the second site. With the carriage there the
wave height H, was measured, and also the alongshore velocity
was measured by videotaping of the floating tracer movements.
In Table 3.1 the wave period T and wave height at both
locations and the mean alongshore velocity V are given.
Table 3.1 Measured data without fluid mud
Test T (s) HA (cm) He (cm) V (cm/s)
1 1.1 3.6 4.0 6.5
2 1.9 4.9 6.6 5.4
3 2.7 5.4 6.2 4.6
4 3.9 4.2 4.8 3.5
5 1.1 3.3 4.3 4.9
6 1.9 5.4 5.8 7.3
7 2.8 3.8 4.2 3.5
8 3.9 3.2 3.5 2.9
9 1.4 6.0 7.0 4.9
10 2.0 4.3 4.7 4.2
11 3.0 3.3 3.7 3.3
12 3.9 3.0 3.5 2.8
3.4.2 Experiments with Fluid Mud
After the experiments without fluid mud were completed
and the data analyzed, fluid mud was placed over the profile
with the help of buckets and distributed uniformly. In this
situation as well, twelve different experiments were
conducted: four different periods for three different
wavemaker displacements. Fluid mud samples at the downstream
sill were collected using a syringe pump. These samples were
used to check the flowing fluid mud density in order to
confirm that the mud was moving without dispersing into the
water column. In Table 3.2 the data obtained in this phase of
the experimental study are given: wave period T wave height
at sites A and B, HA and HB, respectively, and the measured
mean alongshore velocity Vfm.
3.5 Data Preparation and Corrections
The wave data obtained from the set of experiments
without fluid mud were used to calibrate for the correction to
wave height due to shoaling. This correction was needed
because sites A and B had different water depths due to the
sloping mud profile. In order to compute the damping caused by
fluid mud between those two sites, the shoaling effect had to
be considered. Furthermore, the velocity data from these tests
62
were used to calibrate the friction coefficient cf in the
alongshore current due to wave breaking in equation (2.41). In
sections 5.4.1 and 5.4.2 both approaches are explained. It
should be pointed out that wave breaking was observed even
when fluid mud was present, because the incoming wave energy
could not be absorbed completely over the short distance of
wave propagation. It therefore became necessary to account for
wave breaking in calculating the alongshore mud mass transport
velocity due to nonbreaking wave. The procedure by which this
was accomplished is described in section 5.4.4.
Table 3.2 Measured data with fluid mud
Test T (s) HA (cm) HB (cm) Vfm (cm/s)
1 1.4 3.0 3.7 3.7
2 1.8 4.4 5.3 4.5
3 2.7 3.9 4.4 4.2
4 3.9 2.8 3.5 3.9
5 1.2 2.8 3.0 4.1
6 2.3 3.2 3.4 3.9
7 3.1 2.5 3.4 3.8
8 3.9 2.3 2.6 2.7
9 1.2 1.6 2.1 3.1
10 1.9 3.0 3.2 3.2
11 2.6 2.1 2.4 2.7
12 3.9 1.6 2.3 3.2
CHAPTER 4
FIELD DATA ON MUD TRANSPORT
4.1 Introduction
Field information from three different locations already
mentioned in Chapter 1 was used. These locations were selected
because of the good quality data available, and the presence
of large migrating banks of fluid mud at these locations. The
three sites are: 1) the northeastern Atlantic coast of South
America along the Guyanas, 2) The Louisiana coast along the
Gulf of Mexico and 3) The southwest coast of India.
These shorelines are characterized by great quantities of
suspended sediment and by the absence of high waves in the
nearshore mudbank zone. Mudbanks of fine sediment are formed,
in which the waves are highly attenuated. These banks occur
between regions where large waves can cause severe erosion of
sandy profiles.
4.2 Northeastern Coast of South America
The northeastern coast of South America, on the order of
1,500 km of shores between the Amazon and the Orinoco Rivers,
63
64
constitutes one of the world's longest open ocean mud
shorelines. It is formed by the ParaAmazonOrinoco River
plain (Augustinus, 1980), and is characterized as a chenier
plain (Figure 4.1). A chenier is defined by Otvos and Price
(1979) as: "a beach ridge, resting on silty or clayey
deposits, which becomes isolated from the shore by a band of
tidal mudflats." When there are more than one chenier
separated by muddy deposits, a chenier plain is formed. Such
a plain will develop when large amounts of river mud are
available for alongshore transport, and intermittent supply of
sand is available (Augustinus, 1989).
