NEARSHORE WAVE AND SEDIMENT PROCESSES: AN EVALUATION OF
STORM EVENTS AT DUCK, NC
By
JODI L. ESHLEMAN
A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
2004
Copyright 2004
by
Jodi L. Eshleman
This thesis is dedicated to my parents, who provided unconditional support throughout my entire education.
ACKNOWLEDGMENTS
I would like to thank the Army Corps of Engineers Field Research Facility (FRF)
in Duck, NC for providing the data used in this investigation. I extend my greatest
appreciation to the staff at the FRF, who gave me the opportunity to gain some valuable
field experience and were always willing to offer advice and encouragement. I
acknowledge specifically Kent Hathaway, Chuck Long, and Bill Birkemeier, whose input
was vital to this research. I thank all of the FRF for spending countless hours helping me
with everything from analyzing data through interpretation. I would also like to thank
Rebecca Beavers for taking the time to provide additional sediment data.
I thank my supervisory committee chair (Dr. Robert G. Dean) for his continual
support and encouragement throughout this process, and for always finding time for my
questions and concerns. I thank Dr. Robert Thieke for providing the teaching
assistantship that allowed me to continue this research. I also thank Dr. Robert Thieke
and Dr. Andrew Kennedy for serving on my supervisory committee. I also thank Jamie
MacMahan: his patience and insight were invaluable assets to this investigation.
iv
TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................................................................................. iv
LIST OF TABLES............................................................................................................ vii
LIST OF FIGURES ......................................................................................................... viii
ABSTRACT....................................................................................................................... xi
CHAPTER 1 INTRODUCTION ........................................................................................................1
Study Location, Characteristics, and Instrumentation..................................................2 Geographic Location .............................................................................................2 Wave and Weather Conditions..............................................................................3
Bipod Instrumentation ..................................................................................................4 Chapter Contents ..........................................................................................................7
2 GENERAL NEARSHORE CHARACTERISTICS .....................................................8
Introduction...................................................................................................................8 Quality Control ...........................................................................................................10
Data Screening.....................................................................................................10 Representative Data.............................................................................................11
Analysis ......................................................................................................................11 Current Influence.................................................................................................12
Bipod Depth (m) .........................................................................................................13 Bottom Change (cm)...................................................................................................13 Current (cm/s).............................................................................................................13
Wave Influence....................................................................................................17 Combined Waves and Currents ...........................................................................19 Current Direction.................................................................................................19 Wind ....................................................................................................................20 Variance of Total Current Acceleration ..............................................................21
v
3 SEDIMENTS..............................................................................................................24
Introduction.................................................................................................................24 Analysis ......................................................................................................................26
Sediment Characteristics .....................................................................................26 Previous sediment data.................................................................................26 Sonar evaluation...........................................................................................26
Velocity Profile Calculations ..............................................................................29 Critical shear velocity...................................................................................30 Error estimates..............................................................................................33 Combined wave-current influence ...............................................................34 Apparent hydraulic roughness......................................................................37
4 WAVE TRANSFORMATION IN THE NEARSHORE ...........................................42
Introduction.................................................................................................................42 Analysis ......................................................................................................................44
Development of Analytical Spectrum .................................................................44 Dataset .................................................................................................................45 Development of Directional Spectrum from the Data.........................................48 Determination of m Values .................................................................................51
Comparison of Fourier coefficients..............................................................52 Two-sided nonlinear fit ................................................................................53
Comparison of Data to Linear Wave Theory Calculations .................................57 Wave direction .............................................................................................60 Refracted m values .......................................................................................62 Wave height comparisons ............................................................................65 Energy flux comparisons..............................................................................66 Friction factor ...............................................................................................68
Reynolds Stresses ................................................................................................72 Discussion...................................................................................................................74
5 CONCLUSIONS ........................................................................................................77
LIST OF REFERENCES...................................................................................................81
BIOGRAPHICAL SKETCH .............................................................................................85
vi
LIST OF TABLES
Table page 2-1 Erosion events of 3 cm or greater.............................................................................13
2-2 Event-Based comparison of erosion events of 3 cm or greater................................14
4-1 Theoretical Fourier coefficients for different m values ...........................................46
4-2 Measured mean wave directions at peak frequency.................................................51
4-3 Average % energy loss values between bipods........................................................69
4-4 Friction factor estimates from bottom current meter................................................72
4-5 Reynolds stresses for October 1997 .........................................................................75
vii
LIST OF FIGURES
Figure page 1-1 Field Research Facility location.................................................................................3
1-2 Bipod instrumentation ................................................................................................5
1-3 Bipod locations at initial deployment in 1994 ...........................................................6
2-1 November 1997 filtered mean current comparison with sonar ................................15
2-2 May 1998 filtered mean current comparison with sonar..........................................16
2-3 October 1997 mean orbital velocity estimates vs. sonar measurements ..................17
2-4 October 1997 cross-shore orbital velocity estimates vs. sonar measurements.........18
2-5 Wind vectors measured at the Field Research Facility ............................................20
2-6 October 1997 current-wind comparison...................................................................22
2-7 November 1997 current-wind comparison...............................................................23
3-1 Median grain size variation with water depth ..........................................................27
3-2 X-ray images of boxcores ........................................................................................28
3-3 Sonar histogram at 13 m bipod, August 30, 1998....................................................29
3-4 Sonar histogram at 5.5 m bipod, August 12, 1998...................................................30
3-5 Sonar histogram at 8 m bipod, August 31, 1998......................................................31
3-6 Sonar histogram at 13 m bipod, August 29, 1998....................................................32
3-7 Shield’s curve...........................................................................................................33
3-8 Shear velocity vs. sonar at the 13 m bipod, October 18, 1997.................................34
3-9 Shear velocity vs. sonar at the 8 m bipod, August 19, 1998 ....................................35
3-10 Shear velocity vs. sonar at the 5.5 m bipod, October 18, 1997................................36
viii
3-11. Surface roughness variation with mean currents at 5.5 m bipod for October 15-21, 1997..........................................................................................................................38
3-12 Surface roughness variation with mean currents at 8 m bipod for October 15-21, 1997..........................................................................................................................39
3-13 Surface roughness variation with mean currents at 13 m bipod for October 15-21, 1997..........................................................................................................................40
4-1 Coordinate system ....................................................................................................45
4-2 Ratios of Fourier coefficients ...................................................................................46
4-3 Significant wave heights measured in October 1997, November 1997, May 1998, and August 1998.......................................................................................................47
4-4 Bathymetry in vicinity of bipod instrumentation .....................................................48
4-5 Measured spectra for November 7, 1997 time=2200...............................................50
4-6 Error versus m value comparison for October 20, 1997 time=100..........................54
4-7 Comparison of measured and best-fit spectra from matching coefficients for October 20, 1997 time=100......................................................................................55
4-8 Error between spectra fitted from coefficient ratios for October 20, 1997 time=100...................................................................................................................56
4-9 Comparison of measured and best-fit spectra from curve fitting for October 20, 1997 time=100..........................................................................................................57
4-10 Error between spectra from curve fitting for October 20, 1997 time=100...............58
4-11 Variation of m values with frequency range at 8 m .................................................59
4-12 Directional spectrum variation with frequency at 8 m bipod for October 19, 1997 time=700...................................................................................................................59
4-13 Energy density spectral values for October 19, 1997 time=700 ..............................60
4-14 Mean wave direction comparison for October 19, 1997 time=700..........................61
4-15 Measured and calculated wave direction differences for October 19, 1997 time=700 at 5.5 m bipod ..........................................................................................62
4-16 Average measured and calculated wave direction differences for October 1997 ....63
4-17 Measured and refracted directional spectra for November 13, 1997 time=1742.....64
ix
4-18 Comparison of refracted and measured m values over frequency range for November 13, 1997 time=1742................................................................................65
4-19 Significant wave height ratios versus cross-shore position......................................67
4-20 Average measured and predicted energy flux values...............................................70
4-21 Surveyed bathymetry in vicinity of bipod instrumentation......................................71
4-22 Histogram of calculated friction factors at all current meters ..................................73
4-23 Friction factor variation with wave height at the bottom current meter...................74
x
Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science
NEARSHORE WAVE AND SEDIMENT PROCESSES: AN EVALUATION OF STORM EVENTS AT DUCK, NC
By
Jodi L. Eshleman
May 2004
Chair: Robert G. Dean Major Department: Civil and Coastal Engineering
Pressure, sonar, and current measurements were recorded at 5.5 m, 8 m, and 13 m
water depths in the outer surf zone and inner continental shelf region off the coast of
Duck, NC. This unique data set was analyzed to investigate erosion thresholds and wave
evolution. A mean current threshold of 20 cm/s and combined wave and current
threshold of 60 cm/s were identified for bed elevation decrease. Shear velocity was
determined to be a good indicator of bottom elevation change at the 8 m and 13 m bipods,
with erosion beginning 0 to 3 hours after it crossed a movement threshold of 1.17 cm/s.
Surface roughness estimates at these same two water depths decreased with increasing
mean currents.
The combination of measured near bottom pressure and horizontal velocity
components provides the basis for determining a directional spectrum. A simplified
analytical directional spectrum based on a single cosine curve of varying power (m) was
used to approximate these measured directional spectra. A nonlinear least squares curve
xi
fit to each side of the measured directional spectrum proved the most accurate method of
determining the best representation of m values. Refracted mean wave directions were
slightly overestimated by the theory and the decrease in width of the directional spectra
with decreasing water depth was overestimated. Also, energy flux calculations
combining shoaling and refraction theory showed smaller measured than predicted
energy flux values with inshore distance (sometimes by more than one third) emphasizing
the importance of considering energy loss in calculations for engineering design and
planning.
A representative friction factor for each record was determined by accounting for
frictional energy loss in the energy flux calculation, using velocity time series measured
at the bottom current meter. Calculated friction factors varied throughout storm events,
but most fell within a range of 0 to 0.2. A representative value of 0.17 was identified for
this location through the use of average energy flux and energy loss values over all storm
events. Reynolds stresses were calculated and were found to be consistently different at
the current meter at 0.55 m elevation, a result that remains unexplained.
xii
CHAPTER 1 INTRODUCTION
The inner continental shelf off the open mid-Atlantic coast is a wave-driven
environment, where sediment transport and nearshore circulation are primarily forced by
wind-generated ocean surface waves (Wright 1995). This is a friction-dominated region,
where boundary layers may occupy the entire length of the water column, transmitting
effects of wind blowing on the water surface to the seabed (Wright 1995). Wave
propagation is largely characterized by transformation through refraction, diffraction,
energy dissipation, and shoaling. Mean currents are another important component in the
nearshore zone, and can be driven by waves, wind, tides; and gradients in pressure,
temperature, and density, among other things.
Within this dynamic environment, sand movement is not uniform in all directions
and at all locations. Harris and Wiberg (2002) suggest that gradients in bed shear stress
may create gradients in suspended sediment flux. These cross-shelf gradients in sediment
flux will in turn create cross-shelf gradients in sediment size as the higher orbital
velocities on the inner shelf move finer sediment offshore (Harris and Wiberg 2002).
Alongshore sediment flux is also an important component of the sediment transport
discussion. Beach and Sternberg (1996) found that alongshore sediment flux is
dependent on breaker type, and this information should be incorporated into sediment
transport models. Their measurements showed that plunging waves were responsible for
the greatest portion of suspended load and sediment flux, but other breaker types and
1
2
nonbreaking waves combined still contributed almost half of the total suspended load and
sediment flux (Beach and Sternberg 1996).
There is variation in sand transport throughout the water column as well. Many
investigators have found an inverse relationship between distance above the bed and
suspended sediment concentration (Beach and Sternberg 1996; Conley and Beach 2003).
A study conducted in the surf zone during the SandyDuck experiment showed that the
increasing importance of wave-driven transport near the bed might lead to a reversal in
the net cross-shore transport direction in the water column. The directions of transport at
the bed may dominate even if much of the water column has an opposing transport
direction since more than half of the depth-integrated net transport occurs within 5 cm of
the bed (Conley and Beach 2003).
