EFFECTS OF LONGSHORE CURRENTS ON RIP CURRENTS

Transcription

1 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

2 This document is dedicated to my mother and father.

3 ACKNOWLEDGMENTS This research was funded by the - and -6 Florida Sea Grant Program. I would like to thank Andrew B. Kennedy for his academic guidance and support in the research leading to the present thesis. I would also like to thank Robert. J. Thieke for providing financial assistance and giving me the opportunity to be part of a very exciting project, and Robert G. Dean for his help and for serving on my supervisory committee. I would also like to thank Oleg A. Mouraenko for his support and guidance with Matlab, if I have any skills with Matlab it is thanks to him. I also want to thank Jamie MacMahan, who gave me guidance ever since I got here. I would like to thank all of my office mates and friends in Gainesville in general, for making my stay here a very good experience and especially my roommate, Vadim Alymov. I will never forget all of them. Finally I would like to thank my family, especially my parents, who made many sacrifices to provide me with financial support to get my undergraduate degree back in Spain. They have always been there for me. iii

4 TABLE OF CONTENTS Page ACKNOWLEDGMENTS... iii LIST OF TABLES... vi LIST OF FIGURES... vii CHAPTER INTRODUCTION... Problem Statement and Objective... Background: Literature Review... Physical Description of Rip Currents... Forcing Mechanisms of Rip Currents and Longshore Currents...9 Forcing of longshore currents... Forcing of rip currents... Generation of Vorticity... Outline of the Thesis... NUMERICAL MODEL... Theoretical Background... Rip Current Scaling...7 Numerical Description of the Model...8 RESULTS AND ANALYSIS... Mean Peak Offshore Current in the Rip Neck... Parameter (DΓ / Dt) Estimations... Method... Method... Model Comparisons with Lab Data... Effects of Background Longshore Current on the Rip Current... Steady Forcing... Velocities in the rip neck... Vorticity...7 Jet angle evolution with increasing background longshore current... Unsteady Forcing... iv

5 Velocities in the rip neck... Vorticity...6 SUMMARY AND CONCLUSIONS... APPENDIX A MEAN VORTICITY MAPS...9 B LOCATION AND WIDTH OF THE JET DATA...8 LIST OF REFERENCES...8 BIOGRAPHICAL SKETCH...89 v

6 LIST OF TABLES Table page - Unsteady forcing simulations... B- Alongshore location and width of the jet for steady forcing...8 B- Alongshore location and width of the jet for unsteady forcing with amplitude...8 B- Alongshore location and width of the jet for unsteady forcing with amplitude..8 B- Alongshore location and width of the jet for unsteady forcing with amplitude.78 B- Alongshore location and width of the jet for unsteady forcing with amplitude...8 vi

7 LIST OF FIGURES Figure page - Rip current parts: feeders neck and head (from Shepard et al., 9) Time-averaged vorticity... - Definition sketch of the model Sketch of wave breaking over a bar... - Experimental wave basin at the University of Delaware...6. Current meter location Wave height and MWL versus cross-shore distance at the center bar (left) and at the rip channel (right) for test E Cross-shore velocities in the rip: model predictions vs. lab data Cross-shore velocities in the rip: model predictions vs. lab data Snapshots of the simulations for different background longshore currents with steady forcing Computed mean velocities on a longshore profile at x = Mean peak offshore velocities with increasing background longshore currents at different cross-shore locations Mean peak offshore velocity versus background longshore current Mean vorticity maps for steady forcing Mean vorticity maps for steady forcing Jet angle vs. background longshore current... - Comparison of the simulations for steady and unsteady forcing... - Mean peak offshore velocities in the jet at the cross-shore location x = -. for different amplitudes... vii

