Response of mud shore profiles to waves

UFL/COEL-TR/106

RESPONSE OF MUD SHORE PROFILES TO WAVES

by

Say-Chong Lee

Dissertation

1995

RESPONSE OF MUD SHORE PROFILES TO WAVES

By

SAY-CHONG LEE

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1995

ACKNOWLEDGMENTS

First and foremost, I must extend my most profound gratitude to my advisor and the

chairman of my supervisory committee, Professor Ashish J. Mehta, for his tolerant attitude,

probing questions and patient dispensation of timely advice, and unwavering support throughout

the study. It has been a challenging and overall enjoyable and rewarding experience for me.

I wish to thank Professor Robert G. Dean, whose work on sandy beaches has given the

initial impetus for the direction of this study, for his valuable advice and discussion. Thanks are

also due to Professor Brij M. Moudgil, Professor Donald M. Sheppard, and Professor Y. Peter

Sheng for their comments and patience in reviewing this dissertation. Appreciation is extended

to all other teaching faculty members in the department, as well as those in the Departments of

Aerospace Engineering, Mechanics and Engineering Science, Civil Engineering, and Chemical

Engineering whose courses I have attended, for helping to supply the various pieces of knowledge

essential for the pursuit of this study through their creative teaching efforts.

For the experimental phase of this research, the inputs of Mr. Sydney Schofield, Mr. Jim

Joiner (JJ), and other staff members in the Coastal Engineering Laboratory have been valuable.

In particular, the assistance rendered by JJ in ensuring the operational conditions of the sunken

flume is specially acknowledged. So are the many hours of labor put in by both Ishir Mehta and

Ahmad Tarigan, who eased the strenuous work of model slope construction considerably. Special

thanks go to Helen Twedell, John Davis, Cynthia Vey, Becky Hudson, Lucy Hamm, and Sandra

Bivins for freely sprinkling their humor at various times during my stay.

Many fellow student colleagues have given support in various ways: Yigong Li and Jie

Zheng for technical discussions, Al Browder, Mike Krecic, Taerim Kim, Mike Dombrowski, and

Eduardo Yassuda for the verbal sparring sessions that reminded us that there is more to life than

books.

Most important of all, I am deeply indebted to my wife, Bee-Khoon Goh, for providing

me the peace of mind to pursue knowledge and at the same time being close at hand to render

love, comfort, and support. My four children, Wei-Joo, Chea-Yinn, Wei-Teck, and Chea-ee,

have both endured my emotional outbursts and at the same time provided inspiration for the

research. My family has been the source of my perseverance with the research at times when

all seemed lost.

My stay has been made possible through a Federal Scholarship awarded by the

Government of Malaysia. In addition, the study was partially funded by WES Contract DACW39-

93-K-0008. In this respect, special thanks are extended to Mr. Allen Teeter, who was

instrumental in getting the support and supplying the loess for the flume study.

TABLE OF CONTENTS

ACKNOW LEDGMENTS ......................................... ii

LIST OF FIGURES ............................................ vii

LIST OF TABLES ............................................. xi

LIST OF SYM BOLS ............................................ xiii

ABSTRACT ................................................. xxi

CHAPTERS

1 INTRODUCTION ......................................... 1

1.1 Problem Statement and Objective .......................... 1
1.1.1 Problem Statement ............................... 1
1.1.2 O objective ..................................... 3
1.2 Scope and Tasks ...................................... 4
1.2.1 Scope . . . . . . . . . 4
1.2.2 T asks . .. .. .. .. .. . . . . 7
1.3 Outline of Presentation ................................. 8

2 COMPARISON OF COARSE- AND FINE-GRAINED PROFILE BEHAVIORS 10

2.1 Introduction ...................................... .. 10
2.2 Modeling of Sediment Transport ........................... 11
2.2.1 Coarse-Grained Sediments .......................... 11
2.2.2 Fine-Grained Sediments ............................ 11
2.3 Coarse-Grained Profiles ................................ 12
2.4 Fine-Grained Profiles .................................. 13
2.5 Profile Dynam ics .................................... 17
2.6 Concluding Remarks .................................. 22

3 MECHANISMS OF WAVE ENERGY DISSIPATION .................. 23

3.1 Introduction ........................................ 23
3.2 Rigid Bed Mechanisms ................................. 25

3.3 Non-rigid Bed Mechanisms .............................. 26
3.4 Relative Importance of Dissipation Mechanisms ................. 27
3.5 Linkage Between Energy Dissipation Rate and Bed Rheology ......... 31

4 ANALYSIS OF FIELD MUD-SHORE-PROFILE DATA USING POWER AND
EXPONENTIAL PROFILE EQUATIONS ...... ................... 40

4.1 Introduction ........................................ 40
4.2 Power Fit ......................................... 41
4.3 Exponential Fit ..................................... 50
4.4 Concluding Remarks .................................. 52

5 GEOMETRY OF MUD SHORE PROFILES ........................ 55

5.1 Introduction ........................................ 55
5.2 Basic Assumptions ................................... 56
5.2.1 Uniform Wave-Mean Energy Dissipation Rate per Unit Area ..... 56
5.2.2 Adoption of a Dominant Dissipation Mechanism ............. 60
5.3 Analytic Treatment ................................... 60
5.4 Comparisons With Field Profiles ........................... 66
5.5 Significance of Wave Attenuation Coefficient ................... 70
5.6 Role of Mud Rheology .................................. 75

6 LABORATORY INVESTIGATION ............................. 79

6.1 Introduction ........................................ 79
6.2 Experimental Equipment ................................ 80
6.2.1 W ave Flumes .................................. 81
6.2.2 M easuring Carriage .............................. 83
6.2.3 Sediment Coring Device ............................ 84
6.2.4 Other Apparatuses ............................... 84
6.3 Sediments and Fluid .................................. 84
6.4 Test Conditions and Procedures .......................... 88
6.4.1 Test Conditions ................................. 88
6.4.2 Procedures .................................... 91
6.5 Data Analysis ..................................... 100
6.5.1 Correction for Side-Wall Friction ..................... 100
6.5.2 Correction for Linear Shoaling ...................... 102
6.6 Profile Change Data ................................. 106
6.7 Extension of A-w, Curve into Fine Sediment Range .............. 111
6.8 Concluding Remarks ................................. 112

7 APPLICATION OF ANALYTIC MODEL OF PROFILE GEOMETRY ...... 114

7.1 Introduction ....................................... 114
7.2 Nearshore Depth Correction ............................ 115
7.3 Further Comparisons With Field and Laboratory Profiles .......... 124
7.4 Accretionary and Erosional Trends ........................ 133

8 TIME-EVOLUTION OF MUD SHORE PROFILE .............

Introduction .........................
Governing Equations ...................
Scaling of the Transport Equation ..........
Finite Difference Formulation .............
Model Performance ...................
Concluding Remarks ...................

9 DISCUSSION AND CONCLUSIONS ....................

...... 138

...... 138
...... 139
...... 141
...... 143
...... 146
...... 155

....... 157

Introduction ...............................
Linkage Between Shore Profile and Waves and Water Level
A Description of Mud Shore Response to Waves ........
Influence of Tides ...........................
Alongshore Effects ..........................
Summary and Conclusions .....................
Recommendations for Future Studies ...............

APPENDICES

A DERIVATION OF WAVE ENERGY CONSERVATION EQUATION ....... 181

B DERIVATION OF EQUATION 5.13 ........................... 185

C RESULTS OF LEAST SQUARES FITS TO MUD SHORE PROFILES ...... 188

D NUMERICAL ALGORITHM OF
SOLUTION ................

BIBLIOGRAPHY. .................

DOUBLE SWEEP METHOD OF
. . . . . . . 199

. . . . . . . 202

BIOGRAPHICAL SKETCH ........................................ 214

LIST OF FIGURES

page
1.1 A simple system of morphodynamic interactions along a coastline. The
couplings and feedbacks between the primary components are indicated by the
double-headed arrows. ...................................... 5

1.2 Expanded scope of morphological interaction indicating the domain of
investigation. A solid arrow pointing into the shaded area denotes that the
component is considered analytically in the approach while a broken arrow
denotes that the component is incorporated empirically. ............... 6

2.1 Characteristics of some published profile forms. ................... .. 20

3.1 Comparison of various wave dissipation mechanisms in terms of the
magnitude of mean rate of energy dissipation per unit area. .............. 30

4.1 Profile fits using Equations 2.1, 4.1, and 4.10 to a measured profile along
Louisiana coast (Profile No. LMC) ............................. 44

4.2 Profile fits using Equations 2.1, 4.1, and 4.10 to a measured profile along
Malaysian coast (Profile No. K3). ............................... 44

4.3 Profile fits using Equations 2.1, 4.1, and 4.10 to a measured profile along
Chinese coast (Profile No. C1). ............................... 45

4.4 Comparison of histograms of n in Equation 2.1. ................... .. 45

4.5 Comparison of histograms of A for mud profiles obtained using Equations 2.1
and 4.1. ............................................. 48

4.6 Comparison of histograms of A for coarse-grained and mud profiles. ........ 48

4.7 Profile scale parameter, A, as a function of sediment settling velocity, w,. Box
shows the domain of values obtained for field mud profiles. .............. 50

4.8 Sequence of mangrove line retreat showing the formation of erosional scarp.
The vertical lines with circular tops denote mangrove trees and those with
shaded circles denote trees that are at risk of toppling due to erosion. ........ 54

5.1 Definition sketch. ........................................ 61

5.2 Non-dimensional water depth over the profile, h, as function of non-
dimensional cross-shore distance, y, for values of the non-dimensional wave
attenuation parameter, K, ranging from 0.001 to 0.5. . . . ... 64

5.3 Curve of non-dimensional water depth versus non-dimensional offshore
distance showing the mathematical behavior of Equation 5.8. ............. 64

5.4 Comparison between Equation 5.7 and mud shore profile data from Coastal
Louisiana (Profile No. LK81) obtained on 2/13/81. ................... 67

5.5 Estimation of Yo from a measured shore profile. ................... .. 67

5.6 Comparison between Equation 5.7 and mud shore profile data from Coastal
Louisiana (Profile No. LK87) obtained on 6/23/82. ................... 69

5.7 Comparison between Equation 5.7 and mud shore profile data from Coastal
Louisiana (Profile No. LK64) obtained on 10/10/81. .................. 69

5.8 Typical profile change along a coarse-grained shoreline showing profile
alternation. Normal profile refers to a berm profile that occurs during fair
weather conditions while storm profile refers to a barred profile that results
from erosional conditions that accompany storms. ..................... 70

5.9 Superimposed hypsographic curves for two sites in Severn Estuary and Bristol
Channel, England. ....................................... 71

6.1 A schematic of sunken flume. ................................. 82

6.2 A schematic of mud bed coring device. ........................... 85

6.3 Grain size distributions of the component sediments. .................. 87

6.4 A typical plot of profile change measured in preliminary wave flume tests. . 90

6.5 A typical measured (raw) wave height envelope. ................... .. 93

6.6 A typical plot of profile change. ............................... 94

6.7 A schematic of settling column. ................................ 97

6.8 Particle settling curve for AK mud. ............................ 101

6.9 Particle settling curve for BK mud. ............................. 101

6.10 A typical wave height envelope before and after side-wall and linear shoaling
corrections . .. .. .. .. .. .. .. .. . .. . . 105

6.11 A typical result of exponential fitting to measured wave height envelope. . 105

6.12 A plot of spatial changes between periodic profile surveys for AK mud. . 107

6.13 A plot of spatial changes between periodic profile surveys for sand. ........ 107

A plot of spatial changes between periodic profile surveys for loess. ...... 109

A plot of absolute profile changes along profiles for Run 2 with time. ...... 109

A plot of absolute profile changes along profiles for Run 1 with time. ...... 110

A plot of absolute profile changes along profiles for Run 3 with time. ...... 110

A-w, curve with inclusion of laboratory data. ...................... 113

A profile showing a steep slope near the shoreline. .................. 116

A profile showing a break in slope further seaward. . . . ... 116

A profile showing a very mild slope with no visible break in slope. ........ 117

7.4 A schematic showing the patching of solutions at the break point.

........ 118

7.5 Curves of Equation 7.3 for different M values (K = 0.01) in comparison with
the curves of Equation 5.8. .................................

7.6 Curves of Equation 7.5 in comparison with the curves of Equation 5.8 ......

7.7 Curves of Equation 7.12 in comparison with the curves of Equation 5.8. ... ..

7.8 Mud profile as in Figure 5.4 in comparison with Equation 7.13. ...........

7.9 Example of an extensive effect of nearshore depth correction ...........

7.10 Mud profile in Corte Madera Bay (Profile No. TM). ..................

7.11 Mud profile in Coastal Louisiana (Profile No. LK64) .................

7.12 Mud profile in west coast of Peninsular Malaysia (Profile No. K9). .........

7.13 Comparison between Equation 7.13 and AK mud profile from Run 2 ......

7.14 Comparison of wave height envelopes over AK mud profile and sand profile
in Run 6 ............................................

ix

6.14

6.15

6.16

6.17

6.18

7.1

7.2

7.3

7.15 Change in AK mud profile in Run 6. .........................

8.1 Simulation of a hopothetical mud shore profile accretion starting from an
initially planar slope. ..................................... 148

8.2 A plot of spatial changes between periodic profile surveys for a field mud
profile along the Southwest Louisiana chenier plain.. . . . ... 150

8.3 Long-term profile change along a glacial till profile at Grimsby, Lake
Ontario. The dotted line represents the initial profile in 1980 and the full line
represents the profile in 1984. .............................. 150

8.4 Simulated profile as in Figure 8.1, but laterally shifted such that the profile
is pegged at the shoreline position. ............................ 152

8.5 Simulation of a hypothetical erosion episode starting from an initially planar
slope. ............................................... 152

8.6 Simulation of field mud profile accretion along Southwest Louisiana chenier
plain................................................ 154

8.7 Simulation of profile evolution (erosion) of clay bottom profile in a flume. 154

9.1 A simple model of profile transition. After the sea level has risen, the
increased water depth allows more wave energy to erode the convex-upward
profile to that of a concave-upward.............................. 160

9.2 A descriptive model of mud shore response to waves: (a) Profile in calm sea
condition formed under a previous wave episode, (b) surface and mass erosion
of the bottom and turbidity generation in the initial stages of wave motion, (c)
generation and transport of fluid mud under continued wave action; note that
the directions of the two arrows indicating advection of fluid mud are
arbitrary, and (d) reconfigured profile at the end of the episode. .......... 161

9.3 Equilibrium profiles dominated by tidal currents. . . . ..... 168

9.4 Schematic plan view of a lobate and an embayed shoreline. ............. 169

.. 133

LIST OF TABLES

page
Rheological characterization of sedimentary continuum. ............. .13

Published profile forms. ..................................... 18

Some analytical expressions for mean energy dissipation rate, C-, due to rigid
bed mechanisms. ........................................ 25

Some analytical expressions for mean energy dissipation rate, ED, due to non-
rigid bed mechanisms. ................................... 26

Non-dimensional mean rate of energy dissipation, E based on non-rigid bed
processes (shallow water approximation). ........................ 34

Site conditions prevailing at the selected profile locations. ......

Least squares fits of Equation 2.1 to profile geometry. ........

Comparison of best-fit values of the parameters in Equation 4.10.

Loss modulus, storage modulus, and wave attenuation coefficient as
of density for mud from a site near Mobile Bay, Alabama ... .

Sediment groups. ..............................

Composition and properties of sediments. ...............

Chemical composition of well water. ...................

Wave decay and profile change test conditions. .............

Results of density measurement. .....................

. . . 4 1

. . . 46

. . . 52

functions
. . . 77

. . . 86

. . . 89

. . . 89

. . . 90

. . . 96

Comparison of k. values using laboratory data ...................

Values of the coefficients in Equation 7.13 for erosional and accretionary
profile configurations ............. ........ ..............