60' 54 506oo0 55ss '
,h My R. R4. /r
Mud coast
S 6 ATLANTIC OCEAN
Ou t AMAAMONMUD
SHELF MUDS
TURTLE '' lo
o.EBAN A.A.,BNK WIA WIA' l 6R00'
CORANTIJN
RIVER, COPPENAME .I,
Inlertidol and shallow sublidal mudch SURINAM RIVER' F N H 530
o 10 30 O 50' GUIANA
FIG. 2. Northeastern South America showing location of mudbanks from Amazonderived sediments.
Figure 4.1 Northeastern coast of South America (after Wells,
1983).
In the case of the northeastern coast of South America
the muddy sediments are supplied by the Amazon River
65
discharge, while the coarse sediment is from local sources.
The fine sediments from the Amazon River predominate along the
shoreline up to the 20 m bathymetric contour, which on average
is 40 km from the coastline, where relict sands are found
(Augustinus et al., 1989).
The Amazon River discharges yearly an estimate of 11 to
13 x 108 tons of suspended load (Eisma et al., 1991) into the
Atlantic Ocean. About 20% of this discharge is transported
northwesterly along the Guyanas coast, and is deposited mostly
in the Orinoco River delta. This transport takes place partly
in suspension, approximately 1.5 x 108 tons, and partly as
migrating mudbanks attached to the shore, approximately 1 x
108 tons.
Overconcentration of silt and clay in suspension
produces a fluid mud that has a residual movement alongshore.
This mud dissipates the energy of the incoming waves,
protecting the shoreline against erosion and favoring
sedimentation along the western side of each mudbank. Along
the eastern side of the mudbanks, increasing consolidation of
the fine sediment decreases the wave dumping property and
increases wave breaking, thus causing erosion and a steady
westward migration of the mudbank. Over the Surinam coast such
mudbanks move at an average rate of 1.5 km/year, having a
range from 0.5 km/year to more than 2.5 km/year.
66
In the interbank areas, where erosion of the
consolidated clay take place, cheniers are formed. The sand
required for this purpose is supplied by the local rivers
discharges.
The spacing between mudbanks varies from 30 to 60 km and
they are 36 km long on the average, ranging from 20 km to more
than 50 km. The interbank region has the same magnitude, so
that the occurrence of the mudbanks is so regular that they
can be considered as mudwaves with wavelength of 45 km and
period of 30 years. Fluid mud thickness in these regions
varies from 0.5 to 2 m, and its density ranges from 1,030 to
1,250 kg/m3.
The predominant sediment is clayey with 40% illite, 30%
kaolinite, 17% smectite (montmorillonite) and 13% chlorite
(Froidefond et al., 1988) with a mean (dispersed) particle
size of 0.51 um.
The continental shelf near the shore, where the mudbanks
are present, has smooth slopes ranging from 1:1600 to 1:3000.
The tidal range varies from 1 m at neap to 3 m at spring,
averaging 1.8 m, and consequently a very wide intertidal area
with mudflats has developed. Thus the coastal environment is
generally microtidal.
The wave energy is low to medium and waves are generated
by relatively constant NE trade winds. Very few wave
67
measurements are available for this region. From data obtained
during August 1977 (Wells and Kemp, 1985) the root mean square
height, Hms, 22 km offshore was 0.93m, and the wave spectra
showed a combination of swell waves with periods T = 910 s,
and sea waves with periods T = 35 s.