8 -6 Mean peak offshore velocities in the jet at the cross-shore location x = -. for different frequencies Mean vorticity maps for unsteady forcing with amplitude and frequency Alongshore location and width of the jet at x = -. with increasing longshore current for each amplitude Alongshore location and width of the jet at x = -. with increasing longshore current for each frequency... - Alongshore location and width of the jet at x = -. with increasing group amplitude for each frequency... - Alongshore location and width of the jet at x = -. with increasing group frequency for each amplitude... - Mean vorticity map for unsteady forcing with amplitude, frequency.and background longshore current of.7... A- Mean vorticity map for steady forcing...9 A- Mean vorticity maps for unsteady forcing with amplitude. and frequency..6 A- Mean vorticity maps for unsteady forcing with amplitude. and frequency...6 A- Mean vorticity maps for unsteady forcing with amplitude. and frequency..6 A- Mean vorticity maps for unsteady forcing with amplitude. and frequency...6 A-6 Mean vorticity maps for unsteady forcing with amplitude. and frequency..6 A-7 Mean vorticity maps for unsteady forcing with amplitude. and frequency..6 A-8 Mean vorticity maps for unsteady forcing with amplitude. and frequency...66 A-9 Mean vorticity maps for unsteady forcing with amplitude. and frequency..67 A- Mean vorticity maps for unsteady forcing with amplitude. and frequency...68 A- Mean vorticity maps for unsteady forcing with amplitude. and frequency..69 A- Mean vorticity maps for unsteady forcing with amplitude.7 and frequency..7 A- Mean vorticity maps for unsteady forcing with amplitude.7 and frequency...7 A- Mean vorticity maps for unsteady forcing with amplitude.7 and frequency..7 A- Mean vorticity maps for unsteady forcing with amplitude.7 and frequency...7 viii

9 A-6 Mean vorticity maps for unsteady forcing with amplitude.7 and frequency..7 A-7 Mean vorticity maps for unsteady forcing with amplitude and frequency...7 A-8 Mean vorticity maps for unsteady forcing with amplitude and frequency...76 A-9 Mean vorticity maps for unsteady forcing with amplitude and frequency...77 A- Mean vorticity maps for unsteady forcing with amplitude and frequency...78 A- Mean vorticity maps for unsteady forcing with amplitude and frequency...79 ix

10 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 Chair: Robert J. Thieke Major Department: Civil and Coastal Engineering A simplified conceptual representation of a rip current system was used to study the effects of longshore currents on rip currents. The numerical model used for this study is based on a generation of vorticity approach, where oppositely signed vortices are continuously released on either side of the rip channel and let free in the system; and a background constant and uniform longshore current is also added to the system. A flat bed, no bottom friction and no wave-current interaction are assumed in the model. Since generation of vorticity is the only physical process represented in the model, its applicability is strictly limited to the rip neck area, where this process is assumed to be dominant. Velocities in the rip current depend only on the strength of wave breaking and time scales depend both on wave breaking strength and length scales of the system. Performance of the model was tested against measured laboratory data. Despite the high simplicity of the model, very reasonable results were obtained in comparison with the measured data. The model predicts well the trend of the data although overestimates velocities in the rip. The main challenge to apply the results of the model to field or x

11 laboratory data is the estimation of the generation of circulation within the surf zone. Two different methodologies were tested. The model was used to study the effects of longshore currents on rip currents, using both steady and unsteady forcing. Mean velocity fields and vorticity maps within the area of the rip neck were calculated. Steady forcing results showed a very strong effect of the longshore current on the rip current, with nearly constant offshore velocities for small longshore current strengths, decreasing quickly as the longshore current increases in strength. A wide range of unsteady forcing parameters (group amplitude and frequency) was used to test the relative importance of this parameters in the evolution of the rip current with increasing longshore current strengths. Results suggest that the background longshore current strength is the main factor in the behavior of the rip current. Effects of background longshore current are reduced for large group amplitudes. Influence of the frequency is found to be almost negligible, with response decreasing with higher frequencies. The model could be used to evaluate strength of rip currents under a wide range of wave climates, with simple scaling relations for rip current strength and wave breaking strength within the surf zone. Model results could be coupled with existing rip current forecasting indexes. xi

