• TABLE OF CONTENTS
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 Front Cover
 Title Page
 Dedication
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Abstract
 Introduction
 Numerical model
 Results and analyses
 Summary and conclusions
 Mean vorticity maps
 Location and width of the...
 Reference
 Biographical sketch














Title: Effects of longshore currents on rip currents
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Title: Effects of longshore currents on rip currents
Series Title: Effects of longshore currents on rip currents
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Language: English
Creator: Gutierrez, Enrique
Publisher: Coastal & Oceanographic Engineering Dept. of Civil & Coastal Engineering, University of Florida
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Table of Contents
    Front Cover
        Front Cover
    Title Page
        Page i
    Dedication
        Page ii
    Acknowledgement
        Page iii
    Table of Contents
        Page iv
        Page v
    List of Tables
        Page vi
    List of Figures
        Page vii
        Page viii
        Page ix
    Abstract
        Page x
        Page xi
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
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    Numerical model
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
    Results and analyses
        Page 21
        Page 22
        Page 23
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        Page 25
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        Page 53
    Summary and conclusions
        Page 54
        Page 55
        Page 56
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    Mean vorticity maps
        Page 59
        Page 60
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    Location and width of the jet data
        Page 80
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        Page 82
        Page 83
        Page 84
    Reference
        Page 85
        Page 86
        Page 87
        Page 88
    Biographical sketch
        Page 89
Full Text



UFL/COEL-2004/005


EFFECTS OF LONGSHORE CURRENTS ON RIP
CURRENTS







by



Enrique Gutierrez






Thesis


2004
















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.

















TABLE OF CONTENTS
Page

ACKNOWLEDGMENTS ................................................................................................. iii

L IST O F TA B L ES ..................................................... .................................................. vi

L IST O F FIG U R E S .................................................................................................... vii

CHAPTER

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

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
R ip C current Scaling .............................................................................................. 17
Numerical Description of the Model ................................................................ 18

3 RESULTS AND ANALYSIS.............................................................................21

Mean Peak Offshore Current in the Rip Neck.......................................................21
Parameter (DF / Dt) Estimations......................................................................22
M ethod 1 ...................................................... ...................................... . 22
M ethod 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
V orticity ................................................................................................. 37
Jet angle evolution with increasing background longshore current ............40
U steady Forcing...........................................................................................41










Velocities in the rip neck.................................................... ................. 44
V orticity .......................................................................................................4 6

4 SUMMARY AND CONCLUSIONS.........................................................................54

APPENDIX

A MEAN VORTICITY MAPS ............................................... ......... ..........59

B LOCATION AND WIDTH OF THE JET DATA ....................................................80

L IST O F R E FER EN C E S ...................................................................................................85

BIO G RAPH ICA L SK ETCH .............................................................................................89
















LIST OF TABLES


Table page

3-1 Unsteady forcing sim ulations.............................................................................. 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
















LIST OF FIGURES


Figure ge

1-1 Rip current parts: feeders neck and head (from Shepard et al., 1941).....................7

1-2 Tim e-averaged vorticity ..................................................... .............................. 12

2-1 Definition sketch of the m odel. ................................... .................................. 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 m eter 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 am plitudes............................................................................................ 45










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
am plitude for each frequency.......................................................................51

3-21 Alongshore location and width of the jet at x = -0.5 with increasing group
frequency for each am plitude................................................................................52

3-22 Mean vorticity map for unsteady forcing with amplitude 1, frequency 1.5and
background longshore current of 0.75...................................................................53

A-I M ean 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- 1 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










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 I and frequency 2.........78

A-21 Mean vorticity maps for unsteady forcing with amplitude 1 and frequency 2.5......79
















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










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.















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.










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










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










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),

Dronen 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










drifters, which have been used in the field in limited cases [Shepard and Imnan (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










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,












usually a channel can be found which can be as much as Im 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.


* g .- 4'"'- *



HAO 4
IIP
.- *\ ^ '






* 'C


* I o iF *
I .l





~ /^ \ ZON ''.:
,> I r*
,It,
r i j









S irr:4 r
rrr,~~~~r .: :: ".. ..:



-* p -
A
i UHc r c


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










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










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. Dronen 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.










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 Sy,

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,, 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:


F S = [1.1]
ax

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










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), Falqu6s 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 (), creating steady

pressure gradients that balance the gradient of the radiation stress:

d77 1 dS, [1.2]
dx pgh+) dx

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 thrus tbr force. If

there is differential breaking alongshore this will generate differential forces which will

generate circulation.










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).


.7
V






T .








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 Buhler (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










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).

Biihler 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 wave-

breaking 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.










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:

r= 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.









