A mechanism for non-breaking wave-induced transport of fluid mud at open coasts

UFLICOEL-97/004

A MECHANISM FOR NON-BREAKING WAVE-INDUCED
TRANSPORT OF FLUID MUD AT OPEN COASTS

by

Hugo N. Rodriguez

Thesis

1997

A MECHANISM FOR NON-BREAKING WAVE-INDUCED 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 Cross-Shore 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 Surinam-Guyana, 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 two-layered
fluid flow . . . . . . . ... .30

2.3 Forces in s-direction . . . . . . 43

2.4 Schematic sketch for Um determination . . 45

3.1 Plan view of the experimental basin set-up 51

3.2 Elevation view of the experimental basin
set-up . . . . . . . . . . 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 near-bed 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 s-direction

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 s-direction

= Eulerian velocity vector

u, = Irrotational part of u

U = Lagrangian velocity vector

U = Time-average Lagrangian component in the s-
direction

um = Maximum orbital velocity u

uR = Rotational part of u

ux = Velocity component in the x-direction

fx = Oscillatory component of ux

ux' = Turbulent component of ux

uy = Velocity component in the y-direction

Uy = Oscillatory component of uy

uy' = Turbulent component of uy

ui = Velocity in s-direction in the water layer

u* = Dimensionless velocity in s-direction in the
water layer

2 = Velocity in the s-direction in the fluid mud
layer

u2 = Dimensionless velocity in the s-direction 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 s-direction

Ur = Depth-average value of U

Un = Depth-average value of U in the cross-shore
x-direction

U = Depth-average value of U in the alongshore
y-direction

U = Second order mean velocity in the water layer
in the s-direction

Ux = Depth-average velocity in the x-direction

Uy = Depth-average velocity in the y-direction
UM* = Dimensionless depth-average value of U

Um* = Dimensionless depth-average value of U in the
cross-shore direction

U* = Dimensionless depth-average value of U in the
alongshore direction
U = Mean second order velocity inside the boundary
layer in the s-direction

U. = Streaming velocity in the s-direction

U, = Total average mass transport velocity in the
s-direction

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

= Time-average Lagrangian component in the z0
direction

w = Velocity component in the z-direction

wI = Irrotational part of w

WR = Rotational part of w
w, = Velocity component in the water layer in the
z-direction

w = Dimensionless velocity component in the water
layer in the z-direction

w2 = Velocity component in the fluid mud layer in
the z-direction

w = Dimensionless velocity component in the fluid
mud layer in the z-direction

x = Horizontal axis in the cross-shore 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 non-linear 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 NON-BREAKING WAVE-INDUCED 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 non-breaking wave-induced 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 cross-shore 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 one-quarter 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. Wave-mud 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 non-breaking 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 to-and-fro in the cross-shore

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

shore-fast 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 wave-dominated

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

disk-like 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 cross-shore 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 (Surinam-Guyana, 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 Surinam-Guyana, 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 non-breaking wave-induced forcing

may be the dominant cause of transport has been explored. The

action of progressive waves in a two-layered (water-fluid 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 depth-average 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 cross-shore 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 one-layer

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 cross-shore

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 two-layered 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 non-linear

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 t-t0. The

14

relation between these two descriptions of the same physical

phenomenon is

S= K(x0, t-t0) ; x(x, 0) = x0 (2.1)

As the velocity of a fluid element is the time-derivative of

its position,

x (x,, t-to)
1(Xo,t-to) = 0 -t (2.2)

one obtains

t
S= x0 + f u (o, t-to) 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, t-t0)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,t-to) 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

non-zero 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 wave-mean value is zero in that region. However, in

the region between the trough and the crest, the wave-mean

velocity must be obtained by means of Taylor expansion and is

found to have a non-zero 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 no-slip condition at the bottom must be

considered and, therefore, the viscous stress term in the

17

Navier-Stokes 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

s-direction can be neglected with respect to derivatives in

the vertical z-direction, so that the Navier-Stokes 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 Navier-Stokes equation in the z-direction

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 z-direction 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 ks-ot+- 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(ks-ot+ --) -cos (ks-at--
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)
[3-2( z+2 e cos( ) -2( -1 e-W 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 non-zero 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
near-bed 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 Longuet-Higgins (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 wave-induced 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+ ) +Ty-T
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+ 2-Cg1-]
X 2 c c

=s E 2 -'sin2Ct + (2f-c-
yY 2 c

(2.34)

(2.35)

(2.30)

c
= E-g 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

cross-shore 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 cross-shore 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 cross-shore

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

y-direction, 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, Longuet-Higgins (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 ,

Longuet-Higgins 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 two-layered (water-fluid 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, non-breaking 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 cross-shore 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 two-layered

domain, the Navier-Stokes momentum equations are used. With

reference to Figure 2.2, a 2-D problem in a vertical plane is

considered. The s-direction 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 two-layered
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

Navier-Stokes 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 z-directions,

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 = r2-h 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 = zz-P2- 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(1-z) -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+ puwdz-p 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 (r-zi) (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 (Zi-Z) -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 s-direction. 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

TS-at 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(qt-zi)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 temporal-mean 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 s-direction (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 as-h

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
(p2-P))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 (aO-ss-O)dz
s2 i as _h
-h z-h

_-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 (u2-w2)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- + a-z 2I. a a az

_- _S 1 2 p1g (h2 2
-Plgr12 (P2-Pl)gO2--s-Plg(h2+r2) as

The two unknowns in equations (2.63) and (2.66) are the

water surface setup 1l and the mud-water interfacial setup r12

43

In the case of a wave train which approaches the coast at a

certain angle a (Figure 2.1), both cross-shore 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 s-direction.