12 CHAPTER 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.

13 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 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, 9 people have died in Florida per year since 989 due to rip currents (Lascody, 998). 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 (), 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

14 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., ), 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 (97), 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 (), 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 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

15 the difficulty in locating the instruments in the rip channels, which tend to be temporally and spatially unstable [Shepard et al. (9), Shepard and Inman (9, 9), McKenzie (98), Harris (96,96), Sonu (97), Cooke (97), Sasaki and Horikawa (979), Bowman et al. (988), Smith and Largier (99), Chandramohan et al. (997), Aagaard et al. (997), Brander (999), Brander and Short (,)]. 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., 997). Ranasinghe et al. (999) 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 (969), Hamm (99), Oh and Dean (996), Drønen et al. (999, )] 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. (997), Haller and Dalrymple (999), Haller and Dalrymple (), Kennedy and Dalrymple (), Kennedy (), Haller et al. () that provided a comprehensive map of waves and currents, including the details of the mean water level variations, Haas ()] 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

16 drifters, which have been used in the field in limited cases [Shepard and Inman (9), Sonu (97)], and more recently, Schmidt et al. (), 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 () used numerous video tracked Lagrangian drifters with the laboratory setup from Haller et al. (997). The concept of radiation stresses developed by Longuet-Higgins (96) provided a basis to numerically describe the generation of rip currents with alongshore varying wave heights using phase-averaged techniques. Bowen (969) 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 (998) and Chen et al. (999), who used the fully nonlinear extended Boussinesq equations of Wei et al., (99), to model the laboratory setup by Haller et al. (), providing valuable insights into rip current behavior. Slinn and Yu () 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, 998). Rip currents can extent several surf zone widths seaward exchanging water between the nearshore and offshore. They have been observed all over

17 6 the world, on a wide range of beach types but are particularly common on beaches that are dominated by a longshore bar-trough morphology (Wright and Short, 98), 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, 9; Wind and Vreugdenhil, 986) or when longshore currents are directed offshore by a protrusion in the bathymetry or headland (Shepard and Inman, 9). The concept of rip current was first proposed by Shepard (96) (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 (9) 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 (9) 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 -). 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 m/s and. m/s in the rip neck and feeders respectively have been recorded in the field (Brander, 999) 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 9, Brander and Short ). Along the path of the neck,

18 7 usually a channel can be found which can be as much as m 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, 9). The flow separates from the bottom and is mainly confined to the surface, which is supported by no evidence of channels beyond the breakers. Figure -: Rip current parts: feeders neck and head (from Shepard et al., 9). Field observations of rip currents have shown long period oscillations in rips on the wave group time scales (-s) of up to. m/s [Shepard et al. (9), Shepard and

19 8 Inman (9), Sonu (97), Aagaard (997), Brander and Short ()]; 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 (99) and Shepard and Inman (9) 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 (97) hypothesized that rip current pulsations were due to infragravity standing waves. MacMahan et al. (), 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. (9) stated that records obtained off the coast of Southern California showed a clear relation between intensity of rip currents and height of waves. McKenzie (98), 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 (98) 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 (97) off the coast of Redondo Beach, CA observed

20 9 that stationary rip channels were commonly present but well defined rip currents were only present during falling or low tide. Sonu (97) 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 (999) and Brander and Short () conducted experiments at Palm Beach, New South Wales, Australia. Rip current velocities reached maximums at low tide and minimum velocities at high tide, the state of morphological beach evolution was found to be an important factor as well. Drønen et al. () 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 (97) 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 () 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.

