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INTERACTION OF BOTTOM TURBULENCE AND COHESIVE SEDIMENT ON THEMUDDY ATCHAFALAYA SHELF, LOUISIANA, USA
By
ILGAR SAFAK
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2010
c© 2010 Ilgar Safak
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To Besiktas
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ACKNOWLEDGMENTS
I would like to thank my advisor Dr. Alex Sheremet to give me the opportunity to
work with him at the University of Florida. His expertise, readiness for advising, and
suggestions made everything much easier for me. I have also learned from him the real
scientific approach to all sorts of problems, including those in daily life. For me, this is as
valuable as the research work.
This research was supported by Office of Naval Research funding of Contracts No.
N00014-07-1-0448 and N00014-07-1-0756.
Dr. Tian-Jian Hsu from the University of Delaware is greatly acknowledged for
agreeing to share with me the boundary layer model he developed. My research has
benefited substantially from this model and through his guidance. He was initially an
internal member in my committee, but unfortunately he had to be taken out, as he left
the University of Florida.
Special thanks go to Sergio Jaramillo, who is now in the University of Hawaii at
Manoa, and especially Bilge Tutak, who have never hesitated to leave their works aside
to be able to give me a hand for my work.
Dr. Mead A. Allison from the University of Texas, Austin provided extremely valuable
data, help, and guidance at every level of this study.
My committee members Dr. Arnoldo Valle-Levinson, Dr. Jane Mckee Smith,
Dr. Peter N. Adams, and Dr. Donald N. Slinn spent their valuable time to evaluate
the progress of my research work. Dr. Valle-Levinson is further acknowledged for
introducing me some oceanographic concepts that I was not very familiar with.
The data set, used in this study, was collected during a field experiment that
benefited from the efforts of Viktor B. Adams, Sidney Schofield and Jimmy Joiner from
our Coastal Engineering Laboratory, Daniel D. Duncan from the University of Texas,
Austin, and the field support group of LUMCON. Sergio Jaramillo, Uriah M. Gravois, and
Jungwoo Lee also assisted in the deployment and retrieval of the instruments.
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Dr. Clinton D. Winant from Scripps Institution of Oceanography spent his valuable
time on tutoring me and the other graduate students towards our Ph.D. qualifying tests in
2007.
While it is not in the scope of this dissertation, I have had the opportunity to work
on dissipation of surface wave energy while propagating over muddy seafloors, using
a comprehensive data set that was kindly provided by Dr. Steve Elgar and Dr. Britt
Raubenheimer from Woods Hole Oceanographic Institution.
Tracy J. Martz, Chelsea L. Sydow, and Chloe D. Winant kindly agreed to proofread
this dissertation.
I would like to thank my dear friend Hande Caliskan, who graduated from our
program in 2006, and Dr. Aysen Ergin, my advisor during my M.S. studies at Middle
East Technical University. Dr. Ergin made me contact with Hande for applying to
University of Florida, and Hande’s guidance throughout the entire application process
helped substantially.
Although being far away from me, knowing that Irmak Yesilada, Ozan Gokler and
Murat Dilman - the co-starrings if a desperate director would make a biographical sketch
of my life one day- are somewhere in this world always helps me to enjoy life, and get
along with all sorts of difficulties easily.
Cihat and Sukran, you GUYS are the best!
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TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
CHAPTER
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.1 Cohesive Sediments in Marine Environment . . . . . . . . . . . . . . . . . 131.2 Turbulence in Combined Wave-Current Flow . . . . . . . . . . . . . . . . 161.3 Interaction of Turbulent Flow with Cohesive Sediments . . . . . . . . . . . 171.4 Objectives of This Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 FIELD EXPERIMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.1 Experiment Site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3 General Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 DATA ANALYSIS METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1 Wave Spectral Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Estimation of Reynolds Stresses in the Presence of Surface Waves . . . 42
3.2.1 Definition of Reynolds stress . . . . . . . . . . . . . . . . . . . . . 423.2.2 Wave bias in Reynolds stress estimates . . . . . . . . . . . . . . . 433.2.3 Two-sensor methods to reduce wave bias . . . . . . . . . . . . . . 45
3.3 Logarithmic Law of the Wall . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4 OBSERVATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.1 General Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 The LISST Data Set from 2006 . . . . . . . . . . . . . . . . . . . . . . . . 56
5 BOTTOM BOUNDARY LAYER MODELING . . . . . . . . . . . . . . . . . . . . 67
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2.1 Momentum balance . . . . . . . . . . . . . . . . . . . . . . . . . . 685.2.2 Sediment concentration balance . . . . . . . . . . . . . . . . . . . 705.2.3 Turbulent kinetic energy balance and balance of turbulent kinetic
energy dissipation rate . . . . . . . . . . . . . . . . . . . . . . . . . 705.2.4 Sediment definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6
5.3 Analytical Flocculation Model . . . . . . . . . . . . . . . . . . . . . . . . . 725.4 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.5 Model Sensitivity to Floc Size . . . . . . . . . . . . . . . . . . . . . . . . . 77
6 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
APPENDIX
A DIRECT ESTIMATION OF REYNOLDS STRESSES . . . . . . . . . . . . . . . 88
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7
LIST OF TABLES
Table page
2-1 Location, mean depth, and retrieval dates of the instrumented platforms . . . . 32
5-1 Numerical coefficients in the k − ε closure . . . . . . . . . . . . . . . . . . . . . 79
8
LIST OF FIGURES
Figure page
1-1 Noncohesive sand grains and cohesive fluid mud. . . . . . . . . . . . . . . . . 22
1-2 Comparison of spectral evolution of waves over muddy and sandy seafloors. . 23
2-1 Gulf of Mexico . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2-2 Bathymetry of the Atchafalaya Shelf and the locations of the instrumentedplatforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2-3 A typical instrumentation platform . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2-4 The synchronized ADVs and the OBS-5 on the platform before the deployment 36
2-5 Configuration of the instrument array . . . . . . . . . . . . . . . . . . . . . . . . 37
2-6 Wind, wave, and near-bed conditions on the Atchafalaya Shelf throughout the2008 experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3-1 Examples of ogive curves as a function of the dimensionless wavenumber . . . 52
4-1 Tidal variations during the experiment . . . . . . . . . . . . . . . . . . . . . . . 58
4-2 A one-minute segment of the flow velocity, recorded by the ADV array . . . . . 59
4-3 Wind, wave, and current conditions on the Atchafalaya Shelf throughout the2-week experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4-4 Bulk spectral characteristics of waves, and the measurements of suspendedsediment concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4-5 Reynolds stress estimates versus the quadratic drag relation . . . . . . . . . . 62
4-6 Reynolds stress estimates, and the measurements of suspended sedimentconcentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4-7 Wave and current observations during the 2006 experiment . . . . . . . . . . . 64
4-8 Examples of logarithmic layer fits, and grain size distributions from the 2006experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4-9 Observations of particle size distribution and wave-turbulence conditions in2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5-1 An example of time-series in the model simulations . . . . . . . . . . . . . . . . 80
5-2 Comparison of the observations and the model results . . . . . . . . . . . . . . 81
5-3 Analysis of the model representation of the turbulent kinetic energy balance . . 82
9
5-4 Effect of varying floc size D on the model calculations . . . . . . . . . . . . . . 83
10
Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy
INTERACTION OF BOTTOM TURBULENCE AND COHESIVE SEDIMENT ON THEMUDDY ATCHAFALAYA SHELF, LOUISIANA, USA
By
Ilgar Safak
August 2010
Chair: Alexandru SheremetMajor: Coastal and Oceanographic Engineering
Interaction of near-bed wave-induced turbulence and cohesive sediments in muddy
environments is studied based on field observations and a bottom boundary layer
model. Wave, current, and sediment observations were collected with a suite of acoustic
and optical instrumentation at approximately 5-m depth on the muddy Atchafalaya
clinoform, Louisiana, USA. Low wave-bias estimates of near-bed Reynolds stresses are
obtained by a method that is based on differencing and filtering of velocities from two
sensors. The event that is focused on in this study is characterized by moderate waves
with high steepness, currents with speeds sometimes reaching 30 cm/s near bed, and
Reynolds stresses and suspended sediment concentrations reaching to their maximum
values throughout the experiment (0.4 Pa, 3 g/L). In general, Reynolds stresses are
found to be correlated with short-wave near-bed accelerations and suspended sediment
concentration, as previously observed on sandy beaches, where accelerations have
been associated with bed fluidization and sediment transport.
A detailed numerical analysis of the observations is performed with a one-dimensional
bottom boundary layer model for small scale turbulence and sediment transport
processes on cohesive beds. The model accounts for the coupling between the fluid
and the cohesive sediment phases, and uses a floc size that is constant in time and
space. A representative floc size is selected for the experiment site, based on two
independent sources that show consistency. Direct estimates of size distribution of
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suspended sediments in the vicinity of the experiment site show a remarkably stable
floc mode peak under varying wave and turbulence conditions. Indirect estimates of
equilibrium floc size are obtained through calculations of an analytical flocculation
model that uses observation-based parameters. With a floc size input based on the
observations, the model reproduces currents and suspended sediment concentrations
accurately; modeled Reynolds stresses match the low wave-bias estimates, with
better agreement for cases of stronger currents and smaller wave-orbital velocities.
The numerical simulations suggest that sediment-induced stratification effects are the
same order of magnitude as turbulent dissipation, and thus play a significant role in
the turbulent kinetic energy (TKE) balance within the tidal boundary layer. However,
inside the wave boundary layer, the ratio of stratification to shear-induced turbulence
production (i.e., gradient Richardson number) decreases significantly; shear-induced
turbulence production and turbulent dissipation dominate the TKE balance. For these
observations, model results show that the vertical structures of currents and Reynolds
stresses are relatively insensitive to the exact floc size.
Future efforts should include analysis of wider range of conditions (especially events
with higher near-bed concentrations), and comparison of model results with a more
detailed vertical structure of suspended sediment concentration.
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CHAPTER 1INTRODUCTION
1.1 Cohesive Sediments in Marine Environment
Marine sediment transport is a coastal process that affects shoreline change,
methods of coastal protection, design of coastal and offshore structures (see Sterling
and Strohbeck (1973) for an example of failure of oil platforms due to wave-induced
seafloor movements), underwater detection, navigation, water quality, and fate of
pollutants and biomatter due to settling of sediment from surface to deep sea (Hill,
1998). In the conventional scheme, bed sediment is entrained into the water column
by surface waves and is advected by currents. Recently, waves in shallow water were
also noted to cause a net sediment transport (e.g., Hoefel and Elgar, 2003; Hsu and
Hanes, 2004). According to the presence of cohesion between particles, sediments
are classified into two categories: cohesive, which is also commonly known as mud
(silt and clay with primary particle size smaller than 63 µm), and non-cohesive such
as sand grains and coarser particles (Figure 1-1). Mehta (2002) defines mud as
a mixture of water and sediment particles that are predominantly cohesive, which
exhibits a rheological behavior that is poroelastic or viscoelastic when the mixture is
particle-supported, and is highly viscous and non-Newtonian when it is in a fluid-like
state.
The majority of the coasts around the world are covered mostly with sand. However,
there are numerous muddy coasts dominated by cohesive sediments, especially
at river mouths, such as the Atchafalaya Shelf in the Gulf of Mexico (Allison et al.,
2000), continental shelf off the Eel River in Northern California (Traykovski et al.,
2000), southwest coast of India (Jiang and Mehta, 1996), east coast of China (Jiang
and Mehta, 2000), on the Amazon Delta (Cacchione et al., 1995), Po prodelta in the
Adriatic (Traykovski et al., 2007), etc. In muddy environments, there is strong evidence
of coupling between boundary layer turbulence and suspended sediment processes
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(Trowbridge and Kineke, 1994; Allison et al., 2000; Traykovski et al., 2000; Sheremet
and Stone, 2003; Sheremet et al., 2005; Allison et al., 2005; Kineke et al., 2006;
Traykovski et al., 2007; Jaramillo et al., 2009). Waves which generate shear stresses at
the bed greater than the bed strength cause liquefaction of a layer in the muddy bed with
a thickness depending on the properties of the bed material and the wave conditions
(Winterwerp et al., 2007). If the resulting cohesive sediment suspension near the bed
reaches a volume concentration of unity, a space-filling network with both fluid and solid
properties develops (Mehta, 1989). In the literature, cohesive sediment suspensions
with mass concentrations exceeding 10 g/L are commonly classified as fluid mud layers,
and this state is called structural density (Winterwerp and van Kesteren, 2004). The
internal friction within the fluid mud layer can dissipate the surface wave-generated
internal waves at the mud-water interface (Winterwerp et al., 2007). In the presence of
currents, fluid mud (formed and entrained by waves) is transported in the upper water
column, which otherwise remains just above the bed (Mehta, 1989). On the Eel River
continental shelf in Northern California (Traykovski et al., 2000; Wright et al., 2001),
on the Po prodelta in the Adriatic (Traykovski et al., 2007) and on the Yellow River
mouth in the Gulf of Bohai, China (Wright et al., 2001), wave induced fluid mud layers
of thickness on the order of the wave boundary layer thickness were observed to flow
downslope (offshore) in the form of gravity currents. Turbulence, necessary to prevent
deposition of the advected material by this mechanism, was generated by the flow
itself. On the shallow Atchafalaya Shelf, bed liquefaction and sediment resuspension by
surface gravity waves also result in the formation of fluid mud layers. These layers were
observed to flow in the form of a turbidity current (Jaramillo, 2008; Jaramillo et al., 2009).
Beyond the scope of this study, another mechanism through which mud affects
large scale nearshore ocean dynamics is the significant dissipation of wave energy
while propagating over muddy seafloors in shallow waters. Recent studies showed that
fluid mud layers dissipate wave energy not only within the swell band but also within
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the short-wave band. This is due to nonlinear energy transfers from high-frequency
bands towards low-frequency bands, which interact with the seafloor more significantly
(Sheremet and Stone, 2003; Sheremet et al., 2005; Kaihatu et al., 2007; Elgar and
Raubenheimer, 2008; Jaramillo, 2008; Sheremet et al., 2010). Figure 1-2 shows a
comparison of numerical simulation of wave field propagation on sandy and muddy
seafloors. Nonlinear interactions across spectrum are not accounted for. An idealized
case of unidirectional propagation over a distance of 5-km with constant 5-m depth
is set. An initial spectrum is selected based on the parameterizations obtained from
the directional wave measurements during the Joint North Sea Wave Project, i.e.,
JONSWAP (Hasselmann et al., 1980). A significant wave height of 2 m, and a peak
frequency of 0.1 Hz is selected (thick line in Figure 1-2). The material on the sandy
seafloor is represented by 0.2 mm grains. The muddy seafloor is set to have a 15-cm
thick viscous fluid mud layer, with a density of 1.1 g/cm3 and a kinematic viscosity of
10−3 m2/s (Kaihatu et al., 2007; Winterwerp et al., 2007). Bottom friction by the sandy
bed (Jonsson, 1966; Kamphuis, 1975; Dean and Dalrymple, 1991) causes visible
dissipation at only five frequencies around the spectral peak (thin line with squares in
Figure 1-2) and the resulting significant wave height is 1.8 m. The calculations, based
on a formulation which assumes the layer is a viscous fluid (Ng, 2000) show that the
muddy seafloor causes a more significant collapse in the wave field’s energy, in a wide
range of frequencies (dashed line with dots in Figure 1-2). The resulting significant
wave height for this case is 0.6 m. Therefore, while bottom friction by the sandy bed
dissipates 19% of the initial energy of the wave field, the muddy seafloor, characterized
by the parameters given above, dissipates 91% of the initial energy.