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 (DF/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)/(DF/Dt)2

* Time: t' = t (DF/Dt)~/R

* Circulation: F' = /(R(DF/Dt)2)

Therefore, we will need to determine what R and (DF/Dt) are in the field (lab

experiment) and then scale everything accordingly. Estimating R is relatively easy; more










of a challenge is the parameter (DF/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 (DF/Dt)%, but not on the gap width (R). Temporal

response will depend both on wave breaking strength and length scales (R/(DF/Dt)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.










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, Uo = F/2nr, 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)(+a) = (X, Y)() + 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 At, defined as At = N6t, where 20 < N < 125 (runs used in the

present thesis have values for N of 50 and 100) and 6t is the small time step at which

vortices are being displaced. Therefore, there are two kinds of time steps, the "big time

steps" At, which determine the generation of new vorticity at the generation points, and

the "small time steps" 6t, 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 (DF/Dt) has a fixed

value of one in the model. Choosing different values for At will allow some tuning in the

model since the strength of the discrete vortex pairs is dependent on At so that the mean

rate of generation of circulation is kept constant at one. The values used for At in this

thesis were 0.03, 0.04 and 0.05; good convergence was obtained with these values so the










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 (DF/Dt) with a

sinusoidal component, defined in the input file by an amplitude a (values used in the

analysis range from 0
where used with increments of 0.5):

DF) = a sin(ct) [2.3]
Dt

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

dimensionlesss) 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 non-

dimensional 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:










(',') (u,v) [3.1]


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 (DF/Dt)

will be discussed.

Parameter (DF / Dt) Estimations

Estimations of the parameter (DF/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 (DF/Dt) along the wave ray (which is assumed shore normal)

is (e.g., Brocchini et al., 2003):

D) (h, hc) + hC 2 2) [3 2
Dt 16 8










Where y is the ratio of wave height to water depth (y = 0.78) (depth limited

breaking); P is a constant that relates water depth to wave height assuming the breaking

continuous over the bar (H = p8. hc, where 8 = 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 Cgo) y [3.3]

Where Ho and Cgo 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.










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:


r DFB =g(qB A) g(qsdB- 7dA) [3.4]
Dt )A

Where rqB, 7A are the measured mean water surface elevations after and before the

breaking event respectively, and rsdB, rsdA 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.










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 (DF/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 % and % 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










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.


i,~ It








I,)




'4.
SI'1 -j -


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










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.



13 .

12.5
A
II
E 12
o= XXX
p11.5

11

10.5
12 12.5 13 13.5 14 14.5 15 15.5
y (m)


Figure 3.3: Current meter location in Kennedy (2003), ADV 1 (y=13.52m, x=11.8m);
ADV 2 (y=13.72 m, x=l 1.8m); 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











neck velocities for comparisons with the model. For detailed information on location of

instruments see Haller and Dalrymple (1999).


At the center of the bar Rip channel
5 ...........-.....- ---....---- ..........--....... .- .. ........ I-- ...........
1s 4

E3 --- Es

S------------ -------- --------- -------- ....:


01 1
-15 -05 0 0.5 1.5 1 -05 0 05
X' X'

0 0.5
0.3 ------------ ............. ------ 043 .----------- ---- ----------.

U . ........ -. .---............ ........ --------.... -------------.---- ---
S -...... .---- .................-----............ ......................... ....... ... ....---------
i i i








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.




























(Dr/Dt)o (m212)

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,










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 (DF/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.











0.3

0.23---------- ---- --------------------------------------------------------------
.(0.0)
0.2 -----

I / T i IT /- I



0.15 .. ........... -..... ...............

2 i s i T / a ,
S .... .- ... ... .] i. .
C 0 .1 ......... ...... ............... .. ......... -- - - - ---......


0.05 .../. .. ....... ................. ... ... ....... ...... ................................



0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
(Dr/Dt)o (m /s2)


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.










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 (DF/Dt)2.

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 (At).

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.















4

PC 2

0

-2


=0 t=10

4

2


-2
-6 -4 -2 0
t = 20


4

2



-2
-6 -4 -2 0
t=25







-2
-6 -4 -2 0
x/R


-6 -4 -2 0
t=30

4




-2
-6 -4 -2 0
x/R


t=5 (b)v=0.25 t=10


4 ,

-2


-2f
-6 -4 -2 0
t=25


x/R x/R


t=5 (c)v = 0.5 t=10 t = 5 (d)v=0.75 t=10

4 4 4 4

(A2 2 2 2
0 0 0 0

-2 -2 -2 -2
-6 -4 -2 0 -6 -4 -2 0 -6 -4 -2 0 -6 -4 -2 0
t=15 t=20 t=15 t=20


-6 -4 -2 0 -6 -4 -2 0
x/R x/R


-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










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.










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 .. . ... . : . . . . ... . . . . . . . . .







0 i . . . . . . . . .. . . . . .


0 .5 . . . . . . . .. .. . .. . .. . .



-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.










