EFFECTS OF LONGSHORE CURRENTS ON RIP CURRENTS
By
ENRIQUE GUTIERREZ
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
This document is dedicated to my mother and father.
ACKNOWLEDGMENTS
This research was funded by the 2002-2004 and 2004-2006 Florida Sea Grant
Program. I would like to thank Andrew B. Kennedy for his academic guidance and
support in the research leading to the present thesis. I would also like to thank Robert. J.
Thieke for providing financial assistance and giving me the opportunity to be part of a
very exciting project, and Robert G. Dean for his help and for serving on my supervisory
committee.
I would also like to thank Oleg A. Mouraenko for his support and guidance with
Matlab, if I have any skills with Matlab it is thanks to him. I also want to thank Jamie
MacMahan, who gave me guidance ever since I got here.
I would like to thank all of my office mates and friends in Gainesville in general,
for making my stay here a very good experience and especially my roommate, Vadim
Alymov. I will never forget all of them.
Finally I would like to thank my family, especially my parents, who made many
sacrifices to provide me with financial support to get my undergraduate degree back in
Spain. They have always been there for me.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ................................................................................................. iii
LIST OF TABLES............................................................................................................. vi
LIST OF FIGURES .......................................................................................................... vii
CHAPTER
1
INTRODUCTION ........................................................................................................1
Problem Statement and Objective ................................................................................1
Background: Literature Review....................................................................................3
Physical Description of Rip Currents ....................................................................5
Forcing Mechanisms of Rip Currents and Longshore Currents............................9
Forcing of longshore currents ......................................................................10
Forcing of rip currents..................................................................................10
Generation of Vorticity........................................................................................12
Outline of the Thesis...................................................................................................13
2
NUMERICAL MODEL .............................................................................................15
Theoretical Background..............................................................................................15
Rip Current Scaling ....................................................................................................17
Numerical Description of the Model ..........................................................................18
3
RESULTS AND ANALYSIS.....................................................................................21
Mean Peak Offshore Current in the Rip Neck............................................................21
Parameter (DΓ / Dt) Estimations .........................................................................22
Method 1 ......................................................................................................22
Method 2 ......................................................................................................24
Model Comparisons with Lab Data.....................................................................25
Effects of Background Longshore Current on the Rip Current..................................32
Steady Forcing.....................................................................................................32
Velocities in the rip neck..............................................................................34
Vorticity .......................................................................................................37
Jet angle evolution with increasing background longshore current .............40
Unsteady Forcing.................................................................................................41
iv
Velocities in the rip neck..............................................................................44
Vorticity .......................................................................................................46
4
SUMMARY AND CONCLUSIONS .........................................................................54
APPENDIX
A
MEAN VORTICITY MAPS ......................................................................................59
B
LOCATION AND WIDTH OF THE JET DATA .....................................................80
LIST OF REFERENCES...................................................................................................85
BIOGRAPHICAL SKETCH .............................................................................................89
v
LIST OF TABLES
Table
3-1
page
Unsteady forcing simulations...................................................................................42
B-1 Alongshore location and width of the jet for steady forcing....................................80
B-2 Alongshore location and width of the jet for unsteady forcing with amplitude
0.25 ...........................................................................................................................81
B-3 Alongshore location and width of the jet for unsteady forcing with amplitude 0.5.82
B-4 Alongshore location and width of the jet for unsteady forcing with amplitude 0.7583
B-5 Alongshore location and width of the jet for unsteady forcing with amplitude 1....84
vi
LIST OF FIGURES
Figure
page
1-1
Rip current parts: feeders neck and head (from Shepard et al., 1941). ......................7
1-2
Time-averaged vorticity ...........................................................................................12
2-1
Definition sketch of the model. ................................................................................17
3.1
Sketch of wave breaking over a bar .........................................................................23
3-2
Experimental wave basin at the University of Delaware .........................................26
3.3
Current meter location..............................................................................................27
3-4
Wave height and MWL versus cross-shore distance at the center bar (left) and at
the rip channel (right) for test E ...............................................................................28
3-5
Cross-shore velocities in the rip: model predictions vs. lab data. ............................29
3-6
Cross-shore velocities in the rip: model predictions vs. lab data. ............................31
3-7
Snapshots of the simulations for different background longshore currents with
steady forcing ...........................................................................................................33
3-8
Computed mean velocities on a longshore profile at x = -0.5..................................35
3-9
Mean peak offshore velocities with increasing background longshore currents at
different cross-shore locations .................................................................................36
3-10 Mean peak offshore velocity versus background longshore current........................37
3-11 Mean vorticity maps for steady forcing ...................................................................38
3-12 Mean vorticity maps for steady forcing ...................................................................39
3-13 Jet angle vs. background longshore current .............................................................41
3-14 Comparison of the simulations for steady and unsteady forcing .............................43
3-15 Mean peak offshore velocities in the jet at the cross-shore location x = -0.5 for
different amplitudes..................................................................................................45
vii
3-16 Mean peak offshore velocities in the jet at the cross-shore location x = -0.5 for
different frequencies.................................................................................................46
3-17 Mean vorticity maps for unsteady forcing with amplitude 1 and frequency 1.........47
3-18 Alongshore location and width of the jet at x = -0.5 with increasing longshore
current for each amplitude........................................................................................49
3-19 Alongshore location and width of the jet at x = -0.5 with increasing longshore
current for each frequency........................................................................................50
3-20 Alongshore location and width of the jet at x = -0.5 with increasing group
amplitude for each frequency. ..................................................................................51
3-21 Alongshore location and width of the jet at x = -0.5 with increasing group
frequency for each amplitude. ..................................................................................52
3-22 Mean vorticity map for unsteady forcing with amplitude 1, frequency 1.5and
background longshore current of 0.75......................................................................53
A-1 Mean vorticity map for steady forcing. ....................................................................59
A-2 Mean vorticity maps for unsteady forcing with amplitude 0.25 and frequency 0.5.60
A-3 Mean vorticity maps for unsteady forcing with amplitude 0.25 and frequency 1....61
A-4 Mean vorticity maps for unsteady forcing with amplitude 0.25 and frequency 1.5.62
A-5 Mean vorticity maps for unsteady forcing with amplitude 0.25 and frequency 2....63
A-6 Mean vorticity maps for unsteady forcing with amplitude 0.25 and frequency 2.5.64
A-7 Mean vorticity maps for unsteady forcing with amplitude 0.50 and frequency 0.5.65
A-8 Mean vorticity maps for unsteady forcing with amplitude 0.50 and frequency 1....66
A-9 Mean vorticity maps for unsteady forcing with amplitude 0.50 and frequency 1.5.67
A-10 Mean vorticity maps for unsteady forcing with amplitude 0.50 and frequency 2....68
A-11 Mean vorticity maps for unsteady forcing with amplitude 0.50 and frequency 2.5.69
A-12 Mean vorticity maps for unsteady forcing with amplitude 0.75 and frequency 0.5.70
A-13 Mean vorticity maps for unsteady forcing with amplitude 0.75 and frequency 1....71
A-14 Mean vorticity maps for unsteady forcing with amplitude 0.75 and frequency 1.5.72
A-15 Mean vorticity maps for unsteady forcing with amplitude 0.75 and frequency 2....73
viii
A-16 Mean vorticity maps for unsteady forcing with amplitude 0.75 and frequency 2.5.74
A-17 Mean vorticity maps for unsteady forcing with amplitude 1 and frequency 0.5......75
A-18 Mean vorticity maps for unsteady forcing with amplitude 1 and frequency 1.........76
A-19 Mean vorticity maps for unsteady forcing with amplitude 1 and frequency 1.5......77
A-20 Mean vorticity maps for unsteady forcing with amplitude 1 and frequency 2.........78
A-21 Mean vorticity maps for unsteady forcing with amplitude 1 and frequency 2.5......79
ix
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
EFFECTS OF LONGSHORE CURRENTS ON RIP CURRENTS
By
Enrique Gutierrez
May 2004
Chair: Robert J. Thieke
Major Department: Civil and Coastal Engineering
A simplified conceptual representation of a rip current system was used to study the
effects of longshore currents on rip currents. The numerical model used for this study is
based on a generation of vorticity approach, where oppositely signed vortices are
continuously released on either side of the rip channel and let free in the system; and a
background constant and uniform longshore current is also added to the system. A flat
bed, no bottom friction and no wave-current interaction are assumed in the model. Since
generation of vorticity is the only physical process represented in the model, its
applicability is strictly limited to the rip neck area, where this process is assumed to be
dominant. Velocities in the rip current depend only on the strength of wave breaking and
time scales depend both on wave breaking strength and length scales of the system.
Performance of the model was tested against measured laboratory data. Despite the
high simplicity of the model, very reasonable results were obtained in comparison with
the measured data. The model predicts well the trend of the data although overestimates
velocities in the rip. The main challenge to apply the results of the model to field or
x
laboratory data is the estimation of the generation of circulation within the surf zone.
Two different methodologies were tested.
The model was used to study the effects of longshore currents on rip currents, using
both steady and unsteady forcing. Mean velocity fields and vorticity maps within the area
of the rip neck were calculated. Steady forcing results showed a very strong effect of the
longshore current on the rip current, with nearly constant offshore velocities for small
longshore current strengths, decreasing quickly as the longshore current increases in
strength. A wide range of unsteady forcing parameters (group amplitude and frequency)
was used to test the relative importance of this parameters in the evolution of the rip
current with increasing longshore current strengths. Results suggest that the background
longshore current strength is the main factor in the behavior of the rip current. Effects of
background longshore current are reduced for large group amplitudes. Influence of the
frequency is found to be almost negligible, with response decreasing with higher
frequencies.
The model could be used to evaluate strength of rip currents under a wide range of
wave climates, with simple scaling relations for rip current strength and wave breaking
strength within the surf zone. Model results could be coupled with existing rip current
forecasting indexes.
xi
CHAPTER 1
INTRODUCTION
Problem Statement and Objective
Throughout the world, much population concentrates in the coastal regions.
Beaches are one of the preferred recreational areas, attracting a great amount of people
and tourism in general, thus becoming an important economic and social factor for the
coastal regions.
Researchers have been studying rip currents for decades now, and they have
become an area of interest due to their importance in nearshore morphodynamics but also
the increasing public concern with safety at the beaches all over the world. Nowadays,
the general public is more aware of the dangers of rip currents thanks to information
campaigns, and lifeguards are specifically trained to respond to these events; however,
the number of casualties at the beach is still quite large, and a better understanding of the
behavior of rip currents, specially their response to different wave climates, is still
needed.
The nearshore ocean is a very complex region, where many different processes take
place and create a very dynamic system. In a simple approach to the problem, the waves
arrive to the beach and break, losing energy in the process and transferring momentum
into the water column, thus generating currents. Depending on the incident wave angle,
alongshore currents (oblique incidence) or nearshore circulation cells (shore-normal
incidence) will be generated. These currents will drive sediment transport impacting
beach morphology and can cause erosion, potentially affecting coastal real estate.
1
2
When circulation cells form, the seaward directed flow, often observed as a narrow
jet, is known as a “rip current.” Currents can reach up to 2 m/s and they cause thousands
of rescues per year in Florida alone, with more deaths due to rip currents than any other
nature disaster related source combined. On average, 19 people have died in Florida per
year since 1989 due to rip currents (Lascody, 1998). It is then a major concern and of
public interest to understand rip current behavior, what causes them and how they
respond to different factors.