Study Location, Characteristics, and Instrumentation
Geographic Location
Field data were obtained on the inner continental shelf off the coast of the Army
Corps Field Research Facility (FRF) in Duck, North Carolina. The FRF facility is
located on the Outer Banks of North Carolina, on the central portion of the Currituck
Spit, which extends southeast continuously for over 100 km from Cape Henry, Virginia
to Oregon Inlet, North Carolina (Figure 1-1). It is located in the southern portion of the
Middle Atlantic Bight (36 10’ 57”N; 75 45’ 50”W) and bordered by Currituck Sound, a
low-salinity estuarine environment, on the west; and the Atlantic Ocean on the east.
Ocean tides are semi-diurnal, with a mean range of approximately 1 m (Birkemeier et al.
1981).
3
Figure 1-1. Field Research Facility location (from http://www.frf.usace.army.mil)
Wave and Weather Conditions
Wave heights vary seasonally along the Outer Banks, with peak waves occurring
in October and February, and mild conditions prevailing in late spring and early summer
months (Birkemeier et al. 1981). A compilation of wave statistics for the time period of
1985 through 1995 resulted in an average annual wave height of 0.9 + 0.6 m, and a mean
annual wave period of 8.8 + 2.7 s (Leffler et al. 1998). There have been many
observations of water masses that interact with currents in the area, including low salinity
slugs from the Chesapeake Bay and warm, clear Gulf Stream currents (Birkemeier et al.
1981).
4
The majority of storm events that affect the Atlantic coast originate in the middle-
latitude westerly wind belt and are often termed extra-tropical (Dolan et al. 1988).
Tropical storms, including hurricanes, also affect the region, but less frequently. Dolan et
al. (1992) discuss the importance of extra-tropical storms for erosion and note that they
often generate wave heights that are comparable to or greater than those from hurricanes.
A study that examined 1,349 northeast storms on the Atlantic coast found a distinct
seasonality of frequency and duration, with maximum values in the winter and minimum
in the summer (Dolan et al. 1988). When specifically examining extra-tropical storms,
the most significant contribution to erosion is from northeasters. Xu and Wright (1998)
determined that even though comparable winds from the southerly directions sometimes
caused high wave heights, the alongshore current magnitudes recorded during these
storms were only one-fifth of those achieved during northeasters. Cross-shore current
magnitudes were also smaller, but differences were not as large as for alongshore currents
(Xu and Wright 1998).
Bipod Instrumentation
The initial bipod instrumentation was deployed in October 1994 as part of a
multi-year monitoring program to study shore-face dynamics on the inner continental
shelf of the Field Research Facility in Duck, NC (Howd et al. 1994). The instrumentation
consisted of three current meters at varying elevations, a pressure sensor, and a sonar
altimeter, which were all attached to a bipod frame, secured by two 6.4 m pipes jetted
vertically into the seabed (Beavers 1999). The original bipods collected data until
October 1997 using three Marsh-McBirney electromagnetic current meters, which often
experienced significant noise. The current meters were replaced with Sontek Acoustic
Doppler Velocimeters for the SandyDuck experiment in October 1997, and the bipods
5
were redeployed at depths of 5.5 m, 8 m, and 13 m relative to NGVD. They remained
operational at these three simultaneous locations through December 1998. The data used
in this analysis were collected during this second deployment. Figure 1-2 shows the
bipod setup where A, B, and C are electronic housings; P is the pressure sensor; and S the
sonar altimeter. General bipod locations at the time of initial deployment in 1994 are
pictured with respect to local bathymetry in Figure 1-3.
Figure 1-2. Bipod instrumentation – A,B,C = electronic housings, P is pressure sensor, and S is sonar altimeter (from Beavers 1999)
The bipod packages each contained three SonTek Acoustic Doppler Velocimeters
(ADV), which sampled at 2 Hz and were located at elevations of 0.2 m 0.55 m and 1.5 m
above the sea floor. The end of the frame containing the current meters was oriented
toward the southeast to minimize interference of current meters and vertical supports with
orbital velocity measurements, since storm events of interest would have primarily
northeast waves (Beavers 1999). Digital Paroscientific gauges were used for pressure
measurements, operating at 38 k Hz and a sampling rate of 2 Hz (Beavers 1999). A
Datasonics PSA-900 sonar altimeter was used to record bottom elevation. The range was
6
modified from 30 m to 3 m to increase the resolution, sampling at 1 Hz with a beam
frequency of 210 kHz (Beavers 1999). Tests by Green and Boon (1988) of response
characteristics found this model of altimeter to be accurate to one centimeter. Current
meter and pressure data are output in 34-minute segments, with a 10-minute break in data
every 3 h. Average values for each record represent a mean value for a 34-minute burst.
Sonar measurements are determined through a histogram filtering technique, taking the
highest bin value for the burst.
Figure 1-3. Bipod locations at initial deployment in 1994 (from Beavers 1999)
The lowest ADV records bottom measurements for a duration of 9 min every
three hours to provide a second measurement of bottom elevation (Beavers 1999). This
beam has a frequency of 4 MHz and a beam width of one degree, creating a 2 cm
diameter footprint; whereas, the sonar altimeter operates at a frequency of 210 kHz and
has a 10-degree beam width, creating a 20 cm diameter footprint (Beavers 1999). The
differences in beam frequency and footprint size provide different optimum operating
7
conditions. Beavers (1999) suggests that the ADV is more reliable under non-storm
conditions and the sonar altimeter provides a better estimate of bottom elevation under
storm conditions, when suspended sediment concentration within the water column is
high.
Chapter Contents
The purpose of this investigation is to examine sediment movement and wave
evolution within the inner continental shelf and outer surf zone region through the
analysis of field measurements. Chapter 2 focuses on determining general relationships
between waves, currents, and sonar measurements. Chapter 3 describes sediment
characteristics, and discusses some aspects of bottom roughness by analyzing velocity
profiles. Chapter 4 is the heart of the investigation and utilizes the data collected at all
three bipod locations to provide a comparison with analytical predictions of evolution of
wave characteristics.
CHAPTER 2 GENERAL NEARSHORE CHARACTERISTICS
Introduction
Many researchers have attempted to establish a relationship between statistical
properties of velocity measurements and sediment movement. Although there have been
some velocity moments that have seemed more relevant than others, there does not seem
to be any one parameter that shows a consistent significant correlation to sediment
transport. A study by Guza and Thornton (1985) examined velocity moments from
measurements at Torrey Pines Beach in San Diego, CA and found that oscillatory
asymmetries and combined current-wave variance terms are significant to cross-shore
transport. Several studies have shown that the oscillating velocity terms move sediment
onshore and the mean velocities move sediment offshore (Guza and Thornton 1985;
Osborne and Greenwood 1992; Conley and Beach 2003). Measurements of sandbar
migration at Duck, NC showed maximum values of velocity asymmetry and acceleration
skewness near the bar crest (Elgar et al. 2001). Hoefel and Elgar (2003) found that
extending an energetics model to include fluid accelerations resulted in better predictions
of onshore bar migration between storms. Velocity measurements taken in the surf zone
during SandyDuck showed no significant correlation between velocity moments and
wave driven transport, although acceleration skewness showed the strongest relationship
(Conley and Beach 2003). These suggest that velocity asymmetry and acceleration
skewness seem to have the strongest ties to sediment transport in past experimental
results.
8
9
There has been previous work with similar instrumentation done at this location.
Several studies included instrumented tripods deployed during storm and fair-weather
conditions, which also included suspended sediment measurements (Wright et al. 1986;
Wright et al. 1991; Wright et al. 1994). The first tripod deployment was at a single depth
and did not show a relationship between bed level changes and increased mean or orbital
velocities. There was a gradual change in the bottom elevation throughout the middle
and final stages of the storm, followed by significant accretion that was hypothesized to
be the result of a migrating bedform. Suspended sediment measurements did not show a
response to the onset of the storm or peak with mean currents, but peaked with oscillatory
flow, suggesting that waves are the dominant source of sediment resuspension (Wright et
al. 1986). The second study consisted of three separate deployments at the Field
Research Facility and the results suggested that it is the near-bottom mean flows, not
oscillatory components that play the dominant role in transporting suspended sediment
(Wright et al., 1991). Mean flows may also play a role in the direction of sediment
movement. A study conducted at Duck showed the tendency of a mean cross-shore
velocity threshold around 30 or 40 cm/s directed offshore to be the divider between
landward and seaward bar migration (Miller et al. 1999). Both mean and oscillatory
flows are essential to the analysis and they are not independent. The wave boundary
layer creates resistance for the current above and slows down that flow. Waves are often
thought to be more efficient at initiating motion, whereas currents are more efficient at
net transport, but the two interact nonlinearly (Grant and Madsen 1976).
There are two critical differences between previous studies and this dataset.
These include the length of time and number of instrumentation packages deployed.
10
Many other studies have included one or two instrument packages deployed
simultaneously for individual storm events or short periods of fair-weather conditions,
but not three instrument packages with continuous measurements for this length of time.
Some previous investigations have been carried out with this specific dataset that
focused on the sonar data. Sonar Altimeter measurements were compared to surveyed
profiles to discuss discrepancies in depth of closure concepts. The predicted depth of
closure was around 8 m, yet for some storm events, the 13 m bipod showed the greatest
change in bottom elevation. Net and range of seabed elevation changes were examined
during storm events that were defined by wave thresholds. Finally, a comparison of
sonar records to diver collected boxcores served to validate the sonar record and showed
the sonar was capable of monitoring long-term bottom stratigraphy (Beavers 1999). The
current and sonar measurements were also used as forcing and validation for a bottom
boundary layer and sedimentation model (Keen et al. 2003). These analyses have shown
some interesting relationships, but have neglected a major component of the dataset: the
current measurements. The first level of this analysis focuses on the currents and the
manner in which they affect the bottom during all weather conditions, not just storm
events.
Quality Control
Data Screening
Different levels of screening were applied in an attempt to eliminate noise and
assure that the measurements presented here are representative of actual conditions in the
nearshore environment. Spikes were removed using polynomial interpolation. Beam
correlation and intensity values output by the ADV were used to identify low quality
data. The second level of data screening was accomplished through determining several
11
quality control parameters for each record. The quality control parameters included
signal-to-noise ratios for current and pressure measurements, and a z test value based on
a ratio of the wave heights calculated from pressure and current measurements. Data
with signal-to-noise ratios less than 1.5 or z test values outside the range of 0.5 to1.5
were not used.
Representative Data
The following analysis is based on four months during which the described data
standards were satisfied. These months: October 1997, November 1997, May 1998 and
August 1998 were chosen for several reasons. They include a nearly complete data set
that has been successfully edited. They have z-test values near 1, signal to noise ratios of
2 or higher, and wave directions that are consistent for all three current meters,
suggesting that biofouling and problems with current meter rotation were minimal. They
have bottom measurements from the lowest current meter recorded every three hours, so
that trends in the sonar measurements can be validated. In addition to data quality, they
encompass significantly different seasonal variations. Measured significant wave height
values range from less than 1 m to almost 4 m, spanning storm and mild weather
conditions.
It is important to note that the data from these months includes some problems,
but knowledge of data quality can be combined with analysis techniques to obtain results
that account for the limitations of the data.
Analysis
Our knowledge of the dynamics of the nearshore system leads to the recognition
that no single statistical property can explain sediment movement. Sand transport is
governed by a complex combination of many factors. The following discussion of
12
erosion refers to bed elevation decrease, rather than transport initiation, which cannot be
measured by the available instrumentation. The continuity equation gives the following
relationship between bed elevation, z, and gradients in cross-shore, alongshore and
vertical sediment transport components at the bed, qx, qy, and qz respectively.
( )
∂∂
+∂
∂+
∂∂
−=−zq
yq
xq
dtdzp zyx1
where p represents the porosity. It is important to recognize that it is possible to have
sediment transport without bed elevation change; however, if the bottom sediment is
suspended or the gradient in cross-shore or alongshore sediment transport is positive, the
bottom elevation will decrease. This initial analysis is an attempt to discern which
properties appear to play a more significant role when considered individually.