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 background at different x locations
1.4
x=-1
x = 0.75
1.2 ... ... x = -0.5
S- t x =- 0.25
x=0



i d8 in toitf

. . ... .. . . .

0 .. ....... .


0 .2 .. . . .: . . ... .. ..:. .. . : . .



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
background dimensionlesss)


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

At = 0.05.










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 background at location x = 0.5
with different computational resolutions


0 0.1 0.2 0.3 0.4 0.5 0.6
background


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


0.7 0.8 0.9 1












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

S0.5 5

a) 4b


IlII


0.25



0



-0.25



-0.5





0.5



0.25



0



-0.25



-0.5


-2
-2 -1 0 1
Cross shore dimensionlesss)


E 3

= 2


0
r0


-1

vO= 0.2
-2
-2 -1 0 1
Cross shore dimensionlesss)


c3





0





-2 *

Cross shore dimensionlesss)



5
.9










E











P 2


0
u,
C












-1

-2











-2 -1 0 1
Cross shore (dimensioness)


0.5



0.25



0



-0.25



-0.5





0.5



0.25



0



-0.25



-0.5


Figure 3-11. Mean vorticity maps for a)v = 0, b)v = 0.1, c)v = 0.2 and d)v = 0.3


5

4
U)
U)
0
C
3
"o
0 2
E
,'
0

0













Measure of mean vorticity in the system

5 0.5 5

4 4



2 0.25 2
E 3 7 3


4)2 -2


0 s ( e s 0
-0.25



-2 -0.5 -2
-2 -1 0 1 -2 -1 0 1
Cross shore dimensionlesss) Cross shore dimensionlesss)


5




3

. 2
4


0! 1
0





-2
-2 -1 0 1
Cross shore dimensionlesss)


0.5



0.25



0



-0.25



-0.5


5

4
S3
c2
3

2

0



-1

-2 srdeo s
-2 -1 0 1
Cross shore dimensionlesss)


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


0.5



0.25



0



-0.25


0.5



0.25



0



-0.25



-0.5










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 < F < 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











longshore current on the strength of the offshore velocities, thus turning the jet in the

alongshore direction.









0
E 70. .

I 60


0

3 ..... ./ . .

0
20





< 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
background dimensionlesss)


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 At (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 At of 0.05 and N = 50. This leads to











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. At = 0.05, N = 50 and #
of At = 2000.
Group amp. (a) Group freq. (c) v background
(Dimensionless) (Dimensionless) (Dimensionless)
0 _0,0.1, 0.2..., 0.9, 1
0.5 0, 0.25, 0.50, 0.75, 1
1 0, 0.25, 0.50, 0.75, 1
0.25 1.5 0, 0.25, 0.50, 0.75, 1
2 0, 0.25, 0.50, 0.75, 1
2.5 0, 0.25, 0.50, 0.75, 1
0.5 0, 0.25, 0.50, 0.75, 1
1 0, 0.25, 0.50, 0.75, 1
0.5 1.5 0, 0.25, 0.50, 0.75, 1
2 0, 0.25, 0.50, 0.75, 1
2.5 0, 0.25, 0.50, 0.75, 1
0.5 0, 0.25, 0.50, 0.75, 1
1 0, 0.25, 0.50, 0.75, 1
0.75 1.5 0, 0.25, 0.50, 0.75, 1
2 0, 0.25, 0.50, 0.75, 1
2.5 0, 0.25, 0.50, 0.75, 1
0.5 0, 0.25, 0.50, 0.75, 1
1 0, 0.25, 0.50, 0.75, 1
1 1.5 0, 0.25, 0.50, 0.75, 1
2 0, 0.25, 0.50, 0.75, 1
2.5 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).












t=5 a)steady t=10 t-5 b) unsteady t=10
4 4 4 4

2e i 2 i 2 i 2 t-


-2 -2 2 -2 -2
-6 -4 -2 0 -6 -4 -2 0 -6 -4 -2 0 -6 -4 -2 0
t=15 t=20 t=15 t-20





-2 -2----- -2-- -2
-6 -4 -2 0 -6 -4 -2 0 -6 -4 -2 0 -6 -4 -2 0
t-25 t=30 t=25 t=30


-2-
-6 -4 -2 0
x/R


-6 -4 -2 0 -6 -4 -2
x/R 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 o = 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










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 cross-

shore 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.












a)









0
1.2
.o
1i 1
C
E 0.8
-0.6
S0.4
C}
o-0.2

E


" 1.2


E 0.8
v-0.6
- 0.4
(D
S0.2

S0
E


a) amplitude = 0.25











0 0.25 0.5 0.75 1

c) amplitude = 0.75











0 0.25 0.5 0.75 1
v dimensionlesss)


Ca,



E. 0.8
0.6
- 0.4
0-0.2
CO
0
E



.2
O 1

E 0.8
0.6
I 0.4
Ca
0-0.2
C
U 0
E


Figure 3-15. Mean peak offshore velocities in the jet at the cross-shore location x = -0.5
for different amplitudes. (-) Steady forcing, (-o-) c = 0.5, (-o-) W = 1, ( -)
w =1.5,( ) = 2 and( ) =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.


b) amplitude = 0.5











0 0.25 0.5 0.75 1

d) amplitude = 1











0 0.25 0.5 0.75 1
v dimensionlesss)












C)
C 1.2
0
( 1
U)
E 0.8
0.6
0.4
0.2
S0
E

U)
"2 1.2
S1
a)
E 0.8
.0.6
S0.4
U)
0-0.2
E 0
E


a) frequency = 0.50





. . . . . . . .