A considerable amount of effort has been invested in fieldwork and laboratory
experimentation to better understand the behavior of rip currents. Fieldwork has proven
to be very difficult due to unsteadiness of rip currents, both temporal and spatially, also a
great number of instruments are necessary to cover the entire area of interest. New video
techniques and collaborative efforts between several institutions might provide better data
sets in the future. A limited number of laboratory experiments have been conducted over
the years but usually over a single fixed topography, a wider range of rip current
morphology and wave forcing would be desired. Recently, a number of numerical studies
have been conducted to study rip currents generated with alongshore varying wave
heights using phase-averaged techniques.
This study focuses on a barred beach with rip channels, and is a continuation of the
work published by Kennedy (2003), “A Circulation Description of a Rip Current Neck.”
A simple conceptual model, using generation of vorticity and circulation at the edges of
the bar-channel on a barred beach was used to describe the behavior of rip currents in the
area of the rip neck. A simple scaling was introduced which collapses all rip current
topographies to a single form and makes it possible to estimate the relative importance of
3
the different factors that modulate rip current strength. Although the model was
compared to lab data, here we will use additional data and analyze the performance of
different estimations of the mean rate of generation of circulation (using equations from
Brocchini et al., 2004), which is the basic parameter to scale strength of rip currents with
this model.
Longshore currents are almost always present to some degree; when the waves
approach the coast with an angle and break, the transfer of momentum in the surfzone
will drive longshore currents. Sonu (1972), while studying rip currents of the Gulf coast
of Florida, described how if the waves came with an angle, meandering currents would
form within the surfzone as some form of interaction between longshore and rip currents.
Up to date, it is not well understood how important these interactions may be, and what
relative magnitudes of longshore current are required to modify the offshore flow of the
rip and turn it into a meandering current system. This is not only an interesting scientific
topic, but also might be of use for public safety if threshold conditions could be
determined for the formation of rip currents under oblique incident waves. The present
study will explore this, by simply modifying the conceptual model described by Kennedy
(2003), adding a background longshore component to the system. The response of the rip
will be studied under many different forcing conditions (steady and unsteady) and
background longshore currents strengths. Some insight to the problem will be addressed.
Background: Literature Review
Researchers have been studying rip currents for over 50 years now. There have
been a number of field experiments studying rip currents, although many of these
observations were qualitative and very few quantitative measurements were collected.
These measurements are generally limited to the rip area and are difficult to obtain due to
4
the difficulty in locating the instruments in the rip channels, which tend to be temporally
and spatially unstable [Shepard et al. (1941), Shepard and Inman (1950, 1951), McKenzie
(1958), Harris (1961,1964), Sonu (1972), Cooke (1970), Sasaki and Horikawa (1979),
Bowman et al. (1988), Smith and Largier (1995), Chandramohan et al. (1997), Aagaard et
al. (1997), Brander (1999), Brander and Short (2000,2001)].
Another approach to the problem is to use video images from shore. Rectified
video images have been used for the quantification of physical processes in the nearshore
(Holland et al., 1997). Ranasinghe et al. (1999) used averaged video images to study
long-term morphological evolution of the beach under the presence of rip currents and
possible generation mechanisms.
In contrast, the controlled environment of the lab allows experimentation of rip
currents in much more detail, however limited laboratory experiments have been
conducted to date [Bowen and Inman (1969), Hamm (1992), Oh and Dean (1996),
Drønen et al. (1999, 2002)] of interest here is the laboratory setup at the Center for
Applied Coastal Research of the University of Delaware, where a number of rip current
experiments have been conducted [Haller et al. (1997), Haller and Dalrymple (1999),
Haller and Dalrymple (2001), Kennedy and Dalrymple (2001), Kennedy (2001), Haller et
al. (2001) that provided a comprehensive map of waves and currents, including the
details of the mean water level variations, Haas (2002)] on a fixed bar-channel
bathymetry.
The availability of costly instruments like ADV’s (Acoustic Doppler Velocimeter)
or ADCP’s (Acoustic Doppler Current Profilers) is usually limited so it is difficult to
obtain information about the whole flow domain. An alternative to this is the use of
5
drifters, which have been used in the field in limited cases [Shepard and Inman (1950),
Sonu (1972)], and more recently, Schmidt et al. (2001), who applied direct drifter
tracking using Global Positioning System (GPS) but the relatively high cost of the
drifters limited their availability. In the laboratory environment, Thomas (2003) used
numerous video tracked Lagrangian drifters with the laboratory setup from Haller et al.
(1997).
The concept of radiation stresses developed by Longuet-Higgins (1964) provided a
basis to numerically describe the generation of rip currents with alongshore varying wave
heights using phase-averaged techniques. Bowen (1969) theoretically explored the
generation of circulation cells within the surfzone using alongshore-varying radiation
stresses. A number of numerical studies have been conducted, for example Haas and
Svendsen (1998) and Chen et al. (1999), who used the fully nonlinear extended
Boussinesq equations of Wei et al., (1995), to model the laboratory setup by Haller et al.
(2001), providing valuable insights into rip current behavior. Slinn and Yu (2002)
investigated the effect of wave current interaction on rip currents and showed that it
might be an important factor, weakening the strength of rip currents.
In this section, several aspects of rip currents, including vorticity generation within
the surf zone and longshore currents will be reviewed from previous literature.
Physical Description of Rip Currents
Rip currents can be the most visible feature of nearshore circulation systems. They
are strong, narrow currents that flow seaward through the surf zone, often carrying debris
and sediment, which gives the water a distinctly different color and surface texture from
adjacent waters (Komar, 1998). Rip currents can extent several surf zone widths seaward
exchanging water between the nearshore and offshore. They have been observed all over
6
the world, on a wide range of beach types but are particularly common on beaches that
are dominated by a longshore bar-trough morphology (Wright and Short, 1984), which is
the focus of this study. They can also form due to interaction with coastal structures like
piers, groins or jetties (Shepard and Inman, 1950; Wind and Vreugdenhil, 1986) or when
longshore currents are directed offshore by a protrusion in the bathymetry or headland
(Shepard and Inman, 1950).
The concept of “rip current” was first proposed by Shepard (1936) (as opposed to
the popular name of “rip tides,” since they were found not to be related to tides). In the
early twentieth century, it was believed that bathers were pulled out of the surf by a
violent “undertow,” a current beneath the surface that would carry out the water piled up
the beach by the incoming waves. Davis (1925) was the first to challenge this popular
idea. Lifeguards and experienced swimmers were aware of these “rip-tides” that often
carried bathers beyond to depths in which they could not stand.
Shepard (1941) gave a first qualitative description of the rip currents, defining three
main parts, which he called the “feeders,” the “neck” and the “head” (see Figure 1-1).
Feeder currents are flows of water that run parallel to shore just outside the beach from
either side of the rip, one of these currents usually being dominant, that “feed” the main
outward-flowing current or “rip neck,” which moves at high speeds in narrow lanes
through the breakers essentially at right angles to the general coastal trend. Maximum
flow speeds of up to 2 m/s and 1.3 m/s in the rip neck and feeders respectively have been
recorded in the field (Brander, 1999) although the flow was found highly unsteady. Water
enters the neck not only from the feeders but also to some extent from the sides of the rip
channel farther out (Shepard 1941, Brander and Short 2001). Along the path of the neck,
7
usually a channel can be found which can be as much as 1m deeper than the adjacent bar,
indicating that the flow extends through the entire water column. Beyond the breakers,
the current spreads out and dissipates in what is called the rip “head;” along the side of
the advancing head, eddies are often observable, turning to the right on the right side of
the rip, and to left on the left side (Shepard, 1941). The flow separates from the bottom
and is mainly confined to the surface, which is supported by no evidence of channels
beyond the breakers.
Figure 1-1: Rip current parts: feeders neck and head (from Shepard et al., 1941).
Field observations of rip currents have shown long period oscillations in rips on the
wave group time scales (25-250s) of up to 0.4 m/s [Shepard et al. (1941), Shepard and
8
Inman (1950), Sonu (1972), Aagaard (1997), Brander and Short (2001)]; although these
measurements were not accompanied by wave measurements offshore and alongshore the
rip channel, so that relationships between rip currents and wave groups could not be
established. Munk (1949) and Shepard and Inman (1950) suggested that there is a
maximum set up within the surf zone when the largest short waves in a group break,
resulting in a transport of water shoreward that is discharged most efficiently through the
rip channels during the subsequent small short waves of the wave group, Sonu (1972)
hypothesized that rip current pulsations were due to infragravity standing waves.
MacMahan et al. (2004), based on measurements obtained in the Monterey Bay, CA
experiment (RIPEX) concluded that rip current pulsations on that beach were due to
infragravity cross-shore standing waves.
Shepard et al. (1941) stated that records obtained off the coast of Southern
California showed a clear relation between intensity of rip currents and height of waves.
McKenzie (1958), based on observations on sandy Australian beaches, noted that rip
currents are generally absent under very low wave conditions but are more numerous and
somewhat larger under light to moderate swell. This represents an important consequence
for the morphology of the area, since erosional power of rips will significantly increase
under stronger currents.
Another factor that seems to be of importance in modulating the strength of rip
currents is the tide. Numerous field observations in different types of beaches support this
idea. McKenzie (1958) noted a prevalence of rip currents during falling tides and
attributed this to the concentration of the drainage system into the current channels,
resulting in stronger flows. Cooke (1970) off the coast of Redondo Beach, CA observed
9
that stationary rip channels were commonly present but well defined rip currents were
only present during falling or low tide. Sonu (1972) at Seagrove Beach, FL observed
modulation in rip current intensity with the tide which was attributed both to confinement
of rips to narrower regions in the surf zone and stronger breaking during low tide, thus
increasing the transfer of momentum in the surf zone which drives the currents. Brander
(1999) and Brander and Short (2001) conducted experiments at Palm Beach, New South
Wales, Australia. Rip current velocities reached maximums at low tide and minimum
velocities at high tide, the state of morphological beach evolution was found to be an
important factor as well. Drønen et al. (2002) showed in their laboratory experiments that
rip current velocities increased with increasing wave height and decreasing water level.
A third factor in the occurrence of rip currents is the wave angle. Sonu (1972)
observed closed circulation cells only under the presence of shore normal waves while
meandering alongshore currents would form under oblique incidence.
The numerical study by Kennedy (2003) provided valuable insights in the response
of rip current strength to different factors. Temporal response was found to be dependant
both on length scales of the system and the strength of wave breaking. Velocities only
depended on the strength of the wave breaking but not on the channel width. Also, it was
found that rip current response to unsteady wave forcing was strongly dependent on the
group forcing frequency, with stronger response to low frequencies, decreasing quickly
for high frequencies.
Forcing Mechanisms of Rip Currents and Longshore Currents
The first suggestions as to the cause of rip currents were based on the concept of an
onshore mass transport of water due to the incoming waves. This water, piled up on the
beach would provide the head for the out flowing currents.
10
Understanding of the forcing driving the currents within the surf zone was greatly
enhanced when Longuet-Higgins and Stewart (1964) introduced the concept of radiation
stress to describe some of the nonlinear properties of surface gravity waves. Radiation
stress (S ) was defined as the excess flow of momentum due to the presence of waves. It
can be decomposed in three terms: Sxx, radiation stress component in the direction of the
waves; Syy, radiation stress component in the transverse direction of the waves and Sxy,
the flux in the x direction of the y component of momentum.