. 131

. 135

Cl Profile survey information. ..................... . . ... 189

C2 Results of least squares fits to field mud-shore profiles. . . .... 193

C3 Results of least squares fits to laboratory profiles. . . . ..... 198

LIST OF SYMBOLS

a = Wave amplitude

a, = Wave orbital amplitude just outside the wave boundary layer

A = Profile scale parameter in Equations 2.1 and 4.1

A, = Empirical coefficient in Table 2.2

A, = Wave amplitude ratio

A, = Empirical coefficient in Table 2.2

A' = Coefficient in Table 2.2

b = Width of subflume/Orthogonal distance between way rays

B = Coefficient in Equations 4.10 and 7.2/Coefficient in Table 2.2

B, = Empirical coefficient in Table 2.2

B(y) = General function of y in Equation 7.9

c = Sediment concentration

cd, cD = Dimensionless drag coeffcient

Cg = Wave group velocity

Co = Wave speed at closure depth

Cg = Wave group velocity vector

d = Sediment size

D = Actual wave-mean rate of energy dissipation per unit water volume

D. = Equilibrium value of D

e,, = Root-mean square error

e, = Deviatoric component of shear strain

E = Wave energy per unit surface area

f = Wave frequency/Function

fb = Bottom friction coefficient

f = Stress gradient in the y-direction

ft = Stress gradient in the z-direction

F = Bottom slope at the shoreline in Equation 7.8/Wave energy flux

Fr = Froude number

F, = Settling flux

FS = Foreshore slope in Equation 7.5

g = Acceleration due to gravity

G = Shear modulus of elasticity

G' = Storage modulus

G" = Loss modulus

G, = Specific gravity of sediment

G. = Complex shear modulus

h, h, = Water depth

hi = Water depth at seaward limit of effective sediment transport

hi = Corrected wave height

h" = Measured depth

ho = Depth at the seaward terminus of the profile

If = Predicted depth

/ = Non-dimensional water depth

h = Depth ratio of the water and mud layers

H = Wave height

Hb = Breaking wave height

Ho = Incident wave height/Deepwater wave height

H, = Root-mean-square wave height

i = (-l)"/Index for spatial direction

I = Number of measured profile data points

Im = Imaginary part of a complex function

j = Index for spatial direction/Spatial index

k = Complex wave number

ki = Wave attenuation coefficient

ko = Wave number for a rigid bed/Wave number in deepwater

k, = Wave number

k, = Shoaling coefficient

k, = Wave attenuation coeffcient due to side-wall friction

S= Profile averaged wave attenuation coefficient

k = Non-dimensional complex wave number

K = Non-dimensional wave attenuation parameter

Kp = Permeability

Kq = Dimensional rate coefficient in Equation 8.2

KQ = Dimensional rate coefficient in Equation 2.3

K = Ratio of the effective to-and-fro grain movement velocity to sediment net drift
velocity

Lo = Deepwater wave length

M = Proportionality constant between the amplitude of the mud wave and the
wave-induced bottom pressure wave/Ratio of average profile slope to slope
at shoreline

Ma, = Shear Mach number

n = Exponent in Equation 2.1

p = Pressure/Degree of freedom in Equation 4.8

p = Fluctuating component of pressure

p, = Pressure in the water layer

q = Time index

q, = Cross-shore volumetric sediment transport rate per unit length of
shoreline

Q = Probability of occurrence of broken waves

Re = Reynolds number

Re, = Wave Reynolds number

R = Rate of cliff retreat

S9 = Real part of a complex function

So = Beach face slope at the shoreline

t = Time

T = Wave period

u = Instantaneous velocity in the y-direction

u2 = Velocity of the soil layer in y-direction

u = Velocity vector

ui = Instantaneous velocity component in the i direction

u. = Instantaneous outer flow velocity

u, = Mean component of u,

ai = Fluctuating component of u,

U, = Magnitude of bottom wave orbital velocity at offshore boundary

U, = Wave orbital velocity outside the wave boundary layer

v = Instantaneous velocity in the x-direction

vd = Vertical downcutting velocity

V. = Mean component of v,

vi = Fluctuating component of v,

w = Instantaneous velocity in the z-direction

w, = Water content

w, = Sediment fall velocity

w2 = Velocity of the soil layer in the z-direction

W = Maximum strain energy

Wi = Mean component of wi

; = Fluctuating component of wi

x = Horizontal axis parallel to the shoreline located at the mean water level

y = Horizontal axis normal to the shoreline located at the mean water
level/Offshore distance

yo = Offshore distance at the offshore terminus/Profile length

y = Non-dimensional value of the y-coordinate

z = Vertical axis located at the mean water level

a = Wave attenuation parameter in Equation 3.13

= Hydraulic conductivity

xvii

a1, a2, a, ,a4 = Empirical coefficients in Table 2.2

0/ = Parameter characterizing depth correction due to wave breaking and
related effects in Equation 7.8

X = Coefficient in Equation 4.10/Coefficient in Table 3.1

6 = Specific loss/Parameter in Equation 8.19

6, = Phase angle between shear stress and shear strain

ED = Rate of wave energy dissipation per unit area

ED = Wave-mean rate of wave energy dissipation per unit area

g = Equilibrium value of E,

ED = Non-dimensional mean rate of energy dissipation per unit area

e, = Mean rate of energy dissipation per unit volume of the bed

e,/Eb = Ratio of sediment transport efficiency factors

t = Erosion rate

K = Spilling breaker index

/4 = Dynamic viscosity

L" = Second viscosity

u = Complex viscosity

S= Angle of internal friction

p, P, = Water density

pD = Dry density

p, = Mud density

a = Angular wave frequency in radians (= 2rf)

T = Shear stress

xviii

Vh = Horizontal gradient operator

7 = Strain

7 = Strain rate

'o = Ratio of wave height to closure depth

y, = Unit weight of water

i = Instantaneous water surface elevation

72 = Instantaneous elevation of the mud-water interface

f = Dynamic viscosity ratio

v = Kinematic viscosity

v, = Kinematic viscosity of water

/A2 = Dynamic viscosity of the soil layer

/, = Dynamic viscosity of water

v = Viscosity ratio

b = Density ratio

Tb = Instantaneous bed shear stress

r, = Component of stress tensor in thej direction in the plane normal to
i direction

,1j = Deviatoric shear stress component

AW = Energy loss per stress cycle

0 = Angle between the wave approach direction and shore normal/Phase angle
bewteen the crest of the bottom pressure wave and the crest of the mud wave

A = Average beach slope

< 7 > = Average value of the shear strain rate

Az. = Elevation interval in Equation 6.3

Atq = Time increment

Ay = Distance between two adjacent measuring stations

X7 = Log average of sediment concentration

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

RESPONSE OF MUD SHORE PROFILES TO WAVES

By

SAY-CHONG LEE

May, 1995

Chairman: Dr. Ashish J. Mehta
Major Department: Coastal and Oceanographic Engineering

The concept of a spatially uniform rate of wave energy dissipation, dominated by viscous

dissipation due to wave-induced motion of the soft bed, has been used to develop an analytic

model of mud shore profile geometry. In the derivation, exponential wave height decay with

distance and shallow water linear wave theory have been used in the wave-averaged energy

conservation equation with normal wave incidence. The derived analytic profile is capable of

predicting both concave-upward erosionall) and convex-upward accretionaryy) nearshore mud

profiles. The modality of profile change is dependent on the profile-averaged wave attenuation

coefficient, k,, which characterizes the fluidization potential of mud. The parameter ki is shown

to be a function of mud rheology, which, in turn, depends on the incident wave height. A high

k. indicates the presence of a thick fluid mud layer; transport of the fluid mud elsewhere leads

to an erosional profile. Conversely, a lower k, implies low wave height and an accretionary

profile. By varying kI, the analytic profile is shown to reproduce the varying shapes of mud

profiles from several field sites. Comparison of best-fit k, with that obtained from measured wave

height decay in two field applications also shows close agreement.

Profile evolution is then simulated using a closed-loop approach whereby the analytic profile

shape serves as the target profile toward which an initial, non-equilibrium profile eventually

converges under constant wave forcing. The governing dynamic equation for cross-shore sediment

transport, together with the volumetric sediment conservation equation, are numerically solved

for transient profiles using an implicit finite difference formulation. The profile evolution model

is shown to reproduce noteworthy features of the observed seasonal change in profile shape along

a muddy coastline within the Southwest Louisiana chenier plain.

A laboratory investigation using a sunken flume with clayey shore profiles subjected to

monochromatic wave condition was conducted. Comparison of the best-fit k, for a profile at the

end of the test run to that calculated from the measured wave height envelope is shown to yield

fairly close agreement.

CHAPTER 1
INTRODUCTION

1.1 Problem Statement and Objective

1.1.1 Problem Statement

The manner in which water waves mold mud shore profiles is important in predicting

such phenomena of human concern as coastal flooding, impact of sea level change on coastal

wetlands, and coastal turbidity transport. In the simplest physical setting, consider a mud shore

profile of constant slope that is subjected to steady wave action normal to the shoreline. Assuming

that the waves have sufficient fluid power to erode and transport the material, the profile will

change with time. For coarse-grained profiles, an equilibrium profile shape is attained after a

sufficiently long time (Dean, 1990). On the other hand, mud shore profiles may evolve

continuously due to fluidization potential of the bed stratum under waves. The fluidized materials

are easily transported away by other agencies such as currents and are, thus, not likely to remain

within the profile reach from where they are first eroded. However, when the incident wave

condition changes, the profile may respond in a different manner, commencing from the end state

reached during the previous wave episode. Thus, with respect to the prediction of profile

response to waves, the nature of mud shore profiles and their time-rate of change to coastal wave

conditions are processes of interest.

Substantial efforts have been devoted to understanding coarse-grained profile dynamics,

which has led to the development of an array of hierarchical models to simulate two-dimensional

cross-shore shoreline change. These include descriptive beach states and empirical, analytical, and

numerical orthogonal profile change models (Hardisty, 1990). In coarse-grained profile dynamics,

the concept of equilibrium profile has been used in the prediction of beach response to changing

water levels and waves, both long-term as in shoreline response to sea level rise (Titus et al.

1985), and short-term as in storm-induced dune erosion modeling (Kriebel and Dean, 1985) and

profile response to beach nourishment (Dean, 1990). An equilibrium beach profile also has been

used as the target profile in some cross-shore sediment transport models (Larson and Kraus,

1989; Work, 1992).

The study of mud shore dynamics can be facilitated by the knowledge of coarse-grained

profile dynamics. However, marked differences between the two classes of profiles exist and are

taken into account in the development of a modeling approach appropriate for mud shore profiles

in this study. These differences are highlighted in Chapter 2 and further examined based on field

profiles in Chapter 4.

An improved understanding of mud shore dynamics should considerably enhance our

capability to address the following engineering issues.

1) Stability of Coastal Wetlands

Since coastal wetlands are primarily composed of mud, and are slowly becoming

a dwindling resource, an understanding of mud shore profile dynamics should

facilitate the prediction of the stability of coastal wetlands and rational decision

on viable mitigation measures. This issue is especially pertinent due to the

projected rise in sea level, as our present capability to simulate and predict

morphology of coastal marshes and wetlands in response to oceanic forcing is

still lacking. The effort in this study is thus aimed at improving our

understanding.of the physical interactions that govern mud shore profile response

to waves.

I

2) Mud Nourishment

The use of fine-grained sediments such as mud in beach nourishment is a

potentially effective means of shoreline protection due to the significant wave

attenuation associated with mud and its relative abundance at many locations.

This is especially so when sand, the common material used for beach

nourishment, is becoming scarce in certain localities and the cost of hauling from

offshore sources is becoming prohibitive. Attempts to incorporate this aspect into

shoreline protection, though designed for offshore areas, already have been

initiated in the form of underwater geomembrane-encapsulated fluid mud

breakwater (Yamamoto et al., 1991) and underwater mud berms (Mehta and

Jiang, 1993). Mudflat regeneration is also a concept that has been increasingly

proposed as a means of creating and restoring habitats for coastal flora and

fauna. Increased knowledge on mud shore dynamics will provide a rational basis

for using mud shore geometry as a restorative tool in this respect.

1.1.2 Objective

There have been no published attempts to quantitatively evaluate the influence of waves

on mud shore profiles in controlled experiments, though similar works on glacial till have been

reported (e.g., Bishop et al., 1992). Also, available field observations that examined the behavior

of fine-grained shore profiles under waves relate to glacial till (e.g., Nairn, 1992). More recently,

Friedrichs (1993) proposed two analytic models of equilibrium geometry for tidal flat profiles by

considering separately the effects of waves and tides. Therefore, it becomes necessary to examine

the nature of profile geometry of mud shore profiles and their time-dependent profile changes in

both field and laboratory settings in order to gain insight into mud shore profile dynamics under

coastal wave conditions.

The objective of this study was then to develop an appropriate methodology for modeling

the geometry of shoreline profiles composed of mud, and their time-evolution under wave

forcing, using both field and laboratory generated data. In this regard, it was shown that the

geometry and evolution of mud shore profiles in general are not adequately described by available

approaches established for coarse-grained profiles, both in terms of fundamental processes and

field behavior. Hence, there is a need to provide an alternative framework that considers

processes of wave energy dissipation additional to wave breaking and bottom friction, which are

rigid-bed processes. In this study, a combined approach that included analysis of field profiles,

theoretical development, laboratory investigation, and numerical modeling were adopted to attain

the above objective.

1.2 Scope and Tasks

1.2.1 Scope

In the prototype condition, the profile is acted upon by different agents of change with

varying space and time scales, and exhibits a range of responses. In turn, these responses may

invoke a range of feedbacks, and the cycle is repeated. Figure 1.1 represents a simple system

of morphodynamic interactions along a coastline. The three boxes, hydrodynamics, sediment

transport, and morphology, are linked together via the couplings indicated by the double arrowed

pointers to function as a complex system. Each of these interactions is expressed by causal

relations that govern the process-response functions. From this macroscopic picture, each box is

further expanded to highlight the primary components that participate in the mutual adaptations

of the hydrodynamics, the sediment transport processes, and the morphology as given in Figure

1.2.

The domain shown in Figure 1.2 is too wide to adopt in a single study. Hence, for this

study, the domain of investigation was specified such that the selected factors/conditions are

Figure 1.1: A simple system of morphodynamic interactions along a coastline. The couplings and
feedbacks between the primary components are indicated by the double-headed arrows (adapted
from Kroon, 1994).

realistic representations of field situations. On the other hand, the defined domain should also be

such that the associated logistics and available time-frame required for the study still permit

meaningful results to be obtained. Since the focus of this study was on the response of mud shore

profiles under waves, the scope of this study was accordingly defined as follows, which are

indicated in Figure 1.2 by darkened arrows pointing to the domain of interest.

1. The spatial domain of interest is on two-dimensional morphology stretching from

the shoreline offshore to a point where wave-bottom interaction significantly

affects the surface waves. This offshore terminus can be determined analytically,

as a first approximation, by the closure depth based on incident wave statistics

(e.g., Hallermeier, 1981). For measured profiles, it is established using the

pinch-out depth where the depths from periodic profile surveys converge,

Morphology C Sedimentary Environment

3 D 2 D planform 2 D profile fine-grained coarse-graine
swash zone surf zone offshore zone cohesive weakly cohesive
mud glacial till

I

ii

Domain of investigation.

Currents + Tides Waves

(Dynamic Environment')

Figure 1.2: Expanded scope of morphological interaction indicating the domain of investigation (shaded area). A solid
arrow pointing into the shaded area denotes that the component is considered analytically in the approach while a
broken arrow denotes that the component is incorporated empirically.

7

seaward of which little depth change occurs. However, for the surf zone

shoreward of the break point, an empirical profile representation is used in this

study.

2. The study is confined to fine-grained sediments consisting of unconsolidated soft

mud. Mud is considered here to be composed of predominantly inorganic

sediment of median size less than 64 Itm. Hence, the study findings are not

applicable to highly consolidated clays such as glacial till.

3. For causative agents, waves are considered to dominate over tides and currents.

However, effects of tides and alongshore sediment movement are discussed in

qualitative terms.

The rationale for the above scope will be made clear in subsequent chapters, along with

further focus on specific areas of study.

1.2.2 Tasks

Within the scope just defined, the specific tasks designed to complete the study consisted

of the following.

1. Past studies on profile dynamics are reviewed with a view to highlighting

differences in the behaviors of coarse-grained and fine-grained shore profiles,

thereby providing the groundwork for the subsequent tasks.

2. Existing field profile data sets comprising mud sediments (U.S., Malaysia, and

China) are examined to establish the general characteristics of the geometry of

mud profiles. In the process, functional forms developed to describe coarse-

grained profile geometry are examined to assess their adequacy in this respect.