In terms of small scale ocean bottom boundary layer processes, interaction of
turbulent flow with sediment is more challenging to study in muddy environments than in
sandy environments because of the complicated physics related to cohesive sediments.
In a cohesive sediment suspension, rather than bouncing away from each other as
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sand grains do, the collision of sediment particles results in the formation of aggregates
called flocs. These flocs are characterized by high water content (Winterwerp and
van Kesteren, 2004) and a fractal geometry with dimension close to 2 (Kranenburg,
1994). The flocculation process is discussed in more detail in the following sections.
The settling velocity of sand grains can be calculated as a function of grain size and
fluid viscosity through Stokes’ law (e.g., Nielsen, 1992). However, changes in the
water content, geometry and, therefore, density of cohesive sediment flocs cause
their settling velocity to be temporally varying, and higher than those obtained for
primary particles, by Stokes’ law. Therefore, a cohesive sediment sample needs to
be considered site-specific and event-specific in the sense that flocs in each sample
may have varying physical properties such as size, density, and settling velocity
(Mehta, 1989). Gases and organic particles in cohesive sediments (Winterwerp and
van Kesteren, 2004) should also be considered in the analysis of cohesive sediments
interacting with fluid flow. In fact, the interest of the U.S. Navy in the hydrodynamics in
muddy environments has recently increased due to the fact that available Navy models
for waves and circulation were mostly developed for sandy environments but do not work
well with mud.
1.2 Turbulence in Combined Wave-Current Flow
Flow-generated turbulence is a major mechanism in large scale processes such
as momentum balance in the surf zone (e.g., Trowbridge and Elgar, 2001) and small
scale sediment transport processes (Winterwerp, 1998). Numerical models for ocean
bottom boundary layer turbulence are often based on two-equation turbulence closures
and validated with laboratory experiments (e.g., Winterwerp, 2001; Hsu et al., 2007,
2009). On shallow continental shelves, surface wave-induced orbital velocities at the
bed generate a wave boundary layer of a few centimeters thick, much thinner than
the current boundary layer that scales with water depth (e.g., Fredsoe and Deigaard,
1992). Waves cause currents to experience a bottom drag above the wave boundary
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layer larger than that associated with the bottom roughness, deviating from the
logarithmic vertical structure and having a reduced vertical shear near the bed (e.g.,
Grant and Madsen, 1979, 1986). The resulting non-linear friction processes and the
differences in the characteristic length and time scales of waves and currents complicate
modeling of turbulence and sediment transport processes in the bottom boundary layers
(e.g., Styles, 1998; Styles and Glenn, 2000, 2002; Hsu et al., 2009). This is a major
research area in oceanographic studies because sediment entrainment from the bed
is controlled by wave boundary layers where a significant amount of energy is being
dissipated (Trowbridge and Agrawal, 1995) and exchanges of heat and momentum
occur. The flow parameters in these boundary layer models (e.g., eddy viscosity, see
Winterwerp (2001) and references therein) are modified by the turbulence-damping
effect of density-induced stratification, i.e., vertical gradient of suspended sediment
concentration. The turbulence closures on which these models are built have yet to be
evaluated in detail against high-resolution field observations of combined wave-current
flows and sediment transport processes.
1.3 Interaction of Turbulent Flow with Cohesive Sediments
Hydrodynamic properties of cohesive sediments, which differ significantly from
those of the primary particles (e.g., settling velocity, Section 1.1), result in stratification
effects on vertical mixing that are specific to cohesive sediments (e.g., hindered
settling, formation of near-bed fluid mud layers). In hypothetical cases of cohesive
sediment suspensions in tidal flow, Winterwerp (2001) numerically simulated the
formation of these fluid mud layers when the concentration of suspension exceeded
the sediment-carrying capacity of the flow. Simulations in that study showed that
stratification modifies vertical structures of velocity and turbulent flow parameters at
depth-averaged concentrations as low as 0.1 g/L, as long as near-bed fluid mud layers
of at least a few centimeters thick are formed. The one-dimensional boundary layer
model, used by Winterwerp (2001), was validated with data from laboratory experiments
17
of cohesive sediment suspensions of depth averaged concentrations of 1 g/L in tidal
flow (Winterwerp, 2006). Model calculations of Winterwerp (2006) agreed with the
measured velocity and suspended sediment concentration; the model also captured
the decreasing trend of vertical eddy viscosity (estimated through measured data) with
increasing sediment concentration. In other related studies on steady currents in the
tidal boundary layer, field observations (e.g., Trowbridge and Kineke, 1994), laboratory
experiments (e.g., Gratiot et al., 2005), and models (e.g., Trowbridge and Kineke, 1994;
Michallet and Mory, 2004) seem to agree that turbulence is significantly affected by
cohesive sediment in the vicinity of a steep vertical gradient of suspended sediment
concentration. These gradients are commonly known as a lutocline, which separates a
high concentration layer with a mixed and comparatively low concentration layer (e.g.,
Parker and Kirby, 1982; Mehta, 1989; Vinzon and Mehta, 1998).
Flocculation is a combined process of aggregation (floc formation due to collision
of particles by turbulent motions) and floc breakup (disruption of flocs by turbulent
shear), which is governed by turbulent eddies scaled by the Kolmogorov microscale
(Berhane et al., 1997). While the importance of flocculation in altering the flow-sediment
interaction through stratification is well established (Winterwerp, 2001), the role of the
floc size (and therefore settling velocity of the floc) is less understood. Dyer (1989)
hypothesized that floc structure is a function of turbulence levels and availability of
sediment, i.e., suspended sediment concentration. The argument was that flocs should
grow as the concentration of primary particles increases, due to increased probability of
particle collisions. Turbulence (low to moderate shear stresses) is needed to promote
collisions; however, high turbulence levels (high shear stresses) are expected to break
the flocs and limit their size. In turn, floc structure should affect the hydrodynamic
properties of flocs (e.g., settling velocity) with direct effects on the residual time of the
flocs in the water column and, therefore, the vertical structure of suspended sediment
concentration. Following the hypothesis that increasing turbulent shear stress first
18
increases and then decreases floc size (Dyer, 1989), simplified analytical expressions
for equilibrium values of floc size and settling velocity were proposed by Winterwerp
(1998), for a constant fractal dimension of 2. The formulation of Winterwerp (1998)
was later modified by Son and Hsu (2009) to include variable fractal dimension and
variable floc yield strength. The simplified formulation using the floc fractal dimension
2 was used to model cohesive sediment transport in the boundary layer of a tidal
channel (Winterwerp, 2002). Winterwerp (2002), however, did not discuss the details
of the turbulence kinetic energy balance. The flocculation models discussed above
were calibrated for cohesive sediment transport in the laboratory with simple shear
flow or homogeneous turbulence and applied to a tidal boundary layer condition. Their
applicability to wave-induced fluid mud transport is unclear. Moreover, turbulence may
not be the governing mechanism that controls flocculation at every condition. Based
on field observations, Hill (1998) proposed a flocculation mechanism with a different
dependence on turbulence: at low turbulent energy, disruptive stresses on flocs due to
sinking in the fluid may exceed turbulence-induced stresses and limit floc size.
1.4 Objectives of This Study
In this study, the interaction between turbulence induced by combined wave-current
flow and suspended cohesive sediments in bottom boundary layers of muddy environments
is studied based on observations collected during spring of 2006 and 2008, on the
Atchafalaya inner shelf, Louisiana, USA. Wave and current parameters are calculated
using standard data assimilation methods and spectral analysis. In addition, near-bed
Reynolds stresses are estimated through the data with the most advanced of a series of
methods recently proposed to reduce wave bias in turbulence estimates (Feddersen and
Williams, 2007). This is a challenging task, especially in wave-energetic environments
(Trowbridge, 1998). Having Reynolds stress estimates provides the advantage of
designating, merely by analyzing the observations, the events when turbulent fluxes in
the boundary layer are likely to be significant. Based on these observations, small-scale
19
turbulent flow and sediment transport processes are modeled using a one-dimensional
bottom boundary layer model for cohesive beds (Hsu et al., 2007, 2009). The model
uses a constant floc size, and therefore a constant settling velocity, in contrast with other
models (e.g., Winterwerp, 2002) that predict a settling velocity varying over a tidal cycle.
While this may lead to errors when used to simulate the vertical structure of suspended
sediment concentration, the constant floc size assumption seems to be consistent
with the field observations, which suggest a weak relationship between turbulence
and floc size, at least in relatively dilute suspensions. Direct observations of floc size
distribution (but not co-located with these particular hydrodynamic measurements)
are used to estimate a representative floc size. The values used here are of the same
order of magnitude as the equilibrium floc sizes calculated with the model proposed
by Winterwerp (1998) using the observed primary particle size, suspended sediment
concentration and estimated Reynolds stresses.
The description of the field experiment, the details of instrumentation and sampling
schemes, an overview of the sedimentological characteristics of the experiment site,
typical atmospheric and flow conditions for the site, and a brief summary of the wave
and near-bed flow observations during the entire experiment period are presented in
Section 2. In Section 3, the data analysis methods used are presented. A general
overview of the observations during the experiment studied herein is given in Section
4, together with the details of a 1-day event which is characterized by the highest
amount of suspended sediment recorded near bed. Why a relatively stronger local
turbulence-cohesive sediment interaction is expected during this event is discussed.
The bottom boundary layer model is presented in Section 5, together with the
governing equations, description of its execution procedures, and discussions on
the selection of the floc size input based on two independent sources. Capabilities
of the model are tested first by comparing the model results with the measurements.
Model calculations of Reynolds stress are compared with the observation-based
20
estimates. The contribution of different terms in the turbulence kinetic energy balance
and sensitivity of this balance on floc size are evaluated. The results are summarized
and discussed in Section 6.
21
Figure 1-1. Noncohesive sand grains and cohesive fluid mud. (a) Sand grains; and (b)mud sample collected at the Atchafalaya Shelf, Louisiana (Photo courtesy:K. T. Holland, Naval Research Laboratory at the Stennis Space Center).
22
0 0.1 0.2 0.3 0.4 0.5 0.610
−2
10−1
100
101
frequency (Hz)
flux
spec
tral
dens
ity (
m3 )
Figure 1-2. Comparison of spectral evolution of waves over muddy and sandy seafloors.Initial JONSWAP spectrum (thick continuous line), resulting spectra afterpropagating 5-km in 5-m depth over a flat bottom covered with 0.2 mm sand(thin continuous line with squares), and 15-cm thick viscous fluid mud layer(dashed line with dots).
23
CHAPTER 2FIELD EXPERIMENT
2.1 Experiment Site
The main data set on which this study is based was collected from March 25th to
April 7th, 2008. This was the last two-week interval of an experiment which started on
February 22nd, 2008 on the muddy inner shelf fronting Atchafalaya Bay, Louisiana,
USA, in the north-central Gulf of Mexico (Figures 2-1 and 2-2). This experiment was
part of a larger scope study of wave, turbulence and sediment transport processes in
shallow muddy environments that started in 2006. Four instrumented platforms were
deployed on February 22nd by the University of Florida and the University of Texas
(Figure 2-2). Platforms 1-3 were put in a cross-shore transect and Platform 4 was
located in an alongshore transect with the shallow platform 3. Platform 5 was near
Fresh Water Bayou about 60 miles west of the Atchafalaya Bay, near the center of a
transect of current-meters, deployed by the group led by Dr. Steve Elgar and Dr. Britt
Raubenheimer from Woods Hole Oceanographic Institution (Safak et al., 2010b). The
coordinates and retrieval dates of the platforms are given in Table 2-1, together with the
mean depth at the location of the platforms (average of the data collected throughout the
experiment). The experiment site was revisited for each two-week period to retrieve data
and change the batteries of the instruments deployed at platforms 1-4. Therefore, the
three periods February 22nd - March 8th, March 8th - March 25th and March 25th -April
7th were named as experiments A,B and C, respectively. The focus of this study is the
observations collected at Platform 2 near the 5-m isobath during Experiment C.
The experiment site is located on the topset of the Atchafalaya sub-aqueous
feature, which is defined as a clinoform of up to 3-m thick mud layer. The clinoform
extends out to the 8-m isobath at tens of kilometers offshore (Allison et al., 2000; Neill
and Allison, 2005). Since the 1940s, the muddy Atchafalaya inner continental shelf
receives about 30% of the discharge of the Mississippi River, which is the largest river
24
on the North American continent (Mossa, 1996; Allison et al., 2000; Neill and Allison,
2005). The Atchafalaya River leaves the main course 320 km upstream of the Gulf of
Mexico near Simmesport, Louisiana. The annual sediment discharge by the Atchafalaya
River is estimated to be 84 million metric tons. The suspended sediment load into the
bay and the inner shelf is identified by fine grains with median particle diameter D50=2-6
µm, however, includes 17% sand, as well (Allison et al., 2000). West of Marsh Island
(92.5 degrees longitude West), D50 was estimated to be between 2.8-5.9 µm (Allison
et al., 2005). Sheremet et al. (2005) measured D50=6.34 µm on the inner shelf, about
20 km offshore of Marsh Island. These data suggest that D50=2-7 µm is representative
for the area. Size distribution of suspended sediments is available from the previous
experiments and is investigated in the following sections.
The Atchafalaya River is different than the major distributary of the Mississippi 180
km to the east in the sense that the Atchafalaya Shelf is shallower with a milder slope
(e.g., low-gradient where the 10-m isobath is approximately 40 km offshore) and more
wave-energetic. This region is interesting for sediment transport studies due to the
fact that over the last few decades, the coastline has been prograding seaward with
rates of O(10 m/yr) and land accretes vertically at rates reaching O(1 cm/yr) due to
the sediment discharge of the Atchafalaya River in spite of rising sea levels. However,
much of the rest of the Louisiana coastline and the Mississippi Delta are experiencing
significant erosion (Allison et al., 2000; Draut et al., 2005; Neill and Allison, 2005; Kineke
et al., 2006). Despite these high sediment accumulation rates, it is unlikely that the
outer Atchafalaya Bay and the clinoform topset area will accrete to sea level, since
atmospheric conditions force the hydrodynamics to distribute sediments away from
these areas (Neill and Allison, 2005).
Wind is the major forcing that controls circulation, sediment transport, water
level, and salinity changes over the Atchafalaya Shelf. The wave climate is dominated
between December and April (coinciding with the period of high sediment discharge in
25
the Atchafalaya River) by wave fields associated with storms and cold fronts passing
through the area on 3-7 day time scales (Allison et al., 2000; Walker and Hammack,
2000). These fronts are characterized by pre-frontal onshore winds, strong wave
activity and coastal setup, followed by post-frontal offshore winds, set-down, and
formation of large sediment plumes (Allison et al., 2000; Walker and Hammack, 2000).
At the experiment site, these perturbations can generate swells in excess of 1-m
height lasting for several days. In shallow water, such intense wave activity causes
significant variations of the bed state throughout a storm, in a sequence of breaking
down stratification, triggering bed liquefaction, increasing sediment resuspension and
turbidity throughout the water column (Allison et al., 2000), followed by the formation
of fluid mud layers within 1-m near the bed due to settling, and finally consolidation to
a soft bed (Jaramillo et al., 2009). These high concentration fluid mud layers can be
transported over the shelf in different directions by various mechanisms: westward by
residual currents of about 10 cm/s which may account for advection of more than half
of the sediment discharge into the shelf (Wells and Kemp, 1981; Allison et al., 2000);
onshore by coastal upwelling (Kineke et al., 2006); and offshore in the form of a turbidity
current of about 5 cm/s, which is maintained in suspension by wave-induced turbulence
(Jaramillo et al., 2009). On the other hand, the contribution of extreme events such as
tropical cyclones and hurricanes (e.g., Hurricane Lili in 2002) to sediment transport and
organic matter deposition may exceed the input by the Atchafalaya River (Allison et al.,
2005; Goni et al., 2006).
