J.5. Army CERC ee te Gaast Eng. Res ae MR 82-7 (I) Surf Zone Currents VOLUME I ewe AOS State of Knowledge DOCUMENT COLLECTION by David R. Basco MISCELLANEOUS REPORT NO. 82-7 (T) SEPTEMBER 1982 Ay COP OF y <5 LV, SN \ % ® Ny Te distribution unlimited. Prepared for U.S. ARMY, CORPS OF ENGINEERS COASTAL ENGINEERING RESEARCH CENTER as Gal Kingman Building 202, Fort Belvoir, Va. 22060 05S We $2-1 & Reprint or republication of any of this material shall give appropriate credit to the U.S. Army Coastal Engineering Research Center. Limited free distribution within the United States of single copies of this publication has been made by this Center. Additional copies are available from: National Technical Information Service ATTN: Operattons Diviston 5285 Port Royal Road Springfield, Virginta 22161 Contents of this report are not to be used for advertising, publication, or promotional purposes. Citation of trade names does not constitute an official endorsement or approval of the use of such commercial products. The findings in this report are not to be construed as an official Department of the Army position unless so designated by other authorized documents. NOILOST109 INAWNOOd Ruins. MBL/WHO! iilinwanin 0 0301 00 (NU UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) , READ INSTRUCTIONS T. REPORT NUMBER 2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED SURF ZONE CURRENTS Miscellaneous Report Volume I. State of Knowledge 6. PERFORMING ORG. REPORT NUMBER - AUTHOR(a) 8. CONTRACT OR GRANT NUMBER(e) David R. Basco DACW7 2-80-C-0003 10. PROGRAM ELEMENT, PROJECT, TASK - PERFORMING OR ON ZAICN NecwlS 192 ZBORESS AREA & WORK UNIT NUMBERS Department of Civil Engineering Texas A&M University B31672 College Station, Texas 77843 - CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE Department of the Army September 1982 Coastal Engineering Research Center ( CERRE-CO) 13. NUMBER OF PAGES Kingman Building, Fort Belvoir, Virginia 22060 243 ~ MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) | 15. SECURITY CLASS. (of thia report) UNCLASSIFIED 1Sa. DECL ASSIFICATION/ DOWNGRADING SCHEDULE - DISTRIBUTION STATEMENT (of this Report) Approved for public release; distribution inlimited. - DISTRIBUTION STATEMENT (of the abatract entered in Block 20, if different from Report) - SUPPLEMENTARY NOTES - KEY WORDS (Continue on reverse side if necessary and identify by block number) Coastal hydrodynamics Rip currents Longshore current State-of-the-art Nearshore circulations Surf zone currents Numerical models ABSTRACT (Cantinue em reverse side if meceasary and identify by block number) Investigations of surf zone currents have been conducted world wide for more than 60 years. The primary motivation has been to improve methods of understanding sediment and coastal pollution processes. This report (Vol. I) and a companion report entitled, “Annotated Bibliography of Surf Zone Cur- rents” (Vol. II) are part of a major new study of coastal currents initiated by the Coastal Engineering Research Center in 1979. The two reports provide a (continued) DD Fore 1473 ~—s EDI TIonw OF 1 Nov 6515S OBSOLETE UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) state-of-the-art summary of theories and experiments investigated since 1967. The theories before 1967 have been included in previous summaries (e.g., Galvin, 1967). Currents generated by short-period surface gravity waves both within and just beyond the breakers are of particular concern. The major types are longshore currents, nearshore circulations, rip currents, and wind-driven currents. Tidal currents are excluded. The study included the review of physical processes, theories, field and laboratory experiments, numerical models, and the measurement technology and instruments used to conduct the experiments. Related topics included are wave thrust (radiation stress) principles, wave setdown and setup, boundary and lateral mixing stress models, edge waves, wave breaking, and surf zone energy dissipation. There are currently two major theoretical approaches to coastal hydro- dynamics, each in a different stage of development. Both approaches assume uniform currents with depth, require numerical computer methods for general solutions, and both suffer from a limited data base for verification. The older, time-averaged radiation stress theory is in the final refinement stage and is now generally accepted. The new Boussinesq theory, which is just being implemented, follows the instantaneous water surface and current variations to essentially go beyond the time-averaged mean to observe the physics occurring within each wave period. The drawbacks and limitations of both approaches are different. The radi- ation stress approach requires a priori specification of wave height fields by separate means and closure coefficients obtained from time-averaged field data. The proper averaging time is unknown. A major limitation of the Boussinesq theory is the size and speed of computers needed to handle the vast grids and large number of time steps required for a meaningful simulation. Both methods rely on wave breaking and surf zone empiricism that needs con- siderable improvement. Detailed state-of-the-art summaries are presented for both methods. For practical applications in coastal engineering, it is concluded that the knowl- edge gained from future research with the Boussinesq method will best serve to improve the time-averaged radiation stress approach and its eventual coupling to coastal sediment and pollutant transport simulations. 2 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) PREFACE This report (Vol. I) and the accompanying “Annotated Bibliography of Surf Zone Currents” (Vol. II) are published to provide a state-of-the-art summary of research on coastal hydrodynamics and its three main components: longshore currents, nearshore circulations, and rip currents. The two volumes concen- trate on all theoretical aspects since 1967 but include physical descriptions and experimental data made much earlier. The work was carried out under the U.S. Army Coastal Engineering Research Center's (CERC) Nearshore Waves and Currents work unit, Harbor Entrances and Coastal Channels Program, Coastal Engineering Area of Civil Works Research and Development. The report was prepared by Dr. David R. Basco, Associate Professor, Department of Civil Engineering, Texas A&M University, College Station, Texas, under CERC Contract No. DACW72-80-C-0003. The author gratefully acknowledges the help of many individuals who contributed original copies of their journal articles and reports, and the assistance of R.A. Coleman, J. Hyden, and J. Wilkerson in completing this report. Dr. C.L. Vincent, Chief, Coastal Oceanography Branch, was the CERC contract monitor, assisted by M. Mattie, under the general supervision of Mr. R.P. Savage, Chief, Research Division. The author wishes to acknowledge their assistance in providing useful references, technical review, and preparing the final manuscript. Technical Director of CERC was Dr. Robert W. Whalin, P.E., upon publica- tion of this report. Comments on this report are invited. Approved for publication in accordance with Public Law 166, 79th Congress, approved 31 July 1945, as supplemented by Public Law 172, 88th Congress, approved 7 November 1963. TED E. BISHOP , Colonel, Corps of Engineers Commander and Director CONTENTS Page GONVERSION BAGCTORS, © UWieSe CUSTOMARY TO) (METRIC GSM) Kate elelelelelelelelelolsteleloleleleleeterevetenamel (2 SYMBOLS AND DEFINITIONS. ccccccccccccccccccccccscccceccescccccccccccccccccs 13 CHAPTER Ibe LNLRODU GiliONietereretovevetelerekclchelctatotersvetciololetoichelenotchelelelcieneielelelelsiereiorenoreieneners Dak CHAPTER 2. PHYSICAL DESCRIPTION OF NEARSHORE CURRENTS ..cccccecccccccceccee 23 i | Filelid’ ODSCEVAETONS) «1clelolclslelelelelolele|slclele)e oleic clcisic cio clcisiclelsisioicicieee 120) 4. Numerical Models. ...cccccccccccccccccccccccccccccccccccccccccs 123 VI Nonlinear and Irregular WaveS..cecocccccccceccccccccccccccsscccvcee II] dk e Nonlinear Waves @eeeoevoeevoevoeneoeoeeseeeoee eee eeeeeeeeeevoeoeeses eevee eeeeeoeeed 131 Die Irregular Waves. e@eeooeveovoevoevevoeeeeeoeeeeeeeveeeeoee ee eeeeeeveeeee202028080808 133 Vis SE BOUSSH Me SCmMeEhOd Steleleiclelelelelelelolelelolelolelolelelelolelelelelelelciclelelolclelcleletelelclelolevolelelolenn ict 1. Boussinesq Theory. ..ccccccccccccccccccccccccccccccccscccccccce 145 PeomNuMe GA cal Sol itlOmleyers)a (ele) etals/s}elole/ole]e)c/ele/e)olejejls/e/ejele/e)selcle)c/e/s/alelele/e)o/e) wlt)2 3. Simulation of Surf Zone Hydrodynamics... .cccccccccccvcccsccccs LOD VIIL Summary: Theoretical ASPECES cccccccccccccccccccccccccc cc ccccccce US'S) CHAPTER 4. EXPERIMENTAL VERIFICATION OF HM ORYetelctelclclelclelclelcleleleleicielcictelclolctetete iSy7/ WeWaverSetdownl andisSSLEUPlers clolelelcleiololelolslele\clc/olelelolelele) sl slelelsiolelolelclele/eloie)eie\elelclol a l'5)/, ey Regular) Linear ,WaVES \ y/ 58 CONTENTS FIGURES--Continued Page Illustration of the effect of offshore bottom slope on longshore CUEEECMESKfalsislelelelelelsielelelslelololciolsislelclolololelclelerctelelslelelelotelololeielelelereioicleleleicrctetoretercrersmn Lill Calculated longshore current velocity profiles for (a) various mean angles of incidence and (b) various wave steepnessesS..ccccccccee 143 Range of application of the mass and Boussinesq equation system....... 151 Comparison of horizontal and vertical maximum velocity profiles between quasi, 2-D, and linear wave theories for d/L = 0.22...ceccccee 152 Numerical propagation of quasi-long waves in two dimensions using BOUSSINES GME qUAtTONSletelejelsslslereieloleletele ol cleieieleleleleneycreiclelereleleicrelcisicievercieiersicrecicion la Wave setdown and setup, comparison of experiment and theory for Beguilarewaves sOnmdup lane sDeaCh\cstelelciclelelsicls elelelole sieloroieleleleleleleleisiereteteielctersteien nl OG Ratio of wave setup slope to beach slope (K) versuS Yecceccccccccccece 159 Wave setup versus bed slope: All experiments versus theory of van DOVHetelelelsislalelelolelsiclofelelelelolelolotereloleleleleterolelelelolorolololovorciclelcielerevelelelercierekeieisictere 160 Wave setup versus breaker ratio, Yp: Theory from equation (8) versus experiments on slopes around tan B © O.lecceccccccccccccccccee 16] Comparison of measured and calculated mean water level variations Usingeltaser—Doppler VELOCItys MeteI ss slelelels clelelclelefeleietciel cle eleleleicicicicieleielereie el!) LOD. Some comparisons of nonlinear theory for wave setdown and setup with experimental Gate als Sor ora: sie elo uelevei ove clevelelerelare elelerertiore tote evarorate ete ioteenale aks 163 Comparison of cnoidal theory of wave setdown with experiments for regular IWAN C'Sieereleleielelelereieleeieleleleeleleleleleleieleclelalelelolelelelcielelelcleielelcleleleicieleloieieneiene 164 Maximum wave setup dependent upon deepwater wave steepness and beach SMO Pelee fore eiereieiel erate) ence tong oreitoisireve cers evevonevel eteve ele leleveraneveralicrntavetereretetorevereneners 165 Comparison of wave setup for irregular waves and theory of Battjes.... 166 Empirical data for maximum wave SCEUP. <0. ccc ccccccccoccccccceocccccce LO/ Empirical longshore current at midsurf position as deduced by Komar and Inman (1970) based upon sand transport studieS...ccccccccee 169 Comparison of laboratory mean currents with integrated currents Obtained iromystrongy Current seheORy/. <\1erelere elec lolelelelelelelolcloleleleleleleioleiclsierejere) Ly/pl. Original longshore current profile theory versus laboratory data of Galvin and Eagleson, OGD ere etere ob e eieleheratere wisvelsreteve eles areas ave locdioueverelave ete 172 59 60 61 62 63 64 65 66 67 68 69 70 Tal 72 73 74 75 76 CONTENTS FIGURES-——Continued Page Bowen's longshore current profile theory versus laboratory data Of Galvin and Eagleson. ..cccccccccccccccvcceccccccccccccccsccescccces 1/3 Thornton's longshore current profile theory versus laboratory data of Galvin and Eagleson, 1965, test dl—Avtetelelchevchelcieloloteheucierelelelotetercreretclonelere 174 Thornton's Longshore current profile theory versus laboratory data of Galvin and Eagleson, 1965, tests II-2 and ME I5500 00000 00000000000 175 Thornton's longshore current profile theory versus field data by Ingle, LOG Grerererevelekeiovelohelelolevelelovelotelolelelevelelcrelelelie/cleleleleleclelc)elelelclelelelelelereleleielelelerelio 176 Jonsson, Skovgaard, and Jacobsen's longshore current profile versus laboratory and field data...cccocccccccccccvcccccccccccccsecses 1// Skovgaard, Jonsson, and Olsen's longshore current profile versus laboratory Cialtvaleveleboleliclevcielorercheletclerclelclerotelololenelololeholeloiclelstelelelclelelelelelelelclolelelcieloke 178 The model of Kraus and Sasaki compared with laboratory data of Mizuguchi, Oshima, and Horikawa (1978) .cccccccccccccccccccceccesceses 180 Kraus and Sasaki model compared with field data....cccccccccccccccccee 181 Typical vectors of 17.1-minute average currents at Torrey Pines California, from NSTS experimentsS..ccccccccccccccccccscccecccscsceoese 182 Comparison of nonlinear theory of James (1974b) with laboratory data.. 183 Comparison of irregular theory of Collins (1972) with original model of Longuet-Higgins (1970) and laboratory data for regular waves...... 184 Comparison of irregular theory of Collins (1972) with unsteady field Claltiale o cloletelclefetctolcsichelelehoie olelevelclelelelelolelelelclelcfelelctoteclelols elec) evejelelelolelcleiele cle)crolslelcre 185 Bed shear-stress coefficients computed from equation (162) and data taken at Torrey Pines Beach, California, November 1978.cc.ecccceccseee 189 Experimentally determined wave height decay on plane beaches of different SLOPES. cccccccccccccccccccececececcecc reves ecceeveccccc cece 193 Experimental studies of wave height decay in surf zones as function and distance from breakpoint: (a) laboratory; (b) field.....ceeeeeee 195 Wave height versus total water depth and surf zone width from laboratory CXPECTIMENtS occcccccceccccecvecccccceccvecsccsecvecc ccc ceeee 196 Irregular wave height variation across the surf zone on plane beaches: theory versus laboratory data..cceocccccccccccccrccccccocces 19/ Irregular wave height variation across the surf zone for four different beach profiles: theory versus data.cccccccccccccccccecesee 198 10 77 78 79 80 81 82 83 84 85 86 87 88 89 CONTENTS FIGURES--—Continued Page Correlation between rip current spacing Y, and surf zone width, Xpeeee 200 Dimensionless rip current spacing related to Irribaran number and three domains of rip current generation mechanismS....ccccccccccescse 200 Comparisons of field observations of nearshore circulations and analytical (theory, Of BOWE. sc\els\elelelels)e/e sle)e/eicls)s)e)s)0)e\elelelolelele\e\ole|e)e/eiele/s/e\eisie) 204 Comparisons of field observations of nearshore circulation and mummers cally MOdel MOL MSaS AKA) e100) olels) clelolelelclelcl sic lolclsicleloleleliclelsleiecleleleleleisleleieisiclsje) 205 Plan view of experimental site and horizontal coordinate system....... 206 Comparisons of field observations of nearshore currents and numerical modelo ebinkened'ermands Dallrymplieirelsieleielelcielejelelelelelelelelcleleleicleloleleleleleelelclereren 2 O)/) Comparisons of Boussinesq theory and experiments for wave shoaling.... 211 Comparison of numerical computation of shoaling waves obtained using Boussinesq theory and CERC experimental resultS....c.cccccceccee 212 Some results of recent wave shoaling experimentS...cccccccccccccccecce 214 Comparison of wave amplication factors by physical and numerical models in Hantsholm Harbor, Denmark. ..ccccccccccccccccescccccccccecccses 210 Comparison of pure wave diffraction by numerical model and classical diffraction theory for linear, small amplitude waves....... 217 Comparison of pure wave reflection by numerical model and experiments of RaTeTEOW 6 HG DOD DDD 00000005 000000000000000000000000000000000000000600 218 Comparison of pure wave refraction coefficients by numerical model and linear wave CNEOTY cccccccccccccccecc cc cece ccc ese ccccceccscesceccs 219 CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) UNITS OF MEASUREMENT U.S. customary units of measurement used in this report can be converted to metric (SI) units as follows: NT ———————————————_—— — —— Multiply by To obtain inches — 25.4 millimeters — Dobyh centimeters square inches 6-452 square centimeters cubic inches 16.39 cubic centimeters feet 30.48 centimeters 0.3048 meters square feet 0.0929 Square meters cubic feet 0.0283 cubic meters yards 0.9144 meters square yards 0.836 Square meters cubic yards 0.7646 cubic meters miles 1.6093 kilometers square miles 259.0 hectares knots 1.852 kilometers per hour acres 0.4047 hectares foot-pounds 1.3558 hewton meters kilograms per square centimeter millibars 1.0197 x 10> ounces 28.35 grams pounds 453.6 grams 0.4536 kilograms ton, long 1.0160 metric tons ton, short 0.9072 metric tons degrees (angle) 0.01745 radians Fahrenheit degrees 5/9 Celsius degrees or Kelvins! 1To obtain Celsius (C) temperature readings from Fahrenheit (F) readings, use formula: C = (5/9) (F -32). To obtain Kelvin (K) readings, use formula: K = (5/9) (F -32) + 273.15. a tA SYMBOLS AND DEFINITIONS coefficient in Longuet-Higgins' (1970) original theory ec tan By 8f ), Dalrymple and Lozano (1978)®8 wave amplitude; a = H/2 characterizes ene vertical accelerations due to wave steepness (= ae deepwater wave amplitude maximum horizontal excursion amplitude of wave orbital 4 ij ) in shallow water T ™ motion(= 7 coefficients in Longuet-Higgins' (1970) original theory characterizes the vertical accelerations due to bed-slope d2z variations (= ee wave celerity wave celerity at breaking the Chezy friction coefficient a boundary resistance coefficient due to both waves and longshore current, v wave group celerity deepwater wave celerity bed shear-stress coefficient, Huntley (1976) cnoidal theory celerity local rate of energy dissipation per unit area water depth (bed to stillwater level, SWL) depth of water at breaking wave total energy density of the waves mean wave energy per unit area fictitious wave energy per unit area 13 rms complete elliptic integral of the second kind error function general flux of energy toward the shore Darcy-Weisbach type friction coefficient (= 8C.) the mean frequency of the wave energy spectrum current friction factor friction factor for waves and currents relative to total velocity time-averaged friction coefficient wave friction factor acceleration due to gravity wave height the stochastic wave height breaking wave height significant wave height at the breakers mean square value of the fictitious wave height the fictitious root-mean-square (rms) wave height maximum possible wave height in the surf zone deepwater wave height wave steepness deepwater significant wave height root-mean-square (rms) wave height water depth to mean water level (MWL) mean depth at breaking incident wave front Iribarren No. unit vector in the x-direction 14 cu. px unit vector in the y-direction ratio of setup slope to beach slope wave number (2) 21 deepwater wave number G? o) wavelength deepwater wavelength rip current spacing mixing length (Prandtl's hypothesis) Mach stem the corrected beach slope including wave setup a dimensionless, turbulent closure coefficient for lateral wave mixing proposed by Longuet-Higgins (1970) to be of the order N < 0.016 lateral viscosity coefficient wave celerity ratio, GI Longuet-Higgins (1970) parameter showing relative importance of mixing and bed shear a parameter (= T//gd,) in which the magnitude of the longshore current depends, James (1974a) dimensionless mixing parameter pressure at any distance below fluid surface; gage pressure 1 3 3 (=Nog “(tan 8) 12) from Longuet-Higgins (1970) a dimensionless parameter found from wave orbital velocity experiments relating to wave velocity at the bottom to that at the reference elevation discharge per unit width at any d, y, and t (uh) volumetric flow rates per unit width the probability that at a given point a wave height is associated with a breaking or broken wave e lL, constant from Longuet-Higgins (1970) (=(1/7)pC-g *y (tan B) *) discharge per unit width at any x, y, and t (=vh), volumetric flow rate per unit width 15 reflective wave front Reynolds number (=UL/v,) time-averaged, lateral Reynolds number eomsibaay from nein ete ublisiseims (1970) (=*6eaq 2y2 (tan g) 2 Caaae) bottom roughness components of the radiation stress tensor components of the fictitious radiation stress tensor without breaking time-averaged value of the total flux of horizontal momentum across a vertical plane minus the stillwater hydrostatic pressure force principal wave thrust (radiation stress) perpendicular to the wave crest the shear-stress component due to the excess momentum flux. The shear stress is exerted in the y-direction on a fluid surface constant in x component wave thrust parallel to shoreline principal wave thrust (radiation stress) parallel to wave crest the shear-stress component due to the excess momentum flux. The shear stress is exerted in the x-direction on a fluid surface constant in y transformed wave thrust component in x-direction transformed wave thrust component in y-direction transformed shear-stress component wave period deepwater wave period lateral shear force slope of MWL due to wave setup Ursell number 16 ay > vector sum of longshore current velocity v and bottom wave orbital velocity B wave orbital velocity component on horizontal plane wave orbital velocity at the bottom mean wave orbital velocity near the bottom the absolute value, maximum, wave orbital velocity near the bottom for sinusoidal motion dimensionless longshore current velocity (v/v) the wave orbital velocity in the YY-direction parallel to the wave crests a dimensionless longshore current velocity using VR as velocity scale local longshore current velocity mean velocity across the surf zone longshore current velocity the breaker velocity, dimensionless the midsurf velocity, dimensionless maximum V at Se dimensionless horizontal (y) component of local fluid velocity (water particle velocity) time-averaged velocity at breaker line, dimensional time-averaged maximum current velocity, dimensional time-average midsurf velocity, dimensional velocity fluctuation in the y-direction longshore current velocity neglecting lateral turbulent mixing stresses reference longshore current velocity at the breaker line neglecting lateral turbulent mixing modified v* from theory of Longuet-Higgins (1970) 17 1 vertical (z) component of local fluid velocity or current velocity (=x/x,) a x-direction coordinate perpendicular to the wave crest a dimensionless x-direction coordinate using Xp, as the length scale coordinate direction system where the origin is taken at the maximum setup line and a positive x is facing seaward a direction coordinate system where the origin is taken at the maximum setup line and a positive x is facing seaward distance from shore breaker position in x coordinates midsurf position plunge point distance between the breakpoint and plunge point of a curling breaker a y-direction coordinate along the wave crest a dimensionless y-direction coordinate using Yp as the length scale coordinate direction alongshore, positive to right when facing seaward the vertical distance of the bed above an arbitrary datum and always positive the bed elevation above an arbitrary datum angle between wave crest and bottom contour angle between wave crest and the shoreline angle between breaking wave crest and shoreline deepwater wave angle mean deepwater approach angle angle of beach slope with horizontal horizontal (wave) mixing coefficient in surf zone, as introduced by Longuet-Higgins (1970, 1972a) 18 | closure coefficient for the kinematic eddy viscosity ratio of breaking wave height to mean depth at breaking breaker index at breaker a v (measured) - v(predicted) : ( TGEECHIEEEA) ), Madsen, Ostendorf, and Reynolds (1978) scaling parameter (= scaling parameter (= — von Karman Constant = 0.4 mean vertical particle displacement caused by wave passage displacement of water surface with respect to SWL by passage of wave wave surface variation based on second-order cnoidal theory MWL change above or below the SWL magnitude of the wave setdown at the breakpoint Maximum wave setup at the shoreline eigenvalues a angle between u and u, the lateral turbulent eddy viscosity due to waves, time averaged kinematic eddy viscosity (=u,,/) kinematic eddy viscosity in the alongshore direction amplitude of wave orbital particle displacement surf similarity parameter, Battjes (1974b) modified surf similarity parameter, Ostendorf and Madsen (1979) mass density (p=w/g) bottom shear stress a true turbulent Reynolds stress due to random turbulent velocity interactions at scales far less than the wave orbital velocity scale 19 Lxx Glee instantaneous bottom shear the lateral turbulent eddy time-averaged bottom shear shoreline time-averaged bottom shear time-averaged surface wind the shoreline time-averaged surface wind shoreline stress stress due to waves stress perpendicular to the stress parallel to the shoreline shear stress perpendicular to shear stress parallel to the the effective lateral stress components mass transport stream function wave angular frequency 20 SURF ZONE CURRENTS Volume I. State of Knowledge by Davtd R. Basco CHAPTER 1 INTRODUCTION For more than 60 years, investigations of coastal currents have been conducted worldwide with the primary motivation of understanding sediment transport, shoreline migration (erosion and accretion) processes, and the transport and dispersion of pollutants. This report (Vol. I) and its com- panion report entitled "Bibliography of Longshore and Nearshore Currents" (Vol.II) are part of a major new study of nearshore currents initiated by the Coastal Engineering Research Center (CERC) in 1979. Those currents generated by short-period surface gravity waves in water depths generally less than 10 meters are particularly important. These cur- rents are longshore currents, nearshore circulation cells, rip currents, on- shore-offshore flows caused by winds, and wind-driven alongshore currents. Articles and reports on the physical processes, theories, field and laboratory experiments, and numerical models of these coastal flows were reviewed for this study, along with the measurement technology and instruments used to conduct the experiments. In addition, the following related topics were re- viewed: wave thrust (radiation stress) principles, wave setdown and setup, bed shear in oscillatory flow, edge waves, wave breaking, and surf zone hydro- dynamics. Coastal currents induced by tides were excluded. Also, the vast body of literature on fundamental, oscillatory, water wave theory, wave-current interaction, and analytical theories for waves propagating over variable depth bathymetry has been purposely left out. Sediment transport literature is omitted except where coastal hydrodynamic investigations were a major part of the effort. Previous state-of-the-art studies and reviews were important contribu- tions to this report. Galvin's (1967) analysis and conclusions were, no doubt, in some way responsible for the emergence of the wave thrust theory of the of the 1970's. Many explicit summaries of theoretical interest in longshore, nearshore, and rip currents have ensued, including Longuet-Higgins (1972a,b), Meyer and Taylor (1972), Jonsson, Skovgaard, and Jacobsen (1974), Miller and Barcilon (1978), Dalrymple (1978), Komar (1976 a,b), Fisher and Dolan (1977), and Horikawa (1978). Other valuable reviews of the literature can be found in the doctoral dissertations of Bowen (1969a,b), Thornton (1969), Battjes (1974a, b), Sasaki (1974-1975), Meadows (1977), and Gourlay (1978). Researchers con- tinue to investigate various weak links in the field (e.g., Dean, 1976; Collins. 1977). The ongoing Nearshore Sediment Transport Study (NSTS), sponsored by the National Oceanic and Atmospheric Administration, will undoubtedly result in future reassessments of current knowledge. In Europe, a recent mechanics con- ference, Euromech 102 (1978), concentrated on new research efforts in coastal hydrodynamics. And finally, many researchers who have developed numerical 21 simulation models of nearshore currents include extensive summaries of pre- vious efforts (Lui and Mei, 1975; Ostendorf and Madsen, 1979; Ebersole and Dalrymple, 1979; Vreugdenhil, 1980). The objectives of the work reported herein were to review and analyze pertinent literature for the preparation of a state-of-the-art report includ- ing major gaps in knowledge, data, and recommendations for future research. Chapter 2 gives a physical description of coastal flows, including magnitudes of currents measured in the field and the laboratory. The variability of these currents in both space and time is of primary interest. Forcing func- tions and mechanisms responsible for these currents are discussed. Instru- ments and systems devised to measure the currents are also reviewed. In Chapter 3 the fundamental equations, theory of longshore currents, and near- shore circulation are summarized. Details of all theories before 1967 are omitted. Both the time-averaged radiation stress theory and the new "Boussinesq-type" theory are presented in considerable detail; however, full derivations are not presented due to space limitations. Emphasis is on as- sumptions and limitations of each theory. In many cases, assumptions are made to permit analytical solutions of the appropriate equations. Numerical solution methods allow relaxation of these assumptions but at the expense of possible inaccuracies due to the researchers' discretion of the continuum. A111 theories and solution methods rely heavily on many empirical and relatively crude approximations in the surf zone. The chapter also includes the latest re- search efforts to improve on fundamental knowledge in this regard. The ex- perimental verification (or otherwise) of the theories is the subject of Chapter 4. Unfortunately, the number of and complexities of the theories have far outstripped the number and extent of good data sets to confirm (or reject) them, so the state-of-the-art is then presented in this chapter. A brief summary, conclusions, and recommendations are given in Chapter 5. 22 CHAPTER 2 PHYSICAL DESCRIPTION OF NEARSHORE CURRENTS I. FIELD OBSERVATIONS One of the earliest known scientific descriptions of coastal currents was made by Johnson (1919). The longshore current was induced by an oblique angle of wave incidence. Undertows or rip currents were said to occur when waves struck the shore at right angles. The hydraulic currents were thought to be created by translation and asymmetric, shallow wave motion piling water up against the shore. This volume increase concept persisted for more than 50 years and even became the basis for some theories before 1967. bUG Measurements of Mean Values. a. Longshore Currents. The wave-induced current that flows alongshore and is generally confined between the first breaker and the shoreline is termed the longshore current. It was first measured by Putnam, Munk, and Traylor (1949), using traveltimes and distances of floats and dye in the surf on the California coast. Currents up to 1.7 meters per second were recorded. Although, on the average, these currents are usually much smaller, these re- searchers reported 18 measurements that gave a mean of 0.9 meter per second. Shepard (1950) reported the results of more than 2,000 measurements of long- shore currents over the entire southern California coast. Using floats, the maximum value given was only 0.5 meter per second. More detailed measurements taken at Torrey Pines Beach, California, were reported in Inman and Quinn (1955). Measurements by floats positioned near midsurf gave a maximum value of 0.4 meter per second with a mean of about 0.1 meter per second. Moore and Scholl (1961) also reported a maximum value of 0.4 meter per second using dye on an Alaskan beach. Galvin and Savage (1966) reported five measurements using floats at Nags Head, North Carolina. They obtained a maximum current of 1.3 meters per second. Except for Shepard (1950), all these experiments included enough field information to be useful to Galvin's (1967) assessment, and they are still being used today. Since Galvin's study, more data sets have been reported but some failed to record the important independent variables. Many are ex- tremely detailed with the introduction of an array of fixed current meters of the propeller or electromagnetic (EM) type to give continuous velocity read- ings. Appendix A provides a chronological tabulation of 38 experiments re- ported in the literature review. Also given are the range of wave data pre- sent, beach slope, observation method, and location of experiments as reported, if available in the reference. Measurements have been reported from all over the world (South Africa, Mexico, Japan, Australia, the North Sea, England, Egypt, India, Canada, and Poland). Surprisingly, no measurements could be found for the Gulf of Mexico coast of the United States. All other U.S. coasts have reported experiments, including some Lake Michigan measurements. The California coast is the most frequently observed. 23 Field experiments ‘in the past 10 years have generally recognized the variability of longshore currents across the surf zone and down the coast. Mean values distort the true physical process. To date, the maximum long- shore current value reported (see App. A) is less than 2 meters per second. Hom-ma, Horikawa, and Sonu (1962)! reported measurements (methods not stated) of surf zone swash reaching 3 meters per second during a typhoon in Japan (from Gourlay, 1978). b. Nearshore Circulation and Rip Currents. Rip currents are narrow, strong return flows directed through the surf to sea. Together with longshore currents they can create a two-dimensional coastal current system within and beyond the surf zone termed nearshore ctrculation. The first detailed and well-documented measurements of such a system of currents were reported in Shepard and Inman (1950) for La Jolla, California (Fig. 1). Surf floats, drogues, and dye were used to measure the magnitude and directions. Mean values reported ranged from less than 0.1 meter per second outside the surf areas, 0.5 to 1.0 meter per second alongshore in the surf zone, and 0.25 to 1.0 meter per second entering into the rip currents. Detailed current speeds along the rips were not reported. As early as 1936, Shepard reported on the qualitative aspects of rip cur- rents (Shepard, 1936). Shepard, Emergy, and LaFond (1941) cited maximum rip speeds greater than 1 meter per second and occasionally extending more than 500 meters fromthe shoreline. These observations in California were confirmed by McKenzie (1958) in Australia and by Dobson and Draper (1965) on the Atlantic coast of England. The latter researchers used a theodolite to observe the speed of floats on the foam line. A maximum rip current speed of 2.5 meters per second in the surf zone was reported with a mean value of about 1.4 meters per second in the surf and 0.7 meter per second beyond the breakers. Rips were observed more than 300 meters from shore. Horikawa (1978) summarized Japanese field observations and concluded that speeds near the root are about 1 meter per second and can reach 2 meters per second under rough sea condi- tions. Surprisingly, there have been few actual rip current velocity measure- ments reported. Dalrymple (1976) presents a bibliography on rip currents that includes only 14 references to field observations. Many of these concentrated on nearshore circulation patterns as discussed below. A series of rip currents is usually found along the coast with longshore currents feeding the rips and forming independent circulation cells. Spacings on the California coast have been reported as 30 to 100 meters (Inman and Bagnold, 1963)2 and 400 meters on the average (Shepard and Inman, 1951). Horikawa (1978) observed some Japanese coasts with 200 to 300 meters in rip spacings and others with only 80- to 150-meter intervals. Little is known about the actual width of rip currents. Widths of 10 meters or less near their root and 30 meters at the head (beyond the breakers) are crude estimates. 