Current Influence
There appears to be a mean current threshold of approximately 20 cm/s for
erosion in most of the data. There is usually some erosion when the total mean current
reaches 20 cm/s, yet there may be erosion for smaller currents. The currents were filtered
with a cutoff of one day to remove tidal influences and to facilitate a comparison with
sonar altimeter data. Table 2-1 shows events of bed elevation decrease of 3 cm or greater
and the associated currents. There are many times when the mean current is very high
and the bottom change is small and vice versa, indicating the possibility that other forces
may be involved. One important thing to note is that often times erosion occurs when
significant wave heights are fairly low. 54% of the erosion events identified in Table 2-1
occurred when the significant wave height was less than 2 m, which is often considered
as the threshold between storm and calm conditions. This reinforces the need to examine
13
Table 2-1. Erosion events of 3 cm or greater
Bipod Depth
(m) Bottom Change
(cm) Current (cm/s)
Hmo (m) T (s)
Time since last event (days)
Oct-97 5.5 6 25 2.0 12.5 12.5 5.5 20 55 3.0 9.1 3.5 5.5 10 19 1.0 12.5 4.5 5.5 8 21 1.0 10.0 5.0 8 5 22 1.5 6.3 9.0 8 8 28 2.0 12.5 4.5 8 6 58 3.0 9.1 3.0 8 11 18 1.0 12.5 4.0 8 5 19 1.0 10.0 4.5 13 11 56 3.0 9.1 16.0 Nov-97 5.5 5 30 2.5 4.8 10.5 5.5 3 25 3.0 9.1 7.0 5.5 13 28 1.5 3.6 10.5 8 10 15 1.0 8.3 9.5 8 6 30 2.5 4.8 5.0 8 4 28 2.0 11.1 1.5 8 3.5 12 1.5 10.0 1.0 8 5 13 3.0 9.1 6.0 8 10 25 1.5 3.6 9.0 13 4 22 2.5 4.8 19.0 13 3 18 3.0 9.1 7.5 May-98 5.5 10 40 1.5 6.7 9.0 5.5 3 21 2.0 12.5 6.0 8 9 40 1.5 6.7 9.0 8 6 20 2.0 12.5 6.0 8 5 10 1.0 11.1 1.5 13 3.5 35 1.5 6.7 9.0 13 8 40 3.5 9.1 3.5 Aug-98 5.5 4 26 2.0 7.1 1.5
5.5 4 25 1.5 7.1 1.5 5.5 3 15 1.0 6.7 2.5 5.5 5 21 1.5 4.0 6.5 5.5 10 30 1.5 6.3 6.0
5.5 4 18 1.25 11.1 4.0 5.5 10 70 3.2 12.5 3.0 8 8 23 2.0 7.1 1.5 8 11 23 1.5 7.1 1.5 8 8 26 1.5 6.3 15.0 8 16 65 3.2 12.5 7.5 8 11 5 1.0 10.0 4.0 13 4 15 1.5 7.7 4.0 13 25 70 3.2 12.5 22.0
14
the changes occurring during all types of conditions, since it is not necessarily during
storm events that the sediment is moving.
Table 2-2. Event-Based comparison of erosion events of 3 cm or greater Date Current (cm/s) Bottom Change (cm)
5.5 8 13 5.5 8 13 11-Oct 20 22 20 - 5 - 15-Oct 25 28 28 6 8 - 19-Oct 55 58 56 20 6 11 22-Oct 19 18 18 10 11 - 27-Oct 21 19 18 8 5 - 2-Nov 12 15 11 - 10 - 6-Nov 30 30 22 5 6 4 7-Nov 23 28 11 - 4 - 8-Nov 10 12 5 - 3.5 - 13-Nov 25 13 18 3 5 3 23-Nov 28 25 21 13 10 - 9-May 40 40 35 10 9 3.5 12-May 65 58 40 - - 8 15-May 21 20 10 3 6 - 16-May 5 10 8 - 5 - 2-Aug 26 23 17 4 8 - 4-Aug 25 23 15 4 11 - 5-Aug 18 15 15 - - 4 6-Aug 15 15 5 3 - - 13-Aug 21 19 12 5 - - 19-Aug 30 26 28 10 8 - 23-Aug 18 10 9 4 - - 27-Aug 70 65 70 10 16 25 30-Aug 5 5 10 - 11 -
Major erosion events appear to be fairly consistent between bipods, although the
magnitudes of bottom elevation changes are usually different. Table 2-2 presents the data
from Table 2-1 by date, to facilitate a comparison between bipods (- represents < 3 cm of
bottom change). An interesting situation occurs in November 1997 and August of 1998,
where the current reaches one of the maximum values for the month (above the 20 cm/s
15
threshold), but the erosion is not consistent at all three bipods. Both instances show
significant erosion at the 5.5 m and 8 m bipods (ranging from 8-13 cm), and less than
3 cm of erosion at the 13 m bipod. Figure 2-1 shows one case occurring around
November 24, 1997. Currents are positive onshore and south.
Figure 2-1. November 1997 filtered mean current comparison with sonar a) 5.5 m b) 8 m c) 13 m
A situation occurred in May of 1998 that seems to be the reverse of this last
observation and occurs around May 13 (see Figure 2-2). The currents were at their
highest values for the month, in excess of 40 cm/s, causing minimal erosion at the 5.5 m
and 8 m bipods and a more significant change at the 13 m bipod. For this situation, the
ADV bottom measurement showed less erosion, indicating that there may have been
fluffy material at the 13 m bipod that the sonar had trouble penetrating and the higher
16
frequency ADV picked up. Here we see the 5.5 m and 8m bipods showing significant
erosion at the onset of currents over 20 cm/s, but then the erosion leveled off as the
currents continued to increase. The 13 m bipod did not show as much erosion initially,
but as currents continued to increase there was a spike where the sonar based bottom
elevation dropped 15 cm. If this sediment was of the finer type that is sometimes seen at
13 m, a higher current may have been required for movement due to cohesive properties,
and once the current reached that threshold, the entire layer moved. The cause of this
cannot be explained with certainty without more detailed sediment information.
Figure 2-2. May 1998 filtered mean current comparison with sonar a) 5.5 m b) 8 m c) 13 m
17
Wave Influence
The root mean square current speeds were examined in different frequency
ranges. These values were multiplied by the square root of two to obtain the amplitudes
as substitutes for orbital speeds and represent the significant values for orbital speed.
This should provide some insight into which frequency components contribute the most.
A high frequency range from 0.04 to 0.35 Hz was identified to examine sea swell and a
low frequency range of 0.004 to 0.04 Hz to investigate any infragravity contributions.
Figure 2-3. October 1997 mean orbital velocity estimates vs. sonar measurements a) 5.5 m b) 8 m c) 13 m (solid black line represents sonar)
The infragravity orbital speed was always smaller, but reached 20 cm/s at the 5.5 m bipod
during storm events. Results for October 1997 are presented in Figure 2-3, which shows
18
a peak at the maximum erosion event for the month, and slight increase for smaller
erosion and accretion events.
Figure 2-4. October 1997 cross-shore orbital velocity estimates vs. sonar measurements a) 5.5 m b) 8 m c) 13 m (solid black line represents sonar)
This suggests that the infragravity component may be significant for this dataset.
This is an interesting observation because other studies have had conflicting results for
this location in the past. Wright et al. (1994) took similar measurements at this location
at a depth of 13 m during the Halloween storm of 1991 and showed a significant
infragravity component, reaching 20 cm/s near the peak of the storm, which is similar to
these findings. They suggest that roughly half of the infragravity energy emanates from
the surf zone (Wright et al. 1994). An earlier attempt to quantify wave reflection found
no significant quantity of long wave energy, either incident or reflected from
19
measurements taken at a depth of 6.5 m (Walton 1992). One possible explanation for this
difference may be a difference in significant wave heights, since values recorded during
this study were never greater than 3.5 m and those recorded during the “Halloween”
storm reached 6.5 m. The plot of the cross-shore components (Figure 2-4) shows that the
cross-shore is the major component of the R.M.S seaswell velocity, which enforces the
need to consider cross-shore velocity for sediment transport even when mean values are
small. The current amplitude increased at most erosion events for the month, but not all.
Combined Waves and Currents
Another approach to considering erosion causes examined the combined wave
and current maximum velocities. The sum of the amplitude (used to estimate orbital
velocity) and the component of the mean current in the wave direction was calculated.
These are all positive values since they were taken in the wave direction and negative or
positive wave orbital velocities could cause erosion. This was an attempt to consider not
just the mean current or wave orbital velocity, but their combined effect; however, this
did not take into account any nonlinear interactions between waves and currents, but
provided a rough estimate of combined velocity. This analysis did not show any
consistent threshold between maximum velocity and sediment movement, but there was a
relationship between the two. Very high combined velocities in the range of 60 to100
cm/s always seemed to be associated with erosion, but below this range the effect varied.
Current Direction
The maximum erosion events for the month always occurred with a southerly
longshore current and usually a downwelling (seaward) flow in the cross-shore
component. One example is around the 28 of August 1998 in which a northerly
20
longshore current reversed direction and caused the most significant erosion for the
month.
Wind
Measurements of wind magnitude and direction are obtained from FRF wind
gages 932 and 933. Figure 2-5 shows a vector plot of these values. Most peaks in mean
longshore current velocity that are above the 20 cm/s threshold appear to coincide with
peaks in longshore wind velocity. The notable exception to this is the month of
November 1997, where scatter plots show a poor correlation between the longshore wind
velocity and longshore mean current.
Figure 2-5. Wind vectors measured at the Field Research Facility
21
The cross-shore currents showed no significant correlation to cross-shore wind. Current-
wind comparisons for the month of October 1997 are provided in Figure 2-6. Current
and wind measurements are positive onshore and south. Another study by Xu and
Wright (1998) of wind-current correlation at this same general location has shown that it
is dependent on wind direction. The correlations were broken into quadrants, and it was
found that the current and wind speeds are most correlated with winds from the Northeast
or Northwest direction, showing much higher R squared values than winds blowing from
the Southeast or Southwest (Xu and Wright 1998).
Variance of Total Current Acceleration
There seems to be some relationship between the variance of the acceleration of
the total current and bottom change on a month-to-month basis. This follows some
previous observations discussed in the introduction, although the acceleration skewness
did not show any significant trends. There is an increase in this variance at times of
maximum erosion for the month; however, this same trend was seen when considering
current variance. Erosion due to the acceleration variance cannot be distinguished from
erosion due to the current variance, since they differ only by the square of the radial
frequency.
22
Figure 2-6. October 1997 current-wind comparison
23
Figure 2-7. November 1997 current-wind comparison
CHAPTER 3 SEDIMENTS
Introduction
Past observations of bottom change from sonar altimeter measurements have
often been supported by visual observation or other instrumentation since sonar measures
only the elevation at a single point. One limitation of the analysis lies in the inability to
distinguish whether sediment is moving as suspended load or bed load from a single
sonar altimeter. It is possible to have sediment transport without elevation change, yet
the sonar can only capture variation in the bed elevation. Topographic features moving,
including bed forms or ripples, can also cause problems since they may be measured as
the representative bed elevation, but their existence is localized. There have been
observations of non-uniform topography in the vicinity of the bipod instrumentation. An
array of seven sonar altimeters deployed in the surf zone during the SandyDuck
experiment captured mega ripples which ranged from 15 cm to 30 cm high and moved
through the sonar range in a period of about ten hours (Gallagher et al. 1998). The first
tripod deployment by Wright et al. (1986) was supplemented by diver observations,
pictures and side-scan sonar measurements. Side scan sonar images showed sediment
lobes of fine material overlying coarser material which were up to one meter high and
thought to be the cause of the rapid accretion at the 8 m tripod at the end of the storm
event (Wright et al. 1986). Bottom features such as these are difficult, if not impossible
to discern from sonar altimeter measurements alone.
24
25
An important parameter when discussing sediment transport in relation to bottom
topography is the bed shear stress, or equivalently, the shear velocity, ρτ /* b=u . The
von Karman-Prandtl equation
( )
=
0*
ln1zz
uzu
c
c
κ
can be used to estimate shear velocity and hydraulic roughness length values from
measurements of the mean current at different elevations. Madsen et al. (1993)
calculated shear velocity and apparent hydraulic roughness from the log-profile method
for the data collected at the 13 m tripod during the “Halloween” storm of 1991. These
estimates showed shear velocity values on the order of 2 to 3 cm/s and apparent hydraulic
roughness values generally between 0.1 cm and 1 cm. These values were found using
data obtained with similar instrumentation and at a very similar location, therefore the
values found in the subsequent analysis are expected to be the same order of magnitude.
They found a value of 1.5 mm for the Nikuradse sand grain roughness for a movable flat
bed (Madsen et al. 1993).