0 0.25 0.5 0.75 1

c) frequency = 1.50











0 0.25 0.5 0.75 1
v dimensionlesss)


CO
S1.2
0
"F 1
U)
<-
E 0.8
0.6
4 0.4

0
-0.2
C

a-
C 0
E

C,
C 1.2

0.8
E 0.8
0.6
% 0.4


W 0
E


0 0.25 0.5 0.75 1
v dimensionlesss)


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, (- )
a=0.75,( )a=l

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.


b) frequency = 1.00











0 0.25 0.5 0.75 1

d) frequency = 2.00





::I::.:..........













Measure of mean vorticity in the system

5 0.5 5

4 4

0.25 2
3 3
22
2 2
E

00

0 0
-0.25 C
0

-1 -1-0.5 -2

-2 -0.5 -2


-2 -1 0 1
Cross shore dimensionlesss)


-2 -1 0 1
Cross shore dimensionlesss)


0.5



0.25



0



-0.25



-0.5


-2 -1 0 1
Cross shore dimensionlesss)


-2 --
-2 -1 0 1
Cross shore dimensionlesss)


Figure 3-17. Mean vorticity maps for the unsteady case of amplitude 1 and frequency 1.
a)v=O,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


r)
.o
cr



E
o
0)


0.5



0.25



0



-0.25



-0.5


0.5



0.25



0



-0.25



-0.5


-I










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.











a) amplitude = 0.25


S' .. . .
. . . . . .
. . . .


2.5
2

1.5

1

0.5
0

-0.5



2.5

2

1.5

1
0.5

0

-0.5


2

- 1.5
o 1
E 0.5

S0
-0.5


b) amplitude = 0.50











0 0.25 0.5 0.75 1

d) amplitude = 1

. . . . . .. . . . . . . . ...








0 0.25 0.5 0.75 1
v dimensionlesss)


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-)a=0.5, (-x-)
C =1,(-x-) w=1.5,( ) w =2 and( ) o =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.


0 0.25 0.5 0.75 1

c) amplitude = 0.75


S.. . .. . . .. .








0 0.25 0.5 0.75 1
v dimensionlesss)











a) frequency = 0.5 b) frequency = 1


0 0.25 0.5 0.75 1

c) frequency = 1.5
5



0
5 ..
. . 5 : . . . . i . . . . .


5-
0 0.25 0.5 0.75 1
v dimensionlesss)


0 0.25 0.5 0.75 1

d) frequency = 2






) :

5
0 0.25 0.5 0.75 1
v dimensionlesss)


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, ( \ ) a = 0.75 and( )

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.


a) frequency = 0.5


b) frequency = 1


,,











a) frequency = 0.5
5

2
5 i ------ .--- ------ --
5 .

S* .. ..... .. .. .. ... ... ..

0 i T i i

5
0 0.25 0.5 0.75 1

c) frequency = 1.5
5.------------------


t t T T
0 0.25 0.5 0.75 1
a dimensionlesss)


b) frequency = 1


Q)
u,
C
.2

E
:5


0 0.25 0.5 0.75 1

d) frequency = 2










0 0.25 0.5 0.75 1
a dimensionlesss)


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,( ) v = 0.75 and( ) v=l

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 cross-

shore 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.


i 2
7E 1.5
"C 1
E o.5
>, 0
-0.5











a) amplitude = 0.25



. .. . . :.. . . . . . . .


1 1 i


THT-TT


0.5 1 1.5 2 2.5

c) amplitude = 0.75

2 . . . . . . . . . .. . . . .


T T IT


T T


2.5
" 2
" 1.5
0
"^ 1
E 0.5
> 0
-0.5


0.5 1 1.5 2 2.5
f dimensionlesss)


b) amplitude = 0.5


S .... I.....



0.5 1 1.5 2 2.5

d) amplitude = 1


0.5 1 1.5 2
f dimensionlesss)


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,( -) v=0.75 and( ) 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).