1HOM-MA, M., HORIKAWA, K., and SONU, C., "Field Investigation at Tokai, Japan Conducted by Combined Procedure of Macroscopic and Microscopic Approaches," Coastal Engineering in Japan, Vol. 5, 1962, pp. 93-110 (not in bibliography). 2INMAN, D.L., and BAGNOLD, R.A., "Littoral Processes," The Sea, M.N. Hill, ed., Interscience, Vol. 3, New York, 1963, pp. 529-553 (not in bibliography). 24 @) Scripps Canyon 18 November 1948 Wave period 7-6 sec Waves from WNW —= 0 —- $kn => 4-35 kn wep 5; -—1kn ! aa > 1 knot bY Seri —— > Observed current a inetiranon (not measured) oe 50 e Starting position of surface float Hb = Breaker height C— Float recovery area Figure 1. Nearshore current system at La Jolla Beach, California, December 1948 (from Shepard and Inman, 1951). Shepard and Inman's (1950) early observations of nearshore circulation patterns and magnitudes in California were later confirmed and extended by Harris (1964, 1967, 1969) in South Africa and Sonu (1969a) on the northwest Florida coast. Sonu (1972), in a further analysis of these measurements, re- ported shoreward currents of 20 to 30 centimeters per second and rip velocities as high as 2 meters per second. Between these extremes, speeds gradually changed. When circulation velocities were low, the rip current was broad with no apparent root or stem visible. Sonu used weighted floats, dye, and a bi- directional, propeller-type current meter to obtain time-series records of 25 velocities. He also used photos taken from a tethered balloon to record dye trajectories. These innovative techniques produced the first measurements showing velocity variations with depth in the rip currents, and with time at any position in the surf zone. Figure 2 shows the strengths of the circula- tions and rip currents reported as transport velocity (i.e., flow rate per unit width) at Seagrove Beach, Florida (Sonu, 1972). These early field efforts concentrated on measuring mean values of the currents in the nearshore zone and the wave heights, angles, etc. that created them; however, it became evident that space and time variations of the currents must also be measured. Mechanisms, cause-effect relations, and theories stated by these researchers are discussed later in this report. 220 Space Variations of Longshore Currents. a. Across Surf Zone. Ingle (1966) reported current measurements at three to six points across the surf zone and beyond. A fluorescein dye patch was timed and its travel distance noted. Although considerable scatter oc- curred, it was noted that the largest longshore current developed midway be- tween the swash zone and the breaker zone. Currents outside the breaker line were small but in the same downcoast direction and never exceeded 0.3 meter per second. The maximum velocity was 1.3 meters per second at the midsurf position with values dropping rapidly on either side. Galvin (1967) briefly mentions this variation across the surf, but does not mention it as a possible reason for differences between field data and the agen Brooker line Treaspert velocity 0.1 a%Z/soc/m — Root Treaspert otroemling —" Curroal moter @ @ weve gevge x9 meres SEAGROVE, FLA. JULY 31, 1968 meTERs Figure 2. Rip current and nearshore currents at Seagrove, Florida, July 1968 (from Sonu, 1972). 26 theories of that time. Thornton (1969) discusses these early efforts and that of Zenkovitch (1960) as examples of the distribution of longshore cur- rent across the surf zone. Although Thornton measured currents through the surf zone, he did not include enough detail to verify his own theories in this regard. About 1970, the radiation stress theory clearly demonstrated how the time-averaged longshore current velocity must vary across the surf and beyond the breakers. As indicated in Appendix A, some subsequent field measurement programs have concentrated on obtaining data to validate the theory (Sasaki, 1975; Huntley and Bowen, 1974; Manohar, Mobarek, and Morcos, 1974; Wood, 1976; Meadows, 1977; Allender, et al., 1978; Guza and Thornton, 1978, 1980a, 1980b; Kraus and Sasaki, 1979). The magnitudes of the currents were in the same range as those previously discussed. Unfortunately, many of these early data sets were obtained from visual observations. Wave angle observations at the breaker lacked the needed precision. The general trend has been to record even stronger longshore currents in recent experiments than the maximum value of 2 meters per second previously cited (Guza, personal communication, 1980). This is probably due to the use of contiuous recording current meters on solid supports that were operational during storm events. In this regard, no evidence was found of a continuous velocity reading in the surf zone over ex- tended time periods (months or years). b. Along the Coast. It was apparent from even the earliest observa- tions in the late 1940's that the mean current varied along the coast. This is especially true on a scale of the circulation cell with rip currents pre- sent or near coastal obstructions. Even on relatively straight beaches with approximately parallel bottom contours, Inman and Quinn (1951) measured vari- ability along the coast (measured by its standard deviation) that was equal to or greater than the mean longshore current. Keeley and Bowen (1977) measured mean currents over a 1-kilometer length of a Canadian beach and found large-scale variation. Mechanisms and theories to account for the variations will be discussed later. c. With Depth. The most difficult space variable to observe is over the water depth. Dye released in the surf will rapidly mix over the depth and give a depth-integrated and time-averaged current value. To be visible, surface floats must move with surface currents. Drogues help to obtain the depth-averaged current. A vertical array of fixed current meters is the only practical way to measure the current distribution over the depth. Shepard and Inman (1950, 1951) made more than 1,600 observations of surface and bottom current tendencies in the surf zone of southern California beaches. Inside the breaker zone, their observations indicated a net seaward drift along the bottom and a net shoreward movement at the surface. These top-to-bottom, onshore-to-offshore tendencies were not obser- ved beyond the breakers. Galvin (1967) stated that the nonuniformity of currents had not been studied quantitatively at that time except as described above, and it is assumed he meant variations over the vertical. The vertical uniformity of the horizontal currents is implied by many field researchers (e.g., Thornton, 1969; Komar and Inman, 19703; Horikawa and Sasaki, 1972) who never mention it. 3KOMAR, P.D., and INMAN, D.L., "Longshore Sand Transport on Beaches," Journal of Geophysteal Research, Vol. 75, No. 30, 1970, pp. 5914-5927 (not in bibli- ography). 27 Sonu (1972) was the first to publish vertical velocity profiles from field measurements. Figure 3 shows values based on 2-minute averages in a rip current. Other similar profiles generally showed larger currents near the bed at the root (nearshore) with the maximum velocity moving upward farther sea- ward. Variations in current at three depths and three surf zone positions on the eastern Lake Michigan shoreline were measured by Wood and Meadows (1975). A summary of their work (Meadows, 1976) discusses time-dependent currents and says little about the vertical variations measured. A time history is shown in Figure 4. The average current is in the same direction but lower near the bottom. Brenninkmeyer, James, and Wood (1977) used two bidirectional EM current meters in the Massachusetts surf. The onshore-offshore direction was measured at two elevations. Four possible bore-backwash interaction patterns were observed along with the interactions of the bores themselves. Extremely complex three-dimensional flows result that also pulsate in time (Brenninkmeyer, 1978). In one figure in Brenninkmeyer, James, and Wood (1977), the upper current meter showed 6.6 meters per second onshore at the same instant a meter 15 centimeters below read 1.3 meters per second onshore. At other times the directions were reversed. Zenkovich (1967) summarizes Russian thinking through 1960 and argues for a three-dimensional velocity structure in the nearshore zone. Most coasts were thought to have a bottom return flow which resulted in unstable, compli- eated circulation patterns with a horizontal axis. Less stable circulations with a vertical axis produced rip and gradient currents. No general picture of coastal hydrodynamics was felt possible at that time, due to complex and diverse boundary conditions and the instabilities involved. Weter level or? ; —.! 1 0 4 5 ° ‘ : ° 4 0 4 —10 .2) alvec ms ay < : e {-) #2 / 1.0 #3 84 Sa 8S a6 a i AP i Te EL GT a A tte PURI n Te ice NTE LSE rear Sout : “= 20 30 40 50 Met —>Seoward OOD SEAGROVE, FLA. JULY 31, 1968 Figure 3. Velocity profile in a rip current at Seagrove, Florida, July 1968 (from Sonu, 1972). 28 UPPER CURRENT METER 25 SECENDS 8 Lata yin ecto Ann’ MIDDLE CURRENT METER 26_SECONDS oe c 1.6 3 1.0 Pd 0.0 BOTTOM CURRENT ‘METER 26 _ SECONDS 2.0 1.6 1.0 6 0.0 Figure 4. Longshore component current time histories at three elevations in the surf zone of eastern Lake Michigan (from Meadows, 1976). The ongoing NSTS project assumes a uniform vertical velocity structure in the surf zone, in that no vertical current meter arrays are employed. However, the rip currents are being observed by a portable device that per- mits vertical velocity profiles to be taken (Gable, 1979). No results of vertical distributions are available at this writing. Ae) Shc Time Variations By definition, the longshore current is a time-averaged current. Early field measurements with floats, dyes, drogues, etc., measured distance traveled over finite-time increments to obtain a mean value over the time interval employed. These are Lagrangian views of the velocity field. The use of continuous recording velocity meter arrays during the 1970's has permitted local Eulerican velocity fields and their time histories to be constructed. As shown previously (Fig. 4) and in Figure 5 (Dette and Fuhrboter, 1974), the longshore velocity components vary considerably with time and within a wave period. Early researchers gave little thought to the time interval for use in calculating the longshore current, but it has now become the central issue. The velocity recorded is due to the wave orbital motions, local return flows, mass transport by waves, tidal grad- ients, surface wind shears, effects of rip current shears, turbulent eddies, and other factors as discussed in more detail in a later section. Figure 5 also shows a mean longshore current value of V, = 0.97 meter per second which is apparently the average over the 28-second record shown. Why was this time interval employed? Use of a shorter time interval such as the time for wave height repetitions (i.e., the wave period) would clearly give longshore current values that change in time (Fig. 5). Thus, Dette and Fuhrboter (1974) were perhaps the first to claim that longshore currents could not be regarded as steady or quasi-steady flows. They showed fluc- tuations in the range of + 100 percent that occurred with periods from 1/4 to 1/9 of the wave period. A mean longshore current of 1.0 meter per second commonly fluctuated from 0 to + 2 meters per second within shorter time intervals. Somewhat similar results were reported by Wood and Meadows (1975) and Meadows (1976, 1977). Using demeaned (averaging time about 15 minutes) and band pass filtering techniques on the time series (see Fig. 4), Meadows concluded that the mean longshore current velocity varied in time at any point due to the horizontal wave particle velocity and a longer period com- ponent due to nearshore zone-induced wave interactions. An analysis was also made of the phase lag between the water surface time series and the surface longshore current time series. The long-period component was generally in-phase and the wave orbital component (short-period) was out- of-phase. Meadows (1976, 1977) stated that these fluctuations were at times more than 150 percent of the mean longshore current. Variations occurred over time periods of 3 to 80 seconds, i.e., at periods about equal to or greater than the wave height periods present (4.2 seconds). Although primarily interested in the three-dimensional structure of the velocity field in the surf zone, Brenninkmeyer, James, and Wood (1977) also commented on the time histories of their velocity measurements. They found many strong oscillations within any one breaker period, the com- ponents of which appeared to move together. Maximum velocities (different values) were recorded in three component directions at the same time, indicating a dominant velocity vector. Superimposed were smaller, higher frequency oscillations with opposite phases. These were thought to show the secondary water motions (vertices and helices) found within a bore. 30 m WAVE HEIGHTS aa ay ear 0 30sec hat SHORE CURRENT Figure 5. Time history of longshore velocity and wave height in surf zone of North Sea, Isle of Sylt, Germany, November 1973 (from Dette and Fuhrboter, 1974). Six biaxial EM current meters were used to study variability of long- shore currents in the inner surf zone at Torrey Pines Beach, California, in March 1977 (Guza and Thornton, 1978). The ultimate goal was to test theories relating mean longshore current and offshore values of radiation stress (see Ch. 4). Figure 6 shows the meter array locations and some longshore current values based on a 1,024-second (17.07 minutes) means of the continuous records. They concluded that even on a relatively straight beach such as Torrey Pines, with no obvious rip structures or visible variations of the incident wave field, considerable time variations of mean longshore currents can occur; considerable longshore spatial varia- bility of the longshore current is also present. Spectral analysis of more than 100 records over a month period (60 hours of observations) showed the presence of large, low-frequency (long-period) components in the long- shore current similar to those described by Woods and Meadows (1975) and Meadows (1976, 1977). The temporal fluctuations are not site- or wave regime-related since they have been observed in southern California, on the Great Lakes (Woods and Meadows 1975), and also in the Atlantic, Nova Scotia 31 OFFSHORE (M) 80 te) -80 -160 -240 -320 LONGSHORE (M) (A) Plan view of instrument locations. x = pressure sensorse 0 = current meters 100 (b) AI 50 Atl 0 (c) s 100 oR o S 50 5 cl I> 0 100 : (d) b) 50 0 17.1 34.1 51.2 68.3 TIME (min.) (b,c.d) Longshore current at different surf zone locations Figure 6. Temporal variation of averaged (1,024 seconds) longshore currents at five locations in surf zone, Torrey Pines Beach, California, March 1977 (from Guza and Thornton, 1978). (Holman, Huntley, and Bowen 1977/8) nase study further the question of averag- ing time, Guza and Thornton (1978) made additional 256-second (4.3 minutes) averages from five closely spaced current meters (14-meter maximum separation) at Scripps Institute Beach, La Jolla, California. Figure 7 shows how the variations occur together indicating all meters were operating. Since a varia: tion as much as 20 centimeters per second occurs at essentially the same loca- tion, the 4.3-minute averaging time is not representative of the mean for longe' time scales. It is concluded that: "An appropriate temporal averaging time for mean long- shore currents is not known." (p. 768) 4HOLMAN, R.A., HUNTLEY, D.A., and BOWEN, A.J.. "Infragravity Waves in Storm Conditions," Proceedings, 16th Coastal Engineering Conference, Vol. I, Hamburg, 1978, pp. 268-284 (not in bibliography). 32 +50 20 JULY 78 V (cm/sec) -50 7.1 34.1 51.2 68.3 85.3 102.4 119.4 = 136:5 TIME (min.) Figure 7. Time history of averaged (256 seconds) longshore current from five closely spaced current meters, Scripps Beach, La Jolla, California, July 1978 (from Guza and Thornton, 1978). The latest results of the NSTS longshore current measurements are also being analyzed by Guza and Thornton. Some early results of their November 1978 measurements at Torrey Pines, California, were reported in Guza and Thornton (1980a). Again, 17.1-minute averages were used to obtain mean values. The spatial and temporal averaging scales necessary to obtain mean longshore currents will be discussed in a future paper by Guza and Thornton. As indicated in the above discussion, longshore current space variations can extend for considerable distance along the coast (nearshore circulation cells), normal to the coast (surf zone widths), and vertically (water depths), as might be expected. Time scales for velocity variation extend from frac- tions of incident wave periods in seconds to minutes and beyond, depending upon the coastal phenomenon of interest. The very definition of longshore current as a time-averaged velocity is now subject to review since a univer- sally accepted averaging interval is not known. This result may not have been expected, and it follows a general trend in science and especially in physical oceanography. As new instruments are used with better resolution, new phenomena are discovered. Also, the shear volume of the numbers collec- ted (millions for the NSTS project) make analysis time consuming. The variability of field observations in time and space has made the use of controlled laboratory experiments attractive to researchers, as discussed in the next section. 33 II. LABORATORY OBSERVATIONS life Summary of Data. There have been few general laboratory studies of longshore currents and nearshore circulations. The 12 reviewed during this study are tabulated in Appendix B. Monochromatic waves were used in all cases. A maximum long- shore current of 0.7 meter per second was observed, and wave heights ranged up to 14 centimeters, with periods generally between 0.7 and 3 seconds. For space reasons, beach slopes were larger than in nature and about 0.1 for most tests. A wide range of wave approach angles was tested. The major studies before 1967 (Putnam, Munk, and Traylor, 1949; Saville, 1950°; Brebner and Kamphuis, 1963; Galvin and Eagleson, 1965) were thoroughly reviewed and discussed by Galvin (1967). They continue to be used today to validate analytical theories and numerical models (e.g., Ostendorf and Madsen, 1979). . Krumbein's (1944) study on a movable bed used a 30° beach slope and was mainly concerned with sediment transport. Data from it_are not useful for relating to natural beach conditions. Saville's (1950)~ data, although for a 1 on 10 slope, are also of limited value. Circulations in the vicinity of a coastal jetty were determined by Shimano, Hom-ma, and Horikawa (1958), but details were lacking as to how the velocities were measured. Three studies (Putnam, Munk, and Traylor, 1949; Brebner and Kamphuis, 1963; Galvin and Eagleson, 1965) measured longshore currents induced by essentially two-dimensional, horizontally propagating, monochromatic water waves breaking on plane stationary, impermeable fixed-bed laboratory beaches set in basins with constant approach depths. Velocities in the surf zone were measured by the traveltime on dye patches, immiscible fluids, or wooden floats. Surface floats were found to move 1 to 10 percent faster than dye patches (midpoint), as measured by Galvin and Eagleson (1965). These researchers also used a miniature current meter (5/8-inch diameter) to measure velocity profiles across the surf zone and along the beach. Unfortunately, in some cases, the instantaneous water depth was less than the diameter of the meter propeller. The wave tank conditions were steady in time, so that it was tacitly assumed in all these studies that longshore cur¥rents were independent of averaging times (traveltimes). As Galvin (1967) pointed out, velocities were measured at approximately the same (midpoint) position on the beach, thus reducing the influence of nonuniformity along the beach due to the end-wall effects; however, a spatial mean longshore current was never defined. Brebner and Kamphuis (1963) measured dye travel . just inside the breaker line and called this the maximum longshore current. For some unknown reason, it was more than 10 years before the next mean- ingful laboratory investigation took place. Perhaps Galvin's concern with the nonuniformity in profiles along the beach for closed basins and the subse- quent theory of longshore currents profile for infinite beaches made SSAVILLE, T., "Model Study of Sand Transport Along on Infinitely Long Straight Beach," Transacttons Amertcan Geophystcal Unton, Vol. 31, 1950, pp. 555-565 (not in bibliography). 34 researchers leery of laboratory basins for verifying the new theory. Gourlay (1976, 1978) avoided this problem by concentrating on the non- uniform system generated on a beach behind an offshore breakwater. The study employed an idealized geometry, in that the beach planform permitted simultaneous wave breaking crests parallel to the beach at all times. The alongshore gradient of breaking wave height resulted from diffraction in the lee of the offshore breakwater. The wave-generated longshore current and nearshore circulation system studied by Gourlay (1978) is shown in Figure 8. Spherical floats made of sealing wax (19 and 13 millimeters) were photographed with a movie camera to determine surface velocity dis- tributions and circulation patterns (see Fig. 8). Current velocity pro- files normal to the beach varied in shape at various positions along the beach, as anticipated. Figure 8 shows maximum surface current at about 0.4 meter per second along section I and well inside the surf zone. The scale of this model and variables were similar to those previously dis- cussed (wave heights up to 12 centimeters, periods from 0.7 to 1.5 seconds, and a beach slope of 0.1). EXPOSED AREA RUN-UP LIMIT /RIP CURRENT | H,= 69me T= 1.05 ast ai Weve creet Tae Ya/ Ya oes & ©. @ £ ) ) mn a 4 y BREAKWATER Figure 8. Wave-generated current system behind offshore breakwater (from) Gourlay, 1978). Gourlay (1978) also measured the longshore and onshore-offshore vert- ical velocity distributions at section I in Figure 8. A specially con- structed Pitot tube (pitometer) with forward and rear facing total head tubes was developed and calibrated to give results said to be qualitatively 3m satisfactory. No dynamic calibration was attempted. Nevertheless, infor- mation concerning vertical velocity distributions could be obtained as shown in Figure 9. At the inflow region, a helicoidal component was observed with a clockwise circulation with contours shown in Figure 9(a) for the longshore currents. The pitometer only measured velocities below the wave troughs (solid line) and these are connected (dotted line) to surface float values (MWL) to complete the isovelocity patterns. The max- imum longshore current velocity occurred below low water levels near the midsurf area and was somewhat larger than the surface float values. All longshore vertical velocities were in the same direction at each location in the section. This was also true for the onshore-offshore vertical distribution (Fig. 9, b), except for a narrow zone near wave breaking. These velocities were offshore dominated by backwash in the upper layers and had onshore components near.the bottom. It is obvious that the total velocity pattern is extremely complex at this location (section I) and for the circulation patterns induced by the breakwater. Gourlay's research was performed between 1971 and 1972 in Australia. The results of some recent longshore current profile studies in Japan (Mizuguchi, Oshima, and Horikawa, 1978, in Japanese)” have been summarized by Kraus and Sasaki (1979). A 9-meter-long beach of plain concrete with a 1 to 10.4 slope was utilized. A propeller-type current meter measured the current at 15 to 20 points across the surf zone and averages were taken in the vertical direction. Wave period was held constant at 0.8 second, breaker heights were only 3 to 4.5 centimeters, and approach angles ranged from 4° to 15°. Maximum velocity recorded was 22 centimeters per second when the largest breaker angle was present. Kraus and Sasaki (1979) men- tioned that the current velocity profile variation in the alongshore direc- tion was monitored and ". . . no systematic acceleration or other signifi- cant anomaly in the alongshore direction was recorded.'' These researchers were obviously concerned about basin end-wall effects. It is not stated how nor over what alongshore distance this uniformity was obtained. A non- uniform profile created by basin end walls is not the only difficulty faced by laboratory researchers. Dre Laboratory Boundary and Scale Effects. Dalrymple, Eubanks, and Birkemeier (1977) studied the mean wave-induced circulations in enclosed basins. They organized the previous studies and basins employed into three categories: (a) Surf zone openings in both waveguide walls to permit recircula- tion outside the waveguide walls. The recirculation occurs behind and beneath the wave generator or through a pipe beneath the beach (used by Brebner and Kamphuis, 1963). The updrift current is enhanced by pumping to increase the length of the usable test section with uniform current profile (Kamphuis, 1977). SMIZUGUCHI, M., OSHIMA, Y., and HORIKAWA, K., “Laboratory Experiments of Longshore Currents," Proceedings, 25th Coastal Engineering Conference of Japan, 1978, in Japanese (not in bibliography). 36 igure 9. (b) (c) 40 B.P 20|. Velocities are in m/s -60 Distance from t.w.l. in metres 0.6 0.4 0.2 te} -0.2 a) Alongshore vertical velocity distribution at profile Elevation relative to SWL (mn) Distance from s.w.l. in metres 0.2 b) Onshore-offshore vertical velocity distribution at profile Vertical velocity distribution at profile section I for (a) long- shore current and (b) onshore-offshore current (from Gourlay, 1978). A completely enclosed wave basin (used by Putnam, Munk, and Traylor, 1949, and Saville, 1950). One surf zone opening in the downstream waveguide wall and clearance beneath the wave generator board to permit recir- culation (used by Galvin and Eagleson, 1965). 37 They conducted a theoretical analysis using radiation stress theory (Ch. 3) and the various types of boundary conditions for these categories. Although their theory neglected some of the terms in the equations of motion, their results did show that the mean circulation in the basin is strongly affected by the basin geometry. However, in the surf zone near the center of the basin, all three basin types gave similar theoretical results. Dalrymple, Eubanks, and Birkemeier (1977) therefore concluded that if a working recirculation procedure was devised, the type (a) basin would reduce the amount of return flow in the offshore region. The longshore current would be closer to that found on an infinite beach, since wave basin recirculating currents would not be included. The aim of recent research efforts at the Delft Technical University (Fluid Mechanics Laboratory), The Netherlands, is to develop such a test basin (Visser, 1980). The criteria for proper longshore current flows in -a laboratory basin are (a) a uniform current profile along the beach and (b) a zero slope of the MWL in the longshore direction. The width of the longshore current openings in both upstream and downstream waveguides and the recirculation flow rate must be adjusted empirically to determine an optimum combination. However, in a laboratory basin, MWL variation along- shore is difficult to measure. Consequently, the recirculation flow rate offshore and between the waveguides, Q , is used as an alternate criterion. In this method, the wave basin geometry (waveguide openings) and the pumped, recirculating flow rate, Q,;, are varied and a minimum Q_ found. It is then hypothesized that this Qy gives the correct longshore current flow rate, Q, for a uniform current profile. Lower Q_ value will cause the excess Q to return offshore and raise Q.. Conversely, higher Q- values will generate a surplus wave circulation flow and also increase ee The method was verified by a series of well-planned meticulous experi- ments in a 16.6- by 34-meter wave basin with 20.9 meters between the wave- guides. Regular waves are generated on a smooth concrete floor and 10:1 plain beach in 0.4-meter still water. The recirculation is through an 0.8-meter-diameter pipe beneath the beach to a pump on the updrift side. All tests were conducted with waves of 2-second period, 10.5 centimeters in height, and an incidence angle of about 21° at breaking. The depth- averaged mean velocity at a measuring point was calculated as the mean of the surface, bottom, plus twice the middepth velocity. The traveltime of a dye cloud over 0.8 meter was used to calculate velocity in the surf zone. To increase accuracy, two people made independent observations and a minimum of 20 independent readings were averaged to give one surf zone measurement at each depth. Also, the dye was injected at different wave phases during these readings to eliminate the influence of the orbital velocity on the measurement. These stringent procedures were relaxed in regions with slower currents and less turbulence. Flush-mounted piezo- meters on the beach and resistance wave probes measured MLW.. The 38 volumetric flow rate was computed between adjacent points from the mean ve- locity and MWL measured at these points. A maximum, depth-averaged longshore current of 0.67 meter per second was measured about midway in the surf zone. This optimization based on minimizing Q, proved successful. In addi- tion, use of a flow distribution system in the upstream waveguide wall that extended beyond the breaker line increased the region for uniform current profile upstream to section 3, as shown in Figure 10. For con- trast, results for essentially a type (a) basin are shown in Figure 11 as found by Visser (1980). Major differences in profile shape are evident at the midtank position (section 2). For the wave field-basin combination employed, the author concluded that a rational procedure to establish a uniform longshore current profile had been achieved. Thus it is now poss- ible to use other wave fields and basin geometries and the methods devel- oped to obtain uniform current profiles in the laboratory without lengthy trial-and-error efforts. Additional tests are presently underway. These studies also proved that considerable deviation in a uniform profile can occur in nonoptimized basin geometries with improper recirculation flows. Other effects due to scale are also present in laboratory models. These result from improper representations of viscous effects (Reynolds similarity) and surface tension (Weber number) influence on the air entrainment during wave breaking in the surf zone. Viscous effects are for both internal, turbulence decay, and at the boundaries from bed shear (friction) and wind surface shear stresses. For these reasons, Dette and Fuhrboter (1974) indicated surf zone research should be conducted only in the field. However, all the laboratory experiments described were general studies in which no prototype scale existed. These efforts tested the theories available at the time and at the scale of the laboratory inves- tigation conducted, and for this purpose they are entirely valid. The ability to create a steady-state environment in which longshore current measurements at a point in the surf are not dependent upon averaging time is a major advantage of physical model studies. Surprisingly, the use of such models to study longshore currents is not mentioned in a recent state- of-the-art report by Hudson, et al. (1979). Site specific model tests of longshore currents have been made (e.g., Gourlay, 1965/7; Curren and Chatham, 1980) but will not be reviewed in this report. Laboratory studies of: rip currents, which are strongly associated with forcing functions and mechanisms are discussed in the next section. 7GOURLAY, M.R., "Wave Generated Currents - Some Observations Made in Fixed Bed Hydraulic Models," Proceedings, Second Australian Conference Hydraulics and Fluid Mechanics, Auckland, December 1965 (not in bibliography) . 39 zero position wave board wave board wave crest wave guide 0.5 m/sec wave guide a - = toe of slope 1:10 e— 35 I/sec (adjusted) a — —= —— — — =! ——es 1 mecn wave set-up line mean breaker line distribution system Figure 10. Uniform, depth-averaged longshore current profiles and cir- culation velocities under optimum conditions of minimum cir- culations (Q. = 19.2 liters per second), adjusted return flows (Q; = 35 liters per second) and upstream distribution system (from Visser, 1980). zero position wave board wave board wave crest wave guide: id 0.5 m/sec i g seHessoococsescesoesce dh doseascos tee of slope 1:10 e a oom mecn wave set-up line mean breaker line Figure 11. Measurements in wave basin with only surf zone openings in both waveguides and free recirculation (no pump) (from Visser, 1980). 40 IIi. FORCING FUNCTIONS AND MECHANISMS A number of natural forces, geometric features, fluid properties, and sediment characteristics interact to create longshore currents, nearshore circulations, and rip currents. The fundamental cause-effect principles were first stated by Johnson (1919). Waves attacking the coast at oblique angles produced longshore currents while those arriving normal to the coast created rip currents. Thirty years later Shepard and Inman (1950) demon- strated how wave ray convergence and divergence patterns resulting from offshore bathymetry created regions of high and low breakers along the coast and explained nearshore circulation cells that result. Offshore return flows and rip currents occur where low breakers are found. The direction and period of approaching waves were also a factor in whether a longshore current system,-a nearshore circulation system, or a mixed system was observed. These conclusions on forcing functions (wave characteristics, hydraulic water surface gradients) and mechanisms (offshore and local bathy- metry) are still valid in explaining many coastal flows. A general classi- fication of nearshore current systems by Harris (1969) is shown in Figure 12. Oblique wave attack produces longshore currents on plain, straight beaches. Normal wave attack is accompanied by some mechanism that triggers rip currents and also causes water surface gradients, producing currents normal and alongshore, i.e., circulation cells separated by the rips. The asymmetric case corresponds to the mixed system where the offshore flows (rips) are found to meander or move along the coast with a net longshore current translating down the coast. Other forcing functions (tides, surface wind stress, atmospheric pressure) and mechanisms (wave interactions, boundary interactions) have been postulated to help explain the space and time variations observed in the field or laboratory. Continuous recording current meters at a fixed- point location in the surf indicate velocity variations due to many causes. The instantaneous velocity is due to wave orbital velocities, the rollers and eddies from wave breaking, bottom and rip current return flows, mass transport, tidal currents, wind-generated currents, currents due to mean water surface gradients (however induced), density-driven currents, bed shear-generated turbulence, and currents resulting from the excess momen- tum along the coast produced by wind waves breaking at an angle. All these currents can have components directed alongshore or onshore-offshore during some time interval. Some are random or periodic so that no net current results over a standard averaging period. The forces that create these currents are discussed in further detail below. Ihe Forces Causing Currents in Alongshore Direction. a. Wind-Generated Waves. Averaging water wave orbital motion over the wave period will result in a net longshore current under certain conditions. : (1) Momentum Thrust. Longuet-Higgins and Stewart (1962) set the principles by which gravity water waves, when integrated over the water column and averaged over the wave period, produce a net horizontal thrust (force) above the local hydrostatic force. A physical description can be found in Longuet-Higgins and Stewart (1964). The term "radiation 4 MN! y ate Fr aan | Ps eo at ~~ \ (x ae alas NAS / x i eld HE ie Se es ess SYM METRICAL CELLULAR \ \ ne Fionn dN ASYMMETRICAL CELLULAR (MEANDER) ALONGSHORE SYSTEM Figure 12. Nearshore current classification system (after Harris, 1969). 