Forcing of longshore currents
When waves propagate obliquely into the surf zone and break, this will result in a
reduction in wave energy and an associate decrease in S xy , which is manifested as an
applied longshore thrust Fy on the surf zone (Dean and Dalrymple, 1984). For straight
and parallel contours, thrust per unit area is given by:
Fy = −
∂S xy
∂x
[1.1]
The longshore wave thrust per unit area is resisted by shear stress on the bottom
and lateral faces of the water column (Longuet-Higgins, 1970 a, b).
Forcing of rip currents
In general, rip currents are contained within nearshore circulation cells that are
driven by periodic longshore variations in the incident wave field. There have been a
number of theories proposed as to the generation mechanisms for these longshore
variations in the incident wave field. They could be divided into three categories:
•
Wave-boundary interaction mechanisms. Wave refraction over non-uniform
bathymetry can cause convergence in some areas (headlands) while causing wave
divergence in other areas (canyons) thus resulting in high and low waves
respectively in these areas. An example of rip currents generated by this
11
mechanism at La Jolla, CA is described by Shepard and Inman (1951) and Bowen
and Inman (1969).
•
Wave-wave interaction mechanisms. Bowen and Inman (1969) proposed a model
for generation of circulation cells under the presence of edge waves, where rip
currents are located at every other anti-node and rip spacing is equal to the edge
wave length. Dalrymple (1975) used two synchronous wave trains that approach
the beach from different directions to generate longshore variations in incident
wave height.
•
Instability mechanisms. Generation of rip currents on plane smooth beaches can be
explained based on instability theories, where a small initial variation on the wave
field can result in the generation of regularly spaced rip currents [Hino (1975),
Iwata (1976), Dalrymple and Lozano (1978), Falqués et al. (1999)].
However, once the rips erode rip channels in the initially longshore uniform beach,
the wave field becomes topographically controlled and the circulation can last long even
after the initial source of longshore wave field variability has diminished or even
disappeared.
Using momentum balance in the direction of the waves, Longuet-Higgins (1964)
()
showed that radiation stress induces changes in the mean water level η , creating steady
pressure gradients that balance the gradient of the radiation stress:
dS xx
dη
1
=−
dx
ρg h + η dx
(
)
[1.2]
Bowen (1969) exploited the concept of wave set-up to analytically describe the
generation of circulation cells in the nearshore using a transport stream function.
However, as irrotational forcing, wave induced set-up itself cannot generate circulation
(Brocchini, 2003). When the waves break there is a decrease in the radiation stress which
leads to an increase in the set-up, but also this is manifested as a wave thrust or force. If
there is differential breaking alongshore this will generate differential forces which will
generate circulation.
12
Generation of Vorticity
Although generation of vorticity within the surf zone is quite common, direct
observations of vorticity are very difficult to obtain. Smith and Largier (1995), using
acoustic techniques, observed rip current vortices with radii in the order of 10’s of
meters. Schmidt et al. (2001), using direct drifter tracking with GPS technology observed
vorticity within the surf zone. In the laboratory, Thomas (2003) observed time averaged
vorticity in the vicinity of a rip channel, with four distinct macrovortices, two of them
spinning with opposite sign on either side of the rip channel and two more shoreward of
those spinning opposite to them (see figure 1-2).
Figure 1-2: Time-averaged vorticity; contour = 0.1/s; Positive =>Dashed line, Negative
=>Dash-Dot line, and Zero =>Solid line (from Thomas, 2003)
Peregrine (1998) and Bühler (2000) showed theoretically how differential wave
breaking (e.g., at the flanks of wave trains) generates vorticity which re-organizes in the
form of large-scale horizontal eddies with vertical axis or macrovortices. In the case of
alongshore bar-rip channel topography, this generation of circulation is focused at the
edges of the bar, where there is a strong variation in the longshore direction on the wave
13
breaking. There is oppositely signed generation of vorticity on either side of the channel,
which causes mutual advection offshore of the generated vorticity; this mechanism was
showed by Peregrine (1999).
Bühler and Jacobson (2001) conducted a detailed theoretical and numerical study
of longshore currents driven by breaking waves on a barred longshore uniform beach. An
assumed offshore variability in wave amplitude was necessary to generate differential
breaking and thus generate vortices. Strong dipolar vortex structures evolution produced
a displacement shoreward to the bar trough of the preferred location of the longshore
current, a phenomenon that has been often observed on real barred beaches.
Outline of the Thesis
The present Chapter 1 introduces the problem under study and the objectives of the
thesis, and then an extensive literature review is conducted introducing the various
significant concepts relevant to this work as follows: a) physical description of rip
currents, b) forcing of currents in the surf zone, both longshore and rip currents and c)
generation of vorticity in the surf zone.
Chapter 2 analyses the numerical model used in the study. First, a theoretical
background is given to justify the model, which leads to the discussion of the rip current
scaling that forms the basis of the model and links the model predictions with actual
measured data. Finally a detailed numerical description of the model is given.
Chapter 3 is divided in two parts. First, the model predictions are compared with
available laboratory data, for this, results with no background longshore current are used
since all the available laboratory data are based on shore normal waves. Secondly, several
model runs with a range of background longshore currents are analyzed, both for steady
and unsteady forcing. The relative importance of the model parameters (background
14
longshore current and unsteady forcing parameters: amplitude and frequency) is
discussed.
Chapter 4 summarizes all the results and analysis and conclusions will be drawn.
Suggestion for future research and applications of the model will also be given.
CHAPTER 2
NUMERICAL MODEL
The main goal in the development of this numerical model was developing a very
simple model, both to get fast computational times and to study the scaling of the
different processes present on a rip current. This model was originally written by
Kennedy (2003), simple modifications have been included to study the effects of
background longshore currents in the system. This chapter will discuss the theoretical
background that leads to a simple representation of rip currents (conceptual model), the
scaling parameters and dimensional analysis that makes possible this model, and finally a
numerical description of the model.
Theoretical Background
Although rip currents are part of a very dynamic system, with quite complex
forcing, a simple description can be achieved if we focus on an area where one of these
processes is dominant. That is the case of the rip neck, where oppositely signed
circulation and vorticity are the dominant processes.
One of the most common rip current typologies is the one consisting of a longshore
bar with gaps or rip channels on it. This kind of topography induces a differential wavebreaking pattern that is more or less stationary on hydrodynamic scales. Although
migration of rip currents has been observed on the field (Ranasinghe et al, 1999) the time
scales are much larger. Generally, there will be strong wave breaking on the bar and
weak or no breaking at all on the rip channels.
15
16
Peregrine (1998), using the NLSW equations and a bore dissipation model, showed
how differential breaking along a wave crest generates circulation and vorticity, and that
considering a closed material circuit that crosses the bore only once, the instantaneous
rate of change of circulation generated equals the rate of loss of energy by the water
passing through the bore. Vorticity can be defined as:
Γ = ∫ U ⋅ dl
[2.1]
In the case of alongshore bar-rip channel topography, this generation of circulation
is focused at the edges of the bar, where there is a strong variation in the longshore
direction on the wave breaking. There is oppositely signed generation of vorticity on
either side of the channel, which causes mutual advection offshore of the generated
vorticity (Peregrine, 1999). New vorticity will continue to be generated at these locations
and then be self-advected offshore and so on. This is the predominant forcing mechanism
in this area, the so-called “rip neck” and all other mechanisms will be neglected. It will be
the basis for the numerical model therefore the region where the model is valid is limited
to the rip neck area.
Rip currents have been observed combined with longshore currents in the field
numerous times, these observations show either oblique rip currents or the formation of
meandering currents (Sonu, 1972). These different phenomena could certainly affect very
differently the “incautious swimmer;” a better understanding of the formation of either
one would then be useful. In order to study the development of these phenomena and
behavior of rip currents with present longshore currents the original numerical model
(Kennedy 2003) was modified to include a background longshore component.
17
Rip Current Scaling
The model uses three independent parameters to define each different case or run.
These are:
•
The half width of the rip channel (R )
•
The mean rate of generation of circulation (DΓ Dt )
•
The mean background alongshore current (v )
Figure 2-1. Definition sketch of the model.
In order to be able to use the simple representation of a rip current system described
above for a real case (lab experiment or field measured rip), there is a need to relate the
different variables in the real system to the ones present on the model. Using a simple
dimensionless group analysis, the different variables scale as follows:
•
Length scales: (x ′, y ′) = ( x, y ) R
•
Velocities: (u ′, v ′) = (u , v ) (DΓ Dt )
•
Time: t ′ = t ⋅ (DΓ Dt )
•
Circulation: Γ ′ = Γ R(DΓ Dt )
(
1
2
1
2
R
1
2
)
Therefore, we will need to determine what R and (DΓ Dt ) are in the field (lab
experiment) and then scale everything accordingly. Estimating R is relatively easy; more
18
of a challenge is the parameter (DΓ Dt ) (strongly dependant on the local topography and
wave breaking strength), two simple methodologies to estimate it will be discussed in
chapter 3.
Using this scaling has great advantages; the simple length scaling allows us to use a
single configuration for the system, with all different topographies converging to a single
form. In the case of absence of a background longshore current (shore normal waves) all
different possible wave strengths are represented by one non-dimensional case. When
longshore currents are present, the non-dimensional longshore current becomes an
additional parameter to define the model.
The provided scaling suggests that velocities in the rip neck depend on the strength
of wave breaking, scaling with (DΓ Dt ) 2 , but not on the gap width (R ) . Temporal
1
(
response will depend both on wave breaking strength and length scales R (DΓ Dt )
1
2
).
These scaling relations discussed by Kennedy (2003) provide a simple way to scale
strength and temporal responses of rip currents not available in the literature previous to
this paper.
Numerical Description of the Model
As described above the numerical model is based on a circulation-vorticity
approach, no other processes are represented in the model, such as wave-current
interaction, 3-D topography (flat bed), bottom friction, forced and free infragravity
waves, instabilities and others. This leads to a highly numerical simplicity of the model,
but its applicability will be limited to the rip current neck area, where circulation and
vorticity generation are assumed to be the dominant process.
19
Essentially, the system is based on the generation of vorticity at two fixed locations
on either side of the rip channel (x, y) = (0, ±1) using a discrete vortex method. Positive x
coordinates are located shoreward f the generation of circulation and negative cross-shore
locations are seaward of the generation of circulation. The model is written in terms of
mass transport velocities. Using the relation for point vortex velocities, U θ = Γ 2πr , the
model calculates the velocity U at every discrete vortex in the domain as the one induced
by all other vortices present in the domain, then displaces them using a simple Euler
method:
( X , Y )(t +∂t ) = ( X , Y )(t ) + U ⋅ ∂t
[2.2]
Strong interaction between consecutively introduced vortices requires a separation
of time scales in the model, therefore the model releases the discrete vortex pairs in the
system at intervals of ∆t, defined as ∆t = Nδt, where 20 ≤ N ≤ 125 (runs used in the
present thesis have values for N of 50 and 100) and δt is the small time step at which
vortices are being displaced. Therefore, there are two kinds of time steps, the “big time
steps” ∆t, which determine the generation of new vorticity at the generation points, and
the “small time steps” δt, at which the discrete vortices present in the system are
displaced according to the velocity induced by all other vortices in the system at their
location.
When using steady forcing, the mean generation of circulation (DΓ Dt ) has a fixed
value of one in the model. Choosing different values for ∆t will allow some tuning in the
model since the strength of the discrete vortex pairs is dependent on ∆t so that the mean
rate of generation of circulation is kept constant at one. The values used for ∆t in this
thesis were 0.03, 0.04 and 0.05; good convergence was obtained with these values so the
20
less computational demanding value of 0.05 was predominantly used. Also small random
perturbations are added to the strength of each vortex to allow sinusoidal perturbations to
form.