3. The various modes of wave energy dissipation are classified into the rigid bed

and non-rigid bed categories. The relative importance of these mechanisms were

investigated in terms of their relative contributions, thereby enabling the

dominant mechanism, in this case viscous dissipation due to wave-induced bed

motion, to be identified and represented in the formulation outlined in item 4

below.

4. An analytic model is developed that describes geometry of mud profiles using an

exponential decay law for wave height and comparing its performance with field

profile data and laboratory profile data obtained in item 5 below.

5. A laboratory investigation is conducted using wave flumes to examine the

geometry of fine-grained shore profiles and their time-evolution.

6. A dynamic sediment transport relation based on excess rate of energy dissipation

is formulated, and is solved implicitly together with the sediment continuity

equation in a coupled mode by a double sweep algorithm to yield transient profile

response. The results of simulation are compared with both field and laboratory

data.

7. The implications of factors outside the defined scope are assessed in light of the

results obtained as outlined above.

1.3 Outline of Presentation

Chapter 2 reviews previous studies on profile dynamics. The behaviors of coarse-grained

and fine-grained profiles are compared. The various wave energy dissipation mechanisms are

covered in Chapter 3, concluding with the identification and representation of the dominant

mechanism. Available field mud profile data that had been collated are analyzed using various

functional geometrical forms developed for coarse-grained profiles in Chapter 4. Statistics of

parameters from profile fitting of coarse-grained and mud profiles are also compared. The results

of the previous chapters is then used to develop an analytic model of profile geometry appropriate

for mud profiles in Chapter 5. Discrepancies from model comparison with field data are

discussed. To further examine the performance of the analytic model, a laboratory investigation

is carried out, the details of which are given in Chapter 6. It includes measurement of wave

decay, profile change, in-situ density, and fall velocity of component clay sediments. Empirical

correction to drawbacks in the analytic model is proposed in Chapter 7, with additional

comparison with both field and laboratory data. A dynamic model for simulating time-evolution

of mud shore profiles, using a closed loop approach whereby the analytic model is used as the

target profile, is formulated in Chapter 8. The finite difference formulation used to obtain

transient profiles are also detailed in this chapter. The last chapter synthesizes the findings in

previous chapters to present a descriptive model of mud shore response to waves, followed by

discussion of model limitations. The chapter ends with a summary of the findings and

recommendations for future studies.

CHAPTER 2
COMPARISON OF COARSE- AND FINE-GRAINED PROFILE BEHAVIORS

2.1 Introduction

A common feature of coarse-grained beach profile change is the alternating "summer"

accretingg) and "winter" (eroding) profiles. A simple explanation for this phenomenon is that

offshore sediment movement that occurs under storm wave conditions brings about erosion of the

nearshore berm to form an offshore bar, whereas longer period, low amplitude waves that prevail

during fair weather conditions tend to promote onshore sediment movement and, hence, move

the bar shoreward to form the berm. However, when beach profiles are averaged over some

suitable time interval such as a year to remove the effect of seasonality, they can be viewed as

being in dynamic equilibrium. In this equilibrium state, the sediment transporting power due to

fluid motion and the induced sedimentary processes are in balance.

Traditionally, sediment transport studies have been distinguished on the basis of the size

of the sediment. For example, size-based behavioral differences in profile response occur under

waves, as evident from both field observations and laboratory experiments. Retaining this

distinction based on sediment size, this chapter compares the modeling perspectives of sediment

transport and the associated response exhibited by coarse- and fine-grained profiles in order to

highlight some fundamental differences in their behavior. In the process, existing methodology,

primarily developed for coarse-grained sediments, is briefly examined. This treatment serves as

the basis from which a modeling perspective appropriate for simulating the response of mud shore

profiles to waves is first conceived. In what follows, the modeling approaches to coarse- and fine-

grained sediment transport are discussed separately.

2.2 Modeling of Sediment Transport

2.2.1 Coarse-Grained Sediments

Broadly, cohesionless sediment transport is modeled based on two different concepts,

stresses and energetic. Both concepts were originally developed for unidirectional currents and

subsequently extended to include wave effects. Du Boys (1879) is credited with initiating the

stress-based approach, in which the resulting transport is proportional to some power of the fluid-

induced bed shear stress (or excess shear). On the other hand, Bagnold (1963) popularized the

energetics-based approach, in which the concept of work done by the fluid in transporting a

sediment load and energy dissipation by fluid drag at the bed surface are used to parameterize

sediment transport. More recently, both the mechanism of granular flow proposed by Bagnold

(1956) and the kinetic theory of gases have been extended to explain the phenomena of debris

flow, contact load transport, and hydraulic solid transport in pressurized pipes (e.g., Jenkins and

Mancini, 1989).

2.2.2 Fine-Grained Sediments

Similar development for cohesive sediment transport followed later. The erosional process

is modeled either based on stresses (Odd and Owen, 1972; Ariathurai and Krone, 1976; Cole and

Miles, 1983; Nicholson and O'Connor, 1986; Mulder and Udink, 1990) or energy considerations

(Kelly and Gurlarte, 1981). However, the energetic here is considered at the inter-particle bond

level in terms of energy required to cause displacement of particulate flow units based on the rate

process theory commonly used in chemical engineering. Nevertheless, the primary mode of

sediment movement is in suspension, which is simulated using the diffusion-convection equation

written for the sediment concentration. The erosion process is then included either in the bottom

12

boundary condition to account for vertical sediment fluxes in bed exchange processes, or as the

source term if a depth-averaged version of the convection-diffusion equation is used.

At the fundamental level, a counterpart to granular flow research in coarse-grained

sediment dynamics is studies on hyperconcentrated flows such as mud slides and wave-induced

mud transport. An essential element in these studies is the theological characterization of the

sediment, which plays a central role in various attempts to model mud mass transport. In most

reported applications of wave-bed interaction models, the primary approach can be considered

as energetics-based from which the extent of wave attenuation is obtained.

Both theoretical and empirical approaches have been used in characterizing the theological

properties of sediments. Some theoretical approaches employ mechanical analogues (spring,

dashpots, sliders, and their combination) as a prior models whose parameters are evaluated from

specific theological tests. On the other hand, empirical models are based on experimental fits to

measured rheograms. Table 2.1 is a non-exhaustive list of such models.

2.3 Coarse-Grained Profiles

Based on extensive field evidence, a generic equilibrium shape of the profile of the power

form

h(y) = Ayn (2.1)

where h is the water depth at an offshore distance y from the shoreline, A is a profile scale

parameter and n is an empirical exponent, has been identified (Bruun, 1954; Dean, 1977; Hughes

and Chiu, 1978). While the best fit value of the exponent seems to center around 2/3 in these

studies, other efforts at empirical data fit have yielded different n values, e.g., n = 0.55 (Boon

and Green, 1988). Alternate functional forms also have been used in describing profile geometry

along shores composed of cohesionless sediments. Examples are the exponential form (Weggel,

13

1979; Bodge, 1992; Komar and McDougal, 1994) and the logarithmic form (Sunamura, 1992;

Lee, 1994).

Table 2.1: Rheological characterization of sedimentary continuum.

Constitutive models Investigator(s)

Viscous medium Gade (1958); Dalrymple and Liu (1978); Shibayama et al. (1986); Jiang
and Zhao (1989), Sakakiyama and Bijker (1989); Jiang et al. (1990)
Elastic medium Mallard and Dalrymple (1977); Dawson (1980); Foda (1989)
Viscoelastic medium Hsiao and Shemdin (1980); MacPherson (1980); Mehta and Maa
(1986); Suhayda (1986); Shibayama et al. (1989); Jiang (1993); Cueva
(1993); Li and Mehta (1994)
Viscoplastic medium Liu and Mei (1989); Tsuruya et al. (1987)
Poroelastic medium Yamamoto (1983)
Power fluid Feng (1992)
Non-Newtonian fluid Isobe et al. (1992)
Elastic/viscoelastic/viscous medium Chou (1989)
Viscoelastic/viscoplastic medium Shibayama et al. (1990)

Various phenomenological models have been advanced to provide a physical basis to

Equation 2.1. For example, through the equation for the conservation of wave energy, Dean

(1977) correlates the dissipation of energy due to wave breaking to sediment movement, hence

its equilibrium shape. More recently, Lee (1994) used a simple force balance for a single grain

on a slope to obtain an analytical profile model of the logarithmic form applicable seaward of the

breaker zone. Using a similar process-based approach, Leont've (1985) also obtained two

analytical profile models that are individually applicable in the surf and the offshore zones.

2.4 Fine-Grained Profiles

A common feature of the works done thus far, a representative portion of which has been

referred to in Section 2.3, is that they all have been developed for and applied in the case of

cohesionless sediments, i.e., sand. In this connection, a somewhat crude criterion for the

applicability of this approach based on sediment size can be obtained from the suggested

14

empirical curve of the coefficient of A in Equation 2.1 (with n = 2/3) versus d (sediment size)

of Moore (1982). The curve terminates in the region d = 80 pm, which is within the fine sand

range.

Field behavior of fine-grained profiles differs from that of coarse-grained, e.g. sandy,

profiles in two noteworthy ways.

1) The geometric shape of fine-grained profiles differs from that for a sandy beach.

For example, along the Southwest Louisiana chenier plain where the nearshore

profile is dominated by muddy sediments (silt and clay-sized), empirical power

fits to the measured profile data of Morgan et al. (1953) and Kemp (1986)

showed a smaller mean value of n of 0.54 compared with a value of n = 2/3

commonly adopted for coarse-grained profiles (discussed in Chapter 4).

2) The comparatively much more compliant mud bottom absorbs energy more

readily than a sandy beach, thus making it necessary to consider modes of wave

energy dissipation in addition to wave breaking. In fact, at compliant clayey

beaches energy absorption can be so significant as to reduce the wave height to

insignificant values as the wave nears the shoreline (Wells and Coleman, 1981a).

Similar significant wave attenuation exhibited by mud bottom has also been

observed in other sites (Tubman and Suhayda, 1976; Forristall and Reece, 1985)

and in the laboratory (Nagai et al., 1986).

There have been few systematic studies of profile geometry of field profiles composed

of fine-grained sediments. Evidence offered to support the conclusion that beach profiles

composed of cohesive sediment differ from those of sand is overwhelmingly derived from field

data sets. Furthermore, most of the evidence is based on qualitative correlative observations

relating profile geometry with forcing (Dieckmann et al., 1987; Kirby, 1992) and are seldom

15

amenable to quantitative analyses from the perspective of hydrodynamics-sediment transport

interaction. In addition, most laboratory investigations involving cohesive sediments are

conducted on either a horizontal bed, or in quiescent water if an inclined bed is used (e.g., Ali

and Georgiadis, 1991), when the focus is on mud mass transport.

In this respect, the only reported laboratory investigation on cohesive profile response to

waves is that of Bishop et al. (1992), which involves glacial till that is over-consolidated and,

hence, can be characterized as stiff clay. They reported that negligible or minor erosion was

observed in the absence of sand. It is conceivable that the till behaves essentially as a rigid bed

since it has been subjected to great overburden imposed by glaciers. Therefore, its response to

wave forcing approximates that of a sand bed, except that the cohesive bonds augment its erosion

resistance.

Previous studies have indicated that coastal profiles of glacial till seem to exhibit a

"preferred" shape that is similar to sandy profiles, i.e., the 2/3 power form (Kamphuis, 1987).

While glacial till constitutes an over-consolidated sediment bed due to its geological past, most

fine-grained ocean shorelines are composed of unconsolidated mud, which exhibits strong wave

attenuation characteristics. Hence, the field condition along glacial till shoreline is unlikely to be

found along muddy shores where the bottom deposits are likely to be under- or normally

consolidated. However, the differences in the structural makeup of the two types of fine-grained

sediment do emphasize the importance of stress history in modeling cohesive bed response to

wave forcing.

Studies on tidal flat morphology based on hypsometry have correlated convex

hypsometry with large tidal range, long-term accretion, and/or low wave energy (Dieckmann et

al., 1987; Wells and Park, 1992; Kirby, 1992). Conversely, concave hypsometry is linked to

small tidal range, long-term erosion, and/or high wave activity. These linkages are treated in

16

greater detail in Chapter 9. However, the findings from these studies are mainly based on

measurement of changes occurring in the intertidal portion of the shore profile as further

discussed in Chapter 9. In intertidal reaches, tidal forcing tends to dominate over wave forcing.

Hence, the results from these studies, in general, do not cover the seaward portion of the active

profile zone.

Friedrichs (1993) provided an analytic framework to explain observed changes in

equilibrium profile hypsometry of tidal flats based on the concept of spatially uniform bottom

shear stress. On this basis, he was able to explain the transition from convex to concave

hypsometry and vice versa based on the relative strength of tidal and wave activity. Since wave

activity is inherently episodic in nature, the relative strength of tidal and wave forcing is likely

altered by changes in wave activity rather than tidal activity, unless human intervention alters the

tidal regime. The effects of tides are further discussed in Chapter 9.

Based on the action of waves alone, Friedrichs also demonstrated the effects of shoreline

curvature that manifest in both concave and convex hypsometry, depending on whether the

shoreline is lobate or embayed. For a lobate shoreline, the profile is shown to be concave-upward

while for a highly embayed shoreline, a convex-upward profile is shown to be the case. However,

convex topographic profiles have been observed along generally linear shorelines such as

Southwest Louisiana chenier plain (based on profile data of Kemp, 1986). Therefore, the effect

of shoreline curvature is only one of the factors operating in the field that could help explain the

change in profile geometry. In this respect, an alternative framework is advanced in this study

that could explain the transition of profile shape from concave-upward to convex-upward and vice

versa.

17

2.5 Profile Dynamics

The starting point of most of the existing approaches used to support the power form of

the equilibrium beach profile applicable to coarse-grained sediments is the conservation of wave

energy, which is derived in Appendix A,

aE
S+ V, (CgE)= -ED (2.2)

where E = wave energy, t = time, Vh = horizontal gradient operator, -C = group velocity,

and ED = rate of energy dissipation per unit area. This is a common, physically acceptable basis,

even with the absence of a wave generating source term such as wind forcing on the right-hand

side, since the focus in studies of profile dynamics is on waves that have already propagated out

of the wave generating area into the nearshore zone. Frequently, a steady state is assumed to

remove the time dependency. This is justified on the grounds that the primary emphasis is on the

equilibrium and not the transient response in profile evolution. A variation of the steady state

form of Equation 2.2 was used by Dean (1977) to recast the conservation equation in terms of

energy dissipation per unit volume. This was achieved by dividing the left-hand side by the water

depth to arrive at a value of 2/3 for the exponent n in Equation 2.1. Yet another approach is

based on the equivalence of the spatial gradient of wave momentum flux (radiation stress) and

bottom shear stress in the alongshore direction as shown by Dean (1977). Other published

functional forms, both empirical and theoretically derived, are given in Table 2.2, and are

grouped into three functional types,

1) power form: h = aly"',,

2) exponential form: h = a2 (1 e-'"), and

3) logarithmic form: h = 3 ln(c4y + 1),

Table 2.2: Published profile forms.
Coarse-Grained Profile Geometry
Form Sediment Size"
Basis Investigator(s)
A. Power

2 4 theoretical Keulegan & Krumbein (1949)
h(y) y 7
2.468g'"_
.(o, d y,/o -o.a2(nt.y1,
offshore: h(y) = 0.223- y
i Lo,) L, L,O sand laboratory Rector (1954)
S(Ho -0.42( d -1o
foreshore: h(y) = 0..07-) d--/ y
L o Lo
S- theoretical Bruun (1954)
const. 2
h(y) = T y2

2 theoretical Dean (1977)
h(y) = const. y 5
S 2 2 sand theoretical/ Dean (1977)
h(y) = 24-- field

(3.8 1 2 sand theoretical Bowen (1980)

(046 T'2iw5 2 sand theoretical Bowen (1980)
1 y 5____
21 uf, 7 0. sand field/laboratory Vellinga (1983)
h(y)=0.47 6 y+18__2_
7.6 Ho), 0.0268 1
H/ \.17 sand field/laboratory Vellinga (1984)
h(y) = 0.70 w y0Y78
loJ

Table 2.2: Published profile forms (continued)
Coarse-Grained Profile Geometry I
Form Sediment Sizea
Form Sediment Sizea Basis Investigator(s)

B= (B zexp [2B(A*/B):Pyl- [(B/A') _-h]/[(B/A') L+h,
offshore: h(y) =B -[(B--)--hjI[(BIA)'I-hJ
A exp[2B(A-/B)'y]J .+[(BIA-)-hjl[(BIA)t'+hjo sand theoretical Leontfev (1985)
h(y) -h, 2A (hyzh, r+2A21 A-bh12
surf zone: y = h(y +2- h(y) -h' +2 -In -
B B2 B3 A BhA

= 136D. ( A theoretical Creed etal. (1992)

1 2 2 theoretical Friedrichs (1993)
h(y) = -cdu.g .'(3_________

B. Exponential

h(y) = h + he-y sand field Weggel (1979)
h(y) = B(1 e-xY sand field Bodge (1992); Komar and McDougal
(1994)
C. Logarithmic

h(y) = (IB)ln[(ABjRy + 1] rock field Sunamura (1992)
h(y) = (l/B,)ln[(y/A,) + 1] sand theoretical Lee (1994)
Notations. v, = kinematic viscosity of water; g = acceleration due to gravity; Ho = deepwater wave height; L, = deepwater wave length; d = sediment size;
ho = h at y = 0 (i.e., y axis is directed shoreward here) and also demarcates the point of transition from offshore to surf zone; D.= uniform wave dissipation
per unit volume at equilibrium; y,, = unit weight of water; K = spilling breaker index; T = wave period; w, = sediment fall velocity; B = f(es/e)tan 0 where
k = ratio of the effective to-and-fro grain movement velocity to sediment net drift velocity, esl/e = ratio of sediment transport efficiency factors; 0 = angle
of internal friction; A = (8w,1yC,)hfaf and A' = 8w,(yi2,Ch) where y, = Hlh, and CO = wave speed at h = h,; p, = density of water; ; A = profile scale parameter;
c, = dimensionless drag coefficient; U, = magnitude of wave velocity at offshore boundary; h, = depth at seaward limit of effective transport; i = rate of
cliff retreat; a, A,, A, BB,, h,,, = empirical coefficients, B, = -4n2(gf).
'A dash denotes a generic expression.

