A MECHANISM FOR NON-BREAKING WAVE-INDUCED TRANSPORT OF FLUID MUD AT OPEN COASTS by
Hugo N. Rodriguez Thesis
A MECHANISM FOR NON-BREAKING WAVE-INDUCED TRANSPORT OF FLUID MUD AT OPEN COASTS
HUGO N. RODRIGUEZ
A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
First, I wish to express my deepest gratitude to my advisor, Dr. A.J. Mehta, for giving me the opportunity To study under his skillful guidance. His direction and valuable advice contributed greatly to the final form of this work.
I would also like to thank Dr. R.G. Dean and Dr. R.J. Thieke for their participation as supervisory committee members, and for the knowledge given through their interesting lectures and discussions.
Thanks and appreciation are due to the staff and fellow students of the Coastal and Oceanographic Engineering Department, with special acknowledgement to the laboratory staff for helpful assistance during the experimental phase of this study.
Finally, I wish to thank very specially Wally Li and Ismael Piedra Cueva for their invaluable help in reviewing the derivation of the equations and constructive suggestions.
TABLE OF CONTENTS
ACKNOWLEDGMENT..................... .. . .. .. .. . ..
LIST OF FIGURES......................vi
LIST OF TABLES.....................viii
LIST OF SYMBOLS......................ix
1.1 Problem Statement.................1
1.2 Objective and Scope...............9
1.3 Thesis Outline..................10
2 A MECHANISM FOR MUD TRANSPORT............12
2.2 Wave Mass Transport over a Rigid Bottom . . 12
2.2.1 Mass Transport under Wave Flow in
2.2.3 Nearshore Current............23
2.3 Wave Mass Transport over a Fluid Mud Bottom . 29
2.3.1 Interface and Surface Setups........30
2.3.2 Pressure at a Generic Position z
2.3.3 Pressure at the Interface........35 iii
2.3.4 Pressure at a Generic Position z
(Mud Layer) . . .
2.3.5 Vertical Integration of Pressure
(Water Layer) .
2.3.6 Vertical Integration of Pressure
(Mud Layer) .
2.3.7 Free Surface Setup . . . .
2.3.8 Interface Setup .
2.3.9 Mean Current in the Wave Direction
2.3.10 Cross-Shore and Alongshore Mean
3 LABORATORY EXPERIMENTS . . . . . .
3.1 Introduction .
3.2 Experimental Equipment . . . . .
3.2.1 Wave Basin .
3.2.2 Wave Gage .
3.2.3 Velocity Measurement . . . .
3.2.4 Measuring Carriage . . . .
3.2.5 Other Apparatuses . . . .
3.3 Test Preparation .
3.3.1 Profile Sediment .
3.3.2 Profile Preparation . . . .
3.3.3 Profile Measurement . . . .
3.3.4 Fluid Mud Preparation . . .
36 36 38 39
41 43 46
49 49 50 50 53
54 54 55 56 57
3.4 Experimental Procedure and Data . .
3.4.1 Experiments without Fluid Mud
3.4.2 Experiments with Fluid Mud
3.5 Data Preparation and Corrections
4 FIELD DATA ON MUD TRANSPORT . . . .
4.1 Introduction .
4.2 Northeastern Coast of South America .
4.3 Louisiana Coast .
4.4 Southwest Coast of India . . .
. 58 59 61 61
. 63 63 63 68 70
5 DATA ANALYSIS 75
5.1 Introduction 75
5.2 Dimensionless Forms of Mass Transport
5.3 Simplifications of Mass Transport Equations . 80 5.4 Laboratory Data Analysis . . . . . 82
5.4.1 Correction for Shoaling . . . . 83 5.4.2 Friction Coefficient Determination . 85 5.4.3 Determination of Fluid Mud Viscosity . 88 5.4.4 Measured Alongshore Velocity . . . 89 5.4.5 Computed Alongshore Velocity . . . 92 5.4.6 Data Comparison 93
5.5 Field Data Application . . . . . . 94
5.5.1 Input Data Evaluation . . . . 95
5.5.2 Data Comparison 97
6 SUMMARY AND CONCLUSIONS . . . . . . 101
6.1 Summary 101
6.2 Conclusions 102
6.3 Recommendations for Further Studies . . 104 LIST OF REFERENCES 106
BIOGRAPHICAL SKETCH 112
LIST OF FIGURES
1.1 Idealized depiction of seasonally varying,
normally incident waves in the absence of alongshore sediment supply causing cyclic fluctuation of shoreline (A), or recession of "initial" shoreline (B), depending on the distance over which eroded sediment travels in relation to the width of the
1.2 Idealized depiction of obliquely incident
waves in the presence of steady alongshore sediment supply, which can either lead to a stable or prograding shoreline orientation (A), or a receding shoreline (B) .......5
1.3 Undulant low water shoreline associated
with obliquely incident waves and a significantly pulsating sediment load.........6
1.4 Undulant low water shoreline associated
with obliquely incident, seasonally varying waves and a practically steady (in the
mean) sediment load...............7
1.5 Normally incident waves with offshore
sediment supply leading to a crenulate
low water shoreline due to wave
1.6 Global map showing the location of the muddy
coasts of Surinam-Guyana, Louisiana and
Kerala, India (adapted from A.N. Strahler,
2.1 Schematic sketch for an obliquely
2.2 Schematic sketch of a two-layered
2.3 Forces in s-direction...............43
2.4 Schematic sketch for Urn determination.......45
3.1 Plan view of the experimental basin set-up .51
3.2 Elevation view of the experimental basin
3.3 Capacitance wave gage...............52
3.4 Strip chart recorder...............52
3.5 Metric scale ruler for alongshore velocity
3.6 Measuring carriage with video camera and
3.7 Representative bottom profiles.........57
4.1 Northeastern coast of South America
(after Wells, 1983)...............64
4.2 Louisiana chenier plain (after Wells, 1983) ..69
4.3 Southwest coast of India (after Mathew
et al., 1995)..................72
4.4 Schematic mudbank along the southwest coast
of India (after Mehta et al., 1996)........73
5.1 Comparison between measured and shoaled
wave heights at position B without fluid
mud, using the stream function wave theory ..85
5.2 Computed and measured alongshore velocities
5.3 Wave damping over a Surinam mudbank (after
LIST OF TABLES
3.1 Measured data without fluid mud . . . . 60
3.2 Measured data with fluid mud . . . . 62
4.1 Characteristic data for the muddy coast
environment of the Guyanas . . . . . 68
4.2 Characteristic data for the muddy coast
environment of Louisiana . . . . . 71
4.3 Characteristic data for the muddy coast
environment of southwest India . . . . 74 5.1 Characteristic reference parameters . . . 76
5.2 Dimensionless variables 77
5.3 Dimensionless parameters . . . . . 79
5.4 Data for friction coefficient determination . 88
5.5 Data for dynamic viscosity determination . 90
5.6 measured alongshore velocity data . . . 91
5.7 Computed alongshore velocity data . . . 92
5.8 Selected values of relevant parameters for
calculation of field mud streaming velocity 97 5.9 Values obtained from Jiang's model (1993) . 98
5.10 Calculated streaming velocities and measured
characteristic velocity ranges . . . 100
LIST OF SYMBOLS
a = amplitude of near-bed wave orbital motion
b = interfacial oscillatory amplitude
c = Wave celerity
cf = Bed friction coefficient cg = Wave group celerity D Littoral zone width
ds = Increment in the s-direction E = Wave energy density
g = Acceleration due to gravity
h Water depth, or total depth(h+h2)
hb Water depth at breaking
h, = Water depth hi = Dimensionless water depth h2 = Mud layer thickness h2* = Dimensionless mud layer thickness H = Wave height
HA = Measured wave height at position A HB = Measured wave height at position B Hb = Wave height at breaking H0 = Characteristic wave height outside the mudbank k = wave number
ki = damping coefficient kN = bed roughness
m = Profile slope
p = Pressure in the water layer
p2 = Pressure in the mud layer P = Total pressure q = Water discharge per unit width
Re. Boundary layer Reynolds number
Re, = Streaming Reynolds number s = Horizontal coordinate in the direction of wave
s* Dimensionless horizontal coordinate in the
direction of wave propagation S.. = Mean momentum fluxes
S.. = Radiation stresses
/ = Integrated Reynolds stresses
t = Time in an Eulerian coordinate system to = Time in a Lagrangian coordinate system T = Wave period
u = Velocity component in the s-direction
= Eulerian velocity vector
u = Irrotational part of u
ul = Lagrangian velocity vector
u = Time-average Lagrangian component in the sdirection
um = Maximum orbital velocity u uR = Rotational part of u ux = Velocity component in the x-direction x = Oscillatory component of ux
ux' = Turbulent component of ux uY = Velocity component in the y-direction
Y = Oscillatory component of uy
uy' = Turbulent component of uy u = Velocity in s-direction in the water layer
u1 = Dimensionless velocity in s-direction in the
u2 Velocity in the s-direction in the fluid mud
u2 = Dimensionless velocity in the s-direction in
the fluid mud layer
u = Orbital velocity at the edge of the boundary
U = Second order mean velocity in the fluid mud
layer in the s-direction
U = Depth-average value of U
U = Depth-average value of U in the cross-shore
Umy = Depth-average value of U in the alongshore
U = Second order mean velocity in the water layer
in the s-direction
Ux = Depth-average velocity in the x-direction UY = Depth-average velocity in the y-direction U* = Dimensionless depth-average value of U U= = Dimensionless depth-average value of U in the
Umy = Dimensionless depth-average value of U in the
U = Mean second order velocity inside the boundary
layer in the s-direction
U. = Streaming velocity in the s-direction UL = Total average mass transport velocity in the
V = Measured alongshore velocity without fluid mud
Vbreak = Field computed alongshore velocity due to
Vfm = Measured alongshore velocity with fluid mud Vmeas = Measured alongshore velocity due to viscous
= Time-average Lagrangian component in the z0
w = Velocity component in the z-direction
wI = Irrotational part of w wR = Rotational part of w w = Velocity component in the water layer in the
w1 = Dimensionless velocity component in the water
layer in the z-direction
w2 Velocity component in the fluid mud layer in
w Dimensionless velocity component in the fluid
mud layer in the z-direction
x = Horizontal axis in the cross-shore direction
= Space vector in an Eulerian coordinate system
x0 = Space vector in a Lagrangian coordinate system y = Horizontal axis in the alongshore direction
z = Vertical coordinate z0 = Lagrangian vertical coordinate zi = Interface (total) displacement z* = Dimensionless vertical coordinate = Angle between the s- and x- axes
= Modulating coefficient of non-linear wave
energy with respect to linear wave energy
5 = Boundary layer thickness
= Free surface displacement
i = Free surface (oscillatory) displacement
= Free surface (total) displacement
,1i = Free surface setups i = Dimensionless free surface (oscillatory) displacement
= Dimensionless free surface setup
r = Interface (oscillatory) displacement
= Interface (total) displacement
r12 = Interface setup r12 = Dimensionless interface (oscillatory)
12 = Dimensionless interface setup K Wave breaking index
12 = Dynamic viscosity of fluid mud ,\ = Kinematic viscosity of water V = Kinematic viscosity of fluid mud
P,P1 = Water density P2 = Fluid mud density P2 Dimensionless fluid mud density
o = Angular wave frequency
a.- Viscous normal stresses in the fluid mud layer
T1 = Mean bottom stress T = Mean surface stress
Tij = Viscous shear stresses in the fluid mud layer T = Dimensionless mean bottom stress
Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science
A MECHANISM FOR NON-BREAKING WAVE-INDUCED TRANSPORT OF FLUID MUD AT OPEN COASTS
Hugo N. Rodriguez
Chairperson: Dr. Ashish J. Mehta Major Department: Coastal and Oceanographic Engineering
A preliminary study was carried out to examine waveinduced transport of fluid mud at open muddy coasts. Given the large wave damping potential of bottom mud, an analytic formula for non-breaking wave-induced streaming of fluid mud
underf low was developed by considering mud to be a highly viscous continuum.