Together with the studies on nearshore circulation and sediment transport
processes discussed above, the broad-spectrum wave-energy dissipation effect of
muddy seafloors has been studied extensively on the Atchafalaya Shelf and near Fresh
Water Bayou in the last decade (Sheremet and Stone, 2003; Sheremet et al., 2005;
Elgar and Raubenheimer, 2008; Jaramillo, 2008). A recent finding from these wave-mud
interaction studies is that maximum mud-induced dissipation rates are observed not
26
during the input of highest wave energy into the system but in the wane of the storm.
During that period, the seafloor is characterized either as a viscous fluid or a viscoelastic
material, and the boundary layer is expected to be laminar (Jaramillo, 2008; Sheremet
et al., 2010).
2.2 Instrumentation
A typical platform in 2006 and 2008 experiments is shown in Figure 2-3. Currents
in the upper water column and the directional surface wave field were monitored by
upward-pointing Acoustic Doppler Current Profilers (ADCP, Teledyne RD Instruments,
1200 kHz, Figure 2-3 label A). Velocity and backscatter within 1-m of the bed were
continuously measured by downward-pointing Pulse Coherent-Acoustic Doppler
Profilers (PC-ADP, Sontek/YSI, 1500 kHz, Figure 2-3 label B) in bins of either 1.6- or
3.2-cm continuously with sampling rates of either 1- or 2-Hz. The PC-ADPs are also
equipped with built-in pressure sensors. Acoustic Backscatter Profilers (ABS, Aquatec,
Figure 2-3 label C) provided a more detailed profile of the near-bed backscatter at
bins smaller than 1 cm. Optical backscatterance sensors (OBS-3s and OBS-5s, D &
A Instruments, Figure 2-3 label D) measured turbidity. Operational principles of these
acoustic and optical instruments can be found in Lhermitte and Serafin (1984) and
Downing et al. (1981), respectively. Conductivity-Temperature sensors (CT, SeaBird
Electronics) provided salinity estimates and temperature measurements. At the location
of Platform 1, a pressure sensor, located 1-2 m below the surface, sampled at 4-Hz
throughout the entire 7-week experiment. At the location of Platform 3, an Onset, Inc.,
HOBO micro-station located at an elevation of 7-m above the sea surface, provided
30-min averages of wind speed and direction.
The instrumentation used in this analysis comprised a vertical array of two
synchronized Acoustic Doppler Velocimeters (ADV, SonTek/YSI Hydra 5-MHz), and
an OBS-5. An ADV is a pointwise velocity sensor, and it has been tested in laboratories
to estimate turbulence parameters (Voulgaris and Trowbridge, 1998) and to measure
27
instantaneous velocities in concentrated fluid mud (Gratiot et al., 2000). A photograph
of the instrumentation on the platform before the deployment and a schematic of the
platform showing the locations of the instruments are shown in Figures 2-4 and 2-5,
respectively. The sampling volume of the lower ADV (ADV-1) was located at 17 cmab
(cm above bed); the higher ADV (ADV-2) sampled in a volume at 145 cmab. Based
on the measurements in a wave-free environment (Trowbridge et al., 1999), Shaw
and Trowbridge (2001) suggested that two vertically-stacked sensors are optimally
configured to yield uncorrelated cross-sensor turbulent covariances if the distance
between the sensors is greater than 5 times the height of the lower sensor above
the bed. For the application of the two-sensor method of wave-bias reduction from
the Reynolds stress estimates (see Section 3.2.3), the configuration of the ADVs
(Figure 2-5) follows this recommendation. Each ADV was equipped with a built-in
pressure sensor, located at about 60 cm from its sampling volume. The ADVs sampled
pressure and three-dimensional flow velocity (converted to East-North-Up coordinates
in post-processing) at 10 Hz, in 10-min measurement bursts, one burst every hour, for
the entire two-week duration of the experiment. A burst duration of 10-min allows the
ADVs to run for two weeks with a shared battery pack, spans about 75 swell periods
(for a swell period of 8 s, which is typical for the site), and is long enough to provide a
stable mean current estimate (Soulsby, 1980). The OBS-5 was mounted at 12 cmab,
and recorded 1-min averages of backscatter at a rate of 2 Hz. The backscatter signal
from the OBS-5 was calibrated in the laboratory with in-situ sediment and water samples
collected at the site surveys during the deployment and retrieval of the platforms.
Through this calibration, the turbidity records were converted to suspended sediment
concentration estimates. For the directional wave measurements, the data collected
by the ADCP at Platform 3, located 4 km onshore of the experiment site (Figure 2-2)
was used. The ADCP transducer head was located at 1.3 mab and measured pressure,
acoustic surface track, and velocity profiles at 2 Hz, in 40-min measurement bursts, one
28
burst every hour. Wave data were processed using the Teledyne RDI software packages
WavesMon and WaveView, with a frequency resolution of 0.0078 Hz and an angular
resolution of 4 degrees. The ADCP also provided 10-min averaged current profiles in
bins of 20 cm.
For this analysis, the only information about the size distribution of suspended
sediments on the Atchafalaya Shelf was available from a LISST-100X Type-C (Laser In
Situ Scattering Transmissometer, Sequoia Scientific). LISST records the small-angle
scattering distribution of particles in water, which is inverted into size spectra. For further
operational principles, see Agrawal and Pottsmith (1994). Two sets of observations
(Allison et al., 2010) were collected, one from the 2006 experiment between February
28th and March 14th (Jaramillo et al., 2009) at the location of Platform 3 (Figure 2-2)
and one from the 2008 experiment A between February 22nd and March 8th, at Platform
4 (Figure 2-2). As both data sets have time offsets with the analyzed experiment, the
2006 data set was preferred because it was collected at a location closer to the location
of the data set used in the analysis herein. The LISST background was calibrated using
filtered water at the deployment site. The path of the LISST measurements (which
determines the upper limit of sediment concentration at which reliable data can be
obtained) is reduced with an 80% path reduction module. With this setting, the threshold
sediment concentration is about 1 g/L, and reliable data were collected at 120 cmab
between March 1st and March 9th, 2006. The instrument estimates size distributions
of suspended particulates (flocs and primary) in 32 class ranges between 2.5-500
µm size. However, in this experiment, the data is unreliable above 350 µm, and not
reproduced here. The grain-size distribution was recorded every minute (an average
of 100 samples at 2 Hz) in 30-min bursts each hour. On the same platform with the
LISST, a downward-pointing PC-ADP measured velocity and backscatter profiles,
and pressure. The PC-ADP sampled at 2 Hz sampling in 60 bins of 1.6 cm, following
a 10-cm blanking distance. A 10-min burst was started every 30-min. The PC-ADP
29
measurements were missing velocity profiles once in every three or four samples, due to
an unknown instrumentation problem. Therefore, the 2 Hz velocity measurements were
burst-averaged to calculate the vertical structure of mean currents and then estimate
the bottom friction velocity by using the logarithmic law of the wall (Section 3.3). These
estimates, and the spectral wave calculations based on the PC-ADP pressure sensor
data, present a general picture of the variation of size distribution records of LISST
under varying wave and bottom turbulence conditions (Sections 3.3 and 4.2).
2.3 General Conditions
Before focusing on the experiment of interest and the presentation of the related
data analysis methods, a general overview of the conditions (winds, near-bed flows,
and surface waves) throughout the entire seven-week experiment duration is presented
in Figure 2-6. For this, the wind and the PC-ADP data measured at Platform 3 (Figure
2-2), where the local depth was about 3.8 m, and the pressure data from the pressure
sensor near Platform 1 (Figure 2-2), where the local depth was about 7.4 m, are used.
The PC-ADP on Platform 3 was the only near-bed profiler that sampled throughout the
entire experiment, except the retrievals for re-deployments between experiments A-B
and B-C. The pressure sensor near Platform 1 was the only wave gauge which sampled
continuously throughout the entire experiment. In the wave spectral calculations, which
are detailed in the next section, swell (low frequency) and sea (high frequency) bands
are decomposed using a cutoff frequency of fc=0.2 Hz. This arbitrary value is selected
based on the similar evolution trends of spectral wave energy at frequencies larger
than this value, and wind speed. Figures 2-6a and b show that sea waves (black curve
in Figure 2-6b) closely follow the trend of wind speed, which sometimes exceeded 15
m/s and generated sea waves of significant heights exceeding 1.5 m (see the peaks
on February 28th, March 4th, March 8th, and March 20th). Increasing swell energy
seems to be triggered by shifts in wind direction from onshore winds to offshore winds
on February 27th, March 4th, March 8th, and especially March 18th (see the variation
30
in the wind-direction representing color code in Figure 2-6a, from cyan-green sector,
i.e., northward winds, to red-magenta sector, i.e., southward winds). During all four
of these events, near-bed activity was observed in the sense that the location of the
maximum backscatter is recorded to be closer to the PC-ADP sensor head compared
to its initial location at the beginning of the experiment (Figure 2-6c). This indicates
the formation of a high concentration, i.e., fluid mud layer above the consolidated bed.
The backscatter is also showing an increase througout the water column during these
events. The strongest swell event throughout the experiment was recorded on March
18th when swells reached 1.5 m and the near-bed observations indicate formation of
a fluid mud layer of about 20 cm thickness. During the third and last 2-week interval of
this experiment, experiment C, the data set for studying turbulence-cohesive sediment
interaction herein was collected. During that experiment, swell heights were mostly less
than 0.5 m at 7.4 m depth and the backscatter records are not indicating formation of
any fluid-mud layers at 3.8 m depth.
31
Table 2-1. Location, mean depth, and retrieval dates of the instrumented platformsPlatform Latitude(North) Longitude (West) Depth (m) Retrieval date
1 29o11.815’ 91o36.731’ 7.4 March 25th2 29o13.439’ 91o34.807’ 5.0 April 10th3 29o15.574’ 91o34.267’ 3.8 April 10th4 29o22.238’ 91o46.922’ 4.0 April 10th5 29o33.759’ 92o33.800’ 4.0 March 25th
32
−95 −90 −85 −8025
26
27
28
29
30
31
Longitude (deg.)
Latit
ude
(deg
.)
Florida
Atlantic Ocean
Louisiana
Gulf ofMexico
Transect 1Transect 2
Figure 2-1. Gulf of Mexico. The platforms along Transect 1 and Transect 2 werecontaining instrumentation deployed by the field support groups of theUniversity of Florida-the University of Texas, and the University ofFlorida-Woods Hole Oceanographic Institution, respectively.
33
Longitude (deg.)
Latit
ude
(deg
.)
Marsh Island
Atchafalaya River
Louisiana
Atchafalaya Bay
Trinity shoal
Gulf ofMexico
123
4
5
5 m10 m
20 m
30 m
−92.8 −92.4 −92.0 −91.6 −91.228.8
29.0
29.2
29.4
29.6
29.8
30.0
Figure 2-2. Bathymetry of the Atchafalaya Shelf and the locations of the instrumentedplatforms. This bathymetry was based on a data set collected before the2008 experiment and is showing differences with the 2008 bathymetry. Thenumbered red stars indicate the platform locations. The main data set onwhich this study is based was collected at Platform 2. Also from Platform 3which is located 4 km onshore of Platform 2, directional wavemeasurements, wind data, and information on size distribution of suspendedsediments in the area are investigated.
34
Figure 2-3. A typical instrumentation platform. The platforms were equipped with anupward-pointing ADCP (A), a downward-pointing PC-ADP (B), adownward-pointing ABS (C), and pointwise turbidity sensors (D).
35
Figure 2-4. The synchronized ADVs and the OBS-5 on the platform prior to thedeployment.
36
Figure 2-5. Configuration of the instrument array. Circles mark the location of thesampling volumes.
37
Figure 2-6. Wind, wave, and near-bed conditions on the Atchafalaya Shelf throughoutthe seven-week experiment in Spring 2008. (a) Wind speed and direction(the color code shows where the wind flow is towards) measured nearPlatform 3 where the local depth is 3.8 m; (b) significant wave height in sea(black) and swell (red) bands, measured by the near-surface pressuresensor at Platform 1 where the local depth is 7.4 m; and (c) near-bedacoustic backscatter (normalized such that the maximum backscatter isshown by dark red and the minimum backscatter by dark blue) measured bythe PC-ADP at Platform 3. The gaps in the backscatter data on March 8thand March 25th correspond to the intervals between the experiments A-Band B-C, when the data sets from the instruments were being retrieved, andthe batteries and memory cards were being replaced.
38
CHAPTER 3DATA ANALYSIS METHODS
3.1 Wave Spectral Calculations
A wave mode with amplitude a and frequency f has a total average energy (sum of
kinetic and potential energies) per unit surface area (Dean and Dalrymple, 1991):
E (f ) =ρg a2
2, (3–1)
where ρ is the fluid density and g is the acceleration caused by gravity. In the frequency
domain, wave energy is usually represented as:
S(f ) =a2
2 df, (3–2)
where S(f ) is the energy spectral density of the mode with frequency f , and df is
the frequency resolution. The wave field during the analyzed experiment is defined
by statistical parameters which are obtained through spectral calculations. Also, the
cross-spectrum of horizontal and vertical velocity measurements is calculated to
estimate Reynolds stresses (Section 3.2). Therefore, a brief discussion of spectral
analysis is given here. Spectral analysis is used to decompose a time-varying quantity
x(t) into a sum of sine and cosine functions, and calculate the distribution of energy
at modes of different frequencies (Priestley, 1981). For a stochastic process, the
auto-correlation function describes the general dependence of the data values at one
time on the values at another time (Bendat and Piersol, 1971). It is defined as:
Rxx (τ) =∞∫
−∞
x(t) x∗(t + τ)dt , (3–3)
where t is the time and asterisk denotes the complex conjugate. Fourier transform of
x(t) at frequency f is:
39
X (f ) =∞∫
−∞
x(t) e−2π�tdt , (3–4)
where i is the imaginary number. The Fourier transform of the auto-correlation function
gives the energy spectral density:
Sxx (f ) =∞∫
−∞
Rxx (τ) e−2π�tdt = 〈X (f )〉2 , (3–5)
where 〈〉 denotes the expected value operator, i.e., mean. While the auto-correlation
function describes the correlation within a process (equation (3–3)), the cross-correlation
function describes the correlation structure between two stochastic processes (Priestley,
1981), say x(t) and y (t):
Rxy (τ) =∞∫
−∞
x(t) y ∗(t + τ)dt. (3–6)
The Fourier transform of the cross-correlation function gives the cross-spectral density:
Cxy (f ) =∞∫
−∞
Rxy (τ) e−2π�tdt = X (f )Y ∗(f ) , (3–7)
which is decomposed into its real part, i.e., co-spectrum, and imaginary part, i.e.,
quadrature spectrum, as follows:
Cxy (f ) = Coxy (f ) + i Quadxy (f ). (3–8)
Spectral density of pressure (Spp) is estimated using standard discrete Fourier
spectral analysis based on the pressure time series from ADV-2. The pressure time
series from each 10-min burst is demeaned and detrended, divided into 51.2-sec
segments (each containing 1024 samples) with 50% overlap. The purpose of overlapping
is to reduce leakage at the boundaries of the segments. The resulting spectral estimates
are characterized by approximately 22 degrees of freedom and 0.0196 Hz frequency
40
resolution. Spectral density of the free surface elevation (Sηη) is estimated from Spp by
correcting for depth attenuation, using linear wave theory:
Sηη(f ) =[
cosh(kw h)cosh kw (h + zsensor )
]2
Spp(f ) , (3–9)
where kw is the wavenumber, h is the local water depth, and zsensor is the elevation of
the sensor relative to the mean water level. The wavenumber is related to the frequency
through the linear dispersion relation for waves:
(2πf )2 = gkw tanh (kw h) . (3–10)
A cutoff frequency is defined by a variance attenuation with depth of less than 2% from
surface to sensor, then a spectral tail proportional to f −5 (Phillips, 1958) is added to
cover the high-frequency range. The band significant wave height (Hs) is estimated
using the standard relation:
Hs = 4
√√√√√f2∫
f1
Sηη(f ) df , (3–11)
where f1 and f2 are the spectral band limits. Velocity and acceleration spectral densities
are estimated with the same procedure but from the measurements of ADV-1, as this
sensor was placed closer to the bottom. Velocity and acceleration variance estimates
are calculated as:
σ2ξ =
f2∫
f1
Sξξ(f ) df , (3–12)
where ξ represents the parameter of interest, i.e., velocity or acceleration component.