The northwestern migration of the mudbanks and their
linkage with the alongshore mud drift show very complex
dynamics. Oceanic currents, like the Guyana current, as well
as tideinduced and waveinduced flows combine to cause the
general migration of the Amazonborne sediments. However, in
the shallow coastal waters of the Guyanas, waves seem to play
a greater role in fine sediment transport. In this area the
Guyana current deviates from the shoreline, so that its
relative importance diminishes. Furthermore, the difference in
the migration velocities of the mudflats of Surinam and Guyana
can be related to the different angle that the coast at each
location subtends with the direction of prevailing wind. The
mudbanks translate westward along the coast of Guyana at a
rate that varies between 0.4 and 2.0 km/yr. The prevailing
wind (and wave) direction with respect to the normal to the
shoreline is 450 in Surinam and 25 in Guyana (Augustinus,
1987).
In Table 4.1, characteristic data for this region are
presented.
68
Table 4.1 Characteristic data for the muddy coast environment
of the Guyanas.
Factor Range
Tidal range (m) 1.03.0 (1.8 mean)
Wave height (m) 0.51.5
Wave period (s) 510
Slope 1/1,6001/3,000
Fluid mud thickness (m) 0.52.0
Fluid mud density (kg/m3) 1,0301,250
Grain size (pm) 0.51.0
Mudbank crossshore length (km) 1030
Mudbank alongshore length (km) 1050
Mudbank velocity (km/year) 0.52.5
4.3 Louisiana Coast
Along the western coast of Louisiana lies a chenier plain
with similar characteristics as that along the northeastern
coast of South America. Here, the Mississippi River via its
Atchafalaya distributary supplies large amounts of fine
sediment (Figure 4.2).
Fluid mudbanks, backed by salt marshes and chenier
ridges, extend west of the Bayou Canal for approximately 20 km
(Wells, 1983). This fluid mud extends 0.53 km seaward and
beyond it shelf sand occurs. The mudbanks have an alongshore
length ranging from 1 to 5 km, and a wide span of width
varying from as low as 20 cm to as much as a 1.5 m. They are
69
mainly composed of montmorillonitic clays with some illite and
kaolinite, and have a median (dispersed) grain size of 3 to 5
pm. The fluid mud has a bulk density of 1,150 to 1,300 kg/m3
and moves over a denser clay base with a density greater than
1,600 kg/m3. The slope in the seaward direction is on the
order of 1/1,000.
92W 92010'
Fluid muds SOUTHWEST
SFine sands LOUISIANA
SBrackish marsh FRESH WATER O4ENIER RIDGES
BAYOU CANAL
~ Zone of interfidaol mudflats
0 1 21 3 4 5
FIG. I. Louisiana chenier plain showing fluid muds derived from Atchafalaya Bay, to the east.
Figure 4.2 Louisiana chenier plain (after Wells, 1983).
The Gulf of Mexico coast of Louisiana is a microtidal,
storm dominated environment. The tide is diurnal with a mean
range of 0.5 m but its regime is complicated by storms,
including extratropical cyclones during winter and tropical
cyclones in the summer, which generate meteorological tides
cyclones in the summer, which generate meteorological tides
70
superimposed on the astronomical conditions (Penland and
Suter, 1988).
The energy levels caused by winds and waves are low,
except during the passage of storms. The dominant wave
approach is from the southeast with an average wave height of
im and period of 5 to 6 s, offshore of the Mississippi delta
(Penland and Suter, 1988).
The mudbanks show a shift to the west with both
continuous and discrete movements. The average velocity of
this movement is 1 to 3 km/year (Wells, 1983). This movement
is very likely produced mainly by waves affecting the
nearshore circulation during storms. It must be noted that
because tide in the region has a very low range, it produces
weak tidal currents. Furthermore, due to the relatively short
length of the mudbanks in the crossshore direction in the
area, the Gulf Stream current does not affect them
significantly.
Relevant data for the Louisiana coast are presented in
Table 4.2.
4.4 Southwest Coast of India
The mudbanks generated along the southwest coast of India
(Figure 4.3) are very different in nature from those described
before. They form very rapidly during the monsoon season (from
71
about May to August) when high wave energy is available, and
dissipate when the waves decrease towards the end of the
monsoon. These mudbanks do not occur adjacent to large rivers
which could supply sediment. In fact, the fine sediment source
is an offshore mudpool that occurs in depths on the order of
1020 m. High waves occurring during the monsoonal season
transport the sediment as fluid mud towards the shore (Mathew,
1992) over a harder sandy shelf bottom (Figure 4.4). Several
such mudbanks appear along the coast with dimensions of 25 km
alongshore, 0.53 km crossshore and having a mud thickness of
1 to 2 m.