20

where al1, a2, % a4, I and 772 are either coefficients depending on the assumptions made in the

derivations, or empirical constants obtained by fitting to measured profiles. In addition, some of

these profile equations apply only outside the surf zone. A common feature of these profiles is

the typically concave-upward profile shape as exemplified in Figure 2.1.

0.06

-0.5

-1.0

-I.5

-2.0

-2.5
S Dean (1977)
-3.0 --4- Bodge (1992)

5 Weggel (1979)
-3.5
-s- Sunamura (1992)
-4.0 Vellinga (1983)

-4.5 L
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Offshore Distance, y (m)

Figure 2.1: Characteristics of some published profile forms.

As mentioned previously, the equilibrium profile concept has been employed in modelling

profile dynamics and the associated sediment transport in closed-loop models (Swart, 1976;

Kriebel and Dean, 1985; Larson and Kraus, 1989; Work, 1992). The premise of these models

is that sediment transport occurs as a result of wave energy dissipation, being driven by the

disequilibrium in either geometric characteristics of profiles such as the length of profile (Swart,

1976) or mean rate of energy dissipation per unit water volume (e.g., Kriebel and Dean, 1985).

In the latter approach, the dynamic transport equation is represented by

q, = KQ(D D.) (2.3)

where q, = sediment transport rate, Ko = coefficient involving the time scale of profile change,

and D = h'C and D. are the actual mean rate of energy dissipation per unit water volume and

its equilibrium value, respectively. A common characteristic of these models is the eventual

approach to an equilibrium profile under constant wave forcing.

In comparison to studies on coarse-grained sediment dynamics, similar progress has been

slow in the field of fine-grained sediment dynamics. Among the few reported studies, Coakley

et al. (1988) simulated the evolution of an erosional nearshore glacial till profile based on

measured substrate erodibility, which is related linearly to applied shear stress, Tb, established

using laboratory tests. The erosion rate, t, is then used to calculate the vertical downcutting

velocity of the profile, vd, using the relation

V 1 G (2.4)
"v = P G,(I + wc)

where p, = density of water, G, = specific gravity of sediment, and w, = water content,

assuming fully saturated condition. After calculating the total amount of erosion in a specified

duration using the experimentally determined relation linking t and b,, the vertical distance of

erosion is then calculated from Equation 2.4 sequentially at the different water depths seaward

of the break point. The rate of vertical erosion in the breaker zone was assumed to be the same

as that at the water depth just outside the breaker zone. Model application to long-term erosion

over a glacial till profile at Grimsby, western Lake Ontario, showed reasonable replication of

measured profile change from 1980 to 1984. It is interesting to note that in the simulations of till

profile development reported in Coakley et al. (1988), which were over a time scale of years,

there was no indication of a corresponding deposition area to store the eroded material within the

22

profile reach. This aspect is further discussed in conjunction with the simulation of time-evolution

of mud shore profiles in Chapter 8. Later, Nairn (1992) developed a model to simulate profile

dynamics of glacial till in which the observed downcutting of the cohesive profile was simulated

using two empirical coefficients. While the numerical prediction was in reasonable agreement

with experimental results, it was achieved at the expense of substantially varying the two

empirical coefficients. The dependence of these empirical coefficients on other physical

parameters was not established.

2.6 Concluding Remarks

For shore profiles composed of mud, it becomes apparent that energy dissipation should

feature prominently in modeling their response to waves. Hence, it seems appropriate to apply

the time-averaged form of the conservation of wave energy (Equation 2.2) and focus attention on

parameterization of the various mechanisms of energy dissipations. These dissipation mechanisms

are examined in the next chapter, which is then used as a basis to develop an analytic model of

profile geometry in Chapter 5.

CHAPTER 3
MECHANISMS OF WAVE ENERGY DISSIPATION

3.1 Introduction

Equation 2.2 is a general equation, the use of which is not directly contingent upon the

size of sediments forming the profile. For example, Friedrichs (1993) used the same equation to

derive an analytic profile model for tidal flats, which may be composed of fine-grained sediments.

In all of the cases mentioned above, the steady state form of Equation 2.2 was used whereby the

equation is averaged over the wave time scale. As noted in Section 2.5, the assumption of a

steady state is based on the consideration that the wave time scale is considerably shorter than

the time scale of interest, which is the duration required to attain equilibrium.

The steady state form of Equation 2.2 has also been used in wave shoaling problems to

calculate the transformation of wave height (Dalrymple, 1992). By averaging over one wave

period, it is implicitly assumed that wave-induced oscillatory motion is the fundamental motion

in the nearshore zone. In addition, wave-current interaction, the inclusion of which requires the

use of the equation of conservation of wave action defined as Elw, where w is the wave

frequency with respect to a coordinate system moving with the imposed current field, is not

considered. Similar to the above approaches, the steady state form of Equation 2.2 is also used

in this study as the basis to derive an analytic model describing the geometry of mud shore

profiles in Chapter 5.

Hence, Equation 2.2 reduces to

V. (E-g8) a(ECgcosO) +(ECgsinO)
V ay ax = (3.1)

where y and x are coordinates normal and parallel to the shore, respectively, 0 is the angle

between the wave approach direction and the shore normal, and D is the wave-mean rate of

energy dissipation per unit area. When waves propagate into nearshore waters, wave obliquity

is reduced due to wave refraction. Hence, the effect of the /aax term in Equation 3.1 becomes

relatively small compared to the alay term. In addition, the focus in profile dynamics is on cross-

shore motion. Therefore, in the development of profile geometry cited above, Equation 3.1 is

further simplified by considering wave normal incidence to the shoreline, giving

(EC) = -i, (3.2)

Equation 3.2 can be derived rigorously from the Eulerian momentum conservation equation

where " is then an empirical representation of the dissipation and turbulence effects as shown

in Appendix A. Equation 3.2 is also used in Chapter 5 to derive an analytic profile model by

considering a formulation of wave energy dissipation appropriate for mud shore profiles.

For fully developed seas in deepwater, the energy balance is maintained among wind

input, dissipation owing to white capping, and resonant wave-wave interactions. However, as

waves propagate into shallow waters, they begin to "feel" the bottom. Various bottom-induced

mechanisms then operate to dissipate the incident wave energy. In this respect, the different

dissipation mechanisms are next classified into the rigid bed and non-rigid bed categories and

discussed below.

25

3.2 Rigid-Bed Mechanisms

These mechanisms do not depend on deformation of the bed material to bring about the

dissipation of the incident wave energy. For breaking waves in the surf zone, the hydraulic bore

analogy and the concept of surface rollers have been used in characterizing the breaking-induced

turbulence. Energy dissipation in the surf zone takes place mainly in the water column. Seaward

of the surf zone, the primary dissipation mechanisms are associated with boundary effects at the

bottom such as bottom friction and bottom percolation. Published analytical expressions of energy

dissipation rate per unit area include those due to wave breaking (Battjes and Janssen, 1978), to

bottom friction (Putnam and Johnson, 1949), and to bottom percolation (Reid and Kajiura, 1957).

Dean (1977) obtained a simple analytical expression for the rate of energy dissipation per unit

water volume resulting from spilling wave breaking. Table 3.1 lists some of the published

dissipation expressions based on the rigid bed assumption.

Table 3.1: Some analytical expressions for mean energy dissipation rate, due to rigid bed
mechanisms.

Mechanism Expressiona"b Investigator(s)

Bottom turbulent (42/3) p JH3(T3sih3ko) Putnam and Johnson (1949)
layer
Percolation (xcg/4)a.o H/(v ,sinh2kJh) Reid and Kajiura (1957)

Wave breaking (g3/8X)Ap 3/h Battjes and Janssen (1978)

Laminar boundary v _. s k Dean and Dalrymple (1984)
layer vk, 2 Esinh2kh
Interior fluid (1/8)pv ,H2kZ(1 + 2kh/sinh2kh) Dean and Dalrymple (1984)
(l/8)pv" 2Hgkg(l_+ 2krhlsinh2K,h)
'All expressions are in units of wave-mean rate of energy dissipation per unit area.
bp, = water density; f, = bottom friction; H = wave height; T = wave period; ko = wave number in deep water; h =
water depth; a, = hydraulic conductivity; a = wave frequency; v, = water viscosity; g = acceleration due to gravity;
E = wave energy; k, = wave number; I = coefficient; H, = breaking wave height; Q = probability of occurrence of
broken waves = f(H,,JH,); H,,, = root-mean-square wave height andf = function.

26

3.3 Non-rigid Bed Mechanisms

The significant wave attenuation exhibited by mud bottom is a result of the compliant

nature of the bed under wave forcing. The viscous dissipation due to the induced wave motion

can account for the observed wave damping that is beyond that attainable based on a rigid

bottom. Constitutive models proposed for estimating the rate of energy dissipation within mud

generally fall into two groups, 1) models which are specific to the assumed description of mud

rheology, and 2) models which are non-specific with regard to rheology. Some of the published

analytical expressions for D are given in Table 3.2.

Table 3.2: Some analytical expressions for mean energy dissipation rate, due to non-rigid bed
mechanisms.

Bed Type/Mechanism Expressiona'b Investigator(s)

Specific Models

Voigt body 2kosh nhkhMacpherson (1980)
2a (gkcoshkh a2sinhkh1)
(pig/8)oAA, H2I
Sgk(gksinhkh ocosh)
Coulomb friction in 2 Yamamoto and Takahashi
bed (p g/16) o H/(Gkocosh2kohh ) (1985)
Non-specific Models

Interfacial wave (pg/8)oMs(180 )H2/(co2k) Tubman and Suhayda
nriw(pg8) o Msin(180- O) H2lsh2Kh) (1976)

Viscoelastic (,2/8) p/22 Schreuder et al. (1986)
medium _>_
4All expressions are in units of wave-mean rate of energy dissipation per unit area.
bl = water layer; 2 = mud layer; p = water density; or = wave frequency; k = complex wave number = k, + ik, where
k, = wave attenuation coefficient, i = (-1)" and k, = wave number; h = water depth; g = acceleration due to gravity;
A, = wave amplitude ratio; H = local wave height; Im = imaginary part of a complex variable; 6 = Coulomb specific
loss; G = dynamic shear modulus; ko = wave number for a rigid bottom; M = proportionality constant between the
amplitudes of the mud wave and the wave-induced bottom pressure wave ; 0 = phase angle between the crest of the
bottom pressure wave and the crest of the mud wave; VI = real part of complex dynamic viscosity; <.> = average value
of shear strain rate.

In the first group, cohesive mud has been considered variously as a viscous fluid,

viscoelastic medium, or poroelastic medium. Some investigators, e.g., Shibayama et al. (1990),

have also considered cohesive mud to be a Bingham viscoplastic in which, at stresses below the

27

yield value, the material is treated as a viscoelastic, whereas at higher stresses it is a fluid.

Models for silty muds, considered to be poroelastic, incorporate energy loss by Coulomb friction

between clay particles (Yamamoto and Takahashi, 1985). Empirical theological descriptions based

on experimental measurement have also been employed (Isobe et al., 1992; Chou, 1989).

In the second group, the expression of Tubman and Suhayda (1976) requires the

coefficients M and 0, which depend on the character of the water pressure wave near the mud

surface and that of the induced mud surface wave. Since the solutions for both the surface and

the induced mud waves depend on solving the hydrodynamic problem with an assumed

constitutive relationship for mud, M and 0 must be either measured directly or obtained indirectly

via model calibration using experimental data. Likewise, the expression of Shreuder et al. (1986)

involves the mean shear strain rate as an explicit input parameter. Additional information on the

bases of the D expressions are summarized in Mehta et al. (1994).

3.4 Relative Importance of Dissipation Mechanisms

Bottom friction has been found to play a negligible role in wave decay in the surf zone

when compared to shoaling and breaking (Dally et al., 1984). However, it could be significant

in a nearshore region that has a very mild slope or a rough bottom. It has also been found that

energy dissipation due to breaking dominates on steeper beaches (1/20 1/30), whereas for

gently sloping beaches the dissipation due to friction becomes rather significant (Izumiya and

Horikawa, 1984).

On the other hand, while breaking commonly occurs along sandy beaches, waves may

dissipate completely at the shoreline without breaking along mud shores as observed by Wells and

Coleman (1981a). Also, wave damping in excess of that reasonable for the turbulent boundary

layer has been reported at the offshore area of the Mississippi delta (Suhayda 1977; Forristall and

28

Reece, 1985). In a comparison of wave damping due to bottom friction, percolation and Coulomb

friction, Yamamoto and Takahashi (1985) concluded that the soil Coulomb friction is by far the

most important mechanism of water wave damping by soft soils, e.g., clays and silts. In a

parallel study, Yamamoto et al. (1983) noted that for sand beds the wave damping due to the

Coulomb friction is comparable to that due to percolation alone.

To further illustrate the relative amounts of wave energy dissipation based on mechanism,

a comparative evaluation of the individual contributions of the various dissipation mechanisms

was conducted in order to focus attention on the primary mechanisms. This analysis was based

on the computation of assuming linear shallow water wave theory and spilling breaker.

These assumptions lead to Cg = C = (gh)", sinh kh = tanh k,h k,h, cosh kh = 1 and H =

Kh; K = spilling breaker index. These assumptions helped to transform the various published

analytical expressions into a common depth dependence that enabled comparison to be made.

Specifically, the various expressions are as listed below.

(1) ,br = [5p ggl1 -tanA/(2i2 7rH )]h2

(2) g,i = rh

(3) ., = [PhgY- /(16T )]h

(4) -, = [PfjK3g '5/(6 7r)]h32

(5) "p = [rg-Kp2/(8 ,)]h2 3

(6) = gZg'I56(16 G)]h5/2
CD-pb 1~

(7) ,t = (5pwgl.SK2A1.S/24)h

29

In the above list, the subscripts br denotes wave breaking based on the hydraulic jump

analogy (Gu and Shen, 1991), if denotes interior fluid, Ibl denotes laminar boundary layer, bf

denotes bottom friction, p denotes percolation, pb denotes poroelastic bed based on the approach

of Yamamoto and Takahashi (1985), btd denotes breaking induced turbulence based on uniform

energy dissipation per unit water volume (Dean, 1977); all the notations are as defined in the List

of Symbols.