Laboratory basin tests were carried out to examine the validity of the formula for alongshore streaming. The formula showed an order of magnitude agreement with measured alongshore velocities generated by obliquely incident waves over a muddy bottom profile.
The same formula was then applied to microtidal muddy coasts in Surinam, Guyana, Louisiana and India to show that
alongshore and cross-shore mud streaming can be an important mechanism by which sediment is naturally supplied to sustain coastal mudbanks.
1.1 Problem Statement
Along muddy coasts, large amounts of cohesive sediments are present, and these sediments have important interactions
with the hydrodynamics of the littoral zone. However, even though, for example, nearly one-quarter of the total coastline of North and South America is muddy (Wells and Coleman, 1977), and although long muddy shores are present elsewhere in the world, a clear understanding of the sedimentary processes that sustain mudbank dominated coasts is lacking.
The effect of hydrodynamics on coastal mudbanks have been documented, and it is found that in the microtidal (range < 2m) environment, the dominant forcing is the action of wind waves. Free surface wind waves cause liquefaction of the upper stratum of the bottom mud and generate fluid mud. In turn, the interface between the water layer and the fluid mud layer thus formed is forced to oscillate with the same period as that of the surface wave. Wave-mud interaction is further complicated because the generated fluid mud in turn affects the wave by
absorbing and dissipating part of its energy. In fact, in many
situations the high viscous damping potential of f luid mud causes it to absorb wave energy to such an extent that waves tend to disappear entirely before they reach the shoreline.
The generation and transport of fluid mud is a unique and critically important feature that can play a pivotal role in governing bottom profile dynamics. Unfortunately the precise manner in which fluid mud is transported within the littoral zone between the shoreline and the depth of closure of
sediment transport is not well understood. Yet, f ield evidence indicates that, for example, rapid recovery of a storm eroded mud bottom profile can only be explained by taking fluid mud
transport into account (Mehta et al., 1996; Lee and Mehta, 1997). Furthermore, the observation that fluid mud transport also occurs under non-breaking wave conditions suggests that formulations for littoral sand transport based on wave shoaling and breaking over a practically rigid bottom (Bruun, 1983; Kriebel and Dean, 1993) are unlikely to be suitable for explaining muddy coast evolution.
The response of muddy coasts to wave forcing can be described within two main categories. In general, to describe shoreline response in the plan form, the low water line, which is always in contact with water, can be conveniently chosen. Now if, under seasonally variable but normally incident waves,
fluid mud is transported to-and-fro in the cross-shore direction within the littoral zone, the "initial" shoreline will remain stable without any net gain or loss of sediment (case A in Figure 1.1). This type of (first category) response
occurs along the southwest coast of India in Kerala, where fluid mud is transported shoreward during the high monsoonal wave climate in the Indian ocean. This transport results in shore-fast mudbanks, which dissipate at the end of the monsoon as wave activity wanes and a gravity slide returns the mud to the deeper offshore region within the littoral zone ( Mathew, 1992; Mathew et al., 1995; Mathew and Baba, 1995). This type
of response may also explain the stability of some muddy coasts in Indonesia (Tarigan et al., 1996).
When the sediment transport pathway and travel distance are such as to cause the eroded material to go out of bounds
of the littoral zone, the shoreline will recede (case B in Figure 1.1). In some cases, this (second category) response may be actuated at a stable shoreline by drastic changes in the water level, as along the coasts in Lake Ontario, Canada (Coakley et al., 1988).
Dredging can also lead to significant land loss. A dramatic example is that of shoreline recession in excess of 1,200 m between 1972 and 1991 in the Ondo State in Nigeria, possibly due to the dredging of a navigation passage across a
barrier island, which caused strongly scouring tidal currents on both sides of the new entrance (Eedy et al., 1994; Ibe et al., 1989).
--- Receding LW Shoreline Initial LW Shoreline
f Depth of Closure
Normally Incident Seasonal Waves
Figure 1.1 Idealized depiction of seasonally varying, normally incident waves in the absence of alongshore sediment supply causing cyclic
fluctuation of shoreline (A), or recession of "initial" shoreline (B), depending on the
distance over which eroded sediment travels in
relation to the width of the littoral zone.
If waves approach the shoreline obliquely, the alongshore
mud stream they generate can lead to a stable or accreting shoreline provided there is a source of sediment, as depicted in Figure 1.2 (case A) (first category response), or lead to recession if the profile cannot be replenished at the rate at which it is eroded (case B) (second category response).
. . ..- Receding LW Shoreline: B Stable or Prograding LW
Steady Shoreline: A
---------- - -- -Depth of Closure
Obliquely Incident Waves
Figure 1.2 Idealized depiction of obliquely incident waves in the presence of steady alongshore sediment supply, which can either lead to a stable or prograding shoreline orientation (A) or a
receding shoreline (B).
The stability of some of the muddy coasts that are subject to high tidal variations and wave action in South Korea has been attributed to adequate supply of sediment from nearby rivers (Wells, 1983). Along the Selangor coast of peninsular Malaysia, reduced sediment supply to the coast due to fresh water diversion has led to sea encroachment into cultivated lands (Midun and Lee, 1989). In Jiansu Province in China, the long term effects of the shifting of the mouth of the Yellow River has led to prograding beaches where sediment supply from the river is ample, and significantly eroding beaches where the supply is inadequate (Ren, 1992).
- ------ ------- Depth of Closure
Obliquely Incident Waves
Figure 1.3 Undulant low water shoreline associated with obliquely incident waves and a
significant pulsating sediment load.
Muddy coasts exhibit short and long term modulations that
are qualitatively akin to those at sandy coasts; yet the mechanisms that cause these features in these two environments are not always analogous. In the case of muddy coasts,
variations in alongshore supply and wave climate can cause the shoreline to develop a spatially rhythmic or undulating configuration, as in Surinam, Guyana and Louisiana. The simplest way to explain shoreline response in the contiguous
coasts of Surinam and Guyana (Figure 1.3) is in terms of a trade wind induced steady wave action from the northeast in combination with a temporally pulsating sediment supply from
the east derived from the Amazon River. The wave damping
potential of f luid mud in this region is very high. It is reported that as much as 96% of offshore wave energy is dissipated by viscous damping (Wells, 1983).
- - Depth of Closure
Obliquely Incident Seasonally VaryingWaves
Figure 1.4 Undulant low water shoreline associated with obliquely incident, seasonally varying waves and a practically steady (in the
mean) sediment load.
An undulant shoreline may also result if the alongshore supply of sediment is steady, say, on an annual basis, but the
wave intensity and direction change seasonally between the winter and summer months (Figure 1.4) The wave-dominated coastal mudbanks bordering the chenier plain in Louisiana west of the mouth of the Atchafalaya River show features that correspond to this description in an approximate way. Mud is supplied by this river, which receives its sediment load from the Mississippi River. Wave energy losses as high as 48% have
occurred due to viscous damping within fluid mud (Tubman and Suhayda, 1976).
Sand A LW Shoreline
Wave ~ t
-[ - Bathymetry
Figure 1.5 Normally incident waves with offshore sediment supply leading to a crenulate low water shoreline due to wave
The monsoonal mudbanks of Kerala in India tend to be disk-like in plan form, and occur intermittently along a nearly straight sandy shoreline, giving it a crenulate appearance (Figure 1.5). As Mathew and Baba (1995) have shown, the dominant cross-shore mechanism for mud streaming there is influenced by wave refraction due to offshore bathymetry, and this leads to mudbank formation at known locations where wave energy is concentrated. As a result, monsoonal wave breaking activity shows a distinct variability ranging from practically nil in the mudbank areas to plunging breakers along the
intervening sandy beaches. Mathew et al. (1995) have noted that during monsoon, as much as 90% of the wave energy is damped out over the mudbanks.
Three regions of the world (Surinam-Guyana, Louisiana and Kerala, India) that are further described in this study are shown in Figure 1.6.
/ / / / ~&i1 'I L3,LIiiJ~ia / I I / ~'I ~
is. . .90 135 18 5
I / / I I&~~~i~/ / L
Figure 1.6 Global map showing the location of the muddy coasts of Surinam-Guyana, Louisiana and Kerala, India (adapted from A.N. Strahler, 1969).