Table 4.2 Characteristic data for the muddy coast environment
of Louisiana.
Factor Range
Tidal range (m) 0.5 0.8
Wave height (m) 0.11.0
Wave period (s) 56
Slope 1/1,000
Fluid mud thickness (m) 0.21.5
Fluid mud density (kg/m3) 1,1501,300
Grain size (pm) 35
Mudbank crossshore length (km) 0.53
Mudbank alongshore length (km) 15
Mudbank velocity (km/year) 0.53.0
72
Through the entire period of a fully developed mudbank,
the mineralogical composition of fluid mud is about 60%
montmorillonite, 30% kaolinite and 7% illite. The bulk density
spans from 1,080 to 1,300 kg/m3 and the median(dispersed)
grain size varies between 0.5 and 3 pm (Mathew, 1992).
Figure 1. Locations of monsoonal mudbanks along the south
west coast of India.
Figure 4.3 Southwest coast of India (after
Mathew et al., 1995).
During the nonmonsoon period the wave energy is
relatively low, with a significant wave height of less than
0.75 m and wave period ranging from 10 to 18 s. During the
monsoon season the significant wave height varies from 0.05 m
73
to 2 m with a period of 7 to 9 s. As Mathew (1992) has noted
the highest and lowest values of the significant wave height
both occur during the monsoon season. The lowest value occurs
on the mudbank where much of the incoming wave energy is
absorbed. The significant wave height offshore and outside the
mudbank region during the rough monsoon season ranges from 1
to 2 m. The direction of wave propagation is nearly normal to
the coast and the tidal energy in the region is very low with
an average range of 0.6 m (Mathew, 1992).
More Damped
Wave with Less Damped
Smaller Breaker mWave with
soHeight Greater Breaker
Height
Water
Mudbank
Hard Bottom
Figure 4.4 Schematic mudbank along the southwest coast of
India (after Mehta et al., 1996).
The mudbanks rest on a sandy bottom ranging in slope from
1/500 to 1/1,000, and their migration velocity in the cross
shore direction during their propagation towards the shore at
74
the beginning of the monsoon season fluctuates between 1.2 and
2.4 km/day.
Table 4.3 presents significant data for the mudbanks
along the Kerala coast in southwest India.
Table 4.3 Characteristic
of southwest India.
data for the muddy coast environment
Factor Range
Tidal range (m) 0.5 0.7
Wave height (m) 0.052
Wave period (s) 79
slope 1/500 1/1,000
Fluid mud thickness (m) 1.02.0
Fluid mud density (kg/m3) 1,0801,300
Grain size (nm) 0.53.0
Mudbank crossshore length (km) 0.53
Mudbank alongshore length (km) 28
Mudbank velocity (km/day) 0.30.9
CHAPTER 5
DATA ANALYSIS
5.1 Introduction
To assess the validity of equations (2.75) and (2.76)
derived in Chapter 2 for the crossshore and alongshore
velocities, respectively, the data obtained from the
laboratory experiments given in Chapter 3, as well as the
field data available from the sites described in Chapter 4,
were used. This chapter describes the processing of the data
used as input for these equations, inclusive of processing
necessary to obtain comparable "measured" alongshore
velocities from the laboratory experiments.
5.2 Dimensionless Forms of Mass Transport Equations
In order to simplify equations (2.75) and (2.76) so that
their use and interpretation are easily accomplished, they
have been rendered in more useful forms by means of non
dimensionalization. This was achieved by rendering the
variables dimensionless, by dividing them by the appropriate
76
characteristic reference parameters. In general, these
parameters must be chosen in such a way that all the
quantities that include dimensionless variables are
comparable, so that the order of each term is only given by
the dimensionless coefficient that occurs as multiplier of
each variable term.
The selected reference parameters are given in Table 5.1,
and in Table 5.2 the different dimensionless variables are
noted.
Table 5.1 Characteristic reference parameters.