Mechanisms 5 and 6 were applied to sand and clay beds, yielding nine individual curves

for comparison purposes. The input values are the spilling breaker index, K = 0.78; T = 8 s;

Ho = 1 m; the permeability, Kp (sand) = 109 m-'; Kp (clay) = 10-12 m'; Pw = 1,000 kg/m3; v,

= 10-6 m/s; fb = 0.01; and the average beach slope, tan A = 1/500 (a value typical of fine-

grained sediments). While the values of 6 (0.0039) and G (1 x 106 Pa) for sand beds are based

on the empirical expressions listed in Yamamoto and Takahashi (1985), those for clay beds (6

= 1.15 and G = 10 Pa) are based on the quoted values in Yamamoto et al. (1983), which are

assumed to be constant.

Figure 3.1 shows the results of computation as a function of water depth. Based on the

assumed input values, it is seen that the induced bed motion of a soft bed assumed as a

poroelastic medium is by far the most dominant wave dissipation mechanism. On the other hand,

the corresponding loss for a sand bed is much lower than bottom friction, which is opposite to

the conclusions of Yamamoto et al. (1983). The energy loss due to wave breaking as shown by

the curve for e, is comparable to that of Dean (1977), The fact that these two curves are
D,br CD,btd

close to each other may not be surprising as both approaches are based on wave-breaking-induced

turbulence. The mechanisms of percolation and fluid interior result in the least energy dissipation.

For a soft bed composed of fine-grained sediments, there is, as expected, little energy loss due

to percolation as a result of the typically low permeability of the structure matrix.

Water Depth (m)

--a- wave breaking
- interior fluid
--$- laminar boundary layer
- -- bottom friction
-e-- percolation (sand)
-- Dean (1977)
-- -- poroelastic (sand)

-*-- poroelastic (clay)
percolation (clay)

Figure 3.1: Comparison of various wave dissipation mechanisms in terms
mean rate of energy dissipation per unit area.

of the magnitude of

31

Based on Figure 3.1 and the foregoing discussion, it may be concluded that for fine-

grained sediment beds, wave-induced bed motion is the primary mechanism for wave dissipation.

On the other hand, for sand beds, both wave breaking and bottom friction are important. Hence,

for waves traversing over a sandy bottom in the surf zone, wave-breaking-induced turbulence is

the primary dissipation mechanism. Seaward of the surf zone, losses due to bottom friction and

percolation can account for the energy change if a rigid bed is assumed.

After noting the relative importance of the different dissipation mechanisms, the

parametric dissipation functions can be linked via Equation 3.3,

d(EC) + + (3.3)
-___ = -(gb+ f+ E

where the subscripts in the dissipation terms denote br = that due to wave breaking;f = that due

to bottom friction; v = that due to viscous dissipation in the bed, which depends on the

constitutive model adopted for the sediment bed. Different terms then dominate the wave energy

dissipation process as a function of the bed rigidity of the bed as discussed above. This linear

partitioning approach, which neglects non-linear interactions, has also been used by others

(Yamamoto et al., 1983) to obtain an overall estimation of energy dissipation rate. Similarly, the

net source function in the commonly used radiative transfer equation for the wave spectrum,

which is used to calculate the spectral evolution of wave spectrum, is also composed of linear

combinations of various linear and non-linear components (Shemdin et al., 1980). The net source

function and the associated equation are analogous to eD and Equation 2.2, respectively.

3.5 Linkage Between Energy Dissipation Rate and Bed Rheolovg

A common theme of studies conducted on wave attenuation by mud bottom and the

associated mud mass transport is the primary role of bed rheology in governing soft-bed response

32

to wave forcing. In this regard, theological aspects are incorporated in the formulation of the

analytic model describing the geometry of mud shore profiles developed in Chapter 5.

For rigid bed mechanisms, gD is usually calculated based on the mean rate of work done

by the fluid at the water-sediment interface, which is assumed rigid. For example, for the case

of turbulent bottom friction, gD is calculated from

ED = TbUO (3.4)

where rb is the instantaneous bed shear stress and uo is the instantaneous outer flow (outside the

wave boundary layer) velocity. For a wave-soft bed system, the primary energy sink is

deformation in the soil medium, and the associated energy dissipation can be computed in several

ways:

1) Viscous dissipation in Newtonian fluid (Ippen, 1966)

ED = _[2a2 (Ou2/ay)2 + 2 (9w2Oz)2 + (Ou2/8z + aw2lay)2] dz

where the subscripts 1 and 2 denote water layer and soil layer, respectively, ,2

is the dynamic viscosity of the mud layer, the overbar denotes wave-mean value,

and u2 and w2 are the velocity components in the horizontal (y) and vertical (z)

directions in the mud layer. The origin of the coordinate (y, z) system is at the

still water level.

2) Mean rate of work done transmitted through the interface (MacPherson, 1980)

-_2 [Ph,(-h,, t) dt
T dt
-T/2

33

where T is the wave period, p,(-h,, t) is the wave-induced pressure at the

interface and 12(t) is the instantaneous elevation of the interface.

3) Internal friction (Kolsky, 1963)

AWIT = 21rsW/T

where AW is the energy loss per stress cycle, 6 is the specific loss, and W is the

maximum strain energy.

4) Viscous dissipation in viscoelastic material (Schreuder et al., 1986)

-co -0c T/2
Sv dz = Jf Wf 9{7}W{}dt dz
-h, -h -T/2

where v is the wave-mean rate of energy dissipation per unit volume of bed, 5

is the shear rate and 9? denotes the real part of a complex function.

While theological parameters only feature in expression 1 (A) and expression 3 (6), they

are implicitly included in the other expressions as well since they form the link between the

imposed stress and the response (strain or strain rate). For example, in expression 2, bothp,(-h,,

t) and d72(t)/dt are part of the solutions of the wave-mud interaction problem, which requires the

specification of theological parameters for the governing equations to be closed. Hence, it is

evident that the theological parameters bear directly on the dissipation process.

The direct linkage is further made explicit in Table 3.3, which lists some of the

expressions for the non-dimensional mean rate of energy dissipation per unit area, defined

as = /(Eo), based on several theological descriptions. The relation between T and ki was

established from Equation 3.2 based on shallow water condition of constant depth, negligible

34

effects of the wave-induced bed motion on the wave length, and an exponential decay law for the

wave height as shown below.

Table 3.3: Non-dimensional mean rate of energy dissipation, -, based on
processes (shallow water approximation).

non-rigid bed

Constitutive Expression' Investigator(s)
Relation

Inviscid water Gade (1958)
+ viscous where = + [(A + -4(1P)-p l Fr
mud 2(1-p)P

Viscid water Dalrymple and
+ viscous K. where f = 2 JFr and Liu (1978)
mud 1/+1+[(l/'- 1)2+4p/I'] J
[ /(2Fr)]2v/e7- r[,(l/I+ 1) -Fr2/,l2 [1 + 1(/1 p )] 2 Fr/- 2
[1lp vl[l1 +(1/ 1)][1 -( R,/Fr)2]2
Inviscid water MacPherson
+ __1 where = Fr + P/4 and (1980)
viscoelastic 1/Ma, + [Fr2Ma1 (Re)]2
mud
(unbounded) C, =
Fr2IRe + Ma RelFr2
Inviscid water 2 Yamamoto and
+ p Ma, 1 1 Takahashi
poroelastic 4 Fr1( Ma2/2)2 + 82 (1985)
mud
(unbounded)
S= j, + if, = k,ht + ikht' Ma, = gh/ G ', Fr = o hg, Re = o2htlv,, = h,/h,, = pi/P2'
r = 1 tanh(lhjh,)/(iEhyh), i = (-1)"2, v = vl/v,, G = shear modulus, 6 = specific loss.

The exponential decay law for wave height is written as

H(y) = Ho e-ky (3.5)

where Ho is the incident wave height at y = 0 where y is positive shoreward. From linear shallow

water wave theory, C = Cg = alk, = (gh)"2. Also, for constant water depth, Cg can be taken out

of the differential on the left hand side of Equation 3.2. Hence,

d dE a dE
D = (ECg) = -Cd = d (3.6)
dry dy k, dy

Substituting E = pgHf/8 and Equation 3.5 into Equation 3.6 yields

PHa d (e-2k,y) (3.7)
?" T *e- E E(3.7)
8- k, dy k,

where the expression for E has been reinserted.

Normalizing the left hand side of Equation 3.7 by Ea leads to

2ki 2k
o (3.8)
k k,

where k. = k/h, and = k,/h. Equation 3.8 is not specific to any particular theological

characterization of the bed material and the fluid medium, as long as the assumptions as noted

above are used in formulating the problem of wave-soft bed interaction. The role of mud

rheology is reflected through ki as discussed later in Chapter 5.

It is seen from Table 3.3 that the Froude number (Fr) appears in all the expressions since

it characterizes the incident wave and site condition (water depth) while the shear Mach number

(Ma) only appears in models that include shear modulus or energy storage capacity of the

medium. Hence, in general, the functional dependence of -' can be postulated as

E = f(Fr, Re, Ma,, f, ) (3.9)

where Re is the wave Reynolds number, f denotes function and respectively are the

density, dynamic viscosity, and depth ratios of the water and mud layers.

36

In Table 3.3, ki has been obtained from the solution of the equations of motion describing

wave-bed interaction. For a linear viscoelastic material of the Voigt type, whose mechanical

analogue is a spring and a dashpot connected in parallel, it has been shown that the equations of

motion reduce to the form of the linearized Navier-Stokes equations for a Newtonian fluid

(MacPherson, 1980). Hence, the study of waves in a Voigt medium parallels the study of waves

in a viscous fluid. Furthermore, it can be shown that starting from the Cauchy equations, and

using the general constitutive equation written in a differential form for a linear viscoelastic

medium, the above result can be extended to the case of a generalized linear viscoelastic medium

under cyclic loading (Mehta et al., 1994). The result is the following form of the momentum

equation,

Du -Vp + A*V2U (3.10)

where p = pressure, p" = // + ip" is the complex dynamic viscosity, pl = real part, or the

dynamic viscosity, /" = the imaginary part, or second viscosity and i = (-1)"n (Mehta et al.,

1994). In this representation using complex notation, /' is a measure of the viscous response and

AH is a measure of the elastic response of the material under oscillatory forcing.

In this case, the relevant constitutive equation is given by

7, = (L/ + i'")e6 = u'd (3.11)

where r = deviatoric shear stress component, e, = deviatoric component of shear strain and

the dot denotes time derivative. For any particular theoretical model, the associated mechanical

analogue can be used to derive the relations linking / and model parameters (Mehta et al.,

1994). For example, for a Voigt model, which is represented by a spring (G) connected in

37

parallel to a dashpot (i/), subjected to sinusoidal forcing, it can be shown that Al = u and

/" = G/a. For illustration, consider a one-dimensional representation of the tensorial form of

Equation 3.11 given by

r = *7 (3.12)

From force balance, the stresses in the two elements are additive, giving

7 = TE + TV = Gy + t7 (3.13)

where the subscripts E and V denotes the elastic and viscous elements, respectively. The results

on the right hand side of Equation 3.13 are obtained using Hooke's law of elasticity and

Newtonian viscous flow representation. Under oscillatory forcing, the stress and the

corresponding strain can be represented using complex notation as

r = roe-i' (3.14)

7 ='oe-i'< ) (3.15)

where ro = stress amplitude, yo = strain amplitude and 6, = phase shift of strain behind stress.

Differentiating Equation 3.15 with respective to time yields

7 = -iatoe-"( = -lay (3.16)

Hence,

.7 =-_ 7 (3.17)
ia r

Substituting Equation 3.17 into Equation 3.13,

G= G[ -') +7 Py A + i G I (3.18)

Equating the corresponding real and imaginary parts of Equation 3.12 and 3.18, the above

results, /_ = and A" = G/a, are obtained. Hence, the viscoelastic representation of a Voigt

material is

U = + iG (3.19)

Expressions similar to Equation 3.19 can be derived for other linear viscoelastic materials such

as the Maxwell model, which is a spring and a dashpot connected in series, using the same

approach but considering that the strains in the two elements are additive. Hence, the use of

complex dynamic viscosity encompasses a range of linear viscoelastic behavior. However, in

general " does not have to be constrained to any particular theoretical model and can be left to

be determined empirically using rheometric data, an example of which is discussed in Chapter

5.

As noted in arriving at Equation 3.8, the relation between "- and ki have been established

based on an exponential decay law for the wave height. It is further noted that the exponential

decay law follows from the standard harmonic solution that satisfies the equation of motion for

a progressive sinusoidal wave. Examples of such applications that lead to analytical expressions

for ki include laminar bottom boundary layer (Dean and Dalrymple, 1991), porous bottom (Liu,

1973), densely packed surface film (Phillips, 1978) and viscous mud bottom (Dalrymple and Liu,

1978). Other expressions that appear in an implicit form for the complex wave number relate to

wave energy dissipation due to submerged vegetation (Kobayashi et al., 1993; Wang and Torum,

39

1994) and wave-induced bottom motion characterized variously as linear viscoelastic material

(MacPherson, 1980) and poroelastic material (Yamamoto and Takahashi, 1985).

Wave height damping has also been found to vary inversely with distance that deviate

from exponential decay,

H(y) = H 1 (3.20)
1 + ay

where a = wave attenuation parameter. Examples include energy loss due to coastal vegetation

(Dean, 1978), rough bottom with a turbulent boundary layer (Dean and Dalrymple, 1991), and

turbulent friction of flow through trees modelled as vertical cylinders (Dalrymple et al., 1984).

By expanding the exponential term in Equation 3.5 and the right-hand side of Equation 3.20, as

was done by Dalrymple et al. (1984), the two wave height decay expressions can be shown to

be equivalent to the leading order in y while they behave similarly for small values of the

arguments of the two functions as seen from the following. The series expansion of the

exponential term is given by

e-y = 1 + (-ky) + + ( -ki3 -o < -ky < oo (3.21)
2! 3 '

while the binomial expansion for the right-hand side of Equation 3.20 is given by

(1 + ay)- = 1 (ay) + (ay)2 (ay)3... -1 < ay < 1 (3.22)

Hence, to the leading order in y, k, = a, while they behave similarly for small values of the

arguments (< 0.1) of the two functions. Since the exponential decay law for wave height

(Equation 3.5) has been used in many application as discussed above, it is used to develop a

model of profile geometry applicable to mud shore profiles in Chapter 5.

CHAPTER 4
ANALYSIS OF FIELD MUD-SHORE-PROFILE DATA
USING POWER AND EXPONENTIAL PROFILE EQUATIONS

4.1 Introduction

There are very limited data on fine-grained shore profiles, and most published systematic

examinations of their geometric variation relate to profiles composed of glacial till. For example,

Kamphuis (1987) reported that the cohesive shore profiles of the Lake Ontario region, which

consist of fine-grained glacial till, can be described using Equation 2.1 with n = 2/3, but using

the sediment size of sand atop the cohesive profile. He attributed this similarity in profile

geometry between glacial till and coarse-grained profiles to the abrasive and armoring actions of

sand veneer that overlies the cohesive deposit. However, as noted in Chapter 2, mud shore

profiles are likely to respond differently to waves than glacial till profiles due to the difference

in the state of consistency of the bed stratum. In this respect, available profile equations that have

been established using coarse-grained profiles were applied to mud shore profiles with a view to

evaluating their usefulness in describing mud shore profile geometry.

This chapter presents the results of best-fitting the available mud shore profile data using

the power and exponential forms. The choice of the two profile forms is based on their successful

application to coarse-grained profiles, and the availability of statistical data on best-fitting for

comparison. Other forms that have been applied to coarse-grained profiles include the logarithmic

and the rational forms (Sierra et al., 1994).

41

Altogether ninety-six profiles from localities in Western Louisiana coast (71), west coast

of Peninsular Malaysia (22), northeast coast of China (2) and San Francisco Bay, California (1)

were analyzed. Since original survey logs were not available, the profiles were digitized from

hard copies of plotted profiles. Table 4.1 gives the general conditions prevailing at the above

sites.

Table 4.1: Site conditions prevailing at the selected profile locations.