1.2 Oblective and Scope
In this study, a mechanism for fluid mud mass transport at muddy open coasts where non-breaking wave-induced forcing may be the dominant cause of transport has been explored. The action of progressive waves in a two-layered (water-fluid mud)
15 18C 15
V A "_N
~ T '
h H i
60 7 60
system is studied by means of the vertically integrated
momentum equations. The upper layer is considered to be an inviscid fluid and the bottom layer a Newtonian viscous fluid. The free surface and interfacial setups generated as well as the second order depth-average drift are calculated. To examine the validity of the expression for alongshore drift, laboratory basin tests are carried out. Field data from three different muddy coastal areas (Figure 1.6), where good quality data on alongshore and cross-shore drift are available, are analyzed in conjunction with theory.
1.3 Thesis Outline
Chapter 2 presents the derivations of setups and mass transport velocity equations. First, a review of mass transport mechanisms over a rigid bottom, i.e., a one-layer model, is presented. Then, the momentum equations for a twolayered model are integrated in the vertical to obtain the free surface and interfacial setup as well as the cross-shore and alongshore mass transport velocities.
Chapter 3 deals with laboratory basin experiments. In this chapter, laboratory test preparation and performance are mentioned. Wave height measurements at two locations and alongshore currents over a muddy profile built in a basin are described.
Information on the hydrodynamic and sedimentary
environments of the northern coast of South America, Louisiana and the southwest coast of India is reported in Chapter 4.
Chapter 5 gives a comparison and discussion of the
theoretical velocity values with the field and laboratory measured values. For theoretical analysis, a nondimensionalization of the mass transport is carried out.
Finally, in Chapter 6 concluding remarks and recommendations for future studies are made.
A MECHANISM FOR MUD TRANSPORT
At an infinitely long shoreline in the presence of fluid mudbanks, different hydrodynamic mechanisms may cause crossshore and alongshore movements of these banks. In this chapter, an analytical description of such movements caused by waves is presented. First the classical case of a rigid bottom is stated. Then the case of a two-layered fluid, corresponding to the presence of a fluid mud bottom, is analyzed.
2.2 Wave Mass Transport over a Rigid Bottom
The first order solution for the propagation of waves in water of uniform depth h, over a rigid bottom, is given by the linear wave theory. For simplicity a monochromatic sinusoidal perturbation of the water surface is considered. This
assumption does not imply a loss of generality, because a perturbation of the water surface can be reduced to a
summation of sinusoidal functions by means of Fourier
transform. As the first order linear theory is being considered, the different sinusoidal constituents propagate independently, and their linear superposition is valid.
From the linear wave theory, different non-linear properties that are correct to second order can be derived. These properties are the second order momentum flux, the setup and mass transport. They are obtained averaging over the wave period and integrating over depth.
2.2.1 Mass Transport under Wave Flow in Constant Depth
Linear wave theory predicts the trajectories of the water particles to be closed ellipses. However, if the particle displacement under a wave is measured, a net movement in the direction of wave propagation can be observed after a wave length has passed. This movement increases from the bottom towards the free surface. If the Lagrangian description of water motion is used, this second order mass transport, or drift velocity, can readily be derived.
In an Eulerian description of water motion under a wave, the physical properties are given as functions of fixed positions in space k and time t. Therefore, the velocity field is represented as il=U(k,t). The Lagrangian description follows the movement of a particle that, at a given time to, was at a position x0 = (x0 Iz0) so that the independent variables are the initial coordinates x0 and the elapsed time t-t0. The
relation between these two descriptions of the same physical phenomenon is
K = x(:0, t-to); K 0' 0) = xo (2.1)
As the velocity of a fluid element is the time-derivative of its position,
S( O' t-t) = 0t -t) (2.2)
S= K+ f ( I t-to)dt (2.3)
where U = (ul w) is the Lagrangian velocity, which can be equated with the Eulerian velocity,
U( t) = (0+f (0 t-t0)dt, t) (2.4)
Using the Taylor series expansion to the second order, the final expression for the Lagrangian velocity as a function of the Eulerian velocity is obtained as
U11( 0,t) U i (R0,t) + fU(K0,t-t0)dt .Vx0U(x0,t (2.5) to
This expression shows that, to the first order, both the Lagrangian and the Eulerian velocities are the same and that their mean values over a wave period are zero. On the other hand, the second order term of the Lagrangian velocity has a non-zero mean value. If the linear wave theory is used, the second order mean value components of the Lagrangian velocity
with a and k being the wave frequency and the wave number, respectively, H the wave height, u the velocity component in the direction of wave propagation s, and w the velocity component in the vertical direction z. The coordinate axis, s, is horizontal at the still water level and positive in the direction of wave propagation, and z is positive upwards from the still water level. The total water discharge per unit width over the water column associated with this mass
transport velocity is obtained by integrating over the water depth as
where T is the wave period. Due to the linear approximation the Eulerian velocity is periodic under the wave trough and thus its wave-mean value is zero in that region. However, in the region between the trough and the crest, the wave-mean velocity must be obtained by means of Taylor expansion and is found to have a non-zero value in this region. The total water discharge per unit width can therefore be calculated at a given position s, and the same result as equation (2.7) using the Lagrangian approach is obtained. The Eulerian analysis thus yields the same total discharge associated with the net drift, but instead of being distributed over the entire water column, it is confined to the region between the trough and the crest of the wave. Both analyses are different approaches to account for the same phenomenon.
The above results are derived from the linear wave theory considering inviscid irrotational motion, which yields a finite value of the horizontal velocity at the bottom. In a real fluid the no-slip condition at the bottom must be considered and, therefore, the viscous stress term in the
Navier-Stokes equation must be taken into account in the nearbed region.
As noted, the potential theory of irrotational flow predicts a finite value of the horizontal velocity u at the bed. In a real fluid this velocity must be zero and accordingly leads to the development of a boundary layer in which the viscous terms in the momentum equation must be taken into account because they are comparable to the other terms.
A local coordinate system is adopted in the boundary layer with s positive in the direction of wave propagation at the horizontal bottom boundary and z in the upward vertical direction from the bottom.
The continuity equation for an incompressible flow is 8u aw
au 0 (2.8)
with u and w the horizontal and vertical velocity components, respectively. As the boundary layer thickness 6 is much smaller than the wave length L, the spatial derivatives in the s-direction can be neglected with respect to derivatives in the vertical z-direction, so that the Navier-Stokes equation in the s direction can be approximated as,
au auw 1 aP 2u
+ = +- (2.9)
at 8z p as az2
where P is the total pressure and p and v the density and the kinematic viscosity of the fluid, respectively. As w is much smaller than u the Navier-Stokes equation in the z-direction reduces to
0 ---g (2.10)
with g the acceleration of gravity. Noting that p = P pgz is constant in the z-direction according to equation (2.10), the Euler equation, i.e. equation of motion excluding the viscous contribution, applied to the inviscid flow region can be used to define the pressure gradient in the boundary layer.
To the first order, the horizontal velocity component u in the boundary layer can be considered to be the superposition of an irrotational part u1, given by the linear wave theory, plus a rotational part uR, since to the first order, the equations are linear.
The irrotational part satisfies the linearized Euler equation
u 1 p (2.11)
at p as
and, after substitution of equation (2.11) into equation (2.9), the resulting equation is au R a2u R
- uZ (2.12)
with the boundary condition at the bottom
u = uI+uR = 0 i.e. u = -uI 2.13)
The solution of equation (2.12) with the boundary condition (2.13) is
uR - cos ks-ot+- (2.14)
2 sinhkh k
5 2 (2.15)
is the boundary layer thickness.
The flow is consequently oscillatory, and as the horizontal velocity given by equation (2.14) is a function of s, it induces a vertical rotational flow velocity wR in the boundary layer that can be found from the continuity equation
(2.8), along with the boundary condition wR = 0 at z = 0 (Dean 1996b)
wR (z) ____
2 sinhkh (2.16)
e- 6cos(ks-ot+ z )- cos(ks -ot- ) 6 4 4
As observed from equation (2.16), wR does not go to zero outside the boundary layer, that is, as z tends to infinity, wR approaches a finite value. This can be described in physical terms from the continuity equation applied to a control volume including the entire boundary layer depth. As the horizontal discharge per unit width, corresponding to uR, changes with s along the boundary layer, to satisfy mass conservation a vertical discharge associated with w R must exist.
Referring to w_ as the vertical velocity just outside the boundary layer, an additional shear stress uw is induced. This stress is balanced by the viscous stress generated by a mean flow in the boundary layer. The mean momentum equation which states this balance is (Dean 1996b)
- a- 2 (2.17)
az 0z 2
in which U is a mean second order velocity (overbar indicates mean values over a wave period), or
uw UW = va z
uw = (u I+uR) (wI +WR)
and, correct to the second order,
- H2 2k6 uw =
16 sinh2 kh
2( e- sin( ) +2e-( ) cos
5 5 5)
16 sinh2 kh
3 -2 (+2) e-(z) cos ) -2(z -1 e-W sin( )
+ e (
which gives a second order mean streaming velocity, U., outside the boundary layer in the direction of wave propagation equal to
- 3 H2ok
16 sinh2kh (2.22)
*16 sinh2 kh
This mean velocity is independent of the viscosity v even though viscosity must be non-zero for its existence (Dean, 1996b; Phillips, 1966).
The above analysis was carried out considering a laminar boundary layer, for which a Reynolds number can be defined as
Re. 5 (2.23)
where u. is the orbital velocity at the edge of the boundary layer. For values of Re,, > 160 a turbulent boundary layer develops (Phillips, 1966) In this case, an eddy viscosity ve, must be introduced in equation (2.17) in place of the kinematic viscosity v. However, in equations (2.9) and (2.12) the fluid kinematic viscosity v should be retained. A streaming velocity is also found in this case, but its value
is less than that given by equation (2.22). In this case, streaming is no longer independent of viscosity but becomes a weak function of parameter -; a. being the amplitude in the k N
near-bed motion just outside the boundary layer and k, the bed roughness (Fredsoe and Deigaard, 1992).