Past analysis of the dataset examined in this thesis used diver-collected boxcores
to supplement sonar measurements. Beavers (1999) discussed the geologic features of
these cores and the correlation between core layers and sonar records in detail. The
positive correlation between these two records indicates that scour around the pipes was
not appreciable, which will be assumed in the following analysis. Shear stress
calculations showed that when the shear stress is a maximum, seabed elevation decreases
and when shear stress decreases, the seabed elevation increases (Beavers 1999).
26
Analysis
Sediment Characteristics
Previous sediment data
Sediment data are available for the period 1984-1997 from locations adjacent to
the bipod instrumentation. Figure 3-1 compares median grain size versus elevation for
this time period and implies little variation in sediment size at the 5.5 m and 8 m bipods.
Most measurements remain within a ½ φ unit range. There is significantly more
variation at the 13 m bipod with a 3/2φ unit range. The left panel of Figure 3-2 shows an
x-ray of a boxcore taken at a water depth of 13.2 m on August 18, 1997. The white
region shows a section where the material is too fine for the x-ray to register and grain
size analysis determined a median φ value of 4.02 for this section, as opposed to 3.10 for
the rest of the sediment column (Beavers 1999). The right panel of Figure 3-2 shows a
boxcore taken in 1992 at a water depth of 14 m, which also shows a layer of fine, silty
material (Nearhoof 1992). It is important to be aware of the sediment range at the 13 m
bipod when considering sonar measurements, since there are instances when the sonar
may have trouble recognizing fine, silty material.
Sonar evaluation
Bed form migration can register on sonar measurements and changes in elevation
may reflect localized change from large-scale bed forms moving through the sonar
footprint. Tests of ripple fields under a similar sonar altimeter showed that the ripple
could not be resolved if its height above the bed is more than eight times its wavelength
(Green and Boone 1988). The possibility of non-uniform topography is difficult to
resolve and one method of addressing this was to examine sonar histograms. One notable
27
observation is that the raw sonar measurements were very noisy, although the outline of
the bottom could generally be distinguished. A representative sonar value for each
34-minute record was taken as the max bin value of the histogram for that time series.
The hypothesis was that at times when there was silty material at the 13 m bipod, the
sonar might show two peaks, at the top and bottom of this layer. Another possibility was
that during storms, the histograms may show more spread if the sonar was unable to
consistently penetrate the suspended sediment in the water column.
-14
-13
-12
-11
-10
-9
-8
-7
-6
-5
-4
2 2.5 3 3.5 4 4.5
Median Grain Size (phi)
Elev
atio
n (m
,NG
VD)
Boxcore Samples (1994-1997)
Profile Line 62 (1984-1985)
Duck 94 Samples-Oct
Duck 94 Samples-Aug
SandyDuck 97 Samples
Figure 3-1. Median grain size variation with water depth (data from FRF)
Sonar histograms are included for each bipod location during the month of August
1998. In most instances, there is a very well defined peak at a specific value, and smaller
peaks or spreading at depths less than this value (Figure 3-3). Secondary peaks are
within a few centimeters of the major peak. There are times when this histogram
28
deteriorates and the peak is less well defined with a greater spread (Figure 3-4). These
times do not always correlate with storm events as anticipated.
A.
B.
Figure 3-2. X-ray images of boxcores A) h=13.2 m on August 18, 1997 (Beavers 1999) B) h=14 m in 1992 (Nearhoof 1992)
At the 5.5 m and 13 m bipods there is rarely spreading at depths greater than this
peak, which lends confidence to sonar estimates of the bottom where spreading is most
likely an indicator of noise within the water column. One very interesting observation is
that the 8 m bipod shows almost all of the spreading and secondary peaks in the
histograms to be at depths greater than the histogram peak, as evident in Figure 3-5. This
is consistently different from the other bipods and remains unexplained.
Another interesting situation occurred on August 29, 1998 and lasted for
approximately a day, showing the variation in sonar histogram values with time at the 13
m bipod. Here two distinct peaks occurred that are over 20 cm apart and are of similar
magnitude. One peak is at 13.37 m and another at 13.14 m (Figure 3-6). This occurs
immediately after the major storm event for the month. One possible explanation is that
there is a layer of very fine sediment here and the sonar sometimes pings off the top and
29
sometimes penetrates the layer. An observation about this August storm event is that
there are strong northerly alongshore currents reversing direction and reaching 100 cm/s
in the southerly direction, and this is the only time that this alongshore-current reversal
occurred within the four months analyzed. This dual sonar peak phenomenon was not
repeated and with the limited sediment information available it cannot be explained with
any degree of certainty.
Figure 3-3. Sonar histogram at 13 m bipod, August 30, 1998
Velocity Profile Calculations
The velocity profile method discussed earlier is used to estimate shear velocity
and hydraulic roughness length values from the measured mean current values from the
three different current meters. Bed elevation values from ADV measurements are
30
incorporated to account for changing current meter elevations with time and bottom
change.
Figure 3-4. Sonar histogram at 5.5 m bipod, August 12, 1998
Critical shear velocity
One attempt to examine further the shear velocity relationship with bottom change
was to calculate a critical shear velocity value from the range of sediment sizes shown in
Figure 3-1 for each bipod. This method utilized the form of Shield’s curve shown in
Figure 3-7 to determine a value of shear stress ( )*τ for each mean sediment diameter
based on an abscissa value of
( )ρρρ
υζ
gD s −= 2
3
*
The ordinate value of
31
( )gDs
c
ρρτ
τ−
=*
is used to determine a critical shear velocity through the relationship
ρτ 2*cc u=
The next step was to identify the times during the four months analyzed when the bottom
was just beginning to erode. A daily plot was generated for each of these times of erosion
initiation to facilitate a visual comparison of shear velocity with measurements of sonar.
Figure 3-5. Sonar histogram at 8 m bipod, August 31, 1998
The critical range of shear velocity values were overlain on the shear velocity plots to
determine if the bottom elevation began to change around the same time that the shear
velocity crossed the threshold for movement.
32
Figure 3-6. Sonar histogram at 13 m bipod, August 29, 1998
Most situations at the 8 m and 13 m bipod locations show the bottom beginning to
erode zero to three hours after the shear velocity crossed the critical threshold. Figure 3-8
shows a particular time when the two occur almost simultaneously. The blue line
represents shear velocity with the red lines identifying the range of critical values for
different sediment sizes, and the circles marking where the bottom begins to erode and
the shear velocity crosses the threshold. Figure 3-9 shows a particular situation at the
8 m bipod where we see a phase lag between the two, and erosion does not occur until
approximately 3 h later. At the 5.5 m bipod, the shear velocity value is always below the
threshold for movement when sonar measurements show the bottom beginning to erode.
33
One example is included as Figure 3-10. Peaks in shear velocity show some relationship
to peaks in tidal currents.
Figure 3-7. Shield’s curve
Error estimates
It is important to address the error in estimates of shear velocity, since values are
calculated from a velocity profile method that fits a curve to three current measurements,
leaving only two degrees of freedom. Error estimates for shear velocity are calculated for
a 90% confidence interval using the student t distribution and they are very large, around
an order of magnitude higher than most calculated values of shear velocity. This is a
limitation of the method and data available, and most field measurements would
demonstrate similar error estimates when using vertical arrays of individual current
meters for measurements. Another factor is that the chosen critical value of shear
velocity is dependent on sediment size, but the range of values does not vary widely
when considering the range of sediment sizes measured at bipod locations. After taking
34
into account the limitations of the shear velocity estimates the time lags between the
shear velocity reaching a critical value and initiation of sonar change do not appear to be
unreasonably long. The critical shear velocity seems to be a good indicator of when
erosion will begin at the 8 m and 13 m bipods.
Figure 3-8. Shear velocity vs. sonar at the 13 m bipod, October 18, 1997 (Note blue line is shear velocity, black line is sonar, red lines are critical shear velocity)
Combined wave-current influence
Another concern regarding shear stress and shear velocity estimates is that they
are calculated based only on the current values and they do not take into account wave
orbital velocities. The shear stress associated with combined waves and currents is
different than with either alone, because of the turbulence generated by the wave-current
interaction (Grant and Madsen 1979).
35
Figure 3-9. Shear velocity vs. sonar at the 8 m bipod, August 19, 1998 (Note blue line is shear velocity, black line is sonar, red lines are critical shear velocity)
This shear stress would be larger than that given by the mean current. Grant and
Madsen’s model was applied to current measurements taken during the “Halloween”
storm and the wave boundary layer was estimated to be a maximum of 11.6 cm thick,
much lower than their bottom current meter at a 29 cm elevation (Madsen et al. 1993).
This lends confidence to the assumption that our bottom current meter at an elevation of
20 cm above the bed is also outside that wave boundary layer.
Analytical models have been tested which calculate a shear velocity based on both
wave and current influence. These not only account for the individual wave and current
influence, but also any nonlinear interactions between the two. Wiberg and Smith (1983)
36
Figure 3-10. Shear velocity vs. sonar at the 5.5 m bipod, October 18, 1997 (Note blue line is shear velocity, black line is sonar, red lines are critical shear velocity)
compared shear velocity estimates calculated from currents alone to those calculated
from two different models, those of Grant and Madsen, and Smith. The field data used
for the analysis was collected at a similar depth, 18 m, and in an area with a similar
sediment size. They found that the shear velocities calculated using the wave-current
models are similar to the values obtained from the measured average velocity profiles,
although the estimates of surface roughness are very different (Wiberg and Smith 1983).
This suggests that recalculating shear velocities with an added wave influence would not
alter the estimates significantly. It raises the concern that estimates of the surface
roughness may not be characteristic of actual values, and often may be higher by up to an
order of magnitude. This paper also suggested that scour under the instrument frame
37
caused original estimates of surface roughness from the data collected by Drake and
Cacchione to be unrealistically high (Wiberg and Smith 1983). This may allow the
surface roughness calculated through the velocity profile method to be used as a quality
control parameter for the data, indicating situations where settling or scour might be a
concern.
Apparent hydraulic roughness
An investigation into estimates of bottom surface roughness showed that some
values are at the extreme limits for the sediment size present. A general idea of the
magnitude of surface roughness values that are expected is determined from
300Dz =
Choosing a representative sediment size of 3φ (0.125 mm) yields a value of surface
roughness on the order of 10-6 m. Calculated values may range from 10-1 m to 10-20 m
with extreme outliers exhibiting a broader range, reaching 10+150 m at the 5.5 m bipod.
Many of these values do not appear to significant physical meaning, but a relative
comparison yields some interesting observations. One observation that occurs following
a storm event is a decrease in surface roughness values estimated from velocity profiles.
One possible explanation for this is that after the storm event, there is more suspended
sediment, which may inhibit turbulence and subsequently cause greater velocities.
Increased velocities would lower the surface roughness estimate by shifting the velocity
profile. The 5.5 m bipod shows more scatter than the 8 m and 13 m bipods with a
significant number of measurements reaching 10-10 m or 10-20 m. The surface roughness
values calculated here do not always show a decrease with increasing currents. This may
38
be realistic considering that this bipod may be inside the surf zone during peak waves and
currents and may be influenced by breaking waves.
Figure 3-11. Surface roughness variation with mean currents at 5.5 m bipod for October 15-21, 1997
Four different storm events were analyzed, one in each month, and all showed this
decrease in surface roughness at the 8 m and 13 m bipods as mean currents increased.
During periods of consistently large currents and waves there is a gradual decrease in
surface roughness throughout the entire period. Plots of mean currents versus surface
roughness trends for each bipod are included in Figures 3-11,12,13 for a single storm
event in October of 1997. The different colors represent current measurements at three
different elevations. At the 8 m bipod, we see a decrease in surface roughness values
with the initial increase in currents to 40 cm/s, but then the currents increase rapidly to
39
80 cm/s and the values of surface roughness are relatively stable. Perhaps the initial fine
sediment has been removed with the first current increase.
Figure 3-12. Surface roughness variation with mean currents at 8 m bipod for October 15-21, 1997
Comparison of surface roughness estimates at the 8 m and 13 m bipods for the week of
October 15-21 show emerging trends. They both commence decreasing about the same
time and have similar ending values, but there appears to be a phase lag between the two.