"C" 2
u,
a)
S1.5
.o
0
CO 1
r
E 0.5
S0


-U.J








53






Measure of mean vorticity in the system
averaged over 14 group periods
5 0.5


4

S0.25
3





01
0
o

S-0.25
-c
0




-2 -0.5
-2 1 0 1
Cross shore coord. dimensionlesss)



Figure 3-22. Mean vorticity map for the unsteady forcing case with: a = 1, o =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 (DF/Dt), the semi-gap width of the rip channel (R) and the constant










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 (DF/Dt)2, but not on the gap width (R). Temporal response will depend

both on wave breaking strength and length scales (R/(DF/Dt)Y ).

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 (DF/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










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










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 (DF/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


a)


,W -- :- Oi. L ,..

0

-1

-2
-2 -1 0 1


0.5



0.25 A-
c
0

E
0
0 -

0

-0.25



-0.5


Cross shore dimensionlesss)


0
E




-1



-2
-2 -1 0 1
Cross shore dimensionlesss)


0.5



0.25



0



-0.25



-0.5


-2 '
-2 -1 0 1


Cross shore dimensionlesss)


-2 -1 0 1
Cross shore dimensionlesss)


Figure A-i. Mean vorticity map for steady forcing. a)v = 0, b)v = 0.25, c)v = 0.5 and
d)v= 0.75


5

4

c 3

o 2


Co
C
0t


0.5



0.25



0



-0.25



-0.5


0.5



0.25



0



-0.25



-0.5













Measure of mean vorticity in the system


5

4

c 3





o
-




0
-1

-2
-2 -1 0 1
Cross shore dimensionlesss)















-2 -1 0 1







Cross shore dimensionlesss)
5

4

c 3


2



-





-2
-2 -1 0 1
Cross shore dimensionlesss)


0.5



0.25



0



-0.25


-2 -
-2 -1 0 1
Cross shore dimensionlesss)


0.25



0



-0.25



-0.5


-21-- -1 1
-2 -1 0 1
Cross shore dimensionlesss)


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


0.5



0.25



0



-0.25



-0.5





0.5



0.25



0



-0.25



-0.5













Measure of mean vorticity in the system


-2'-'
-2 -1 0 1
Cross shore dimensionlesss)


5

4
U)

c 3
o
E
aI 2
E

21
-C
C 0
_o


-2 '- --
-2 -1 0 1
Cross shore dimensionlesss)


0.25



0



-0.25




-0.5






0.5



0.25



0



-0.25



-0.5


." 3 ..:I
.A .
U)

0

E

1

0
-c
o

o
-1

-2
-2 -1 0 1
Cross shore dimensionlesss)




5


4

7 3
.2
v-
e 2
E

1
o
0
0

-1


-2
-2 -1 0 1
Cross shore dimensionlesss)


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


0.5



0.25



0



-0.25



-0.5






0.5



0.25



0



-0.25



-0.5













Measure of mean vorticity in the system


U,
I,
S3
0o

2
E


0
o

<0
,c


a).


I '2 'V'= u JI
-2'-
-2 -1 0 1
Cross shore dimensionlesss)




5

4

S
0
)2
(D 2
.0

t-





v,. ..v=*0 .5-
-2
-2 -1 0 1
Cross shore dimensionlesss)


0.5



0.25



0



-0.25



-0.5






0.5



0.25



0



-0.25



-0.5


-2 -1 0 1
Cross shore dimensionlesss)


-2 -1 0 1
Cross shore dimensionlesss)


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


0.5



0.25



0



-0.25



-0.5






0.5



0.25



0



-0.25



-0.5


_ i;


t













Measure of mean vorticity in the system


5


I) .
~aa
4


C 3i
c3








- iC



-2 *
Ct





-2 -1 0 1
Cross shore dimensionlesss)


-2 -1 0 1
Cross shore dimensionlesss)


0.5



0.25



0



-0.25



-0.5






0.5



0.25



0



-0.25



-0.5


-1 1. 0.,1

-2
-2 -1 0 1
Cross shore dimensionlesss)



5



_ A
c3

C,2
E
0



0



-2
-2 -1 0 1
Cross shore dimensionlesss)


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 andd)v=0.75


0.5



0.25



0



-0.25



-0.5






0.5



0.25



0



-0.25



-0.5













Measure of mean vorticity in the system


c 3
o

e 2
E

)1
0
o

S0
t-
O
<


-2''
-2 -1 0 1
Cross shore dimensionlesss)


S3

0 2
E

01
0
o
0)
0


v'='0.5
-2
-2 -1 0 1
Cross shore dimensionlesss)


0.5



0.25



0



-0.25



-0.5


-2 -1 0 1
Cross shore dimensionlesss)


0.5 5



0.25 (
c3

e 2
E
0
o1


-0.25 C

-1

-0.5 -2
-2 -1 0 1
Cross shore dimensionlesss)


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 andd) v = 0.75


0.5



0.25



0



-0.25



-0.5


0.5



0.25



0



-0.25



-0.5












Measure of mean vorticity in the system


4-

D .