42 stress'' was employed to describe this wave-induced thrust even though the units were force per unit length and not a true stress. About the same time, Lundgren (1962, 1963) discussed these same principles as a wave- induced thrust and derived similar expressions which were later corrected (Danish Technical University, 1969)”. The term radiation stress is now generally accepted for this forcing function. It has significantly helped to unravel the mystery of how oblique wave attack can generate longshore currents that baffled scientists and engineers up to 1970. The general principles and equations associated with the radiation stress theory are summarized in Chapter 3. The local streamline curvature of rapidly varied, free-surface flows gives rise to vertical accelerations that also change the hydrostatic pressure distribution assumption. In 18/72, Boussinesq derived expressions to account for this effect in the horizontal equations of motion for nearly horizontal free-surface flows, and an additional term (or terms depending upon the order of accuracy of the derivation) that permits frequency disper- sion and wave propagation of permanent form. No time averaging is involved. Called the "Boussinesq Theory," it offers an alternative way to introduce the lateral momentum thrust due to gravity waves. (2) Mass Transport. Time-averaging nonlinear waveforms result in a net drift velocity because particle orbits are not closed. The current that results is very small and can be considered negligible in the surf zone and beyond the breakers. After breaking, mass transport by translation of hydraulic bores through the surf can not be neglected. The wave’ characteristics of interest are height, period, and angle of crest with the shoreline. The values at the first breaker line are reportedly the most significant. Surf zone transformations are discussed separately. 4 b. Tides. The tides create currents alongshore that are relatively small except near inlets. These currents have been omitted from this review. However, as mentioned by Sonu (1972), the slow water level changes induced by tides change the wave breaker location, hence the surf zone width, and ultimately affect the strength of wave-induced longshore currents. Also rip currents were found to be stronger at low tide and pulsated with a 25- to 50-second beat. The weaker rips at high tide fluctuated at the swell period. c. Wind Surface Shear. The wind moving across the water surface has a vertical velocity distribution that gives rise to a surface shear stress. As a result the water surface layers move at a speed of about 3 percent of the mean windspeed above the boundary layer. Viscous processes (both molecular and eddy type) transport this momentum down into the water column. In large water bodies, Coriolis forces will cause the current to deflect to the right of the wind direction (facing down- wind, Northern Hemisphere) and spiral away from the wind direction (Ekman motion). These classic deepwater oceanographic concepts can be 8DANISH TECHNICAL UNIVERSITY, ISVA, "Index to Reports," Report No. 20, ‘Dec. 1969, Lyngby, Denmark (not in bibliography). 43 misleading in the surf zone when winds blow overland from the shore, and shallow depths allow bottom boundary layers to interact with wave breaking turbulence at the surface. The importance of winds was mentioned by Shepard and Inman (1950). Field experimenters, however, find it difficult to differentiate effects of wind shear from short-period wind waves and the currents they generate. Linear multiregression analysis has been attempted to separate the relative importance of independent variables to generate the longshore current. Harrison and Krumbein (1964) found windspeed to rank ahead of offshore wave direction. If breaker angle was used instead, the result would be reversed which points out difficulties with regression studies if the wrong variables are employed. Sonu, McCloy, and McArthur (1967) found wind to be the second most important field variable influencing longshore current. Field tests of wind data should be recorded if com- parisons are to be made. A recent study by Nummedal and Finley (1978) evaluated whether or not the inclusion of additional physical parameters (other than wave field characteristics) could significantly improve current predictions. More than 250 observations were made over 1 year at Debidue Island beach, South Carolina, using the Littoral Environmental Observation (LEO) program. Four, widely varied equations for longshore current prediction were used to evaluate the linear combinations of wave and other physical parameters, including the longshore component of wind velocity. It was invariably found that this wind component explained most of the observed variance in the current velocity. Consequently, their statistical data analysis suggested that wind stress is an impor- tant factor in longshore current generation. Further strong evidence supporting this conclusion came when Dette and Fuhrboter (1974), in their North Sea experiments using EM current meters located at middepths, concluded that high longshore current velocities (up to 1.5 meters per second, means) could be expected during two different weather patterns: (1) Heavy storms with high breakers (>3 meters) and small wave breaking angles (<25°). (2) Winds blowing parallel to shore with low breakers (<1 meter) but large breaking angles (>25°). When the wind blew parallel to the shoreline and shifted from south to north, the mean longshore current changed almost at the same time to create mean currents in opposite directions of more than 1 meter per second. Between these patterns winds blew from land to sea, small waves were present and longshore currents were near zero. Windspeeds were 5 to 10 meters per second. Fox and Davis (1971, 1976) included the barometric pressure which causes the winds in their empirical analysis. They directly related local time variations (days) of barometric pressure to a longshore current velocity which varied in time (days) as well. The correlations 44 were based on experiments in Lake Michigan where local storm fronts control the wave heights and windspeeds. This work made no effort to separate the wave- or wind-generated currents. Fundamental studies on boundary layer profiles and wind stress in surf zones (e.g., Hsu, 1972) and wind-induced drift currents (e.g., Tang, et al., 1978) are available. Because of the significance of winds as a forcing function, it is surprising that few specific investiga- tions have been conducted. d. Planform and Bathymetry. The shape of the coastline, the slope of the beach face, the nearshore profile, bar formation, and offshore bathymetry all influence longshore currents. These geometric features primarily dictate the variation along the coast of the type, height, and location of breaking waves. These are the primary mechanisms for near- shore circulation and rip currents formation, as described in further detail in the section on rip currents. Beach-face slope and nearshore profile influence variations normal to the coast and can be used to distinguish two broad extremes of beach conditions. A reflective beach system is characterized by the typical profile shown in Figure 13(a) (after Wright, et al., 1979). Much of the incident wave energy is reflected from the beach face. Other distin- guishing features summarized by Wright, et al. include: (1) surging breakers with little setup, (2) well-developed beach cusps, and (3) the rare appearance of inshore circulation cells and rip currents. Shore- normal current spectra also have dominant peaks at incident wave periods (IT) or subharmonics (2T). Increasing breaker heights are accompanied by an increase in the strength of seaward flows which pulse at the subhar- monic period. This subharmonic resonance will also dominate longshore current oscillations. At the other extreme, a dissipative beach system has a wide surf zone with complex and varied topography (Fig. 13, b). One or more bars, three-dimensional features, and different scales of circulation cells and rips are frequently present. Subregions with contrasting turbulent mixing intensities can be present. Wright, et al. (1979) classified dissipative beaches into six basic subtypes. Each is dominated by a different combination of surf zone processes and by different scales and frequencies of resonant phenomena. The complexity of the nearshore planform and the bathymetry asso- ciated with dissipative beach systems is partly the reason for the wide number of hypotheses advanced as mechanisms for triggering rip currents and circulation cells. Zo Mechanisms Causing Nearshore Circulations and Rip Currents. These flows are treated together simply because they are physically separate parts of circulation cells that occur in the nearshore zone. 9TANG, F.L.W., et al., "Wind-induced Water Surface Set-up and Drift Currents," Proceedings, 16th Coastal Engineering Conference, Vol. I, Hamburg, 1978 (not in bibliography). 45 rs @ = a SF surging _breaker slow shoaling linear SteepsNe Noe 7 Se/ N Sir We OF beach face #38 MW L— (tan\B 125920) Nea Ginir! ghactirueret ya =lpeedciantn pete MLW - linear low-gradient nearshore profile (tan B = -012 - -02) a 4 100 150 200 250 Distance seaward (m) [Pose sensors ‘INSHORE Sek RE We Ny CON NEARSHORE Elevation relative to MWL (m) dissipative surf zone break storm rapid with setup : -2| concave upward in beach profile" (Gosition : aN ~ 93) (position a BO oo ax st concave ard fee 02 Oi} highly ide, hore profi heaehore profile variable) (tan 8 = -01) (tan f = -04) Elevation relative to MWL (m) 1 o- x 50 100 150 200 250 300 350 Distance seaward (m)) Figure 13. Beach classification system, showing (a) a reflective beach and (b) a dissipative beach (after Wright, et al., 1979). Why do rip currents form? This question has been the subject of much debate for 40 years. The summary of various mechanisms proposed to date is taken from an excellent state-of-the-art report by Dalrymple (1978); also, Miller (1977) provides excellent summary. This section concen- trates on the physics, although Dalrymple goes much further into the theory and equations supporting each model. Rip current spacing is a product of the theory. There is no single theoretical model that can account for the presence of rip currents. Rip current generation mechanisms are divided into two main categories: (a) wave interaction and (b) structural interaction. The difference is that wave interaction mechanisms can occur on plain, infinite 46 beaches, and the second category mechanisms cannot. Table 1, as developed by Dalrymple (1978), lists these two categories and the major subclasses of mechanisms as first proposed in the literature. Table 1. Proposed mechanisms for rip current generation and primary investigators since 1969 (after Dalrymple, 1978). Wave Interaction Models Primary Investigators Incident edge wave Synchronous Bowen (1969), Bowen and Inman (1969) Infragravity Sasaki (1975) Intersecting wave trains Dalrymple (1975) Wave-current interaction LeBlond and Tang (1974), Dalrymple and Lozano (1978) Structural Interaction Bottom topography Bowen (1969), Noda (1974) Coastal boundaries Breakwaters Liu and Mei (1976) Islands Mei and Angelides (1977) Barred coastlines Dalrymple, Dean, and Stern (1976) a. Structural Interaction. The subclasses listed in Table 1 reveal that these structural interaction models are closely associated with the planform and bathymetry features previously discussed. Structural interaction probably accounts for the majority of rip currents present on the world's coastlines. (1) Bottom Topography. Although Shepard and Inman (1950, 1951) correctly stated that rip currents could be related to variations in breaking wave height along the beach, it was Bowen (1969b) who explained why, using radiation stress principles. The excess momentum thrust of the waves normal to the beach produced a tilt or wave setup of the mean water surface from the breakers to the shore. Where the breakers were large the setup was large, and vice-versa. Longshore variations in breaker height created MWL gradients within the surf zone that produced . currents flowing from positions of highest breaker height to positions of lowest breakers. Here, the longshore currents converge and turn sea- ward as rip currents, i.e., taking the path of least resistance and main- taining a mass and momentum balance. Outside the breakers a weak return flow is needed to complete a nearshore circulation pattern very similar to observed cell circulation. The bottom contours offshore causing wave 47 refraction was offered as one mechanism for producing the longshore varia- tion in breaker height. (2) Coastal Boundaries. Natural headlands or manmade lateral barriers (breakwaters, jetties, groins) can also cause nearshore circula- tions. The wave fields are diffracted and reflected by these structures (some wave refractions also) causing wave breaking variations in complex patterns. Rubble-mound structures also absorb wave energy present. Figure 8 from Gourlay's (1978) laboratory study is a good example of a circulation pattern in the lee of a breakwater. (3) Barred Coastlines. From field observations and measurements in an area with irregular bottom topography at Seagrove, Florida, Sonu (1972) found that shoreward currents in a cell occurred over the shoals while rips were observed over the troughs (Fig. 2). Paradoxically, breaker heights were uniform down the coast so that the observed circula- tion must be due to another mechanism. Over the shoals spilling breakers were observed which continuously dissipated energy and created some setup in the surf zone. Waves entering the rip current areas broke by plunging, re-formed, and created little MWL change at the trough positions along the beach. Thus, this difference in wave breaking type produced mean surface gradients in the surf zone that were said to create the nearshore circula- tions and rip currents present. Sonu (1972) took actual measurements and used radiation stress theory to demonstrate theoretically that cell circulations can be created by these conditions, i.e., a barred coastline with seaward-directed troughs and no variation in breaker height. An additional forcing for the circulation is suggested to stem from wave reflection off the submerged bar (Dalrymple, 1978). Investigations listed in Table 1 and not mentioned here will be discussed in Chapter 3. b. Wave Interaction Models. The other major mechanism category in Table 1 is more subtle. Circulation cells can exist on long, straight, beaches with plain profiles as noted by Shepard and Inman (1950, 1951). Three subcategories are presented by Dalrymple to explain how this may occur. (1) Incident Edge Wave. This was the earliest mechanism pro- posed (Harris, 1967; Bowen, 1969) and first required an understanding of edge wave modes (Eckart, 1951)°~, standing edge waves, and proof that they actually existed on natural beaches. Waves propagating along the shore- line or standing waves along the shoreline are termed edge waves. Huntley and Bowen (1974) measured edge waves on the southern coast of England with a period of 10 seconds that was twice (subharmonic) the incident wave period. The theory states that incident swell waves can generate standing edge waves of the same period or the moving type (synchronous) of the same period on the beach. The linear superposition of these wave fields pro- duces alternately high and low breakers along the shoreline. A regular 10ECKART, C., "Surface Waves in Water of Variable Depth," The Theory of Edge Waves, Report No. 100, S10, Ref. 51-12 (not in bibliography). 48 pattern of equally spaced rip currents separating the circulation cells results from the same reasons as discussed for refraction. A series of controlled laboratory test results was reported by Bowen and Inman (1969) and offered as confirmation of the theory. Regular, normally incident waves were made on a smooth concrete beach of slope 0.075. A 7.3-meter working section bounded by vertical barriers which extended seaward gave well-defined boundary conditions for the experiments. A stand- ing edge wave was quickly formed in the basin. It was found for both swell- type waves, which did not break, and plunging-type breakers that the addi- tion of these incident waves and the standing edge wave at the breaker position gave a longshore variation in observed breaker height. The net height was greatest where incident and edge waves were in-phase and lowest where they were 180° out-of-phase. Thus, every other edge wave antinode produced small breakers and rip currents appeared at these positions along the coast as shown in Figure 14 (from Komar, 1976a). The key aspect of these experiments and theory was that rip currents could not form unless both incident and edge waves have the same period. Their efforts to con- firm the theory in the field at El Moreno Beach, Gulf of California, proved inconclusive. Synchronous edge waves could not be measured directly in the moving cell circulation system they observed (Bowen and Inman, 1969). Guza and Davis (1974)11 showed theoretically, and Guza and Inman (1975) and Harris (1967) showed experimentally, that the most likely resonant edge wave is the subharmonic edge wave. It does not allow a rip current to form. But, Dalrymple (1978) points out that this model first proposed by Bowen (1969a) is only for steep, reflective beach systems with no wave breaking. It also requires some means such as reflection of the incident waves by a structure or headlands to create the synchronous edge wave mode. As pointed out by Wright, et al. (1979), flat dissipative-type beaches have more circulation cells than the steep reflective-type. Tait (1970) and Tait and Inman (1969) showed that many of these beaches (east coast of the United States) have rip spacings far greater than the fundamental edge wavelength associated with the dominant, incident wavelength. Thus, there must be an additional mechanism that triggers longer rip spacing on flat dissipative-type beaches. For wide surf zones and dissipative-type beaches (e.g., Silver Strand Beach, California), Bowen and Inman (1969) suggested that some type of surf beat phenomena of the incident waves could create longer period edge waves to produce the greater rip spacings observed. Sasaki (1977) made an exten- sive summary of the literature on rip current spacing including many field experiments conducted on Japanese beaches. Using surf zone width, beach slope, breaker type, and other parameters of the surf zone together with a number of empirical, dimensionless plots of the data, Sasaki defined three domains where he felt a hydrodynamically different mechanism produced the rip currents. At one extreme were the steep, reflective beaches dominated by synchronous edge waves. At the opposite extreme, for wide, flat aiGuzAe R.T, and DAVIS, R.E., “Excitation of Edge Waves by Waves Incident on a Beach," Journal of Geophystcal Research, Vol. 79, No. 9, 1974, pp. 1285-1291 (not in bibliography). 49 uniform incoming wave “node edge wove | | Figure 14. Positioning of rip currents at alternate antinodes of standing edge waves on a plain beach with regular incident swell waves (from Komar, 1976). dissipative beaches, Sasaki (1977) proposed that infragravity waves 30-second to 55-minute periods) present in the surf zone trigger the rip currents with corresponding large spacings. In between, and in- stability domain was said to exist. This middle type of mechanism was classed as wave-current interaction by Dalrymple (1978). Huntley (1976a) and Sasaki and Horikawa (1978) reported field obser- vations of edge waves with periods in the infragravity domain that are a major component of low-frequency energy. It remained to show where the edge waves with large periods come from. Gallagher (1971) }2 developed a 50 theory which suggested that surf beating between particular pairs of incom- ing waves led to the resonant growth of long-period edge waves. However, wave breaking was not permitted in this model. To study resonant inter- action conditions in the presence of breaking waves, Bowen and Guza (1978) conducted a series of carefully designed laboratory tests. It was found that theoretical resonance can be produced in a wave basin with incident wave breaking present. When resonance conditions for edge wave growth from the theory were satisfied in the experiments, the response at the beat frequency was in the form of the theoretically predicted edge wave mode. It was concluded that these results suggest (strongly) that surf beat is predominantly an edge wave phenomenon. The reverse is also true to provide a mechanism for generation of long-period edge waves with resulting large rip spacings on flat dissipative beaches. (2) Intersecting Wave Trains. A second wave interaction mech- anism to produce rips is the intersection of incident wave trains with the same periods, but from different directions (Dalrymple, 1975; Dalrymple and Lanan, 1976). Large rip spacings are possible from the theory (no maximum limit). However, as pointed out by Dalrymple (1978), the amount of time natural conditions (separate distant storms, refrac-— tion, diffraction, local reflection) produce wave intersections at real beaches is unknown. Synchronous, progressive cross-waves can also be excited in a wave basin with a sloping beach as demonstrated by the lab- oratory experiments of Maruyama and Horikawa (1977). The cross-waves are shown to be component waves, with angles different from 90° to the wave maker, and not edge waves. Numerical simulations, agreed well with observed flow patterns which included rip currents. It was concluded that the interaction of the incident and cross-waves produced the longshore variation in breaker height causing nearshore circulations and rip currents. (3) Wave-Current Interaction. Finally, rip currents have been observed where there is no wave topography, wave wave, or wave structure interaction. There is no obvious external reason for a longshore variation in breaker height. Bowen and Inman (1969) stated this case clearly for perpendicular wave incidence on a plain noneroding beach: "In theory, the set-up at the beach can be in equilibrium with the momentum input of a uniform wave train over any width of beach, provided that the conditions are uniform in the longshore direction. However, as this condition is not observed in nature and not observed even in the laboratory unless the beach width is quite small, it has been suggested that the equilibrium might not be stable. That is, a small temporary disturbance might cause a com— plete breakdown of the two-dimensional equilibrium." (p. 5480) They then presumed that a small disturbance could excite edge waves in the region and their incident edge wave theory ensued. It is also conceivable 12 GALLAGHER, B., Generation of Surf Beat by Non-linear Wave Interactions, Journal of Fluid Mechanics, Vol. 47, No. 1, Sept. 1971, pp. 1-20 (not in bibliography). 51 that a purely hydrodynamic instability, i.e., an initial disturbance (or perturbation) of the proper wavelength, can extract potential energy from the setup regime and convert it into horizontal circulation patterns and rip currents. Thus, perhaps a better title for Dalrymple's mechanism (Table 1) is hydrodynamic instability theory. Hino (1974) was the first to postulate and use this mechanism to dev-— elop a theory of rip current spacing. Details of the theory are reviewed in Chapter 3. When the instability grew out of an allowed, movable bottom boundary, Hino found rip currents theoretically formed with an alongshore spacing about four surf zone widths apart. This was close to those obser- ved in nature. However, when instabilities were purely hydrodynamic, the spacings were far too low and the analysis untenable. lLeBlond and Tang (1974) using similar analysis procedures investigated the possibility that a wave-current feedback mechanism may cause a preferential spacing for rip currents (see also Iwata, 1976,.1978). But these models did not predict rip current spacings (Dalrymple and Lozano, 1978) unless an extra condition was introduced. LeBlond and Tang (1974) invoked the condition that rip currents will be found where the relative rate of energy dissipation is a minimum, i.e., a path of least resistance approach (Miller, 1977). Theo- retical rip spacings were still far too small. Mizaguchi (1976) intro- duced (rather arbitrarily) a longshore variation in bottom friction as the extra condition. Dalrymple and Lozano (1978) showed that if the extra effect from current refraction of incident waves by the outgoing rip currents is included, reasonable rip current spacings are predicted. Finally, Miller (1977) and Miller and Barcilon (1978) postulated that the dominant rip current instability occurs at those wave numbers for which a balance can be steadily maintained between the kinetic energy dissipated by friction, wave breaking, and the potential energy released. Their model will be discussed further in Chapter 3 as will the key, empirical assump- tions about the surf zone. In retropsect, many of these arguments and theories about mechanisms are of the "chicken or egg” variety and somewhat academic. As summarized by Komar (1976), the rip currents probably come first as generated by some mechanism and cause sediment transport to produce the local bathy- metry. (The analogy with popular "mechenisms" for winds to begin to generate surface water waves is appropriate.) These bottom irregularities probably maintain hydraulic control over the nearshore circulation and rip currents observed in today's field experiments. It is undoubtedly true that two or more factors could be present together and the unsteadi- ness of nature may prevent being absolutely sure. However. there is the need to sort out and understand the forced-type mechanisms (structural and wave intersection types) in relation to the free, intrinsic insta- bility type for purposes of correct numerical modeling of nearshore systems. (4) Other Factors. Fluid properties and sediment character- istics also play a role in wave interaction models. Water temperature changes its viscosity so that viscous shear effects are different on Alaskan versus Gulf of Mexico beaches. Density differences can create density currents. It is usually assumed homogenous fluids are present 2 at the coast. Also, density effects are generally ignored when comparing test results from the Great Lakes (freshwater) with ocean (saltwater) beaches. Surface tension plays a role when waves break, especially for large-scale differences as found on laboratory versus natural beaches. Turbulence intensity and resultant turbulent shear stresses must also be considered. When generated by wave breaking, the type of breaker and the scale of the observation (again, laboratory versus field) produce differ- ent internal turbulence intensities. Boundary shear-generated turbulence (oscillatory-type) can be significantly different over rigid versus mov- able beds. Movable bottoms form rippled or duned beds that locally gener- ate more vorticity. Sediment concentration distributions in the vertical suppress turbulent fluctuation intensities near the bed. As discussed in Chapter 3, both the bottom boundary resistance coefficient and lateral turbulent mixing parameter play significant roles in theories on longshore currents. They are both strongly influenced by the fluid properties. It would be a mistake to assume that sediment characteristics (compo- sition and weight, grain-size distribution, roundness, shape) are only of interest for sediment transport studies. They play a role in coastal currents that is yet to be fully understood or appreciated. For example, the weight and size of the grains on the beach can influence the size of ripples formed and the concentration distribution of suspended sediment. Both properties thus change the turbulence intensity present in the water column. If more wave energy is expended to keep sediment in suspension, less is available to generate currents. Also many studies have shown how grain size and beach slope are related (e.g., Bascom, 195113). The major differences between reflective- and dissipative-type beach systems and their slope dependence are discussed earlier. Size, distribution, and shape also affect beach porosity which. in turn, influences the extent of backwash in the swash zone (Kemp, 1975). Wave energy is absorbed in the surf zone by porosity effects as well as boundary resistance. Finally, perhaps the key factor is the wave breaking phenomenon. When, where and why do waves break? Because the physical understanding is so weak, empirically based information is heavily relied upon. The surf zone empirism that results plays a critical role in all theories of longshore currents and nearshore circulations. Consequently, wave break- ing processes are discussed in Chapter 3. IV. INSTRUMENTATION AND MEASUREMENTS A number of methods and problems have been discussed that relate to making current measurements in the field and laboratory. This section elaborates further on the instruments employed and on a number of measure- ment systems devised for surf zone applications. The section concentrates on the EM-type current meter and the so-called Syy gage for estimating momentum flux and wave direction. A number of anes instruments are used 13 Bascom, W.H., "The Relationship Between Sand Size and Beach Face Slope," Trans. Am. Geophys. Unton, Vol. 32, 1951, pp. 866-874 (not in bibliography). a3 in the field to measure water surface variations, bottom pressures, winds, etc., but were felt outside the intended scope of this review (e.g., see Horikawa, 1978a; Gable, 1979). An additional method to develop steady uniform profile longshore currents in the laboratory is also discussed. Ihe Velocity Instruments. The section relies primarily on the published literature of researchers making field measurements in the surf zone; useful information was also obtained from Teleki, Musialowski, and Prins (1976) and Woodward, Mooers, and Jensen (1978). The most recent source is the ongoing: Nearshore Sediment Transport Study (NSTS) and publications from it (e.g., Gable, 1979). a. Lagrangian. Before 1968, currents were measured by timing the travel distance of dye, surface floats, or drogues. The main advantage is the simplicity of use. A mean current is obtained representative of the distance and traveltime used in the measurement. Most early researchers fail to mention these values in their publications. Surface floats give surface readings. Dye and drogues (weighted floats) give some type of depth-averaged value. The primary advantage of these methods is that when a series of photos are taken from overhead, nearshore circulation patterns, rip currents, and mean velocities (magnitude and direction) can be obtained. Such measurement systems are described in further detail below. The Japanese (Horikawa and Sasaki, 1972; Sasaki, 1977) have continued the use of drogues as part of their tethered float technique (Sasaki, Igarashi, and Harikai, 1980). They began with polyurethane foam boxes (33 by 25 centimeters) as floats and evolved to the 20-centimeter cube drogue (Fig. 15). This size was found to be a minimum for viewing on ordin- ary color photographic film at 200- to 500-meter altitudes. Surfboarding can be a major problem with the drogue configuration when large spilling breakers are present (Sasaki, 1977). Larger and heavier drogues can be used but are not as readily nor swiftly deployed. Thus current magnitudes and directions are very questionable in the breakers but assumed to give reasonable estimates of depth-averaged longshore velocities and rip flows at other locations (Sasaki, 1977). Dye also continues to be used. Breaking waves travel rapidly across the surf zone to quickly disperse a spot of dye in the onshore-offshore direction until it extends completely across the surf zone. The less intense longshore diffusion then spreads the patch laterally as it moves with the longshore currents (Bowen: and Inman, 1974). What group of dyed water particles are used to measure longshore currents? Most researchers simply state that movement of the resulting dye-patch was measured and timed from shore. Clearly, the location of the dye-patch center or centroid is somewhat subjective and leads to errors. At what depth and when in the wave cycle the dye is released are also important factors. Laboratory researchers at the Delft Technical University (Visser, 1980) released the dye at three depths (repeated twice in middle) to obtain a depth-averaged value. They also released the dye at different phases in the wave cycle and made 20 independent measurements to arrive at one mean value. Such careful procedures would help to increase field accuracy but can be applied only under steady-state field conditions. 54 Polyurethane foam coated by PVC sheet (colored orange ) Drain hole ABS resin Figure 15. Schematic of drogue used by Japanese (from Sasaki, 1977). Further advantages and SGN eile oe of dye and float methods are discussed by Galvin and Savage (1966) 14, b. Eulerian. Propeller-type current meters were first used in the surf zone by Sonu (1969b). These bidirectional meters used a three-bladed impeller mounted within a duct. Paired reed switches sensed both rotation speed and flow direction through magnetic coupling in the blades. Thres- hold velocity was 5 centimeters per second. The meter's response is non- linear near the threshold speed due to impeller inertia and thus not truly bidirectional. This type of meter is also phase-dependent in that there is always a real timelag between forcing function and meter response. In CERC's development of the Towed Oceanographic Data Acquisi-— tion System (TODAS), the ducted impeller type (mechanical sensor) was chosen over acoustic EM and force meters (Teleki, Musialowski, and Prins, 1976). The reasons for this choice were the simple but rugged construc-— tion, zero drift, and antifouling characteristics (suspended sediments, . seaweed etc.) of the propeller type (Teleki, Musialowski, and Prins. OWS)e A 4-inch duct (8 inches long) directed the flow through a five-bladed impeller. The measurement range was 0 to 2.6 meters per second (0 to 5 knots) with a threshold speed of 2 centimeters per second. They found 14 GALVIN, C.J., and SAVAGE, R.P., “Longshore Currents at Nags Head, North Carolina,” Bulletin No. 2, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Washington, D.C., 1966 (not in bibliography ). 55 considerable phase lag and much scatter depending on flow direction for 10- to 20-second swell. It would seem that this type is also readily fouled by kelp or seaweed. Electromagnetic (EM) current meters are based on inductance prin- ciples. They are currently the most popular type since first being introduced for surf zone measurements by Thornton (1969). A magnetic field (a.c.) is created by an electromagnet within the probe. Electrodes on the probe measure the voltage induced by the conductor (moving seawater) to generate the output signal. A popular brand is the two- axis, Marsh-McBirney EM current meter probe and electronics package. The Model 512 OEM has a 4-centimeter-diameter spherical probe with four middiameter electrodes, 90° apart and protruding 0.5 centimeter. The EM meter is rugged, has good cosine response, a wide frequency band width, and offers less chance for fouling than the propeller-type meter. During recent extensive use in the NSTS experiments at Torrey Pines, California (Gable, 1979), 22 EM meters withstood forces large enough to bend one support rod but not damage the probe. Entrapment of kelp and seaweed will alter the calibration, so EM probes must be frequently inspected and cleared of debris and marine fouling attachments. Calibration problems have recently been reported for the EM probe under both steady-state and oscillatory~-flow regimes. LaVelle, et al. (1978)15 showed the meter to have a transition (change in slope) in steady flow (d.c. gain) at about 80 centimeters per second. This could possibly be due to nonlinear boundary layer effects including flow sepa- ration as suggested by Cunningham, Guza, and Lowe (1979) and also by Guza and Thornton (1980). Aubrey (NSTS Workshop, Scripps Institute of Oceanography, La Jolla, California, personal communication, February, 1981) found.a break in the steady-flow calibration at 60 centimeters per second and attributed it to the electrodes jutting out from the sphere to disturb the boundary layer. This meter has a nominal output d.c. gain of 1 volt at 1 meter per second and calibration is usually made by towing in still water at this speed. Calibration of the 22 meters used in the NSTS experiment at Torrey Pines, California, made both before and after the experiment, showed less than 5 percent variation in d.c. gain for most meters (Gable, 1979). But such meters deployed in the surf zone are subject to oscillatory motions over a wide range of frequencies and should be dynamically cali- brated. For this purpose, the Shore Processes Laboratory of the Scripps Institute of Oceanography at La Jolla, California, has recently developed a hydraulically driven mechanical calibration device. A detailed description of the equipment can be found in Cunningham, Guza, and Lowe (1979). A horizontal arm with rack gear is driven back and forth by a hydraulic servo-controlled motor. The probe is rigidly posi- tioned beneath the arm near its midpoint and kept about 20 centimeters below the water surface. Various types of arm motions were used to study 1 3 ; F LAVELLE, J.W., et al., "Near Bottom Sediment Concentration and Fluid Velocity Measurements on the Inner Continental Shelf, New York," Journal of Geophystcal Research, Vol. 82, No. C12, 1979 (not in bibliography). 