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 +r-r2) p,+ (h2+r2) P2g a
(2.68)
+[(h2+nr+hl+n,) p2-(hl+ri-n)p ]g-2

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 Cross-Shore and Alonashore Mean Currents

When a wave train approaches a muddy shoreline at an

angle a as shown in Figure 2.1, both a cross-shore component Ux

and an alongshore component Uy are present. In the cross-shore

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,

shore-fast 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 non-reflecting 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)p2-hpl]g cosa (2.75)

h =a+[(h+h 8)p-hpg1sin

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 2-D 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 cross-shore

and alongshore mass transport velocities in a two-layer 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 set-up 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 piston-type 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 capacitance-type 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 wave-induced mean velocities were measured

using colored styrofoam tracers and recording their movements

with a video-camera. 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
set-up.

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 video-camera 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 carriage-mounted 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 greenish-white 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 set-ups 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
Cross-shore 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 in-situ 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 1-2 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 set-up 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. Horse-hair 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 video-taping 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 non-breaking 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 Para-Amazon-Orinoco 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 6-R00'

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 Amazon-derived 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.

Over-concentration 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 inter-bank 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 inter-bank 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 0.5-1 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 = 9-10 s,

and sea waves with periods T = 3-5 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 tide-induced and wave-induced flows combine to cause the

general migration of the Amazon-borne 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) 1.0-3.0 (1.8 mean)

Wave height (m) 0.5-1.5

Wave period (s) 5-10

Slope 1/1,600-1/3,000

Fluid mud thickness (m) 0.5-2.0

Fluid mud density (kg/m3) 1,030-1,250

Grain size (pm) 0.5-1.0

Mudbank cross-shore length (km) 10-30

Mudbank alongshore length (km) 10-50

Mudbank velocity (km/year) 0.5-2.5

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 0.5-3 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 extra-tropical 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 cross-shore 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 mud-pool that occurs in depths on the order of

10-20 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 2-5 km

alongshore, 0.5-3 km cross-shore 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) 0.1-1.0

Wave period (s) 5-6

Slope 1/1,000

Fluid mud thickness (m) 0.2-1.5

Fluid mud density (kg/m3) 1,150-1,300
Grain size (pm) 3-5

Mudbank cross-shore length (km) 0.5-3

Mudbank alongshore length (km) 1-5

Mudbank velocity (km/year) 0.5-3.0

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 non-monsoon 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) 0.05-2

Wave period (s) 7-9

slope 1/500 1/1,000

Fluid mud thickness (m) 1.0-2.0

Fluid mud density (kg/m3) 1,080-1,300

Grain size (nm) 0.5-3.0

Mudbank cross-shore length (km) 0.5-3

Mudbank alongshore length (km) 2-8

Mudbank velocity (km/day) 0.3-0.9

CHAPTER 5
DATA ANALYSIS

5.1 Introduction

To assess the validity of equations (2.75) and (2.76)

derived in Chapter 2 for the cross-shore 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 non-dimensionalized, 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 non-dimensionalized expression is given

by equation (5.2):

78

ar2 1 ar1 1 2 r 1 2
as* p2-1 as* h (p2-1) 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 (P-1) 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)p2-h 31 (5.3)
3 V12T as* Jas*

and the corresponding cross-shore and alongshore components

are obtained as

79

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 non-dimensionalization three dimensionless

numbers result, which are given in Table 5.3. The first is the

well-known 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

80

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

81

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 cross-shore and

alongshore velocities in the mud layer that are finally

obtained are

i P1Ho h2 p (2-p2) +h; (p2-1) +p;h;h; al (s)
UL(s ) r- 1 -- 2 S)
3 1 2T h* (p*-1) as
(5.6)
h p2 (2-p2) a r.(s) 1 ;(S)
*--- - 1(s) (h2+hi) p2-h ln(s) -- cos
h (p2-1) as* as*

S pH 2 h2 p2 (2-p) +h* (p;-1) +p2h *h2 ( a) r (s)
U,(s) r (s)
3 2T h (p2-1) as
_____ (5.7)
h2p2* (2-p2*) all s) [ 1 9^ {(S)
2h-p-- 1) 1(s) *(h +h1) p2-h;] a2( s)
* (I +h) sin*I
h,* (p2-1) 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(ks-at) (5.8)
1 2

12 =bei(ks-at) (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

83

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 non-linearities 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

84

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.

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