Quantity Reference
parameter
vertical length Ho
horizontal length gT2
time T
density Pi
velocity H0/T
shear stress T P2gT/H0
First, using the dimensionless variables presented in
Table 5.2, the equations for the free surface (2.63) and
interfacial setup (2.65) are nondimensionalized, thus
obtaining equations (5.1) and (5.2) after linearization.
Table 5.2 Dimensionless variables.
Quantity Definition
ar* a,b /H
hi* hi/Ho
s* s/gT2
z* z/Ho
__ To oHo/P2__gT
P2 P2/P1
ui* uiT/Ho
i* wiT/Ho
a Superscript denotes a dimensionless variable.
b Subscript I is 1 for water and 2 for fluid mud.
a, 1* 1 a ( *2 *2
hi (  (I1)  (u wl ) dz
s as gT 2 s* h
h
H a2 0 HO 0
H Ho al2 a
( dz *I wdz* w i dz
gT 2 as *2 f) t 9 2 s
S hi z ih
(H a2r2* 0 H) ar 0
H0 Ho 2 0 g 2 a a ,
wl dz wl dz
gT2 at*asT2 as t* f*
h h
(5.1)
Equation (5.1) is for the free surface setup. For the
interfacial setup, the nondimensionalized expression is given
by equation (5.2):
78
ar2 1 ar1 1 2 r 1 2
as* p21 as* h (p21) as' 2h as
p2 o a 2 *2 ,
h2 (p ) gT2 h
P ( aH2 2 h hi
2 dz U2 W2 dz*
h2 (2 1) gT2 as*2h z (5.2)
M 2 1 h 2 Z 9 U
P2 1 Ho a2 H aw au2
2 gT2 I
2 H*o 1 2 1 _ 2 2 Iz=h,
PlHogT2 2 ) gT as 2 2 gT2 as az*
S2112 H_ 2 z=ha r _
H0 au2 aw2
pHOgT2 h2 (P1) as gT2 as az*
After linearizing equation (2.68) for the bottom shear
stress and substituting it in (2.72) for the mean mass
transport velocity in the mud layer, an expression for the
dimensionless velocity Um(s)in the mud layer as a function of
the dimensionless setups given by equations (5.1) and (5.2)
can be found as
(h2 1_ 2rl2
Um(s) h PH (h+p2h2)+ (h+h)p2h 31 (5.3)
3 V12T as* Jas*
and the corresponding crossshore and alongshore components
are obtained as
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h pH a8i2 (s)(s)
Sl2(h2+ph) aI(S) [h;h; P*h rl(s) cosa (5.4)
h =H (s) as2*
UH (s) (h1 +p2h2) a+p ~ s) + (h +h')p;h  s*ina (5.5)
u3(s ,T a: asss
From this nondimensionalization three dimensionless
numbers result, which are given in Table 5.3. The first is the
wellknown wave steepness parameter, which relates the wave
height to the wave length. The other two numbers, as
rearranged in Table 5.3, are seen to be characteristic
Reynolds numbers.
Table 5.3 Dimensionless parameters.
Name Definition
Wave steepness Ho/gT2
parameter
Wave Reynolds H gT/(P2/P1)
number
Streaming Reynolds Ho (Ho/T)/(p(2/pi)
number
The second number (wave Reynolds number), in which the
velocity is related to the wave celerity, gives the ratio
between viscous diffusion and wave propagation, and the third
number, which characterizes velocity related to particle
movement, compares viscous diffusion in the mud layer to
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particle movement generated by the wave. This last number,
herein called streaming Reynolds number Res, appears in
equation (5.3) as the only multiplicative coefficient, hence
it can be used as the characteristic flow parameter for mass
transport.
5.3 Simplifications of Mass Transport Equations
As far as this study is concerned, recognizing the range
of values of the dimensionless parameters of interest, some
simplifications to equations (5.1) through (5.5) can be made.
An inspection of these equations shows that the wave steepness
parameter is the most important coefficient in equations (5.1)
and (5.2), because in these equations only the wave Reynolds
numbers appears, and is always associated with the wave
steepness.