Unsteady forcing can be used in the model by modulating
(DΓ
Dt ) with a
sinusoidal component, defined in the input file by an amplitude a (values used in the
analysis range from 0<a<1 with increments of 0.25) and a group frequency ω (0.5<f<2.5
where used with increments of 0.5):
 DΓ 

 = 1 + a ⋅ sin (ω ⋅ t )
 Dt  g
[2.3]
An additional parameter of the model is a constant background longshore current,
this value is added to the longshore component of the computed velocity of each point
vortex. No interaction between this current and the vortices in the domain is assumed.
Several values have been used to study the effects of longshore currents on the rip
strength.
The output files obtained from the model provide location in time and strength
(dimensionless) off all vortices introduced in the domain in three separate files, one for
the x coordinate, one for the y coordinate and finally another one for the vortices
strength. Using these data files is easy to calculate velocity fields using the point vortex
velocity relations discussed above. A series of Matlab codes were used in the post
processing and analysis of the data in the present thesis.
CHAPTER 3
RESULTS AND ANALYSIS
In the first part of the chapter, model results will be compared to available
laboratory data in an attempt to test the model performance compared with measured
data. Since most of the available lab data is based on experiments with shore normal
waves (due to the difficulty of avoiding reflection at the lateral walls of the basin), only
model results with no background longshore currents could be compared to lab data. In
the second part of the chapter, the effects of longshore currents on rip currents will be
studied using the model results; no lab or field data were available for comparison. Both
steady forcing and group forcing were used.
Mean Peak Offshore Current in the Rip Neck
In the following subchapter the ability of the numerical model (described in chapter
2) to obtain reasonable results will be discussed. In order to evaluate its performance
comparisons with measured data will be performed.
Two sets of available lab data will be used, the first set from Kennedy and
Dalrymple (2001) and other experiments performed by Kennedy and the second set from
Haller et al. (2002).
The model is written in terms of mass transport velocities and works with nondimensional quantities; therefore the lab data will be scaled using the scaling
relationships provided in chapter 2. Velocities scale with the square root of the mean rate
of generation of circulation as follows:
21
22
(u ' , v') =
(
(u , v)
DΓ
)
1
Dt
2
[3.1]
O
In order to compare measured velocities with model predictions, these will be
transformed into mass transport velocities (using measured local wave height to calculate
short wave mass transport) and then will be plotted against estimated mean rate of
generation of circulation. Two methodologies to evaluate the model parameter (DΓ Dt )
will be discussed.
Parameter (DΓ / Dt) Estimations
Estimations of the parameter (DΓ Dt ) , mean rate of generation of circulation, are
necessary to compare model predictions with measured data in the lab as stated above.
This is not an easy task since measurements of vorticity are quite complicated to obtain.
Therefore, a methodology that allows the use of alternative measurements (wave heights,
topography, mean water levels…) is necessary. Brocchini et al. (2003) proposed two
different methodologies to estimate the rate of generation of circulation due to differential
breaking along a wave ray, they are based on the wave forced Non Linear Shallow Water
Equations (NSWE).
Method 1
The first methodology needs as inputs topography and offshore wave conditions,
and is based on assumptions on the breaking over the bar. The estimated mean rate of
generation of circulation (DΓ Dt ) along the wave ray (which is assumed shore normal)
is (e.g., Brocchini et al., 2003):
2
gh
 DΓ  5 gγ
(hB − hC ) + C (γ 2 − β 2 )

=
16
8
 Dt 
[3.2]
23
Where γ is the ratio of wave height to water depth (γ = 0.78) (depth limited
breaking); β is a constant that relates water depth to wave height assuming the breaking
continuous over the bar ( H = β ⋅ hC , where β = 0.45 ), hC is the water depth at the bar
crest and hB is the water depth at the start of breaking, which can be estimated using the
following equation for shore-normal waves breaking in shallow water (e.g., Dean &
Dalrymple, 1984):
(
hB = H O2 C gO
)
2
5
γ
−4
5
g
−1
5
[3.3]
Where H O and C gO are the wave height and group velocity in deep water
respectively. Figure 3.1 shows a sketch of breaking over the bar (from Brocchini et al.,
2003).
Figure 3.1: Sketch of wave breaking over a bar (from Brocchini et al., 2003).
The main characteristic of this approach is the use of predicted processes instead of
using measured local quantities, probably limiting the accuracy of the methodology.
However, it is easier to apply because it doesn’t require measurements in the surf zone,
especially if field data were to be compared with the model.
24
Equation [3.2] estimates the generation of circulation due to the breaking over the
bar, assuming there is no breaking in the rip channel. This is probably not true in many
cases, especially in the field, where some breaking will take place in the channel.
Therefore this methodology only considers the maximum possible generation of
circulation in the system due to the differential breaking in the alongshore as discussed by
Peregrine (1998). A possible fix to this problem would be the introduction of a parameter
(less than 1) that would diminish this estimated generation of circulation.
Method 2
Wave-induced setup is an irrotational forcing and therefore cannot generate
circulation. However, the same breaking wave forces generate setup and circulation so
setup may be used to estimate the generation of circulation across a breaking event in
some mildly restrictive situations (Brocchini et al. 2003). In their paper, Brocchini et al.
came up with an expression that links the change in mean water surface elevation across
a breaking event (with corrections for the irrotational pressure setdown associated with
the waves) with the rate of generation of circulation across that breaking event:
B
 DΓ 

 = g (η B − η A ) − g (η sdB − η sdA )
 Dt  A
[3.4]
Where η B , η A are the measured mean water surface elevations after and before the
breaking event respectively, and η sdB , η sdA are the irrotational pressure setdown
associated with measured wave heights at those same locations.
This second methodology is then based on locally measured data, both wave and
setup fields, which suggests a better accuracy in the prediction than the first
methodology.
25
Method 1 assumes that there is no breaking in the channel thus gives the maximum
possible rate of generation of circulation due to the breaking over the bar. Equation [2]
estimates (DΓ Dt ) in one line so as long as we have data available in the channel we can
account for the breaking in the channel diminishing the generation of circulation in a
more realistic manner.
Model Comparisons with Lab Data
Using the methodologies previously discussed, model predictions will be compared
to available lab data sets. Two data sets will be used, both from experiments performed at
the directional wave basin at the Center for Applied Coastal Research of the University of
Delaware:
•
Data set 1: Kennedy and Dalrymple (2001) and Kennedy (2003).
•
Data set 2: Haller et al. (2002)
The wave basin is approximately 17.2 m in length and 18.2 m in width, with a
wave-maker at one end that consists of 34 paddles of flap-type. The experimental setup
consists of a fixed beach profile with a steep (1:5) toe located between 1.5 m and 3 m
from the wave-maker and a milder (1:30) sloping section extending from the toe to the
shore of the basin opposite to the wave-maker. The bar system consists of three sections
in the longshore direction with one main section approximately 7.2 m long and centered
in the middle of the basin (to ensure that the sidewalls were located along lines of
symmetry) and two smaller sections of approximately 3.66 m. placed against the
sidewalls. This leaves two gaps of approximately 1.82 m wide, located at ¼ and ¾ of the
basin width, which serve as rip channels. The edges of the bars on each side of the rip
channels are rounded off in order to create a smooth transition and avoid reflections. The
26
seaward and shoreward edges of the bar sections are located at approximately x = 11.1 m
and x = 12.3 m respectively (Figure 3-2). The crest of the bar sections are located at
approximately x = 12 m with a height of 6 cm above their seaward edge.
Figure 3-2: Experimental wave basin at the University of Delaware, (a) Plan view and (b)
cross section (from Haller et al., 2002)
If the experiments are considered as an undistorted Froude model of field
conditions with a length scale ratio of 1/30, then the experimental conditions correspond
to a rip spacing of 270 m and rip channel width of 54 m. The Haller dataset would
27
correspond to breaking wave heights of 0.8-2.3 m, wave periods of 4.4-5.5 s, and mean
offshore velocities of 0.8-1.7 m/s in the rip neck.
The first dataset was analyzed by Kennedy (2003); here it will be analyzed together
with Haller’s dataset, which expands the range of wave conditions (larger waves). It
consists of measurements of offshore wave conditions and cross-shore velocities in three
different locations (in the alongshore) of the rip neck. Although the 3 ADV’s were
located relatively close together, (see figure 3-3) there are significant differences in the
measured velocities probably due to jet instability of the rip current. (Haller and
Dalrymple, 2001). To address this, the largest and smallest means of the three ADV’s are
plotted together with the mean of all three ADV’s as an error bar plot.
Figure 3.3: Current meter location in Kennedy (2003), ADV 1 (y=13.52m, x=11.8m);
ADV 2 (y=13.72 m, x=11.8m); ADV 3 (y=13.92m, x=11.8m)
Haller’s dataset consists of mean values of the measured cross-shore velocities in
the rip, and wave heights and mean water levels along cross-shore profiles on the bar and
the channel (see figure 3-4). Although velocities were measured in a wide range of
locations within the rip channel these mean velocities will be assumed to be the mean rip
28
neck velocities for comparisons with the model. For detailed information on location of
instruments see Haller and Dalrymple (1999).
Figure 3-4. Wave height and MWL versus cross-shore distance at the center bar (left) and
at the rip channel (right) for test E (from Haller et al., 2002)
Both data sets and the model predictions are shown in figure 3-5. Model
predictions are shown at two locations, at the cross shore location where the two fixed
generation points are (0,0), where the model predicted quantitatively well velocities at
startup (see Kennedy, 2003) and at a shoreward location from that point (0.5,0), which
seems to better fit the lab data. The error bars in red represent Kennedy’s data and
Haller’s data is shown in blue symbols.
29
Figure 3-5: Cross-shore velocities in the rip: model predictions vs. lab data. Model
predictions at (0,0) and (0.5,0); Kennedy, 2003 (red error bars); Haller, 2002
[method 1 (squares), method 2 with no breaking in the channel (circles),
method 2 with breaking in the channel (triangles)
The availability of mean water levels and wave heights along profiles on the bar
and channel allows us to use method 2 to estimate the generation of circulation. Due to
very shallow water there are some gaps in the data on the profiles on the crest of the bar
(see figure 3-4), therefore it is difficult to determine the end of the breaking event on the
bar and the channel profiles. For comparison reasons, the farthest offshore point in the
profile is taken as point A, and the measurement just shoreward of the bar is taken as the
point B to apply equation [3.4] for all the experiments. Method 2 is applied both
considering the breaking at the channel thus diminishing the generation of circulation
(triangles) and neglecting the breaking in the channel (circles) so that it can be compared
to method 1 (squares), which neglects any breaking in the channel. Looking at figure 3-4,
30
it seems like method 1 overestimates by almost a factor of two the generation of
circulation (some estimates are out of range in the figure) as compared to method 2,
probably due to an overestimation of the breaking over the bar. Method 2 applied to both
the bar and the channel seems to concentrate all the data in a narrow range of rates of
generation of circulation, which doesn’t make much sense since larger waves should
generate larger circulation. A reason for this could be the relative location of the
circulation cells, which would move offshore for larger wave experiments, but since
equation [3.4] is applied at the same cross shore locations for all the cases we could be
looking at locations situated too far shoreward where a second circulation cell of opposite
sign forms (as described in Haller, 2002). It seems like the lack of data in the bar crest
area mentioned before limits the applicability of this methodology.