56 the dynamic response, including single-frequency sinusoids, broad band random noise, and asymetrical (ramp) functions, all with various amplitude- frequency combinations. No net flow rate was present in the basin during the tests. The results were somewhat disconcerting as shown in Figure 16 (from Cunningham, Guza, and Lowe (1979). For the broad band response, the gain was found to be dependent upon both the frequency present and the spectral content. A random-noise generator running in three different modes was employed (curves I, II, and III) to vary the spectrums (Fig. 16). The black dots in the figure show results for the single-frequency tests where it was learned that the gain was also amplitude dependent. Recognizing these limitations and based on further experience gleaned from the Santa Barbara experiments for NSTS, Guza (Scripps Institute of Oceanography, La Jolla, California, personal communication, February 1981 stated that the uncertainty associated when using a single gain factor (d.c.) for all frequencies is roughly +10 percent or +5 centimeters per second, whichever is larger. Air bubbles in the water did not seem to alter this conclusion as stated by E.B. Thornton (Scripps Institute of Oceanography, La Jolla, California, personal communications, February 1981). Further research is needed to improve this accuracy, especially if velocity spectra are of interest. @ Single Frequency Pris. X DC Calibration (towed) 0 25 50 ac) 1.0 1.25 1.5 1.75 2.0 2.25 2.5 FREQUENCY, Hz Figure 16. Gain response of flow meters for pseudorandom noise runs, single-frequency runs, and towed (d.c.) values (after Cunningham, Guza, and Lowe, 1979). Because of these difficulties other meters for continuously record- ing current components in the surf zone are being developed and tested. These included the use of radar to track reflectors, doppler radar, and acoustic current meters (e.g., see Appell, 1978) such as the Neil Brown 57 system. These meters are not without their own problems. For example, the acoustic type is very sensitive to air bubbles in the water column to make it relatively intractable in the surf zone without further develop- ment (van der Graff, NSTS Workshop, Scripps Institute of Oceanography, La Jolla, California, personal communication, February 1981). All these types have not been extensively used in field experiments of surf zone currents, to date. Articles by Kalaitis (1975), Appell (1977). and Mero and Appell (1977) provide further discussions of dynamic calibration tests. Turbulence effects are discussed by Bivins and Appell (1976). Other design shapes are also being developed for the EM type (Crump, 1976; Aubrey, personal communication, February 1981) that may lead to more uniform frequency response characteristics. Dae Measurement Systems. It can be reasonably argued that the scale of the phenomena observed (or perceived) at the coast depends on the observation method. Sasaki (1972) divided the coastal scale into (a) the spacing or rip currents, i.e. a unit cell scale, and (b) the velocity field within a cell. He then made (somewhat subjective) tables of desirable features and com- parisons of observation methods for both scales. Color aerial photos taken at low tide were judged best for the larger scale phenomena. Horikawa and Sasaki (1972) also developed two measurement systems for taking successive overhead photos of floats to quantitatively describe the nearshore currents and rip currents at the smaller scales. One system, SHIELS, used two hovering helicopters to take successive stereo photos but was very expensive. The second, simpler and far cheaper method, BACS, uses a single balloon-borne, motor-driven camera but wave height determinations are impossible. This system is still used today by the Japanese. Figure 17 schematically illustrates the BACS system. The pier is used to tether the balloon and for workers to readily cross the surf zone while tossing the drogues. Wind drift of the balloon between pictures contributes measurement errors and limits use to calm days. They normally photograph rip currents usually expected near piers. This could be a valuable research tool but needs further development. Nonsurfboarding drogue shapes, tethering without piers, and electrical systems to precisely locate the camera at all times for position corrections must be developed. An excellent discussion of the use of these remote sensing systems can also be found in Sasaki (1977). Sled systems have been used extensively in the surf zone. They can be instrumented heavily with wave staffs, pressure sensors, current meters, sediment concentration devices, etc. Also, they can be winched out into deeper water beyond the breakers and they follow the bottom contours. One of the first sleds developed is described by Lowe, Inman, and Brush (1972)16, Others that followed are reported by Teleki, Musialowski, and 1 6 LOWE, R,L., INMAN, D.L., and BRUSH, B.M., "Simultaneous Data System for Instrumenting the Shelf," Proceedings, 13th Coastal Engineering Conference, ASCE, Vol. I., 1972, pp. 95-112 (not in bibliography). 58 BALLOON APPROX. CAMERA 200M. RELEASE CORD PIER f>. BEACH \V44 we ay, ‘ eet aa eee OAT Si BOAT ap tn Seah ° Re ci : er ‘ OS 5 ! 2 ans SURF ZONE ‘ t) ‘ Cd é x O -- rt ld Figure 17. Schematic of BACS nearshore current system (from Sasaki, 1977). Prins (1975, 1976), Coakley, et al. (1978), Bradshaw, et al. (1978), and Allender, et al. (1978). The newest U.S. sled has been developed for simultaneous bottom profiling, wave height, and two-axis current measure-— ments with one EM meter (Sallenger, et al., 1980). Field tests of the latter device at Monterey, California, in 6-meter breakers over a 100- meter wide surf zone measured bottom current greater than 3.5 meters per second (Sallenger, et al., 1980). The major problem with use of sleds is data interpretation. Some type of temporal stationarity of the currents must be assumed (Guza and Thornton, 1978, p. 768). If not continually moving, how long should the sled remain at each station to record information? Figure 6 showed considerable tem- ' poral variation between successive 17.l-minute ((1,024 seconds) averages at a plain beach for each gage location. If 17.1-minute records are taken at each station, it could be several hours between data taken at the outer and inner reaches of the surf zone. Can this data be combined to yield an instantaneous profile of the 17.1-minute mean longshore current? To eliminate the need to make arbitrary assumptions regarding temporal stationarity, NSTS researchers choose to employ a large number of fixed 59 current meters in perpendicular arrays as shown in Figure 18 for the Torrey Pines Experiment. Twenty-two EM current meters were alined normal and along the beach with some bunching up in the surf zone. A similar configuration was established for the Santa Barbara experiments where a total of 25 EM current meters were employed. These alined arrays proved to be very successful in obtaining instantaneous and simultaneous longshore and cross- shore velocity time histories through and beyond the entire surf zone and down the beach near the midsurf position (Guza and Thornton, NSTS Workshop, Scripps Institute of Oceanography, La Jolla, California, personal communica- tion, February 1981). All the raw data from both_the Torrey Pines (Gable, 1979) and Santa Barbara, California (Gable, 1981!7) experiments were made available on magnetic tape. Data were recorded at 64 samples per second, then later low pass-filtered and reduced to 2 samples per second. These then became the raw data tapes. They can be obtained for nominal cost from the National Oceanic Data Center (NODC) in Washington, D.C. Some of the early results will be discussed in Chapter 4. These data sets form an extensive and excellent data source for future analysis. Unfortunately, however, no EM current meters were alined in the ver- tical for either of the NSTS field experiments. Plans to do so at Santa Barbara were scrapped due to the storm conditions encountered. Thus it was not possible to observe the vertical current structure nor know if circulations about horizontal axes were present. 3 Wave Direction. Wave direction is an important parameter in surf zone current genera- tion. It is also difficult to determine for irregular wave trains striking the coast. Linear arrays of pressure sensors have been quite successful in measuring directional wave data at the coast (e.g., Pawka, 1974). Other devices have been proposed (Hallermeier and James, 1974). The most prac- tical and successful for measuring wave direction is the slope array, as developed by the Scripps Institute of Oceanography (Seymour, 1977; Seymour and Higgins, 1977; Higgins, Seymour, and Pawka, in preparation, 1981). Besides wave direction, pressure measurements from the device can also be used to calculate the longshore component of shoreward-directed momentum flux (S,.) due to the waves. The device is also called an Srey meter. As such it provides a valuable tool for comparison with theories which use sea-surface variation and wave angle to calculate Sxy- The slope array device is very simple. Four pressure transducers are mounted at the corners of a square frame, 6 meters on a side. The orientation of the sides on the bottom of the frame is determined rela- tive to a reference longshore direction. Coordinate rotation yields the desired surface slope components from any array alinement. The real part of the cross-spectrum of the sea-surface slope components in the longshore and cross-shore directions is then used to compute a significant direction 17 ; GABLE, C.G., Report on Data from the NSTS Experiment at Ledbetter Beach, Santa Barbara, California, Jan-Feb., 1980,"' IMR Ref. No. 80-5, Scripps Institute of Oceanography, Mail Code-A022, La Jolla, Calif., 1981 (not in bibliography). 60 Ww “Z39NVLSIO 3YOHSII0 o b oie § Oo —% ; uJ oO LUE 8 a ues 6 S06e (ISA ©} eanops W) H1d30 - 7.00 61 8 a 4 : g- ° TEEN ER ERE ae REE 40 : 120 200 LONGSHORE DISTANCE, m -40 (A) ancaowe ree @ TRANseT TER —— 3-4 MEAN SEA LEVEL PCM ENCODER STATION % CURRENT AND SUSPEROEO SEDIMENT STATION © PRESSURE (0.0) © CURRENT (22) N WAVESTAFE (7) ® SURVEY RANGE LINES Location of EM current meters and other instruments in NSTS, Torrey Pines, California,experiment (from Gable, 1978). Figure 18. and the spectrum of the momentum flux components, Sxy- Details are beyond the intended scope of this review. The calculation method stems from Longuet-Higgins, Cartwright, and Smith (1963)!8 and complete details can be found in Higgins, Seymour, and Pawka, in preparation, 1981.. They also made laboratory tests with irregular waves and field comparisons with a large linear array of five pressure sensors. Comparisons were judged to be excellent and quite good, respectively. Use of the slope array device during the NSTS experiments showed its sensitivity to alinement errors along the coast relative to wave direc- tionality. At Torrey Pines, California, alinement inaccuracy with the straight coastline was judged to be about 3° and greater than the wave directions observed (Seymour, 1981, NSTS Workshop, Scripps Institute of Oceanography, La Jolla, California, personal communication, February 1981). Consequently, a linear array of five pressure sensors more than 360 meters long was employed to measure the directional spectrum. The pro- nounced variability in bottom topography and large incident wave angles due to narrow wave windows present at Santa Barbara, California, made use of the slope array device ideal for this location. 4. Laboratory Systems. There is another way to create a uniform longshore current profile (i.e., infinite beach) in the laboratory other than the method developed at the Delft Technical University (Visser, 1980). End effects are elimina- ted, and a zero mean slope in the longshore direction results. The tech- nique, described by Dalrymple and Dean (1972), involves a spiral wavemaker generating waves in the center of a circular basin. A vertical right- circular cylinder oscillates in a small circle about its vertical axis to generate the waves. Wave crests move out in an Archimedian-type spiral and crests impinge on the circular beach everywhere at the same angle of incidence. A theory for the amplitude of generated waves based on shallow- water theory was confirmed in the laboratory. Dalrymple and Dean (1972) conducted littoral drift experiments with the device, and they recommend it for further sand transport studies. However, scale effects preclude obtaining quantitative results for this purpose. It could be an excellent tool, however, for confirmation of longshore current theory. V. SUMMARY More than 60 years of observations and measurements of longshore currents, nearshore circulations, and rip currents worldwide has been briefly reviewed. Magnitudes and directions of these currents depend upon the following factors: 1 | ; 8 | ONGUET-HIGGINS , M.S., CARTWRIGHT, D.E., and SMITH, N.D., "Observa- tions of the Directional Spectrum of Sea Waves Using the Motions of a Floating Buoy," Ocean Waves Spectra, Conference Proceedings, Prentice- Hall, Englewood Cliffs, N.J., 1963 (not in bibliography). 62 (a) Wave characteristics: Evaluation point (usually the breaker line) Regular or irregular forms Height Direction (number of incident wave trains) Period Type of breaker (spilling, plunging. or surging) Wave energy transfer to other scales (nonlinear effects) Soliton formation (harmonic scales) Subharmonics (longer period scales) Breaker location Presence of edge waves or infragravity waves Wave-current interactions Surf zone wave energy decay (b) Wind characteristics: Strength Direction Duration Atmospheric pressure gradients (c) Tidal influence Mean water depths Width of surf zone (d) Planform and bathymetry Coastal boundaries (natural headlands, jetties, breakwaters) Coastal planform Offshore bathymetry (refraction, diffraction) Local bathymetry (bars, troughs) Nearshore profile Beach face slope Reflections Bottom roughness (smooth, ripples, dunes) (e) Fluid Gravity (density, saltwater, freshwater) Temperature (viscosity) Surface tension (air entrainment) Turbulence intensities (wave breaking, bottom shear, sediment effects) (£) Sediment Size, gradation, shape, unit weight Porosity, permeability Current variations in space and time result from the interplay of many of these features plus the fact that most are unsteady in nature. Many simplifications and assumptions are necessary to evolve theories from all these factors. 63 CHAPTER 3 THEORETICAL DESCRIPTION I. HISTORICAL SUMMARY alt Before 1967. All the relevant theories proposed to predict longshore current velocity before 1967 have been thoroughly reviewed by Galvin (1967). At least 12 different equations existed and were derived by the following methods: (a) Continuity of water mass, (b) energy flux, (c) momentum flux, and (d) empirical correlations. Galvin (1967) concluded that all were oversimplified models and the empirical methods lacked sufficient data. For example, the Inman-Quinn (1951) theory used momentum conservation principles modified by empirical data. It was shown to be based on an untenable assumption, supported by inappropriate data and found to be only a fair predictor for just one of the three data sets then available. No adequate theory was felt to exist at that time (Galvin, 1967). A summary of most of these longshore current formulas is included as Ap- pendix C (from Thornton, 1969). It lists an additional empirical equation based on multiregression analysis as found by Harrison (1968) at Virginia Beach, Virginia. Such equations are only roughly tenable for the location and range of variables observed. Equations based on the conservation laws of physics are much preferred for general usage. Additional analysis and discus- sion of the theories before 1967 can be found in Horikawa (1978a) who includes some Japanese formulas not previously referenced. The formula of Shadrin (1961), based on measurements at a barred coastline of the Black Sea (Anapa coast), is also discussed by Horikawa (1978). There are also recent original summaries by Komar (1976) and Gourlay (1978) of most theoretical attempts befor 1967. In retrospect, all these theories were doomed to failure for a number of reasons. Those based solely on mass conservation and kinematics omitted the crucial dynamics and interplay among forces and fluid accelerations. The amount of energy dissipation in the surf zone is a significant percentage of the total available and difficult to estimate. Energy flux theories rely on that small fraction remaining which represents a second-order phenomenon (Galvin, 1967). Momentum conservation principles were simply not applied pro- perly and neglected the influence of the streamline curvature present in short waves on the vertical pressure distribution. They all predicted one mean longshore current velocity for the entire surf zone width. They neglected important factors such as breaker type, beach profile, and bottom roughness. They were all for longshore current estimates and no theories to handle near- shore circulations and rip currents existed at that time. 64 Die Modern Approaches. Since 1969, two fundamentally different yet related theoretical ap- proaches to predict coastal hydrodynamics have emerged. Both rest on solid physical laws of mass (continuity) and momentum (motion) conservation that form the basis of Newtonian fluid mechanics. Most importantly, both assume the velocity profile to be uniform over the water depth for all points in the flow. No flow direction reversals with depth can be resolved (Fig. 9,b) so that circulations about horizontal axes are impossible. a. Radiation Stress Theory. The equation of continuity and two horizon- tal momentum equations are depth-integrated and time-averaged to account for the excess lateral momentum thrust present in wind waves on the coast. Local net velocity components and the average MWL are the dependent variables of interest. The theory is now more than 12 years old and has undergone consid- erable development and refinement. It requires a priori specification of wave height fields throughout the area of interest. This is usually accomplished by normal wave refraction procedures (Snell's Law), combined with diffraction, reflection, and wave-current interaction estimates, as required. Most of this chapter (Secs. IL to VI) is devoted to a thorough review of the extensive literature that has been published since 1969 on this subject. A key aspect is the time-averaging process inherent in the theory so that this could be labeled a time-average theory. For simple geometries, analytical (closed form) solutions are possible. b. Boussinesd Theory. Based on ideas that go back much further (Boussinesq, 1872)*", the vertical acceleration and streamline curvature effects in wind waves give rise to lateral momentum fluxes. They appear as additional, mixed and higher derivative terms in the horizontal, long wave momentum equations that are depth-averaged but not time averaged. Local instantaneous velocity components and the instantaneous water surface fluctuations are the dependent variables of interest. The first engineering applications to coastal swell wave propagation in two dimensions appeared in 1978, and only outside the breaker zone. Considerable research and development work remains to use the method for calculation of coastal currents, circulations, and rip currents. No additional wave field specifications are required since the waves propagate in space and time as a fundamental part of the solution. The limited amount of published information on this method is reviewed in Section VII. Numerical integration methods are required to obtain solutions. For further insight into the physics and mathematics surrounding waves in the coastal zone, Lundgren (1976) is recommended. He refers to the Boussinesq approach as T-Methods (time-step methods). 19 BOUSSINESQ, M.J., "Theory of Waves and Surges Which Propagate the Length of a Horizontal Rectangular Canal, Imparting to the Fluid Contained Within the Canal Velocities That Are Sensibly the Sare from the Top to the Bottom," Journal of Pure and Applied Mathématics, Vol. 17 (2nd Series), Feb. 1872. For English translation, see VASTANO, A.C.J., amd MUNGALL, J.C.H., Refer- ence /76-2-T, Dept. of Oceanography, Texas A&M University, College Station, Tex., Mar. 1976 (not in bibliography). 65 II. PRINCIPLES OF RADIATION STRESS In a series of papers beginning in 1960, Longuet-Higgins and Stewart (1962, 1964) laid down the principles and gave the physical meaning behind the concepts of radiation stress for water waves. About the same time, Lundgren (1962, 1963)29 described how water waves can exert an extra wave thrust in the horizontal direction. The term radiation stress is borrowed from EM wave theory where a radiation pressure impinges on a surface. It is actually a misnomer in that depth integration gives a wave-induced excess pressure force per unit length (in excess of the hydrostatic pressure force) and not a true stress (force per unit area). However, the term stress implies a directional quantity which is true for this wave-induced thrust and radiation stress is now an accepted term in the literature. Transformations applicable to true stresses can be used. As will be shown later in this chapter, these radiation stresses resulting from time-averaging gravity wave orbital motions are also completely analogous to Reynolds stresses resulting from time-averaging turbulent flow motions. q The radiation stress principle has been used to develop theoretical, ana- lytic expressions for the following coastal phenomena: (a) Wave setdown and setup (Sec. IIT) (b) Uniform longshore current profiles (Sec. IV) (c) Nonuniform longshore current profiles (Sec. IV) (d) Nearshore circulation systems and rip currents (Sec. V) In addition, it is the basis for numerical integration schemes over variable bathymetry and boundaries (jetties, breakwaters) where analytic solutions are not available (Sec. V). Finally, principles of radiation stress theory have been extended to nonlinear and irregular waves as discussed in Section V1. ibe Progressive Waves in Uniform Water Depths. a. Radiation Stress Components. The summary that follows is essentially that from the work of Longuet-Higgins and Stewart (1964). A monochromatic wave is shown propagating in the X-direction in Figure 19. A relatively standard set of nomenclature is adopted and defined at the beginning of this report. (1) Normal Stress. The component Syx of the radiation stress along a wave ray is defined as the time-average value of the total flux of horizontal Momentum across a vertical plane minus the stillwater hydrostatic pressure force. Thus n O Syy = (ptpu2)dz - pidz (1) =d -d 20See some corrections in Danish Technical University, ISVA, Index to Reports, Rept. No. 20, Dec. 1969, Lyngby, Denmark (not in bibliography). 66 Syy ey Sxx +X Syy Shoreline yo Bottom z=-d Figure 19. Definitions and nomenclature for radiation stress components. where the overbar indicates a time-averaged mean. For convenience in further evaluation, Syy is rearranged and divided into three parts as follows: n 5 fo) n = a ) Syx putdz + (p Pp, dz + pdz @ -—d -d fo) or a CU) rue) aC), Sie = She t Sige he Sey (3) The assumptions now begin. In the first term gold. the integration above the stillwater level to z = n produces a third-order term (proportional to a3) and is neglected. S x is generally positive and can be considered a Reynolds stress apeeeeaveeel from the bottom to the stillwater level. (e) sD = | zie (4) = 67 (2) The second term Syx can be shown equal to ° (@) 2 2 Syy = | -pw-dz (5) -d « Longuet-Higgins and Stewart (1964) considered the time mean flux of vertical momentum across a horizontal plane balanced by the weight of water above that plane. This is equivalent to integrating and time averaging the vertical mo- mentum equation over the stillwater column. This term is obviously generally negative. The third term S poses some difficulties when n is below z = 0. To get around this, gaeaey Aeedine and Stewart (1964) assumed the pressure to fluctuate in-phase with the surface elevation, i.e., a hydrostatic distribution for pressure, p. This gave Sey) = Zpan? (6) which is also generally positive. Combining gives fo) Syy = + 0 (u2-w2)dz + Span? (7) -d for the principle radiation stress. Equation (7) is completely general. Further simplification depends upon what classical wave theory is used for the three variables involved, n, u, and w and the averaging time employed. How- ever, in deep water the particle orbits are almost circular so that the inte- gral term approaches zero. Conversely, in shallow water u>>w so that the hori- zontal velocity stress dominates the integral term. (2) Transverse Stress Component. Analogous to equation (1) the transverse radiation stress, Syy> is defined n ) = 2 e Sing (ptpu*)dz Pp dz (8) -d -d where v is the wave orbital velocity in the YY-direction parallel to the wave crests. For long-crested gravity waves, vy equals zero. The transverse stress Syy thus becomes ) IL Syy ow-dz + 5p en (9) —l if the same assumptions are the same as for Syx: 68 (3) Shear Component. Since v is zero for long-crested waves, the shear component resulting from the cross products uv is identically zero everywhere, therefore n Syy = puvdz = 0 (10) -d for wave propagation in the X-direction. Consequently, S and S$ are prin- cipal stresses. It must be emphasized that S and S$ are horizontal forces per unit width acting normal to and parallel es the wave crests, respectively (see Fig. 19). Subsequently these principal radiation stresses will be trans- formed to a more convenient coordinate system oriented in the alongshore (y- coordinate) and shore-normal (x-coordinate) directions. In general, for waves making some angle, a with the shoreline, Sxy is not identically zero in this transformed coordinate system (Fig. 19). For additional physical interpreta- tions of these stress components see Svendsen and Jonsson (1976). b. Components Using Linear Wave Theory. For small-amplitude waves of sinusoidal forn, nN = a cos(kx-wt) (LiL) where _ # , _ 20 _ 20 a> De k = eee, ae 22 and the particle orbital velocities u and w are defined in the usual manner“. Inserting this definition and equations for n, u and w into equation (7) for S and equation (9) for S|, using the linear theory dispersion relation w? = at eentin. performing the integration, and time-averaging over one wave period yields p Dice Meal Sxona” s8el 2 ie) eal | Be fe Sant oe where E is the usual total energy density of the waves given by 21SVENDSEN, I.A., and JONSSON, 1.G., "Hydrodynamics of Coastal Regions," Den Private Ingenirfond, Technical University of Denmark, Lyngby, 1976 (not in 225 ibliography). See, e.g., WIEGEL, R.L., Oceanographical Engineering, Prentice-Hall, Inc,, 1964, Englewood Cliffs, N.J., p. 15, for equations when equation (11) is in terms of the sine function (not in bibliography). = 69 1 1 E = 5pga° = ge gH* (14) for linear theory. Use of the ratio n between group celerity, c_, and wave celerity, c, defined as 8 c ge Bgl __2kd_ ie c 2 io sinhkd (15) gives the commonly found forms ep) i] E(2n - 5) (16) Syy Ga. >) (17) where the units of radiation stresses as force per unit length are now obvious. In deep water, kd>>1 and n +s so that iL Bap = 5 E (18a) Syy = 0 (18b ) In shallow water, kd<<1 and n > 1 so that S = 2 E (19a) OG? su 2 ie (19) Yan ec. Coordinate Transformations. The usual strength of material method of plane stress analysis is applicable to transform the principle stresses into equivalent stresses on any other orthogonal coordinate system. The most convenient is a y-direction parallel to the coastline and positive to the right facing seaward, and the x-direction normal to the coast and positive toward the shoreline (Fig. 19). From either the Mohr's Circle or the stress element shown in Figure 20, the mathematical transformation equations become S + S S - § Se Cy (EEO bs Im (20) xx 2 2 70 Shear Stress E/2 Normal stress b.STRESS ELEMENT a MOHR’S CIRCLE Figure 20. Mohr's Circle analysis. S oF S - $ ag XX YOu XX YO" Say = € 5 —) ( 7 ) eos 20 (21) S - $ Mpa XO gn cP Gr: Sy = ( 5) ) sin 20 (22) where a is the angle between the wave crest and the shoreline. S is the shear stress component in the longshore direction due to the excess momentum flux of oblique wave incidence. This shear stress is exerted in the + y- direction on a fluid surface constant in x. The subscript nomenclature and order is consistent with that normally employed for shear stresses in fluid mechanics. An equal and opposite Say also exists. For linear wave theory, the transformed components become i = E(Gn - 5) + E(5) cos 2a, (23) sae i n ai EGn 7) = E() cos 20 (24) Say = En sina cosa (25) 71 These equations form the basis for all subsequent applications of radiation stress principles that has been termed the first-order theory. The key as-— sumptions are linear wave theory and a time-average over one wave period. If a steady state prevails, the averaging time can be over many wave periods. Des Other Waveforms. The principle stress equations (7) and (9) can be employed with other waveforms and expressions for n, u and w. For example, for standing waves in water of uniform depth and again using linear wave theory, Longuet-Higgins and Stewart (1964) obtained S_. = E(4n - 1) (26) Se = FE 2 @s-ercos 2kx) = il] (27) for the principle stress components. S is again identically zero. Although linear theory has been used extensively and forms the basic theory, various nonlinear wave theories and irregular waves with known spectral characteristics have also been studied. These special theories are discussed in Section VI of this chapter. The linear theory is reviewed in Sections III, IV, and V. III. MEAN WATER LEVEL CHANGES The radiation stress components are directly expressed in terms of the wave parameters: wave height H, wavelength L, crest angle a, and the still- water depth d. Waves approaching a sloping coastline or near structures will undergo modifications in these parameters resulting from shoaling, refraction, diffraction, reflection, and breaking processes. Spacial changes (gradients) in the radiation stress components must result. Under steady-state influence of the incident wave field and the modified wave field, a new time-averaged equilibrium will be established for the time-averaged water level and time- averaged currents present. For this steady-state situation, the equilibrium equations are the momentum balance equations perpendicular to and parallel to the shoreline and the continuity (mass conservation) equation. All forces and stresses in these equations are mean values over the wave period T. Forces are depth-integrated values per unit horizontal width. al Normal Wave Incidence. When waves approach the coast at right angles with crests parallel to the coastline, the principles of radiation stress theory are readily demonstrated in terms of MWL changes. In this case for a = 0, the x-direction stress S reduces back to principle stress Syy and no shear-stress component exists. a. Momentum Balance. Consider a plain sloping beach with bottom contours parallel to the wave crests as shown in Figure 21. For steady-state time-mean conditions, the rate of change of the radiation stress must create a rate of 72 Wave Helght, H Breaking Wave Helght, Hy, Osep Water Wave Height HOALINS Ho WAVE _S' on see Horizontal Distance Vertical Elevation (Distorted) Wave Peaking and Breoking Point Swash Zone Run-up —— -BREAKERS WAVE ENVELOPE i) Maximum SWL Set-up 4x = > WAVE SET- DOWN Horizontal Distance Breok Point Deep Wolter Oceon Boundory Figure 21. Schematic of wave setdown and setup due to normal wave incidence on a plain beach. MWL change on a sloping bottom, 8. If no net shear stresses are assumed at the bottom or on the free surface, the momentum balance gives —— + pgh P= 0 (28) where h = d + nN, and n is the MWL change above or below the stillwater level (SiN) Ibe Sx, is known, equation (28) can be integrated to yield a MWL setdown outside the breaker line and a MWL setup in the surf zone. Such an analysis was first conducted by Longuet-Higgins and Stewart (1963, 1964) and also by Bowen, Inman, and Simmons (1968) for plain sloping beaches and using S,, from linear theory (ea. 23). However, equation (28) applies to an arbitrary bottom profile (still assuming straight and parallel contours) as long as the depth, d is monotonously decreasing. b. Wave Setdown. Seaward of the breakers, it is reasonable to neglect wave reflections, percolation, bed-shear and internal turbulence dissipation so that the waves propagate with constant energy flux, Ecn = Ecg = Constant. (29) Using equation (29), Longuet-Higgins and Stewart (1963) integrated equation (28) over a plain sloping beach to obtain sii pe ae a ae n= ~ 8 sinh(2kd) (0) ¢® Thus a lowering of the MWL below SWL takes place since n is negative and this wave-induced change is called setdown. The integration is not straightforward since all three variables k, d, and E can be dependent upon the horizontal coordinate x for this problem. It is also possible to express @ ae @ given depth d as a function of the deepwater conditions Ho and Ly (or aoe k) as discussed by Gourlay (1978). His results are —_ 1 “oo coth#kd WoT Baa Gaye Stn Del GY or H2 = fe) d is Se fC) (32) Oo Oo with the form of the function f(d/L_) shown in Gourlay (1978) as plotted against a normalized setdown. Wave setdown is zero in deep water and in- creases rapidly as the depth increases. The maximum setdown is limited by wave breaking in shallow water which voids the assumption of constant energy flux. Using shallow-water approximations and a breaking criteria as discussed later, the maximum setdown at the breaker line is approximate 5 percent of the wave height at the breaker. The most questionable assumptions in this theory are the use of Sx based on linear wave theory and the neglect of bed shear in the region approaching wave breaking. c. Wave Setup. Shoreward of the breakers, internal turbulence from wave breaking and bottom shear both drain energy from the waves in the surf zone. Since S,. varies directly with E in shallow water @e >) eonmas H2, the re- duction of wave height toward the shoreline means a negative gradient of Shee must be balanced by a positive gradient of n in the x-direction. This is called wave setup. The key question is how the wave height H varies from the first breakpoint to the shoreline. (1) Dissipative-Type Beaches. Spilling-type breakers are found on wide, flat dissipative beaches. After breaking, the wave height continuously decreases as the waves and bores propagate shoreward as schematized in Figure 21. No truly satisfactory theory exists to explain when, where, and why waves break. Also, very little is known about energy dissipation rates in the surf zone for various types of breakers and beach profiles. Consequently, simpli- fying assumptions primarily based upon experimental evidence have been em- ployed to develop wave setup theories. For spilling-type breakers on dissipa-— tive beaches, the assumption commonly employed is that the breaker index, y (ratio of breaking wave height to mean depth at breaking) remains a ike Gee Foy eet (33) Gide ser 74 fixed ratio throughout the entire surf zone. This assumption is also important in longshore current theory. A complete review of surf zone empiricism will be presented in the section. Now if it is again assumed (Longuet-Higgins and Stewart, 1964) that linear wave theory is applicable to compute S__, that shallow-water conditions prevail so ae = 3/2E, and that y is constant in the surf zone, then bh 33 Dm 2 a = 76 °8Y (n + d) (34) Using equation (34) in the momentum balance equation (28), where h = (n + da) is retained in the second term, Bowen, Inman, and Simmons (1968) showed that for a plane beach of slope, tan 8 dn = 1 ae eS OGVeY B (35) For a given constant index y, this meant that the mean water surface slope (setup) was proportional to the beach slope as illustrated in Figure 21. Integration of equation (35) to find n on a plain beach reduces to a simple trigonometric analysis. All that must be specified is the magnitude of the breaker index y, the location _of the breakpoint, and the magnitude of the wave setdown at the breakpoint, n_. Again, using shallow-water theory and equations (30) and (33), " becomes == alt Th, = Seals (36) For the maximum wave setup nN. at the shoreline, Battjes (1974a,b) used equation (36) and simple geometry to Show that nh, = ae Vy (37) m This means that the MWL at the shoreline is predicted to rise about 25 percent of the breaker wave height due to wave setup. Horizontal distances to locate setup values of interest can easily be determined from the geometry involved. The key assumptions are use of linear theory for S and a constant y in the surf zone. oe (2) Reflective-Type Beaches. Plunging-type breakers are found on steep reflective beaches with narrow surf zones. Most laboratory beaches are of this type. Two theories have been advanced for setup primarily based on v2 observations by laboratory researchers. Swart (1974)%3, for complete plung- ing, assumed that all the energy of the approaching wave immediately trans-— formed at the outer edge of the breaker zone. An abrupt water level change occurs to balance the change in radiation stress at the breaker line and the mean water surface is assumed level across the surf zone. Maximum setup was found to be lower than for spilling breakers. This was cited as a limiting case for natural beaches which have some combination of plunging- and spilling- type breakers. Gourlay (1974) made allowance for the effect of a plunge point distance X_, between the breakpoint and plunge point of a curling breaker, where he postulated that energy dissipation began. Assuming a constant wave setdown n, over this distance, no abrupt rise in setup at the plunge point and a con- stant index y from the plunge point to the shoreline on a plain beach, n,=apya-?