21 Understanding of the forcing driving the currents within the surf zone was greatly enhanced when Longuet-Higgins and Stewart (96) 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: S xx, radiation stress component in the direction of the waves; S yy, radiation stress component in the transverse direction of the waves and S xy, the flux in the x direction of the y component of momentum. Forcing of longshore currents When waves propagate obliquely into the surf zone and break, this will result in a reduction in wave energy and an associate decrease in S xy, which is manifested as an applied longshore thrust F y on the surf zone (Dean and Dalrymple, 98). For straight and parallel contours, thrust per unit area is given by: F y S xy x = [.] The longshore wave thrust per unit area is resisted by shear stress on the bottom and lateral faces of the water column (Longuet-Higgins, 97 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

22 mechanism at La Jolla, CA is described by Shepard and Inman (9) and Bowen and Inman (969). Wave-wave interaction mechanisms. Bowen and Inman (969) 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 (97) 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 (97), Iwata (976), Dalrymple and Lozano (978), Falqués et al. (999)]. 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 (96) showed that radiation stress induces changes in the mean water level ( η ), creating steady pressure gradients that balance the gradient of the radiation stress: dη = dx ρg ds ( h + η) dx xx [.] Bowen (969) 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, ). When the waves break there is a decrease in the radiation stress which leads to an increase in the set-up, but also this is manifested as a wave thrust or force. If there is differential breaking alongshore this will generate differential forces which will generate circulation.

23 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 (99), using acoustic techniques, observed rip current vortices with radii in the order of s of meters. Schmidt et al. (), using direct drifter tracking with GPS technology observed vorticity within the surf zone. In the laboratory, Thomas () 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 -). Figure -: Time-averaged vorticity; contour =./s; Positive =>Dashed line, Negative =>Dash-Dot line, and Zero =>Solid line (from Thomas, ) Peregrine (998) and Bühler () 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

24 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 (999). Bühler and Jacobson () 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 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 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 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

25 longshore current and unsteady forcing parameters: amplitude and frequency) is discussed. Chapter summarizes all the results and analysis and conclusions will be drawn. Suggestion for future research and applications of the model will also be given.

26 CHAPTER 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 (), simple modifications have been included to study the effects of background longshore currents in the system. This chapter will discuss the theoretical background that leads to a simple representation of rip currents (conceptual model), the scaling parameters and dimensional analysis that makes possible this model, and finally a numerical description of the model. Theoretical Background Although rip currents are part of a very dynamic system, with quite complex forcing, a simple description can be achieved if we focus on an area where one of these processes is dominant. That is the case of the rip neck, where oppositely signed circulation and vorticity are the dominant processes. One of the most common rip current typologies is the one consisting of a longshore bar with gaps or rip channels on it. This kind of topography induces a differential wavebreaking pattern that is more or less stationary on hydrodynamic scales. Although migration of rip currents has been observed on the field (Ranasinghe et al, 999) 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.

27 6 Peregrine (998), using the NLSW equations and a bore dissipation model, showed how differential breaking along a wave crest generates circulation and vorticity, and that considering a closed material circuit that crosses the bore only once, the instantaneous rate of change of circulation generated equals the rate of loss of energy by the water passing through the bore. Vorticity can be defined as: Γ = dl U [.] 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, 999). 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, 97). 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 ) was modified to include a background longshore component.

28 7 These are: Rip Current Scaling The model uses three independent parameters to define each different case or run. The half width of the rip channel ( R ) The mean rate of generation of circulation ( DΓ Dt) The mean background alongshore current ( v ) Figure -. Definition sketch of the model. In order to be able to use the simple representation of a rip current system described above for a real case (lab experiment or field measured rip), there is a need to relate the different variables in the real system to the ones present on the model. Using a simple dimensionless group analysis, the different variables scale as follows: Length scales: ( x, y ) = ( x, y) R Velocities: ( u, v ) = ( u, v) ( DΓ Dt) Time: t = t ( DΓ Dt) R Circulation: Γ = Γ R( DΓ Dt) ( ) Therefore, we will need to determine what R and ( DΓ Dt) are in the field (lab experiment) and then scale everything accordingly. Estimating R is relatively easy; more