oo
0)
c 3







-1
r 2 :0 .,


)= 0 K-


-2 -1 0 1
Cross shore dimensionlesss)
-2 -


W-1

-2
-2 -1 0 1
Cross shore dimensionlesss)


0.25 W
c 3

2
E
0 1

0
C 0
-0.25 c

-1

-0.5 -2
-2 -1 0 1
Cross shore dimensionlesss)


0.5



0.25



0



-0.25



-0.5


4

c3
.2
C
E








-2
-2 -1 0 1
Cross shore dimensionlesss)


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


0.5



0.25



0



-0.25



-0.5


0.5



0.25



0



-0.25



-0.5













Measure of mean vorticity in the system


0
c 3


C 2

S1
o
u-
nO 0
c-


a)





.. ., .


-2
-2 -1 0 1
Cross shore dimensionlesss)




5 .


4
S4
7 3


E
:5
0 1
-O





-2
-2 -1 0
Cross shore dimensionlesss)


0.5




0.25




0




-0.25




-0.5






0.5



0.25




0



-0.25



-0.5


-2
-2 -1 0 1
Cross shore dimensionlesss)









%-4.
4
U -
S3


E



0





-2
2 -1 0 1
Cross shore dimensionlesss)


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


0.5




0.25



0




-0.25




-0.5






0.5



0.25




0



-0.25



-0.5













Measure of mean vorticity in the system


(0

S3
0

S2
E

01
c 0
0)0


a)


W.


--1



-2
-2 -1 0 1
Cross shore dimensionlesss)




5

4

c 3

( 2
E_


r-
-1
0





-2
-2 -1 0 1
Cross shore dimensionlesss)


0.25



0



-0.25



-0.5






0.5



0.25



0



-0.25



-0.5


o
-_1

-2
-2 -1 0 1
Cross shore dimensionlesss)




5



.0


a) 2

02
E








2 1 0 1
Cross shore dimensionlesss)
-2
-2 -1 0 1
Cross shore dimensionlesss)


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


b)

' "i


0.5



0.25



0



-0.25



-0.5






0.5



0.25



0



-0.25



-0.5














Measure of mean vorticity in the system


c 3







) 0
c
o


<


a)


I- -w =< Ij
-2 I
-2 -1 0 1
Cross shore dimensionlesss)




5

4
ci3


2
E


0
t--

C
0
c


-v'= 0.5
-2
-2 -1 0 1
Cross shore dimensionlesss)


0.25



0



-0.25



-0.5






0.5



0.25



0



-0.25



-0.5


U)
U)
c 3
o
.2



01
o

C


-2
-2 -1 0 1
Cross shore dimensionlesss)


5

4
U)
U)
c 3

I 2
E

Pi


o
0
C


-2 1' Ii 1
-2 -1 0 1
Cross shore dimensionlesss)


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


b)


0.5



0.25



0



-0.25



-0.5






0.5



0.25



0



-0.25



-0.5


P













Measure of mean vorticity in the system


U)

3
c 3
0

e 2
E

) 1
o
0
tC
O 0


a):


w


-2
-2 -1 0 1
Cross shore dimensionlesss)




5

4

c 3
o

4) 2
E


0


-21


-2
-2 -1 0 1
Cross shore dimensionlesss)


0.25



0



-0.25



-0.5






0.5



0.25



0



-0.25



-0.5


U)
U)
0)
7 3
o
C
( 2
E



0
0


-21
-2 -1 0 1
Cross shore dimensionlesss)




5

4
U)
c 3
o
C





- 2




-2
-2 -1 0 1
Cross shore dimensionlesss)


Figure A- 1. 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



0.25



0



-0.25



-0.5






0.5



0.25



0



-0.25



-0.5













Measure of mean vorticity in the system


a.)


-2
-2 -1 0 1
Cross shore dimensionlesss)




5

4
U,

c 3

2
E


o

S0

i--

v'=-0.5
-2
-2 -1 0 1
Cross shore dimensionlesss)


0.25



0



-0.25



-0.5






0.5



0.25



0



-0.25



-0.5


5 0.5

4

S3 0.25
" 3




, 1
0

o

0 -0.25
0
-1

-2 ~ -0.5
-2 -1 0 1
Cross shore dimensionlesss)


-2 -1 0 1
Cross shore dimensionlesss)


0.5



0.25



0



-0.25



-0.5


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













Measure of mean vorticity in the system


-2 1
-2 -1 0 1
Cross shore dimensionlesss)



5

. 4


.2
2
E
S1
0
c-
CO
C 0
.-



-2
-2 -1 0 1
Cross shore dimensionlesss)


0.5



0.25



0



-0.25



-0.5





0.5



0.25



0



-0.25



-0.5


-2 -1 0 1
Cross shore dimensionlesss)


-2 -1 0 1
Cross shore dimensionlesss)