Figure 3-6 shows the model predictions plotted versus the lab data using
methodology 1 for both data sets, so that they can be compared. Haller’s experiments
consists of larger waves, therefore the predicted (DΓ Dt ) is larger. In general the model
predicts very well the trend of the data, although it seems to over predict the velocities on
the rip. For larger waves (Haller’s data), the trend for the velocities seems to flatten out,
obtaining lower velocities than expected, the reason for this could be the relative location
of the velocity measurements with respect to the generation points being displaced
shoreward since larger waves will break farther offshore. Also, when breaking occurs
past the bars there will be breaking in the channel as well, diminishing the amount of
circulation generated.
31
Figure 3-6: Cross-shore velocities in the rip: model predictions vs. lab data. Model
predictions at (0,0) and (0.5,0); Kennedy, 2003 (red error bars); Haller, 2002
(blue squares) (Both applying method 1).
Kennedy (2003), showed a good quantitative prediction of the velocities at startup
at the location (0,0), although past the initial peak measured velocities decayed more than
in the model. This was attributed to the effects of wave-current interaction starting to take
part. This could be one of the reasons why the model over predicts the velocities, but
other neglected physical processes like bottom friction, 3D effects, etc are probably not
negligible here. Also, relative location of the measurement points with respect to the
generation points is certainly difficult. This could be adjusted with some kind of constant
parameter to decrease predicted velocities, however if the model is used as a predictive
tool it would give predictions that are conservative.
32
Effects of Background Longshore Current on the Rip Current
As described on chapter 2, the original numerical model used in Kennedy (2003)
was modified to include an additional parameter v, which represents a constant and
uniform background longshore current in the system. This background non-dimensional
current scales as any other velocity in the system with (DΓ Dt ) 2 .
1
In this subchapter, the effects of different background alongshore-current strengths
on the rip current will be studied. First, steady forcing will be used to analyze the
evolution of the jet with increasing background longshore currents. Estimations for the
angle of the jet and mean offshore peak velocities will be given. In the second part,
unsteady forcing will be used with a number of different parameters for the group
forcing. The effects and relative importance of the different parameters will be analyzed.
Steady Forcing
A number of different background current strengths were used with steady forcing
in the model ranging from 0 to 1 dimensionless unit at increments of 0.05. Model runs of
50 dimensionless time units were used with different computational resolutions (∆t).
Once the data files were obtained from the model, a series of matlab routines were
used to obtain velocity fields within the area of the rip neck and also maps of mean
vorticity. Velocities at any location were calculated as the sum of all the induced
velocities by each vortex present in the system using the relation for point vortex
velocities. The vorticity with time was defined as the sum of the strength of all vortices
present in predefined boxes in the system divided by the area of the box.
33
(a) v = 0
t = 10
2
2
2
2
0
0
−2
−6
−4
−2
0
0
−2
−6
−4
−2
0
−6
t = 15
4
2
2
2
2
−2
0
−2
−6
−4
−2
0
y/R
4
y/R
4
0
−4
−2
0
−4
−2
0
−6
t = 25
2
2
2
2
0
−2
−6
−4
−2
0
−4
(c) v = 0.5
−2
0
−4
−2
x/R
x/R
t = 10
t=5
0
−6
(d) v = 0.75
2
2
2
0
−2
−6
−4
−2
0
y/R
2
y/R
4
y/R
4
−2
−4
−2
0
−4
−2
0
−6
t = 15
2
2
2
2
0
−2
−6
−4
−2
0
y/R
4
y/R
4
y/R
4
−2
t = 25
−4
−2
0
−4
−2
0
−6
t = 25
2
2
2
2
0
−2
−6
−4
−2
x/R
0
y/R
4
y/R
4
y/R
4
−2
−4
−2
x/R
0
−4
−2
0
0
−2
−6
0
t = 30
4
0
−2
−2
−6
t = 30
0
−4
0
−2
−6
t = 10
t = 20
4
0
0
−2
−6
t = 20
0
−2
0
−2
−6
t = 15
−4
x/R
4
0
0
−2
−6
4
0
−2
0
−2
−6
x/R
t=5
y/R
4
y/R
4
y/R
4
−2
−4
t = 30
4
0
0
−2
−6
t = 30
0
−2
0
−2
−6
t = 25
−4
t = 20
4
y/R
y/R
−2
t = 20
0
y/R
−4
t = 10
0
−2
−6
t = 15
y/R
y/R
4
y/R
4
−2
y/R
(b) v = 0.25
4
0
y/R
t=5
4
y/R
y/R
t=5
−2
−6
−4
−2
x/R
0
−6
−4
−2
0
x/R
Figure 3-7. Snapshots of the simulations for different background longshore currents with
steady forcing, positive vortices are plotted in red and negative vortices in
blue: a) v = 0 , b) v = 0.25 , c) v = 0.50 d) v = 0.75
34
The introduction of a background longshore current in the system has dramatic
effects as can be seen in the in figure 3-7, where snapshots of the system in time from the
start of the simulations are shown.
In figure 3-7 a) a simulation with no background current is shown. The sequence
shows how two macrovortices form at the start of the simulation and then, as they
continue to increase in strength they interact with each other and get advected offshore.
After this vortex couple leaves the generation area, the rip neck behaves as a turbulent jet.
This behavior is observed for all the different background longshore current strengths in
figure 3-7, although the size of the initial macrovortices is decreased with increasing
longshore current strength as the macrovortices are pushed downstream before they can
reach higher strength. These initial macrovortices result in a peak in the offshore velocity
in the rip neck, and a decline in the velocity follows once the turbulent jet-like flow is
established. Looking at the stronger longshore current cases, it seems apparent an
increase in the interaction between the vortices from either side of the rip channel would
cancel each others effects as they have opposite signs (vortices shown in two colors to
indicate opposite signs).
Velocities in the rip neck
Velocity time series were obtained for alongshore profiles located at different
cross-shore locations (x = 0, x = - 0.25, x = - 0.5, x = - 0.75, x = - 1). In order to avoid
turbulent jet instabilities and possible local interaction with passing point vortices,
averages of the velocities were performed without taking into account the initial peak in
cross-shore velocities due to the formation of the macrovortices.
35
The location of the jet was defined by the one-third highest offshore velocities for
each alongshore velocity profile. Once the velocities within the jet were located, their
average was defined as the mean peak offshore velocity (see figure 3-8).
v = 0.25
3
2.5
2
y/R
1.5
1
0.5
0
−0.5
−1
−1.5
−3
−2
−1
0
1
x/R
Figure 3-8. Computed mean velocities on a longshore profile at x = −0.5
Figure 3-9 shows the mean peak offshore velocities plotted against increasing
background longshore currents. Peak offshore velocities increase farther offshore as the
jet narrows for small background longshore currents, however as the longshore current
strength increases this pushes the vortices downstream causing more interaction between
opposite sign vortices (canceling their effects), thus decreasing the offshore velocities
more quickly. The velocities were plotted only as far as x = −1 because the applicability
of the model is limited to the rip neck area and other physical processes would become
dominant that far offshore.
36
Velocities at x = 0 are influenced by the presence of two discontinuity points, the
source points where the new point vortices are inserted in the system. From now on we
will be looking at the velocities at the location x = −0.5 since it is far enough from those
discontinuities but close enough so that generation of vorticity and circulation remains
the dominant physical process.
upeak vs vbackground at different x locations
1.4
x=−1
x = − 0.75
x = − 0.5
x = − 0.25
x=0
upeak (dimensionless)
1.2
1
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
vbackground (dimensionless)
Figure 3-9. Mean peak offshore velocities with increasing background longshore currents
at different cross-shore locations
Figure 3-10 shows the evolution of the mean peak offshore velocities with
increasing background longshore currents at the cross-shore location x = −0.5 for
different computational resolutions (e.g., time increment at which a new vortex pair is
introduced in the system). Good convergence is obtained with the model for all different
resolutions, which allows us to use the less computationally demanding resolution of
∆t = 0.05 .
37
The plot shows a strong effect of the background longshore current (v ) on the rip
current strength. Almost constant values of the mean peak offshore velocity are observed
for low values of the background longshore current (v < 0.3) with a rapid decrease for
larger values v. For background longshore currents of v > 0.8 the offshore currents
become very small. As stated above, since there are no other dissipative mechanisms the
main reason for this rapid decrease in the offshore velocity is the higher interaction
between oppositely signed vortices, which cancels out their effects as the longshore
current strength increases.
upeak vs vbackground at location x = − 0.5
with different computational resolutions
1.4
res = 0.03
res = 0.04
res = 0.05
upeak (dimensionless)
1.2
1
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
vbackground
Figure 3-10. Mean peak offshore velocity versus background longshore current
Vorticity
Since the model is based on the generation of point vortex pairs within the rip neck
area, it is very straightforward to obtain measures of vorticity in the system. Mean
38
vorticity values at any time for predetermined boxes (of size 0.25 by 0.25 dimensionless
units) are the sum of the strength of all the individual point vortices contained within
each particular box divided by the area of the box. Mean vorticity maps (see appendix A)
were obtained by averaging with time the instantaneous measures of vorticity in the
domain.
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
4
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.1
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.2
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.3
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
−0.5
Cross shore (dimensionless)
Figure 3-11. Mean vorticity maps for a) v = 0 , b) v = 0.1 , c) v = 0.2 and d) v = 0.3
39
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
4
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
2
0
1
0
−1
−0.25
−1
v = 0.4
−2
−2
0.25
3
v = 0.5
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.6
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.7
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
−0.5
Cross shore (dimensionless)
Figure 3-12. Mean vorticity maps for a) v = 0.4 , b) v = 0.5 , c) v = 0.6 and d) v = 0.7
Figures 3-11 and 3-12 show mean vorticity maps for increasing values of the
background longshore current, with increments of 0.1 dimensionless current from no
current up to v = 0.7 , for which the offshore current becomes very small. The vorticity
maps for v < 0.3 show no mixing between vortices of opposite signs with the jet getting
40
pushed downstream thus narrowing farther offshore and increasing the peak velocities in
the jet farther offshore (see figure 3-9). For stronger background longshore currents
(v > 0.3)
mixing between oppositely signed vortices starts to occur, as the upstream
vortices are pushed into the downstream source of vortices. Some of these point vortices
go right through the source point and might change their trajectory radically, resulting in
some very small mean values of vorticity ( − 0.1 < Γ < 0 dimensionless units) away from
the actual jet. When the longshore current becomes very strong the mechanism with
which the vortex pairs get advected offshore weakens and the longshore current becomes
dominant in the system.
Jet angle evolution with increasing background longshore current
Since the jet gets displaced downstream with the background longshore current, in
order to obtain a representative angle of the jet the angle of the velocities within the jet
was averaged with time. The angle was measured relative to the shore-normal, therefore
for no longshore background current the angle should be close to 0 degrees. As before the
jet was defined by the one-third highest offshore velocities for each alongshore velocity
profile. To be consistent, the velocity profile at the cross-shore location x = −0.5 was
used for the angle calculations.
Figure 3-13 shows the evolution of the jet angle with increasing background
longshore current strength. The jet angle increases almost linearly with the current.
Offshore velocities and background longshore current scale the same way so if the
longshore current had no effect on the jet an angle of about 45 degrees should be
expected for a background longshore current of 1. However, the plot shows how the
angle reaches 80 degrees for that value of v, indicating a strong effect of the background
41
longshore current on the strength of the offshore velocities, thus turning the jet in the
Angle with respect to the shore−normal (degrees)
alongshore direction.