Where convenient, swell (long waves) and sea (short waves) bands are decomposed by
using a cutoff frequency of fc=0.2 Hz, such that the swell band is defined by f1=0.0196
Hz, f2 = fc .
41
3.2 Estimation of Reynolds Stresses in the Presence of Surface Waves
3.2.1 Definition of Reynolds stress
In the Reynolds-averaged Navier-Stokes equations, the contribution of the turbulent
motion to the mean stress is described with the components of the symmetric Reynolds
stress tensor τij (also referred to as turbulent covariance tensor), which includes
correlations of turbulent fluctuations (Tennekes and Lumley, 1972):
τij = −ρ
u′u′ u′v ′ u′w ′
v ′u′ v ′v ′ v ′w ′
w ′u′ w ′v ′ w ′w ′
, (3–13)
where u and v are the horizontal components and w is the vertical component of
velocity, a prime denotes turbulent fluctuations, and an overbar denotes a time-averaged
quantity. The diagonal components are normal stresses, i.e., pressures. These normal
stresses contribute little to the mean momentum transport (Tennekes and Lumley,
1972), however, half of the sum of these stresses is defined as the turbulent kinetic
energy (k), which is one of the most common quantities used in two-equation turbulent
flow modeling (Section 5):
k =12
(u′2 + v ′2 + w ′2
). (3–14)
The off-diagonal components of equation (3–13) are Reynolds shear stresses, which
play a dominant role in momentum transport in turbulent motion (Tennekes and Lumley,
1972). For the one-dimensional bottom boundary layer modeling study herein, the
Reynolds shear stresses of interest are those that appear in the horizontal momentum
equations:
τxz = −ρu′w ′ ; τyz = −ρv ′w ′ . (3–15)
42
Due to their significant effects in boundary layer flows at several scales such as
large-scale surf zone flows (Trowbridge and Elgar, 2001) and small-scale sediment
transport processes (Winterwerp, 1998), estimating Reynolds stresses accurately using
field observations in the absence of models (uniform eddy viscosity model, mixing length
model, two-equation models, etc.) is an important objective in oceanographic studies. In
addition, as mentioned in Section 1.3, the closure schemes implemented in turbulence
models to calculate the Reynolds stresses need to be evaluated against observations of
combined wave-current flows.
3.2.2 Wave bias in Reynolds stress estimates
The main difficulty in estimating Reynolds stresses to characterize flow turbulence
is due to the contamination of the turbulent signal by surface waves, which typically
dominate the variance of horizontal and vertical velocities (Trowbridge, 1998). The
discussions in the rest of this section are going to be for a two-dimensional model
velocity vector u = (u, w ). Therefore, the quantity of interest is u′w ′ (equation (3–15)),
however, the analysis is easily adaptable to v ′w ′ as well. u is decomposed as:
u = �u + ~u + u′ , (3–16)
where an overbar denotes a time-averaged quantity, i.e., mean current, a tilde denotes
wave-induced fluctuations, and a prime denotes turbulent fluctuations, respectively.
Assuming that wave-induced and turbulence-induced fluctuations are uncorrelated, i.e.,
statistically orthogonal (Kitaigorodskii and Lumley, 1983), the quantity of interest u′w ′
can be obtained as:
u′w ′ = uw − uw . (3–17)
The first term on the right hand side is the covariance of the measured horizontal
and vertical velocities. However, uw needs to be separated in the presence of energetic
wave motions. Several approaches have been developed to reduce the wave bias in
43
the Reynolds stress estimates. Ensemble averaging is implemented in the time domain
and wave-induced fluctuations are estimated by averaging the same point in the wave
phase over several consecutive regular waves. High-pass filtering can be applied by
assuming that waves and turbulence scales are clearly separated, and then specifying
a cutoff frequency. The accuracy of this approach depends on how carefully the cutoff
frequency is selected, since effects of large-scale turbulent eddies may be neglected.
These two techniques can be used in tightly controlled environments such as laboratory
experiments (Scott et al., 2005), but are less useful for random wave fields. In a class
of methods, the wave signal is identified by velocity fluctuations that are coherent with
pressure, whereas, fluctuations that are not coherent with pressure are considered to
be due to turbulence (Benilov and Filyushkin, 1970; Agrawal and Aubrey, 1992; Wolf,
1999). The coherence between the pressure and the wave-induced velocity fluctuations
is calculated as:
γ2 =CpuC ∗
pu
SppSuu, (3–18)
where Cpu is the cross-spectrum of the pressure and velocity (equations (3–7) and
(3–8)), Spp and Suu are power spectra of pressure and velocity, respectively. The power
spectrum of turbulence-induced velocity fluctuations can be calculated as (Agrawal and
Aubrey, 1992):
Su′u′ = Suu[1− γ2] . (3–19)
The only additional requirement of this method is to have pressure measurements
synchronized with velocity measurements. However, this approach requires accurately
resolving wave nonlinearities and directional spread (Herbers and Guza, 1993).
Therefore, its practical applicability for field studies is questionable. The Phase method
(Bricker and Monismith, 2007), on the other hand, requires a single pointwise velocity
sensor to remove wave bias from Reynolds stress estimates by assuming equilibrium
44
turbulence and no wave-turbulence interaction. Wave bias in a single-sensor Reynolds
stress estimate is calculated as:
~u ~w =fmax∑
fj =fmin
~U∗j
~Wj =fmax∑
fj =fmin
|~Uj || ~Wj | cos(
�Wj −�U
j
), (3–20)
where Uj and Wj are the Fourier transforms of u(t) and w (t), respectively, at frequency
fj . [fmin, fmax ] represents a frequency band dominated by waves, and is determined
by visual inspection of the spectra. ~|U j | and ~|W j | are obtained from the difference
between the spectra of the raw signals and the spectra obtained from the linear least
squares fit based on the assumption of the equilibrium turbulence. The phases of the
raw signal(
�Uj , �W
j
)are used to calculate the wave bias, because waves are expected
to dominate the spectrum of the raw signal under the wave peak. This simple method,
however, is likely to overestimate low-frequency turbulence in shallow water due to
infragravity wave bias, which is an effect of wave nonlinearities.
3.2.3 Two-sensor methods to reduce wave bias
Wave bias in Reynolds stress estimates can also be reduced by measuring
the flow field with two sensors, so that optimal distances can be found between the
measurement points and the bed, that will yield cross-correlated signals for waves and
uncorrelated signals for turbulence (Trowbridge, 1998). If the cross-sensor correlation
is low for turbulent fluctuations, but high for waves, cross-sensor differencing of the
measurements will not affect turbulent terms but will reduce the wave bias. Having
uncorrelated turbulent fluctuations at spatially separated points (i.e., two sensors) has
an empirical basis in the literature. The region over which turbulent fluctuations are
correlated is called an eddy, and this scale was found to be proportional to the height
of measurement above bed (Grant, 1958). Trowbridge and Elgar (2003) observed that
alongshore scales of turbulence contributing to near bottom Reynolds stresses range
from 0-4 times the height of measurement above the bottom. An empirical guideline was
presented by Shaw and Trowbridge (2001), based on the measurements obtained from
45
a vertical array of five current-meters in the relatively wave-free lower Hudson estuary
(Trowbridge et al., 1999). The decay of turbulence correlation terms was calculated as a
function of vertical separation:
DT =Ruw (r )Ruw (0)
, (3–21)
where Ruw (r ) = u(zl ) w (zl + r ), r is the vertical separation of the two sensors,
and zl is the height of the lower sensor above the bottom. Shaw and Trowbridge
(2001) concluded that for the turbulence to be considered uncorrelated between
two sensors, i.e., DT less than 0.1, and to have the Reynolds stress estimates not
biased by turbulence correlation, r has to be greater than 5zl . Due to having only two
current meters in the vertical array and the presence of waves in the data, the decay
of turbulence correlation terms can not be checked for the data set analyzed here.
Therefore, the constraint given by Shaw and Trowbridge (2001) was followed in the
experimental setup (Section 2.2).
The two-sensor method requires an elaborate experimental setup, is most
effective for well-separated spatial scales of waves and turbulence, and is likely to
alias low-frequency turbulence as waves, but seems better suited for shallow ocean
environments. Based on this approach, the variance method (Lohrmann et al., 1990;
Stacey et al., 1999), which was initially proposed to estimate Reynolds stress profiles
using ADCPs in the absence of waves, has recently been modified to reduce wave-bias
in the estimates (Whipple et al., 2006; Rosman et al., 2008). The two-sensor method
(Trowbridge, 1998) is briefly explained here and the calculations are detailed in the
Appendix.
In the 2-D model coordinate system, let U1 and W1 be the horizontal and vertical
components of velocity, measured in instrument coordinates and accounting for their
small misalignment θ1 from the model coordinate system:
46
U1 ≈ ~u1 + u′1 + θ1( ~w1 + w ′1) , (3–22)
W1 ≈ ~w1 + w ′1 − θ1(~u1 + u′1) . (3–23)
The subscript “1” (and subscript “2” given below) identifies the position of a measurement
or an estimate. The covariance of the measured velocities gives:
Cov(U1, W1) = u′1w ′1 + ~u1 ~w1 + θ1( ~w1
2 − ~u21) + θ1(w ′
12 − u′12) . (3–24)
The first term is the Reynolds stress at the sensor location, which is the quantity
of interest, the second and the third terms are wave biases, and the last term is a
turbulence bias. Assuming that all the components of the turbulent covariance tensor
are of the same order of magnitude (Tennekes and Lumley, 1972), and recalling that
θ1 is small, turbulence bias is smaller than the quantity of interest in equation (3–24).
However, wave biases need to be reduced in the presence of energetic wave motions.
By taking the difference of the velocity components measured at two spatially-separated
sensors, Trowbridge (1998) reduced wave contamination at the Reynolds stress
estimates, and calculated an average of Reynolds stresses at the two sensors:
u′1w ′1 + u′2w ′
2
2=
⟨u′w ′⟩ ≈ Cov {�U, �W }
2, (3–25)
where
�U = U1 − U2 ; �W = W1 −W2 . (3–26)
U2 and W2 are defined analogously to equations (3–22) and (3–23), but with subscripts
“1” replaced with “2”. Shaw and Trowbridge (2001) improved this raw differencing
method, such that wave bias can be reduced further by differencing after mapping the
horizontal component of velocity at one of the sensors, using the velocities measured at
47
the other sensor, i.e., with linear filtration techniques. Also, the estimate at the sensor
location can be calculated, instead of an average of the Reynolds stresses at the two
sensor locations:
u′1w ′1 ≈ Cov
{�U, W1
}, (3–27)
where
�U = U1 − U1 . (3–28)
U1 is the horizontal velocity estimated at position “1” using the horizontal velocity
measured at position “2” with least squares linear adaptive filtering:
U1(t) =∞∫
−∞
h(t ′)U2(t − t ′)dt ′ , (3–29)
where h(t) is a filter that represents the relationship between the wave-induced
fluctuations at the two sensor locations. Here, low wave-bias estimates of Reynolds
stresses are obtained by the method presented by Feddersen and Williams (2007). This
method is actually a refinement of the approach developed by Trowbridge (1998) and
Shaw and Trowbridge (2001). Feddersen and Williams (2007) mapped both horizontal
and vertical components, and reduced the wave bias even further. The minus of the
product of density and the integral of the co-spectrum of the velocity differences gives a
nearly wave-free estimate of the Reynolds stress:
τ1 = −ρu′1w ′1 ≈ −ρCov(�U, �W ) = −ρ
∫ ∞
0Co�U,�W (f )df . (3–30)
Subscripts “1” and “2” specifically denote measurements of ADV-1 at 17 cmab and
ADV-2 at 145 cmab (Figure 2-5), respectively, hereinafter. For the vertical component of
velocity, W1 and �W are calculated through a procedure analogous to equations (3–28)
and (3–29). Co�U,�W (f ), the co-spectrum (equation (3–8)) of the velocity differences,
48
is estimated based on the same Fourier analysis parameters as those used for wave
spectrum analysis (Section 3.1). This method can be used in principle to estimate
Reynolds stress at both ADV locations. However, away from the bed, the effect of the
bottom boundary decreases, and turbulent fluctuations become slower, increase in
scale, weaken, and are likely increasingly modulated by wave motion (Monismith and
Magnaudet, 1998; Bricker and Monismith, 2007). Therefore, Reynolds stresses are
estimated for the lower sensor (ADV-1 at 17 cmab, Figure 2-5).
The effectiveness of the two-sensor method (with differencing and filtering) in
reducing wave-bias varies with each measurement burst. Following atmosphere and
ocean boundary layer studies (Kaimal et al., 1972; Soulsby, 1977), Feddersen and
Williams (2007) proposed to accept as valid Reynolds stress estimates for which the
non-dimensional integrated co-spectrum (Og, also called ogive curve) satisfies the
condition:
− 0.5 ≤ Ogu′w ′(f ) =∫ f
0 Cou′w ′(s)dsu′w ′ ≤ 1.6 , (3–31)
for 0.1 < kN < 10, where kN = 2πfzV is the non-dimensional wavenumber, with z
the vertical distance above bed, and V the average along-stream velocity. Examples
of estimates considered relatively wave-biased or bias-free are shown in Figure 3-1.
Approximately half of the measurement bursts satisfy the condition in equation (3–31).
3.3 Logarithmic Law of the Wall
As mentioned in Section 1.3, floc formation is controlled by turbulent eddies which
are scaled by Kolmogorov micro-scale λ = 4√
ν3/ε (Berhane et al., 1997), where ν is the
kinematic viscosity of water, and ε is the dissipation rate of turbulent kinetic energy. ε is
related to the bottom friction velocity (u∗) as follows:
ε =u3∗
κ z, (3–32)
49
where κ=0.41 is von Karman’s constant and z is a reference elevation above the bed.