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



0.25



0



-0.25



-0.5





0.5



0.25



0



-0.25



-0.5













Measure of mean vorticity in the system


U)

S3


E

01

O 1
0 0
o-
O,


a)


-i--i
-2
-2 -1 0 1
Cross shore dimensionlesss)


-2' -
-2 -1 0 1
Cross shore dimensionlesss)


0.25



0



-0.25



-0.5






0.5



0.25



0



-0.25



-0.5


0
c 3
.o
c
(D 2
E
o

0
'-
0
o

~-1p


-2
-2 -1 0 1
Cross shore dimensionlesss)


a)
S3
0
r-
I 2
E

2 1
0
o
00
o
<


-2' -----i
-2 -1 0 1
Cross shore dimensionlesss)


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



0.25



0



-0.25



-0.5






0.5



0.25



0



-0.25












Measure of mean vorticity in the system


4

c 3

02
E

a1

-c
o
0




-2
-2 -1 0 1
Cross shore dimensionlesss)


-2 -1 0 1
Cross shore dimensionlesss)


0.5 5

4

0.25
c 3

2
E
0

o
0
-0.25 C

-1

-0.5 -2
-2 -1 0 1
Cross shore dimensionlesss)


0.5



0.25



0



-0.25



-0.5


U


-2
E



0




-2 s ( i
-2 -1 0 1
Cross shore dimensionlesss)


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



0.25



0



-0.25



-0.5


0.5



0.25



0



-0.25



-0.5













Measure of mean vorticity in the system


-2
-2 -1 0 1
Cross shore dimensionlesss)




5

4C

E
I.0



1

,-0




-2
-2 -1 0 1
Cross shore dimensionlesss)


0.25



0



-0.25



-0.5






0.5



0.25



0



-0.25



-0.5


c3
0
a) 2
E

2 1
0
o
0
o o

-1

-2
-2 -1 0 1
Cross shore dimensionlesss)









3
o
4



0
2




0




-2
-2 -1 0 1
Cross shore dimensionlesss)


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



0.25



0



-0.25



-0.5






0.5



0.25



0



-0.25



-0.5













Measure of mean vorticity in the system


5 0.5
a)
4

Q 0.25
c 3

2
E
0

0

C -0.25
<-1


-2 -0.5
-2 -1 0 1
Cross shore dimensionlesss)



5 0.5

4(

a) 0.25
-E 3


E






1
-0.25



-2 -0.5
-2 -1 0 1
Cross shore dimensionlesss)


3
i-

o



o
-1
S0



-1

-2
-2 -1 0 1
Cross shore dimensionlesss)


, 2
E

2 1
o
0,
t-
S0
o
a


-2' =
-2 -1 0 1
Cross shore dimensionlesss)


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


0.5



0.25



0



-0.25



-0.5





0.5



0.25



0



-0.25













Measure of mean vorticity in the system


0
Cn
c 3

C
a 2
E
"0
S1
o
0

o
<


I a)


-2'1
-2 -1 0 1
Cross shore dimensionlesss)


-2 -1 0 1
Cross shore dimensionlesss)


0.5



0.25



0



-0.25



-0.5


0.25



0



-0.25



-0.5


-2'
-2 -1 0 1
Cross shore dimensionlesss)


S2


21
o
0
c
UO
0


-2' 1
-2 -1 0 1
Cross shore dimensionlesss)


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


0.5



0.25



0



-0.25



-0.5






0.5



0.25



0



-0.25



-0.5













Measure of mean vorticity in the system


S3
0

S2
E


o
(0
S0
_o
<


a)


-21 INi
-2 -1 0 1
Cross shore dimensionlesss)


v0
c- 3

) 2
E
S1
ol
0





-2-
v 0




-2 -1 0 1
Cross shore dimensionlesss)


0.5



0.25



0



-0.25



-0.5


0.5



0.25



0



-0.25



-0.5


-2' *
-21 -1 1I
-2 -1 0 1
Cross shore dimensionlesss)




5

4

&0
c 3





0
0 2
E




o

-1

v,=0.115
-2
-2 -1 0 1
Cross shore dimensionlesss)


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


0.5



0.25



0



-0.25



-0.5


0.5



0.25



0



-0.25



-0.5













Measure of mean vorticity in the system


5
a).
4

c 3
o

E

I
0


I l c . : .j
-2
-2 -1 0 1
Cross shore dimensionlesss)


4

c 3
U) \
) 2
E

01
oC
0 0
a0


-21'
-2 -1 0 1
Cross shore dimensionlesss)


0.25



0



-0.25



-0.5






0.5



0.25



0



-0.25


&-
Co





0

0




-2 '
-2 -1 0 1
Cross shore dimensionlesss)
-2

Cross shore dimensionlesss)


-2 -1 0 1
Cross shore dimensionlesss)