The 13 m surface roughness values seem to increase first and decrease sooner than those
at 8 m. Surface roughness values vs. mean currents at the 5.5 m bipod show much
greater variation and do not follow the same relationships at the other two bipods.
Surface roughnesses for the entire month of October were found to be largest
during times of low currents and waves. These are on the order of one centimeter and
40
would indicate unrealistically large bedforms, although this observation is consistent at
all three bipods. These are on the same order of magnitude as the surface roughness
estimates that were calculated by Smith and Wiberg (1983) when they used velocity
profiles to estimate roughnesses and did not account for wave-current interaction.
Figure 3-13. Surface roughness variation with mean currents at 13 m bipod for October 15-21, 1997
The possibility of error in shear velocity and surface estimates has already been
addressed, but another potential problem is that the presence of bed forms would alter the
shape of the velocity profile. Extensive research has examined bed forms in rivers and
their effect on velocity profile estimates. Smith and McLean (1977) conducted a study in
the Hood River in Oregon, which found that the velocity profile over a bedform has a
41
convex shape in a semi-log plot, which is different than the traditional linear shape. This
potential problem cannot be addressed with the limited dataset available here.
CHAPTER 4 WAVE TRANSFORMATION IN THE NEARSHORE
Introduction
There is extensive research and theory in the field of Coastal Engineering that
focuses on the evolution of wave properties with onshore propagation. Of particular
interest due to implications for engineering design, are changes in wave direction and
energy. Linear wave theory is generally accepted to provide reasonable estimates of
these changes, with the understanding that there are many other nonlinear interactions
involved in the process. The most basic application of linear wave theory considers a
single direction and frequency, when in reality waves originate from many different
directions with many different frequencies. Directional characteristics can be represented
by a directional spectrum, describing, for each frequency, a range of directions with a
mean wave direction. Based on wave refraction theory, the width of this directional
spectrum should decrease with proximity to shore.
Longuet-Higgins et al. (1991) developed a method for calculating directional
spectral estimates from field data using measurements recorded by a floating buoy. Since
then, many researchers have developed other formulations (Capon et al. 1967; Long and
Hasselmann 1979; Herbers and Guza 1989). Borgman (1969) tested different models for
design use, including what he termed a circular-normal, wrapped-around Gaussian, and a
wrapped-around Hermite series expansion. There have been many other models of
varying complexity developed, including some that adapt to specific properties of the
data. Herbers et al. (1999) considered cross-shore evolution of the mean wave
42
43
propagation direction and a directional spreading parameter. They tested measured
values of these parameters at the spectral peak frequency against those calculated from
linear wave theory and found that linear theory predicted mean wave direction and
spreading well, except for the region inshore of the bar crest where waves were breaking.
Calculations showed additional directional spreading of wave energy in this region, but
consistency in mean propagation directions suggests that this spreading was nearly
symmetric.
Small amplitude wave theory assumes irrotational flow, and an impermeable and
horizontal bottom, which are not realistic in a natural setting. Waves propagating over
real seabeds will be affected by porosity and permeability of bottom sediment, bottom
slope, and bottom surface roughness. They will also experience energy dissipation from
bottom friction, due to nonlinear shear stresses created by a turbulent boundary layer at
the bottom (Dean and Dalrymple 1991). White capping is an additional mechanism of
energy loss, here considered to be secondary to bottom friction. There has been
significant effort directed toward determining friction factors based on bottom velocity
measurements. Jonsson (1966) related the friction factor to maximum bed shear stress
and developed relationships with Reynolds number and bottom roughness parameters.
Whitford and Thornton (1988) applied a momentum balance approach to determine bed
shear stress coefficients from surf zone measurements taken at the FRF. Madsen (1994)
gives explicit formulas for wave friction factors that are dependent on the relative
magnitude of the current shear stress. Several recent studies have used turbulence
measurements to determine near-bottom turbulent shear stress and friction factors
(Trowbridge and Elgar 2001, Smyth and Hay 2002,2003). Although accomplished
44
through many different computational techniques, these all apply linear wave theory to
account for energy dissipation.
One focus of the present study is to compare measured values of wave height,
energy flux, and wave direction to linear wave theory estimates. Guza and Thornton
(1980) investigated differences in energy density spectra predicted from horizontal
velocity and calculated from pressure, and also compared measured and shoaled elevation
spectra assuming onshore propagation. They found reasonably good agreement between
measured and predicted spectra from measurements out to 10 m depths collected at
Torrey Pines Beach in San Diego, California (Guza and Thornton 1980). Error estimates
were around 20% in energy density and variance calculations, and even less for wave
height comparisons, although differences determined from shoaling theory were more
frequency dependent (Guza and Thornton 1980).
Analysis
Development of Analytical Spectrum
The first step toward representing directional wave properties was to develop an
analytical directional spectrum whose width could be varied to represent different wave
directional spreads. The theoretical distribution that was used for the following analysis
was
( ) ( ) ( ) ( ) ( ) ( ) (( ))[ ]022011002 2coscoscos θθθθθθθ −+−+≈−= mdmdmdAAD mm
mp
where 0=θ is directed normal to shore and θ in both is limited to 22πθπ
≤≤− to
include only onshore wave directions (Figure 4-1). The above representation will be used
to approximate the so-called “measured” spectra, determined from velocity and pressure
45
measurements. ( )θD is normalized such that . The subscript “p”
represents the predicted spectrum for future nomenclature. Initially, integer values were
chosen for m, but the realization that other values could be useful in fitting the data led to
an extension of the analysis to non-integer values. The Fourier coefficients for several m
values that are used to approximate the theoretical spectrum are included in Table 4-1.
Figure 4-2 presents the ratios of the Fourier coefficients,
( )∫−
=2/
2/
1π
π
θθ dD
0
1
dd and
0
2
dd , for different values
of m. As m is increased, the theoretical spectrum becomes narrower and the ratios
approach those for the delta function; which is applicable for a single direction.
Figure 4-1. Coordinate system
Dataset
The analysis focuses on estimates of wave direction, transformation, and energy
dissipation during storm conditions in the vicinity of the FRF pier at a longshore position
of approximately 900 m. The dataset consists of current and pressure measurements
recorded in three different water depths (nominally 5.5 m, 8 m, and 13 m), which are
46
located at relatively similar alongshore locations, thus establishing a cross-shore array of
instrumentation. Choosing records from the dataset that contain significant energy helps
to assure that the measurements are meaningful.
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2
n
d n/d
0
m=0m=0.5m=1m=2m=3m=4m=5m=6delta fcn
Figure 4-2. Ratios of Fourier coefficients (curves are fit to 3 points at n = 0,1,2)
Table 4-1. Theoretical Fourier coefficients for different m values m Am d0 d1 d2 d1/d0 d2/d0 0 0.318 0.500 0.637 0.000 1.274 0.000
0.5 0.500 0.318 0.500 0.212 1.572 0.667 1 0.637 0.250 0.424 0.250 1.696 1.000
1.5 0.751 0.212 0.375 0.255 1.769 1.203 2 0.848 0.188 0.340 0.250 1.809 1.330
2.5 0.936 0.170 0.312 0.243 1.835 1.429 3 1.019 0.156 0.291 0.234 1.865 1.500
3.5 1.090 0.146 0.273 0.226 1.870 1.548 4 1.164 0.137 0.259 0.219 1.891 1.599
4.5 1.234 0.129 0.246 0.212 1.907 1.643 5 1.293 0.123 0.235 0.205 1.911 1.667 6 1.408 0.113 0.217 0.193 1.920 1.708
( fcn)δ∞ 1.000 0.159 0.318 0.318 2.000 2.000
47
There are six significant storm events that occur within the four-month period discussed
in previous chapters. Figure 4-3 shows a histogram of the significant wave height values
measured in a water depth of eight meters for each thirty-four minute record of the four
months. To obtain representative results, three samples were analyzed from each of the
six storms. The range of significant wave heights included in this analysis is 1.75 m < Hs
< 3.5 m.
One assumption employed here is Snell’s Law considering bathymetry consisting
of straight and parallel bottom contours. Figure 4-4 presents bathymetry in the area
around the time that these data were collected and the red circles represent the
approximate locations of the 5.5 and 8 m bipods. The bipods are located near profile line
66 in the plot, which is an area where this assumption is reasonable.
Figure 4-3. Significant wave heights measured in October 1997, November 1997, May 1998, and August 1998
48
Figure 4-4. Bathymetry in vicinity of bipod instrumentation (13 m bipod is located 690 m offshore of 8 m bipod; http://www.frf.usace.army.mil)
Development of Directional Spectrum from the Data
Directional spectral estimates were determined from the measured data during
significant storm events using the p-u-v method, which follows the work of Longuet-
Higgins et al (1961). The following equations used to calculate the directional spectrum
are based on pressure represented in “feet of water.” The coordinate system is taken as
positive onshore and south for the measured velocity components, u and v (Figure 4-1).
For simplicity, this analysis considers the pressure sensor and current meter to be located
at the same horizontal location and distance above the bottom. The auto and cross
spectra are calculated with a segment length of N=128, which gives a resolution of
49
0.015 Hz and 64 degrees of freedom. The actual number of degrees of freedom is
somewhat greater due to the fact that half-lapped segments are used.
The energy density spectrum is represented by
( ) ( ) ( )θθθ ηηηη ,, fDSfS =
where the directional spectrum is approximated by the Fourier series
( ) ( ) ( ) ( ) ( ))2cos()cos(, 0220110 θθθθθ −+−+= fCfCfAfD
The coefficients are calculated as follows (Longuet-Higgins et al., 1961)
( )π21
0 =fA
( ) ( )( ) ( ) ( )zkzkfS
fSfA
pu
pu
ηηπ=1
( ) ( )( )( )kh
zhkfzku sinhcosh2 +
=π
( ) ( )( )( )kh
zhkzk p coshcosh +
=
( ) ( ) ( )( ) )(22 zkfS
fSfSfAu
vvuu
ηηπ−
=
( ) ( )( ) ( ) ( )zkzkfS
fSfB
pu
pv
ηηπ=1
( ) ( )( ) ( )zkfSfSfBu
uv22
2
ηηπ=
The values of C and n n0θ are determined as
( ) ( ) ( ) 2,1,22 =+= nfBfAfC nnn
( ) 2,1,tan 10 =
= − n
AB
fn
nnθ
50
The mean wave direction ( 0 )θ for each frequency to be used in calculations is taken as
the location of the maximum value of the computed directional spectrum.
The directional spectrum as determined above will be referred to as the
“measured” directional spectrum in future discussion, although this is an estimate from
the data. Figure 4-5 presents an example of the measured directional spectra at the
bottom current meter for the 5.5, 8, and 13 m bipods at the peak frequency of 0.094 Hz.
Time is measured in hour-minutes, where 100 represent 1 h and 1 represents 1 min.
Figure 4-5. Measured spectra for November 7, 1997 time=2200
Table 4-2 presents measured mean wave direction values in degrees for the peak
frequency of each storm event. These values are averaged over the three current meters at
each location. The last column includes corresponding peak direction values that were
51
calculated by the FRF using the iterative maximum likelihood method from
measurements recorded at the 8m array of 15 pressure gages. The 8 m array values are
often different than the measured mean wave directions at the 8 m bipod, sometimes by
up to 14 degrees. These are calculated for longer records of 2.5 h versus 34 min, but the
differences bring up a concern for the accuracy of measured mean wave direction
estimates. Measurements are expected to show mean wave directions approaching closer
to onshore ( 00 = )θ at shallower water depths, and this is generally the case, with several
exceptions.