Location Mean Tidal Mean Sediment Mineralogical Composition Data Source
Range (m) Size (Am)

Western Louisiana 1.2 1 5 illite, smectite and Morgan et al.
coast kaolinite (1953); Kemp (1986)
West coast of 2.0 2.5 2 15 N.A. Malaysian EPU
Peninsular (1986); Hor (1991)
Malaysia
Northeast coast of a 4 N.A. Yu et al. (1987)
China
San Francisco Bay, 1.3 5 > 50% clay consisting of Liang and Williams
CA illite, kaolinite and (1993)
montmorillonite
N.A. = information not available.
'The maximum computed tidally-induced shear velocity is less than 1 cm/s, implying that tidal action can be reasonably
neglected.

4.2 Power Fit

The power profile forms is as expressed by Equation 2.1 and the coefficients, A and n,

were determined using the method of least squares. To further assess the applicability of the

power form to mud profiles, the least squares fit using a variation of Equation 2.1,

h(y) = Ay213 (4.1)

which has had relative success in describing coarse-grained profile geometry (Bruun, 1954; Dean,

1977), was also attempted.

Denoting the measured and predicted water depths as h'I and hf,, respectively, for each

seaward distance y, from the shoreline, the predicted depth is then

h? = Ay7 (4.2)

where the subscript i denote the distance index across the profile. Equation 4.2 is first linearized

by taking the natural logarithm of both sides to yield

In hf = In A + nln yi (4.3)

The objective function then becomes

In n [ln hf In hi (4.4)
In A, n (4.4)

where I is the total number of profile points. Differentiating Equation 4.4 successively by the two

parameters, In A and n, in turn, while keeping the other constant, and setting both resulting

expressions to zero lead to a pair of simultaneous equations in two unknowns, In A and n. The

simultaneous equations can be solved analytically to yield

I I I
SIn y, (In h") I (In yi)(ln hi)
n i=i i=1 (4.5)
2

isI i=l

1 1 1
n I I I hm~l

In y,. (In yi)(ln hi) E (In yi)2 (In h")
A = exp i -2 i=- i=1 (4.6)

In yi I (In y)2
i=1 i=i

Applying the same procedure to Equation 4.1 yields

2 ini L ,nyi (4.7)
A = exp i. -- 1

With the values of A and n thus established for each profile, the theoretical profile was computed

using these "best fit" values, and the root-mean-square error, e,, for each profile determined

as follows,

SI 1/2
S(h,. hfi)2 (4.8)
erm I-p

where p is the degree of freedom, which is equal to the number of free parameters used in best

fitting. Both Dean (1977) and Hughes and Chiu (1978) defined e, without p while Sierra et al.

(1993) used Equation 4.8. In this case, p = 2 when Equation 2.1 was used and p = 1 when

Equation 4.1 was used. Some typical results of the profile fitting using both Equations 2.1 and

4.1 are shown in Figures 4.1 (U.S.), 4.2 (Malaysia), and 4.3 (China).

Based on empirical fitting to field mud profiles, it was found that these profiles are

reasonably described by a power form similar to Equation 2.1, but with different exponent

values. In Figure 4.4, the histogram for the exponent n is shown together with that for coarse-

grained profiles of Dean (1977). The values on the abscissa are the mid-points of the class

interval (0.10) used. Additional profile fitting comparisons using Equation 2.1 are summarized

in Table 4.2.

Notwithstanding the fact that the analysis for mud profiles is based on fewer profiles than

for coarse-grained profiles, three observations are pertinent. Firstly, convex profiles (n greater

than 1) do occur on both coarse-grained and muddy shorelines, although with much less

frequency than their concave counterparts.

20 40 60 80 100 120 140 160

Offshore Distance, y (m)

180 200 220

Figure 4.1: Profile fits using Equations 2.1, 4.1, and 4.10 to a measured profile along Louisiana
coast (Profile No. LMC).

0 0 I- 4 : 0 5 m

-0.5

-1.0

-1.5

E -2.0

-2.5

C1 -3.0

-3.5 Data of Malaysian EPU (1986)

-4.0 Eq. 2.1: A=0.24,n=0.41

- Eq. 4.1: A = 0.053 m'i/3
-4.5
Eq. 4.10: B = 4.25 m, X= 0.0029 1/mI
-5.0 I I I I I .5
0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000

Offshore Distance, y (m)

Figure 4.2: Profile fits using Equations 2.1, 4.1, and 4.10 to a measured profile along Malaysian
coast (Profile No. K3).

-0.3

-0.4

-0.5

Data ofMorgan et al. (1953)

Equation 2.1: A = 0.05, n= 0.52

-- Equation 4.1: A = 0.025 mn1
- Equation 4.10: B= 0.76 m, .= 0.014 1/m

240 260

-3.0

-3.5

-4.0

-4.5

-6.0 i'
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

Offshore Distance (m)

Figure 4.3: Profile fits using Equations 2.1, 4.1, and 4.10 to a measured profile along Chinese
coast (Profile No. Cl).

0.25
averaged n for mud shore profiles = 0.54
averaged n for coarse-grained profiles = 0.66
0.20

o 0.15
0

0.10

0.05
11 TFf

Figure 4.4: Comparison of histograms of n in Equation 2.1.

Data of Yu et al. (1987) 0
Eq. 2.1: A= 0.02, n= 0.96
- Eq.4.1;A=0.09

- Eq. 4.10: B = 22.7 m, X= 0.0008 1/m

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35
n

mud shore profiles (96) 0 coarse-grained profiles (502)

II(_

V_

-1.5 F

46

Secondly, the median value of n for most coarse-grained profile data lie within the range

0.6 0.7 (the average ranges from 0.63 to 0.68). On the other hand, the median value of n for

mud profiles is in the range 0.5 0.6 (average 0.54). Since a larger n value results in a larger

water depth at the same offshore distance for a given A, Figure 4.4 suggests that mud profiles

are flatter (having a smaller average slope) than coarse-grained profiles, an observed feature of

shore morphology as well as in accord with the statistical correlation between sediment size and

beach slope (e.g., Bascom, 1951).

Table 4.2: Least squares fits of Equation 2.1 to profile geometry.

Coarse-grained
Type of
profile Mud Dean (1977) Hughes and Boon and Stockberger Moutzouris
Chiu (1978) Green (1988) and Wood (1991)
(1990)

Geographical As in U.S. East and Florida Sint Maarten/ Southeast Greece
location Table Gulf Coast (Nassau to St. Martin shore of Lake
4.1 (Long Island Martin (Netherlands Michigan
to Texas/ County, Gulf Antilles/ (Indiana)
Mexico to Franklin French West
Border) County), Indies)
Lake
Michigan
(Indiana)

Number of 96 502 464 11 99 > 70
profiles

Offshore 100 360' 900' 1201 800e
distance (m) 300

Offshore 0.5 3 -8 7 12 4 6 7.5
depth (m) 5

Sediment size silt/clay quartz sand quartz sand carbonate quartz sand sand
sand

S0.54 0.66 0.67 0.55 0.63 0.68

0.14b 0.36c 0.23c (0.07 0.171 0.181
0.30)b.d

(m) 0.11 0.67 0.73 0.14
'An over-bar denotes average value; e, = root-mean-square error;
expressed in metric units; Cexpressed in British units; dOnly the range is given;
maximum offshore distance; A dash denotes information is not available.

47

Thirdly and more importantly, the distribution of n for sand is generally bell-shaped, and

hence the use of a single value of n, e.g., 2/3, is reasonable. On the other hand, while a single

peak does appear in the histogram for mud profiles, the distribution is more rounded and skewed,

implying that the value of n has a wider spread than that given by the normal distribution. Hence,

the use of a fixed value of n, for example 2/3 as in the case of coarse-grained profiles, cannot

replicate a sizeable portion of profile geometry observed along muddy coastlines. The implication

is that the adoption of a single value of n such as the median/average value to characterize mud

profile geometry becomes tenuous.

Figure 4.5 shows the histogram for the coefficient A obtained using Equations 2.1 and

4.1. Clearly the A value is most frequent toward the low end of the range, a trend significantly

accentuated by the use of Equation 4.1. This trend is similar to the results of Dean (1977), but

the A value for mud shore profiles is generally smaller as seen from the comparison shown in

Figure 4.6. However, it is important to note that the root-mean-square error, e,,, resulting from

best-fitting using Equation 4.1 (average e, = 0.22 m) is much larger than that using Equation

2.1 (average e,, = 0.11 m), which further reinforces the argument against adopting a constant

n to describe mud profile geometry. Hence, the outcome of the comparison indicates that

Equation 4.1 is not adequate for describing mud profile geometry. Nevertheless, it may be

instructive to speculate as to the direction the empirical A versus w, curve of Dean (1987) given

by

A = 0.067w044 (4.9)

where A is in m"3 and w, in cm/s, extends into the fine sediment range. It is noted here that even

though Equation 4.1 is obtained empirically, Vellinga (1984) and Wang et al. (1994) have

obtained an exponent of 0.44 and 0.40, respectively, when their results are expressed in the

1.00

r 0.80

S0.60
0

S0.40

0 0.20
I-

Averaged A (Eq. 2.1) = 0.139
Averaged A (Eq. 4.1) = 0.042

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35
A (metric unit)

Eq. 2.1 Eq. 4.1

Figure 4.5: Comparison of histograms of A for mud profiles obtained using Equations 2.1 and
4.1.

1.0

u 0.8

o 0.6
0
0
o 0.4

02

0.034 0.101 0.168

0.235 0.302
A (mA1/3)

0.369 0.436 0.503

E Mud profiles (96) Coarse-grained profiles (502)
I_^ ^ ^ -- ^ ^ ^- ^ -

Figure 4.6: Comparison of histograms of A for coarse-grained and mud profiles.

49

power form, using dimensional reasoning and similitude considerations. However, the

corresponding n values are higher (0.78 and 0.80, respectively).

The curve of Equation 4.9 shown in Figure 4.7 terminates at w, = 1 cm/s (corresponding

to a sediment size of about 80 /m). Since fine-grained sediments tend to form flocs due to the

presence of cohesive forces, it is the floc settling velocity that is of concern here. However,

available data on settling velocity of flocs are scarce, and non-existent for the locations of the

profiles included in the analysis. For illustrative purposes, the settling velocity data of Lin (1986),

which included two fine-grained sediments, Vicksburg loess and Cedar Key mud, are considered

to encompass the range of likely falling velocity for the fine-grained sediments forming the above

profiles. Using the best fit values of A obtained by fitting Equation 4.1 to fine-grained profiles,

the variation of A against w, is shown in the form of a box in Figure 4.7. It is seen that the trend

of Equation 4.9 toward the low settling velocity region seems to flatten out to a constant A value.

At any rate, the deviation from the suggested curve of Dean (1987) is apparent. Further insights

will have to await additional data collection.

Several explanations for the variation of the exponent n for sandy profiles, as indicated

in Table 4.2, have been suggested using physical arguments. One reason given is that the

assumption of a profile having been developed from uniform (sand) material will not be met

(Dubois, 1993). It is usually observed that sediment size generally grades seaward from coarse

to fine and at least for sand, each size is expected to respond to the same hydrodynamic forcing

differently. Field studies have found that both sediment sorting and profile variation are strongly

related (e.g., Medina et al., 1993). In this respect, various investigators have just begun to

address the issue of non-uniformity of A across the profile (Larson, 1991; Dean and Charles,

1994) and in time (Pruszak, 1993). However, the change in the exponent due to the change from

sand to mud is likely a reflection of the basic difference in their response to waves, and it is felt

50

that unless this basic difference in process response is built into the approach, it is unlikely that

the profile response of mud shore profiles can be adequately described by the power form of

profile geometry.

1:00 .
Data compiled by Dean (1987)
Equation 4.9

Sediment Settling Velocity, ws(cm/s)

Figure 4.7: Profile scale parameter, A, as a function of sediment settling velocity, w,. Box shows
the domain of values obtained for field mud profiles.

4.3 Exponential Fit

The exponential profile form is expressed as
I-_

h(y)= B(1 e-) (4.10)
6-

where the coefficients B and are coefficients to be determined from profile fitting. Due to the

non-linear nature of Equation 4.10, it cannot be linearized by taking natural logarithm as was

done in Section 4.2. Therefore, e, for each profile was computed using a range of B and .
values encompassing the published range of the best-fit values for coarse-grained profiles, and

non-linear nature of Equation 4.10, it cannot be linearized by taking natural logarithm as was

done in Section 4.2. Therefore, e aa for each profile was computed using a range of B and X

values encompassing the published range of the best-fit values for coarse-grained profiles, and

51

the pair of (B, X) values that yielded the least e,, was taken as the best-fit values. Some typical

results are shown in Figures 4.1, 4.2, and 4.3 mentioned in Section 4.2.

In applying Equation 4.2 to the same data set of sandy profiles used by Dean (1977),

Bodge (1992) found that the exponential shape more closely approximated the data when

compared to the 2/3 power form. Bodge (1992) interpreted the coefficient B to be a depth which

is asymptotically reached offshore and concluded that the adoption of a fixed value for X is ill-

advised due to the wide spread of best-fit X values (3 x 10- m-' to 1.16 x 10- m-'). However,

while he suggested possible linkages between the parameter B to a functional depth of closure and

X to sediment size and/or its offshore gradation, no attempts were made to relate them. Komar

and McDougal (1994) recast the exponential form in a slightly different form whereby B = So/X

where So is the beach face slope at the shoreline. By doing so, they transformed the exponential

form from a purely diagnostic mode to a more prognostic (predictive) mode in that So is

determined from known relationships as a function of sediment size and wave parameters. Hence,

only one parameter, X, needs to be best-fitted from the measured profile depths or bottom slope

variations across the profile. In applying the exponential form to the a beach profile from the Nile

Delta coast of Egypt, they found that it performed better than the 2/3 power form. They ascribed

the better agreement with data to the ability of the exponential form to replicate the greater profile

concavity, which is characteristic of a more reflective beach, than is allowed for by the y?3

dependence.

Table 4.3 compares the results of the best-fitting analysis using Equation 4.10. It is seen

that the best-fit values of X for mud profiles spread over a considerable range and are higher than

the corresponding values for coarse-grained profiles. To the extent that X governs the degree of

profile concavity, larger X values signify greater reduction in the overall slope from the shoreline,

which tends to be associated with reflective profiles with steep slopes. Hence, this interpretation

52

seems to run counter to the expectation that mud profiles are more dissipative. Also, both Bodge

(1992) and Komar and McDougal (1994) applied the exponential form in a diagnostic mode and

did not present any basis for its predictive use. This lack of theoretical foundation precludes the

use of the exponential form in describing mud profiles as envisioned. Nevertheless, as will be

seen in Chapter 7, the ability of the exponential form to replicate strong concavity in the

nearshore region makes it a potential candidate as an empirically introduced nearshore depth

correction term.

Table 4.3: Comparison of best-fit values of the parameters in Equation 4.10.

Coarse-grained profiles

Parameters US Atlantic and Gulf Coast Nile Coast, Egypt (Komar Mud profiles (96)a
(Bodge, 1992) (504)' and McDougal, 1994) (1)'

X (1/m) 5.1 x 10" [4.1 x 10-]b 7.3 x 102 5.1 x 102 [1.4 x 10-1]b
B or S,/; (m) 12.5 [18.1]b 4.0 15.5 [63.8]b
aThe number in parentheses refers to the number of profiles.
'The entry combination denotes mean [standard deviation].

4.4 Concluding Remarks

Two observations emerge from the above comparisons of profile fits using the power and

exponential forms between mud and coarse-grained profiles. Firstly, the parameters in the profile

equations obtained for mud shore profiles vary over a range much larger than that for coarse-

grained profiles as indicated by the wider distribution of n in Figure 4.4 and higher standard

deviations for B and X in Table 4.3 for fine-grained profiles. While empirical evidence based on

fitting to coarse-grained profiles in the field supports the adoption of a fixed parameter value,

e.g., n = 2/3, the same is found to be inappropriate for mud profiles as discussed above. In this

respect, Inman et al. (1993) introduced a compound power form wherein the profile zone is

divided into two sub-zones separated by the breaker line with a different power form applicable

in each. They found that their compound form showed a better fit to measured profiles along the

53

Californian coast compared to Equation 4.1. However, the better agreement was achieved at the

expense of introducing seven parameters, which drastically reduces the prognostic capability of

the approach.