¢ >) yean8) i, (38) of Mp For surging breakers, X_ was zero and equation (38) reduces to equation (37). The dimensionless plunge distance (X_/H,_) must be found empirically, as by Galvin (1969), and is related to the’ beach slope. Maximum setup is again lower than for spilling breakers. More assumptions are employed in the theory for plunging breakers. These points are discussed when comparing theories to the observations in Chapter 4. Die Oblique Wave Incidence. The more general case is when waves approach the beach at an angle. Wave refraction causes changes in wave height and length. The magnitudes of theo- retical setdown and setup can be shown to depend on wave angle and all other factors that influence surf zone wave heights such as the induced longshore current and resulting wave-current interaction processes. All the analytic theories to date neglect the feedback of the current on the wave motion. The partial differential equations are decoupled in this way to become two ordinary differential equations. The momentum equation per- pendicular to the shore (eq. 28) is solved independently so that calcula- tion of 1 is independent of longshore current. Such a theoretical solution for a profile with straight and parallel bottom contours but arbitrary shape (monotonously decreasing depth toward shore) was given by Jonsson and Jacobsen (1973). The shore-normal radiation stress is found from equation (23). a. Wave Setdown. Outside the breaker line, the wave height is deter- mined by assuming a constant energy flux between orthogonals and Snell's law (c/sin a = constant) to reference conditions to deep water. Interestingly, 23 SWART, D.H., “Offshore Sediment Transport an Equilibrium Beach Profiles," Pub- lication No. 131, Delft Hydraulics Laboratory, The Netherlands, 1974 (not in biblography) . aS wave setdown is identical to that given for perpendicular wave incidence, equation (30). In terms of deepwater wave height Ho» and deepwater wave angle a this becomes 2) mio!) ol cothewkn soo e 1 5 16 nm sinh 2kh cosa (39) Since the ratio n/h is very small outside the breakers, the stillwater depth d can replace h in equation (39) for ease in computation. b. Wave Setup. The solution for spilling-type breakers with y constant across the surf zone follows closely to that outlined above with normal inci- dence. All the same shallow-water assumptions are employed. The solution for wave setup obtained by Jonsson and Jacobsen (1973) is dn il dd ae 1 8/ (3y2cos2a) dx G0) The setup slope is no longer a constant proportion to the beach slope as with normal wave incidence. For a given set of deepwater conditions, H_andT , refraction will cause less setup at a given x due to the cos*a term in equation (40) and the fact that waves break at a smaller water depth. The theoretical maximum setup Te, was also determined by Jonsson and Jacobsen (1973) to be ae om abo (Ly = 413 yH, “5 vile sin a) (41) and verified by Gourlay (1978) who put it in this form. Here the subscripts b and o mean breaker point and deepwater conditions, respectively. Equation (41) was found by integrating equation (40) and use of the appropriate boundary conditions and not the trigonometric manipulations for a plain beach as before to obtain equation (37). Surprisingly, therefore, this model shows that maxi- mum setup is independent of the bottom profile in the surf zone. The oblique maximum setup is less than normal setup and equation (41) reduces to equetion (37) for a = 0. Further parameter study by Jonsson and Jacobsen (19,3) showed that wave steepness (H_/L_ and y have much less MEL GEMS than wave angle on maximum setup which varied approximately as (cos a) 5, All the above are based on linear wave theory and regular waves. Wave setdown and setup theories when nonlinear and irregular waves are present will be reviewed in Section VI. 3. Other Factors. A tilt of the MWL at the coast can also be due to the wind stress over a long fetch distance inducing a wind setup. The horizontal distance involved al is much longer than wave setup which is a pure coastal phenomena. The direc- tion, fetch distance, windspeed, and duration are all factors influencing the magnitude of wind setup. A quadratic wind-stress law is generally applied where surface stress is proportional to the square of some reference windspeed. The wind stress can be added to the momentum balance equations to theoretically predict its influence on mean water surface gradients and nearshore currents as discussed further in Section V. For normal wave incidence to the coast, wave breaking also induces a vertical circulation current (about a horizontal axis) in the surf zone. This is due to a vertical distribution of radiation stress which is greater near the surface than near the bottom since it is proportional to orbital wave motion. The result is schematized in Figure 22 (from Bijker and Visser, 1978)**, where it is theorized that a net shoreward force at the surface and a net seaward force near the bottom results. The hypothetical circulation pattern resulting is also shown in Figure 22. There is no known theoretical attempt to solve this wave-induced circulation problem using radiation stress principles. IV. LONGSHORE CURRENTS As summarized at the end of Chapter 2, there are a large number of factors influencing longshore currents. The complexity of the forcing field, geometry, and fluid must be reduced to an idealized level to allow theoretical treatment that permits analytic solution. The basic theory described in this section is for the longshore current induced by simple waves striking an infinite, plane Resulting forces and current Sse) 4 {| = | eet 1 Distribution of excess static pressure “bottom pressure Distribution of distribution Momentum Flux ea te Figure 22. Circulation current in breaker zone (not to scale) (from Bijker and Visser, 1978). 24 BIJKER, E. W., and VISSER, P.J., “Wave Set-Up, "Coastal Engineering,” W.WI Massie, ed., Vol. II, Harbor and Beach Problems, Department of Civil Engineer-: ing, Delft University of Technology, The Netherlands, 1979 (not in bibliogra-—' phy). 78 beach at an angle. Table 2 (modified from Ostendorf and Madsen, 1979) lists the idealized environment along with those stresses and accelerations that have been neglected. This list is long but many of these restrictions have been relaxed in later generalizations of the basic theory, which will be re- viewed separately, mostly in Sections V and VI. In addition, the basic theory for uniform longshore current profile has undergone considerable modification from the original models first introduced about 1970. These modifications primarily reflect the influences of wave incidence angle, bottom shear stress, and lateral turbulent mixing approximations and assumptions in the surf zone. The original models and all subsequent modified theories for uniform longshore current profile are reviewed in this section. Table 2. Idealized environment for longshore current theory. WAVE FIELD Simple, monochromatic gravity wave trains Steady-state, incident wave field Two-dimensional, horizontally propagating Linearized theory and radiation stresses Oblique angle of incidence, long wave crests Spilling-type breakers Constant breaker ratio in surf zone BEACH Infinite length, straight and parallel contours Plane bottom slope Gentle slope Impermeable bottom FLUID Incompressible Homogeneous (no air entrainment) CURRENT Depth-integrated, parallel to coastline Time-average (one wave period) NEGLECTED STRESSES AND ACCELERATIONS No surface wind stress No atmospheric pressure gradient No Coriolis acceleration No tides No local (time-average acceleration, i.e., steady flow No wave-turbulence interaction stresses No bed shear stress outside the surf zone No rip currents present No wave-current interaction stresses 13) ee Conservation Law Balances. a. Momentum. Consider the idealized setting defined in Table 2. A schematic and definition sketch is shown as Figure 23, where the plan view shows two refracted wave rays a unit distance apart (exaggerated). The wave height and MWL variations shown are taken along any wave ray which are all identical for a plane, infinite beach. Clearly, there can be no gradient in S in the y-direction for this case since time-average normal stresses are identical in the y-direction. The radiation shear stress S,. Gie., flux of y-direction momentum across plane perpendicular to the x-direction) is not the same on two sides of a differential element as shown in Figure 23. This is because all three factors (E, n, and a) in equation (25) for S$ vary in the x-direction. The gradient dS,__/dx thus becomes the driving stress in the y-direction momentum balance and is resisted only by the bed shear stress, T,. The overbar is for a time-averaged value. An additional y-direction stress is due to the gradient of the lateral shear force over the total depth, T._ due to turbulent mixing, i.e. wave orbital velocity interactions in the x- and y- directions. For the direction of wave incidence and coordinate system shown in Figure 23, the y-direction momentum balance becomes ds ae dT, ae Pipe ergata © (42) The longshore current velocity, v appears in the time-averaged bed shear- stress term T,, and in the lateral shear force term T,. With appropriate expressions for these quantities and for Syy», 1t is possible to integrate equation (42) to derive an expression for the distribution of longshore cur- rent v(x) across the nearshore zone as schematized in Figure 23. This pro- cedure again means a decoupling of the x-direction equation (28) from the y- direction equation (42). b. Energy Balance. It is informative to also consider the energy balance equations in the nearshore zone. A good summary is found in Longuet- Higgins (1972a,b) for the idealized case in Figure 23. The general flux of energy per unit length of shoreline toward shore is given by F_ = EC cosa = EC cosa (43) g n where E is the local energy density of the waves (eq. 14) and Cos the group celerity. Assuming negligible wave reflection and no wave-current interac-— tions, then the energy balance in the x-direction gives dF = + da ; (44) where D is the local rate of energy dissipation per unit area. But earlier, using linear wave theory, the shear component of the radia- tion stress was given by equation (25), S = En sina cosa 80 Wave Height, H (refracted) Hp S| EELING BREAKERS 0 b Wave : Ray Le aes | NAVE_SET- i ae Wave Down rade = Ray cr eer +y W z zy WW ws o = ae fn aay n 2 ah = i = = x n a Ww = s 5 < te yy oF 3) a Ww 7) > 2 : a 2 | = > (Sxy +T,) B a 4 = z i = = = wW Cy x 9 z ° Syy = REFRACTED do WAVE CRESTS 7 +X Longshore_ Current, V Figure 23. Schematic of refracted oblique wave incidence on a plane infinite beach showing resulting longshore current profile and MWL changes. 8| so that comparing equations (43) and (25) gives S_ = E (45) where sina/c is also constant due to Snell's law. Longuet-Higgins (1972) proved that equation (45) is independent of the wave theory involved. Conse- quently, the driving force in the y-direction momentum balance equation (42) for longshore current becomes dy sina ox sina aan Pare asrennene Reciae Go) and is thus directly proportional to the energy dissipation. "If there were no dissipation, there would be no current" (Longuet-Higgins, 1972). Outside the breakers, if the energy flux is assumed constant, no energy is dissipated (D = 0) and consequently the longshore current must be identi- cally zero. Inside the surf zone, continuous spilling-—type breakers dissipate large amounts of wave energy to generate a longshore current. F, is needed to be a continuous function of x in the surf zone to validate equation (44). Longuet-Higgins (1972) argued that most of the energy dissipation is due to wave breaking and not due to the bottom shear stresses. Thus, the shape of the wave height decay curve in the surf zone, as shown in Figure 23 and em- pirically determined from the breaker ratio, y, plays a key role in the theo- retical longshore current profile. Wave breaking, internal turbulent shears, and resulting energy dissipation as heat loss to the surroundings are the main mechanism driving the longshore current. From the point of view of the momentum balance equation (42), the common belief is that the bottom shear stress dictates the strength of the longshore current. The lateral turbulent stresses only redistribute or smooth the velo- city normal to the shore (e.g., see Horiwaka, 1978, p. 210). The question of the relative importance of internal turbulence dissipation and bottom shear is discussed further by examining their relative magnitude in the theoretical equations. 2. Original Model. The use of radiation stress principles in the development of a theory for uniform longshore current PeOeLLee was first made by Bowen (1969a), Thornton (1969), Bakker (1970)2°, Iwata (1970), and Longuet-Higgins (1970). All used a longshore momentum balance as equation (42) but with some differ- ences in the formulations for the three terms involved. Here the model by Longuet-Higgins (1970) has been designated as the original model primarily 25 BAKKER, W.T., “Littoral Drift in the Surf Zone," Study Rept. WWK70-16, Directorate for Water Management and Hydraulic Research, Coastal Research Department Rijkswaterstaat, The Hague, The Netherlands, 1970 (not in bibliog- raphy). 82 because his results were put in dimensionless form, and further elaboration was presented by Longuet-Higgins (1972a). It is convenient to introduce a new x-direction coordinate system (after Bowen, 1969a) where the origin is taken at the maximum setup line and a posi- tive x is facing seaward (Fig. 23). This simply gives the longshore current v=o at x=o, and eliminates the uncertainty as to where the x-direction coor- dinate begins in deep water. The breaker position is x,. For the three terms in the y-direction momentum balance equation (42) Longuet-Higgins (1970, 1972a) derived the following. a. Driving Stress dS,,/dx. Differentiating equation (25) or equation (45) in terms of energy flux gave (1) Outside the Surf Zone SL = (47) (2) Inside the Surf Zone ds 3 : wry: Do ey2 (gh) I (eS dh (48) ae Soe dk where in equation (47) a constant energy flux is assumed. In equation (48) it is assumed that in the shallow-water surf zone, n=l; the wave angle is small, cosa=1; and Snell's constant is still applicable. Longuet-Higgins (1970) also neglected the effects of wave setdown and setup in further simplification of equation (48) although he recognized their effects. Wave setup is included in later modifications of this original theory. Therefore, within the surf zone he simply derived for a plane sloping beach 8} = S (ees) ee x he tang agi dx so that equation (48) became ds 6 Ys =: pg ¥22 (tang) 7 (San), Ie (49) dx From equation (46), therefore “hs Yo D = oe oe (tang) (50) mggnene the local rate of energy dissipation D due to wave breaking varies as 2 from zero at the maximum setup line (on a plane beach with Ones 17) 6 Other forms for equation (49) stem from using the celerity C=/gh in the surf zone. 83 b. Resisting Stress, T For the time-average bottom shear stress, T TP onieuce tie guns alo TOVGEE ISTE = 7, = = C,olu, |v (51) where C. = a boundary resistance coefficient due to both waves and longshore fe oA current, v the absolute value, maximum wave orbital velocity near the bottom for sinusoidal motion lu, Longuet-Higgins (1970) obtained equation G1) from the usual quadratic stress law in real time for the vector value of co oo e 2 olelm, > Golan (52) B 2 two?! Up! UB £° "BIB where Up is the wave orbital particle velocity just above the bottom boundary. To go from equation (52) for 7, to the time-average value in equation (51) is not trival. Longuet-Higgins (1940) made the additional assumptions that (1) the longshore current velocity v is small in comparison with the wave orbital velocity, » and (2) the wave incident angle a is very small in the surf zone so that v is roughly normal to Up: Taking the time-averaged mean value of lu | over one wave period and assuming Ce represents a constant mean value over this same period, the component of Tp in the y-direction becomes that given by equation (51). Assumption (1) above essentially makes this a linearized bed stress term or a weak current theory. Removal of assumptions (1) and (2) has been a significant achievement in modifying the original theory as described below. Further approximations are needed to put equation (51) in a more usable form. Using linear theory to obtain Una in shallow water and taking y = H/h give so that equation (51) becomes if wave setup is again neglected ss = eee i, = “oc Cee y(tan8) x 2y (54) This form clearly shows how the, longshore current v is introduced and how the bottom shear stress varies as x. 84 ec. Lateral Turbulent Mixing Stress, dT, /dx. For the time-averaged lateral mixing force over the water depth, T, Longuet-Higgins (1970) derived L See Reedy =p y4y dv we = ht, = hu, == h(Npx(gh) *) iz (55) where Tr = the lateral turbulent eddy stress due to waves uy, = the lateral turbulent eddy viscosity due to waves N = a dimensionless, turbulent closure coefficient for lateral wave mixing proposed by Longuet-Higgins (1970) to be of the order N<0.016 The term tg is reserved for a true, turbulent Reynolds stress due to random turbulent velocity interactions at scales far less than the wave orbital velo- city scale. Ime If wave setup is again neglected taking h = d = xtanf8 gives Sy) ee me Nog (tan) 2 ee (56) so that the lateral turbulent mixing stress becomes ily. 3 =) = == = Nog (tang) I a [x Io Sa (57) dx dx dx d. Dimensional Longshore Currents. In summary, Longuet-Higgins (1970) derived the momentum balance equation (42) ds ae, je 3° ae - with equations (49), (54), and (57) ds MASHT IS). . 43 p ~sina,-% _ =% Aa euniG oa (tang) (as = TOS =o alee. lee le T= 7 Ces ‘y (tang) WS Oey dT - - eee Wy Gistp Gi. GO tp oly ge ° Mog Came) © gels” Gel = 2 ae ES ag where we define three new constants 3 5 ; r = > pg 2y2 (tang) 2 (S22) (58a) 85 L lL a 1 eC 8 *y (tan8) * (58b) q = ie 3 p = Nog *(tang) ” (58c) to give the second-order differential equation for v 5 2= —s) Fe =e —§) po 2 2X43 PW = 8? (59) dx? dx with the right-hand side vanishing outside the surf zone. Thus v(x) is the solution of two second-order linear, ordinary differential equations which must match in magnitude and gradient of v(x) at the breaker line and vanish at x=0 and the ocean boundary. (1) Neglecting Lateral Turbulent Mixing. Taking p=0 greatly simplifies equation (59) to give as an approximate solution ve == x= Sut 5. g (tang)? (2="*)x inside surf zone (60a) q 16 Ce C i} S vs outside surf zone (60b) On a plane beach with y, C., and (sina/c), all constants in the surf zone, the longshore current profile is triangular in shape, reaching a peak at the breaker line and dropping to zero outside the breakers (see Fig. 24, P = 0). (2) Reference Longshore Current Velocity. Again neglecting later- al turbulent mixing stresses, the longshore current velocity at the breaker line v* can be used as_a reference velocity. It becomes from equation (60a) after again taking hy = d., = tanpx, Na. Eee aye sina Wauciaia Ge (ed) CG 2 tang (61) Other forms result from taking C = Ved, in equation (61). In reality, the velocity discontinuity at the breaker line cannot occur so that the lateral turbulent mixing smooths the velocity profile. The solution with lateral mixing is simplified using dimensionless variables with ve as the reference velocity. e. Dimensionless Longshore Currents. Longuet-Higgins (1970) introduced the dimensionless variables, X and V as and Ve= (62) into equation (59) to obtain 86 0.5 WZ aS fol] \ \ N x Z P=10 ENG ——————— 0 9 “a SSS ke) 05 10 15 20 x Figure 24. Dimensionless longshore current profiles as function of P-parameter (from Longuet-Higgins, 1970, 1972). Sp a2 3 ts 3 pee Clg 8 ye ey aoe (63) dx? where again the right-hand side vanishes outside the surf zone and, vik a PYG _ Natanp (64) Epab SEN The dimensionless parameter P now represents the relative importance of lateral turbulent mixing of the wave orbital motion to the bottom frictional resistance. Taking P = 0 for no lateral mixing gives V = 1 at KX = 1(x = x) and the triangular solution as before with V = 0 outside the breakers. Longuet-Higgins (1970) solved equation (63) using the boundary conditions that V>0 as X+0 and X+= and internally that V and dV/dX are continuous at X = 1. He obtained the following results. 87 (1) In general, P # 0.4. Pp V = AX + B,X i O< X<1 (inside breakers) (65a) Po WS BAX X>1 (outside breakers) (65b) where 1 AS ESTER (66a) Glo = > P) 3 v9 1 Sseo2u ce = Pay 4 Tae 5 (66b) 3 v9 all pa Wate eS Ae a 6 Po 4 16 * P (66c) Po-t By SN (66d) Pit PD p,-1 Bit> apa A (66e) Pais) so that all the constants (A, Py> Po» B » and B,) in equation (65) depend upon 2 AL (2) Singularity, P = 0.4. The coefficient A becomes infinite for P= 0.4. For this singularity, Horikawa (1978a) gives the solution Vie= ax -2 X1nxX- O< X<1 (inside breakers) (67a) ZS We “ x Ip X>1 (outside breakers) (67b) The family of solutions for some representative P values is shown in Figure 24 (after Longuet-Higgins, 1972a). Taking larger P values gives more lateral turbulent mixing to smooth and spread the theoretical longshore cur- rent profile across the surf zone and beyond the breakers. Using lower P values shifts the maximum longshore current toward the breakers. The theory requires estimates for three parameters (y, C¢, and N) for a solution on a given plane beach slope. It neglects wave setdown and setup effects in mean water depth but includes the maximum setup location as the new shoreline. The radiation stress theory of longshore currents as summarized by equa- tions (65), (66), and (67) provided the needed breakthrough in 1970. Since 88 then, many modifications attempting to improve the generality and accuracy of the original model have appeared in the literature. These are summarized in the next section along with the original contributions of Bowen (1969a) and Thornton (1969). 3) Modified Models. The modified theories for longshore currents since 1970 are listed in Table 3. These theories still retain linear’ wave theory to calculate the radiation stresses for regular waves. Nonlinear and irregular wave theories are discussed in Section VI. In some cases, closed-form analytic solutions are no longer possible due to the beach profile or bed shear-stress formulation employed. Rather than discuss each model separately, the major modification areas involved are reviewed. In most cases, actual theoretical results will be discussed in Chapter 4 and compared with experimental measurements. a. Beach Profile and Wave Setup. Those theories that are for a beach profile with straight and parallel bottom contours but with depth in the surf zone that is monotonously decreasing (arbitrary profile) require numerical integration methods for a solution. Thornton (1969) was the first to provide such an analysis. Wave setdown effects on the MWL are neglected by all theories except the numerical integrations by Jonsson, Skovgaard, and Jacobsen (1974) and Skovgaard, Jonsson, and Olsen (1978). However, the influence of wave-induced setup has been incorporated in most subsequent theories because of its influence upon y. Bowen (1969a) and Komar (1975) simply took the resulting wave setup slope equation for normal wave incidence on a plane beach (see eq. 35) which for the x-coordinate system becomes dn 1 = cane = —Ktane ax Mees) Gy) KGS) Letting Machine ida an = az - + ie = tang Ktang or 7 GS eens — a (69) 1+3y7/8 then the following modified equations result, neglecting wave setdown. For the driving stress term, equation (49) now becomes ds iy 5 tang 3 sino = = = py*—— _ (gh) *% (—) cosa (70) dx 16 14+3y2/8 Cc 89 (AudeaZottqtq UT 30U) 6/61 ‘SPUPTIS4ION PUL ‘kZo0Touyoe] jo AQTSsieATUp AyTeq ‘ZupazveuT3uyq [TTATD Fo Juewj1edeq ‘suetTqoig yoeog pue 1t0q71ey Corr STON, ope Satssen “M°M ‘buzaaourbug 7p4spo0g ‘sjueaing erzoyssuoT,, “*f “dayvad pue “°M°a “WAALTIgz 061) SULBDIH jenbuo] o19ALeuy ee pa73.1uo aueld —— LaPOW LeULB E40, “[apoll UOLZOPAy W0FZOq AedU} |-UOU a[buy absey + yUasIN) Hu0s7Sex *(LG) UoLQeNba Lapow uoLzDJ4y wWOZ320q pazfseaut| ajBuy [pews + JUaduND yeamx “ezep platy sayozew Asoayr quauund eam “OUuLL uayoug JO apLs yoea uo YUaaJSJLP sassauzs BuLxty o1yALeuy Pats tPow (6261) !yeseS y snery ees ezeg a_qeyLeay Leotuawny Pals LPow Yai 344 aAung a ES (8261) “ahquy vaso y ‘uossuop ‘pueebAoys “upos dLaALeue lwuad 07 S}Ua.uNd Huouzs uo} pasn Sura [PILUL dw 8/61) uewhay 3 J4opuazsg “uaspeW S}uadund Buouzs 40} pautnbau “uLOS LeoLuaUny OLA euy (8261) etdwéapeg § ALT a Leo, 4auiny 5 P2LSLPOW | xx26107 | ¥x6u0135 pa792[62N Squasund Bu0rzs AO} pautnbau *ujos LedLvauny 97(8261) jgeeug “pra B 4ayfta AO OLZAjeuy LLeus dn-72S aAeM UL (GS) ; : > Jjau aAPM $790 ban ILyALeuy *ba peurBbL4g *ba LeuLbLuo SA (9261 ‘SZ6T) sewoy dn- as QAM UL papnouL (pZ6T) UOLQIEUJaA BALM [eoLauny pats Lpow LL ews 6u013S5 Sa, “abquy uasqooser y ‘pueeBAoys ‘uossuor 40292} UOLZILAS LeoLUaUny “TEMY uoL}oW ahem pasn | RB L3ALeuy PaLsrpow | x LLews sa, | 9 aueld (OZ6 ‘696T) voIUAOUL JPOYS pag paryitdupsuang | o13Ajeuy | pats tpon Pats tPow Aeaut] S2A (6961) uamog syueuiay | “ULOS “wn | aybuy | +yoeazau | aLbyOud Jdauoueasay yorag * £20943 ssgi4s uoTzeLrpert uo pseseq 696T FVIUTS satsoeyy quszAAND aloyssuoyT °"€ 9TGeL 90 where the incident wave angle is no longer assumed small. Using the same bed shear-stress formulation as before (eq. 54), except including wave setup, and neglecting lateral turbulent mixing stresses, the following modified equation is obtained tang sino, 1+3y2/8 C —~ ie 5 nye We aig C, gh cosa (71) for longshore currents. This reduces to equation (60a) inside the surf zone when h*d. Equation (71) is essentially that given by Bakker (1970)2°, Thornton (1969,1970a), and confirmed by Gourlay (1978) and others. At the breakpoint, it becomes the modified reference velocity (using C_ = Ygh, for shallow linear waves) ve = Se )? rane — sino, cosa (72) > 9 Wah Ge by)” T3y2/8 b b : For y values from 0.5-1.2, the term [ (1/ (1+3y2/8) ] in equation (72) varies from 0.91 to 0.65, so that the wave setup modification is not insignificant. Komar (1975a, 1976b) obtained the additional term 1/(1+y)2 in equation (72) that was shown to be inconsistent with previous assumptions for dS,,/dx and Tt, by Gourlay (1978). Komar (1975a, 1976b) also had the term (1432/8) 2 instead of the first power as in equation (72). The present report and Gourlay (1978) have been unable to verify Komar's form. Using this modified ve in equation (64) gives for Longuet-Higgins, P-parameter (modified) Coorg Shunt tanp Pam Gay) ae 3y2/8 (73) so that all of Longuet-Higgins' (1970) dimensionless results are still appli- cable (eqs. 65, 66, and 67) if V* and P are replaced by the modified versions (eqs. 72 and 73). However, it must be remembered that this setup correction assumes normal wave incidence and hence also neglects wave refraction effects in the surf zone. All other modified models listed in Table 3 include some form of Snell's law refraction in the surf zone to additionally modify the wave setup. As shown earlier (eq. 40), the setup slope is no longer a constant proportion to the beach slope and refraction results in less wave setup. For the numeri- cal solution methods, the decoupled motion equations (28) and (42) are solved together. The N solution from equation (28) includes effects of geo- metry refraction in the dS ,,/dx term and is in turn used in equation (42) to determine the longshore current profile. Numerical accuracy is important so that careful numerical integration procedures are needed (e.g., see Jonsson, Skovgaard, and Jacobsen, 1974). In all such models to date, current-refraction effects on the calculated n values have not been incorporated to make the equations a coupled set. 25Tbid. 91 Finally, Kraus and Sasaki (1979), following Liu and Dalrymple (1978), include wave refraction in their analytic development. Inside the breaker line they obtained the following result for the driving stress ds a2 5 257 6S A _ tealals ; he. fae peur an, 8 ce 1G OVS Fase Sey ae [Goines bh, oa A Sins ae ee Gente Be) =] (74) eye b hy Note that neglecting wave setup, refraction and for small a,, equation (74) reduces to that employed by Longuet-Higgins (eq. 48). These researchers also modified the form of the bed shear stress and lateral mixing terms, so their final results are deferred to the subsection on lateral mixing. b. Modified Bottom Shear Stress. The most important modifications to the original model have been in relaxing the assumptions of a weak longshore current and small wave incidence angle. This is because the longshore cur- rent is inherently related to the bed shear-stress model employed. The major weakness of Bowen's (1969a) model was the overly simplistic, linearized, shear- stress term, tT, = pC,v. All other models in Table 3 begin with a quadratic form as given Ey equation (52). Bottom shear stress is a vector oscillating in both direction and magni- tude. The major difficulty is to find an expression for the effective bed stress (and friction factors involved) in terms of a time-averaged current. The coordinate axes and velocity vectors are shown in Figure 25. The instan- taneous bottom shear stress Tt, is assumed to be in the direction of the re- sultant velocity U of the vecfor sum of longshore current velocity v and bottom wave orbital velocity Up (eq. 52). 4 |U|U > >)\> =, = c.e|U|U =f B > = > where U = vt+> U,> and C ohne are combined current and wave friction factors. fee The ratio of theoretical breaker current with no lateral mixing to maxi- mum wave orbital velocity near the bed, v*/ has been shown to exceed unity except for small wave angles. This ratio was used to argue the inconsistency of the weak original model of Longuet-Higgins (1970) as discussed by Huntley (1976a), Madsen, et al. (1978), and Liu and Dalrymple (1978). But v* is not a physical velocity. Therefore Kraus and Sasaki (1979) used the ratio Vin/UBm instead, where v_ is the maximum current from experiments or theory. Surpris- ingly, they showed that most laboratory experiments are invalid to test the Longuet-Higgins (1970) model since Vf Up t-3s indicating strong longshore currents are present. 92 Figure 25. Schematic of wave and longshore current velocity vectors in bed shear-stress models. (1) Strong Currents at Small Angles. Following the earlier work of Jonsson (1966a) for waves and currents in the same direction, Jonsson, Skovgaard, and Jacobsen (1974) proposed a simple interpolation formula for the instantaneous friction factor f __ when the waves are roughly normal to the current direction (i.e., a = 0 in Fig. 25). Sa Sie a CE, = f )sinu (75) This made f identical to the wave friction factor fw with no current (u = 0°) and also made f __ equal to the current friction factor f_ with no waves present (uy = 90°). e combined friction factor f,,, is time dependent, since varies over the wave period. Using linear wave theory for Tio Jo..sson., Skovgaard, and Jacobsen (1974) derived the following expression using equa- tion (75) in equation (52) or a ae (76) for the time-averaged bottom shear stress parallel to the coast. The time- averaged friction coefficient see is found from 2730NSSON, I.G., "The Friction Factor for a Current Superimposed by Waves," Basic Research Program, Report 11, 2-12, ISVA, Technical University, Denmark, 1966a (not in bibliography). 93 fj -f,+ Gvi+ @, 2 + Bm) - DE, (77) The expression E(m) is a complete elliptic integral of the second kind, i.e., 1/2 E(m) = (1 - msin*y) du (78a) Oo with the parameter m given by ne (78) For weak currents (San), equations (76) and (77) reduce to = mp uy Es Gree 6 9a (79) which is identical to equation (51) used by Longuet-Higgins (1970). The scientists' friction coefficient C¢ could then be defined as fw/2 for the weak current-small angle theory. This form is also identical to that employed by Thornton (1969) who similarly began from the wave-current approach of Jonsson (1966a)27. Equation (77) requires determination of both cevureint deta coefficients. For steady free surface flows, £,. depends on both water depth and bottom roughness. The experimental determination of fw in oscillatory water tunnels is still in progress. Details are beyond the intended scope of this review. The nonlinear nature of both Tt, and f_ prevents simple analytic formulation of the current profile. Jonsson,’Skovgaard and Jacobsen (1974) also included a different lateral mixing formulation (see below). The com- plete final results are discussed in Chapter 4. Liu and Dalrymple (1978) gave a complete analysis for the strong current large-angle model with small angles as a special case. When U/u, >> they found uZ apyls =O) ge peebi T 05 fly + Z ] (80) but emphasized that to be applicable, up_ must also be very small if v dimin- ishes for small ao. They neglected lateral turbulent mixing stresses in all longshore current formulations. Their work is primarily of interest for the large-angle modifications on longshore current theory as described below. 27Tbid. . 