Table 4-2. Measured mean wave directions at peak frequency
Date Time f (Hz) ( )130θ ( )80θ ( )50θ ( )mArrayp 8θ
19-Oct-97 700 0.141 35.3 36.5 32.7 22.0 19-Oct-97 1216 0.125 39.7 33.7 32.0 20.0 20-Oct-97 100 0.094 18.4 19.2 16.5 12.0 7-Nov-97 2200 0.094 18.2 16.0 10.9 10.0 7-Nov-97 2342 0.094 17.7 11.3 14.5 10.0 8-Nov-97 208 0.094 18.7 20.8 12.8 10.0 13-Nov-97 1742 0.133 22.2 19.4 12.0 12.0 13-Nov-97 2008 0.125 12.2 16.0 14.5 10.0 13-Nov-97 2200 0.117 13.2 9.7 7.8 5.0 13-May-98 852 0.086 8.7 9.8 10.2 14.0 13-May-98 1442 0.078 12.4 7.5 12.7 6.0 13-May-98 1816 0.078 8.7 9.8 10.2 12.0 2-Aug-98 916 0.133 8.9 5.0 3.7 8.0 2-Aug-98 1300 0.133 4.2 3.5 1.7 -10.0 2-Aug-98 1516 0.133 6.8 1.9 3.2 -10.0 27-Aug-98 1408 0.094 -36.7 -31.8 -29.5 -32.0 27-Aug-98 1634 0.086 -35.2 -33.6 -31.4 -28.0 27-Aug-98 1816 0.086 -32.4 -23.2 -26.4 -28.0
Determination of m Values in ( ) ( )02cos θθθ −= m
mp AD
The m value that gives the best-fit analytical spectrum to the measured data is
determined through implementation of two different approaches. The first approach
52
deals directly with the coefficients of the measured and predicted spectra, while the
second employs a curve fitting technique. The primary purpose of discussing the first
approach is to emphasize that the second approach gives a greatly improved fit, although
it has some limitations.
Comparison of Fourier coefficients
The spectrum ( ) ( )02cos θθθ −= m
mp AD was determined by matching Fourier
coefficient ratios 0
1
dd and
0
2
dd . This approach was based on determining m to minimize
as follows ( fmz , )
( )
( )( )
( )( )
( )( )
( )( )
+
−
+
−
=2
0
2
2
0
1
2
0
2
0
2
2
0
1
0
1
,
fAfC
fAfC
dd
fAfC
dd
fAfC
fmz mm
A cutoff of 6 was used for m in recognition that the measured spectra were generally
wide and including higher m values would not provide an improved fit. Generally, the
central frequencies give higher best-fit m coefficients on the order of 4 to 6, producing a
narrower theoretical spectrum. The higher frequencies, and sometimes even the lower
ones, give a smaller best-fit m coefficient on the order of 2. Figure 4-6 a, b show this
trend. The 5.5 m bipod measurements often determine almost all best-fit values to be the
maximum allowed in this analysis (mmax=6). Plot c of Figure 4-6 is one example. These
values follow the general trend expected; with larger m values at inshore bipods, yet the
fit to measurements is not very good. Figure 4-7 compares the measured (solid line) and
fit (dashed line) directional spectra for the same three runs at a frequency of 0.13 Hz.
Predicted spectra are much too narrow. Error values determined from
53
( ) ( )[ ]∑=
−=n
iipim DD
n 1
22 1 θθε
are presented in Figure 4-8 over the entire frequency range and they are extremely high.
Minimization of the error between coefficients does not imply that the error between the
spectra is at a minimum.
Two-sided nonlinear fit
This second approach attempts to provide a better fit by focusing on direct least
squares fit rather than matching directional spectral coefficient ratios. The measured
spectrum is often very wide, extending past the offshore directional limit of 2πθ ±= .
This makes it difficult for a spectrum ( ) ( )02cos θθθ −= m
mp AD based on a single m
value to give an accurate prediction, since it is forced between the limits of 2π− and
2π .
This problem was overcome by fitting a different m value to each side of the measured
spectrum. The modified formulation for the analytical spectrum is
( )02cos)( θθθ −= Lm
pL AD , 02θθπ
≤≤−
( ) ( )02cos θθθ −= Rm
pR AD , 20πθθ ≤≤
The subscripts “L” and “R” denote values for the left and right side of the directional
spectrum respectively, and A is the same for both sides. The best-fit values of mL, mR,
and A are determined through a nonlinear least squares data fitting technique, utilizing
the Gauss-Newton method.
Plots are included (Figures 4-9,10) of measured and predicted spectra and error
calculations for the same data included in Figures 4-7 and 4-8 to facilitate an appreciation
54
Figure 4-6. Error versus m value comparison for October 20, 1997 time=100 a) 13 m b) 8 m c) 5.5 m
55
Figure 4-7. Comparison of measured and best-fit spectra from matching coefficients for October 20, 1997 time=100 a) 13 m b) 8 m c) 5.5 m
of the quality of the fit. It is evident that the fit is greatly improved, with significantly
lower error values by more than an order of magnitude. Note that the predicted spectra
are forced to zero at 90 degrees, because allowing waves to come from onshore would be
unrealistic.
The best-fit m values for each side of the spectrum vary with frequency.
Representative plots are presented in Figure 4-11 for measurements from the 8 m bipod
bottom current meter. There is a general trend of increasing separation between left and
right m values with increasing frequency. The decrease of mR for 00 ≥θ is a result of the
increasing percentage of the measured spectrum that is outside of the 22πθπ
≤≤− range
56
with increasing frequency. This trend can be observed in Figure 4-12, which includes
measured and predicted directional spectra at different frequencies, for the run included
in plot c of Figure 4-11. The low frequency spectrum has the entire width contained
within the range of onshore directions, since this energy probably represents storm swell
that has traveled a significant distance and is dominantly oriented onshore. Higher
frequency plots show an increasing portion of the width on the right side to be outside of
this range as the mean direction increases, so best-fit m coefficients that provide the
smallest error are not necessarily representative of the measured spectrum.
Figure 4-8. Error between spectra fitted from coefficient ratios for October 20, 1997 time=100
The left side m values increase slightly with increasing frequency. The right side
values show a decrease, but it is primarily due to a limitation of the method and probably
not a meaningful trend. This problem arises because the width of the measured spectrum
57
is very large. One way to overcome this would be to use a more accurate method of
predicting the measured spectrum, which would most likely decrease the width and
eliminate or minimize the percentage of the directional spectrum outside of the 2π
±
range.
Figure 4-9. Comparison of measured and best-fit spectra from curve fitting for October 20, 1997 time=100 a) 13 m b) 8 m c) 5.5 m
Comparison of Data to Linear Wave Theory Calculations
The bipod measurements provide a basis for testing linear wave theory by
comparing theoretical values at inshore water depths to the associated measurements by
the bipod instrumentation. In the following analysis, calculated directional spectrum
values are determined by refracting data from the13 m bipod and will have a subscript
58
“c”. Measured values are those determined directly from bipod instrumentation at the
water depth of interest and will have a subscript “m”.
Figure 4-10. Error between spectra from curve fitting for October 20, 1997 time=100
These comparisons are conducted within the frequency range of 0.05 to 0.2 Hz,
which is the frequency range of significant energy evident from pressure measurements.
The high frequency cutoff is chosen as 0.2 Hz because the water surface energy density
spectral ( )( )fSηη values that are calculated from the pressure response factor ( )( )zk p
begin to increase unrealistically at higher frequencies at the 13 m bipod, and are not
representative of field conditions. This is evident in Figure 4-13. This cutoff is also
consistent with linear wave theory since 0.2 Hz represents a wave period of 5 s, and at a
59
water depth of 13 m waves with a period of less than 4 s would be deep-water waves and
would not interact with the bottom.
Figure 4-11. Variation of m values with frequency range at 8 m a) May 13, 1998 time=852 b) November 7, 1997 time=2200 c) October 19, 1997 time=700
Figure 4-12. Directional spectrum variation with frequency at 8 m bipod for October 19, 1997 time=700
60
Figure 4-13. Energy density spectral values for October 19, 1997 time=700 a) 13 m b) 8 m c) 5.5 m
Wave direction
A first comparison is of measured and calculated central wave direction ( 0 )θ values
from directional spectral estimates. Snell’s Law is applied to find refracted wave
direction values at the two inshore bipods. The calculated wave direction is defined as
( )( ) ( )
( )
= −
13
13818
sinsin
CC m
c
θθ
and the measured wave direction ( )( )8mθ as the direction of the maximum value of the
computed directional spectrum. Figure 4-14 includes a plot of measured vs. calculated
wave directions at the 5.5 m and 8 m bipods on October 19, 1997. The different symbols
represent measurements from each of the three current meters. Measured mean wave
61
directions at the 8 m bipod appear to be well predicted, while those at the 5.5 m bipod
show a slight offset. Figure 4-15 compares these same values at the 5.5 m bipod in a
different format. Plot a presents measured angles in degrees and plot b presents the
difference in measured and calculated angles ( )cm θθ − in degrees. There is a
deterioration of measured and calculated directions in the low frequency range since there
is not significant energy present and directional spectral estimates become less
meaningful. This specific plot shows an overall slight offset of approximately seven
degrees.
Figure 4-14. Mean wave direction comparison for October 19, 1997 time=700
Average values between the three current meters for each of 3 records in October
1997 are presented in Figure 4-16. These values represent the difference in measured and
calculated angles ( cm )θθ − in degrees. It is important to note that the large variation
between measured and calculated wave directions at low frequencies is a result of the
wave direction deterioration shown in Figure 4-15. In the frequency range with
significant energy, the differences fluctuate around zero, suggesting that angles are
predicted fairly well by linear theory. When there does appear to be an offset, it is
usually positive, which represents greater measured values. This suggests that linear
62
theory refracts the angles too much toward normal incidence in these
instances.
mθ ( )cm θθ −Figure 4-15. Measured and calculated wave direction differences for October 19, 1997
time=700 at 5.5 m bipod a) , degrees b) , degrees
Refracted m values
An extension of the refracted angle analysis is to determine how well the
directional spectrum at inshore bipod locations can be predicted from refraction of
directional spectral estimates from the offshore bipod. This is accomplished by
comparing best-fit m values for a refracted directional spectrum versus those for a
measured directional spectrum at 8 m and 5.5 m water depths. The refracted directional
spectrum is (Lee et al. 1980)
63
( ) ( ) ( ) ( )( ) ( )
=
fkfkfDfD pc
1313 ,, θθ
where has been normalized to have an area of one. The corresponding ( )13pD θ values
for the refracted spectrum are
( )
( )
= −
13
131 sinsin
CC
c
θθ
The refracted spectrum is also normalized to one before fitting m values using
Figure 4-16. Average measured and calculated wave direction differences for October 1997 a) ( )cm θθ − , degrees 8 m b) ( )cm θθ − , degrees 5.5 m
64
( ) ( )( )
=
∫−
2/
2/
,
1,, π
π
θθθθ
dfDfDfD
c
cc
Best-fit mL and mR values for the refracted spectra are determined using the method
discussed previously. The curve fitting procedure was used since this was determined to
provide a much better fit than the alternate approach.
Figure 4-17. Measured and refracted directional spectra for November 13, 1997 time=1742
Sample refracted and measured spectra for November 13, 1997 at a frequency of
0.13 Hz are presented in Figure 4-17. The refracted spectra are narrower than those
measured at both inshore bipod locations. This suggests that linear theory predicts more
refraction than associated with the measurements. This comparison is further reflected in
the values of m coefficients. Figure 4-18 shows that refracted m values are greater than
measured values, and show a slight decrease with increasing frequency as opposed to the
slight increase of measured m values. The measured right side coefficients are much
lower for reasons discussed previously. The difference between measured and refracted
65
spectral widths may also be affected by the method chosen to estimate the measured
directional spectra.
Figure 4-18. Comparison of refracted and measured m values over frequency range for November 13, 1997 time=1742 a) 8 m b) 5.5 m
Wave height comparisons
The previous analysis focusing on direction leads to a comparison of significant
wave height values by applying the concept of conservation of energy flux. Energy flux
is defined as
( ) mgCgSf θρ ηη cos=ℑ
A calculated water surface spectrum was obtained at the 8 m inshore bipod location using
the equation
66
( ) ( ) ( ) ( )
( ) ( )
=
8
)13(
8
13)13(8 cos
cos
c
m
g
gmc C
CfSfS
θθ
ηηηη
and similarly for the 5.5 m inshore bipod. In the above equation the value of mθ was
taken as the location of the maximum value of the computed directional spectrum. This
leads to slightly larger energy flux values than if the entire range of directions was
considered. The zero moment of the water surface spectrum gives a significant wave
height value of
04 mHmo =
( )dffSmHz
Hz∫=2.0
05.00 ηη
and calculated and measured values are obtained by using the respective water surface
spectra.
Ratios of measured to predicted values of significant wave height for the 18
records considered are plotted in Figure 4-19. This shows most measured values being
less than predicted since almost all of the ratios at the inshore bipods are less than 1,
ranging from a 5% increase to a 25% decrease. This decrease is expected since friction
losses have not yet been considered.