29 8 of a challenge is the parameter ( DΓ Dt) (strongly dependant on the local topography and wave breaking strength), two simple methodologies to estimate it will be discussed in chapter. Using this scaling has great advantages; the simple length scaling allows us to use a single configuration for the system, with all different topographies converging to a single form. In the case of absence of a background longshore current (shore normal waves) all different possible wave strengths are represented by one non-dimensional case. When longshore currents are present, the non-dimensional longshore current becomes an additional parameter to define the model. The provided scaling suggests that velocities in the rip neck depend on the strength of wave breaking, scaling with ( ) DΓ Dt, but not on the gap width (. Temporal response will depend both on wave breaking strength and length scales R) ( DΓ Dt) (R. These scaling relations discussed by Kennedy () 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, -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. )

30 9 Essentially, the system is based on the generation of vorticity at two fixed locations on either side of the rip channel (x, y) = (, ±) using a discrete vortex method. Positive x coordinates are located shoreward f the generation of circulation and negative cross-shore locations are seaward of the generation of circulation. The model is written in terms of mass transport velocities. Using the relation for point vortex velocities, U = Γ πr, the θ 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: ( ) ( t+ t X Y ) ( X Y ) ( t, =, ) + U t [.] Strong interaction between consecutively introduced vortices requires a separation of time scales in the model, therefore the model releases the discrete vortex pairs in the system at intervals of t, defined as t = Nδt, where N (runs used in the present thesis have values for N of and ) and δt is the small time step at which vortices are being displaced. Therefore, there are two kinds of time steps, the big time steps t, which determine the generation of new vorticity at the generation points, and the small time steps δt, at which the discrete vortices present in the system are displaced according to the velocity induced by all other vortices in the system at their location. When using steady forcing, the mean generation of circulation ( DΓ Dt) has a fixed value of one in the model. Choosing different values for t will allow some tuning in the model since the strength of the discrete vortex pairs is dependent on t so that the mean rate of generation of circulation is kept constant at one. The values used for t in this thesis were.,. and.; good convergence was obtained with these values so the

31 less computational demanding value of. was predominantly used. Also small random perturbations are added to the strength of each vortex to allow sinusoidal perturbations to form. Unsteady forcing can be used in the model by modulating ( DΓ Dt) with a sinusoidal component, defined in the input file by an amplitude a (values used in the analysis range from <a< with increments of.) and a group frequency ω (.<f<. where used with increments of.): DΓ Dt g = + a sin( ω t) [.] An additional parameter of the model is a constant background longshore current, this value is added to the longshore component of the computed velocity of each point vortex. No interaction between this current and the vortices in the domain is assumed. Several values have been used to study the effects of longshore currents on the rip strength. The output files obtained from the model provide location in time and strength (dimensionless) off all vortices introduced in the domain in three separate files, one for the x coordinate, one for the y coordinate and finally another one for the vortices strength. Using these data files is easy to calculate velocity fields using the point vortex velocity relations discussed above. A series of Matlab codes were used in the post processing and analysis of the data in the present thesis.

32 CHAPTER 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 ) 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 () and other experiments performed by Kennedy and the second set from Haller et al. (). The model is written in terms of mass transport velocities and works with nondimensional quantities; therefore the lab data will be scaled using the scaling relationships provided in chapter. Velocities scale with the square root of the mean rate of generation of circulation as follows:

33 ( u, v) ( u', v' ) = [.] ( DΓ ) In order to compare measured velocities with model predictions, these will be transformed into mass transport velocities (using measured local wave height to calculate short wave mass transport) and then will be plotted against estimated mean rate of generation of circulation. Two methodologies to evaluate the model parameter ( DΓ Dt) will be discussed. Parameter (DΓ / Dt) Estimations Estimations of the parameter ( DΓ Dt), mean rate of generation of circulation, are necessary to compare model predictions with measured data in the lab as stated above. This is not an easy task since measurements of vorticity are quite complicated to obtain. Therefore, a methodology that allows the use of alternative measurements (wave heights, topography, mean water levels ) is necessary. Brocchini et al. () 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 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 ( D Γ Dt) along the wave ray (which is assumed shore normal) is (e.g., Brocchini et al., ): Dt O DΓ gγ ghc = ( hb hc ) + Dt 6 8 ( γ β ) [.]