For the investigation of the LISST data set from 2006, having u∗ estimates based on the
PC-ADP measurements from the same platform would allow the variation of floc size
distribution with a turbulent flow parameter to be seen. For this, the PC-ADP velocity
profiles are fit to logarithmic profiles, based on the logarithmic-law of the wall (called
log-law hereinafter). By following Nielsen (1992), the derivation of the log-law is shown
below, starting from the two-dimensional horizontal momentum equation:
∂u∂t
+ u∂u∂x
+ w∂u∂z
= −1ρ
∂p∂x
+ ν
(∂2u∂x 2 +
∂2u∂z 2
). (3–33)
The parameters in this equation are defined in this section previously. A steady flow (i.e.,
∂∂t =0) which is horizontal in the boundary layer (i.e., w ≈0) and horizontally uniform (i.e.,
u = u(z)) is assumed:
ν∂2u∂z 2 =
1ρ
∂p∂x
. (3–34)
In addition to these assumptions, substituting the viscous shear stress τ = ρν ∂u∂z
simplifies equation (3–34) to:
∂τ
∂z=
∂p∂x
. (3–35)
Assuming hydrostatic pressure, i.e., the pressure gradient is constant over the depth:
∂τ
∂z=
∂p∂x
= ρg∂η
∂x. (3–36)
A bottom boundary condition is set by parameterizing the bottom shear stress as
τ(0) = τb = ρu2∗ . The velocity profile obeying the log-law in a turbulent boundary layer is
obtained as follows:
u(z)u∗
=1κ
ln(
zzo
), (3–37)
50
where zo is a function of bottom roughness that describes a finite elevation above the
bed, where the velocity profile goes to zero.
The PC-ADP data set was collected in the presence of surface waves with
significant heights exceeding 1 m (Section 4.2). Since near-bed turbulence within
the wave boundary layer enhances bottom roughness experienced by currents (Grant
and Madsen, 1979, 1986), as discussed in Section 1.2, fitting a logarithmic profile to
the current velocity profiles in a least-squares sense (Lueck and Lu, 1997) would cause
errors in the resulting u∗ estimates. If there were not missing profiles in the PC-ADP
data set (Section 1.2), the method proposed by Rosman et al. (2008) could be applied
to estimate Reynolds stresses and then relate these to u∗. However, this method would
still require a modification due to different beam geometries of ADCPs and PC-ADPs.
Therefore, u∗ is estimated here from the mean current velocity profiles outside of the
wave boundary layer, following the approach of Lacy et al. (2005). The missing velocity
profiles also preclude obtaining accurate profiles of root-mean-square velocities, which
would give estimates about the thickness of the wave boundary layer. Therefore, a
constant thickness of 5 cm is assumed. The first three bins above the bottom are
assumed to be within the wave boundary layer (bin size was 1.6 cm, Section 2.2) and
not included in the analysis. With linear least-squares regression, a logarithmic profile
is fit to the velocity measurements at the first three bins just above this layer. A fit is
accepted to be valid if the correlation coefficient, r 2, between the measurements and the
fit is greater than 0.8. The bins above the first three bins are added into the logarithmic
fit, within 3-bin groups, as long as the resulting r 2 remains to be greater than 0.8.
51
10−1
100
101
−0.5
0
0.5
1
kN
Og u’ w
’
Figure 3-1. Examples of ogive curves (cumulative integrals, equation (3–31)) as afunction of the dimensionless wavenumber kN . A low-bias estimate,corresponding to the 10-min measurement burst starting at 15:00 UTC onMarch 31st (thick line), and an estimate with significant wave-bias,corresponding to the burst starting at 05:00 UTC on April 1st (thin line).
52
CHAPTER 4OBSERVATIONS
4.1 General Overview
At the experiment site, the water depth, averaged over the experiment duration, was
about 5 m (Figure 4-1a). Tidal ranges were mostly around 60 cm with an extreme of 80
cm, recorded close to the end of the experiment. Based on current-meter records
covering up to 30 months (Di Marco and Reid, 1998), and the results during the
2006 experiment in this area (Jaramillo, 2008), the dominant tidal constituents on
the Atchafalaya Shelf were identified to be K1, O1 and M2, with periods of 23.93 hrs,
25.82 hrs, and 12.42 hrs, respectively. Depth variation during this experiment was
controlled mainly by a diurnal signal, and the spectrum has a relatively weak peak near
the semidiurnal frequency, as well (Figure 4-1b). A more comprehensive tidal analysis is
beyond the scope of this study.
A 1-minute segment from the ADV measurements is shown in Figure 4-2. Good
synchronization of the instruments can be seen in all three of the velocity components,
especially in the horizontal velocity records (Figures 4-2a-c). The quality of the velocity
observations is illustrated in Figure 4-2d. Throughout the experiment, along-beam
correlations between successive acoustic returns from the scatterers in the water (Elgar
et al., 2005) stayed above 90% (Figure 4-2d), well within the acceptable limits, where the
lower limit recommended by the manufacturer is 70% (SonTek/YSI, 2001).
The first 4 days of the experiment (Figures 4-3 and 4-4, label 1) were characterized
by relatively calm weather. Winds exceeded 5 m/s only briefly at the beginning of
this period (Figure 4-3a), generating seas of about 0.5-m height (Figure 4-4a), and
decayed to about 3 m/s at the end of this period. Swell height never exceeded 0.5-m,
and remained below 0.3-m for most of this period.
The most energetic event observed during the experiment occurred on March
31st (Figures 4-3 and 4-4, label 2), characterized by steady winds blowing towards
53
North-Northwest (Figure 4-3a). Wind speed reached 10 m/s for about a day on March
31st, forcing a detectable mean Northward-Northwestward current component (reaching
30 cm/s, Figure 4-3d). The atmospheric perturbation was likely local (relatively small
fetch) judging by the large seas generated (up to 1-m height, Figure 4-4a), but relatively
weak swell (about 0.25-m height).
The last part of the experiment (Figures 4-3 and 4-4, label 3) had the characteristics
of an atmospheric front passage, including the typical drop in wind velocity as the front
passed over the site (April 5th), and the rotation of wind and wave direction (onshore to
offshore) followed by the post-frontal wind intensification (Figures 4-3a and 4-3c). Wave
fields associated with this event also suggest a larger fetch: swells were the strongest
recorded during the experiment (Figure 4-3b), reaching up to 0.5-m height (Figure 4-4a)
and their arrival at the site was delayed with respect to the maximum winds.
The valid Reynolds stress estimates (Section 3.2.3) are evaluated with the
calculations based on the quadratic drag relation. The bottom drag coefficient is
calculated from its relation with the bottom stress, τb = ρu2∗ = ρCd
(u2 + v 2), equivalent
to Cd = u2∗/
(u2 + v 2). This can be rewritten separately for North-South and East-West
components as, τN−S = ρCdN−S u√
u2 + v 2, and τE−W = ρCdE−W v√
u2 + v 2 where u and
v are set hereinafter to denote North-South and East-West components, respectively.
Linear least square fits based on the Reynolds stress estimates (Figure 4-5) yield the
average drag coefficients for the entire duration of the experiment: CdN−S =2.3x10-3 with
a correlation coefficient of r 2=0.72 (Figure 4-5a), and CdE−W =2.9x10-3 with a correlation
coefficient of r 2=0.77 (Figure 4-5b), which are both within the standard expectations.
Figure 4-4e shows the response of near-bed flow and suspended sediment
concentration to surface flow forcing. The highest near-bed concentrations of suspended
sediment throughout the experiment were recorded during March 31st, with an average
of about 2 g/L and sometimes exceeding 3 g/L (Figure 4-4e, label 2). In general,
near-bed sediment resuspensions should respond to a number of forcing mechanisms,
54
such as mean current velocity, current direction and divergence, and long-wave energy.
Short waves are typically not included in this list because of the strong attenuation
with depth. In the observations, seas consistently dominate swells in the surface
measurements throughout the experiment (Figure 4-4a). The two frequency bands
contribute almost equally to near-bed RMS orbital velocity (Figure 4-4b). However,
steeper short waves (seas) dominate near-bed accelerations (sometimes exceeding 0.5
m/s2) and exhibit a stronger correlation with sediment concentration values (compare
Figures 4-4c, 4-4d and 4-4e). This suggests that, for water depths characteristic for
this experiment, the short wave band is the dominant forcing mechanism that maintains
near-bed sediment suspensions. This is in agreement with previous observations and
numerical simulations of sand transport (Hoefel and Elgar, 2003; Hsu and Hanes, 2004).
Because the present observations were collected at one location in the horizontal
plane, it is not possible to assess the effect of horizontal advection on the observed
levels of suspended sediment concentration. The observed values of suspended
sediment concentration (Figure 4-4e) do follow the general trends in current velocity and
direction, increasing typically for stronger westward currents (Figure 4-3d). However,
due to the platform location (Figure 2-2), it is unlikely that the site was directly affected
by the sediment discharge at the mouth of the Atchafalaya River. Rather, several
characteristics of the observations seem to support the alternative hypothesis, that (on
the time and energy scales of this experiment) near-bed sediment suspensions are of
local origin and are maintained by local wave-induced turbulence. Bottom core samples
collected before and after the experiment suggest an effectively inexhaustible local
reserve of mobilizable, soft bottom sediment. Suspended sediment concentration values
are well correlated with the wave-induced accelerations (Figure 4-4c). The evolution
of Reynolds stress estimates and suspended sediment concentration (Figure 4-6) is
consistent with the assumption that local turbulence is the main forcing mechanism
driving near-bed suspensions.
55
The strong correlation between Reynolds stresses and suspended sediment
concentration in Figure 4-6 suggests a nearly-monotonic relation between the two
parameters and, implicitly, a weak dependence of the turbulence-sediment processes
on the third parameter – the floc size. This unexpected result could be an expression of
a general weak dependence on the floc size; however, the available data set does not
cover a wide enough range of concentrations to support this. Instead, we adopt here
the conservative interpretation that, for the relatively dilute suspension conditions in
this experiment, floc size variability is small enough to justify neglecting its effects on
turbulence-cohesive sediment interactions.
4.2 The LISST Data Set from 2006
This hypothesis about floc size variation is supported by the available information
about suspended sediment size distribution on the Atchafalaya Shelf, i.e., LISST
observations, and the co-located PC-ADP measurements of waves and currents,
collected in 2006 (Figure 2-2). A comprehensive analysis of the near-bed flow
measurements of the PC-ADP was done by Jaramillo et al. (2009). For the analysis
herein, only the mean current profiles are analyzed to estimate the bottom friction
velocity, and the significant wave heights are calculated from pressure spectrum (with
17 degrees of freedom). Figure 4-7 shows the significant heights of seas and swells
(Figure 4-7a), vertical structure of current speeds (Figure 4-7b), and the estimates
of bottom friction velocity (Figure 4-7c), based on log-law (Section 3.3). For 82% of
the measurement bursts, valid logarithmic layers (r 2 greater than 0.8, Section 3.3)
are obtained. While the vertical range of the estimated logarithmic layers vary (black
“x”s in Figure 4-7b), a logarithmic layer is fit to almost the entire profiling range of
the PC-ADP for majority of the cases. The average r 2 of the valid logarithmic fits is
0.97, with more than 90% of them having r 2 greater than 0.9. Therefore, for this data
set, the log-law assumption describes the near-bed current structure very well. Two
examples of logarithmic fits to the current profiles are shown in Figure 4-8a, together
56
with the corresponding LISST records. Size distribution is given as fractional volume of
particles in µL/L. The distributions for both of these cases have a major peak between
200-230 µm and a smaller peak near 50µm, which is in agreement with the general
trend throughout the experiment, as discussed below.
Throughout the experiment, the size distribution exhibits two modes (Figure 4-9a).
The dominant mode is within the 150-300 µm range, with the peak position between
200-230 µm. A weaker mode, which is centered around 50 µm, appears to be sensitive
to the amount of suspended sediment in the water column. The position of the peaks
of two modes are remarkably stable, given the varying bottom turbulence (Figure 4-9b)
and surface wave conditions (Figure 4-9c). The dominant grain-size is in the range of
coarse sand. However, because there is not coarse sand in this region, it is likely to
represent the floc mode. The weaker mode is probably due to either coarse silt particles
in suspension or smaller flocs.
To examine the bottom turbulence and cohesive sediment interaction in more
detail, and the effects and uncertainties due to floc size (i.e., settling velocity), a bottom
boundary layer model for cohesive beds is used in the next section. The fundamental
assumption above is critical in allowing for numerical analysis, as existing numerical
models of flow-sediment interaction cannot yet handle floc aggregation-breakup
processes. In this study, the floc size will be assumed to vary on a time scale longer
than the 10-min duration of a measurement burst. Therefore, it will be approximated as
constant for the duration of a measurement burst, but it will be assumed to vary slowly
from burst to burst.
57
03/28 04/02 04/074.4
4.6
4.8
5
5.2
5.4D
epth
(m
)
time (month/day 2008)
(a)
0 1 2 3 40
0.01
0.02
0.03
Spe
ctra
lde
nsity
(m
2 /cpd
)
Frequency (cpd)
(b)
Figure 4-1. Tidal variations during the experiment. (a) Time evolution of the mean waterdepth at the measurement location; (b) Fourier spectrum of the depthvariations, with 10 degrees of freedom.
58
0
0.2
0.4N
orth
−co
mpo
nent
of v
eloc
ity (
m/s
)(a)
−0.2
−0.1
0
0.1
Eas
t−co
mpo
nent
of v
eloc
ity (
m/s
)
(b)
−0.1
0
0.1
Ver
tical
−co
mpo
nent
of v
eloc
ity (
m/s
)
(c)
5 10 15 20 25 30 35 40 45 50 55 6090
95
100
Cor
rela
tion
(%)
(d)
time (second)
Figure 4-2. A one-minute segment of the flow velocity, recorded by the ADV array. (a)North component of velocity; (b) East component of velocity; (c) verticalvelocity; and (d) 3-beam average of the ADV signal correlation. Thin andthick lines show the records of ADV-1 at 17 cmab and ADV-2 at 145 cmab,respectively. The measurements correspond to the first minute of the burst,starting at 11:00 UTC on March 30th, 2008.
59
Figure 4-3. Wind, wave, and current conditions on the Atchafalaya Shelf throughout the2-week experiment. Time evolution of: (a) wind speed and direction(color-coded); (b) spectral density, normalized between 0 and 1, based onthe ADCP measurements; (c) peak wave propagation direction for eachfrequency band in the power spectrum, based on the ADCP measurements(directions are shown only for frequencies with spectral density greater than10-2 m2/(Hz.deg)); and (d) current speed and direction at 17 cmab (ADV-1).The direction indicated is the direction of the flow (e.g., N means flowingnorthward).
60
0
0.5
1
Wav
ehe
ight
(m
) (a)
0
0.05
0.1
0.15
RM
S
velo
city
(m
/s) (b)
0
0.2
0.4
RM
Sac
cele
ratio
n (m
/s2 )
(c)
0
0.02
0.04
0.06
Ste
epne
ss (d)
03/27 03/29 03/31 04/02 04/04 04/060
1
2
3
Sus
pend
ed s
edim
ent
conc
entr
atio
n (g
/L)
time (month/day 2008)
(e)
1 2 3
Figure 4-4. Bulk spectral characteristics of waves, and the measurements of suspendedsediment concentration. Time evolution of: (a) significant wave height at thesurface; (b) RMS velocity estimated at 17 cmab (ADV-1); (c) RMSacceleration estimated at 17 cmab (ADV-1); (d) surface wave steepness; and(e) suspended sediment concentration (10-min averages) measurements at12 cmab (OBS-5). In panels a,b,c and d, swell and sea frequency bands aredenoted by a thick line and a thin line, respectively.
61
−50 0 50 100
−0.2
−0.1
0
0.1
0.2
0.3τ
nort
h (P
a)
(a)
ρu
√
u2
+ v2
−100 −50 0−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
τ ea
st (
Pa)
(b)
ρv
√
u2
+ v2
Figure 4-5. Reynolds stress estimates versus the quadratic drag relation. (a)North-South components (CdN−S =2.3x10-3, r 2=0.72); (b) East-Westcomponents (CdE−W =2.9x10-3, r 2=0.77). The straight lines show the linearleast square fits.