For laminar fluid mud flow relevant to this study, the
total mass transport velocity just outside the boundary layer,
U'correct to the second order, is therefore the sum of the irrotational expression given by (2.6) at z0 =-h and U_; i.e.,
5 H2 k
UL 16 sinh2kh (2.24)
2.2.3 Nearshore Current
In the previous section the mass transport generated by waves over a constant depth of water was analyzed. However, coastal areas generally have a bottom slope that causes other effects, such as breaking, that affect mass transport in other ways.
The radiation stress is another mean wave quantity introduced by Longuet-Higgins (1953), which represents the sum of the momentum flux and the mean pressure effects, and which must be accounted for describing mass transport in the littoral (surf) zone.
The equations of conservation of mass and momentum are considered, but instead of using their differential forms, they will be used after integration over the total instantaneous depth h + p (where p is the water free surface displacement) and averaging in time over a wave period.
The mean wave-induced velocities in the horizontal horizontal directions, x and y (Figure 2.1), can be defined as (Thornton and Guza, 1989)
Ux- fuxdZ ;
x h+ -h
U u dz
where r is the mean deviation of r from the still water level. The corresponding velocity components are
u = U ++u ; x x x x
U = U + +u y y y Y
in which denotes oscillatory component and the prime denotes turbulent component.
Figure 2.1 Schematic sketch for an obliquely
After integrating the continuity equation over depth and averaging the resulting expression in time, the following equation is obtained:
8Ux (h+r) BUy (h+i) = 0 (2.27)
Integrating the momentum equation, the components of the mean horizontal momentum per unit area in the x and y directions are obtained as:
au_ au _a asx
X X X XX
p(h+E) --+Ux +U +y]
aS -8+ s b
+ =y' -pg(h+-) x +Tx- x
au au au_1 as
p (h+E) Y +U '+U Y + ax p(h+nl 8t x 8x +ey J 8x
+ YY = -pg(h+E) + -i1
In equations (2.28) and (2.29), Ts and Ib are the mean surface and bottom stresses, respectively, and the first term on the right hand side is the horizontal force per unit area caused by the slope of the mean water level. The mean momentum flux Sij (with ij = xx, yy or xy) is defined as
S -S +.+S
1] 2] i
The contribution due to integrated Reynolds stress eddy viscosity. The wave radiation stresses and are
turbulent motion, S'ij, is the and can be parameterized using the motion contributions are called given by
Sxx = P fUxdz + Pdz 2 pgh
S = p f ydz + Pdz- 2 pgh2
S= p ffaydz
(2.31) (2.32) (2.33)
where P is the total pressure. If the linear wave theory is used, the following expressions are obtained (Dean and
E 2 C' Cos 2U+ 2 CH-1 xx 2 2c cs2+2c
= 2 g sin2 + 2 c -1 7y 2 c c
-2 sina cosa (2.36)
where E is the energy density, c9 the group velocity, c the wave celerity and u~ the angle of wave incidence with the cross-shore direction. Equations (2.34) through (2.36) show that the wave momentum fluxes (radiation stresses) are proportional to the wave energy. Therefore any change in the energy, such as by dissipation due to breaking, will cause the radiation stresses to change. These changes must be balanced by external forces so that the momentum equations (2.28) to (2.29) are satisfied. That is, inside the surf zone forces are induced that in turn account for water mass transport in this zone.
In the cross-shore direction, as there is a boundary at the shore, the balancing force is a pressure gradient (associated with the wave setup n~). Given x the cross-shore direction, the final equation derived from (2.28) is
___ ___ XX (2.37)
dx pg(h+-) ax
In the case of an open and unbounded shore in the alongshore direction, there is no adverse pressure gradient capable of balancing the radiation stress. As a result, a mean current is
produced that generates a bottom shear stress which balance the radiation stress.
For a long and straight shoreline that can be considered to be infinite in extent, all derivatives with respect to y must be zero, so that the balance of forces in the alongshore y-direction, when the coupling term is neglected, is
- b (2.38)
Given the turbulent nature of flow in the surf zone, the mean bottom shear stress (due to turbulent flow) is usually expressed by the quadratic law: b = PCf (2.39)
where cf is the bed friction coefficient and U is the total velocity. For the case of small angle of incidence of the waves, Longuet-Higgins (1970) simplified and linearized the expression for the bed shear stress to
u PCfumU (2.40)
by 4 T
where us is the maximum orbital velocity. Solving for UM, Longuet-Higgins gave the following expression:
U (x) 5 ng KM(h + l)(sic
2 K 2 C) (2.41)
2 cf 1+ 3 l
where K is the wave breaking index, m the bottom slope and c the wave celerity. The index K is given by the breaker
criterion of McCowan(1894), which states that the breaker height H b =Kh b with K 0.78; the subscript b denoting the values at breaking (Dean and Dalrymnple, 1991). This criterion will be used for accounting for the wave breaking effect in laboratory measurements in Chapter 5.
2.3 Wave Mass Transport over a Fluid Mud Bottom
A two-layered (water-fluid mud) model is now introduced in order to calculate the momentum fluxes and mass transport
when a fluid mud bottom occurs under a water column. An inviscid water column and a viscous fluid mud are assumed, subject to a progressive, monochromatic, non-breaking wave at
the free water surface (Figure 2.2). For simplicity, both layers are considered to be of constant depth. First, the derivation will be carried in the wave propagation direction s, and the setups at the free surface and the interface will be derived. Then the bottom stress and mean velocity in the wave propagation direction in the absence of a setup will be
derived. Finally the cross-shore and alongshore velocities under an obliquely incident wave will be considered.
2.3.1 Interface and Surface Setups
In order to obtain mean quantities within the two-layered domain, the Navier-Stokes momentum equations are used. With reference to Figure 2.2, a 2-D problem in a vertical plane is considered. The s-direction is horizontal at the still free surface along wave progression and z is the vertical
direction, positive upwards from the still free surface.
Figure 2.2 Schematic sketch of a two-layered fluid flow.
Subscript 1 refers to the water region and 2 to the fluid mud layer. The overbar denotes mean value in time due to second order effects. Also, Tli and r12 are the oscillatory components of the free surface and interface displacements, respectively, and r and i2 are the total free surface and interface displacements, respectively. The water layer depth h, and the mud layer thickness h2 are considered constants. Water is considered to be an inviscid fluid, thus neglecting its viscosity V1 in the momentum equations. Fluid mud is considered to be a Newtonian viscous continuum with kinematic viscosity V2"
The horizontal and vertical components of the general Navier-Stokes equations, after introducing the continuity equation for an incompressible fluid, are
au 0u2+ Ouw -1 Op +( -2u + a2U(2.42)
Ot 7s Oz p Os 0s2 0z2
O + Ouw+ 1 Op ---+ (2.43)
at as z Sz as2 az2)
where u and w are the velocities in the s- and z-directions, respectively.
In order to integrate equations (2.42) and (2.43) in both the water and fluid mud layers, the following kinematic and dynamic boundary conditions are required:
At the water free surface the kinematic condition is
at 1(slt)--as -wi(S'rlt) 0 (2.44)
The dynamic condition at the free surface states that the pressure at the free surface is the atmospheric pressure p0, which, without any loss of generality, can be considered to be equal to zero.
At the interface between the two layers the kinematic condition states that
alt+u2 (S zit)-a -s -w2(szi~t) = 0 (2.45)
where zi hi, is the total displacement of the interface for the coordinate system chosen (Figure 2.2). Also at the bottom z = -h = (hi+h2) both u2 and w2 are zero.
The dynamic condition at the interface asserts that the stresses in both directions for both fluids must be the same, as follows:
P i- = (-Gss +p2) as + sz
-PI = z-P2 Tsz as
in which o.. and T.3 respectively are the normal and shear stresses given by the viscous stress tensor:
au2 iaw2 au2
s = 212 (2.47)
Lsx ]zz 1( s2 u 2)
a2 az az
After having established the appropriate governing equations and boundary/interfacial conditions, the necessary integral values of the pressure gradients that are required to solve for the free surface and interfacial setups will be now determined. The pressure terms are derived first, followed by setups.
2.3.2 Pressure at a Generic Position z (Water Laver)
In order to obtain the pressure at a generic point z within the water layer, equation (2.43) is integrated over the vertical between z and the free surface. For this layer,
neglecting the kinematic viscosity of water vj, this integration yields
p dz+ 1 W 2 (2.48)
fP1 adz +f Pl dz +p~w1 'Z = -P~g (rllt-z) p1' (248
Then, making use of Leibnitz rule,
a b(x) b(x) OF(x.,x.)
Sf F(xi,x) dx.= x,
ax. f ax
1 a(xi) a(xi) (
8b (x.) Ba(xi)
+F [xi.,b(x.) ] -F [xi.,a(xi) ]
1 1ax. ax.
and taking into account the kinematic condition given by equation (2.44), as well as noting that the pressure is zero at the free surface, the pressure at a generic position z in the water region is obtained as
p (z) = Pw dz+ PU Wdz -P1 wl2 +P1g( _-z) (2.50)
To obtain the pressure at the interface it cannot be simply stated that z = z., because the generic position z is a fixed coordinate point that does not vary with time or s, whereas the interface is a function of time and the propagation direction s, i.e., zi = z.(s,t). Thus, in order to obtain the
pressure at the interface, a different integration must be carried out.
2.3.3 Pressure at the Interface
To obtain the pressure at the interface, equation (2.43) is integrated between the interface zi = zi(s,t) and the free surface I :
P1 dz + p sdz+pw2 I' =p1g (r -z) p |l (2.51)
Then, making use of the Leibnitz rule given by equation (2.49), introducing the dynamic condition p = 0 at the free surface, and considering the kinematic condition (2.44) at the free surface and the kinematic condition (2.45) at the
interface, the pressure at the interface is obtained as
P1(zi) 1w dz+- pul w dz+plg(i -zi) (2.52)
atf zii T fPUW
As can be seen by comparing equations (2.52) and (2.50), the introduction of the interfacial kinematic condition (2.45) eliminates the term -pw2 in the interfacial pressure. If the pressure at the interface were computed by considering z = zi in equation (2.50), that term would have remained because
interfacial variation with time and s, and the condition (2.45) would not have been considered.