34 Where γ is the ratio of wave height to water depth (γ =.78) (depth limited breaking); β is a constant that relates water depth to wave height assuming the breaking continuous over the bar ( H = β h, where β =. ), is the water depth at the bar C h C crest and h B 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, 98): ( ) H C hb = O go γ g [.] Where H and C are the wave height and group velocity in deep water O go respectively. Figure. shows a sketch of breaking over the bar (from Brocchini et al., ). Figure.: Sketch of wave breaking over a bar (from Brocchini et al., ). 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.

35 Equation [.] 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 (998). A possible fix to this problem would be the introduction of a parameter (less than ) that would diminish this estimated generation of circulation. Method 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. ). In their paper, Brocchini et al. came up with an expression that links the change in mean water surface elevation across a breaking event (with corrections for the irrotational pressure setdown associated with the waves) with the rate of generation of circulation across that breaking event: B DΓ Dt A = g ( η B η A ) g( ηsdb ηsda ) [.] Where η B, η A are the measured mean water surface elevations after and before the breaking event respectively, and η sdb, η sda are the irrotational pressure setdown associated with measured wave heights at those same locations. This second methodology is then based on locally measured data, both wave and setup fields, which suggests a better accuracy in the prediction than the first methodology.

36 Method 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 [] estimates ( DΓ Dt) in one line so as long as we have data available in the channel we can account for the breaking in the channel diminishing the generation of circulation in a more realistic manner. Model Comparisons with Lab Data Using the methodologies previously discussed, model predictions will be compared to available lab data sets. Two data sets will be used, both from experiments performed at the directional wave basin at the Center for Applied Coastal Research of the University of Delaware: Data set : Kennedy and Dalrymple () and Kennedy (). Data set : Haller et al. () The wave basin is approximately 7. m in length and 8. m in width, with a wave-maker at one end that consists of paddles of flap-type. The experimental setup consists of a fixed beach profile with a steep (:) toe located between. m and m from the wave-maker and a milder (:) 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. 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.66 m. placed against the sidewalls. This leaves two gaps of approximately.8 m wide, located at ¼ and ¾ of the basin width, which serve as rip channels. The edges of the bars on each side of the rip channels are rounded off in order to create a smooth transition and avoid reflections. The

37 6 seaward and shoreward edges of the bar sections are located at approximately x =. m and x =. m respectively (Figure -). The crest of the bar sections are located at approximately x = m with a height of 6 cm above their seaward edge. Figure -: Experimental wave basin at the University of Delaware, (a) Plan view and (b) cross section (from Haller et al., ) If the experiments are considered as an undistorted Froude model of field conditions with a length scale ratio of /, then the experimental conditions correspond to a rip spacing of 7 m and rip channel width of m. The Haller dataset would

38 7 correspond to breaking wave heights of.8-. m, wave periods of.-. s, and mean offshore velocities of.8-.7 m/s in the rip neck. The first dataset was analyzed by Kennedy (); 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 ADV s were located relatively close together, (see figure -) there are significant differences in the measured velocities probably due to jet instability of the rip current. (Haller and Dalrymple, ). To address this, the largest and smallest means of the three ADV s are plotted together with the mean of all three ADV s as an error bar plot. Figure.: Current meter location in Kennedy (), ADV (y=.m, x=.8m); ADV (y=.7 m, x=.8m); ADV (y=.9m, x=.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 -). 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

39 8 neck velocities for comparisons with the model. For detailed information on location of instruments see Haller and Dalrymple (999). Figure -. 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., ) Both data sets and the model predictions are shown in figure -. Model predictions are shown at two locations, at the cross shore location where the two fixed generation points are (,), where the model predicted quantitatively well velocities at startup (see Kennedy, ) and at a shoreward location from that point (.,), 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.