62
0
0.1
0.2
0.3
0.4
Rey
nold
sst
ress
(N
/m2 ) (a)
03/27 03/29 03/31 04/02 04/04 04/060
0.5
1
1.5
2
2.5
3
3.5
(b)
Sus
pend
ed s
edim
ent
conc
entr
atio
n (g
/L)
time (month/day 2008)
1 2 3
Figure 4-6. Reynolds stress estimates, and the measurements of suspended sedimentconcentration. Time evolution of: (a) the magnitudes of the Reynolds stressestimates (filled circles and squares denote magnitudes of North-South andEast-West components, respectively); and (b) suspended sedimentconcentration.
63
03/02 03/04 03/06 03/08 03/10 03/12 03/140
0.01
0.02
0.03
0.04
0.05
(c)
Bot
tom
fric
tion
velo
city
(m
/s)
time (month/day 2006)
0
0.5
1(a)
Wav
e he
ight
(m
)D
ista
nce
from
se
nsor
hea
d (m
)
(b)
0.2
0.4
0.6
0.8
1
Cur
rent
spee
d (m
/s)
0
0.2
0.4
Figure 4-7. Wave and current observations during the 2006 experiment. (a) Significantwave height at the surface in the sea (f >0.2 Hz, thin line) and swell (f ≤0.2Hz, thick line) bands; (b) vertical profiles of mean horizontal current speeds(“x”s indicate the elevations where the logarithmic-layer of the profilesreach); and (c) bottom friction velocity estimates based on the log-law.
64
0 0.1 0.2 0.3
0
0.2
0.4
0.6
0.8
Dis
tanc
e ab
ove
botto
m (
m)
(a)
0 0.1 0.2 0.30
0.2
0.4
0.6
0.8
Current speed (m/s)
Dis
tanc
e ab
ove
botto
m (
m)
(c)
0 50 100 150 200 250 300
0
5
10
15
Fra
ctio
nal
volu
me
(µL/
L)
(b)
0 50 100 150 200 250 3000
5
10
15
Particle size (µm)
Fra
ctio
nal
volu
me
(µL/
L)
(d)
Figure 4-8. Examples of logarithmic layer fits, and grain size distributions from the 2006experiment. In panels a and c, circles denote the PC-ADP measurements ofcurrent speeds, and the thick lines show the logarithmic fits to themeasurements outside of the wave boundary layer. In panels b and d, sizedistribution records of LISST are given in fractional volume of each particlesize. Panels a and b correspond to the burst starting at 06:00 UTC on March4th, panels c and d correspond to the burst starting at 12:00 UTC on March8th.
65
Par
ticle
si
ze (
µm)
(a)50
100
150
200
250
300
log 10
(Fra
c.vo
l.)
(µL
/L)
0
1
2
03/02 03/04 03/06 03/080
0.5
1
W
ave
heig
ht (
m)
(c)
time (month/day 2006)
0
0.01
0.02
0.03
Bot
tom
fric
tion
velo
city
(m
/s)
(b)
Figure 4-9. Observations of particle size distribution and wave-turbulence conditions in2006. (a) Fractional volume of each particle size, calculated fromhourly-averaged LISST data; (b) estimates of bottom friction velocity; and (c)significant wave height at the surface in swell (thick line) and sea bands (thinline).
66
CHAPTER 5BOTTOM BOUNDARY LAYER MODELING
5.1 Introduction
Analysis of turbulent flow interacting with cohesive sediments can be strengthened
by numerical modeling, since flow-related parameters, which can not be directly
measured or estimated, can be calculated. The numerical analysis of the observations
herein is based on the one-dimensional bottom boundary layer model developed by Hsu
et al. (2007) for cohesive beds and improved by Hsu et al. (2009). The model assumes
a constant floc size (also constant floc density and settling velocity) and integrates the
two-phase (fluid-sediment) Reynolds-averaged equations based on a turbulent kinetic
energy (TKE, k , equation (3–14)) - dissipation rate of turbulence kinetic energy (ε)
closure. The model is different than a single-phase model in the sense that sediment is
not considered to be passive, but the effect of sediment on fluid turbulence is accounted
for (Hsu and Liu, 2004). The details of the numerical model and the derivations can
be found in Hsu et al. (2007). The governing equations for momentum, sediment
concentration, k − ε balances, and the boundary conditions are cited in Section 5.2
for convenience. Then, numerical simulations are done for the measurement bursts,
which were collected on March 31st 2008 (within event 2 in Figures 4-3, 4-4, and 4-6)
and provided valid Reynolds stress estimates (Sections 3.2.3 and 4.1, Figure 4-6a).
This is the interval when the most energetic flow conditions were observed and these
conditions caused the amount of suspended sediment to reach its maximum values
throughout the experiment.
5.2 Governing Equations
In order to make the presentation of the governing equations simpler, the numerical
coefficients in the k − ε closure are given in advance (Table 5-1), and the overbars that
represent ensemble-averaged quantities in the equations are dropped.
67
5.2.1 Momentum balance
The coordinate system is defined such that x and y are the horizontal directions
and z is the vertical direction normal to the bed. The cohesive fine sediments of
interest were noted to have particle response time of O(0.001 sec), which is very
small compared to the timescale of flow forcing, therefore the sediment is assumed to
follow the fluid velocity (Hsu et al., 2007). Therefore, the two-phase flow velocity can be
described by a single mean flow velocity, i.e., one momentum equation is necessary to
describe the flow. The horizontal momentum equations (equations 1 and 2 in Hsu et al.
(2007)) read:
∂uj
∂t=
−1ρ(1− φ)
∂p∂xj
+1
ρ(1− φ)
[∂τw
jz
∂z+
∂τ sjz
∂z
]+ gj
[(s − 1)φ
1− φ
], (5–1)
where uj is the j-th (j=1,2) horizontal component of velocity along the horizontal
dimension xj , φ is the volume concentration of sediment floc, τjz is the element of the
stress tensor, and gj is the j-th horizontal component of the gravitational acceleration.
The superscripts “s” and “w” denote the quantities corresponding to sediment and
water, respectively, and s = ρm/ρ is the specific gravity of cohesive sediment floc with
density ρm. The right-hand side terms in equation (5–1) describe forcing by free-stream
pressure gradient in the horizontal, momentum transport caused by fluid and sediment
shear stresses (including both laminar and turbulent components), i.e., diffusion, and
transport of momentum by gravity-driven fluid flow. The model assumes a horizontally
uniform bottom boundary layer, therefore the advective acceleration terms are neglected
in this formulation. Due to the mild slope at the experiment site which is about 10-3, the
implementation of the equation (5–1) for this study neglects the terms of gravity-driven
transport of momentum. Horizontal pressure gradient is defined as the prescribed flow
forcing of the bottom boundary layer, i.e., velocity time series which includes oscillatory
forcing and a mean current component as follows (equation 23 in Hsu et al. (2007)):
68
− ∂p∂xj
= ρ∂ �Uj (t)
∂t+ f c
j , (5–2)
where �Uj (t) is the de-meaned time-series of velocity which represents oscillatory forcing
and prescribed into the model directly as input. As addressed in Section 5.4, an iterative
procedure is followed to calculate the forcing f cj (function of the bottom friction velocity
u∗j ), that is required to simulate the measured currents. The fluid shear stress in the
equation (5–1) is calculated as:
τwjz = ρ (ν + νt)
∂uj
∂z, (5–3)
where νt is the eddy viscosity, given as:
νt = Cµk2
ε(1− φ) . (5–4)
Calculations of k and ε are given in Section 5.2.3. The sediment shear stress in equation
(5–1) is calculated as:
τ sjz = ρνr
∂uj
∂z. (5–5)
νr is the relative viscosity, and is calculated as a function of the sediment volume
concentration (Zarraga et al., 2000):
νr = νexp(−2.34φ)(1− φ/φo)3 , (5–6)
where φo is the gelling concentration for mud flocs (Winterwerp and van Kesteren, 2004)
at which a space filling network develops and settling velocity becomes zero.
The vertical velocity shear (e.g., equations (5–3) and (5–5)) is calculated by
following log-law. As the bottom boundary condition, the shear stress at the bed (z = 0)
is calculated as τjz (0) = ρu2∗j .
69
5.2.2 Sediment concentration balance
The evolution of sediment concentration is given as (equation 3 in Hsu et al.
(2007)):
∂φ
∂t= − ∂
∂z
[φ(1− φ)Tp(1− s−1)g − νt
σc
∂φ
∂z+
Tp
ρs
∂τ szz
∂z
], (5–7)
where Tp is the particle response time. The first term on the right-hand side describes
gravitational settling. The second term is the closed form of turbulent mass flux,
which is, in fact, the covariance of concentration fluctuations and vertical velocity
fluctuations, i.e., φ′w ′. The last term in equation (5–7) is the vertical gradient of sediment
intergranular normal stresses, i.e., rheology, which serves as an additional suspension
mechanism. For the low values of suspended sediment concentration measured
(Figures 4-4e and 4-6b), the resuspension effect of rheological stress is assumed to be
negligible. Therefore, suspended sediment concentration is established by a balance of
gravitational settling and turbulent mass flux.
The bottom boundary condition for downward flux is approximated by continuous
deposition. Upward flux at the bottom boundary is determined by defining a resuspension
coefficient (γo), and a critical shear stress to initiate sediment motion (τc ). The
resuspension coefficient controls sediment availability from the bed and has to be
defined, together with the critical shear stress, in the numerical simulation process by
matching measured and simulated suspended sediment concentration values. More
advanced erodibility formulations (e.g., Stevens et al., 2007) are not used in this study,
as they require information about bed erodibility parameters that are not available for this
experiment.
5.2.3 Turbulent kinetic energy balance and balance of turbulent kinetic energydissipation rate
The equations for turbulent kinetic energy (k) and its dissipation rate (ε) are
(equations 9 and 10 in Hsu et al. (2007)):
70
(1− φ)∂k∂t
= νt
[(∂u∂z
)2
+(
∂v∂z
)2]
+∂
∂z
[(ν +
νt
σk
)∂(1− φ)k
∂z
]− (1− φ)ε
− (s − 1) gνt
σc
∂φ
∂z− 2φsk
Tp + TL, (5–8)
and
(1− φ)∂ε
∂t= Cε1
ε
kνt
[(∂u∂z
)2
+(
∂v∂z
)2]
+∂
∂z
[(ν +
νt
σε
)∂(1− φ)ε
∂z
]
−Cε2ε2
k(1− φ) + Cε3
ε
k
[− (s − 1) g
νt
σc
∂φ
∂z− 2φsk
Tp + TL
], (5–9)
where TL is the turbulent eddy timescale. The first three right-hand side terms in
equations (5–8) and (5–9) describe shear-induced turbulence production, diffusion,
and turbulent dissipation. The fourth terms represent the damping effect of suspended
sediment on fluid turbulence by density-induced stratification. This term accounts for
the vertical gradient of suspended sediment concentration, i.e., density throughout
the water column, which reduces vertical turbulent mixing due to buoyancy flux within
the stratified layer. The stratification effects are discussed in detail in Sections 5.4 and
5.5. The last terms in equations (5–8) and (5–9) represent the effect of suspended
sediment on turbulence by fluid-sediment velocity interaction through viscous drag.
For the low values of suspended sediment concentration measured (Figure 4-6b), the
sediment effects on turbulence due to viscous drag are assumed negligible. This leaves
stratification as the only damping effect of sediment in turbulence closure.
At the bottom boundary, a no-flux condition is set for k , and a standard near-wall
approximation is made for ε (Hsu et al., 2007):
∂k∂z
= 0 ; ε =C
34µ k
32
κz. (5–10)
71
5.2.4 Sediment definition
The sediment phase is defined by a primary particle size (Dp) with density
ρs = 2.65ρ, a floc size (D), and the fractal dimension of the floc (nf ), all of which are
independent of time and position. Density (ρm) (Kranenburg, 1994), mass concentration
(c), and settling velocity (ws) of flocs are then calculated as:
ρm = ρ +(
Dp
D
)3−nf
(ρs − ρ) , (5–11)
c = ρsφ
(Dp
D
)3−nf
, (5–12)
ws =(s − 1)D2g
18ν(1− φ)q . (5–13)
In the presentation of the model, q was given equal to 4. The selection of floc size
input is based on the LISST data (Section 4.2), and the results of a flocculation model
(Section 5.3) that uses the Reynolds stress estimates (Section 3.2.3, Figure 4-6a) and
the sediment concentration measurements (Section 4.1, Figure 4-6b) as inputs.
5.3 Analytical Flocculation Model
Winterwerp (1998) proposed a model to calculate an equilibrium floc size and
settling velocity for given turbulence conditions and sediment information, for a constant
fractal dimension of 2. In the model, flocculation is described by a linear combination of
formulations for simultaneous aggregation and floc breakup due to turbulent motion. The
aggregation is formulated by assuming that eddies larger than the Kolmogorov scale
bring particles smaller than the Kolmogorov scale together (Levich, 1962). The fact that
not all collisions will result in flocculation is included in the model through an empirical
coefficient in the aggregation formulation. The effects of collision between particles due
to Brownian motion and overtaking of particles with small settling velocity by those with
large settling velocity, i.e., differential settling, are neglected. Floc breakup by turbulent
72
shear is formulated as a function of the ratio of the shear stress and the floc strength. An
empirical coefficient is implemented in the formulation due to the uncertainties related to
the floc strength. The formulation for simultaneous aggregation and floc breakup is given
as:
dDdt
= kAcaveGD2 − kBG 3/2D2 (D −Dp) , (5–14)
where kA and kB are calibration coefficients, cave is the depth-averaged suspended
sediment concentration, and G is the dissipation parameter, which is a measure of the
turbulent shear in the flow. G is related to the shear stress, such that τ = ρνG in the
viscous regime at the Kolmogorov scale. The terms on the right hand side of equation
(5–14) are growth and breakup terms, which scale with D2 and D3, respectively.
Therefore, the growth term dominates for small flocs, and the breakup term dominates
for large flocs. The equilibrium floc size (De) is obtained for dDdt = 0 as:
De = Dp +kAcave
kB√
G. (5–15)
For details of the derivation, see Winterwerp (1998). Winterwerp (1998) obtained the
calibration coefficients from a series of laboratory experiments. In a settling column,
sediment samples with Dp=4 µm and varying concentration were mixed homogeneously.
Then, a homogeneous turbulence field was generated in the settling column by a grid,
oscillating at varying frequencies, i.e., varying G . Based on the floc sizes estimated
for the sediment samples withdrawn from the column, Winterwerp (1998) obtained the
empirical coefficients in equations (5–14) and (5–15) as kA=14.6 m2/kg and kB=14x103
s1/2/m2.
Through equation (5–15), equilibrium floc sizes are predicted for the simulated
bursts (Section 5.4). The primary particle size, representative for the experiment site,
is taken as Dp=5 µm (Section 2.1). The measured concentrations, and the estimated
Reynolds stresses for these bursts are used as inputs. Equation (5–15) yields floc size
73
values between 73 µm (ρm=1.14 g/cm3) and 335 µm (ρm=1.05 g/cm3), with an average
of 182 µm (ρm=1.07 g/cm3). These floc densities are calculated with nf =2, following the
assumption made by Winterwerp (1998). The calculated floc sizes are consistent with
the dominant peak in the LISST observations of grain-size distribution (Section 4.2,
Figure 4-9a). The reader is cautioned, however, that equation (5–15) is an idealized
model, and has limited applicability in a field study. Therefore, consistent with the LISST
observations (Section 4.2, Figure 4-9a), a representative constant floc size D=200 µm
is used for all the numerical simulations, with a fractal dimension of nf =2.3 (Khelifa and
Hill, 2006). This yields ρm =1.15 g/cm3 (equation (5–11)) for a primary particle diameter
Dp=5 µm (Section 2.1). The resulting sediment settling velocity, based on Stokes’ law
(equation (5–13)), is ws=3.3 mm/s.