90
80
70
60
50
40
30
20
10
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
vbackground (dimensionless)
Figure 3-13. Jet angle vs. background longshore current
Unsteady Forcing
As described in chapter 2, the forcing in the model, fixed at value 1 for steady
forcing, can be modulated using a harmonic representation with a group amplitude and
frequency (see equation 2-3). In the next section, an analysis of the relative importance of
these parameters used to define the unsteady forcing will be conducted.
Figure 3-10 shows good convergence for the model with three different resolutions
or ∆t (large time step). Since unsteady forcing is obviously more unstable than steady
forcing, it was decided to use longer model runs with a number of dimensionless time
units of 100 with the less computational demanding ∆t of 0.05 and N = 50 . This leads to
42
a total number of large time steps of 2000 and a total computational time of about 10
hours for each run. A compromise between computational time and widest range of
parameters needed to be achieved so a smaller number of background longshore currents
were used.
Table 3-1. Unsteady forcing simulations. Definition parameters. ∆t = 0.05 , N = 50 and #
of ∆t = 2000 .
Group amp. (a)
(Dimensionless)
0
0.25
0.5
0.75
1
Group freq. (ω)
(Dimensionless)
0.5
1
1.5
2
2.5
0.5
1
1.5
2
2.5
0.5
1
1.5
2
2.5
0.5
1
1.5
2
2.5
v background
(Dimensionless)
0, 0.1, 0.2…, 0.9, 1
0, 0.25, 0.50, 0.75, 1
0, 0.25, 0.50, 0.75, 1
0, 0.25, 0.50, 0.75, 1
0, 0.25, 0.50, 0.75, 1
0, 0.25, 0.50, 0.75, 1
0, 0.25, 0.50, 0.75, 1
0, 0.25, 0.50, 0.75, 1
0, 0.25, 0.50, 0.75, 1
0, 0.25, 0.50, 0.75, 1
0, 0.25, 0.50, 0.75, 1
0, 0.25, 0.50, 0.75, 1
0, 0.25, 0.50, 0.75, 1
0, 0.25, 0.50, 0.75, 1
0, 0.25, 0.50, 0.75, 1
0, 0.25, 0.50, 0.75, 1
0, 0.25, 0.50, 0.75, 1
0, 0.25, 0.50, 0.75, 1
0, 0.25, 0.50, 0.75, 1
0, 0.25, 0.50, 0.75, 1
0, 0.25, 0.50, 0.75, 1
Table 3-1 shows all the different runs that were used in the analysis. Four different
amplitudes, with five frequencies each were used with five different background
longshore current strengths ranging from 0 to 1. Also a base case with no groupiness was
used for comparison reasons. Results for this case compared well with the ones used in
the steady forcing section (smaller number of time steps and N=100).
43
a) steady
t = 10
2
2
2
2
0
0
−2
−6
−4
−2
0
y/R
4
y/R
4
−2
−4
−2
0
−2
−6
t = 20
−4
−2
0
−6
t = 15
2
2
2
2
−2
0
−2
−6
−4
−2
0
y/R
4
y/R
4
y/R
4
0
t = 25
−4
−2
0
−4
−2
0
−6
t = 25
2
2
2
2
0
−2
−6
−4
−2
x/R
0
y/R
4
y/R
4
y/R
4
−2
−4
−2
x/R
0
−2
0
0
−2
−6
−4
t = 30
4
0
0
−2
−6
t = 30
0
−2
0
−2
−6
−4
t = 20
4
0
t = 10
0
−2
−6
t = 15
y/R
b) unsteady
4
0
y/R
t=5
4
y/R
y/R
t=5
−2
−6
−4
−2
x/R
0
−6
−4
−2
0
x/R
Figure 3-14. Comparison of the simulations for steady and unsteady forcing with a
background longshore current of v = 0.25 . Unsteady forcing with amplitude
a = 1 and frequency ω = 1
Figure 3-14 shows a comparison of the simulations for steady and unsteady forcing
with the same background longshore current. The amplitude for the unsteady case is
equal to 1, so this is the limiting case where the forcing strength changes with time from
strength 0 to 2. The figure shows a much wider spreading of the jet and higher
interactions between oppositely signed vortices. This would suggest smaller offshore
velocities in the jet since these interactions would cancel their effects, however the
strength of the vortices gets up to double during one group period, so both effects may
counteract each other. We will try to address this here. Also, it is noticeable how the
direction of the jet changes with time as the relative strength of the forcing to the
background longshore current changes. This effect resembles a hose being swung back
44
and forth and becomes more evident with increasing background longshore currents and
group amplitudes.
Since the behavior of the rip current is very unsteady with time the analysis will be
conducted using mean quantities (averaged over an integer number of wave periods). As
with the steady forcing cases, mean velocity fields within the area of the rip neck and
mean vorticity maps were obtained using a series of matlab routines. Comparison with
the base case (no groupiness) will be used when possible.
Velocities in the rip neck
For comparison reasons, velocities at the alongshore profile at x = −0.5 will be
used. The two basic parameters that define the unsteady forcing are the amplitude and the
group frequency. The data will be grouped together to study the effects of these
parameters separately.
Figure 3-15 shows the mean peak offshore velocities in the rip neck at the crossshore location x = −0.5 . The different colored lines on each plot represent all the
frequencies. As could be expected, the smaller amplitudes compare better with the steady
case (black line), with the larger amplitudes separating from it. In general, the velocities
are smaller than in the steady case for smaller background longshore current strengths,
and larger for larger background longshore current strengths, resulting in a change of
shape of the steady case plot. Basically this indicates that higher amplitudes have smaller
responses to higher background longshore current strengths, therefore the periods in the
forcing where the intensity of the vortices is above 1 seem to dominate over the periods
where it is smaller than 1.
1.2
1
0.8
0.6
0.4
0.2
0
0
0.25
0.5
0.75
1
c) amplitude = 0.75
1.2
1
0.8
0.6
0.4
0.2
0
0
0.25
0.5
0.75
v (dimensionless)
1
mean peak u (dimensionless)
a) amplitude = 0.25
mean peak u (dimensionless)
mean peak u (dimensionless)
mean peak u (dimensionless)
45
b) amplitude = 0.5
1.2
1
0.8
0.6
0.4
0.2
0
0
0.25
0.5
0.75
1
d) amplitude = 1
1.2
1
0.8
0.6
0.4
0.2
0
0
0.25
0.5
0.75
1
v (dimensionless)
Figure 3-15. Mean peak offshore velocities in the jet at the cross-shore location x = −0.5
for different amplitudes. (-) Steady forcing, (-o-) ω = 0.5 , (-o-) ω = 1 , (-o-)
ω = 1.5 , (-o-) ω = 2 and (-o-) ω = 2.5
Figure 3-15 shows some spreading on the lines on each plot, which correspond to
the different frequencies. In order to study the effect of these, the mean peak offshore
velocities in the rip neck at the cross-shore location x = −0.5 are plotted on figure 3-16
for each frequency. The different colored lines on each plot represent the different
amplitudes. The differences between the base case and the unsteady results are smaller
for the higher frequencies with higher responses for the smaller frequencies. This
behavior agrees with the results presented by Kennedy (2003), where it was determined
that for dimensionless frequencies greater than 1 the response decreases very quickly.
1.2
1
0.8
0.6
0.4
0.2
0
0
0.25
0.5
0.75
1
c) frequency = 1.50
1.2
1
0.8
0.6
0.4
0.2
0
0
0.25
0.5
0.75
1
v (dimensionless)
mean peak u (dimensionless)
a) frequency = 0.50
mean peak u (dimensionless)
mean peak u (dimensionless)
mean peak u (dimensionless)
46
b) frequency = 1.00
1.2
1
0.8
0.6
0.4
0.2
0
0
0.25
0.5
0.75
1
d) frequency = 2.00
1.2
1
0.8
0.6
0.4
0.2
0
0
0.25
0.5
0.75
1
v (dimensionless)
Figure 3-16. Mean peak offshore velocities in the jet at the cross-shore location x = −0.5
for different frequencies. (-) Steady forcing, (-o-) a = 0.25 , (-o-) a = 0.5 , (-o-)
a = 0.75 , (-o-) a = 1
Vorticity
Mean vorticity maps were obtained for each one of the unsteady cases with the
same resolution (boxes of 0.25 by 0.25 dimensionless units). In general, the “area of
influence” of the jet, or the areas with presence of vortices widens with increasing
amplitude. As a representative case, the vorticity maps for the case of amplitude 1 and
frequency 1 are shown on figure 3-17. This is the consequence of having vortices
strength changing from 0 to 2 over a group period resulting on the behavior described
before as a hose being swung back and forth.
47
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.25
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.5
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.75
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
−0.5
Cross shore (dimensionless)
Figure 3-17. Mean vorticity maps for the unsteady case of amplitude 1 and frequency 1.
a) v = 0 , b) v = 0.25 , c) v = 0.5 and d) v = 0.75
In order to being able to compare the relative importance of each parameter, it is
necessary, once again, to represent as many cases together as possible in plots sorted by
the different values of the parameter under study. In order to do that, alongshore vorticity
48
profiles were calculated and the one at the cross-shore location x = −0.5 was chosen to
be consistent with the velocity analysis. The zero vorticity crossing is a good indicator of
the location of the center of the jet and was obtained for each case. Also it was
determined the alongshore location of the –0.1 and +0.1 vorticity values as an indicator of
the width of the jet, although this value might be high for the largest background
longshore currents were offshore velocities, and therefore mean vorticity values, are
small (see appendix B).
On figure 3-18, the alongshore location of the (-0.1, 0, 0.1) values of the vorticity
(error bars) are plotted against the background longshore currents for different values of
the amplitude. The different lines on each plot represent all the frequency values.
Observing figure 3-18, it appears that the frequency has very little influence on the
location of the jet since all the different lines are very close together except for the
amplitude 1 case (figure 3-18d) and larger background longshore current, however the
location of the zero crossing becomes noisier as the longshore current increases. It is
noticeable how the width of the jet increases with increasing amplitude (larger error
bars), also the location of the positive crossing (downstream of the background longshore
current) is generally closer to the zero crossing, indicating that the vorticity gradient is
larger in the downstream side of the jet, thus inducing larger velocities on that side of the
jet (as can be seen on the mean velocity profile on figure 3-8).
Similarly, figure 3-19 shows the alongshore location of the (-0.1, 0, 0.1) values of
the vorticity plotted against the background longshore currents but for different values of
the frequency. The different lines on each plot represent all the amplitude values.
49
a) amplitude = 0.25
b) amplitude = 0.50
2.5
y (dimensionless)
y (dimensionless)
2.5
2
1.5
1
0.5
0
−0.5
0
0.25
0.5
0.75
2
1.5
1
0.5
0
−0.5
1
0
c) amplitude = 0.75
0.75
1
2.5
y (dimensionless)
y (dimensionless)
0.5
d) amplitude = 1
2.5
2
1.5
1
0.5
0
−0.5
0.25
0
0.25
0.5
0.75
v (dimensionless)
1
2
1.5
1
0.5
0
−0.5
0
0.25
0.5
0.75
1
v (dimensionless)
Figure 3-18. Alongshore location of the (-0.1, 0, 0.1) values of the vorticity at x = −0.5
with increasing longshore current for each amplitude. (-x-) ω = 0.5 , (-x-)
ω = 1 , (-x-) ω = 1.5 , (-x-) ω = 2 and (-x-) ω = 2.5
From figure 3-19 it can be inferred that the location of the zero vorticity crossing
(location of the jet) for the larger longshore current strengths has a dependence on the
group amplitude, with the jet being pushed farther downstream for the lower amplitudes.