Energy flux comparisons
Measured energy flux values are found for each bipod location and compared by
considering the percent loss between bipods. The percent energy loss between the 13 m
and 8 m bipods is determined from
67
( ) ( )
( )
%100*)(
)()(% 2.0
05.013
2.0
05.0
2.0
05.0813
ℑ
ℑ−ℑ
=
∫
∫ ∫Hz
Hzm
Hz
Hz
Hz
Hzmm
dff
dffdffloss
and similarly for the distance between the 8 m and 5.5 m bipods. Table 4-3 shows the
average energy loss values of the three current meters at the bipod for each record, with
negative percent values representing cases that had greater measured energy flux values
at inshore bipods. There are many instances of significant loss, with some reaching more
than one third of the total energy flux. This enforces the need to consider energy loss in
Figure 4-19. Significant wave height ratios versus cross-shore position (FRF coordinate system)
68
engineering design and planning, since neglecting this component would lead to a
significant overestimation of energy flux values in many cases, especially for propagation
over long distances.
Figure 4-20 shows the average energy flux variation over frequency range for all
18 storm events, and includes percent loss of average energy flux values. The dashed
lines in plots b and c indicate the measured energy flux at the offshore bipod, which is the
same as that which would be predicted with no energy loss. These are compared to the
measured values shown by the solid lines. There are different cross-shore separation
distances between the three bipods. The 13 m and 8 m bipods are separated by 690 m in
the cross-shore direction, whereas the 8 m and 5.5 m bipods are separated by only 333 m.
Of interest is that the average percent energy loss between the 8 m and 5.5 m bipods is
greater than between the 13 m and 8 m bipods (Table 4-3), even though the separation
distance for the smaller percentage energy loss is twice as long. The percent energy
losses of the average values included in Figure 4-20 also indicate this trend. Of course,
the reason is that bottom friction is more effective in causing energy loss in shallower
water.
Friction factor
The wave height and energy flux analyses showed over-prediction from linear
theory since energy loss was not considered. This analysis assumes friction is the only
cause of energy change between bipods. In reality there are many other contributing
factors, including the possibility of energy growth due to wind, energy loss due to white
capping, or the redistribution of energy within the spectrum due to non-linear
interactions. By accounting for frictional energy loss in the energy flux calculation, a
representative friction factor can be determined for the site.
69
Table 4-3. Average % energy loss values between bipods
Year Month Day Time % Loss (13 - 8)
% Loss (8 - 5.5)
1997 10 19 700 17.12 1.11 1997 10 19 1216 25.76 -1.18 1997 10 20 100 -7.11 10.13 1997 11 7 2200 2.45 3.28 1997 11 7 2342 35.57 -8.18 1997 11 8 208 -12.34 5.51 1997 11 13 1742 10.74 8.61 1997 11 13 2008 -17.13 27.13 1997 11 13 2200 17.58 22.75 1998 5 13 852 20.70 16.55 1998 5 13 1442 3.47 21.01 1998 5 13 1816 20.26 21.08 1998 8 2 916 10.87 0.84 1998 8 2 1300 -5.77 2.95 1998 8 2 1516 -2.99 14.12 1998 8 27 1408 14.65 28.41 1998 8 27 1634 25.24 27.74 1998 8 27 1816 15.61 25.81
Average 9.70 12.65
The friction factor is estimated using
[ ] xdfdf DbDa
Hz
Hzb
Hz
Hza ∆+=ℑ−ℑ ∫∫ εε
212.0
05.0
2.0
05.0
where is the total cross-shore distance between the 13 and 5.5 m bipods, andx∆ Dε is the
energy loss term expressed as
3
8 bbxyD uffu ρτε =⋅=
which accounts for energy damping by bottom friction. In this expression, u is the
velocity time series measured at the bottom current meter, located approximately 20 cm
above the bottom, and represents the friction factor. Figure 4-21 shows surveyed
bathymetry that was collected near the bipod locations and around the same general
b
ff
70
timeframe that the data were collected, giving some insight into the bottom
characteristics between instruments. These survey lines were measured at the FRF using
the Coastal Research Amphibious Buggy (CRAB) and Lighter Amphibious Resupply
Cargo (LARC). Specific dates are included in the legend.
Figure 4-20. Average measured and predicted energy flux values a) 13 m b) 8 m c) 5.5 m
The calculated friction factors of interest are those resulting from total velocity
measurements taken at the bottom current meter, since this velocity is the most
representative of the bottom velocity. This total velocity is calculated as the magnitude
of the instantaneous velocity vector. These friction factors are included in Table 4-4.
Most values from the bottom current meter are in the range of 0 to 0.2, although there is a
considerable spread from 0 to 0.32. The mean and standard deviation values for the
71
bottom current meter are 0.116 and 0.105 respectively. The negative value on August 2,
1998 is probably not real, since the bottom current meter had a flag for low beam
correlation. The other current meters gave estimates of 0.009 and 0.014 for that record,
which are more reasonable. Friction factor estimates from velocities measured at the
higher current meters were found to check for consistency. A histogram of all of the
calculated values is shown in Figure 4-22. Combining values from all current meters
gives a mean of 0.094 and a standard deviation of 0.08.
Figure 4-21. Surveyed bathymetry in vicinity of bipod instrumentation
It does not seem unreasonable to find a range of factors during storms and for
different storm conditions since friction factors can change as currents increase and
sediment is displaced. Figure 4-23 shows that friction factors generally increase with
significant wave height values. It is important to note that if the highest waves were
72
breaking before reaching the 5.5 m bipod, then friction factors would be unrealistically
high, since the calculation assumes that all energy dissipation is from bottom friction.
Table 4-4. Friction factor estimates from bottom current meter Date Time ff Hmo
10/19/1997 700 0.101 2.28 10/19/1997 1216 0.111 2.86 10/20/1997 100 0.025 2.22 11/7/1997 2200 0.056 1.88 11/7/1997 2342 0.283 2.09 11/8/1997 208 -0.029 1.63 11/13/1997 1742 0.154 2.81 11/13/1997 2008 0.1 2.82 11/13/1997 2200 0.318 3.01 5/13/1998 852 0.165 3.31 5/13/1998 1442 0.094 2.99 5/13/1998 1816 0.246 3.43 8/2/1998 916 0.016 2.1 8/2/1998 1300 -0.083 2.19 8/2/1998 1516 0.042 2.11 8/27/1998 1408 0.129 3.35 8/27/1998 1634 0.177 3.45 8/27/1998 1816 0.186 3.13
Average 0.116 Standard Deviation 0.105
It seems useful to determine a single representative friction factor for this specific
location. This was accomplished by using average energy flux and energy loss values
over all 18 storm events in the above equation. This analysis resulted in a friction factor
of 0.170 between the 13 m and 8 m water depths and 0.177 between 8 m and 5.5 m water
depths. These two values are consistent and 0.17 is determined as a representative
friction factor in the vicinity of the bipod instrumentation.
Reynolds Stresses
Another approach to examine shear stress was through calculation of the
Reynolds stresses. These are related to shear stress by the following equations
73
' 'wuzx ρτ −=
''wvzy ρτ −=
where positive values are onshore and south. The time series were filtered to include only
the frequencies within the range 0.05 to 0.2 Hz. It was expected that these values would
be comparable to shear stress values calculated from the velocity profile method
discussed in Chapter 3. Those values generally ranged from 0 to 0.5 N/m2, as calculated
based on mean velocities. The stresses based on Reynolds stresses are larger, but they
represent shear stress values determined from the oscillatory velocity component.
Examples of values for October 1997 are presented in Table 4-5. The notation X1, X2,
and X3 represents the bottom, middle, and top current meters respectively.
Figure 4-22. Histogram of calculated friction factors at all current meters
74
-0.15-0.1
-0.050
0.050.1
0.150.2
0.250.3
0.35
0 1 2 3 4
Hmo m
ff
Figure 4-23. Friction factor variation with wave height at the bottom current meter
One disturbingly consistent feature of these results is that the values at the middle
current meter often have a different sign and magnitude than the other two current
meters, at all three bipod locations. If there was flow reversal within the water column,
these values would be expected to show a trend. If there was a problem with the current
meter at a specific bipod, the anomalous value should be present only at one location.
Neither of those situations occurs, and this consistency is present throughout all four
months. At present, this feature cannot be explained.
Discussion
To reinforce the analysis presented in this chapter, this section provides a brief
overview of primary discussion points. The analytical spectrum establishes a simplified
representation of directional properties of the wave data that can be used to approximate
the measured spectrum, obtained through a direct Fourier Transform method. Later
comparisons are facilitated by considering the m value, or power of the cosine curve as a
measure of the properties of the spectrum. The first approach matched Fourier
75
coefficients to the data, but this resulted in a poor fit to the directional spectra. The
second approach employed a curve fit to individual sides of the spectrum, which gave a
vastly improved fit. One limitation of this approach is that measured spectra often extend
beyond the ± 90º limits, giving unrealistically low right side m values for 00 ≥θ .
Table 4-5. Reynolds stresses for October 1997 (X1, X2, X3 represent bottom, middle and top current meters respectively)
Date Time Gage ''wuzx ρτ −= ''wvzy ρτ −=
10/19/1997 700 13X1 1.66 0.68 13X2 -2.34 -0.36 13X3 2.23 1.43 8X1 2.50 1.69 8X2 -6.79 -2.81 8X3 4.67 4.67 5X1 -0.96 -0.51 5X2 -15.27 -7.37 5X3 -1.34 -1.01
10/19/1997 1216 13X1 2.20 1.44 13X2 -8.20 -3.74 13X3 3.25 2.72 8X1 0.88 1.74 8X2 -10.23 -4.91 8X3 7.33 3.91 5X1 5.66 4.76 5X2 -15.23 -5.18 5X3 0.97 1.93
10/20/1997 100 13X1 3.77 0.74 13X2 -7.67 -2.87 13X3 3.77 1.79 8X1 6.68 2.44 8X2 -15.03 -5.58 8X3 3.88 3.15 5X1 -0.93 -0.59 5X2 -17.12 -4.23 5X3 -3.65 0.34
A second focus was to compare measurements with linear wave theory
predictions. Refracted mean wave direction angles were similar to measurements. If
76
offsets were present, they usually represented too much refraction by linear wave theory.
Interestingly, when refracting the entire directional spectrum, the width was narrower
than that which was measured, representing an overestimation by refraction theory on the
whole. This trend is also reflected in the refracted and measured m value comparisons.
The wave height and energy flux calculations combined shoaling and refraction theory
and showed smaller measured values, as expected when energy losses are not accounted
for. Friction factors were estimated by accounting for energy losses, and most values
were in the range of 0 to 0.2, although these appear to vary with storm conditions. A
representative value of 0.17 was identified for this location using average energy flux and
energy loss values. Calculated Reynolds stresses were very strange with the middle
current meter yielding significantly different values at all three water depths; this effect
remains unexplained.
CHAPTER 5 CONCLUSIONS
A knowledge of wave characteristics, sediment characteristics, and bed elevation
within the nearshore zone are imperative for engineering design and planning purposes.
Predictions of these processes are utilized when making decisions regarding coastal
development and beach preservation. Improving our knowledge of nearshore processes
and the accuracy of current prediction results is required for coastal planners to make
better decisions. This study presents analysis of sonar, pressure, and current
measurements to evaluate erosion thresholds, wave evolution and bottom friction results.
Comparisons between mean current and sonar measurements in Chapter 1 defined
a mean current threshold of 20 cm/s for bed erosion. In addition, this comparison
highlights the need to consider fair-weather conditions for sediment transport, since many
significant bed elevation changes occurred when wave heights were less than 2 m. No
orbital velocity threshold was determined, but combined waves and currents always
caused a bed elevation change when velocities reached 60 cm/s. Alongshore currents
appeared to coincide with wind velocities, though cross-shore currents did not.
Sonar histograms at the 5.5 m and 13 m bipods generally showed a well-defined
peak with minimal spreading, lending confidence to sonar estimates of bottom elevation.
The 8 m bipod consistently showed spreading at depths greater than the peak, an
observation that remains unexplained. One situation in late August 1998 at the 13 m
bipod shows two peaks following a significant storm event. This may be an indication of
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the presence of fine, silty material, although this hypothesis cannot be validated with
confidence.