5.4 Application
The model domain is set to extend between the bed and the sampling volume
of ADV-2 at 145 cmab. At the top of the domain, the model is constrained by the
de-meaned horizontal velocity measurements. A trial and error process is used to
estimate the components of the friction velocity and resuspension coefficient (γo) for
which the model predictions of the mean flow at 145 cmab and suspended sediment
concentration at 12 cmab (OBS-5 location) match the observations. Previous numerical
studies, based on the model used here (Hsu et al., 2009), show that wave-averaged
values of suspended sediment concentration are not sensitive to the critical shear stress
(τc ), because the variation in the critical shear stress is compensated by the variability of
the resuspension coefficient during the model initialization procedure. The simulations
presented here used a constant critical bottom shear stress of 0.4 Pa, within the range
of 0.05-1.0 Pa investigated by Hsu et al. (2007, 2009). In the numerical experiments
with several τc -γo pairs, the model results did not indicate a sensitivity on τc in this
formulation. The procedure gradually “spins up” the model from a zero-flow state to
a steady state (such that the Reynolds-averaged flow parameters become constant
74
in several consecutive runs with the same set of inputs), achieved typically within
a time duration that scales with the model domain height and eddy viscosity, which
controls the mixing time. For a model domain of 1 m, and νt= 10−4-10−3 m2/s, the time
necessary to reach a steady state is of the order of 103-104 sec in model time. Below,
we discuss numerical simulations for 13 of the most energetic 10-min measurement
bursts, collected on March 31st (within event 2 in Figures 4-3, 4-4, and 4-6) with low
wave-bias Reynolds stress estimates.
Examples of time series of model simulations are shown in Figure 5-1, for the
10-min measurement burst starting at 11:00 UTC on March 31st. Figures 5-1a and b
show the North-component of velocity simulated at the top of the model domain (dashed
line in Figure 5-1a denotes the northward current at 145 cmab, the location of the
sampling volume of ADV-2), and the suspended sediment concentration simulated at the
lower 20 cm of the model domain, respectively. The numerical results match observed
values: suspended sediment concentrations at 12 cmab, the location of the OBS-5
(Figure 5-1c) agree well with “sub-burst” (1-min average) OBS-5 observations (thick line
in Figure 5-1c). This agreement is remarkable given that the model is “spun up” only
through matching the 10-min averaged values, and shows that the model captures the
small-scale trends of the process. The North-component of the Reynolds stress at 17
cmab, the location of the sampling volume of ADV-1 (Figure 5-1d), is of the same order
as the estimate based on the observations (dashed line in Figure 5-1d). A more detailed
analysis of the performance of the model can be found in Hsu et al. (2007, 2009).
Figure 5-2 compares the calculations of the numerical model with measurements
and Reynolds stress estimates. The model reproduces the 10-min averages of
suspended sediment concentration observed at the OBS-5 location (Figure 5-2a)
and the mean currents measured at the locations of the two ADVs (Figures 5-2b and
5-2c). Suspended sediment concentration, vertically-averaged over the model domain,
varies between 0.44-1.41 g/L, well above the threshold 0.1 g/L suggested by Winterwerp
75
(2001) and Winterwerp (2006) for sediment-induced stratification effects to modify
the turbulence field. The agreement between the model and the estimated Reynolds
stresses (Figures 5-2d and 5-2e), albeit less good, is encouraging, considering the
involved processing and the multi-layered hypotheses on which both the model
results and the estimates are separately based. The model agrees better with the
estimates of the West components of the Reynolds stress (Figure 5-2e) probably due to
a higher mean to fluctuation ratio for the West velocity component (Figure 5-2f), usually
translating into a lower wave-bias estimate (Feddersen and Williams, 2007).
The overall model-observations agreement justifies a more detailed analysis of
the model representation of the bottom turbulence-cohesive sediment interaction.
The vertical structures of the four terms in the governing TKE equation (5–8) are
calculated. Figure 5-3a shows Reynolds-averaged vertical profiles of four terms in the
governing TKE equation (5–8) calculated by the model for a 10-min measurement burst
characterized by one of the highest suspended sediment concentrations (2.83 g/L)
observed during the experiment. In the upper 10 cm of the model domain, vertical shear
due to velocity is relatively small and therefore TKE production is mainly due to turbulent
diffusion (stars in Figure 5-3a) balanced by sediment-induced stratification (“x”s in
Figure 5-3a). Below this upper layer of the model domain, the shear-induced turbulence
production (circles in Figure 5-3a) becomes the dominant turbulence generating term
due to increasing bottom boundary effect. At 100 cmab, sediment-induced stratification
(“x”s in Figures 5-3a and 5-3b) is the dominant turbulence damping term in most of
the cases, however, shear-induced turbulence production is balanced by the combined
effects of sediment-induced stratification and turbulent dissipation (Figures 5-3a and
5-3b). At 30 cmab, these two effects are of the same order of magnitude (“x”s and
squares in Figures 5-3a and 5-3c). Below 10 cmab, just above the bed, the TKE
balance is dominated by shear-induced turbulence production and turbulent dissipation.
Although overall sediment-induced stratification damping decreases with increasing
76
distance above the bed, because the other terms are decreasing faster, paradoxically,
the role of stratification in the TKE balance becomes more important with increasing
distance above the bed. This suggests that, for the conditions of this experiment,
sediment-induced stratification term can be neglected in the wave boundary layer, but is
essential within the tidal boundary layer. Compared to the other three terms in the TKE
balance, vertical variation of sediment-induced stratification is smaller, as it varies within
two orders of magnitude throughout the model domain (Figure 5-3a).
One quantity of interest that the numerical model calculates is the turbulent mass
flux term φ′w ′ (equation A11 in Hsu et al. (2007)) in the sediment concentration
balance (5–7). In principle, this quantity could be estimated from the observations
by calibrating the acoustic backscatter records of an ADV with co-located suspended
sediment concentration measurements (e.g., Fugate and Friedrichs, 2002). However,
uncertainties due to application of this method in the presence of waves, and not having
the ADV-1 and the OBS-5 sampling at the same vertical level, make it impossible to
control the errors of such an estimate.
5.5 Model Sensitivity to Floc Size
To investigate the sensitivity of the model to floc size, additional numerical
simulations were carried out for one burst, with floc sizes of D=150 µm (ρm=1.18
g/cm3, ws =2.2 mm/s), and D=300 µm (ρm=1.12 g/cm3, ws=5.9 mm/s), while keeping
all other parameters constant (the resuspension coefficient γo varies in each test
run to match the model and the measured suspended sediment concentration). In
the LISST measurements (Figure 4-9a), the interval [150,300] µm brackets the main
distribution mode and has a time-averaged probability of 85%. The results (Figure 5-4)
suggest that decreasing floc size (i.e., decreasing settling velocity) decreases near-bed
suspended sediment concentration, produces a well-mixed concentration profile (Figure
5-4a), and increases the total amount of suspended sediment. The corresponding
vertically-averaged suspended sediment concentration values are 1.37 g/L for 150 µm,
77
1.15 g/L for 200 µm, and 0.96 g/L for 300 µm. This is consistent, for example, with the
saturation concentration proposed by Winterwerp (2001) (equation 3 therein). Mean
currents are largely insensitive to floc size variation between 150 µm and 200 µm flocs,
however, 300 µm flocs result in a slight over-prediction of currents at 17 cmab (Figures
5-4b and c). The vertical structure of Reynolds stresses is not modified significantly
and, therefore, not shown here due to overlapping vertical profiles. To see the effect
of floc size variation on stratification effects, the gradient Richardson number (Ri )
associated with sediment-induced stratification is calculated. This is the ratio between
the stratification term and shear-induced turbulence production term in equation (5–8):
Ri = −(s−1)g
σc
∂φ∂z[(
∂u∂z
)2 +(
∂v∂z
)2] . (5–16)
The vertical structure of the gradient Richardson number is similar for all three
floc sizes tested (Figure 5-4d). It exceeds the critical value of 0.25 above a height
of 5-20 cmab (depending on the floc size), and then increases slowly with height
above the bed, slightly faster for the smaller floc size (i.e., for higher total amount of
suspended sediment). Relatively stratified profile near bed, obtained with larger flocs,
causes stratification to be important, i.e., Ri >0.25, in a wider domain (dashed lines in
Figures 5-4a and d). This increase in turbulence damping enhances mean current flow
throughout the model domain (dashed lines in Figures 5-4b and c) which is known as
drag reduction. The higher values of Ri at the top of the model domain are due to the
boundary condition of zero velocity shear at the top of the domain.
78
Table 5-1. Numerical coefficients in the k − ε closureCµ Cε1 Cε2 Cε3 σk σε σc
0.09 1.44 1.92 1.20 1.00 1.30 1.00
79
0
0.2
0.4
Nor
th−
com
pone
ntof
vel
ocity
(m
/s)
(a)
Dis
tanc
eab
ove
bed
(cm
)
(b)5
10
15
20
log 10
(SS
C)
(g/L
)
0.4
0.8
1.2
2
2.5
3
3.5
4
Sus
pend
ed s
edim
ent
conc
entr
atio
n (g
/L)
(c)
0 1 2 3 4 5 6 7 8 9 100.1
0.15
0.2
0.25
0.3
Nor
th−
com
pone
ntof
Rey
nold
s st
ress
(P
a)
time (minute)
(d)
Figure 5-1. An example of time-series in the model simulations. (a) North-component ofvelocity at 145 cmab (ADV-2 location); (b) suspended sedimentconcentration at the lower 20 cm above bed; (c) suspended sedimentconcentration at 12 cmab (OBS-5 location); and (d) the North-component ofReynolds stress at 17 cmab (ADV-1 location) for the 10-min measurementburst which started at 11:00 UTC on March 31st. In panel (a), the dashedline represents the North-component of the (10-min averaged) currentmeasured at 145 cmab. In panel (c), thick lines denote the 1-min time spansof the OBS-5 measurements of suspended sediment concentration. In panel(d), the dashed line marks the value of the (10-min averaged) estimate forthe North component of Reynolds stress at ADV-1.
80
1 2 30.5
1
1.5
2
2.5
3
3.5
Sus
pend
ed s
edim
ent
conc
entr
atio
n−m
odel
(g/
L)
Suspended sediment concentration−meas (g/L)
(a)
0.1 0.2 0.3 0.4
0.1
0.2
0.3
0.4
|u| m
odel
(m
/s)
|u|meas
(m/s)
(b)
0 0.2 0.40
0.1
0.2
0.3
0.4
0.5
|v| m
odel
(m
/s)
|v|meas
(m/s)
(c)
0 0.2 0.40
0.1
0.2
0.3
0.4
|τu−
mod
el| (
Pa)
|τu−est
| (Pa)
(d)
0 0.2 0.40
0.1
0.2
0.3
0.4
0.5|τ
v−m
odel
| (P
a)
|τv−est
| (Pa)
(e)
0.06 0.08 0.10.05
0.1
0.15
0.2
0.25
0.3
σx (m/s)
|x| (
m/s
)
(f)
Figure 5-2. Comparison of observations and model results. (a) Suspended sedimentconcentration (at 12 cmab, the location of OBS-5, Figure 2-5); (b) Northcomponent of current velocity (“x” – 17 cmab, location of ADV-1; circles –145 cmab, location of ADV-2, Figure 2-5); (c) West component of currentvelocity (same symbols as in b); (d) North component of Reynolds stress at17 cmab; (e) West component of Reynolds stress at 17 cmab; (f) mean vs.standard deviation of velocity measurements at 17 cmab (crosses – Northcomponent; stars – West component).
81
10−7
10−6
10−5
10−4
10−3
0
20
40
60
80
100
120
140(a)
D
ista
nce
abov
e be
d (c
m)
Contribution to TKE balance (m2/s3)
0 3 6 9 12 15 18 21
10−6
10−5
10−4
Con
trib
utio
n to
TK
E b
alan
ce (
m2 /s
3 )
(c)
time (hour on March 31st)
10−7
10−6
10−5
Con
trib
utio
n to
TK
E b
alan
ce (
m2 /s
3 )
(b)
Figure 5-3. Analysis of the model representation of the turbulent kinetic energy. (a)Vertical structures of the four terms in the turbulent kinetic energy (TKE)balance (equation (5–8)) for the 10-min measurement burst which started at11:00 UTC on March 31st; (b) TKE balance at 100 cmab; and (c) at 30cmab, vs. time, for all modeled cases. In the order in which they appear inthe right-hand side of equation (5–8), the terms are represented by: circles –shear-induced turbulence production; stars – turbulent diffusion, squares –turbulent dissipation; and “x” – sediment-induced stratification.
82
0 5 10 150
20
40
60
80
100
120
140
(a)
D
ista
nce
abov
e be
d (c
m)
Suspended sediment concentration (g/L)
0 0.1 0.2 0.3
(b)
North−componentof velocity (m/s)
0 0.2 0.4
(c)
West−componentof velocity (m/s)
10−1
100
101
(d)
Gradient Richardsonnumber
Figure 5-4. Effect of varying floc size D on the model calculations. Vertical structures of:(a) suspended sediment concentration; (b) North- and (c) West-componentsof the mean current; and (d) gradient Richardson number (thick line – 150µm; thin line – 200 µm; dashed line – 300 µm). Observations are marked bycircles. In panel d, the gray line corresponds to the critical Richardsonnumber value of 0.25. The numerical simulations correspond to the 10-minmeasurement burst starting at 11:00 UTC on March 31st (see Figure 5-3a).
83
CHAPTER 6CONCLUSION
The interaction between bottom turbulence, suspended sediment concentration, and
sediment floc size in cohesive sedimentary environments (e.g., Dyer, 1989; Winterwerp,
1998) was investigated by analyzing field observations and using a bottom boundary
layer model. The data sets were collected during a large scope experiment, conducted
on the muddy Atchafalaya inner shelf, Louisiana, in 2008. During the event which was
focused on in this study, wave energy was moderate (significant height of order of 1 m),
currents were mostly westward and reached speeds of 30 cm/s near bed, and sediment
suspension was relatively dilute with an average concentration of about 2 g/L near bed.
Together with the calculation of wave and current parameters, near-bed Reynolds
stresses were estimated using the differencing-filtering method (Trowbridge, 1998; Shaw
and Trowbridge, 2001; Feddersen and Williams, 2007), which is a challenging task
in wave-energetic environments. In this study, Reynolds stress estimates allowed the
intervals, when bottom turbulence-suspended cohesive sediment interaction was likely
to take place, to be designated, merely by analyzing the observations, even before using
the boundary layer model. These estimates played a key role also in evaluation of the
available information about floc size, as discussed below.
Several elements of the data concur in supporting a weak floc size variability during
the experiment: suspended sediment concentrations show a strong correlation with
short-wave near-bed accelerations, and at the same time, with estimated Reynolds
stresses. A LISST data set, which was collected in the vicinity of the site in 2006,
was evaluated as an independent source of information on suspended sediment size
distribution. Bottom friction velocity was estimated from the fit of logarithmic layers to
the current profile measurements that were co-located with these LISST records. Size
distributions of suspended sediments estimated by LISST, showed remarkable stability
for wave conditions similar to those of the experiment which was focused on here, and
84
for varying turbulent flow conditions. The major floc mode in the 2006 LISST data was
also consistent with the equilibrium floc sizes (Winterwerp, 1998), calculated using
Reynolds stress estimates and suspended sediment concentration measurements in
2008. It is hard to assess, based on this data set, whether the experimental conditions
were simply not energetic enough to lead to higher concentrations of suspended
sediment, or if a more fundamental physical process might be involved. However, for this
experiment site, and these wave-current conditions, the simplifying assumption that the
floc size is approximately constant seems justified.