94 (2) Weak Currents at Large Angles. As observed in the laboratory and field (see Ch. 2), it is the occurrence of relatively large wave incidence angles that drives the longshore current (see Fig. 12). Liu and Dalrymple (1978) concluded from studies of available field data (Inman and Quinn, 1951; Balsillie, 1975; Komar, 1976a) that a weak current (U/ <<1) large wave angle model for bottom shear stress was consistent with field Observations. Using this assumption they derived the expression = pC. - > ia Das Thy Gh aret up al (vsin2a)4 + 2v(1 + sin“a)j] (81) with i, 7 the unit vectors in the x and y directions, respectively. For small angles a, the original model formulation of Longuet-Higgins (1970), equation (51) is recovered, as expected. Of major interest was the theoretical longshore current velocity profile that resulted in comparison with that derived by Longuet-Higgins (1970), ne- glecting lateral mixing stresses and corrected for wave setup due to normal wave incidence, i.e., equation (71). Liu and Dalrymple (1978) derived the fol- lowing complicated expression for the longshore current without lateral mixing pac . . * ie ve = ont gheang C9) (5-6 (28) “gh] (1-2) “gh *. f ; 2 Dahe seme ly il { (143y2/8) + (1-5y2/16) G28) gh - E48) 97h?) (82) Equation (79) uses Snell's Law to include wave setup from refraction. Conse- quently, the large-angle velocity profile that results from equation (82) is different from the small-angle modified form (eq. 71) for two reasons; namely, the bottom shear-stress formulation and the inclusion of wave refraction in the wave setup. Comparison of the results of the two theories must be viewed with this in mind. The results of two comparisons made by Liu and Dalrymple (1978) for their weak current large-angle theory are shown in Figure 26. Here, X is the di- mensionless surf zone distance such that X = 1 at the breaker. The symbol vx, means the modified theory of Longuet-Higgins given by equations (71) in general and equation (72) at the breaker. Figure 26 (a and b) demonstrates a significant deviation between the two theories as incident wave angle in- creases. Even at a = 10°, approximately a 20-percent difference is observed at the breaker line (Fig. 26,a) and across the surf zone (Fig. 26,b). It was also shown that the v*/ ratio depended upon two factors, (tan§/C.) and wave angle a. When the ratio tang /C¢ was small (mild slopes or large bottom friction as ustially found in the field), the v* ratio was also relatively low across the surf zone for 0 < a <45°. This gives further credence to the weak current large-angle theory for dissipative-type beaches. Liu and Dalrymple also developed a validity diagram (see Fig. 5 in Liu and Dalrymple, 1978) for longshore current theories that depended upon the ratio v¥/uy bp at the breaker line to separate the weak and strong current 95 Figure 26. Comparison of theoretical longshore current velocity for weak cur- rents, small and large-angle theories (from Liu and Dalrymple, 1978) theories. Although the effort is to be commended, the results are not very meaningful since lateral turbulent mixing stresses are neglected in the theories. The reference velocity vit is not a true physical velocity as discussed above. Kraus and Sasaki (1979) extended the weak current large-angle theory to include lateral mixing stresses. Their contributions are fully examined in the subsection on lateral mixing. (3) Strong Currents at Large Angles. For steep (reflective) beaches with narrow surf zones (or smooth bottoms), a bed shear-stress formulation for both strong currents and large incidence angles is required. Complete expressions for Tp, and vk (neglecting lateral mixing) are given by Dalrymple and Liu (1978). The value of v* is questionable. A constant y = H/h ratio is used in the surf zone even though plunging—type breakers are normally found on steep beaches. The theory is of interest for comparison of laboratory ob- servations but requires modification by lateral mixing and plunging-type breaker formulations in the surf zone. 96 A different approach was suggested by Bijker (1966), modified by Swart (1974)23 and fully discussed by Bijker and v.d. Graaff (1978)2°. This theory is unique from those above in that the location above the bed where the wave orbital velocity and current velocity are specified is explicitedly defined. The elevation chosen was equal to the laminar sublayer thickness which is dependent upon the bed roughness height. In this way, modern turbulent boun- dary layer theory is incorporated into the bed shear-stress model. aes sions for the shear stress at this elevation used the resultant velocity U as defined in Figure 25. They then obtained the result T/4 r - = iE 2V20 EVs {[1 + peo sinwt sina]. By C2 Vv c -T/4 2 1 [1 + (ee sinwt) + 2€ SB sinwt sind] “}dt (83) where p = a dimensionless parameter found from wave orbital velocity experi- ments (e.g., Jonsson, 1966a)?® relating the wave velocity at the bottom to that at the reference elevation kK = the von Karman constant = 0.4 C_ = the Chezy friction coefficient ie = the wave friction coefficient from TS bef pug T = the wave period e Bijker (1966) numerically integrated equation (78) for a range of realistic values of v, Upm? — anda. Curve fitting the results for inte 20° gave “bm L513} Tp = BY" 9, Is) 25 (05) (3 =) ] (84) y C2 z (e For weak currents and small incident wave angles equation (80) can be inte- grated directly to yield a form identical with equation (51) taking (85) A numerical example is shown in Figure 27 adapted from Bijker and v.d. Graaff (1978). Here, turbulent mixing is neglected along with wave setup effects. The result labeled v¥ is for weak current small-angle model but each velocity is calculated from a local friction term. This makes the profile 2 Sriipsicl. *6Tbid. 28 JONSSON, I.G., "Wave Boundary Layers and Friction Factors," Proceedings, 10th Coastal Engineering Conference (Tokyo), Vol. I, Ch. 10, 1966b, pp. 127-148 (mot in bibliography). 97 EXAMPLE VELOCITY PROFILES NO, SYMBOL REFERENCE 0.8 BATTJES (1974) LONGUET HIGGINS (1972) BATTJES (1974) 0.6 LONGSHORE CURRENT VELOCITY (m/s) OAUNhWH — BPpedgno0 DISTANCE FROM SHORE (m) Figure 27. Effects of nonlinear bed shear stress on theoretical longshore cur- rent velocity profile neglecting lateral mixing (from Bijker and v.d. Graaff, 19782°). deviate from an exact triangular shape where the bed friction factor is as— sumed constant in the surf zone (eq. 61). Profile v5 used the full nonlinear equation (80) for bed shear and full expression for dS,,/dx with no shallow- water approximations. The results differ up to 20 percent near the breakpoint. Use of vay as a reference velocity in the dimensionless equation with lateral mixing would then reflect this difference in the magnitude of the longshore currents calculated. (4) Empirical Formulation. Finally, in addition to the above theoretical modifications of the original bed-stress formulation by Longuet- Higgins (1970), the empirical, curve fit approach of Madsen, Ostendorf, and Reynolds (1978) must be included (see also Ostendorf and Madsen, 197) q LO remove the weak current and small-angle assumptions, they first postulated that the form of the longshore current profile (with lateral mixing) is to remain that given by equations (65), (66), and (67) from the original model theory. A scaling factor is introduced between the characteristic reference velocity v* (with setup) used by Longuet-Higgins (1970) and that for full nonlinear bottom stress proposed. This scaling factor is also proportional to the ratio Vp/up . Curve-fitting procedures are used to obtain an expres-— sion for the scaling factor from a surf zone force balance. The modified model also includes new formulations for lateral turbulent mixing and the breaking criteria, y. The resulting modified model requires appropriate values to be selected for the bottom stress coefficient and mixing parameter, as usual. Madsen, Ostendorf, and Reynolds (1978) used the laboratory data of Galvin and Eagleson (1965) to calibrate the model. It was recognized that plunging breakers present in these experiments did not match the y = constant 98 assumption in the surf zone, but the data were used anyway. An explicit ex- pression for the combined, nonlinear friction factor was obtained that pro- perly reflected the weak and strong current theories. It is not clear how the effect of wave approach angle is included by this calibration approach. The magnitudes of the bottom shear-stress coefficients required in all the above theories are discussed in Chapter 4. (QE Modified Lateral Turbulent Mixing Shear Stresses. Theoretical know- ledge is very weak regarding the horizontal transfer of momentum due to tur- bulent mixing processes. Turbulence length scales represented here are on the order of the water depths, wave particle excursions, or surf zone widths, and generated by wave breaking shears, boré-bore interactions, and swash zone mixing. These processes in the surf zone are not sufficiently understood to permit detailed models of the effective stresses that result. Consequently, recourse has been universally made to some type of eddy viscosity model to linearly relate the time-averaged lateral mixing shear stress, Ty to the long- shore current gradient dv/dx. If the velocity fluctuations (i, ¥) in the x- y directions are considered respectively due to wave orbital motions and inter- actions, then for one-dimensional motion, as in equation (55) with Uy, the time- averaged lateral eddy viscosity t =o i & au (86) The tilde symbol here means velocity fluctuations on the scale of wave motion. The prime symbol is reserved for true, random velocity fluctuations due to turbulence, however generated. The kinematic eddy viscosity is We = uy /P. Prandtl's mixing length hypothesis bol yay VS ba ; (87) could also have been employed giving = — dv ae -—p ur aE (88) so that the eddy viscosity is simply u. = Ow, & o |u| -2 (89) The eddy viscosity is related to the reference velocity t and length scale 2 in some time-averaged sense. A rigorous and physically defensible deriva- tion of equation (89) can be found in Battjes (1975). Longuet-Higgins (1970, 1972a) chose to take the characteristic velocity proportional to the maximum bottom wave orbital velocity (u,_) and the refer- ence length scale proportional to distance from shore (x). e closure coef- ficient N in his equation (eq. 55) incorporated both proportionality factors plus assumed that turbulence velocities t were about 10 percent of mean velo- cities, as in normal turbulence (Longuet-Higgins,1972a). 99 All other models for lateral mixing shear stress, Tino use some different combinations of characteristic velocity and length scales to approximate the eddy viscosity. They are summarized in Table 4 along with the resulting ex- pression for vy, Three distinct categories of thought have emerged for both the velocity t, and length 2 scales employed. (1) Reference Velocity, u. Thornton (1969, 1970a) and Jonsson, Skovgaard, and Jacobsen (1974) used time averaged (over one wave period) values of the maximum orbital velocity near the bottom ug, as the reference velocity ti. To reduce its influence outside the breakers where turbulent mixing is smaller, Jonsson, Skovgaard, and Jacobsen (1974) used the mean overdepth, u_, in place of the bottom value. a Madsen, Ostendorf, and Reynolds (1978) used the maximum orbital velocity predicted by linear long wave theory. For shallow water this reduces to equation (53) and is identical to the original model theory used by Longuet- Higgins (1970). It should be noted that (90) and thus up, is related to the surf zone celerity. Longuet—-Higgins (1970, 1972a) never stated that the reference velocity taken was the wave celerity as reported by Thornton (1976, Table 3). Equation (90) also relates the time-averaged and maximum values of these two approaches. Kraus and Sasaki (1979) took slightly different models inside and outside the surf zone and followed the original model of Longuet-Higgins (1970). A completely different approach was taken by Battjes (1975). He felt surf zone turbulence was generated by wave breaking so that the velocity scale chosen should reflect this fact. Since kinetic energy transport is propor- tional to velocity cubed, Battjes took the one-third Down - of the wave energy dissipation per unit area and per unit mass, i.e., (D/p) 3 as the reference velocity. Here, D, the rate of energy dissipation per unit area,was found from equation (50). Skovgaard, Jonsson and Olsen (1978) took the same result inside the surf zone but a simple proportion of vy, (at the breakers) to give less mixing outside the breakers. Inman, Tait, and Nordstrom (1970) related the characteristic velocity to the breaker height and number of waves in the surf zone. The end result was simply H/T. (2) Characteristic Length Scale, 2. , Thornton (1969, 1970a) and Jonsson, Skovgaard, and Jacobsen (1974) used the excursion amplitude to get results for v, that were twice those derived by Thornton. The discrepancy is discussed by Moneconskouseeuisend Jacobsen (1974), Nielsen (1977), and Gourlay (1978) but without resolution. Fortunately, such differences in v, produce relatively small (10 percent) changes in v magnitude and little shape change as indicated by Jonsson, Skovgaard, and Jacobsen (1974). Inman, Tait, and Nordstrom (1970) used solitary wave theory to calculate the horizontal excursion distance employed in their model. Further discussion can be found in Longuet-—Higgins (1972a). 100 (€S) *ba : é (2261 ‘SULBBLY-Jenbu07) (b°0 )queqsu09 5) U9 85 AAS aay SANLeA abuel AL4aAo saath uewey UOA=¥ z/,(uP) sa zy (yb) /X/N (0261) een pue auoz juns aptszno pattdde os|y /X/=% 4A9zZeM MOL LEYS u Zpe wW0730q 3e a8, Jazem MOL Leys UL aire = UOLJOUW [PJLqUO AAEM JO apNyL{dwe uoLSunoxa LezUOZLoY wnutxeu=e suayeauq apisqno 4/4 74, (48)/x/2/1 *S4ayeaug apisjno bULXLW 45M0] 03 (8/61) “Le 38 ‘UaSpPey SMOo| [04 /x/ 7, (46) 2="8n Stones tqyeprsuts 7) (UP) /X/27 Te |a(ec6)) bxeses, pues sneuy suayeauq aptsino Ty z0°9 *suayeauq aplszno pue 4 APLSUL S[apow AZLSODSLA Appa azeuedas 4 e/ (0/0) eRESUt 27, (UP) /x/e , ued op £LT°0 (6Z6L) “Le 38 SpueeBaors ( Step wg wg : [PULBLUO UL YY 02 UELLWLS JUeJSUOD e J /x/ n n/X/1 (8Z6L) “Le 32 Suaspey 2 0 ) San a, g y, ue} z/ (91/2 G)W=#N (0g °be) disstp ABuaua jo azeu | ed0[=g y (°7a) 4 T (yB)/X/.N juapuadap adols paq ‘A10ay} 8° T>W>e°0 Os th ~1X (92461 ‘SZ6T) satazeg *9uoz wg juns aplsuL “e asm ‘abue, se 991M wg wg, Wa, zl : qdaoxa (0/61) uoJUsoY) 032 Ue LWLS ee ae BSOI 5) 7} ee Y (vZ6L) °Le 3a Suossuor *AZLUN Ynoge QUaLILJya0d est y FC S3APM SOAPM J4NS JO °O} (LZ6L) “Le 38 ‘uewuy WYUBLay wayeaug swu st 44 UOLSUNDX aLILJued 4], eY-aUO YOY % POluad BAPM BUD UBAO pabesaav-auly nsoo # att 4H (O61 ‘696T) uozUsoYL queysuo) (6961) uamog o av) ayeos yqbuay A}190 [3A 1 1. Results from equation (90) are shown in Figure 28. As o4%0 (Note that a, = 0 means theory reduces to original order (zero order) solution given by Longuet-Higgins, 1970, the triangular solution is regained. Increasing causes the rela- tive profile to be lowered across the surf zone. Figure 28 is directly com- parable with the results of Liu and Dalrymple (1978) in Figure 26 since both normalized currents by equation (72) as reference velocity, v. Although the general shapes are similar, the results of Kraus and Sasaki (2979) are gener- ally 50 percent greater near the breaker line (a, < 30°). The only difference is in treatment of wave setup wherein Liu and Dalrymple (1978) included full setup effect for large wave incidence. Hydraulically, this trend for free surface flows is in the right direction. As shown by equation (40) oblique angle wave setup is léss than that caused by normal incident waves. For con- stant bed resistance and friction slope, velocity decreases as water depth decreases. The only surprise is in the magnitude of the difference. The full implications of the influence of wave refraction and wave setup on the longshore current theory need further research. Kraus and Sasaki (1979) ar- gued that wave setupinthe field was more complex than given by equations (35) or (40) to justify their result. Also, they point out that including the cosa, term in Yh would reduce the magnitude of their results. (2) General Solution With Lateral Mixing. The y-direction momentum balance equation (42) with lateral mixing is solved by expanding v in a power series. The unknown coefficients are determined from the boundary conditions, namely _ v and dv/dx continuous at the breaker line and finite within the bounds where x*o and infinity. In the usual dimensionless terms, the longshore cur- rent profile is co ZE (AX + Bex) Xa, within surf zone 0l n=o 103 Figure 28. Dimensionless theoretical current profile inside the breaker line as a functiori of incident breaker wave angle, neglecting lateral mixing (from Kraus and Sasaki, 1979). Solution includes refraction and angle-dependent bottom friction force. (a, = 0 means angle correction to zero-order solution is zero.) so that the solution form is consistent with the original model (eq. 65). However, the coefficient expressions and definitions are quite lengthy and involved, so they are included as Appendix D. The key dimensionless parameter P* for this derivation is defined as Tt tang Roe SS a ES B C2 1432/8 (95) where T is the closure coefficient for the kinematic eddy viscosity. P* is slightly different than that defined by Longuet-Higgins (eq. 73); however, this distinction is not fundamental and will be disregarded in the discussion. The factor y must now be specified along with P* to obtain a solution. Dimensionless profiles with mixing parameter P* = 0.5, 0.1 and 0.05 (y= 1.0) and various incident breaker angles (o lone 10 x10" ax10! absorption reflection progressive wave TunO is Me! Some standing wave Set-up predominant run-up predominant *) number of waves in surf zone Recently, Ostendorf and Madsen (1979) defined a modified Battjes breaker parameter as tanf cosa (97) a at (HL /L,) for waves of oblique incidence and using the wave height evaluated at break- ing, for convenience. It is then demonstrated that the longshore current theory discussed above, which neglects such surf zone phenomena as wave runup, wave reflection, edge waves and air entrainment but includes a linear y ratio inside the breaker line, is reasonable when 0.3<é<0.7. The data employed are somewhat different than that used to establish Table 5. Interestingly, they conclude from laboratory data that for y linear in the surf zone, & >0.3, which a contradicts the belief, in this opinion, that the theory holds for all spilling- type breakers. b. Wave Breaking Criteria. The longshore current models simply take 0.5 NUMERICAL W = 0.50 (NO MIXING) = | w (O) 2 | S ©.23 NUMERICAL (WITH MIXING) 0 30 60 90 120 150 180 210 240 270 DISTANCE OFFSHORE (M) a. Numerical model results (Ebersole and Dalrymple, 1980). 1.50 @\ 1.28 i) SS = Si Le = oO sj OTS > uJ ° So 0.50 n [o) S 3 0.25 (0) 30 60 90 120 150 180 210 240 270 DISTANCE OFFSHORE (M) b. Analytical model results (Longuet-Higgins, 1970). Figure 35. Numerical and analytical model results for longshore current, with and without mixing. 128 height, and angle at the shoreline. The original model of Longuet-Higgins (1970), modified by U.S. Army, Corps of Engineers, Coastal Engineering Research Center, (1977) 38 is then used to calculate a mean longshore current. Although local winds are a dominant feature of the field data displayed, no term to include surface wind shear-generated currents is included. e. Others Outside the United States. In Japan, Sasaki (1977) essen- tially followed the efforts of Noda, et al. (1974) and neglected all the terms in the basic equations except mean water surface gradient, radiation stress gradient, and bottom shear. The weak current small-angle friction model is employed. The wave height field is calculated numerically but neglects wave-current interactions. A successive overrelaxation (SOR) nu- merical method solved the resulting equations after first being put in transport stream-function form. Because of the large number of omitted terms and neglect of wave setup, the model is only valuable as a general indicator Of trends (see Ch. 4). In response to the need to investigate currents near a proposed cooling water intake basin on the coast, Bettess, et al. (1978), in England, de- veloped a steady-state finite-element model. It included all terms in the motion equations plus the Coriolis accelerations. Wave-current refraction effects in the wave height field calculations were neglected. Wave height fields were also calculated using the finite-element method for solution of Berkhoff's (1972)22 modified form of the shallow-water equations. An example of their results for currents calculated using a constant eddy viscosity co- efficient is shown as Figure 36. The size and strength of the large eddy in the lee of the breakwater compares favorably with physical model results (dotted lines). The authors call for an improved means to simulate surf zone energy decay, radiation stresses with standing waves, and lateral mixing eddy coefficients in order to improve the numerical simulation. Finally, Vreugdenhil (1980) carefully outlines methods presently being implemented at the Delft Hydraulics Laboratory to develop a numerical model for unsteady wave-driven currents. The primary purpose of this model is to better understand physical processes such as migrating rip currents. The equations employed are written in conservation form (eqs. 107, 108, and 109) with all terms included. Modules are introduced to permit easy variation or suppression of submodels for lateral mixing stress, bottom friction, the wave theory in the radiation stress, wave breaking, and the surf zone energy loss criteria. Wave height fields are computed from linear theory to include refraction from both depth and current variations, and diffraction effects can also be included. The numerical method selected as the finite-difference method of the implicit type. Together with a transformed coordinate system to readily handle curved breaker lines and boundaries, the finite-difference method was felt superior to the finite-element method where little is known about 38y.s. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, Shore Protection Manual, Vol. I, U.S. Government Printing Office, Washington, D.C., 1977 (mot in bibliography) . 39RERKHOFF, J.C.W., "Computation of Combined Diffraction-Refraction," Proceed- ings of the 13th Coastal Engineering Conference,Vancouver, 1972 (not in bibliog- raphy). 129 / / lela ae | ( “bed ddas . . > ~ — —— —- — — — =_— —_ Nearshore Circulation Pattern, constant viscosity Physical Model Pattern shown dotted Figure 36. Nearshore circulation pattern, constant viscosity; physical model pattern shown dotted (from Bettess, et al., 1978). accuracy for propagation-type problems. The locally one-dimensional implicit method (Mitchell and Griffiths, 1980)*° was employed to efficiently and numeri- cally integrate the equations. The weighting coefficient between upper and lower time levels 6 of the two level scheme is adjustable for efficient steady- state solutions (@ = 1) or accuracy (6 = 4) in unsteady flows, All the space derivatives are centered. Considerable discussion is presented by Vreugdenhil (1980) regarding stability, numerical accuracy, and the boundary conditions. Complete details are beyond the scope of this report but such analyses are critical to the quantitative success of any numerical simulation. For example, the direction- al nature and repetition of the solution procedure for the x-direction and y-direction sweeps (using either locally one-dimensional or alternating direc~ tion implicit methods) will affect the amount of numerical viscosity or dif- fusion generated. Numerical viscosity can easily be much greater than the eddy viscosity even for schemes using centered space derivations for the con- vective acceleration terms. For steady-state solutions, spatially oscillating solutions (wiggles) in the velocity fields can be generated even in linearly stable schemes. These come from the nonlinear convective acceleration terms and can be artificially damped by using large eddy coefficients at the expense of numerical accuracy. The relationships between boundary conditions, bed friction, internal, lateral eddy viscosity, numerical viscosity (truncation *OMITCHELL, A.R. and GRIFFITHS, D.R., Ihe Finite Difference Method tn Partial Differenttal Equations, Wiley-Interscience, J. Wiley and Sons, New York, 1980 (not in bibliography). 130 errors), and boundary geometry for the proper simulation of flow vorticity, circulations, eddies, etc. in numerical models are still an active research area. The modeling effort described by Vreugdenhil (1980) is by far the best effort to model nature because it includes all the physically important terms and simulates them numerically with the most accuracy. VI. NONLINEAR AND IRREGULAR WAVES All the analytic and numerical methods described previously in this chapter were with radiation stresses computed from linear wave theory for regular sinusoidal waves. The stress components are then simply given by equations (23), (24), and (26) or first-order theory. Because of the rela- tively crude assumptions needed for modeling the surf zone energy dissipa- tion, it could be argued that higher order radiation stress terms were not warranted. On the other hand, since irregular waves naturally occur and break at different offshore locations, the stochastic approach to longshore current modeling may be more realistic. Both nonlinear and irregular wave theories of MWL change and longshore currents are reviewed in this section. Differences and similarities with the linear regular wave theories are noted. Ibe Nonlinear Waves. a. MWL Change. James (1973, 1974a) used third-order Stokes theory in deep water and a modified (Iwagaki, 1968)*! cnoidal wave theory nearshore to compute the higher order radiation stress needed to define wave setdown and setup for spilling breakers on plane, gentle slopes. Wave setdown was less than that found by linear theory especially near the breaking point. Theo- retical wave setup is‘also less and the gradient is not a constant proportion of the beach slope as in linear theory. Numerical methods are employed to integrate the resulting ordinary differential equations. The theory for cnoidal waves over a gently sloping bottom (Svendsen, 1974)*2 is used by Svendsen and Hansen (1976) to derive analytic expressions for wave setdown. Near breaking both nonlinearity and vertical acceleration effects must be included in the wave theory. Using the actual bottom velocity in cnoidal waves given by 32n uy =e a - 2) +e cd (115) ox setdown became 4] TWAGAKI, Y., “Hyperbolic Waves and Their Shoaling," Voastal Engineering in Japan, Vol. I1, 1968, pp. 1-12 (not in bibliography). 42 SVENDSEN, I.A., "Cnoidal Waves Over a Gently Sloping Bottom," ISVA, Series Paper No. 6, Technical University of Denmark, Lyngby, 1974,) (not in bibliography). 131 bee 2 = 3 Spayece (Sai ie (116) | where c is cnoidal theory celerity, and n is wave surface variation based on second-order cnoidal theory. The results again give less wave setdown than linear theory and are closer to experiments in this regard as discussed in Chapter 4 where other even closer approximations are also shown. Maximum wave setup, n_, is determined by Jonsson and Buhelt (1978), using a series of solitary waves just outside the breaker line to calculate the mean water depth at the breakers, a -Y, -"/ Bag OL 77 ie (Hi om a (117) When inserted into equation (37) using Hy = yh, they obtained the result Y/, 3 = 0.107y Io Ga): (118) oO This expression includes setdown effects and holds for any beach profile where depth decreases continuously. These results are for normal wave in- cidence and have been found to explain how beach slope and wave steepness influence maximum setup in experimental data (Ch. 4). b. Uniform Longshore Current Profile. James (1973, 1974b) was pri- marily interested in how a nonlinear wave theory for the radiation stresses would affect the longshore current profile. The same wave theories as re- viewed for wave setup were employed. He took the full nonlinear bed fric- tion expression (eq. 52) and Longuet-Higgins (1970) expression for lateral mixing stress (eq. 55) in his numerical solution methods. All results are for plane, flat beaches with spilling breakers and y = 0.85. A comparison between the linear and nonlinear theories is shown in Figure 37 (after Gourlay, 1978) where it is important to recognize that the nonlinear solu- tions include both the nonlinear bed stress and wave setup effects while the linear solution does not (original model, Longuet-Higgins, 1970). With this in mind, it is observed that the linear theory required significantly larger bottom friction values (to match experimental surf zone currents) and is more sensitive to eddy viscosity variations than the nonlinear theory. James (1974b) concluded that the results for longshore current in the surf zone are of the same order of magnitude for both linear and nonlinear theories. Perhaps this is the reason his research remains the only one known of this nature. All such efforts require numerical solution methods which, in light of their use for two-dimensional solutions, is not an additional problem. Dutch researchers are planning to vary the wave theory in the radiation stress terms of their numerical model (Vreugdenhil, 1980). Their future results and others of a similar nature will be of considerable interest. 132 offshore zone surf zone (a) £ = 0.004, N = 0.016 Serle =| 0.0040) N= OF Ol wc wc nonlinear theory flame linear theory cocvccccceeccsceccce offshore zone oce e* . (b) Linear theory 90.9 9019 000006 Eom = 0.004, N = 0.01, P = 2.09 809 ——— 0.005 0.01 167 ----- 0.004 0.016 3.34 —_- — 0.014 0.01 0.60 SS 0.004 0.05 10.44 Figure 37. Comparison between linear and nonlinear theories of uniform longshore current profiles (from Gourlay, 1978). Zo Irregular Waves. In a wide range of circumstances, wave breaking is a local phenomenon in an irregular (random) wave field that is qualitatively similar to individual wave breaking. Both the type of breaker and the wave height at breaking are similar. However, in the irregular wave surf zone there is no one given breaker line since at each location only a percentage of waves passing have broken, and this percentage varies gradually toward the shore giving rise to average gradual variations of energy density, energy flux, momentum flux, and other wave parameters. The dissipation of wave energy and resultant decrease in radiation stress for irregular waves that break in the surf zone has been the subject of extensive research by Collins and Wier (1969), Collins (1972), and Battjes (1972, 1974). JSS) Two approaches to mathematicaly describe irregular waves in shallow water exist. Ijima, Matsuo and Koga (1972)*3 used an equilibrium linear spectral model (frequency domain) related to the deepwater spectrum. McReynolds (1977) used a two-component frequency spectrum to simulate a nar- row spectrum. Battjes (1974a) argued that this approach gives an upper saturation limit on the spectral density values which depends on the width of the spectrum -- this is not realistic. Also, wave breaking is highly non- linear and occurs to individual waves in physical space (space domain) and not to individual spectral components. For these reasons, Collins (1972) and Battjes (1974a) both adopted irregular models based on a wave-by—wave height theoretical and empirical probability distribution for individual waves in the space-time domain. In this model, it is the integral of the spectrum which has an upper bound in shallow water and not the spectral density. No rigor is claimed, only a rational approach where nonlinear pro- cesses are important. Collins' (1972) approach was to consider the irregular sea as the en- semble average of periodic components, each with its own H,, Lo, and ao in deep water. The energy, energy flux, momentum flux, radiation stresses, and longshore current velocities are assumed expressible in terms of Ho> Lo, and a, for regular waves. The mathematical expectation of these quantities is then calculated assuming that the joint probability density function of the stochastic ensembles of H,, L,, and 4, is known. Suitable functions for the joint probability density function are determined by empirical means from field data. Implied in this approach is the assumption that various non- linear inte~actions between the waves and the mean motion (e.g., wave setup changes, wave characteristics) are properly represented by ensemble averages of individual waves. Battjes (1974a) presents arguments to show that this assumption is incorrect. In the highly nonlinear surf zone, the contribution of a wave with certain characteristics to wave setup, longshore current velo- city, etc. is affected by the presence of waves of different characteristics. Therefore, Battjes took a different approach. For irregular waves on gentle slopes with spilling breakers, Battjes made the basic assumption that ",..at each depth a limiting wave height H, can be defined (which may also depend on the wave period), which cannot be exceeded by the individual waves of the random wave field, and that those wave heights which in the absence of breaking would exceed Hj, are reduced by breaking to the value H,." (Battjes, 1974a, p. 125). The energy variation in the surf zone thus results from clipping a fictitious wave height distribution which is present (theoretically) if breaking did not occur. This upper bound is found from the regular wave breaking ratio Y. In this way, energy varies gradually due to shoaling, refraction, bottom friction, and because of the increasing number of breaking waves in shallow water. 4+3IJIMA, T., MATSUO, T., and KOGA, K., "Equilibrium Range Spectra in Shoaling Water, " Proecsedinge. l2th Camenall Engineering Conference, Vol. I, Washington, D.C., 1972, pp. 137-149 (not in bibliography). 134 The fictitious wave heights He are assumed to follow the Rayleigh prob- ability distribution. Battjes (1974a) presented additional arguments and empirical evidence to support this choice even where waves are definitely nonlinear and do not possess a narrow spectrum. Clipping this fictitious wave height distribution at H = H, gave the following approximation to the true height distribution, F(H) 0 ah xs (0) F(H) = P{H < H} = {1-exp(-H?/H2) OSS Ta (119) 1 lel 22 Jal ™m where H = the stochastic wave height H = the wave height of interest He = the mean square vaiue of the fictitious wave height Ho = the maximum possible wave height in the surf zone, i.e., Hy: In the shallow-water surf zone, it is further assumed that the effects of variability of wave period and wave direction on breaker heights are neglig- ible. These factors are important, however, in calculations of the ficti- tious wave heights from the complete two-dimensional spectrum. The mean energy per unit area at a fixed location, taking breaking into account is then calculated from linear theory as Be Spall? (120) where H2 = | H2dF (H) (121) The radiation stresses which can be found (to second order) as the weighted integral of the two-dimensional spectral density are of interest. But again, in the shallow-water surf zone, the only frequency-dependent weighting factor is n which is approximately unity. Consequently, Battjes (1974a) assumed that the radiation stresses are reduced by breaking in the same proportion as the total energy pees Se ee Siy E, i, {1-exp ( Hy / He) } ih, (122) where Si. = components of the radiation stress tensor Se = components of the fictitious radiation stress tensor J¢ without breaking such that 135 Soy = Soe = transformed component in x-direction S =S = transformed component in y-direction 22 yy S = S$ = = transformed shear stress component Woes sto and given by equations (23), (24), and (25), respectively. The validity of this approach improves for narrow frequency and directional spectra. Another model for predicting irregular wave height distributions near- shore on continuously decreasing depths was proposed by Goda (1975). The following assumptions are made: (a) the equivalent significant wave height and peak spectral period in deep water are known; (b) the Rayleigh distri- bution applies in deep water for wave heights; (c) average beach slope is known; (d) empirical formulas for wave setup, breaking limits, etc. are applicable; (e) wave shoaling is nonlinear; and (f) broken waves can re- form at smaller heights. A numerical procedure to use this model to predict nearshore conditions, maximum wave heights, and critical water depths has recently been developed (Seelig and Ahrens, 1980'"; Seelig, 1980*°). Goda's model is similar to Battjes (1974a) but provides a smoother cutoff at break- ing by use of a varying probability for the breaking ratio, y. Finally, as briefly mentioned earlier in Section IV of this chapter, Battjes and Janssen (1976) used hydraulic jump bore theory to calculate the energy loss rate in the surf zone for irregular waves. The same probability theory as described above (Battjes, 1974a) is utilized except the probabil- ities (Q) are expressed in terms of H i and He to give a clearer physical meaning. The local value of H is found by integrating the fundamental surf zone energy equation (44) = oF x = aE +D=0 with F.. = ECgcosa ad B= Spake (123) To close the system of equations for H,,., the rate of energy dissipation per unit width D was determined from classical hydraulic jump theory with the depth across the jump approximately the local wave height. This gave isd LS OEAAHe D=K7 Q fogHe (124) for irregular waves with £ the mean frequency df the energy spectrum and K, a constant, near unity if the model is acceptable. 