Velocity profile analyses provided estimates of shear velocity and surface
roughness. The shear velocity proved a good indicator of bottom elevation change at the
8 m and 13 m bipods, with erosion beginning zero to three hours after it crossed the
threshold for movement. Shear velocity estimates at the 5.5 m bipod always remained
below this threshold. Surface roughness values at the 8 m and 13 m bipods decreased
with increasing mean currents. The 5.5 m bipod experiences more scatter in surface
roughness estimates and does not always follow this relationship. Comparisons of the
variation in surface roughness values at the 8 m and 13 m bipods indicate a phase lag,
with 13 m values increasing first and decreasing sooner than 8 m.
Chapter 4 focused on evolution of wave characteristics. Measured directional
spectra were obtained for each bipod location using the Direct Fourier Transform
method. A simplified analytical spectrum based on a cosine curve of varying power (m)
was used to approximate measured values. A nonlinear least-squares curve fit to each
side of the measured spectrum proved the most accurate way of determining best-fit m
values. Limiting wave directions to onshore introduced some problems with mR
predictions being unrealistically low when measured spectra were outside of this range.
Unfortunately, this could not be avoided since allowing waves to come from onshore
would not be realistic.
The comparison with theoretical calculations employed shoaling and refraction
theory. Refracted mean wave directions were similar to those measured, with a slight
overrefraction by the theory. On the whole, the theory predicted a narrower spectrum
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than that measured, representing overrefraction. This may be expected at the 5.5 m bipod
if it were inside the surf zone, since there can be directional spreading associated with
wave breaking. This same observation at the 8 m bipod seems more surprising. It is very
possible that this result represents a limitation of the method used to estimate measured
directional spectra. Energy flux calculations emphasized the need to consider energy
loss, which reached as high as one third of measured energy flux values. Friction factors
showed considerable variation with storm conditions, although most values fell within the
range of 0 to 0.2. This is reasonable, as storm conditions encompass varying current
intensities that interact with bottom sediment, affecting roughness and subsequently
energy loss from bottom friction. Through utilization of average energy flux and energy
loss values, a representative friction factor of 0.17 was determined for the area in the
vicinity of bipod instrumentation.
Reynolds stress calculations yielded unexpected results. The bottom and top
current meter shear stress estimates were consistent in magnitude and direction. The
middle current meter values were inconsistent at all three bipod locations. The
systematic occurrence of this feature in measurements recorded by different instruments
suggests validity, yet the cause remains unexplained.
One specific area where these results could be improved for future discussion is in
the representation of measured directional spectra. This analysis utilized a simple
technique, and much work has been done to improve upon this method and develop
techniques that are data adaptive. The measured spectra were very wide, which caused
problems when fitting the analytical representation and may have influenced results from
the refraction comparison. Another limitation of this analysis deals with the erosion
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discussion, since the sonar measurements only record bottom elevation change. Similar
studies have incorporated suspended sediment measurements or side scan sonar images,
which provide a more complete basis for sediment transport determination.
LIST OF REFERENCES
Beach, R. A. and R. W. Sternberg (1996). “Suspended-sediment transport in the surf zone: response to breaking waves.” Continental Shelf Research 16(15): 1989-2003.
Beavers, R. L. (1999). "Storm Sedimentation on the Surf Zone and Inner Continental Shelf, Duck, North Carolina." Thesis. Geology, Duke University.
Beavers, R. L., P. A. Howd, W. A. Birkemeier, and K. K. Hathaway (1999). "Evaluating profile data and depth of closure with sonar altimetry." Coastal Sediments 1: 479-490. Long Island, NY, ASCE.
Birkemeier, W. A., A. W. Dewall, C. S. Gorbics, and H.C. Miller (1981). "A user's guide to CERC's Field Research Facility." CERC Miscellaneous Report 81-7, US Army Corps of Engineers (US ACE), Coastal Engineering Research Center, Fort Belvoir, Va.
Borgman, L. E. (1969). "Directional spectra models for design use." Offshore Technology Conference 1: paper no. OTC-1069. Houston, Texas.
Capon, J., Greenfield, R. J. and Kolker, R. J. (1967)."Multidimensional maximum-likelihood processing of a large aperture seismic array." Proc. IEEE 55 192-211.
Conley, D. C. and R. A. Beach (2003). “Cross-shore sediment transport partitioning in the nearshore during a storm event.” Journal of Geophysical Research 108(C3): 3065.
Dean, R. G. and R. A. Dalrymple (1991). Water Wave Mechanics for Engineers and Scientists, World Scientific, Singapore.
Dean, R. G. and R. A. Dalrymple (2002). Coastal Processes with Engineering Applications. Cambridge, MA, Cambridge University Press.
Dolan, R. and R. E. Davis (1992). “Rating northeasters.” Mariners Weather Log 36: 4-16.
Dolan, R., H. Lins, and B. Hayden (1988). “Mid-Atlantic coastal storms.” Journal of Coastal Research 4: 417-433.
Elgar, S., E. L. Gallagher, and R. T. Guza (2001). “Nearshore sandbar migration.” Journal of Geophysical Research 106(C6): 11,623-11,627.
81
82
Gallagher, E. L., S. Elgar, and R. T. Guza (1998). “Observations of sand bar evolution on a natural beach.” Journal of Geophysical Research 103(C2): 3203-3215.
Grant, W. D. and O. S. Madsen (1979). “Combined wave and current interaction with a rough bottom.” Journal of Geophysical Research 84(C4): 1797-1808.
Green, M. O. and J. D. Boon (1988). “Response characteristics of a short-range, high-resolution digital sonar altimeter.” Marine Geology 81: 197-203.
Guza, R. T. and E. B. Thornton (1980). “Local and shoaled comparisons of sea surface elevations, pressures, and velocities.” Journal of Geophysical Research 85(C3): 1524-1530.
Guza, R. T. and E. B. Thornton (1985). “Velocity moments in nearshore.” ASCE Journal of Waterway, Port, Coastal and Ocean Engineering 111: 235-256.
Harris, C. K. and P. Wiberg (2002). “Across-shelf sediment transport: Interactions between suspended sediment and bed sediment.” Journal of Geophysical Research-Oceans 107(C1): Art. No. 3008.
Herbers, T. H. C., S. Elgar, and R. T. Guza (1999). “Directional spreading of waves in the nearshore.” Journal of Geophysical Research-Oceans 104(C4): 7683-7693.
Herbers, T. H. C. and R. T. Guza (1989). "Estimation of wave radiation stresses from slope array data." Journal of Geophysical Research 94: 2099-2104.
Hoefel, F and S. Elgar (2003). "Wave-induced sediment transport and sandbar migration." Science 299: 1885-1887.
Howd, P. A., R. L. Beavers, and K. K. Hathaway (1994). “Shoreface morphodynamics I: Cross-shore scales of fluid flows.” EOS, Transactions, American Geophysical Union: 339.
Jonsson, I. G. (1966). "Wave boundary layers and friction factors." Proceedings of the 10th International Conference on Coastal Engineering 1: 127-148. Tokyo, Japan, ASCE.
Keen, T. R., R. L. Beavers, P. A. Howd, and K. K. Hathaway (2003). “Shoreface sedimentation during a northeaster at Duck, North Carolina, U.S.A.” Journal of Coastal Research 19(1): 24-40.
Lee, Y. K., F. H. Wu, W. Wier, P. Parnicky, and D. DeMers (1980). "Methodology for computing coastal flood statistics in southern California." Pasadena, CA, Tetra Tech Inc.
Leffler, M. W., C. F. Baron, B. L. Scarborough, P. R. Hodges, C. R. Townsend (1998). "Annual data summary for 1995 CHL Field Research Facility." Technical Report CHL 97-14, USACE, Waterways Experiment Station, Vicksburg, MS.
83
Long, R. B. and K. Hasselmann (1979). "A variational technique for extracting directional spectra from multi-component wave data." Journal of Physical Oceanography 9: 373-381.
Longuet-Higgins, M. S., D. E. Cartwright, and N. D. Smith (1963). "Observations of the directional spectrum of sea waves Using the motions of a floating buoy." Ocean Wave Spectra Easton, MD, Prentice-Hall.
Madsen, O. S., Y. Poon, and H. C. Graber (1988). "Spectral wave attenuation by bottom friction: Theory." Proceedings of the 21st International Conference on Coastal Engineering 1: 492-504. Delft, Netherlands, ASCE.
Madsen, O. S., L. D. Wright, J. D. Boon, and T. A. Chisholm (1993). “Wind stress, bed roughness and sediment suspension on the inner shelf during an extreme storm event.” Continental Shelf Research 13(11): 1303-1324.
Miller, H. C., S. J. Smith, D. G. Hamilton, and D. T. Resio (1999). "Cross-shore transport processes during onshore bar migration." Coastal Sediments 2: 1065-1080. Long Island, NY, ASCE.
Nearhoof, S. L. (1992). Box core data set - survey line 62. Duck, NC, CERC Field Research Facility.
Osborne, P. D. and B. Greenwood (1992). “Frequency dependent cross-shore suspended sediment transport. 1. A non-barred shoreface.” Marine Geology 106: 1-24.
Smith, J. D. and S. R. McLean (1977). “Spatially averaged flow over a wavy surface.” Journal of Geophysical Research 82(12): 1735-1746.
Smyth, C. and A. E. Hay (2002). “Wave friction factors in nearshore sands.” Journal of Physical Oceanography 32: 3490-3498.
Smyth, C. and A. E. Hay (2003). “Near-bed turbulence and bottom friction during SandyDuck97.” Journal of Geophysical Research 108(C6): 3197.
Stauble, D. K. (1992). "Long term profile and sediment morphodynamics: Field Research Facility case history." Technical Report CERC-92-7, US ACE, Waterways Experiment Station, Vicksburg, MS.
Stauble, D. K. and M. A. Cialone (1996). "Sediment dynamics and profile interactions: Duck94." Proceedings of the 25th International Conference on Coastal Engineering 4: 3921-3934. Orlando, FL, ASCE.
Trowbridge, J. and S. Elgar (2001). “Turbulence measurements in the surf zone.” Journal of Physical Oceanography 31: 2403-2417.
US ACE. SandyDuck '97 sediment samples, CHL Field Research Facility, http://frf.usace.army.mil/geology/sediments.stm. Accessed November, 2003.
84
Whitford, D. J. and E. B. Thornton (1988). "Longshore current forcing at a barred beach." Proceedings of the 21st International Conference on Coastal Engineering 1: 77-90. Costa del Sol-Malaga, Spain, ASCE.
Wiberg, P. and J. D. Smith (1983). “A comparison of field data and theoretical models for wave-current interactions at the bed on the continental shelf.” Continental Shelf Research 2: 147-162.
Wright, L. D. (1995). Morphodynamics of Inner Continental Shelves, Boca Raton, FL, CRC Press.
Wright, L. D., J. D. Boon, S. C. Kim, and J. H. List (1991). “Modes of cross-shore sediment transport on the shoreface of the Middle Atlantic Bight.” Marine Geology 96: 19-51.
Wright, L. D., J. D. Boon, III, M. O. Green, and J. H. List (1986). “Response of the mid shoreface of the southern mid-Atlantic bight to a "Northeaster".” Geo-Marine Letters 6: 153-160.
Wright, L. D., J. P. Xu, and O. S. Madsen (1994). “Across-shelf benthic transports on the inner shelf of the middle Atlantic bight during the "Halloween Storm" of 1991.” Marine Geology 118: 61-77.
Xu, J. P. and L. D. Wright (1998). “Observations of wind-generated shoreface currents off Duck, North Carolina.” Journal of Coastal Research 14(2): 610-619.
BIOGRAPHICAL SKETCH
Jodi Eshleman was born in Pittsburgh, Pennsylvania on May 22, 1980. After
graduating high school in 1998, she attended Lehigh University, located in Bethlehem,
Pa. There she developed a passion for addressing water-related issues worldwide. She
completed a bachelor’s degree in Civil Engineering in the spring of May, 2002. Many
summer vacations spent along the Atlantic coast gave her a love of the beach; and recent
trips to the Outer Banks of North Carolina brought an awareness of the sensitive balance
between coastal development and beach sustainability. This led her to the Coastal
Engineering program at the University of Florida, to pursue a Master of Science degree.
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