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


0.5



0.25



0



-0.25



-0.5






0.5



0.25



0



-0.25



-0.5












Measure of mean vorticity in the system


C,,
c 3

0 2
E


0
0) 0
o
;Z


a)


-2-
-2 -1 0 1
Cross shore dimensionlesss)


C,
c3
o




0"
0
0
ro




-2
-2 -1 0 1
Cross shore dimensionlesss)


0.5



0.25



0



-0.25



-0.5


0.5



0.25



0



-0.25



-0.5


-2
-2 -1 0 1
Cross shore dimensionlesss)


-2 -1 0 1
Cross shore dimensionlesss)


I

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


0.5



0.25



0



-0.25



-0.5


0.5



0.25



0



-0.25



-0.5


~er
















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
No groupiness cases
v y- yO y+
0 -0.38 0 0.4
0.25 0.89 0.93 0.96
0.5 1.1 1.14 1.21
0.75 1.33 1.59
1 1.97













Table B-2. Alongshore location and width of the jet for unsteady forcing with amplitude
0.25
f=0.5 f=1.0
v y- yO y+ y- yO y+
0 -0.4 0 0.38 -0.4 0 0.35
0.25 0.88 0.92 0.96 0.91 0.97 1.01
0.5 1.03 1.06 1.1 1.03 1.08 1.13
0.75 1.31 1.37 1.52 1.23 1.36 1.44
1 1.86 1.83
f=1.5 f=2.0
v y- yO y+ y- yO y+
0 -0.39 0 0.39 -0.38 0 0.39
0.25 0.89 0.94 0.98 0.89 0.94 0.97
0.5 1.03 1.07 1.11 1.02 1.06 1.1
0.75 1.27 1.34 1.45 1.29 1.36 1.47
1 1.7 _1.68_
f=2.5
v y- yO y+
0 -0.4 0 0.36
0.25 0.9 0.95 1
0.5 1.05 1.09 1.13
0.75 1.3 1.36 1.5
1 1.67











Table B-3. Alongshore location and width of the jet for unsteady forcing with amplitude


f=0.5 f=1.0
v y- yO y+ y- yO y+
0 -0.36 0 0.4 -0.39 0 0.4
0.25 0.79 0.9 0.96 0.88 0.98 1.04
0.5 1 1.05 1.1 0.93 1.03 1.1
0.75 1.24 1.31 1.37 1.13 1.27 1.36
1 1.68 1.6
f=1.5 f=2.0
v y- yO y+ y- yO y+
0 -0.39 0 0.37 -0.38 0 0.37
0.25 0.89 0.95 1 0.88 0.93 0.98
0.5 1.02 1.06 1.11 1.01 1.04 1.08
0.75 1.15 1.29 1.42 1.32 1.46
1 1.56 1.59
f=2.5
v y- yO y+
0 -0.38 0 0.38
0.25 0.89 0.94 1
0.5 1.05 1.1 1.18
0.75 1.3 1.38 1.64
1 1.53











Table B-4. Alongshore location and width of the jet for unsteady forcing with amplitude
0.75
f=0.5 f=1.0
v y- yO y+ y- yO y+
0 -0.39 0 0.41 -0.4 0 0.39
0.25 0.6 0.82 0.91 0.8 0.94 1.02
0.5 0.92 1.03 1.08 0.92 1.03 1.11
0.75 1.11 1.24 1.33 1.04 1.2 1.34
1 1.56 1.4
f=1.5 f=2.0
v y- yO y+ y- yO y+
0 -0.39 0 0.39 -0.39 0 0.38
0.25 0.86 0.93 0.99 0.87 0.93 0.99
0.5 1.04 1.09 1.14 1 1.04 1.08
0.75 1.22 1.35 1.35
1 1.4 1.68
f=2.5
v y- yO y+
0 -0.39 0 0.38
0.25 0.84 0.92 0.99
0.5 1.08 1.12 1.23
0.75 1.29 1.37 1.63
1 1.56











Table B-5. Alongshore location and width of the jet for unsteady forcing with amplitude


f=0.5 f=1.0
v y- yO y+ y- yO y+
0 -0.41 0 0.4 -0.4 0 0.42
0.25 0.39 0.72 0.84 0.61 0.88 0.98
0.5 0.87 1.01 1.07 1.01 1.09 1.61
0.75 1.04 1.15 1.26 1.16
1 1.3 1.38 1.56 1.33
f=1.5 f=2.0
v y- yO y+ y- yO y+
0 -0.41 0 0.41 -0.41 0 0.42
0.25 0.8 0.9 0.96 0.81 0.9 0.96
0.5 1.11 1.22 1.36 1.01 1.05 1.09
0.75 1.36 1.44
1 1.34 1.43
f=2.5
v y- yO y+
0 -0.39 0 0.44
0.25 0.82 0.88 0.97
0.5 1.12 1.29 1.44
0.75 1.29 1.42 1.7
1















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