44+ SEELIG, W.N., and AHRENS, J., "Estimating Nearshore Conditions for Irregular Waves ,''TP 80-4, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., June 1980 (not in bibliography). =I SPRING. W.N., “Maximum Wave Heights and Critical Water Depths for Irregular Waves in the Surf Zone," CZTA 80-1, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., Feb. 1980 (not in bibliography) 136 . Equation (124) is the key result of Battjes' (1978) paper and when in- serted into equation (44) permitted H,,,, to be calculated across the surf zone for sloped or bar-type beach profiles. The closure parameters are K and y, and it is important to point out that y is only used as a breaking criteria and not to estimate the wave height decay. The main interest is in the resulting mean water surface changes calculated from the momentum balance as discussed below. a. Wave Setdown and Setup. Figure 38 displays example theoretical mean water surface profiles for two beach profiles using the theory of Collins (1972). Less wave setdown is evident for the irregular waves than regular (monochromatic) waves with the same energy content. The wave setup profile is also highly nonlinear in the surf zone and maximum setup is less for irregular waves. Similar results are shown in Figures 39 from Battjes (1974a) for a plane beach. Note the horizontal axis is stillwater depth, d, normalized by deep- water wave height, H_, and could be related to horizontal distance offshore. Details of the solution method are omitted here but include shoaling and bot- tom refraction effects. Numerical integration procedures are employed and the results reduced to the regular wave setup as a check (eq. 35) for normal wave incidence. Figure 39(a) shows how wave refraction reduces wave setup, as expected and the mean water surface profile is almost linear near the Lo =500 ft. =20 ft® @ =30° WAVE SET UP, ft. DISTANCE OFFSHORE, ft. Figure 38. Illustration of wave setup on offshore slopes of 1:25 and 1:50, broken lines indicate results for periodic waves having the same energy content (from Collins, 1972). 137 Ho/Lg= 0.02 y =0.8 calcs. I (narrow spectrum) Q = 15° y =08 calcs.I (narrow spectrum) Ho/L, = 0.005 of -0 WA (b) Figure 39. Calculated setup curves for (a) various mean angles of incidence and (b) various wave steepnesses (from Battjes, 1974a). 138 the maximum setup line, in contrast to Collins (1972). Wave steepness effects are shown in Figure 39(b). Battjes (1974a) used the breaking criterion of Miche. (1951)*© which included wavelength effects so that decreasing steepness (longer waves) made a wider surf zone to increase setup. In both these exam- ples, y = 0.8 In Goda's (1975) model, the effect of wave steepness on setup is similar to that obtained by Battjes (1974a). However, the magnitudes of both setdown and setup are relatively greater and depend more on beach slope since y is variable and increases as beach slope increases. This causes a greater maxi- mum setup than the Battjes (1974a) model. The more sophisticated hydraulic jump model of the surf zone by Battjes and Janssen (1978) produced the MWL variations shown as Figure 40. In all cases the theoretical wave height variation across the surf zone is also shown, The steep plane beach (Fig. 40,b) has less setup than the flatter pro- file (Fig. 40,a). A smooth transition from setdown to setup is again evident. The results for two bar-trough profiles are given in Figure 40 (c and d). In one case (Fig. 40,c) two wave setdown regions are apparent while in another case (Fig. 40,d) a broad flat setup region is created. The crosses (x) in- dicated laboratory measurements, and a detailed discussion of the comparison is given in Chapter 4. Finally, based on extensive measurements in the field off the Isle of Sylt (Germany) in the North Sea, Hansen (1978b) offered the following empi- rical relations for the maximum wave setup due to irregular waves on a flat plane beach n = 0.3) il max 0,s (125) =} i] max Doe SDE where H, , is the deepwater significant wave height and H a is the same but at the breakers. ; b. Longshore Current Profile. Collins (1972) and Battjes (1974a) were primarily interested in the effect of irregular waves on longshore currents. In a random sea the breaking line for regular waves becomes a zone for irre- gular waves since the higher steeper waves break farther offshore than the gentle waves which break near the beach. The stochastic description of a real sea state (either spectral or probablistic) will find waves breaking in different locations at slightly different angles and with a range of wave heights and lengths. The net result is that irregular waves provide a de- gree of lateral mixing to distribute the longshore currents and produce a smooth profile across the surf zone similar to that found in the theory of regular waves when the lateral turbulent mixing stress term is included (see 46MICHE, R., "Le Pouvoir Réfléchissant Des Ouvrages Maritimes Exposes a L'action De La Houle," Amn. des Ponts et Chaussées, 121 é Année, 1951, pp. 285-319 (not in bibliography). 139 (a) Run 3; experimental values (x); (b) - Run 2; experimental values (x); theoretical values for theoretical values for a=1, y = 0.8 ( ) a=1, y = 0.8 ( ) Bun 13; experimental values (x); Run 15; experimental values (x) ( ) theoretical values for d theoretical vaiuea for c a=, y= 0.8 ( ) (d) a-l, 7° 0.8 ( ) Figure 40. MWL variations for irregular waves (a),(b) plane beach (c), (d) bar-trough profiles (from Battjes and Janssen, 1978). 140 Fig. 24). In their original contributions, both Collins (1972) and Battjes (1974a) chose to omit the lateral mixing term since their irregular wave theories provided the same effect. They recognize, however, that such a term should also be included in future analysis. Some example results from Collins (1972) are presented in Figure 41 for two beach profiles. The flatter slope produces a wider surf zone and broader longshore current profile. The dashlines are for regular waves with the same total energy content from the original theory of Longuet-Higgins (1970). No lateral mixing stress terms are involved. The extent and magni- tude of the longshore current outside the breaker line is excessive and per- haps due to the lack of nonlinear interactions in Collins' approach. The results obtained by Battjes (1974a) are more realistic in this re- gard. A weak current small-angle bed shear-stress model (eq. 51) was em- ployed. For irregular waves with a narrow spectrum, Battjes took the time mean wave orbital velocity near the botton, up rather than the maximum. This gave 3 | 10%) Ga! (126) LONGSHORE CURRENT, ft. /sec. (e) cS [——- (9) 200 400 600 DISTANCE OFFSHORE, ft. Figure 41. Illustration of the effect of offshore bottom slope on longshore currents, deepwater wave angle = 30°, deep- water wavelength = 500 feet, broken lines show longshore currents for periodic waves having the same total energy content (from Collins, 1972). 141 of — als where Ww = =" - (gk tanh kh)* = a mean frequency fo) T = the deepwater mean wave period (i.e., peak period for narrow spec-— = trums) and H = the mean wave height q= oy erf ( ) (127) ZO H rms £ rms with H. = the fictitious root mean square wave height rms the local breaker height om erf = the error function. Small errors (<10 percent*in deep water) result if H is replaced by H cane in equation (126). Using a bed shear-stress model of the form tT, = C_pu.v (128) in the longshore momentum balance equation (42) without lateral mixing gave ds = —___—_. 1 k me: 'A v=t 5 c. = sinh kh dz (129) with x again defined a positive in the ocean direction. When the wave setup is included as expressed in terms of the bottom slope for a plane beach (i.e., see eq. 68) this expression for v becomes a ap es 1 ds. lk v= = tang sinh kh ah (1 + SE Gln ) (130) 2o0c -H ie where the wave setup has yet to be determined. Battjes (1974a) defined a normalized current velocity — Cel, = >1 Nonlinear long wave equations u { 0e(1) Boussinesq equations (136) < = S|] ays h dxoy dt a gn SOmyds) , 88 (Pa on * eo Gy) @ ie Sho 7 SB oe _ ae on cl erg eOSIETAD 5 Oellien) 1 Daaare we Ce) S3GREEN, A.E., and NAGHDI, P.M., "A Derivation of Equations for Water Propagation in Water of Variable Depth," Journal of Flutd Mechanics, Vol. 78, Pt. 2, 1976, pp. 237-246 (not in bibliography). S4MASS, W.J., and VASTANO, A.C., "An Investigation of Dispersive and Nondis- persive Long Wave Equations Applied to Oceans of Variable Depth," Reference 78-8-T, Department of Oceanography, Texas A&M University, College Station, Texas, July 1978 (not in bibliography). SSABBOTT, M.B., and RODENHUIS, G.S., "On the Formation and Stability of the Undular Hydraulic Jump,'' Report Series No. 10, International Courses in Hydraulic and Sanitary Engineering, Delft, Netherlands (not in bibliography) . S6LIGGETT, J.A., "Basic Equations of Unsteady Flow," Ch. 2, Unsteady Flow in Open Channel, Vol. 1, K. Mahmood and V. Yevjevish, eds., Water Resources Publications, Fort Collins, Colo., 1975 (not in bibliography). 149 where p = uh and q = vh are the volumetric flow rates per unit width. Numeri- cal integration methods, testing, and some application examples of these equations are reviewed below. Recently, Hauguel (1980) obtained somewhat different results by assuming that the vertical velocity increases linearly through the water column and averaging the Navier-Stokes equations over this depth. It should be noted here that such an assumption for the vertical velocity, w is equivalent to assuming a uniform horizontal velocity profile based on continuity principles. If the bed stress term is neglected, the x-direction motion equation is Dynes (Sy pe Ay 8) BHD, acne To em oe Joe a Ay, 92 =(g+bt+ ph a (147) where p, q, and h are as previously defined, z is the bed elevation above an arbitrary datum, and the new terms a and b characterize the vertical accel- erations due to wave steepness and bed-slope variations. They are defined as 2 ge deby (148) dt2 2 ba Se (149) dt2 where the total or substantive derivative is given by ws 08 Pein Gy Oe dt ot 'h ox h dy 0) The derivation is said to follow that given by Serre (1953)°7 for cnoidal waves. A similar expression is given for the y-direction momentum. This approach may permit steeper waves to propagate farther over irregular bathy- metry. e. Limitations of Boussinesq Theory. The range of application of equations (144), (145), and (146), based on a comparison with first-order cnoidal-wave theory and Dean's stream-function wave theory (Dean, 1974)°°, / 57SERRE, F., "Contribution a letude des ecoulements permanents et variables dans les canaux,'"' La Houtlle Blanche, 1953, pp. 374-388, 830-872 (not in bibliography). 8 DEAN, R.G., "Evaluation and Development of Water Wave Theories for Engineering Application,"SR-1. Vols. I and IL, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., 1974 (not in bibliography). 150 as determined by Abbott,et al. (1978), is presented in Figure 43. In shallow water, the wave celerity approaches the group velocity or speed of energy propagation. the theory. breaking limit. Hence celerity is a good indicator of usefulness of An error of 2 percent is indicated for a wide range at the The expected application limit shown is for d/L = 0.2 (d/Lo = 0.17). This is in the intermediate wave theory range and near the engineer's limit of d/Lo = 0.25 for deepwater waves. Even near this limit, the assumption of a uniform velocity profile is still reasonable as shown in Figure 44, The Unax profile based on classical linear wave theory for d/L = 0.22 is shown for comparison. variation with depth. The vertical wy4x profile is very close to a linear The exponential decay of the horizontal velocity com- ponent is roughly analogous, in some respects, to the fully developed log- arithmic modeled as depth-averaged flow. LES 257 H lo gt2 10° a PERFORMANCE ENVE- LOPE FOR "CIN THE >T SINUSOIDAL (AIRY ) APPROACH (PERIODIC WAVES ON CONSTANT DEPTH. ) BREAKING LIMIT -Hs,.8xD 102} LESS THAN 14% ERROR IN C boundary layer profile of open channel flows which are routinely 6 LIMIT OF MOST | PRACTICAL PROBLEMS H = WAVE HEIGHT Leer TO CREST ) ; D = MEAN WATER DEPTH i, L = cT= WAVE LENGTH Lge fy t?: L FOR D-» o9 AND H+ Oo c = PHASE SPEED T = WAVE PERIOD g = GRAVITY ACCELERATION Figure 43. (from Abbott, et al, 1978). 151 Range of.application of the mass and Boussinesq equation system. Nn (Linear Wave Theory) Relative 95 Depth 2—u (Depth Averaged) 04 Linear Wave Theory ---- Quasi, 2-D Theory O : OQ OZ “Oe @S Oss LO 1.4 I< Depth Factor Figure 44. Comparison of horizontal and vertical maximum velocity profiles between quasi, 2-D, and linear wave theories for d/L = 0.22. The practical limitations for engineering purposes of the Boussinesq theory are still open to question and require further research. This is because for variable bathymetry, the precise form of the continuum equations involved is not known. In addition, accuracy errors in the numerical solu- tion procedures produce additional numerical amplitude and frequency disper- sion effects. Numerical integration procedures are very important in use of the Boussinesq theory. Die Numerical Solution. The finite-difference method has been the only numerical integration method emptoyed to date to solve the Boussinesq equations. Early numerical efforts to study one-dimensional wave shoaling and transformations over 152 steps (e.g., Peregrine, 1967, Camfield and Street, 1969°9, Madsen and Mei, 19699, Chan and Street, 1970°9) must be reviewed with caution since the numerical accuracy of the results is questionable. This is because the truncation errors from the acceleration terms were of the same order or lower order than the Boussinesq terms and essentially masked its effect. These researchers were interested in the physical aspects of wave trans- formations which were qualitatively simulated so that the value of these early efforts from this. aspect is not diminished. In addition, all numerical methods must’ use only a discrete number of points per wavelength (N) to define each wave. Finite-amplitude (cnoidal) waves when decomposed by Fourier analysis consist of many, superimposed sinusoidal components. All harmonic components are described by fewer and fewer grid points, and this coarse description can lead to amplitude and phase errors. Thus, solutions of the Boussinesq equations are extremely sensitive to numerical errors (Abbott and Rodenhuis, 1972°°, Hauguel, 1980). a. Abbott, Petersen, and Skovgaard (1978a, 1978b). These researchers of the Danish Hydraulics Institute were the first to successfully develop a two-dimensional numerical model to accurately propagate quasi-long waves over variable bathymetry to near the breaking limit. They integrated equa- tions (144), (145), and (146) using a staggered, implicit finite-difference scheme. One key aspect of their contribution (see also Abbott, 1978, 1979) is the method to remove all truncation error terms of order comparable to the Boussinesq term so the resulting finite-difference equations are third- order accurate. This was done (following the observations by Long, 196461) by rewriting all higher order truncation error terms using the linearized wave equations with no loss of accuracy or generality. In one dimension these linearized equations are on , 9p aS dt Ox. | e en) Sp on _ 152 ve + gd a 0 ( ) In this way, the truncation errors are finite-differenced, combined with the finite-difference Boussinesq term and subtracted out of the equations together in an efficient manner. It is theoretically possible to use higher order accurate finite-differences initially to accomplish this goal i but Abbott, Petersen, and Skovgaard (1978a) claim this leads to "...algo- rithmically intractable difference forms of the Boussinesq equations." SICAMFIELD, F.E., and STREET, R.L., "Shoaling of Solitary Waves on Small Slopes," Journal of the Waterways and Harbors Division, Vol. 95, No. Wwl, Feb. 1969, pp. 1-22 (not in bibliography) . 60CHAN, R.K.C., and STREET, R.L., "Shoaling of Finite-Amplitude Waves on Plane Beaches," Proceedings of the 12th Coastal Engineering Conference, American Society of Civil Engineers, 1970 (not in bibliography). S1LONG, R.R., "The Initial Value Problem for Long Waves of Finite Amplitude," Journal of Flutd Mechanics, Vol. 20, Pt. 7, 1964, pp. 161-170 (not in bibliography) . A number of performance envelope tests were conducted on the numerical scheme to determine the extent to which it came close to the continuum limitation shown in Figure 44. These were all for nonbreaking waves and included the following one- and two-dimensional tests: (1) One-dimensional: (a) Finite-amplitude, periodic wave propagation on a horizontal bottom; (b) Full and partial wave reflections (transmission) ; (c) Wave shoaling on plane beach. (2) Two-dimensional: (a) Wave diffraction, and (b) Comparison with physical model results of harbor resonance. An example perspective view of two-dimensional short wave propagation on the computer is shown in Figure 45. A comparison of the numerical results with physical experiments and other theories is presented in Chapter 4. For practical applications, l-second time steps and 10-meter space steps are estimated to give reasonable results for engineering purposes. = Za ; ee Pea >= eI, SPF VIS Figure 45. Numerical propagation of quasi-long EES in two dimensions using Boussinesq equations (from Abbott, 1979°*). 154 b. Others. Hauguel (1980) employed a different numerical solution strategy with the finite-difference method for equations (147) to (150). A predictor-corrector scheme which uses the method of characteristics in the prediction phase and a three time level implicit scheme in the corrector phase is employed. Complete details are forthcoming since only the abstract is presented in Hauguel (1980). Additional efforts in one dimension using the techniques developed by Abbott, Petersen, and Skovgaard (1978a) for truncation error removal but with an implicit three level box or Preissmann®? scheme have been made by McCowan (1978)®? or are in progress by Jensen (personal communication, 1981). 3. Simulation of Surf Zone Hydrodynamics. Research is currently in progress to extend the Boussinesq approach to include wave breaking, surf zone energy dissipation, and lateral turbulent mixing effects in an effort to simulate the hydrodynamics of the surf zone. The main features of the flow are of interest which will be the instantaneous depth-integrated velocity components and water depth. Time-averaging these results would produce longshore currents, nearshore circulations, rip cur- rents and MWL changes, if desired to compare with the many radiation stress theories. Such numerical results are a few years in the future. In summary, the primary advantages of the Boussinesq approach are (a) the insight into physical processes taking place within each wave period, and (b) the elimination of the need to specify the wave height field by other calculation procédures throughout the area of interest. On the other hand, the approach still requires empirical surf zone simulations that are just being developed plus large efficient computers for the extensive com- putations. VIII. SUMMARY: THEORETICAL ASPECTS A comprehensive review and summary of relevant theories since 1967 has been presented. The time-averaged radiation stress approach has been ex- tended in considerable detail since first introduced. Mean water level variations and net currents are linked together by the conservation laws of mass and momentum. The primary driving force is created by the excess momentum flux due to the waves, i.e. the radiation stress gradients. Major modifications to the original theory have taken place primarily in three areas: 62PREISSMANN, A., "Propagation des Intumescenses Dans Les Canaux et Rivieres," Cer Congres de 1'Assoc., Francaise de Calcul, Grenoble, 1961, pp. 433-442 (not in bibliography). 62McCOWAN, A.D., "Numerical Simultation of Shallow Water Waves," 4th Australian Conference on Coastal and Ocean Engineering, Adelaide, Nov. 1978, pp. 132-136 (not in bibliography). 155 (a) Bed shear-stress formulations, (b) extensions to irregular waves, and (c) use of two-dimensional numerical models. Extensive use of empirical surf zone models remains a weak link in the theory, expecially in determining, when, where, and why waves break. Use of the method also requires prior specifications of wave heights throughout the region of interest. This aspect poses difficulties in its own right due to wave shoaling, refraction, diffraction, reflection, transmission, and wave-current interaction computations required. Additional research is needed; e.g., the modified bed shear-stress and lateral mixing stress models used with an accurate two-dimensional numerical formulation to study both regular and irregular wave conditions. Time-averaging masks the physical processes that occur at scales within each dominant wave period. The Boussinesq approach unlocks this information and allows finite-amplitude waves in the nearshore zone to propagate and interact with the surroundings in a fundamental way. Solutions to the gov- erning equations are only possible with the aid of large high-speed computers. A new research tool is thus emerging to perhaps raise the level of theoreti- cal understanding above that obtainable from radiation stress theory. In either case, only depth-integrated flow characteristics are consid- ered. No truly three-dimensional model of coastal hydrodynamics has been attempted. Which theory is correct? All the physical observations (Ch. 2) have purposely been separated from the theory (Ch. 3) and left for a comparison of each in Chapter 4. It will become apparent that the number, extent, and detail of the theory far exceed the number of good data sets available for comparison and verification. 156 CHAPTER 4 EXPERIMENTAL VERIFICATION OF THEORY Again, all theory before 1967 which has been thoroughly discussed by Galvin (1967) has been omitted in this report. Results of comparisons between laboratory and field measurements and theory as original presented in the literature are reviewed. No new analysis of the available data base is attempted. Only those equations from Chapter 3 that have experimental backing or are seriously questioned are repeated in this chapter. I. WAVE SETDOWN AND SETUP les Regular Linear Waves. Some classic results of MWL change as determined in a 0.5-meter-wide by 0.75-meter-deep by 40-meter-long wave flume by Bowen, Inman, and Simmons (1968) are shown in Figure 46. The wave height in deep water was 6.45 centimeters and the beach slope 1 on 12. For normal wave incidence on a plane beach, wave setdown is shown to match the theory given by equation (30), except near the breaker line. Wave setup was found to be almost linear over the entire surf zone and proportional to the beach slope as expected from equation (35), repeated here dn _ 1 s ie 1 OIRe tan 8 = K tan 8 and letting | Le ee SS tan B (hina? Ny), (153) K versus y as plotted in Figure 47. Included are data from Putnam, Munk and Traylor (1949) for other slopes and the agreement was characterized as Me - quite good." Gourlay (1978) reanalyzed the Bowen, Inman, and Simmons data and con- cluded that values of maximum wave setup mass in equation (153) were over- estimated (see App. A4.43 in Gourlay, 1978) due to inclusion of wave runup effects. Both plunging- and spilling-type breakers were produced in the tests of Bowen, Inman, and Simmons (1968). Similar results were obtained by Smith (1974) for breaking waves of the same types. Gourlay (1978) also adjusted these data to account for total water volume and include a discussion of the results. 157 MEAN WATER LEVEL, 7 THEORY Vi > S.W.L: ° a ° EXPERIMENT +} BREAK POINT PLUNGE POINT ENVELOPE OF WAVE: HEIGHT ue at x x* BEACH WAVE CREST x xx**" a x xXx xxw Xx xxxxx% Xxx x* XXX KX KKK XX * x XoXo) Ke AKIN RUIN xX x x % WAVE TROUGH ‘400 300 200 100 (e} DISTANCE FROM STILL WATER LINE ON BEACH, x (cms) Figure 46. Wave setdown and setup, comparison of experiment and theory (eqs 3 and 8) for regular waves on a plane beach (after Bowen, Inman, and Simmons, 1968). Additional experiments in the same wave basin as originally used by Bowen, Inman, and Simmons (1968) but with three different beach slopes (tan B = 0.022, 0.040, 0.083) have been reported by van Dorn (1976). The object was to distinguish wave setup from the dynamic shoreline motions called runup due to partial reflections. The empirical result was that the setup slope was found approximately linear across the surf zone (as before) but proportional to the square of the beach slope. This relation 158 O tan B+0 082 S10 (1966) + tan B+ 0072 O4- PUTNAM (1945) e@ tonB+0 054 ° K “THEORY O2- OIF ee SS eee O06 08 10 12 Figure 47. Ratio of wave setup slope to beach slope (K) versus y (after Bowen, Inman, and Simmons, 1968). is shown in Figure 48 (after Gourlay, 1978) with the equation dn _ nies aay ane BoA) eehli= (8 4 (154) given as line of best fit to the data. The vertical error bar lines indicate possible variatioh due to wave periods of the experiments. Results of all experimenters, including that by Gourlay (1978) are also shown. Van Dorn (1976) concluded that on relatively steep beach slopes (tan 8 = 0.1), with some reflection present, setup slope increased with wave period. Conversely, for tan 8 < 0.04 the setup was independent of wave frequency (within experimental error). The reason offered by van Dorn (1976) for this difference is the assumed constant breaker ratio y in the theory. Data are presented to approximately confirm this assumption on the 0.083 slope, but the flatter slopes show a nonlinear variation with abrupt wave height decay near the breaker point and little frequency correlation. Thus, Bowen, Inman, and Simmons (1968) may have reached different conclusions if flatter beach slopes had been tested. Gourlay (1978) based on his own data and a thorough reanalysis of Bowen, Inman, and Simmons (1968) and Smith (1974) also concluded that experimental data do not show very good agreement with equation (35) as shown in Figure 49. All data here are for relatively steep beaches (tan 8 > 0.083) and a wide range of wave periods. 159 0.1 Gourlay 0.05 if Bowen et al. % 0.02 A 0.005 0.002 Van Dorn 0.00I 0.01 002 0.05 0.1 0.2 tan 8 BED SLOPE Figure 48. Wave setup versus bed slope: all experiments versus theory of van Dorn (1977) (after Gourlay, 1978). Because of these observed differences between theory and observation, the assumptions for the theory are now being studied further. Wind (1978) used 23 MWL readings on a 1:40 slope in the-:laboratory to compute S,, by direct integration of equation (28) from the shoreline toward deep water. In the horizontal (deepwater) section, linear wave theory gives radiation stress levels about 10 to 20 percent greater than found from the data. On the slope, in the surf zone, the linear wave theory was found to give estimates for SNe? sr Sas (i.e., the orbital velocity components) about three times smaller than measured values, found by subtracting the observed potential energy density, Sy from the total radiation stress, S,.. 160 0.04 0.03 dn /dx 0.02 Eqn 3.3-|6 @T= | | | |.5s 0.01 © Bowen et al.(I in 12) © Smith (1 in 10) & Smith (1 in|!) Figure 49. Wave setup versus breaker ratio, y,: Theory from equation (8) versus experiments on slopes around tan 8 = 0.1 (after Gourlay, 1978). A large difference in observed and calculated wave setup was also reported. It was concluded that the linear wave theory only roughly followed the trend of the observed results. Finally, Stive (1980) briefly discusses preliminary results of -the first known direct laboratory measurements of internal velocity and pressure fields to directly calculate momentum and energy flux in breaking waves. All previous experimental studies of the surf zone were literally on the surface, i.e., wave gage surface variations or MWL readings. The use of a two-component laser-Doppler velocity system, minature pressure transducers, and electronic data recording and analysis systems greatly facilitated the work. A sample from one experiment is summarized in Figure 50. The radia- tion stress for a number of surf zone locations was directly calculated from equation (1), and the values used in equation (28) are close but slightly above the measured 7 values. Use of linear wave theory and H,;ms measured gave excessive wave setdown and setup. It will be extremely helpful to study the published results of these experiments. In summary, it can be concluded that wave setup for normal wave inci- dence of regular waves on plane beaches as given by equation (35) is 161 i measured (cm) calculated from Sx». measured | an ‘ calculated from Hrms measured SW.L.=0 -2.0 Figure 50. Comparison of measured and calculated mean water level varia- tions using laser-—Doppler velocity meter (after Stive, 1980). questionable. The use of linear wave theory for S,, and constant y across the surf zone is a major assumption being addressed. Wave setdown based on the same theory is definitely incorrect at the breaker line. These results are important in that investigators have used this theory to correct for setup effects in the longshore current theory. Experiments to verify the theory for oblique wave approach (eqs. 39 and 4) are not known. They reduce to the normal incidence forms and hence are based on the same questionable assumptions. 2\. Nonlinear and Irregular Waves , James (1974a) only included the two comparisons of his nonlinear theory and experiments shown in Figure 51. No real conclusions can be drawn from this limited comparison. Predicted setup is sometimes greater and sometimes less than measured by Galvin and Eagleson (1965), and of the correct order of magnitude. Setdown is much less near the breaker line and more in agree- ment with observations by Bowen, Inman, and Simmons (1968) than linear theory. James' model is said to be limited to spilling breakers on gently sloping beaches and for the case of large P values (P = T/Vegd,) where non- linear effects are more pronounced. The cnoidal theories for wave setdown (Svendsen and Hansen, 1976) were compared with experimental results on a 1:35 slope using a regular wave generator that eliminates free second harmonics. Example results are . reproduced in Figure 52 where the dark line is experimental data. Linear wave theory (eq. 30) is shown to greatly overestimate the magnitude of set- down (labeled eq. 4 in figure). The curve (eq. 9 in figure) does a better 162 x 08 O7 Corals ae oe BellSioM S O5¢ \ Se i bese = aso \ : ~X a4 Se | yet ets 2228 Ns ee ee 03+ | ) (@) ik {o} | Pe _ Janta en — ——— oe 1 2 Sha G) | 20 O 4050 100 H,=d/d, 4 oe 2 Ly ips (a) 2 a SS re a BR oe (crest) N pe se PID | Meon water level a Se Wove envelope (trough) 5 a) ~ = (b) + N ae ERI eS uSEUEL MEDEA weed iy Ue Bt 2 x 3 Figure 51. Some comparisons of nonlinear theory for wave setdown and setup with experimental data (after James, 1974a). 163 0.04 006 hilo 005 OO trite Breakin | as Ho/Lo= 0.0297 Figure 52. Comparison of cnoidal theory of wave setdown with experiments for regular waves (after Svendsen and Hansen, 1976). job but also gives incorrect values (too small) near breaking. These results are from the theory given in equation (116). This poor fit near breaking is greatly improved by modifying the approximation employed (1-n/d) in the bottom velocity up as given by equation (115). The result, consistent with all other cnoidal approximations, is 2 2 Ae = 8 ese ae aN ae n Ne is 0.35] ae © lhe a | (155) where both c and n are determined from second-order cnoidal wave theory. Equation (155) (shown as eq. 14 in Fig. 52) gives excellent results right up to the breaker limit. Finally, the work of Jonsson and Buhelt (1978) on maximum wave setup has been included here since the motion just outside the breaker line is calculated as a series of solitary waves. Figure 53 compares the results of equation (118) with some experimental results. Maximum setup does indeed vary with deepwater wave steepness to the one-third power. Also, y (calcu- lated values shown as constant across the surf zone and values in parentheses with setdown neglected) increases with beach slope as expected from Battjes' (1974a) analysis (see Table 5). Unfortunately, due to other assumptions involved, Figure 53 cannot alone justify whether y is a constant in the surf zone. 164 jo-1|- © Beach slope 1:12 (Bowen, Inman & Simmons) ae fi oO - - > Vy 4] ae r % 1-30! (Saville) | [ J LL * Spilling breaking l | (eS ip Sapte L ! eens 1 | Ome 10-2 1071 Revita Figure 53. Maximum wave setup dependent upon deepwater wave steepness and beach slope (after Jonsson and Buhelt, 1978). In summary, for nonlinear regular wave theory, it is tentatively con- cluded that excellent results are attainable for wave setdown calculated by equation (155) from cnoidal theory, and that the theory of James (1972, 1974a) requires further comparison with experimental results. All wave setup theories still require assumed values for the wave breaking ratio, y. Only the laboratory scale experiments by Battjes (1972, 1974a) and Goda (1975) are available to test the theory for irregular waves. The theory of Collins (1972) has never been compared with measured values. Battjes used a 100-meter-long and 2-meter-wide basin with a water depth of 0.55 meter in the horizontal section. Beach slope was 1:20. Two examples of measured and computed wave setup (based on theory of Battjes, 1972, 1974a) are shown in Figure 54. A systematic difference between theory and measured laboratory data was observed in both cases, (a) mean period Te. = 1.2 s; rms, incident height, Hp = 8.2 centimeters, (b) mean period LS = 2.0 s; rms, incident height, H, = 8.5 centimeters, with the theory always giving too high a wave setup. Wave setdown values were very close 165 T= 12s H= @2 cm + meas. calc. (y= 08) = ws Ho= @Scm + meas. cale. (y= 09) Figure 54. Comparison of wave setup for irregular waves and theory of Battjes (after Battjes, 1974a). to measured data. A thorough investigation of possible reasons for the dis- crepancy was made by Battjes (1974a). Air entrainment was ruled out after separate tests proved this effect was not significant. Battjes concluded that because of the uncertainty regarding the laboratory system to measure the setup, it is not known whether the differences are real or apparent. 166 In addition, Battjes (1974a) obtained the field measurements of Dorrestein (1962) to verify his theory. Differences between measured and calculated values for two separate cases when averaged over the surf zone proved to be less than experimental error. It was concluded that the field measurements lend support to the computational model. Goda's (1975) model for irregular waves is very similar except it uses a varying probability function for the wave breaking criteria. It was checked against experiments in a 30-meter basin with a 1:50 slope and a water depth of 35 centimeters in the uniform end. Also on 1:10 slope where the H, = 50 centimeters in constant depth region. However, no comparison of theory versus experimental results was presented by Goda (1975) for wave setdown or setup. Finally, in support of the purely empirical relations for hee (eq. 125) put forth by Hansen (1978a), Figure 55 presents the field data (black dots) along with some measured or theoretical values. Generalizations for other locations are pure speculation. oy 8 al 2 i | Figure 55. Empirical data for maximum wave setup (after Hansen, 1978). 