For the range of values of wave characteristics
corresponding to the field data, including the three cases
selected, the wave height Ho varies from 0.1 to 2.0m and the
wave period T has a range of 5 to 10s. With these values the
range of the wave steepness parameter is between 0.001 to
0.04. This demonstrates that even the highest values of the
field wave steepness are very small, and that the terms that
include it as a coefficient can safely be neglected. It must
be recognized that the range of wave steepness was obtained
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using the extreme combination of wave height and period, which
in reality is not the case, and in fact the real values will
be around 0.01 or less.
In the laboratory experiments the steepness parameter was
never greater than 0.003, so that the terms that include it as
a coefficient can be neglected in that case as well.
Based on the above considerations, the simplified
respective expressions for the dimensionless crossshore and
alongshore velocities in the mud layer that are finally
obtained are
i P1Ho h2 p (2p2) +h; (p21) +p;h;h; al (s)
UL(s ) r 1  2 S)
3 1 2T h* (p*1) as
(5.6)
h p2 (2p2) a r.(s) 1 ;(S)
*  1(s) (h2+hi) p2h ln(s)  cos
h (p21) as* as*
S pH 2 h2 p2 (2p) +h* (p;1) +p2h *h2 ( a) r (s)
U,(s) r (s)
3 2T h (p21) as
_____ (5.7)
h2p2* (2p2*) all s) [ 1 9^ {(S)
2hp 1) 1(s) *(h +h1) p2h;] a2( s)
* (I +h) sin*I
h,* (p21) as as *
Equations (5.6) and (5.7) will be used hereafter for
application to laboratory and field data.
The interfacial oscillatory movement and the damping
coefficient k. are obtained from Jiang's model (1993). This
1
82
model considers that the free surface and the interfacial
oscillatory movements are of the form
= ei(ksat) (5.8)
1 2
12 =bei(ksat) (5.9)
where k = (kr,ki) is the complex wave number,kr is the wave
number and b the interfacial amplitude. Using expressions
(5.8) and (5.9), the mean correlation terms in equations (5.6)
and (5.7) can be computed as
a k 2 2kis(5.10)
I e (5.10)
as* 8
all kigT2b 2kis
r 12  e (5.11)
as* 4Ho
a, 2b* kigT2b2 2kis
r12 2 e (5.12)
as* 2 H2
5.4 Laboratory Data Analysis
The data obtained from the experiments described in
Chapter 3 are analyzed in the sections below. First, with the
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data obtained from the set of experiments without fluid mud,
a correction for shoaling is developed, and the friction
coefficient cf in equation (2.41) is obtained. Using the wave
height data measured with fluid mud present and considering
shoaling, a value of the dynamic viscosity of the fluid mud p,
is calculated. Finally, the alongshore velocity due to mud
effect is obtained as the measured value minus the breaking
component. Then, making use of equation (5.7) the measured and
calculated values are compared.
5.4.1 Correction for Shoaling
The wave height data were corrected for shoaling using
the stream function wave theory. This was done because the
linear wave theory had to be discarded as a result of the
significant nonlinearities present in the wave records, and
the poor results obtained with this theory. For shoaling it is
required that (Dean, 1974)
E g = constant (5.13)
where
E = Elinear (5.14)
Here c, is the group celerity, B is a correction coefficient
and E and Elinear are the wave energy from the stream function
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and linear wave theories, respectively. Then, the wave height
at a given location as a function of the wave height at
another location may be computed as
H = H A CgA (5.15)
B A B cgB
In order to calculate 3BCgB the wave height HL is needed.
Hence, an iterative procedure must be used. The first value
used in this procedure was the one obtained with the linear
wave theory. Using this value of the wave height, PBCgB was
obtained using the stream function wave theory and HB
corrected. PACgA was calculated using the stream function wave
theory using the wave height HA, the period T and the water
depth hA as inputs.
In order to check the procedure, data obtained without
fluid mud at sites A and B were used, choosing site A to be
the characteristic offshore position for computations. Figure
5.1 shows a comparison between the measured wave height at
position B, i.e. H and H'B, obtained by using the values
measured at position A as input to the stream function theory
for shoaling the wave between A and B. The computed and
measured values agree reasonably well with one exception.