2.3.4 Pressure at a Generic Position z (Mud Layer)
For the fluid mud layer, equation (2.43) is integrated between a generic position z and the interface zi:
Z i aW2 zi u 2 WZ2 2
f 2 dz+fp a dz +pw2 2 i= 2 Zi-)
-p2z + -S dz+ozzz
Z f das ZZ
and again making use of Leibnitz rule (2.49) and the
kinematic condition at the interface (2.45), the expression for pressure in the mud layer is obtained as
p2(z) p1 (Zi) +2g (Zi-Z) -P2W2
zi zi zi (2.54)
+ P22 dz+ p2 u2w2 dz -- Tszdz +ozz
z z z
2.3.5 Vertical Integration of Pressure (Water Layer)
The pressure p, obtained in equation (2.50) must be integrated over the vertical between the interface zi and the
free surface ni in order for it to be used in the integrated momentum equation in the s-direction. This integration yields
t t t
,1 1 1 a Il
p1(z)dz = f -piwldz dz
zi zi z
4 ut t t
+ u-s Pulw dz dz p wy dz
zi z zi
t z r
+pg z- 2 zi
After taking the derivatives outside the integral and differentiating with respect to s,
+ asat fPwldz
8zi 8 [ + a pu
as as Ju
t t t
Il dz pw dz + as pwldz
LZi Z z
r~l r~l a2 Z. Il
dzjpiuwidz +as puwldz
z z z
a p1g a(1-zi)2
a f pw 2dz +-as 1 2 as
Noting that the mean value of the oscillatory terms on the left hand size of (2.56) over a wave period is zero, and neglecting terms of order greater than two, becomes
fp1(z)dz Oz a2z i
t s plwldz + f PWldz
-Z atsat z
a2 z llwd (2.57)
a l p1g (_-zi)2
pw 2dz +
as fJ1 2 8s
In this expression the overline stands for a temporal average value.
2.3.6 Vertical Intearation of the Pressure (Mud Layer)
As was done for p, in section 2.3.5, in this section the pressure p2(z) given by equation (2.54) is integrated in the mud layer from z = -h = -(h1+h2) to the interface z = zi. After taking the derivatives outside the integrals, the following expression is obtained:
zi z zi zi zi zi
fp2 (z) dz = dz P2 dz + i dz p2u2wdz -p28w dz
-h -h z h z -h (2.58)
P2 i zi ]I zi ( 8
+ 2g(z.+h) a fdzfTdz +f ozzdz+p (z) (z +h)
-h z -h
Next, as was done for p1(z), equation (2.58) is differentiated with respect to s and the mean value over a wave period is
obtained. The final temporal-mean value of the pressure term in the mud layer thus is given by
0z. 0 z. z. z.~wd
2 P(z) dz = 2 dz p2u2w2dz p2w dz
h -h z -h
SP2g 8(zi+h)2 a2 Z a d (2.59)
+ (dZ (szdz +- a dz"
2 as as 2 s zz
-h z -h
a(zi +h) apl (z)
+p (z) + (zi+h) as
2.3.7 Free Surface Setup
In order to obtain the free surface setup, the equation of momentum in the s-direction (2.42) is integrated in the water layer from the interface zi to the free surface rf:
" uI 11 Bu2 t i p
f P dz+ pf P1s dz +uw Iz a dz (2.60)
zi zi zi
If the derivatives are taken outside the integrals and the kinematic conditions at the free surface and interface are introduced, it follows that
] t t
a i II nllud ra [azi
dz + u =- p a dz -pl (zi) (2.61)
uldz +7 1 as U as
zi zi zi
After inserting the expression for p1(zi) from equation (2.52)
and averaging over time, the following expression is found:
p u1d p d-p, g (p -z.) 8zi a dz (2.62)
- 1rd -~fdZ a as at f-Zi
Bsz. 8sz. *B s8
S31 31 O~ z (62
When equation (2.57) is introduced and the terms of order
greater than second are neglected, the final expression for
the free surface setup is obtained:
a80 a 0 2 2
g(h +n-np2) 0 a (u -w1)dz as as J
a2 0 z wal 1
S 2 f dzf uwdz -g (1 -2) as (2.63)
fw2 0 a2 a 0 O2 w dz
at as /wdz -- w dz fidz tas f1dz
-hI as at -hi
2.3.8 Interface Setup
Next, in order to obtain the interfacial setup, equation (2.42) have to be integrated in the mud region from the total depth -h to the interface zi. This time, the fluid is considered to be a viscous fluid. Therefore, the viscous terms must be taken into account. After taking the derivatives outside the integrals and considering the kinematic and dynamic conditions at the interface, the following equation results:
P2 U2 dz +P2- U 2zP2dzj
h h h
az. a z
+p1(z.)- + foss dz
Recall a similar procedure as the one used in the water
region, that is, the methodology proceeded to insert p1(zi) from equation (2.52) and then averaging in time over a wave period, taking into consideration equation (2.59). The equation for the interfacial setup is obtained as
(p2- pl) g (h+2) a2
-h 2 2 r2 -h
2 (U2w 2)dz 2 OS2 fdz fu w dz
_h as h z
a2 zi zi z
+ d2 z IszdZ + (Cs ozz)d
-h z -h
- PO 2 1 2 _1g (h2 a
- Os -(P2-P)g2- P Pg(h2+r2) Os
If the relations given in equation (2.47) for the viscous stresses are considered, after keeping the second order terms, the final expression for the interfacial setup is
( 2 1 2 2
(p2-p1)g(h2+ 2) Os-p2 (u 2-w 2) dz -p2 ~2-h fdz f u w dz
+ 2 h 2 2 u =2 +2p2 -2 z =-h2
+ aS2 r-r a aT lz _h 21 as az z=-2
-Pg) 2 P2 -PO1s2 2g(h +r2)
The two unknowns in equations (2.63) and (2.66) are the water surface setup fl and the mud-water interfacial setup 12"
In the case of a wave train which approaches the coast at a certain angle a~ (Figure 2.1), both cross-shore and alongshore components of waves forces will be present. At an unbounded coast, in the alongshore direction, a setup cannot occur and
the mean forces must be balanced by a shear stress at the bottom. This shear stress can only be generated by a mean current.
2.3.9 Mean Current in the Wave Direction
When a setup cannot occur, the mean shear stress produced is equivalent to the setup pressure gradient. With reference to Figure 2.3, the following steady state force balance can be stated:
Figure 2.3 Forces in s-direction.
-- -- 2l
Tods = hl+ + ds--r- dS
0B a 12 8s 2
+h2 2 ds +2 h +r + ds h2++ 2 ds 2g (2.67)
as as s 2
(h + )2 12 g
h i +p lg -[h2 +12+ 2hi +p i](h2 2fr 2
Neglecting the quadratic terms in ds, one obtains
T0(s) = (h, +p- ) p 1+(h+ 2 2 1g
+[(h,+ +hl+ )p2 (h, +p-2 12
The mean bottom stress T0, which in effect is equivalent to the setup pressure gradient, must be generated by a mean velocity, as shown in Figure 2.4. If a steady state, fully developed viscous flow is considered in the mud layer, the differential equation to be satisfied by the mean velocity U is
d2U 1 dP
dz2 p2 ds (2.69)
This is a Poiseuille flow, in which the total pressure P is the driving force and a function only of s. Since U does not
vary with s, is a constant and can be expressed as a
function of To*
Figure 2.4 Schematic sketch determination.
Also, the following boundary conditions must be satisfied:
at z = (h1+h2)
- 0 at z = -hi 6z
To 2 dz lz=-(h1+h2)
the average mean velocity in the mud layer can readily be
U (s) 2 (2.72)
and the mean velocity in the water layer is
U (s) T0(s)h2 (2.73)
w 2 2
2.3.10 Cross-Shore and Alonashore Mean Currents
When a wave train approaches a muddy shoreline at an angle a as shown in Figure 2.1, both a cross-shore component UX and an alongshore component U Y are present. In the cross-shore direction, when the fluid mud is far from the coast, the wave effect will move it towards the shoreline until the shore will prevent further movement of the mud and a setup will build up in that direction, thus leading to the formation of a convex, shore-fast mudbank (Jiang and Mehta, 1996). In the alongshore direction the second order effects considered will produce an alongshore current because in at open coast no setup is possible. For small angles of incidence of the wave train and considering a non-reflecting shore, the mean velocities in both directions can be expressed as
U (S) = U (s)cosa;
Umy(S) = Um(s)sina
or, introducing the linearized equation (2.68) and equation(2.72):
Un- h2 (hlpl+h2p2)g- +[(h2+hl)p2-hlp1l]g --sc (2.75)
U- 3 2 (hlpl+h2 2)g-as + [(h2+hl)p2-h1p1lg sin (2.76)
It should be again emphasized that equations (2.75) and (2.76) are valid only for small angles a. Higher values of the angle of incidence introduce modifications due to refraction that were neglected in the above derivations. Furthermore, equations (2.75) and (2.76) were obtained based on the assumption of a 2-D model in the vertical. Thus, no variations in wave energy were considered in the horizontal direction perpendicular to s, i.e. along the wave crest. If a is not small, this assumption is no longer valid because at a given time t the wave front travels different distances over the mudbank in the direction perpendicular to s and the wave height is no longer constant in that direction. In this case, momentum transfer along the wave crest must be taken into account.
Equation (2.75) and (2.76) yield the desired cross-shore and alongshore mass transport velocities in a two-layer system inclusive of the irrotational and rotational second order effects.
To examine the validity of equation (2.76) derived in Chapter 2, which gives the alongshore mass transport velocity
in the mud layer, laboratory experiments were conducted in order to compare the theoretically predicted results with experimental measurements. The tests were carried out in a wave basin at the Coastal Engineering Laboratory of the University of Florida. The experimental set-up and conditions
together with the test data obtained are presented in this chapter. Experimental results along with field data are analyzed and discussed in Chapter 5.