This reinforces the fact that the periods during which the forcing is higher than 1
dominate over the ones where it is smaller than 1, as concluded for the offshore velocity
profiles. Although the frequency has a smaller influence on the location of the jet, the
cases for higher frequencies on figure 3-19 show less separation from the steady forcing
case.
50
a) frequency = 0.5
b) frequency = 1
2.5
y (dimensionless)
y (dimensionless)
2.5
2
1.5
1
0.5
0
−0.5
0
0.25
0.5
0.75
2
1.5
1
0.5
0
−0.5
1
0
c) frequency = 1.5
0.75
1
2.5
y (dimensionless)
y (dimensionless)
0.5
d) frequency = 2
2.5
2
1.5
1
0.5
0
−0.5
0.25
0
0.25
0.5
0.75
v (dimensionless)
1
2
1.5
1
0.5
0
−0.5
0
0.25
0.5
0.75
1
v (dimensionless)
Figure 3-19. Alongshore location of the (-0.1, 0, 0.1) values of the vorticity at x = −0.5
with increasing longshore current for each frequency. (-x-) Steady forcing,
(-x-) a = 0.25 , (-x-) a = 0.5 , (-x-) a = 0.75 and (-x-)
Figure 3-20 shows the mean location and width of the jet as defined before but
plotted against increasing group amplitudes. It shows that for larger values of the
amplitude and the background longshore current the jet is not pushed downstream as
much, as stated before, and that for the smaller longshore currents the amplitude has no
influence. The background longshore current is the predominant factor on the location of
the jet.
51
a) frequency = 0.5
b) frequency = 1
2.5
y (dimensionless)
y (dimensionless)
2.5
2
1.5
1
0.5
0
−0.5
0
0.25
0.5
0.75
2
1.5
1
0.5
0
−0.5
1
0
c) frequency = 1.5
0.75
1
2.5
y (dimensionless)
y (dimensionless)
0.5
d) frequency = 2
2.5
2
1.5
1
0.5
0
−0.5
0.25
0
0.25
0.5
0.75
a (dimensionless)
1
2
1.5
1
0.5
0
−0.5
0
0.25
0.5
0.75
1
a (dimensionless)
Figure 3-20. Alongshore location of the (-0.1, 0, 0.1) values of the vorticity at x = −0.5
with increasing group amplitude for each frequency. (-x-) v = 0 , (-x-)
v = 0.25 , (-x-) v = 0.5 , (-x-) v = 0.75 and (-x-) v = 1
Figure 3-21 shows that frequency has almost no influence on the location of the jet.
For the higher values of the frequency and amplitude some of the zero crossing are
missing in the figure, this is due to the fact that along the alongshore profile at the crossshore location x = −0.5 the vorticity is always negative and never becomes positive. This
indicates that the jet is parallel to the shoreline. Figure 3-22 shows one of these cases
where the jet, defined by the oppositely signed vorticity on either side is completely
parallel to the shore.
52
a) amplitude = 0.25
b) amplitude = 0.5
2.5
y (dimensionless)
y (dimensionless)
2.5
2
1.5
1
0.5
0
−0.5
0.5
1
1.5
2
2
1.5
1
0.5
0
−0.5
2.5
0.5
c) amplitude = 0.75
2
2.5
2.5
y (dimensionless)
y (dimensionless)
1.5
d) amplitude = 1
2.5
2
1.5
1
0.5
0
−0.5
1
0.5
1
1.5
2
2.5
2
1.5
1
0.5
0
−0.5
0.5
f (dimensionless)
1
1.5
2
2.5
f (dimensionless)
Figure 3-21. Alongshore location of the (-0.1, 0, 0.1) values of the vorticity at x = −0.5
with increasing group frequency for each amplitude. (-x-) v = 0 , (-x-)
v = 0.25 , (-x-) v = 0.5 , (-x-) v = 0.75 and (-x-) v = 1
Although it seems like for these cases cross-shore velocities in the jet should be
almost zero, this does not agree completely with the results obtained with the velocity
profiles where a small residual offshore current remains for the same cases. However, the
mean peak offshore velocities calculated at the cross-shore location x = −0.5 have a peak
in the rip neck area before the jet is turned parallel to the shore while going around the
downstream fixed source of vorticity (see figure 3-22).
53
Measure of mean vorticity in the system
averaged over 14 group periods
0.5
5
Alongshore coord. (dimensionless)
4
0.25
3
2
0
1
0
−0.25
−1
v = 0.75
−2
−2
−1
0
1
−0.5
Cross shore coord. (dimensionless)
Figure 3-22. Mean vorticity map for the unsteady forcing case with: a = 1 , ω = 1.5 and
v = 0.75
CHAPTER 4
SUMMARY AND CONCLUSIONS
A simplified conceptual representation of a rip current system was used to study the
effects of longshore currents on rip currents. Rip currents are part of a very complex
circulation system within the near shore, where many different processes interact with
each other. By focusing on the rip neck area, where generation of circulation and vorticity
was assumed to be the main physical process, a simplified representation of a rip current
was achieved.
The numerical model used for this study (discussed in detail in chapter 2), first
introduced by Kennedy (2003), is based on a generation of vorticity approach, where
oppositely signed vortices are continuously released on either side of the rip channel and
let free in the system. Flat bed and essentially no energy dissipation (no bottom friction,
no wave-current interaction) are assumed in the model. Since generation of vorticity is
the only physical process represented in the model, its applicability is strictly limited to
the rip neck area, where this process is assumed to be dominant. A constant background
longshore current was added to the model to study its effects on the generated jet-like rip
current. Both steady and unsteady forcing were used. A wide range of unsteady forcing
parameters were used in conjunction with a number of increasing background longshore
current strengths.
There are three parameters that define the model. A mean rate of generation of
circulation which depends mainly in the strength of wave breaking and the local
topography
(DΓ
Dt ) , the semi-gap width of the rip channel
54
(R )
and the constant
55
background longshore current
(v ) .
Scaling of the different processes is very
straightforward. Velocities in the rip neck depend on the strength of wave breaking,
scaling with (DΓ Dt ) 2 , but not on the gap width (R ) . Temporal response will depend
1
(
both on wave breaking strength and length scales R (DΓ Dt )
1
2
).
The model was tested against measured laboratory data in the first section of
chapter 3. Two datasets were available, both from experiments conducted at the Center
for Applied Coastal Research of the University of Delaware. The experimental setup
consisted of a fixed topography of a barred beach with rip channels. In order to scale
measured velocities in the experiments, estimates of the scaling parameter (DΓ Dt )
where obtained using two different methodologies proposed by Brocchini et al. (2003).
The first methodology, based on breaking assumptions over the bar uses offshore wave
data and water depth at the bar crest as inputs. The second methodology uses local
measured wave heights and setups. It was found that the first methodology over estimates
the generation of circulation by a factor of two as compared to the second methodology
(see figure 3-5), which was believed to be more accurate since it uses local measured
data. On the other hand, the first methodology could easily be applied to field data since
offshore wave measurements and water depth at the bar crest are easy to obtain, whereas
obtaining local setup measurements is very complicated.
The model predicted quite well the trend of the data, although it seemed to over
predict velocities in the rip neck. This was attributed to the difficulty in determining the
relative cross-shore location of the measuring points with respect to the generation of
vorticity in the surf zone. Also, the estimations of the mean rate of generation of
circulation are probably high since the methodologies applied consider the maximum
56
possible generation of circulation (ignoring any breaking in the channel). Ignored
physical processes like wave-current interaction, 3D effects or bottom friction are
probably not negligible. This could be adjusted with a parameter to decrease predicted
velocities.
In the second section of chapter 3, the model was used to study the effects of
longshore currents on rip currents, using both steady and unsteady forcing. Velocity
fields and mean vorticity maps within the rip channel area were used in the analysis. The
introduction of a background longshore current in the systems was found to have
dramatic effects on the rip current behavior (see figures 3-10). The mean peak offshore
velocities within the jet are approximately constant for small background dimensionless
longshore currents (v < 0.3) but decrease quickly once the relative strength of the
longshore current becomes stronger (v > 0.3) . This was found to be due to the mixing of
oppositely signed vortices from either side of the rip once the current became strong
enough to push the upstream vortices into the downstream vortices, canceling their
effects. This can be observed in figures 3-11 and 3-12.
The angle of the jet was estimated using the direction of the velocities with the 1/3
highest offshore component of velocity at the dimensionless location x = −0.5 . The jet
angle, relative to the shore normal, became almost 80 degrees for dimensionless
longshore current strengths of 1, but would be expected to be around 45 degrees (figure
3-13) since offshore velocities and background longshore current scale the same way.
This is another indicator of the strong effect of the longshore current on the rip current.
A wide range of values was used to test the relative importance of the parameters
defining the unsteady forcing (amplitude and frequency) with increasing background
57
longshore current strengths. Unsteady forcing resulted in a very unsteady behavior of the
rip current or jet (see figure 3-14), thus only mean quantities could be studied.
Mean peak offshore velocities in the jet were found to be mainly dependent on the
strength of the background longshore current (see figures 3-15 and 3-16). Higher
amplitudes (of unsteady forcing) were found to have smaller responses to higher
background longshore current strengths, therefore the periods in the forcing where the
intensity of the vortices is above 1 seem to dominate over the periods where it is smaller
than 1. For dimensionless frequencies greater than 1 the response was very small. This
agrees with the results presented by Kennedy (2003).
The alongshore location of the jet along a mean vorticity profile at x = −0.5 was
determined by the zero vorticity crossing and estimations of the jet width were obtained
by the locations of –0.1 and 0.1 mean vorticity values. The location of the jet was found
to be mainly dependent on the strength of the background longshore current. Higher
values of the group amplitude resulted in a smaller displacement of the jet downstream
(figure 3-20), supporting the argument that the periods during which the forcing is higher
than 1 dominate over the ones where it is smaller than 1. The frequency was found to
have almost no influence on the jet location (figure 3-21). The jet narrows for stronger
background currents as the flow is confined closer to the downstream source of vorticity.
Higher amplitudes induce wider mean jet widths (figure 3-20) although its influence is
very small compared to the background longshore current.
Despite the high simplicity of the model, it has proven to obtain very reasonable
results in comparison with measured data from laboratory experiments. The main
challenge to apply the results of the model in the field is the estimation of the model
58
parameter
(DΓ
Dt ) ; however if the model were used as a predictive tool the
methodologies used in this thesis would give predictions on the safe side.
For future research, the model results could be coupled with existing rip current
forecasting indexes as a predictive tool. Further comparison with laboratory and field
data would be desired, especially with longshore current data.