Secondly, the parameters have been linked empirically to other measurable quantities,

such as A as a function of sediment fall velocity (Dean, 1987) and So as a function of beach

sediment size at the shoreline (Komar and McDougal, 1994). However, field observation along

fine-grained shorelines frequently reveals the existence of a near-vertical scarp at the shoreline.

A typical scenario leading to the development of an erosional scarp drawn from the experience

of mangrove line retreat along the muddy coastline of Malaysia is shown in Figure 4.8. This

scarp formation is more likely the manifestation of a collective response of soil mass in bulk to

gravitational forces rather than the stability of individual sediment grains. Hence, it is doubtful

that So can be realistically determined from sediment size information alone. As discussed in

Chapters 2 and 3, an important process that has substantial control over the response of mud

profiles to waves is wave-induced bottom motion and the associated energy dissipation. None of

the above parameters lend themselves easily to incorporation of the effects of wave-induced

bottom motion.

In summary, an alternative framework is required to describe the geometry of mud

profiles that takes into account the operative dominant processes in accord with field observation.

Such a framework is developed in the next chapter. The same data set is then used to evaluate

the suitability of the derived profile form.

54

0;7/7

Figure 4.8: Sequence of mangrove line retreat showing the formation of erosional scarp (after
Malaysian EPU, 1986). The vertical lines with circular tops denote mangrove trees and those
with shaded circles denote trees that are at risk of toppling due to erosion.

CHAPTER 5
GEOMETRY OF MUD SHORE PROFILES

5.1 Introduction

In Chapter 1, it was noted that mud shore profiles will likely continue to erode under

continuous wave attack since the exposed mud bed is prone to fluidization under wave action.

Hence, the concept of an equilibrium profile attained after the remolding of an initial profile,

which, by definition, changes little with time when subjected to an unchanging incident wave

field, seems tenuous. However, there are two scenarios in the field that may render the concept

worthy of closer examination. Firstly, the incident wave field in nature changes continuously.

Therefore, it is not difficult to visualize the existence of an end-state profile at the cessation of

a particular wave episode. The assumption of an end-state profile does not rule out subsequent

profile evolution, possibly at a reduced rate as postulated based on the second scenario considered

next, if the same wave conditions persist.

Secondly, once fluid mud is generated, it plays a significant role in absorbing and

dissipating turbulent kinetic energy, with consequent reduction in the wave energy level in the

shoreward direction. In addition, the lutocline, which is a zone of high concentration gradient

representing the top surface of fluid mud, tends to damp the turbulent intensity resulting from

wave action that reaches the mud bottom. The lutocline is formed as a result of buoyancy

stabilization of the high concentration suspension (Mehta, 1991). If profile evolution is viewed

as an adjustment to competing destructive forces and constructive forces similar to that

conceptualized for the stability of coarse-grained profiles (Dean, 1990), the reduction in the wave

56

energy due to energy adsorption by fluid mud and near-bed turbulence damping can be considered

as a diminution of destructive forces. Hence, the erosional condition, the magnitude of which can

be expressed as a function of applied shear stress, may diminish with consequent reduction in

erosion of the bed material. This reduced erosion may, in comparison, be a small proportion of

profile erosion that occurs during the onset of the wave episode.

Hence, the consideration of an end-state profile for mud shore profiles is a dynamic one

and the "final" profile attained at the end of a particular wave episode may be approximated by

a characteristic geometry from practical considerations. In Chapter 4, the analysis of field mud

shore profiles has shown that mud shore profiles exhibit a range of characteristic shapes.

Therefore, in this chapter an analytic expression has been derived that can describe the

characteristic geometry of mud shore profiles, assuming that the end-state profile can be

approximated by an equilibrium profile.

5.2 Basic Assumptions

Two main physical premises used in developing an analytic model of profile geometry

relates to the use of Equation 3.3 with a uniform wave mean rate of energy dissipation per unit

area, and the adoption of a dominant dissipation mechanism to the exclusion of other

mechanisms. These premises are examined separately below.

5.2.1 Uniform Wave-Mean Energy Dissipation Rate per Unit Area

The first physical premise is that the profile adjusts in a manner consistent with the

dissipation of incoming wave energy when equilibrium is reached. This premise can be viewed

as a more quantitative interpretation of our earlier understanding of nearshore equilibrium

morphology as defined by Johnson (1919): "At every point the slope is precisely of the steepness

required to enable the amount of wave energy there developed to dispose of the volume and size

of debris there in transit." There are two considerations that need to be examined: the use of

energy dissipation rate as the agent of change, and that, at equilibrium, the energy dissipation rate

is spatially uniform.

For coarse-grained profiles, Dean (1977) viewed the equilibrium profile conceptually as

the result of a balance of destructive versus constructive forces. Destructive forces in this context

relate to those that move sediment offshore while constructive forces move the sediment

shoreward. If either of these two forces are altered, there results a force imbalance with the larger

force dominating. Stability is restored after a period of time when profile evolution brings the

forces back into balance.

Dean (1991) postulated, in general terms, the nature of these forces. The list of

destructive forces include gravity and wave breaking-induced turbulence. Net onshore shear

stresses, bottom streaming velocities, and intermittent suspensions and selective transport of the

particles under the crest where water particles velocities are in the shoreward direction, are all

constructive forces. The net onshore shear stresses acting on the bottom arises from the nonlinear

(asymmetric) form of a shallow water wave while bottom streaming velocities are induced by

energy dissipation within the bottom boundary layer and the resulting local momentum transfer.

Since the present state of knowledge does not permit quantification of individual

destructive and constructive forces, Dean (1977) proceeded to focus attention on one identifiable

destructive force, the energy dissipation rate per unit water volume by breaking waves, without

attempting to identify the coexisting constructive forces. He reasoned that if a sediment of a given

size is considered to be able to withstand a given level of energy dissipation per unit water

volume, then the energy dissipation per unit volume may be taken as representative of the

magnitude of turbulent fluctuations (destructive forces) per unit volume. Hence, he used the mean

rate of energy dissipation per unit water volume, D., to parameterize the destructive forces that

dislodge sediment particles and transport them offshore. He offered two other representations of

58

destructive forces in place of constant wave energy dissipation rate per unit water volume, viz.,

constant bottom shear stress in the alongshore direction under oblique wave incidence and

constant wave dissipation per unit area, to obtain different expressions for the equilibrium profile.

The concept of the wave energy dissipation rate per unit area has also been used by Bruun

(1989) to derive an analytic equilibrium profile. For tidal flats composed mainly of fine-grained

sediment, Friedrichs (1993) used the concept of constant cross-shore maximum bottom shear

stress under normal wave incidence to obtain an equilibrium profile shape.

In an analogous manner, the concept of wave energy dissipation rate per unit area is

applied to mud shore profiles to obtain an analytic model of profile geometry in this chapter.

However, this application entails a different formulation of the causative agent than discussed

above. In this case, the fluidization potential of the mud bed is the primary physical factor

influencing mud shore profile adjustment, which, in turn, is related to the energy dissipation rate

per unit area. The existence of the linkage was supported by the laboratory evidence of Feng

(1992).

Using a mud bed consisting of an equal mixture of kaolinite and attapulgite subjected to

regular waves in a wave flume, Feng (1992) calculated both the energy dissipation rate based on

measured wave attenuation coefficient over a horizontal mud bed from wave height envelope, and

the rate of bed fluidization obtained from the slope of the curve of bed elevation versus time.

The cohesive bed level was determined as the level in the mud layer where the measured effective

stress equalled a value of 5 Pa. Above this level, a (fluid-supported) fluid mud layer occurred,

whose thickness was given by the difference in elevation between the water/mud interface and

the cohesive bed level. Hence, the bed elevation was found from the effective stress curves

obtained using a vertical array of total and pore pressure tranducers flush-mounted on the side.

wall of the flume. Typically, it was found that the energy dissipation rate gradually increased

59

under wave action to a maximum value, and decreased again to approach some constant value

as the fluid mud thickness approached a constant value as well. The constant fluid mud thickness

is reflected in the decrease in the rate of bed fluidization to zero as the energy dissipation rate

decreased. The energy dissipation rate at the stage when the rate of fluidization goes to zero

corresponded with the retention of a constant fluidized mud layer in suspension. This attainment

of a steady state condition implies that the rate of energy input equalled the rate of energy

dissipation due to viscous dissipation in the mud, for a given state of the bottom mud consisting

of a bed and a fluid mud layer of constant thickness.

The extension of the above scenario to the nearshore zone with a sloping configuration

may not be straightforward. For example, the effect of gravity becomes progressively important

as the profile slope steepens. In addition, shoreward of the wave break point, energy dissipation

in the water column due to turbulence resulting from wave breaking becomes important.

However, as will be discussed in Chapter 7, these additional effects can be incorporated

empirically through a nearshore depth correction term to the profile geometry developed in this

chapter. In this respect, it is appropriate to use the energy dissipation rate per unit bed area as

the agent of profile change since the process of energy absorption occurs mainly within the bed.

The second consideration of uniform forcing when equilibrium is reached can be

envisaged as follows. A deviation from the uniform value of the forcing causes local gradient in

the forcing, thereby causing a local change in sediment erosion and deposition pattern. The

profile then responds to these local changes by adjusting its shape such that a uniform spatial

distribution of the forcing is restored. Therefore, in the subsequent development, it is considered

that an equilibrium profile exists when the wave-mean rate of energy dissipation per unit bed area

is uniform across the mud shore profile.

60

5.2.2 Adoption of a Dominant Dissipation Mechanism

As discussed in Chapters 2 and 3, significant wave attenuation that characteristically

occurs over shallow mudbanks is beyond that attainable over a sandy bottom. This effect is

mainly due to viscous dissipation within the soft, oscillating mud, since mud viscosity can be two

to four orders of magnitude greater than the viscosity of water. For example, using a capillary

viscometer, Kemp (1986) obtained a viscosity value of 5.2 Pa.s for a newly deposited mudflat

sediment taken from the Southwest Louisiana chenier plain (bulk density = 1,260 kg/m3), which

is about three orders of magnitude larger than the viscosity of water (v, = 103 Pa.s).

For coarse-grained profiles, wave breaking is identified as the primary energy sink within

the surf zone (Dean, 1977). As pointed out in Section 5.1, significant energy absorption by soft

mud bed can reduce the nearshore wave energy level markedly. Since nearshore wave breaking

to the first order is dependent on the breaker index, i.e., the ratio of the breaker wave height to

the local water depth, a reduced wave height implies that waves break in shallower water, and

hence, a smaller breaker zone. Hence, as a first order approximation, the case of absorption of

wave energy by mud without involving the wave breaking process is considered. For the

development of the analytic model of profile geometry then, only the V term on the right hand

side of Equation 3.3 is retained, which is then taken as in the subsequent development.

5.3 Analytic Treatment

As discussed in Chapter 3, the exponential decay law for wave height has been found

useful in characterizing wave energy dissipation process in a variety of situations. Hence, wave

height decay is represented by an exponential function as follows,

L

H(y) = Hoe-k,(y. ) (5.1)

where Ho is the incident wave height at y = yo, ki is the wave attenuation coefficient and yo is the

seaward end of the active profile length with y taken as positive offshore as defined in Figure 5.1.

It has been observed in the field that waves traversing over mud bed assume a solitary wave

profile (Wells and Coleman, 1981). To the extent that solitary waves are shallow water waves,

ho,

Figure 5.1: Definition sketch.

it seems appropriate to use a shallow water wave theory to describe wave kinematics. However,

non-linear shallow water wave theory, for example, cnoidal wave theory, is analytically difficult

to deal with since they contain transcendental and complicated functions. Therefore, in line with

the goal of developing an analytic model and considering that linear wave theory does reproduce

some of the gross features of wave kinematics (but not the asymmetry in wave form and orbital

Y

62

velocities, which is of little direct relevance to the approach adopted), linear shallow water wave

theory is used.

From linear shallow water wave theory, the wave group velocity is given by Cg = (gh)",

where g is the acceleration due to gravity. Substituting Equation 5.1 into Equation 3.3 leads to

pg3nH d[-2y (5.2)
P9 31 [e-(Y, )hln] = .(5.2)
8 dy

where p is the density of water. Note that the minus sign on the right hand side of Equation 3.3

has been made positive since y is directed offshore here. Equation 5.2 is integrated in the seaward

direction to yield

yh y -(
[d[e-2(Y Yh"2] -= 8 dy (5.3)
!ct opg/nH0

The left hand side of Equation 5.3 represents the integration of a perfect differential, the result

of which is not dependent on the path of integration, but on the end points, i.e, the integration

limits. On the other hand, is considered uniform and can be taken out of the integration of

the right hand side, leading to

e-(y- )h/2 8- y (5.4)
pg3/2H2

where the superscript y denotes a function of y. Hence, ki should vary with y across the profile.

However, k, and are related as shown in Section 3.5. Therefore, a constant E,' i.e., gE

implies a single ki value. To maintain the one-to-one correspondence between ki and a profile-

averaged ki, i.e., k., is introduced such that it is interpreted as a representative mean value over

the active profile length, yo, along which waves measurably influence the bottom. Hence,

-2k,(yo _)h_/2 _8D
e(y-y)hn = E y (5.5)
pg3/2 H2

Satisfying the boundary condition at the offshore terminus (ho, yo), Equation 5.5 reduces to

h12 / 8". yo (5.6)
pg3H2

Substituting Equation 5.6 into Equation 5.5, the following profile geometry is obtained,

2
h = hoe4kYo y Y (5.7)

Equation 5.7 can be conveniently non-dimensionalized according to

h= e4K( y9)2 (5.8)

where y = y/yo, A = h/ho, and K = k.yo is a non-dimensional wave attenuation parameter,

which scales ki by the length of the profile, yo.

In Figure 5.2, h is plotted against y for a range of values of K from 0.001 to 0.5.

Observe that as K approaches zero the convex-upward profile shapes become practically affine,

and at K = 0, Equation 5.7 represents a parabola. With increasing K the shape becomes

concave-upward. For values of K slightly greater than 0.5, the profile close to the seaward

terminus develops a trough with h > 1.0 as shown in Figure 5.3. This is a mathematical

behavior that can be established by differentiating Equation 5.8 with respect to y and setting the

resulting expression to zero to locate the points where the maximum and minimum occur along

the f y curve. Hence,

0.0

-0.1 *

S-0.2

-0.3

-0.4

-0.5

0
S-0.6
.4

E
-0.8

I -0.9

-1.0
0.0 0.1 0.2

Figure 5.2: Non-dimensional
cross-shore distance, y, for
ranging from 0.001 to 0.5.

0

a

E:
I

4)
II

0
I-

4)

0
Z

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Non-dimensional offshore distance, 9 = y/yo

water depth over the profile, as function of non-dimensional

values of the non-dimensional wave attenuation parameter, K,

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Non-dimensional offshore distance, 9 = y/Yo

Curve of non-dimensional water depth versus non-dimensional offshore distance
mathematical behavior of Equation 5.8.

Figure 5.3:
showing the

dA = 2ye4K(1 -(1 2Ky) 0 (5.9)
d9

The solutions of Equation 5.9 are y = 0, Ky = 1/2, and large y where the function approaches

zero asymptotically. The last solution is not of interest since for y > 1, the active profile length

is exceeded. By differentiating Equation 5.9 with respect to y again, it can be shown that y =

O corresponds to a point of minimum (valley) and Ky = 1/2 corresponds to a point of maximum

(hill). Taking h as negative as was conventionally plotted in representing water depth below the

water line, Ky = 1/2 then represents a minimum point. Since the maximum value of y is 1, the

minimum point will occur within 0 < y 5 1 for K > 1/2 and beyond for K 1/2.

Therefore, the physically meaningful solution domain is constrained such that 0 < K < 1/2,

in which case a profile sloping downward monotonically offshore is obtained as shown in Figure

5.3.

It is noted that given the exponential decay in wave height as per Equation 5.1, k. does

not appear to depend on wave height (e.g., Jiang and Mehta, 1992). However, a wave-height

dependence of is implicit as explained next. The wave induced shear stresses acting on the

bottom and within the mud layer is calculated from the velocity distribution, which is a function

of wave height. The resulting change in shear strain then alters the theological properties of the

bed. However, k. is directly dependent on the theological properties of the bed material as given

by the expressions listed in Table 3.3. Through this feedback mechanism, ki varies with wave

height via the solution of the equations of motion for wave-soft bottom interaction discussed in

Chapter 3.