3.2 Experimental Eauipment
3.2.1 Wave Basin
The wave basin used for the experiment measured 3.1 m wide, 0.9 m deep and had a total length of 33.5 m. A
monochromatic wave generator was located at one end of the
basin. It was a piston-type wavemaker consisting of a vertical
paddle driven by a 7 HP motor. The wave amplitude could be adjusted by changing the piston displacement, and the wave period by changing the angular frequency of the transmission from the motor to the piston.
Only a portion of the basin was used for the experiment. A muddy bottom profile, obliquely placed in relation to incident waves, was built 16 m from the wavemaker (Figures 3.1 and 3.2).
3.2.2 Wave Gaae
A capacitance-type wave gage was used to measure the wave height (Figure 3.3). The data were recorded on a strip analogue chart recorder (Figure 3.4). Wave height and period were measured at two locations situated along the centerline
of the basin. The first, called A (Figure 3.2) had a still water depth of 0.144 m and was located 12.0 m from the wavemaker. The water depth at the second site, called B, was
0.114 m and was 1.2 m shoreward of location A.
3.2.3 Velocity Measurement
Alongshore wave-induced mean velocities were measured using colored styrofoam tracers and recording their movements with a video-camera. For the purpose of calibration of tracer movement a linear scale ruler was fixed on the beach (Figure
3.5). The tracers used were flat cylinders with a diameter of 2.5 cm and 1 cm width. They were colored in order to facilitate the measurements.
S- 17.3 m
I4K~ S ill .2 12.0Om
Figure 3.1 Plan view of the experimental basin setup.
W avem aker,
Figure 3.2 Elevation view of the experimental basin set-up.
Figure 3.3 Capacitance wave gage.
figure 3.4 Strip chart recor
Figure 3.5 Metric scale ruler alongshore velocity calibration.
3.2.4 Measuring Carriaqe
Both the video-camera and the wave gage were mounted on a measuring carriage (Figure 3.6). This carriage was of steel
construction mounted on wheels that moved on iron rails placed on both side walls of the basin.
3.2.5 Other ADaratuses
Other apparatuses to ease the preparation and running of the experiment were used. Some examples are a water pump, buckets, spades, rulers, etc. For setting the final profile of the mud bottom(at a slope of 1:40), a carriage-mounted point gage with a flared tip was used.
3.3 Test Preparation
3.3.1 Profile Sediment
In order to run the experiments a mud profile was built in the basin. The sediment used for the profile was a clayey mixture of an attapulgite and a kaolinite in equal proportions by weight. The mixture was made with commercially available clays: attapulgite from Floridian Company in Quincy, FL and kaolinite from Feldspar Corporation in Edgar, FL. Kaolinite is a pulverized kaolin, which is a light beige powder, and attapulgite is a greenish-white powder. The median (dispersed) size for both clays was approximately 1.5 pm.
Well water was used to mix with the powder mixture in order to generate mud for the profile. This mud had a
relatively high density of 1,355 kg/n3 to ensure that it could sustain itself in a sloping configuration.
Figure 3.6 Measuring carriage with video camera and wave gage.
3.3.2 Profile Preparation
mud profile preparation was carried out by first building a wall with concrete blocks at the shoreward end of the slope
to be constructed (Figure 3. 1) This wall was placed at an angle of 701 with the sides of the basin so that the incident waves would have an angle of 20' with the beach. This angle was chosen so as to generate a measurable alongshore velocity without introducing a high degree of distortion due to wave refraction. Two sills 10 cm wide were placed on each side of the back wall in order to allow flow recirculation in a back
return flow channel constructed for this purpose. The back channel was made with another concrete brick wall 58.5 cm apart from the previous one (Figure 3.1). In general, due to the comparatively short length of the mud profile, the
alongshore current cannot reach its equilibrium unless certain procedures are followed in order to avoid the boundary effects
generated by a finite length basin. Visser (1991) studied different experimental set-ups previously used in longshore currents studies, and found that layouts similar to the one
used in the present study usually work adequately provided the return flow in the offshore region is properly reduced by the recirculation procedure as devised.
3.3.3 Profile Measurement
Profile depths were measured in a rectangular grid pattern with 20 cm spacing. Profile depth was measured at each grid position by lowering the point gage until its flared tip touched the muddy bottom. Using the moving carriage, depths
were determined throughout the gridded area. With these data three representative profiles were obtained, one at the center and the two others 50 cm on each side. In Figure 3.7 the three profiles are shown. As can be seen in each case there is a nearshore segment of steeper slope and a offshore segment with a mean slope of 1/40. The steeper nearshore segment was built in order to reduce any surf zone that would have occurred invariably, and to have the desired water depth in the basin.
E -Western profile
10 --- Center profile
Still water level
0 50 100 150 200 250 300
Cross-shore distance (cm)
Figure 1.7 Representative bottom profiles.
3.3.4 Fluid Mud Preparation
Due to limitations in the basin size, wave characteristics and test durations, fluid mud could not be generated in-situ by the action of waves on the mud bottom.
Therefore, fluid mud was prepared in large containers and then poured over the profile layout. The final fluid mud layer was 2. 5 cm thick in the mean with a density of 1, 150 kg/n3. Additional fluid mud forming a 1-2 cm thick layer over the 2.5
cm layer was placed along the updrift end to serve as a sediment source (Figure 3.1).
3.4 Exyerimental Procedure and Data
Once the bottom profile was prepared, the basin was filled with well water. Some preliminary tests without fluid mud were carried out in order to check for the effectiveness of model design and measuring procedures. Waves with periods ranging from about 1 to 4 s and heights of 3 to 6 cm at a water depth of 14.4 cm at the toe of the profile were used.
The flow recirculation set-up proved to work generally without the need of forcing the flow. However, it was observed that generating a wave train over durations exceeding a few waves
led to return flows and basin circulation which interfered with the alongshore flow. As a result, it became necessary to restrict the duration of each test to a few waves only. The
procedure was to run five to six waves and to use the last three for the measurements. The use of floating tracers for the velocity measurement proved to work reasonably well. Data
scatter in these cases was not overly significant and with different tests the results obtained were similar.
The procedure devised to obtain the alongshore velocity was to carefully place the colored tracers at a distance of approximately 40 cm from the shore. This distance guaranteed that the complete littoral zone was covered, and that tracer movements were within the range of the video camera. During the processing of the data three tracers that moved within the
camera range were chosen, and a representative alongshore water layer velocity V was obtained by averaging the velocities of these three floats.
During the preliminary tests some wave reflection from
the basin walls as well as from the shore was observed. Lateral reflection was produced along the walls mainly due to
a slight wave refraction caused by the oblique wave incidence. The shore reflection was due to the steeper profile slope near the shore. Horse-hair was put against the basin walls along the profile and at the shoreline to minimize wave reflection.
3.4.1 Ex-periments without Fluid Mud
The first set of tests were run without the addition of fluid mud. The purpose was to determine the effect of any wave breaking on the alongshore fluid mud current in the subsequent
tests with f luid mud. Twelve dif ferent tests were perf ormed in this situation following the procedure explained below.
Three different wavemaker displacements and, for each, four different periods were run. For each test condition the measuring carriage was positioned at the deepest wave measuring point and the offshore wave height HA was measured there. Then, the carriage was moved so that the wave gage was
positioned at the second site. With the carriage there the wave height H, was measured, and also the alongshore velocity was measured by video-taping of the floating tracer movements. In Table 3. 1 the wave period T and wave height at both locations and the mean alongshore velocity V are given.
Table 3.1 Measured data without fluid mud
Test J T (s) H, (cm) H, (cm) V (cm/s)
1 1.1 3.6 4.0 6.5
2 1.9 4.9 6.6 5.4
3 2.7 5.4 6.2 4.6
4 3.9 4.2 4.8 3.5
5 1.1 3.3 4.3 4.9
6 1.9 5.4 5.8 7.3
7 2.8 3.8 4.2 3.5
8 3.9 3.2 3.5 2.9
9 1.4 6.0 7.0 4.9
10 2.0 4.3 4.7 4.2
11 3.0 3.3 3.7 3.3
12 3.9 3.0 3.5 2.8
3.4.2 Experiments with Fluid Mud
After the experiments without fluid mud were completed and the data analyzed, fluid mud was placed over the profile with the help of buckets and distributed uniformly. In this situation as well, twelve different experiments were conducted: four different periods for three different wavemaker displacements. Fluid mud samples at the downstream sill were collected using a syringe pump. These samples were
used to check the flowing fluid mud density in order to confirm that the mud was moving without dispersing into the water column. In Table 3.2 the data obtained in this phase of the experimental study are given: wave period T wave height at sites A and B, H A and HB 1 respectively, and the measured mean alongshore velocity Vfm*
3.5 Data Preparation and Corrections
The wave data obtained from the set of experiments without fluid mud were used to calibrate for the correction to wave height due to shoaling. This correction was needed because sites A and B had different water depths due to the sloping mud profile. In order to compute the damping caused by fluid mud between those two sites, the shoaling effect had to be considered. Furthermore, the velocity data from these tests
were used to calibrate the friction coefficient cf in the alongshore current due to wave breaking in equation (2.41). In sections 5.4.1 and 5.4.2 both approaches are explained. It should be pointed out that wave breaking was observed even when fluid mud was present, because the incoming wave energy could not be absorbed completely over the short distance of
wave propagation. It therefore became necessary to account for wave breaking in calculating the alongshore mud mass transport velocity due to non-breaking wave. The procedure by which this was accomplished is described in section 5.4.4.
Table 3.2 Measured data with fluid mud
Test T (s) H, (cm) H, (cm) _T Vfm (cm s)
1 1.4 3.0 3.7 3.7
2 1.8 4.4 5.3 4.5
3 2.7 3.9 4.4 4.2
4 3.9 2.8 3.5 3.9
5 1.2 2.8 3.0 4.1
6 2.3 3.2 3.4 3.9
7 3.1 2.5 3.4 3.8
8 3.9 2.3 2.6 2.7
9 1.2 1.6 2.1 3.1
10 1.9 3.0 3.2 3.2
11 2.6 2.1 2.4 2.7
12 3.9 1.6 2.3 3.2
FIELD DATA ON MUD TRANSPORT
Field information from three different locations already
mentioned in Chapter 1 was used. These locations were selected because of the good quality data available, and the presence of large migrating banks of fluid mud at these locations. The three sites are: 1) the northeastern Atlantic coast of South America along the Guyanas, 2) The Louisiana coast along the Gulf of Mexico and 3) The southwest coast of India.