While the sediment concentrations recorded were probably not high enough
to modify grain geometry, the values observed are expected to result in flow-field
modifications due to sediment-induced stratification (Winterwerp, 2001, 2006). The
constant floc size assumption allowed the investigation of these effects by using
numerical tools to analyze the observations. Here, a one-dimensional bottom boundary
layer model that was developed for small-scale turbulence and sediment transport
processes on cohesive beds was used (Hsu et al., 2007). The constant floc size input
of the model was obtained from the size distributions in the LISST records. The vertical
structures of flow and suspended sediment concentration were reconstructed. The
model was validated using several test parameters: mean flow, suspended sediment
concentration, and Reynolds stresses. Flows and suspended sediment concentrations
were reproduced accurately; modeled Reynolds stresses matched the estimates based
on observations, with better agreement for cases of strong currents and weak waves,
that were consistent with the low wave-bias formulation of Feddersen and Williams
(2007).
Model simulations show that the turbulence-damping effect of suspended sediment
(i.e., sediment-induced stratification) is important for the turbulence kinetic energy (TKE)
balance throughout the model domain, except in the vicinity of the wave boundary layer,
where shear-induced turbulence production and turbulent dissipation dominate the
85
balance. In the tidal boundary layer, sediment-induced stratification is of the same order
of magnitude as turbulent dissipation, even without a detectable lutocline (steep gradient
of sediment concentration). In terms of turbulence production, shear-induced turbulence
production appears to dominate over turbulent diffusion in most of the model domain.
The effect of floc size on the numerical results was investigated in the quasi-equilibrium
sense, by carrying out numerical tests with floc size values (within the limits of
observations) under otherwise identical conditions. These tests resulted in virtually
identical Reynolds stress structures. However, smaller floc sizes in the model caused an
increase in the total amount of suspended sediment in the water column (Winterwerp,
2001), that, in turn, resulted in a lower gradient Richardson number increase in the wave
boundary layer.
The results of this study suggest that sediment-induced stratification is an important
mechanism affecting hydrodynamics in muddy environments and therefore has to be
accounted for in practical 3-D circulation models for continental shelf areas. The model
limitations, the dilute concentrations in these observations, i.e., the absence of fluid mud
layers and relatively small contribution of sediment-induced stratification in turbulence
damping near bed, are not allowing an in-depth investigation of mud-induced damping
of surface wave energy. To achieve a better understanding of bottom turbulence -
cohesive sediment interaction and its large scale implications on ocean circulation and
wave dynamics, analysis on a wider range of conditions (more energetic wave action,
lutocline formation, higher suspended sediment concentration values measured) is
essential. The implementation of flocculation models into bottom boundary layer models
(e.g., Winterwerp, 2002) is also likely to improve the modeling capabilities in cohesive
sedimentary environments. Another important contribution to the understanding of
these processes in combined wave-current flow would be to compare model results
with a more detailed vertical structure of suspended sediment concentration and current
(and even Reynolds stresses which could be possible with a current-profiler) near
86
the bed. This is the topic of an ongoing study in which the backscatter of acoustic
profilers (e.g., ABS and PC-ADP, Figure 2-3) is being calibrated to estimate suspended
sediment concentration throughout the profiling range of the instrumentation (Sahin
et al., 2010). In terms of data analysis methods, direct estimates of the turbulent mass
flux term (φ′w ′), that could be based on the approach of the Reynolds stress estimation
method used herein, would also be useful to evaluate the related closure schemes of the
boundary layer models (equation (5–7)).
The overall findings of this study are summarized in Safak et al. (2010a).
87
APPENDIX ADIRECT ESTIMATION OF REYNOLDS STRESSES
In this section, derivation of direct estimates of Reynolds stress based on
measurements from one sensor, and from two sensors are detailed. The calculations
are shown for a 2-D coordinate system, let u and w be the horizontal and vertical
components of velocity, respectively. Therefore, the turbulent covariance term of interest
is u′w ′. u and w can be decomposed as follows:
u = u + ~u + u′ , (A–1)
and
w = w + ~w + w ′ , (A–2)
where an overbar denotes a time-averaged quantity, i.e., mean current, a tilde denotes
wave-induced fluctuations, and a prime denotes turbulent fluctuations, respectively. Let
U and W be the horizontal and vertical components of velocity, measured in instrument
coordinates and accounting for their small misalignment θ from the model coordinate
system:
U = (u + u + u′) cos (θ) + (w + w + w ′) sin (θ) ≈ u + u + u′ + θ (w + w + w ′) , (A–3)
and
W = (w + w + w ′) cos (θ)− (u + u + u′) sin (θ) ≈ w + w + w ′ − θ (u + u + u′) . (A–4)
For a single-sensor estimate of Reynolds stress, u′w ′ can be estimated from the
covariance of the horizontal and vertical measurements, i.e., Cov (U, W ) which is equal
to:
88
Cov(U, W ) =⟨[
U − U] [
W −W]⟩
, (A–5)
where 〈〉 denotes the expected value operator. Substituting demeaned velocities from
equations (A–3) and (A–4) gives:
Cov(U, W ) = 〈[u + u′ + θ (w + w ′)] [w + w ′ − θ (u + u′)]〉 , (A–6)
Cov(U, W ) = 〈uw + uw ′ − θuu − θuu′ + u′w + u′w ′ − θu′u − θu′u′ + θw w + θw w ′
−θ2uw − θ2u′w + θw ′w + θw ′w ′ − θ2uw ′ − θ2u′w ′⟩ . (A–7)
By assuming θ is small, and therefore, by keeping only terms up to order O(θ):
Cov(U, W ) = 〈uw + uw ′ − θuu − θuu′ + u′w + u′w ′ − θu′u − θu′u′ + θw w + θw w ′
+θw ′w + θw ′w ′〉 . (A–8)
A critical step at this point is to assume that wave-induced fluctuations and turbulence-induced
fluctuations are statistically orthogonal (Kitaigorodskii and Lumley, 1983). In other
words, waves and turbulence are assumed to be uncorrelated. Therefore, while
calculating the expected value at the right hand side, terms that include multiplication of
a wave-induced fluctuation and a turbulence-induced fluctuation are assumed to have
zero mean. This gives:
Cov(U, W ) = 〈uw − θuu + u′w ′ − θu′u′ + θw w + θw ′w ′〉 , (A–9)
which can be rewritten as:
Cov(U, W ) = u′w ′ + uw − θ(
uu − w w)− θ
(u′u′ − w ′w ′) . (A–10)
89
This is the raw estimate of Reynolds stress based on a single sensor’s measurements,
as given in equation (3–24). The first term is the turbulent covariance term of interest.
As explained in Section 3.2.3, assuming that all the components of the turbulent
covariance tensor are of the same order of magnitude (Tennekes and Lumley, 1972), the
last term in (A–10) is O(θ) with respect to the quantity of interest. However, the second
and third terms in equation (A–10) are wave biases and have to be reduced.
Shaw and Trowbridge (2001) presented only the essential steps of the three
versions of the two-sensor method, i.e., by differencing horizontal component, by
differencing vertical component, and by differencing both components. Here, the
two-sensor method that calculates the difference of horizontal components is detailed.
The calculations for the other two versions are similar and the basic steps can be
followed from Shaw and Trowbridge (2001). In the 2-D coordinate system presented
above, the horizontal and vertical velocity measurements from two sensors can be
written as:
U1 ≈ u1 + w1θ1 ; W1 ≈ w1 − u1θ1 , (A–11)
U2 ≈ u2 + w2θ2 ; W2 ≈ w2 − u2θ2 . (A–12)
Subscript “1” is set to refer to the sensor at which the Reynolds stress is estimated.
Since waves and turbulence are assumed to be uncorrelated (Kitaigorodskii and Lumley,
1983), the covariance of the horizontal velocity difference between the two sensors,
and the vertical velocity at the sensor of interest, can be decomposed into its wave and
turbulence components:
Cov(�U, W1) = Cov(� ~U, ~W1) + Cov(�U ′, W ′1) , (A–13)
90
where � denotes differencing operator (herein, subtraction of velocity measurements at
sensor 2 from those at sensor 1). The turbulence component is given as:
Cov(�U ′, W ′1) =
⟨[�U ′ − �U ′] [
W ′1 −W ′
1
]⟩, (A–14)
Cov(�U ′, W ′1) =
⟨[U ′
1 − U ′2 − U ′
1 − U ′2
] [W ′
1 −W ′1
]⟩. (A–15)
By definition, turbulence-induced fluctuations have a zero mean, which simplifies (A–15)
as:
Cov(�U ′, W ′1) = 〈[U ′
1 − U ′2] W ′
1〉 , (A–16)
Cov(�U ′, W ′1) = 〈[u′1 + w ′
1θ1 − u′2 − w ′2θ2] [w ′
1 − u′1θ1]〉 , (A–17)
Cov(�U ′, W ′1) =
⟨u′1w ′
1 − θ1u′1u′1 + θ1w ′1w ′
1 − θ21u′1w ′
1 − u′2w ′1 + θ1u′1u′2
−θ2w ′1w ′
2 + θ1θ2u′1w ′2〉 . (A–18)
By keeping only terms upto order O(θ):
Cov(�U ′, W ′1) = 〈u′1w ′
1 − θ1 (u′1u′1 − w ′1w ′
1)− u′2w ′1 + θ1u′1u′2 − θ2w ′
1w ′2〉 , (A–19)
which can be rewritten as:
Cov(�U ′, W ′1) = u′1w ′
1 − θ1(
u′1u′1 − w ′1w ′
1
)− u′2w ′1 + u′1u′2θ1 − w ′
1w ′2θ2 . (A–20)
The first term on the right-hand side is the quantity of interest. The second term is an
order of magnitude smaller than the quantity of interest (Tennekes and Lumley, 1972).
91
The last three terms describe cross-sensor turbulence correlations, which bias the
Reynolds stress estimates. The most significant of these terms is u′2w ′1, because the
other two terms are O(θ). In order to reduce the effect of this turbulence bias term, the
distances between the measurement points and the bed need to be adjusted (Section
3.2.3). Assuming that the experimental setup is prepared properly, Cov(�U ′, W ′1) yields
an estimate of the quantity of interest, with reduced wave bias compared to the single
sensor estimate (Equation (A–10)).
Wave component in equation (A–13) is calculated as:
Cov(� ~U, ~W1) =⟨[
� ~U − � ~U] [
~W1 − ~W1
]⟩, (A–21)
Cov(� ~U, ~W1) =⟨[
~U1 − ~U2 − ~U1 − ~U2
] [~W1 − ~W1
]⟩. (A–22)
By definition, wave-induced fluctuations have a zero mean, this yields:
Cov(� ~U, ~W1) =⟨[
~U1 − ~U2
]~W1
⟩. (A–23)
Velocities in instrument coordinates are substituted through equations (A–11) and
(A–12):
Cov(� ~U, W1) = 〈[~u1 + ~w1θ1 − ~u2 − ~w2θ2] [ ~w1 − ~u1θ1]〉 , (A–24)
Cov(� ~U, W1) =⟨
~u1 ~w1 − θ1~u1~u1 + θ1 ~w1 ~w1 − θ21~u1 ~w1 − ~u2 ~w1 + θ1~u1~u2 − θ2 ~w1 ~w2 + θ1θ2~u1 ~w2
⟩.
(A–25)
Keeping only the terms up to order O(θ) gives:
Cov(� ~U, W1) = 〈~u1 ~w1 − θ1~u1~u1 + θ1 ~w1 ~w1 − ~u2 ~w1 + θ1~u1~u2 − θ2 ~w1 ~w2〉 , (A–26)
92
which can be compiled as:
Cov(� ~U, ~W1) = ~w1�~u + θ1( ~w1� ~w − ~u1�~u) + �θ( ~w 21 − ~w1� ~w ) . (A–27)
At a stationary point in the horizontal plane, a sinusoidal wave of height H and
radian frequency ω at a water depth h induce the following horizontal and vertical orbital
velocities:
~u(t) =Hω
2cosh(kw z)sinh(kw h)
cos (ωt) , (A–28)
~w (t) = −Hω
2sinh(kw z)sinh(kw h)
sin (ωt) , (A–29)
where z is the height above the bed, and kw is the wavenumber corresponding to ω at
depth h. Vertical gradients of these orbital velocities are calculated as:
∂~u(t)∂z
=Hωkw
2sinh(kw z)sinh(kw h)
cos(ωt) = ~u(t)kw tanh(kw z) , (A–30)
∂ ~w (t)∂z
= −Hωkw
2cosh(kw z)sinh(kw h)
sin (ωt) =~w (t).kw
tanh(kw z). (A–31)
Vertical gradients are represented as first order expansions (Shaw and Trowbridge,
2001) for vertical separation �z = r :
�~u = r ~ukw tanh(kw z) ; � ~w =r ~w kw
tanh(kw z). (A–32)
Substituting these terms into equation (A–27) gives:
Cov(� ~U, ~W1) = ~w1r ~u1kw tanh(kw z) + θ1
(~w1
r ~w1kw
tanh(kw z)− ~u1r ~u1kw tanh(kw z)
)
+�θ
(~w 2
1 − ~w1r ~w1kw
tanh(kw z)
), (A–33)
93
Cov(� ~U, ~W1) = rkw tanh(kw z) ~u1 ~w1 + θ1
(~w 2
1rkw
tanh(kw z)− rkw tanh(kw z)~u2
1
)
+�θ ~w 21
(1− rkw
tanh(kw z)
), (A–34)
Cov(� ~U, ~W1) =rz
[kw ztanh(kw z) ~u1 ~w1 + θ1
zr
(~w 2
1rkw
tanh(kw z)− rkw tanh(kw z)~u2
1
)
+�θ ~w 21
zr
(1− rkw
tanh(kw z)
)]. (A–35)
Near the bottom, i.e., for small z , tanh(kw z) ≈ kw z . This yields:
Cov(� ~U, ~W1) =rz
[k2
w z 2 ~u1 ~w1 + θ1 ~w 21 − k2
w z 2θ1~u21 + �θ ~w 2
1zr− �θ ~w 2
1
]. (A–36)
Near the sea bed, the first and third terms on the right hand side are expected to be
larger than the other three terms. Therefore, the wave component of the Reynolds stress
estimate can be approximated as:
Cov(� ~U, ~W1) ≈ rk2w z
[~u1 ~w1 − θ1~u2
1
]. (A–37)
Applying the differencing to the horizontal velocity components reduces the wave bias in
the one-sensor estimate (equation (A–10)) with a factor of rk2w z .
94
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BIOGRAPHICAL SKETCH
Ilgar Safak was born in 1982 in Ankara, Turkey. He graduated from T.E.D. Ankara
College in 1999. He earned his B.S. in Civil Engineering in 2003, and M.S. in Coastal
Engineering in 2006, both from Middle East Technical University in Ankara. During his
M.S. studies, he worked on physical modeling of performance of floating breakwaters,
and numerical modeling of wind-wave induced longshore sediment transport. Since
August 2006, he has been working with Alex Sheremet at the University of Florida.
His research project is on the interaction of hydrodynamics and sediment transport
processes in wave-energetic muddy environments. After he earns his Ph.D. degree, he
would like to enrich his understanding of coastal and oceanographic processes.
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