APPENDIX A
MEAN VORTICITY MAPS
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
4
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.25
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.5
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.75
1
−0.5
Cross shore (dimensionless)
−2
−2
−1
0
1
−0.5
Cross shore (dimensionless)
Figure A-1. Mean vorticity map for steady forcing. a) v = 0 , b) v = 0.25 , c) v = 0.5 and
d) v = 0.75
59
60
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.25
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.5
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.75
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
−0.5
Cross shore (dimensionless)
Figure A-2. Mean vorticity maps for unsteady forcing with amplitude 0.25 and frequency
0.5. a) v = 0 , b) v = 0.25 , c) v = 0.5 and d) v = 0.75
61
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.25
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.5
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.75
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
−0.5
Cross shore (dimensionless)
Figure A-3. Mean vorticity maps for unsteady forcing with amplitude 0.25 and frequency
1. a) v = 0 , b) v = 0.25 , c) v = 0.5 and d) v = 0.75
62
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.25
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.5
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.75
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
−0.5
Cross shore (dimensionless)
Figure A-4. Mean vorticity maps for unsteady forcing with amplitude 0.25 and frequency
1.5. a) v = 0 , b) v = 0.25 , c) v = 0.5 and d) v = 0.75
63
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.25
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.5
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.75
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
−0.5
Cross shore (dimensionless)
Figure A-5. Mean vorticity maps for unsteady forcing with amplitude 0.25 and frequency
2. a) v = 0 , b) v = 0.25 , c) v = 0.5 and d) v = 0.75
64
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.25
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.5
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.75
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
−0.5
Cross shore (dimensionless)
Figure A-6. Mean vorticity maps for unsteady forcing with amplitude 0.25 and frequency
2.5. a) v = 0 , b) v = 0.25 , c) v = 0.5 and d) v = 0.75
65
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.25
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.5
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.75
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
−0.5
Cross shore (dimensionless)
Figure A-7. Mean vorticity maps for unsteady forcing with amplitude 0.50 and frequency
0.5. a) v = 0 , b) v = 0.25 , c) v = 0.5 and d) v = 0.75
66
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.25
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.5
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.75
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
−0.5
Cross shore (dimensionless)
Figure A-8. Mean vorticity maps for unsteady forcing with amplitude 0.50 and frequency
1. a) v = 0 , b) v = 0.25 , c) v = 0.5 and d) v = 0.75
67
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.25
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.5
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.75
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
−0.5
Cross shore (dimensionless)
Figure A-9. Mean vorticity maps for unsteady forcing with amplitude 0.50 and frequency
1.5. a) v = 0 , b) v = 0.25 , c) v = 0.5 and d) v = 0.75
68
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.25
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.5
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.75
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
Cross shore (dimensionless)
Figure A-10. Mean vorticity maps for unsteady forcing with amplitude 0.50 and
frequency 2. a) v = 0 , b) v = 0.25 , c) v = 0.5 and d) v = 0.75
−0.5
69
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.25
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.5
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.75
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
Cross shore (dimensionless)
Figure A-11. Mean vorticity maps for unsteady forcing with amplitude 0.50 and
frequency 2.5. a) v = 0 , b) v = 0.25 , c) v = 0.5 and d) v = 0.75
−0.5
70
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.25
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.5
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.75
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
Cross shore (dimensionless)
Figure A-12. Mean vorticity maps for unsteady forcing with amplitude 0.75 and
frequency 0.5. a) v = 0 , b) v = 0.25 , c) v = 0.5 and d) v = 0.75
−0.5
71
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.25
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.5
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.75
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
Cross shore (dimensionless)
Figure A-13. Mean vorticity maps for unsteady forcing with amplitude 0.75 and
frequency 1. a) v = 0 , b) v = 0.25 , c) v = 0.5 and d) v = 0.75
−0.5
72
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.25
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.5
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.75
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
Cross shore (dimensionless)
Figure A-14. Mean vorticity maps for unsteady forcing with amplitude 0.75 and
frequency 1.5. a) v = 0 , b) v = 0.25 , c) v = 0.5 and d) v = 0.75
−0.5
73
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.25
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.5
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.75
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
Cross shore (dimensionless)
Figure A-15. Mean vorticity maps for unsteady forcing with amplitude 0.75 and
frequency 2. a) v = 0 , b) v = 0.25 , c) v = 0.5 and d) v = 0.75
−0.5
74
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.25
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.5
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.75
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
Cross shore (dimensionless)
Figure A-16. Mean vorticity maps for unsteady forcing with amplitude 0.75 and
frequency 2.5. a) v = 0 , b) v = 0.25 , c) v = 0.5 and d) v = 0.75
−0.5
75
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.25
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.5
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.75
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
−0.5
Cross shore (dimensionless)
Figure A-17. Mean vorticity maps for unsteady forcing with amplitude 1 and frequency
0.5. a) v = 0 , b) v = 0.25 , c) v = 0.5 and d) v = 0.75
76
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.25
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.5
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.75
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
−0.5
Cross shore (dimensionless)
Figure A-18. Mean vorticity maps for unsteady forcing with amplitude 1 and frequency
1. a) v = 0 , b) v = 0.25 , c) v = 0.5 and d) v = 0.75
77
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.25
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.5
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.75
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
−0.5
Cross shore (dimensionless)
Figure A-19. Mean vorticity maps for unsteady forcing with amplitude 1 and frequency
1.5. a) v = 0 , b) v = 0.25 , c) v = 0.5 and d) v = 0.75
78
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.25
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.5
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.75
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
−0.5
Cross shore (dimensionless)
Figure A-20. Mean vorticity maps for unsteady forcing with amplitude 1 and frequency
2. a) v = 0 , b) v = 0.25 , c) v = 0.5 and d) v = 0.75
79
Measure of mean vorticity in the system
5
0.5
5
0.5
b)
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
a)
4
4
2
0
1
0
−1
−0.25
−1
v=0
−2
−2
0.25
3
v = 0.25
0
1
−0.5
Cross shore (dimensionless)
5
−2
−2
−1
0
0.5
5
0.5
0.25
3
2
0
1
0
−0.25
−1
Alongshore (dimensionless)
Alongshore (dimensionless)
d)
4
4
0
0.25
3
2
0
1
0
−0.25
−1
v = 0.5
−1
−0.5
Cross shore (dimensionless)
c)
−2
−2
1
v = 0.75
1
Cross shore (dimensionless)
−0.5
−2
−2
−1
0
1
−0.5
Cross shore (dimensionless)
Figure A-21. Mean vorticity maps for unsteady forcing with amplitude 1 and frequency
2.5. a) v = 0 , b) v = 0.25 , c) v = 0.5 and d) v = 0.75
APPENDIX B
LOCATION AND WIDTH OF THE JET DATA
The following tables contain the alongshore locations of the vorticity values used in
figures in chapter 3 to define location (zero vorticity crossing) and width ([-0.1,0.1]
vorticity values) of the jet. All quantities are dimensionless and were obtained from the
mean vorticity maps of each case (see Appendix A).
Table B-1. Alongshore location and width of the jet for steady forcing
v
0
0.25
0.5
0.75
1
No groupiness cases
yy0
-0.38
0
0.89
0.93
1.1
1.14
1.33
1.59
1.97
80
y+
0.4
0.96
1.21
81
Table B-2. Alongshore location and width of the jet for unsteady forcing with amplitude
0.25
v
0
0.25
0.5
0.75
1
y-0.4
0.88
1.03
1.31
v
0
0.25
0.5
0.75
1
y-0.39
0.89
1.03
1.27
v
0
0.25
0.5
0.75
1
y-0.4
0.9
1.05
1.3
f=0.5
y0
0
0.92
1.06
1.37
1.86
f=1.5
y0
0
0.94
1.07
1.34
1.7
f=2.5
y0
0
0.95
1.09
1.36
1.67
y+
0.38
0.96
1.1
1.52
y-0.4
0.91
1.03
1.23
y+
0.39
0.98
1.11
1.45
y-0.38
0.89
1.02
1.29
y+
0.36
1
1.13
1.5
f=1.0
y0
0
0.97
1.08
1.36
1.83
f=2.0
y0
0
0.94
1.06
1.36
1.68
y+
0.35
1.01
1.13
1.44
y+
0.39
0.97
1.1
1.47
82
Table B-3. Alongshore location and width of the jet for unsteady forcing with amplitude
0.5
v
0
0.25
0.5
0.75
1
y-0.36
0.79
1
1.24
v
0
0.25
0.5
0.75
1
y-0.39
0.89
1.02
1.15
v
0
0.25
0.5
0.75
1
y-0.38
0.89
1.05
1.3
1.53
f=0.5
y0
0
0.9
1.05
1.31
1.68
f=1.5
y0
0
0.95
1.06
1.29
1.56
f=2.5
y0
0
0.94
1.1
1.38
y+
0.4
0.96
1.1
1.37
y-0.39
0.88
0.93
1.13
y+
0.37
1
1.11
1.42
y-0.38
0.88
1.01
1.32
y+
0.38
1
1.18
1.64
f=1.0
y0
0
0.98
1.03
1.27
1.6
f=2.0
y0
0
0.93
1.04
1.46
1.59
y+
0.4
1.04
1.1
1.36
y+
0.37
0.98
1.08
83
Table B-4. Alongshore location and width of the jet for unsteady forcing with amplitude
0.75
v
0
0.25
0.5
0.75
1
y-0.39
0.6
0.92
1.11
v
0
0.25
0.5
0.75
1
y-0.39
0.86
1.04
1.22
v
0
0.25
0.5
0.75
1
y-0.39
0.84
1.08
1.29
1.56
f=0.5
y0
0
0.82
1.03
1.24
1.56
f=1.5
y0
0
0.93
1.09
1.35
1.4
f=2.5
y0
0
0.92
1.12
1.37
y+
0.41
0.91
1.08
1.33
y-0.4
0.8
0.92
1.04
y+
0.39
0.99
1.14
y-0.39
0.87
1
1.35
f=1.0
y0
0
0.94
1.03
1.2
1.4
f=2.0
y0
0
0.93
1.04
1.68
y+
0.38
0.99
1.23
1.63
y+
0.39
1.02
1.11
1.34
y+
0.38
0.99
1.08
84
Table B-5. Alongshore location and width of the jet for unsteady forcing with amplitude
1
v
0
0.25
0.5
0.75
1
y-0.41
0.39
0.87
1.04
1.3
v
0
0.25
0.5
0.75
1
y-0.41
0.8
1.11
1.36
v
0
0.25
0.5
0.75
1
y-0.39
0.82
1.12
1.29
f=0.5
y0
0
0.72
1.01
1.15
1.38
f=1.5
y0
0
0.9
1.22
1.34
f=2.5
y0
0
0.88
1.29
1.42
y+
0.4
0.84
1.07
1.26
1.56
y-0.4
0.61
1.01
y+
0.41
0.96
1.36
y-0.41
0.81
1.01
1.44
1.43
y+
0.44
0.97
1.44
1.7
f=1.0
y0
0
0.88
1.09
1.16
1.33
f=2.0
y0
0
0.9
1.05
y+
0.42
0.98
1.61
y+
0.42
0.96
1.09
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BIOGRAPHICAL SKETCH
Enrique Gutiérrez Diez was born in Palencia, in the region of Castilla y León,
Spain, to Mr. Enrique Felix Gutiérrez Ozámiz and Mrs. María Cruz Diez González. The
author and his family used to spend the summer vacations in Cadiz, in the south of Spain,
where he enjoyed the beach and later on he had the opportunity to sail along the south of
Spain. These experiences instilled in him the passion for the ocean and everything related
to it.
After graduating from Victorio Macho High School in 1995, he moved to
Santander to pursue the degree in civil engineering at the University of Cantabria. In
early 2000 he was accepted to the exchange program to attend the University of Miami
during the academic year 2000/2001, during which he took his first coastal related
course. After going back to Santander for his final year, he decided to specialize in
coastal engineering, during this year he worked as a student assistant in the Department
of Coastal and Oceanographic Engineering (GIOC) at the University of Cantabria. After
being accepted to a number of universities, he decided to go to Gainesville, Florida, to get
his Master of Science in coastal engineering at the University of Florida. Upon obtaining
the degree the author plans to go back to Spain to work on this exciting field.
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