These shorelines are characterized by great quantities of
suspended sediment and by the absence of high waves in the nearshore mudbank zone. Mucibanks of fine sediment are formed, in which the waves are highly attenuated. These banks occur between regions where large waves can cause severe erosion of sandy profiles.
4.2 Northeastern Coast of South America
The northeastern coast of South America, on the order of 1,500 km of shores between the Amazon and the Orinoco Rivers, 63
constitutes one of the world's longest open ocean mud shorelines. It is formed by the Para-Amazon-Orinoco River plain (Augustinus, 1980), and is characterized as a chenier plain (Figure 4.1). A chenier is defined by Otvos and Price (1979) as: "a beach ridge, resting on silty or clayey deposits, which becomes isolated from the shore by a band of tidal mudflats." When there are more than one chenier separated by muddy deposits, a chenier plain is formed. Such a plain will develop when large amounts of river mud are available for alongshore transport, and intermittent supply of sand is available (Augustinus, 1989).
6b 4 56'00 5500 500
Orun,, R. A44
" Md -fl oo,
GUAN t0 ATLANTIC OCEAN U3
TURTIE "/ ""
r, or ,,KVISERBANKANI( WIA WIA
BUACOPP AA:,? R
COPNMB' BN, KWE:RIMANBANK
SURINAM RIVERMAROWIJNE 53SIniertidal and shallow subtidal muds SURINAM RIVER ENCH
S10 20 30 O GUIANA
FIG. 2. Northeastern South America showing location of mudbanks from Amazon-derived sediments.
Figure 4.1 Northeastern coast of South America (after Wells, 1983).
In the case of the northeastern coast of South America the muddy sediments are supplied by the Amazon River
discharge, while the coarse sediment is from local sources. The fine sediments from the Amazon River predominate along the shoreline up to the 20 m bathymetric contour, which on average is 40 km from the coastline, where relict sands are found (Augustinus et al., 1989).
The Amazon River discharges yearly an estimate of 11 to 13 x 108 tons of suspended load (Eisma et al., 1991) into the Atlantic Ocean. About 20% of this discharge is transported northwesterly along the Guyanas coast, and is deposited mostly in the Orinoco River delta. This transport takes place partly in suspension, approximately 1.5 x 108 tons, and partly as migrating mudbanks attached to the shore, approximately 1 x 10' tons.
Over-concentration of silt and clay in suspension produces a fluid mud that has a residual movement alongshore. This mud dissipates the energy of the incoming waves, protecting the shoreline against erosion and favoring sedimentation along the western side of each mudbank. Along the eastern side of the mudbanks, increasing consolidation of the fine sediment decreases the wave dumping property and increases wave breaking, thus causing erosion and a steady westward migration of the mudbank. Over the Surinam coast such mudbanks move at an average rate of 1.5 km/year, having a range from 0.5 km/year to more than 2.5 km/year.
In the inter-bank areas, where erosion of the consolidated clay take place, cheniers are formed. The sand required for this purpose is supplied by the local rivers discharges.
The spacing between mudbanks varies from 30 to 60 km and they are 36 km long on the average, ranging from 20 km to more than 50 km. The inter-bank region has the same magnitude, so that the occurrence of the mudbanks is so regular that they can be considered as mudwaves with wavelength of 45 km and period of 30 years. Fluid mud thickness in these regions varies from 0.5 to 2 m, and its density ranges from 1,030 to 1,250 kg/M3.
The predominant sediment is clayey with 40% illite, 30% kaolinite, 17% smectite (montmorillonite) and 13% chlorite (Froidefond et al., 1988) with a mean (dispersed) particle size of 0.5-1 pm.
The continental shelf near the shore, where the mudbanks are present, has smooth slopes ranging from 1:1600 to 1:3000. The tidal range varies from 1 m at neap to 3 m at spring, averaging 1.8 m, and consequently a very wide intertidal area with mudflats has developed. Thus the coastal environment is generally microtidal.
The wave energy is low to medium and waves are generated by relatively constant NE trade winds. Very few wave
measurements are available for this region. From data obtained during August 1977 (Wells and Kemp, 1985) the root mean square height, Hrs, 22 km offshore was 0.93m, and the wave spectra showed a combination of swell waves with periods T = 9-10 s, and sea waves with periods T = 3-5 s.
The northwestern migration of the mudbanks and their linkage with the alongshore mud drift show very complex dynamics. Oceanic currents, like the Guyana current, as well as tide-induced and wave-induced flows combine to cause the general migration of the Amazon-borne sediments. However, in the shallow coastal waters of the Guyanas, waves seem to play a greater role in fine sediment transport. In this area the Guyana current deviates from the shoreline, so that its relative importance diminishes. Furthermore, the difference in the migration velocities of the mudflats of Surinam and Guyana can be related to the different angle that the coast at each location subtends with the direction of prevailing wind. The mudbanks translate westward along the coast of Guyana at a rate that varies between 0.4 and 2.0 km/yr. The prevailing wind (and wave) direction with respect to the normal to the shoreline is 450 in Surinam and 25 in Guyana (Augustinus, 1987).
In Table 4.1, characteristic data for this region are presented.
Table 4.1 Characteristic data for the muddy coast environment of the Guyanas.
Factor [ Range
Tidal range (m) 1.0-3.0 (1.8 mean)
Wave height (m) 0.5-1.5
Wave period (s) 5-10
Fluid mud thickness (m) 0.5-2.0
Fluid mud density (kg/m3) 1,030-1,250
Grain size ( im) 0.5-1.0
Mudbank cross-shore length (km) 10-30
Mudbank alongshore length (km) 10-50
Mudbank velocity (km/year) 0.5-2.5
4.3 Louisiana Coast
Along the western coast of Louisiana lies a chenier plain with similar characteristics as that along the northeastern coast of South America. Here, the Mississippi River via its Atchafalaya distributary supplies large amounts of fine sediment (Figure 4.2).
Fluid mudbanks, backed by salt marshes and chenier ridges, extend west of the Bayou Canal for approximately 20 km (Wells, 1983). This fluid mud extends 0.5-3 km seaward and beyond it shelf sand occurs. The mudbanks have an alongshore length ranging from 1 to 5 km, and a wide span of width varying from as low as 20 cm to as much as a 1.5 m. They are
mainly composed of montmorillonitic clays with some illite and kaolinite, and have a median (dispersed) grain size of 3 to 5 im. The fluid mud has a bulk density of 1,150 to 1,300 kg/m3 and moves over a denser clay base with a density greater than 1,600 kg/m3. The slope in the seaward direction is on the order of 1/1,000.
fluid muds SOUTHWEST Fine sands LOUISIANA
B Rakish marsh FRESH WATER CHENIER RIDGES
290 Zone of intertidal mudflats BAYOU CANAL
0 1 2 3 4 .5
Fig 10 i.
LOUISIANA Afrchfl ara EASTERN River
UL.CO AtchaalriVa Bav'
.290 -,92O km
FIG. I. Louisiana chenier plain showing fluid muds derived from Atchafalaya Bay, to the east.
Figure 4.2 Louisiana chenier plain (after Wells, 1983).
The Gulf of Mexico coast of Louisiana is a microtidal, storm dominated environment. The tide is diurnal with a mean range of 0.5 m but its regime is complicated by storms, including extra-tropical cyclones during winter and tropical cyclones in the summer, which generate meteorological tides
superimposed on the astronomical conditions (Penland and Suter, 1988).The energy levels caused by winds and waves are low, except during the passage of storms. The dominant wave approach is from the southeast with an average wave height of Im and period of 5 to 6 s, offshore of the Mississippi delta (Penland and Suter, 1988).
The mudbanks show a shift to the west with both
continuous and discrete movements. The average velocity of this movement is 1 to 3 km/year (Wells, 1983). This movement is very likely produced mainly by waves affecting the
nearshore circulation during storms. It must be noted that because tide in the region has a very low range, it produces weak tidal currents. Furthermore, due to the relatively short
length of the mudbanks in the cross-shore direction in the area, the Gulf Stream current does not affect them significantly.
Relevant data for the Louisiana coast are presented in Table 4.2.
4.4 Southwest Coast of India
The mudbanks generated along the southwest coast of India (Figure 4.3) are very different in nature from those described before. They form very rapidly during the monsoon season (from
about May to August) when high wave energy is available, and dissipate when the waves decrease towards the end of the monsoon. These mudbanks do not occur adjacent to large rivers which could supply sediment. In fact, the fine sediment source is an offshore mud-pool that occurs in depths on the order of 10-20 m. High waves occurring during the monsoonal season transport the sediment as fluid mud towards the shore (Mathew, 1992) over a harder sandy shelf bottom (Figure 4.4). Several such mudbanks appear along the coast with dimensions of 2-5 km alongshore, 0.5-3 km cross-shore and having a mud thickness of
1 to 2 m.
Table 4.2 Characteristic data for the muddy coast environment of Louisiana.
Factor J Range
Tidal range (m) 0.5 0.8
Wave height (m) 0.1-1.0
Wave period (s) 5-6
Fluid mud thickness (m) 0.2-1.5
Fluid mud density (kg/m3) 1,150-1,300
Grain size ( im) 3-5
Mudbank cross-shore length (km) 0.5-3
Mudbank alongshore length (km) 1-5
Mudbank velocity (km/year) 0.5-3.0
Through the entire period of a fully developed mudbank, the mineralogical composition of fluid mud is about 60% montmorillonite, 30% kaolinite and 7% illite. The bulk density spans from 1,080 to 1,300 kg/m3 and the median(dispersed) grain size varies between 0.5 and 3 um (Mathew, 1992).
Figure 1. Locations of monsoonal mudbanks along the southwest coast of India.
Figure 4.3 Southwest coast of India (after
Mathew et al., 1995).