66

5.4 Comparisons With Field Profiles

By considering K, or equivalently k., to be a profile-fitting parameter, Equation 5.7 can

be compared with data, as shown in Figure 5.4. The measured profile was obtained from a Gulf

of Mexico coast site near Cheniere au Tigre, Louisiana (Profile No. LK61) where the mean tidal

range is 1.2 m (Kemp, 1986). The shore had a mudflat morphology with sediment diameter in

the 1 /m to 5 Ctm range, and dominated by fluid mud in the region between the shoreline to about

100 m offshore. Equation 5.7 was fitted to the data by minimizing e,, the root-mean-square

(rms) error as defined by Equation 4.8. In this case, p = 1 since only one fitting parameter, k,

was involved. The best fit value of k obtained was 0.0026 m' with e,,, = 0.051 m.

The terminal depth, ho, is analogous to the depth of closure of sandy profiles in the sense

that seaward of that depth the influence of waves on the bottom can be ignored. For a sandy

bottom, ho depends on the grain size, the wave height and the period, although it is commonly

selected on the basis of wave height alone (e.g., Hallermeier, 1981). For mud, ho depends on

the fluidization potential of the bottom in addition to wave characteristics. Since mud properties

are influenced by wave action, the task of determining ho is considerably more complicated than

the depth of closure of sandy profiles. Therefore, judgement is often required in selecting the

offshore terminus of the mud profile. For example, if the profile becomes abruptly steep with

distance offshore and forms part of a deep trough or a channel, ho must be terminated at that

point where the break in slope occurs. This is illustrated in Figure 5.5, which shows a mud

profile measured in Corte Madera Bay, San Francisco. Corte Madera Bay is one of the several

embayments along the eastern margin of Marin peninsula in the northern part of San Francisco

Bay. The mean tidal range is 1.3 m in this area and the median diameter of mud there is of the

order of 5 /m. In this case, the change in profile shape to a steeper slope starting at y = 1,300

I

S Data of Kemp (1986)

S Eq. 5.7: ii = 0.0026 1/m, er = 0.051 m, yo = 198 m, y =(

0 20 40 60 80 100 120 140 160

Offshore Distance, y (m)

180 200 220 240 260

Figure 5.4: Comparison between Equation 5.7 and mud shore profile data from Coastal Louisiana
(Profile No. LK81) obtained on 2/13/81 (Kemp, 1986).

-0.1

-0.6

-1.6
' -1.1

5-

.1.

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800

Offshore Distance, y (m)

Figure 5.5: Estimation of yo from a measured shore profile.

68

m, may be caused by slumping due to its proximity to the deeper navigation channel and, hence,

may not be primarily wave-induced. Therefore, the profile is truncated at y = 1,300

m.

In Figure 5.6, the field profile, which exhibits a nearly linear offshore profile, was

measured at the same location as that in Figure 5.4, but on a different day (6/23/81). The best-fit

k. is 0.0019 1/m with a higher e, of 0.045 m. On the other hand, Figure 5.7 displays a convex-

upward profile, which is located about 8 km to the east of the above site. It is seen that the

discrepancy between the best-fit curve (k. = 0.0007 1/m) and the measured profile becomes

progressively larger (e,. = 0.082 m).

In all three cases, the agreement at the nearshore portion of the profile is rather poor. The

primary cause for the significant departure is the zero slope predicted by Equation 5.7 at the

shoreline. This aspect is addressed and a nearshore depth correction term introduced in Chapter

7.

The other apparent drawback of Equation 5.7 is the prediction of monotonic profile where

the profile slopes continuously in the offshore direction. However, this feature is likely to be less

problematic as reasoned next. The equilibrium beach profile is an idealized profile that has

adjusted to the sediment, wave, and water-level fluctuations at the site of interest (Dean, 1990).

The nearshore depth correction term only provides a finite profile slope as well as simulates a

steep-sided scour in the vicinity of the waterline, and does not, in general, preclude the formation

of a monotonic profile as discussed and shown in Chapter 7. Along coarse-grained coastlines, a

barred profile is quite common. The offshore bar characteristically stores material eroded from

the profile and the receding shoreline during storms conditions as shown in Figure 5.8. During

fair-weather conditions, the stored material contributes to beach buildup, with consequent

0.0

-0.1

-0.2

S-0.3

-I

. -0.4

-0

-^^

* Data of Kemp (1986)

Eq. 5.7: ic-= 0.0019 1/m, yo = 114 m, ho= 0.37 m, e = 0.045 m

20 40 60 80 100 1

Offshore Distance, y (m)

Figure 5.6: Comparison between Equation 5.7 and mud shore profile data from Coastal Louisiana
(Profile No. LK87) obtained on 6/23/82 (Kemp, 1986).

o.U

-0.1

-0.2 *

-0.3

-0.4

-0.5

-0.6

-0.7

-0.8

-0.9

-1.0

-1.1

-1.2 *Data of Kemp (1986)

-1.3 Eq. 5.7: .i= 0.0007 1/m, yo = 234 m, ho = 1.06 m, es = 0.082 m
-1.4 I I I I

0 20 40 60 80 100 120 140 160 180

Offshore Distance, y (m)

200 220 240 260

Figure 5.7: Comparison between Equation 5.7 and mud shore profile data from Coastal Louisiana
(Profile No. LK64) obtained on 10/10/81 (Kemp, 1986).

70

diminution of the offshore bar under the action of the prevailing longer period swells. The

process then repeats itself in the next season of stormy weather. This scenario depicts the gross

features of cross-shore sediment redistribution occurring within the active coarse-grained profile

zone. However, this sediment recycling, which leads to alternate bar and berm profiles according

to seasons, is less likely along fine-grained shorelines due to the predominant suspended mode

of sediment movement as discussed in Chapter 9.

Normal profile

Equal areas

-Storm profile

Figure 5.8: Typical profile change along a coarse-grained shoreline showing profile alternation.
Normal profile refers to a berm profile that occurs during fair weather conditions while storm
profile refers to a barred profile that results from erosional conditions that accompany storms.

5.5 Significance of Wave Attenuation Coefficient

Previously, convex mud shore profiles have been designated as accretionary and concave

as erosional as illustrated in Figure 5.9 (Kirby, 1992). The two sites, which are predominantly

or entirely muddy, are located in Severn Estuary/Bristol Channel, England, and are characterized

as a hypertidal environment (mean tidal range = 8.55 m). Using mainly aerial survey data, the

area of shore at sequential heights was measured to permit area/height (hypsographic) curves to

-10
0.

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Relative Constant Level Area (A/Am)

Figure 5.9: Superimposed hypsographic curves for two sites in Severn Estuary and Bristol
Channel, England (after Kirby, 1992).

be plotted. The sedimentary state of the profiles was established through empirical assessments

based on physical, biological and chemical indicators of long-term change. Examples of indicators

of long-term erosion are a coast-aligned sequence of mud and sand flats across the shore

(physical), entombed remains of hard-shelled, burrowing animals under rising tidal flat surface

(biological) and anthropogenic chemicals found in tidal flat sediments (chemical). On the other

hand, over-consolidated nature of the deposits and mud cracks (physical), scoured out examples

of boring bivalves (biological) and the absence of anthropogenic chemicals (chemical) are some

of the indicators of long-term accretion.

The two curves in Figure 5.9 show two distinct shapes. The curve for Clevedon, which

has been classified as undergoing long-term erosion through the use of the indicators mentioned

- CardiffBay: MTR = 8.6m
- -- Clevedon: MTR = 9.4 m
Mean Low Water (MLW). -
I Mean High Water (MHW)

MTR = Mean Tidal Range
Am = Constant level area at MHW

I I 1 1 I I I

72

above, displays a concave-upward shape. On the other hand, the curve for Cardiff Bay, classified

as accretionary, shows a convex-upward shape.

Convex mud profiles characteristically occur where comparatively low wave heights and

a supply of sediment favor accumulation of fine-grained material over the profile. If in such a

locality the wave action becomes inclement or the sediment supply is measurably reduced, thus

favoring erosion of the accumulated material, the profile can assume a concave configuration.

According to Equation 5.7, for a given yo, high k. values are associated with the

erosional profile, and as k. decreases and approaches zero the profile becomes accretionary.

While this relationship between ki and profile shape follows mathematically from Equation 5.7,

this trend can be explained in terms of the physics of mud shore response to wave forcing. Thus,

a high ki indicates the presence of a correspondingly thick fluid mud layer, which has the

potential for transport through advection by alongshore currents, Stokes' drift and bottom

streaming as discussed in Chapter 9, or gravity-induced flow downslope, thus leading to erosion.

In laboratory experiments, the thickness of the fluid mud layer has been shown to

increase with increasing wave-mean rate of energy dissipation (Feng, 1992), hence increasing the

wave attenuation coefficient as previously elaborated in Section 5.2.1. In turn, therefore, a

higher k. is associated with a larger wave height. This is consistent with the laboratory

investigation of Yamamoto et al. (1985) who also observed increasing wave attenuation with

increasing wave height in a wave flume study using a horizontal bed of bentonite clay.

Conversely, lower k. implies low wave heights and an accretionary profile, and as ki decreases

the bed becomes increasingly rigid. However, this trend does not ultimately lead to the

characteristically concave sandy profile, because in the latter case wave breaking, and to some

73
extent bottom friction, are more important energy sinks than absorption of wave energy by the

bed. As discussed in Chapter 2, equilibrium profile modeling using breaking wave-induced

turbulence (Dean, 1977) and bottom friction (Bruun, 1989) lead to a characteristic concave-

upward shape for coarse-grained profiles.

To gain further insight into the significance of this parameter with respect to incident

wave condition and profile properties, a simple approach is considered herein starting from the

analytic solution of Equation 5.19 discussed later with regard to the role of mud rheology in

Section 5.7. Equation 5.19 is first non-dimensionalized as follows,

K = 4 [ P ][PgYo] 1 I) (5.10)
L-- A'J 1 + (A

where the subscripts fand m denote fluid and mud, respectively. Further grouping leads to

K = 4 p [1 (ho/g) 1/ (5.11)
P. / 1/2 h oyo. _1 +(A" )2

Using shallow water linear wave theory, a(ho/g)" = kho from linear dispersion relation where

kho is the relative depth. Also, holyo = tan A can be considered as the average profile slope and

pmahI/Al is recognized as the Reynolds number. Hence the above equation can be represented

parametrically as

K = f(PfI/P, "//'/, Re, tanA, kho) (5.12)

Furthermore, it can be shown that (see Appendix B)

2
/ G/Pm Pmahh 1 (5.13)
7 gh, I (holg)1/2

74

i.e., the ratio includes most of the dimensionless numbers already identified in addition to a new

dimensionless group termed the Mach Shear number (the first group on the right-hand side),

which is similar to the Mach Shear number defined by Yamamoto et al. (1983). Hence, the

parametric form can be more compactly represented as

K = f(p/pm, j /ig', tanA) (5.14)

Two further observations can be made. Firstly, the first two dimensionless groups on the

right hand side of Equation 5.14 are not independent of each other since mud rheology is

dependent on mud density. A high-density mud tends to be more rigid and vice versa (e.g., Jiang

and Mehta, 1995). From theological measurement, the measured changes in both viscosities

reflect changes in mud density. Secondly, the effects of incident waves (wave height and wave

period) are incorporated in the test protocol used to measure the response of theological

parameters in rheometric tests by varying the applied stress or strain and frequency. Hence the

parametric representation is finally,

K = f(I"/', tanA) (5.15)

From the analytic result for K shown in Equation 5.10, it is seen that for large values of

//"lt/, the square term dominates and K is inversely proportional to /tyi, i.e., a relatively rigid

bed implies small K values, which in turn translates into convex geometry. For small values of

A //1, the last term on the right-hand side of the analytic solution becomes unity and K is seen

to reduce in magnitude with increasing mud viscosity. This implies that at intermediate values

of A/// l/, a maximum K occurs as supported by laboratory measurement. Since larger waves have

higher mud fluidization potential as discussed in Section 5.2.1, a less rigid (and more viscous)

bed results under large wave condition. This in turn enhances bottom motion that augments wave

75

energy dissipation, i.e., a higher K value. In this way, large wave heights are associated with

concave profile geometry and vice versa.

5.6 Role of Mud Rheology

The magnitude of ki varies with the constitutive theological properties of the bed

material. For illustration, consider the general constitutive equation for a linear viscoelastic

material subject to small-strain cyclic loading represented by Equation 3.12 discussed in Section

3.5. The coupled equations of motion under small-strain oscillatory loading of the water and mud

layers, with i' characterizing the viscous term for mud, have been solved by using different

expressions for p*. Thus, Dalrymple and Liu (1978) considered mud to be a viscous medium,

Suhayda (1986) assumed a viscoelastic representation of mud and Mei and Liu (1987) explored

the response of a viscoplastic mud.

In what follows, the role of mud bed rheology is demonstrated via a simple approach

using the analytic results of MacPherson (1980). He considered water to be inviscid and mud a

Voigt viscoelastic medium given by Equation 3.13 discussed previously in Section 3.5. Using the

linearized form of Equation 3.10 by neglecting the non-linear convective acceleration term, i.e.,

P = -Vp + I'V2 (5.16)
Qt

he obtained the following explicit solution for the complex wave number, k,

g Pg + gG pwg. (5.17)
k= + +i
(gh)112 402(/t2 + G2/a2) 4o('U2 + G2/2)

where h = water depth and p, = density of water. The explicit solution applies to the case when

either the viscosity or the elasticity of a bed of infinite depth is large, and the wavelength is long

compared to the water depth, ki is then given by the imaginary part of Equation 5.17 as

k. = wg (5.18)
S 4o(i2 + G2)

However, for a generalized representation of linear viscoelastic material, Al = t and p" = G/a

as shown in Section 3.5. Substituting these results into Equation 5.18 yields

k. = g (5.19)
S 4a(I2 + /I2)

Jiang and Mehta (1995) determined Al and /ll via creep and oscillatory shear tests in a

controlled-stress rheometer using bottom sediment from the Gulf of Mexico near Mobile Bay,

Alabama, having a median diameter of 15 um. A creep test is a static test whereby the material

is subjected to a constant stress. The strains during the constant stress application (creep) and the

unloading stage (recovery) were measured. On the other hand, an oscillatory shear test is a

dynamic test where the input is the oscillatory stress (since a controlled stress rheometer was

used) and the response is the oscillatory strain. For a viscoelastic material, the oscillatory stress

always exhibits a phase lead over the oscillatory strain. The oscillatory tests were conducted by

feeding into the rheometer a digital, computer-generated sinusoidal wave signal based on the

torque, which then applied the corresponding stress wave to the sample. The corresponding

displacement was measured by an optical encoder that scanned the movement of the measuring

geometry. The stress and strain data were obtained from the information on the amplitude of the

measured displacement curve and the phase difference between it and the input torque wave via

sin6, (5.20)
170o

"= o cos (5.21)
a3o

where To = amplitude of the input oscillatory stress, y0 = amplitude of the measured

displacement curve, a = radian frequency of the applied stress wave, and 6, = measured phase

lead of stress curve over displacement curve. Muds of different densities were subjected to the

test using a range of input frequency to obtain the frequency dependence of the viscoelastic

parameters.

For an input stress wave frequency = a/2r = 0.1 Hz, the values of il and AH were

first calculated using the relations expressing the frequency dependence of p/ and /A for three

typical mud densities, p,, obtained by Jiang and Mehta (1995). ki values were then calculated

from Equation 5.19 and given in Table 5.1.

Table 5.1: Loss modulus, storage modulus and wave attenuation coefficient as functions of
density for mud from a site near Mobile Bay, Alabama.

P, (kg/m3) /l (Pa.s) H" (Pa.s) Al'/1 ki (l/m)

1,139 0.11 140 1,273 0.0230

1,204 2.51 1,687 672 0.0035

1,302 13.18 10,485 796 0.0005

It is observed that both /1 and A// increase rapidly with a comparatively small increase

in density. The increased rigidity imparted by increasing l'/ with density caused ki to decrease

by three orders of magnitude. At the density of 1,302 kg/m3, mud tends to be a solid with a

structured matrix, whereas at 1,139 kg/m3 it is often fluid-like (Jiang and Mehta, 1995). Thus,

these calculations indicate that bottom fluidization is essential for wave damping to increase above

that over a harder mud.

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