During the non-monsoon period the wave energy is relatively low, with a significant wave height of less than 0.75 m and wave period ranging from 10 to 18 s. During the monsoon season the significant wave height varies from 0.05 m
to 2 m with a period of 7 to 9 s. As Mathew (1992) has noted the highest and lowest values of the significant wave height both occur during the monsoon season. The lowest value occurs on the mudbank where much of the incoming wave energy is absorbed. The significant wave height offshore and outside the mudbank region during the rough monsoon season ranges from 1 to 2 m. The direction of wave propagation is nearly normal to the coast and the tidal energy in the region is very low with an average range of 0.6 m (Mathew, 1992).
Wave with ULss Damped
Smaller Breakert"" Wave with
x % Height
nd (Water M
The.,...,. #: : i : : : = :@ li aagbank,
. ..,.................E . :!:
::it~iii. ....iii~i ...............
i::! . .:.'. . .:. .. .
Figure 4.4 Schematic mudbank along the southwest coast of India (after Mehta et al., 1996).
The mudbanks rest on a sandy bottom ranging in slope from 1/500 to 1/1,000, and their migration velocity in the crossshore direction during their propagation towards the shore at
the beginning of the monsoon season fluctuates between 1.2 and
Table 4.3 presents significant data for the mudbanks along the Kerala coast in southwest India.
Table 4.3 Characteristic of southwest India.
data for the muddy coast environment
Factor [ Range
Tidal range (m) 0.5 0.7
Wave height (m) 0.05-2
Wave period (s) 7-9
slope 1/500 1/1,000
Fluid mud thickness (m) 1.0-2.0
Fluid mud density (kg/m3) 1,080-1,300
Grain size (lam) 0.5-3.0
Mudbank cross-shore length (km) 0.5-3
Mudbank alongshore length (km) 2-8
Mudbank velocity (km/day) 0.3-0.9
To assess the validity of equations (2.75) and (2.76) derived in Chapter 2 for the cross-shore and alongshore velocities, respectively, the data obtained from the
laboratory experiments given in Chapter 3, as well as the field data available from the sites described in Chapter 4, were used. This chapter describes the processing of the data
used as input for these equations, inclusive of processing necessary to obtain comparable "measured" alongshore velocities from the laboratory experiments.
5.2 Dimensionless Forms of Mass Transport Eauations
In order to simplify equations (2.75) and (2.76) s0 that
their use and interpretation are easily accomplished, they have been rendered in more useful forms by means of nondimensionalization. This was achieved by rendering the variables dimensionless, by dividing them by the appropriate
characteristic reference parameters. In general, these parameters must be chosen in such a way that all the quantities that include dimensionless variables are comparable, so that the order of each term is only given by the dimensionless coefficient that occurs as multiplier of each variable term.
The selected reference parameters are given in Table 5.1, and in Table 5.2 the different dimensionless variables are noted.
Table 5.1 Characteristic reference parameters.
vertical length H0
horizontal length gT2
shear stress T 2gT/H0
First, using the dimensionless variables presented in Table 5.2, the equations for the free surface (2.63) and interfacial setup (2.65) are non-dimensionalized, thus obtaining equations (5.1) and (5.2) after linearization.
Table 5.2 Dimensionless variables.
rI* a,b p:/Ho
S0* T H0/412gT
a Superscript denotes a dimensionless variable. b Subscript I is 1 for water and 2 for fluid mud.
Sar 1 8 *2 *2 ,
hi (Il -r2)- 2 (ux -w1 ) dz
as' as gT2 -aS
2 0 0 0
Ho 82, Ho 802 8 ,
- a2 dz *utw:dz -- wdz
g2 a*2 f f2 f S
T g-hl z T at *as*-h
Ho a2 0 0 a
- wldz* widz
gT2 at *as* gT2 as* at*
Equation (5.1) is for the free surface setup. For the interfacial setup, the non-dimensionalized expression is given by equation (5.2):
22 H a 1 *2
2 0 a *2
hS* 2 -2_1) S 2 (P2-i) S 2h2 s
h~p-1)~ gT2 aJ(u2 -W2 ) dz*
P2 H 2 -h -hi
2 dz u2 W2 dz
h2 (p -1) gT2 as*2-h z (5.2)
H2 i Ho 2 a2 H aw2 + u2
PHo2gT (-1) T 2 2 s c z=-h
292 1P Ho a Ho u2 -W2l
p1HogT 2 h 2 _1) gT2 as 2 z=-h2
2 2 -1 ) gT 2 as* hZ
After linearizing equation (2.68) for the bottom shear stress and substituting it in (2.72) for the mean mass
transport velocity in the mud layer, an expression for the dimensionless velocity Um(s)in the mud layer as a function of the dimensionless setups given by equations (5.1) and (5.2) can be found as
h2* 2 1 arel
Um(S) h(h+p2h )-- + (h2+h*) p2-h (5.3)
3 2 as* as*
and the corresponding cross-shore and alongshore components are obtained as
h2 p1HO2 ar1(S is ar02(S)
U (s) (h +p2h2) + (h2 +h*) p2 -h cosa (5.4)
Mx3 v2 TS 2 s1 2 s*
3 T ns on pi oas dasi*
From this non-dimensionalization three dimensionless numbers result, which are given in Table 5.3. The first is the well-known wave steepness parameter, which relates the wave height to the wave length. The other two numbers, as rearranged in Table 5.3, are seen to be characteristic Reynolds numbers.
Table 5.3 Dimensionless parameters.
Wave steepness H0/gT2
Wave Reynolds H0 gT/ (P2/pl)
Streaming Reynolds H0 (H0/T) / (2/P1)
The second number (wave Reynolds number), in which the velocity is related to the wave celerity, gives the ratio between viscous diffusion and wave propagation, and the third number, which characterizes velocity related to particle movement, compares viscous diffusion in the mud layer to
particle movement generated by the wave. This last number, herein called streaming Reynolds number Re., appears in equation (5.3) as the only multiplicative coefficient, hence it can be used as the characteristic flow parameter for mass transport.
5.3 Simiplifications of Mass Transoport Eauations
As far as this study is concerned, recognizing the range of values of the dimensionless parameters of interest, some simplifications to equations (5.1) through (5.5) can be made. An inspection of these equations shows that the wave steepness parameter is the most important coefficient in equations (5.1) and (5.2) because in these equations only the wave Reynolds numbers appears, and is always associated with the wave steepness.
For the range of values of wave characteristics
corresponding to the field data, including the three cases selected, the wave height H0 varies from 0.1 to 2.Om and the wave period T has a range of 5 to l0s. With these values the
range of the wave steepness parameter is between 0.001 to 0.04. This demonstrates that even the highest values of the field wave steepness are very small, and that the terms that include it as a coefficient can safely be neglected. It must be recognized that the range of wave steepness was obtained
using the extreme combination of wave height and period, which in reality is not the case, and in fact the real values will be around 0.01 or less.
In the laboratory experiments the steepness parameter was never greater than 0.003, so that the terms that include it as a coefficient can be neglected in that case as well.
Based on the above considerations, the simplified
respective expressions for the dimensionless cross-shore and alongshore velocities in the mud layer that are finally obtained are
1 H2 hp (2 p) +h (p-1) +phh 8pf (s)
UL (S) n I rh2(s)
3 2T hl (p -1) as
hp2 (2-p2) a n(S) r 1 (S (S)
2 02 (S) -(h2+h *) p2-h 2(s) Cos (
h (p -1) rs* L s
1(2 h1Ho h p (2-p) +h (p2-1) +p2h h2 a (s)
3 P2 T h' (p2 -1) as
h p (2 -p2 ) 8 (s) [ 1 8p2(s)
S a 6(s) I (h2 +h;) p-hj r(s) sina
h' (p2-1) as* as
Equations (5.6) and (5.7) will be used hereafter for application to laboratory and field data.
The interfacial oscillatory movement and the damping coefficient k. are obtained from Jiang's model (1993). This
model considers that the free surface and the interfacial oscillatory movements are of the form
- = ei(ks-ot) (5.8)
2 = bei(ks-at) (5.9)
where k = (krki) is the complex wave number,kr is the wave number and b the interfacial amplitude. Using expressions (5.8) and (5.9), the mean correlation terms in equations (5.6) and (5.7) can be computed as
al* kig 2 -2kis
1l e (5.10)
,8pyl kigT 2b -2kis r2 -e (5.11)
* aT2 kigT2b2 2~
, 8 ki9 2b2 -2kis
12 e (5.12)
as* 2 HO2
5.4 Laboratory Data Analysis
The data obtained from the experiments described in Chapter 3 are analyzed in the sections below. First, with the
data obtained from the set of experiments without fluid mud, a correction for shoaling is developed, and the friction coefficient cf in equation (2.41) is obtained. Using the wave height data measured with fluid mud present and considering shoaling, a value of the dynamic viscosity of the fluid mud 12 is calculated. Finally, the alongshore velocity due to mud effect is obtained as the measured value minus the breaking component. Then, making use of equation (5.7) the measured and calculated values are compared.
5.4.1 Correction for Shoalina
The wave height data were corrected for shoaling using the stream function wave theory. This was done because the linear wave theory had to be discarded as a result of the significant non-linearities present in the wave records, and the poor results obtained with this theory. For shoaling it is required that (Dean, 1974)
E c- constant (5.13)
E Elinear (5.14)
Here cg is the group celerity, 1 is a correction coefficient and E and Elinear are the wave energy from the stream function
and linear wave theories, respectively. Then, the wave height
at a given location as a function of the wave height at another location may be computed as
H HA Ag (5.15)
In order to calculate %Bc gB the wave height H; is needed. Hence, an iterative procedure must be used. The first value used in this procedure was the one obtained with the linear wave theory. Using this value of the wave height, %3cgB was obtained using the stream function wave theory and H/1
corrected. %?cgA was calculated using the stream function wave theory using the wave height H A' the period T and the water depth h A as inputs.
In order to check the procedure, data obtained without fluid mud at sites A and B were used, choosing site A to be the characteristic offshore position for computations. Figure
5.1 shows a comparison between the measured wave height at position B, i.e. H., and H'/ B 1 obtained by using the values measured at position A as input to the stream function theory
for shoaling the wave between A and B. The computed and measured values agree reasonably well with one exception.