167 In summary, it is concluded that all irregular wave theories for MWL change in the nearshore area suffer from a lack of concrete experimental confirmation of their validity. This situation is unfortunate since the theories of Battjes and Goda show much promise in this regard. If success-— ful comparisons exist in the published literature, they have escaped the review in this study. Several possible reasons have been offered to explain why the experi- mental measurements of wave setup disagree with the theory. Air entrain- ment by breaking waves is one possibility since the mixture has a lower density. Another is the neglect of the instantaneous bed shear stress resulting from the asymetrical wave orbital motion which can produce a nonzero average stress (Bijker and Visser 1978) .24 II. LONGSHORE CURRENTS Of primary interest is a comparison of laboratory and field measure- ments with radiation stress theory for uniform longshore currents. In most cases the accuracy of both laboratory and field data is relatively low due to observational difficulties in measurement of breaker heights and wave angles. Also, nonuniform boundary conditions, alongshore varia- tions in breaker height, winds, etc. can contribute to the uncertainties in the measured values. Current variation with water depth is another factor to be considered regarding the available data base. ie Mean Currents. Before 1967, all theories were for a mean longshore current (Galvin, 1967). Horikawa (1978a) provides a more recent summary which includes Japanese research efforts. What was meant by mean velocity in the theory and what was measured is not always clear. For example, Komar (1976b, p- 184) used v, for the average longshore current in a review of all theories and also the current at midsurf (p. 190) when referring to data in Figure 56. The data by Harrison et al. (1968) in Figure 56 were said to plot higher than the others because the data were measured just shore- ward of the breakers as a maximum longshore current (Komar, 1976b). In the theory (Ch. 3), the following time-averaged longshore currents are defined: <| Il b velocity at breaker line v = maximum current velocity Vv = the midsurf velocity ¢4BIJKER and VISSER, op. cit. 168 Ie aa T 1 T ir KOMAR AND INMAN_ (1970) n pole & Putnam, Munk and Traylor (1949) Z O Inman and Quinn (195!) * Galvin and Savage (1966) ® Harrison et al. (1968) | 140 }~ @ Komor and Inman (1970) @ '20 4 4 ” SS = oO _ 100 | > : 2 5 oO o =) ° is v” a 6 ° 4 + | | L 40 60 80 Um sin'a@p COS ap, cm/sec Figure 56. The empirical longshore current at midsurf position as deduced by Komar and Inman (1970) based upon sand transport studies (after Komar and Inman, 1970). v = the mean or average velocity across the surf zone. Even here the meaning is somewhat ambiguous since the mean could be determined by including that portion seaward of the breaker line (also a local value when profile is given) and their dimensionless counterparts (using V). It is assumed here (follow- ing Komar, 1969, 1975b, 1976b) that all field data, except as noted, were for the midsurf position, Xl, A comprehensive discussion and detailed analysis of all available lab- oratory and field data led Komar and Inman (1970) , 64 and Komar (1975b, 1976b) S4KOMAR, P.D., and INMAN, D.L., "Longshore Sand Transport on Beaches," Journal of Geophysteal Research, Vol. 75, No. 30, 1970, pp. 5914-5927 (not in bib- liography) . 169 to propose the following empirical equation (see Fig. 57) Yi, = 2.7 Ue sin a cos a (156) Considerable discussion of equation (156) can be found in Longuet-Higgins (1970). Gourlay (1978) reanalyzed the same data using higher average 65 values of y estimated from the wave breaking criterion of Weggel (1972) which includes beach-slope effects. He obtained the empirical result Vv, = 3.727 gH tan 8 sin 2a 1 2 b @ia7) When using new data by Lee (1975) from western Lake Michigan, the coeffi- cient in equation (157) was 2.87 and the differences attributed to the coarser materials present to give a rougher bed. The empirical expression by Komar (eq. 156) in revised form (using eq. 53 and taking y = 0.78) could be written Vv, = 0.607eH, sin 20, (158) Komar argued that the ratio of beach slope-to-bed friction coefficient was essentially constant so that tan 8 did not appear in his result. Dette (1974a) measured longshore currents on the west coast of the Island of Sylt in the North Sea for comparison with all the available formu- las at that time, including those based on radiation stresses. He concluded that the best agreement was obtained with the relation (in dimensionless form) v= 0.32/gH, sin 20, (159) If it is assumed this average longshore current is equivalent to the mid- surf velocity, then these empirical results for sand beaches (like Komar's) are also independent of beach slope. Kraus and Sasaki (1979) analyzed why equation (156) or (158) satisfied the field data over a wide range of beach slopes, breaker angles, and beach materials. As demonstrated in Figure 30 based on their radiation stress theory for large wave angles, it is seen that the midsurf velocity, v1, remains almost constant for 0.01 WEGGEL, J.R., ‘Maximum Breaker Height," Journal of Waterways, Harbors, and Coastal Engineering Diviston, Vol. 98, No. WW4, 1972, pp. 529-548 (not in bibliography). 170 meas (m/s) Figure 57. Comparison of laboratory mean currents with integrated currents obtained from strong current theory (after Liu and Dalrymple, ILO)7A3))'¢ current data over a wide range of variables. Stated in another way, the field data and radiation stress theory both demonstrate that the midsurf velocity is insensitive to changes in independent variables. It is a poor choice to compare theory and experiment for this reason. Consequently, Kraus and Sasaki (1979) used the maximum longshore current velocity, Vp in their analysis, which will be discussed later. Theoretical results of the large-angle strong current theory of Liu and Dalrymple (1978) were compared with the data set of Putnam, Munk, and Traylor (1949) which had large breaking angles. An average longshore cur- rent velocity was computed from the theoretical profile by intergrating over the surf zone and dividing by the surf zone area. Their results (Fig. 57) show computed values consistently larger than measured for all three bed slopes tested when a friction factor, f., = 0.01 was employed. The friction factor could be adjusted to improve the agreement. Liu and Dalrymple (1978) concluded that their large-angle strong current model is primarily valid for steep beaches and large incidence angles as commonly occur in laboratory studies. Comparisons with other data sources were not made. Similar comparisons of integrated mean theoretical values and observations can be found in Thornton (1969, p. 88), Sasaki (1977, p. 147), and Sonu (1975, p. 63). 171 Ze Longshore Current Profile. The original model of Longuet-Higgins (1970) for longshore current profile in dimensionless form was compared with the experiments by Galvin and Eagleson (1965). Figure 58 reproduces the results. The small numbers in Figure 58 represent different wave periods tested (e.g., plotted point 2 = 1 second wave period; see Longuet-Higgins, 1972b for key). The labora- tory observations generally lie between P = 0.1 and P = 0.4 which corres- ponded to a lateral mixing coefficient, N in the range between 0.0024 and 0.0096, respectively. The bottom friction coefficient, Cf was assumed constant at 0.01. As shown, large variations in P produce relative small changes in longshore current velocity; however, it is very dependent upon C¢ which directly influences the reference velocity v}*. Longuet-Higgins (1972b) felt the lateral mixing and bottom friction effects were too large outside the surf zone. In addition, the neglect of wave setup and use of y = 0.82 in the theory when the experiments by Galvin and Eagleson (1965) were on a steep beach (tan 8 = 0.11) must be noted when analyzing these results. Series IL Vin (loci) Figure 58. Original longshore current profile theory versus laboratory data of Galvin and Eagleson, 1965 (after Longuet-Higgins, 1970). 172 Table 3 provides a list of modified theories and additional details. Bowen (1969a) also compared his theory against the experiments of Galvin and Eagleson (1965), as shown in Figure 59. The 6 parametér here related mixing coefficient, friction, etc., but in a different manner than P. B is a grouping of independent variables to give a dimensionless, relative longshore current velocity. The results shown are for K = 0.4 (eq. 68) and include normal wave incidence and setup effects. Because of the relatively simplistic shear stress and lateral mixing models employed, Bowen's theory has not been subjected to further analysis. Figure 59. Bowen's longshore current profile theory versus laboratory data of Galvin and Eagleson, 1965 (after Bowen, 1969). Thornton (1969, 1970a) included wave setup in his theory that in all aspects except lateral mixing was very similar to Longuet-Higgins (1970). A comparison with one experiment of Galvin and Eagleson (1965) is shown in Figure 60. Also shown are distributions of lateral mixing 1S Test I-4 (Galvin) Hp = O.13ft. Y = 0.954 6 QiaeS35)9 5 ; 3 —O— measured T = 1.25sec. = theoretical z oy r 0.0033 ft. 5 fyp= 0.023 V (fps) “1, = x (ft) V_ (ft/sec) 0.06 Figure 60. Thornton's longshore current profile theory versus laboratory data of Galvin and Eagleson, 1965, test II-4 (after Horikawa, 1978). coefficient (vz) and bed friction coefficient (f,) employed in his theoretical analysis. Only bottom roughness (here r = 0.0033 foot for a concrete beach) is chosen in Thornton's model. Comparisons for two other cases with larger waves at greater angles are shown in Figure 61. In all cases, the theory predicts too strong a current outside the breaker line. Some field data by Ingle (1966) taken at Trancas Beach, California, are used to compare the theory as shown in Figure 62. The beach profile was represented by a fourth degree polynomial. The magnitude of the eddy viscosity is much larger in the field than in the laboratory but values employed are too large and overly smooth the profile. 174 TEST M - 2 (Galvin) Hy © 0.145 ft Y = 0.95 Vitps) a, = 15.0° MS T r 2» 1.0 sec = 0.0033 ft Woes 0.052 tt¥sec fap 2 0.024 -3.0 -2.0 ie} x (ft) —e— Measured —— Theoretical TEST I - 3 (Galvin) Hy = 0.198 tt Vi 8 Oar ay = 6.3° Teeemlileousec r 2 0.0033 ft Vinaxt 0-096 t1¥/sec fey 2? 0.021 1150) x (it) Figure 61. Thornton's longshore current profile theory versus laboratory data of Galvin and Eagleson, 1965, tests II-2 and II-3 (after Thornton, 1969). Jonsson, Skovgaard, and Jacobsen (1974) were the first to include a strong current formulation for bed shear in the theory. Depth refraction and a different lateral mixing formulation was also incorporated. Some numerical results are compared in Figure 63 against (a) the laboratory data of Galvin and Eagleson (1965) and (b) some limited field measurements of Ingle (1966). The theory with y = 1.42 gave a better fit within the surf zone and also matched the breaker location for the laboratory data. A poor comparison outside the breaker line was possible due to an exces-— sively large eddy viscosity in this region. The field data shown in Figure 63(b) are too incomplete to make a true analysis possible. Jonsson, Skovgaard,and Jacobsen (1974) chose to use y = 0.8 for the field comparison which gave a rather good agreement for the Vm value. In one example, the combined bed friction coefficient, f,.,, across the surf zone was about 175 Trancas Beach, Calif. |I8 Oct, 1961 Hp 2 4.5¢t y = 0.78 By 8 CeO. T = [5.6 sec t 2 O.I ft 2.0 V (tt/sec) XN AEE bees lactuding Iaterael Sheer Stress" ROSS —O— Measured Theoretical -250 -200 “150 -100 -50 fo) x 20.0 te’? A ( sec) 10.0 Figure 62. Thornton's longshore current profile theory versus field data by Ingle, 1966 (after Horikawa, 1978). fow = 0.05 and rose rapidly nearshore and beyond the breakers as the current approached zero. The nonlinear longshore current profile model developed by Madsen, Ostendorf, and Reynolds (1978) (see also Ostendorf and Madsen, 1979) used the laboratory data of Gavlin and Eagleson (1965) to determine appropriate values for bed friction and lateral mixing coefficients. An explicit empirical equation for oe resulted with average f., = 0.01 and decreased as ¥ increased for the Galnippation runs on smooth concrete (ce = 0.001 foot). The lateral eddy viscosity parameter, I was found to be 0.013. To test this model a parameter, 6, defined as 176 BREAKER LINES V (m/s) Vmox 0.5 MEASURED ¥, — BREAKER LINE PRESENT MODE N 0.0 === -200 -150 -100 -50 O 50 x(m) Figure 63. Jonsson, Skovgaard, and Jacobsen's longshore current profile versus laboratory and field data (after Jonsson, Skovgaard, - and Jacobsen, 1974). so v(measured) - v(predicted) iy ¥ (predicted) (USO) was employed and its mean and standard deviation used in the analysis. For the Putnam, Munk, and Traylor (1949) data v = , the average over the surf zone; for the data for Brebner and Kamphuis (1963) V = V,, the maximum longshore current; and for Galvin and Eagleson (1965) data, only the denominator was . In all cases tested, the standard deviation was such that an accuracy of + 15 percent was expected for their nonlinear longshore current model (Ostendorf and Madsen, 1979, p. 147). The model tended to underpredict current magnitudes for beach slopes flatter than 1 on 10.- A weak current small-angle (linear) model was also tested and and found to be less accurate. 177 The eddy viscosity model in Jonsson, Skovgaard, and Jacobsen (1974) was improved by Skovgaard, Jonsson, and Olsen (1978). Different models and lateral mixing coefficients were employed within and outside the breaker line. An example of the result is shown in Figure 64 for two different cases of lateral mixing. Both show much smaller tails beyond the breaker zone which is closer to reality where " .. . observations indicate a vir- tually complete absence of turbulent mixing outside the breaker zone" (Battjes, 1978). The experimental data are again from Galvin and Eagleson (1965). The tail was found to diminish further as bed slope decreased. { Galvin and Eagleson (1965), Ser. III Test No.3 4 V_= 0.220 3/3 @’? Ix, Von Gh in SZ pe lvgh in sz V7 = $5(°/_ a2 cos*Q) in NBZ = "ho hpVg in NBZ V(m/s) Measured breaking me 0.5 Calgulated breaking line ,Measu red set - Figure 64. Skovgaard, Jonsson, and Olsen's longshore current profile versus laboratory data (after Skovgaard, Jonsson, and Olsen, 1978). The most recent longshore current profile model by Kraus and Sasaki (1979) was verified with new laboratory and field measurements taken in Japan. The theory is discussed in complete detail in Chapter 3 and essen- tially extends the weak current large-angle theory to include lateral mix- ing stresses. A key dimensionless parameter in the theory is P* which indicates the relative importance of the lateral mixing stress to the bed stress and also includes wave setup effects. Laboratory measurements of 178 of Mizuguchi, Oshima, and Horikawa (1978) °° were performed in a 15- by 15-meter. wave basin with a 9-meter beach section of slope 1:10.4. Table 7 summarizes the results for four separate test cases. Current measurement was by propeller-type meter and values recorded represented averages taken in the vertical direction. A test section with no systematic acceleration in the longshore direction was found to exist and data reported for this location. The intersection of the mean setup and setdown lines was used to define the breaking point. The wave breaking ratio, y was described as approximately linear in the surf zone. Results of the laboratory experiments when compared with the theory (solid line) are presented in Figure 65. Velocities are normalized by the maximum current V, and distance by x,. By requiring the theoretical location of the maximum velocity to match the experimental results, all curves were fitted to the data in this regard. Agreement for profile shape is excellent in the region shoreward of Vv, in all four cases. The dotted line is the original model of Longuet-Higgins (1970) which gives very similar results in this region since small angles are present. The model of Kraus and Sasaki(1979) gives better agreement beyond Vm and the breaker line where large wave angles are present. In two cases (1 and 4) the theoretical tail dropped off too rapidly to fit the data. In three of the four cases studied, P* values were less than 0.1. The eddy viscosity coefficient, I ranged from 0.0062 to 0.037 with higher values associated with large wave angles. The bed friction coefficient, C¢ was in the range between 0.011 and 0.024. All these values were com- puted and were not initially specified since fitting the maximum velocity was used in their determination. To do this, y and a, must also be initially specified. Thus by adding the extra condition available through Xp, all parameters of the theory not usually obtainable by measurement, Us@pg Wig I grovel Ce, gan be obtained by fitting the theoretical and experi- mental maximum velocities. The model is made predictive by inputting expec- ted values of Xm, y, tan 8, and Op Results of the plane beach theory compared with limited observations in the field (mear Niigata, Japan) on a step-type beach are presented in Figure 66 which also shows the beach profile with average slope of 1 to 40. Each velocity is the average of five measurements for the same loca- tion. Reasonable agreement is now found seaward of vie but the stepped beach profile produced secondary breaking and a nonlinear y ratio to give the disparity of results shown shoreward of Vy. Kraus and Sasaki stated that a refined wave height description (requiring numerical solution) in this shoreward region should produce better agreement. For this one field _ observation, P* = 0.072, T = 0.015 and C, = 0.0061. Other field data also give P* < 0.1 and such low values are in agreement with observed rapid decrease in longshore current profile tails outside the breaker zone. Kraus and Sasaki (1979) also use this as evidence in support for their weak current model. 66MIZUGUCHL, M., OSHIMA, Y., and HORIKAWA, K., "Laboratory Experiments on Longshore Currents," Proceedings of the 25th Conference on Coastal Engt- neering, Japan (in Japanese) (not in bibliography). 179 Table 7. Summary of the preliminary laboratory data of Mizuguchi, Oshima, and Horikawa (1978). The last six rows contain parameters cal- culated from theory; subscript "L'' refers to the model of Longuet-Higgins: (1970); other calculated values from the present work. ey Poe | ose: | 4.5 4.8 Breaker Angle Mean water line Breaker distance Depth at breaking Maximum velocity Nondim. location of maximm velocity Beach slope, including setup Mean wave height to water depth ratio Wave steepness Wave period Mixing parameter (including wave angle) Priction coefficient (including wave angle) Lateral viscosity coefficient (including wave angle) 10 = = Ti ee we eo CASE -2 e’ © @% @ Observed(F21 15) YL. @ Observed(F=112) 08 Oy 2 4 5° (P #0 067) °° \ —— 1%. BP: 0058) t e@ —---— t-4 (P=0 049) 4 eo —-- tH /P:0043) eo \ x 06 We 4 / ° 04 / e e 02 Oo 4 4 4 fl Dh fee AS Observed(¥+099 ) %p7154°(P20170 ) Ofp* 0" (P#0175) L-H (PeO12 ) Observeaig 126 ) OH=184°(P20077) J 20° (P20081, t-H — (P=0055) | | (S22 I0) ) Figure 65. The model of Kraus and Sasaki compared with laboratory data of Mizuguchi, Oshima, and Horikawa, 1978. 180 G 12) — S 2 1.0 Breaker zone r (ee oo = Breaker o~ 08 : a [ e- -® line ee OW ewe e @ Cc 06 F ) e a > 0.4} eae | Ok 9 u —— OS oO | > ow q> fo} 055 -10 Bottom “1.0m | =1.5 a ui St L el) 0 5 10 15 20 25 30 35 40 45 50 Offshore distance, X (m) Average depth, h(m) 0 0.2 0.4 06 08 1.0 1,22 1.4 X Figure 66. Kraus and Sasaki model compared with field data (after Kraus and Sasaki, 1979). Finally, some preliminary data from the NSTS field study at Torrey Pines, California, have been presented by Guza and Thornton (1980). Vec- tors of 17.l1-minutes average currents are shown in Figure 6/7 for one shore- normal section and along the coast. Also shown is the theory of Longuet-— Higgins (1970) by putting the measured offshore radiation stress into an equivalent regular, unidirectional wave at the measured spectral peak. The dashline includes lateral mixing stress. Complete details regarding this and other extensive comparisons were not available but said to be forthcoming. Bijker and Visser (1978)2* and Liu and Dalrymple (1978) did not make comparisons of their longshore profile theoretical results with the data available. The NSTS field data, the laboratory data of Mizuguchi, Oshima, and Horikawa (1978), and the ongoing extensive laboratory data sets 24BIJKER and VISSER, op. ctt. BeiiD nals 181 OFFSHORE (m) LONGSHORE (m) Figure 67. Typical vectors of 17.1-minute average currents at Torrey Pines, California, from NSTS experiments (after Guza and Thornton, 1980). currently being obtained at the Delft Technical University (see Visser, 1980) should provide the needed information to verify the theory in the near future. The data of Galvin and Eagleson (1965) and others are simply inadequate for this purpose. This is true for both regular (linear and nonlinear) and irregular theories. 3H Nonlinear and Irregular Waves. The nonlinear theory of James (1974b) is shown in Figure 68 along with four experiments from Galvin and Eagleson (1965) and their data. Transition between the hyperbolic and Stokes wave theories is indicated by the arrow. Various combinations of friction coefficient Cr and eddy coefficient N are presented. The values indicated for these coefficients give relatively small variations in velocity and are said to give good agreement with the measured velocity shown by crosses. Friction coefficients varied between 0.001 and 0.0025 and eddy coefficients between 0.01 and 0.016 in these cases. Figures 38 and 59 showed how the original linear theory required significantly larger bottom friction values (C, = 0.01) to match the surf zone measurements. The nonlinear theory is also apparen- tly less sensitive to variations in eddy coefficient. Collins (1972) and Battjes (1974a) did not attempt any comparisons with laboratory or field data for their irregular wave theories. Sonu (1975) prepared the results presented here as Figure 69 where the random sea model (heavy solid line) is from Collins (1972). It was assumed that the monochromatic wave height is equal to a rms wave height in the irregular sea. For comparison, the original model theory of Longuet-Higgins (1970) 182 0-25 Longshore current profiles for the conditions of experiment (a), 1 for C = 0-002, N = 0-016: 2 for C = 00025, N = co16. Longshore current profiles for the conditions of experiment (a), 1 for G@ = 0-002, Ni = 0:036- 2 for G = 0:002, N = o:01- Longshore current profiles for the conditions of experiment (b), 1 for @= o02, N = 0-01; 2 for C = ¢:9525, N = o-o1. Xx Longshore current profiles for the conditions of experiment (c), 1 for C = o-0ar; N = 0:016;.2 for C = 0-002, N = 0-016; 3 forC = o.002,N =o 01. Figure 68. Comparison of nonlinear theory of James (1974b) with laboratory data (after James, 1974b). 183 1.0 “| NO MOMENTUM MIXING MODEL RANDOM-SEA xs MODEL MOMENTUM MIXING MODEL SHORELINE BREAK POINT X/X Figure 69. Comparison of irregular theory of Collins (1972) with original model of Longuet-Higgins (1970) and laboratory data for regular waves (after Sonu, 1975). at P = 0.4 and 0.135 is plotted along with the experimental data of Galvin and Eagleson (1965). It is stated that the irregular wave model represents more closely the mean trend of the experimental data in the surf zone. This conclusion is debatable and the irregular wave theory clearly over- estimates the laboratory data outside the breakpoint. As discussed in Chapter 2, field measurements of longshore currents generated by irregular waves also show time variations. To illustrate this fact, Meadows prepared the results shown in Figure 70 where the theory is by Collins (1972) for various deepwater approach angles. The error bars indicate the magnitudes of the variations in currents measured. It was therefore concluded that time-averaged theories for plane beaches, based upon radiation stress principles, are inappropriate for expressing natural surf zone water motions that are three dimensional and unsteady (see also Wood, 1976; Wood and Meadows, 1975). These are the only known comparisons between nonlinear and irregular wave theories and experiments available in the literature. 4. Bed Friction and Eddy Coefficients. Different coefficients of bed friction and eddy viscosity appear in these investigations. Comparisons are complicated by the use of different stress models, laboratory and field data sets, theories of longshore current profile, and different methods of analysis including time-averaging. For example, most researchers selected Cr (or f,, or f) and N (or T or M) 184 COLLINS 73 LONGUET-HIGGING DISTANCE OFFSHORE (H) ¥ (M/SEC) Figure 70. Comparison of irregular theory of Collins (1972) with unsteady field data (after Meadows, 1977). to fit the data set employed, whereas Kraus and Sasaki (1979) matched Vm so that Cr and T could be computed. Some previous efforts to summarize values have appeared in the literature. Table 8 lists the major formu- lations and results. a. Bed Friction Coefficients. The coefficients found for bed shear- stress models must be approached with caution. No satisfactory formulation exists and the user must apply coefficients consistent with the model type, beach slope, laboratory or field situation, etc. under consideration. In any event, values ranging from 0.001 to 0.06 have been applied. For the weak current small-angle (linear) model used in early investigations, it is generally true that C- decreased with flatter beach slopes as found in the field. In fact,Komar (1975b, 1976b) argued that the ratio tan B/C, is approximately constant. This argument has been attacked on physical grounds by Longuet-Higgins (1972b) and also experimentally by Huntley (1976) and Kraus and Sasaki (1979). 185 Table 8. Summary of bed friction and eddy viscosity coefficients. Bed Friction Coefficient Eddy Viscosity Closure Coefficient Researcher Experiments (L or F) | Symbol | Equation | Model Closure Model By Symbol Galvin and Eagleson (1965) (L) Remarks Longuet- Higgins (1970) Thornton (1969) | Galvin and Jonsson (varies) Eagleson (1965) (1966) small 4 0.02-0.06 Rone See Table 4 Ingle (1966) (F) Jonsson | Bed (varies) (1966) roughness] 0.02-0.6 James (1972) Galvin and Eagleson 0.001- n | (55) | Different (1965) (L) 0.0025 Jonsson, Putnam, Munk, and | Bed friction coef fi- Skovgeard, and Traylor (1949)(L) (varies) j} clent function of £. Jacobsen (1974) | Galvin and Eagleson 0.05- i None See Table 4 and fy for uniform (1965) (L) 0.1 currents and waves, Ingle (1966)(F) respectively. Romar (1975b, Others prior to 0.0172 nN (55) Same 1976b) 1967 0.0225 Battjes (1975, None made none 4 See Table 4 | 0.3-1.8 M depends upon bed 1978) specified (Lab) slope. 21-45 (Field) Huntley (1976) Own experiments (S51) Suy 0.0026 tan B £ 0.01 Devon, England measured | ¢.0006 Includes lateral directly mixing in bed stress in field NOTTAFELICABLE coefficient. non- 0.0019 0.0019 linear 2.0006 +.0006 Liu and Dal- Putnam, Munk, and non- strong/ {0.04 NOT APPLICABLE tryzple (1978) Traylor (1949) linear large Skovgaard, Galvin and Eagleson Same as Jonsson, Skovgaard, c See Outside 0.02- Jonsson, and (1965) (L) and Jacobsen (1974) Table | Surf Zone | 0.04 Olsen (1978) 4 M Inside 0.177- Same as Battjes (1975) Surf Zone | 0.220 in surf zone. Madsen et al. Galvin and Eagleson Empi- curve (varies) r See Table 4 | 0.013 (1978) (1965) (L) tical fitting {0.01 Ostendorf and Madsen (1979) Different Formulation Weak v |0.011- |, Different | 0.0062 Kraus and Mizuguchi, Oshima, Sasaki (1979) and Horikawa (1978 Large a, 0.026 0.037 for Vy in and outside = r surf zone. Weak ¥ 0.0061 0.015 Large % Thornton (1980) | Own experiments, Ss. 0.0062 es Talv. Torrey Pines, mesoured See NOT APPLICABLE Th TeCeluly Calif. (1978) (F) directly | Fig. 71 A OGRE in field COO Os 186 Huntley (1976) showed that the assumption for using equation (51) based on V << up, is invalid for his data so that C¢ values so deduced are not a true bottom friction coefficient. He defined another CR in the equation of motion (neglecting lateral mixing stress) as 2— [puv (n + h)] =- pcas| uly ‘ (161) where u, v include both wave orbital velocities and turbulence. It must be noted here that the right-hand side of equation (161) is not the same as equation (52) where U is the total velocity vector. A two-component current meter on a relatively flat beach (tan 8 = 0.01) was employed to directly measure u and v, and hence U and Vv. From equation (161) Huntley directly computed the values of C* shown in Table 8. By the same procedure, C,- values were calculated using u m 12 place of U in equation (161). It is not clear why the term 2/7 in (eq. 5) was dropped from the later calculation by Huntley. The value of Cr is generally larger than C* found from equation (161). The difference was found to vary across the surf zone and with approach angle. It is fairly clear that the wide range of Ce values deduced in the literature is partly due to the incorrect weak current small-angle model employed. Finally, C* values based upon equation (161) must also be considered somewhat questionable for the following reasons. The left-hand side of equation (161) is also an approximation of the true gradient of the radia- tion stress in that (a) the mass flux term is neglected, and (b) the term uv is assumed independent of water depth, Only single position measurements were made with a meter in the water column. In addition, lateral turbulent mixing stresses are combined into C* and steady uniform flow conditions must be present in the field. Hunt fey (1976) presented methods and arguments in support of his contention that wave-induced turbulence effects were relatively small compared with the wave orbital interactions. Recently, Thornton (1980) overcame some of these questions by directly measuring the radiation stress component, S,., offshore and by indirectly computing it inshore (in the surf zone). Thus, a simple steady-state wave- induced current model for a straight and parallel contour beach is employed to estimate Ce. Ay = - Ty =< pce |U|v (162) The right-hand side is identical to equation (161) but the left hand side is not. Outside the breakers, a linear array of five pressure sensors (in 10-meter water depth) was used to measure S,, for a series of experiments at Torrey Pines Beach, California (tan 8 * 0.023), in connection with the NSTS (see Guza and Thornton, 1979, for details). In the surf zone, wave angle measurement directional errors made this procedure inaccurate 187 for Torrey Pines Beach. Consequently, inshore S sion of equation (10), i.e., xy was computed from a ver- oO 1 Sg = | pu(£,z)v(f£,z)dz = E(£,0)n(f£) sina cosa da (163) = —T where n is given by equation (15);.f£, frequency; and E, the energy density per unit frequency and direction. Further details are beyond the intended scope of this review. The method relies heavily upon the fact that linear wa\~ theory can be employed to relate velocity measured at a single elevation to the energy density spectrum for u (see Guza and Thornton, 1980). The bed shear-stress coefficient, C* found in this fashion from equation (162) is shown in Figure 71 for various locations in November 1978. Seventeen-minute averaging times were used for each point and eight such values when averaged produced the solid line. The average of all such calculations gave C* = 0.006. Considerable scatter occurs outside the breakers due to the onshore velocity variability in space and time. In the surf zone, less scatter was observed because the variability is related to local water depth. Thornton (1980) also found that C* was consistently less inside the surf zone when comparing the results of other days. It should be noted that this single value of 0.006 is consistent with that deduced by Kraus and Sasaki (1979) from Japanese field data and their model. Some negative C* values are noted in Figure 72 outside the breaker line. Thornton (1986) believes this means the currents are driven in the wrong direction. However, Huntley (1976) states that C* can possibly become negative beyond the surf zone due to neglect of the lateral turbu- lent mixing stress gradient in equation (161) or (162). Electromagnetic current meters with a gain response cutoff around 2 hertz (see Fig. 16) behave as filters to distort all higher frequency turbulence signals. Their use is limited for studying wave breaking turbulence and wave-turbulence interactions. The methods described by Thornton (1980) and applied to additional field data in the NSTS from Santa Barbara, California, will be of consider- able interest in the near future. Estimates of the bed roughness patterns, heights, etc., both beyond and within the surf zone, are needed to corre- late the data for future applications. b. Eddy Viscosity Closure Coefficients. ‘Even less is formally known about magnitudes and variability of eddy viscosity closure coefficients. As discussed in Chapter 3, eddy viscosity is normally calculated from the expression (eq. 86) where considerable disagreement exists as to the proper expressions to’ 188 Q.03 Breaker-line + wy ~ E = *< £ Qa i-* _< 188 120 148 160 180 208 220 Distance offshore (m) Figure 71. Bed shear-stress coefficients computed from equation (162) and data taken at Torrey Pines Beach, California, November 1978 (after Thornton, 1980). employ for the characteristic reference velocity t and reference length scale, 1. A closure coefficient (N, I, M, etc.) then is inserted in some expressions (see Table 4) such as that first proposed by Longuet-Higgins (1970) Vv. = NxV/gh (164) or later by Madsen, Ostendorf, and Reynolds (1978) Oe eles (UE (165) Consequently, comparisons of various theories of turbulent mixing in the surf zone and beyond must be in terms of Vz, (or uy) and not the closure coefficients involved. 189 The earliest effort to compare nearshore mixing theories was by Bowen and Inman (1974). Table 9 presents estimates of Vj, (Ay) from the limited available field and laboratory studies by Inman, Tait, and Nordstrom (1970) and Harris, et al. (1962). Also included are \j, values that Thornton (1970a) used to reproduce longshore current observations with his theory. Although the eddy viscosity varies over almost four orders of magnitude from laboratory to field, the closure coefficient N shows a relatively small range of variation from 0.009 to 0.065 with an average N # 0.03 across the surf zone. This value is considerably larger than that used by Longuet-Higgins (1970) to verify his profile theory (N = 0.0024 to 0.0096 from Galvin and Eagleson, 1965, see Table 8). One explanation is that his model overestimates V;, beyond the surf zone (Longuet-Higgins, 1972a) so that it much smaller Vy, values are used outside the breakers, a larger mean coefficient is needed to reproduce observed profiles (Bowen and Inman, 1974). Paradoxically, because direct measurements of eddy viscosity by dye diffusion in the field are difficult, Bowen and Inman (1974) state that estimates of Vz, by fitting observed velocity distributions to theory may be a more accurate method. Nielsen (1977) used the available field and laboratory data in Table 9 to test four different theories for V,. These were by Longuet-Higgins (1970) and Jonsson, Skovgaard, and Jacobsen (1974), and were similar to Thornton (1969, 1970a), Inman, Tait, and Nordstrom (1970), and Battjes (1975, 1978). The comparison is summarized in Table 10 and only for surf zone mixing. Based on Nielsen's analysis of mean values of eddy viscosity across the surf zone v*, the model by Longuet-Higgins is preferred. The resulting empirical constant of 0.007 gives N = 0.018 which is near the maximum value of 0.016 predicted by Longuet-Higgins (1970, 1972a). It is not clear why these results by Nielsen for N, using essentially the same data, differ from those by Bowen and Inman (1974) who obtained N : 0.03. Also, the average eddy viscosity (¥*) is not indicative of how the y variation across the surf zone and beyond produces the proper longshore current profile. Figures 61 and 62 give an idea of how widely v, can change in this regard. Finally, as discussed by Nielsen (1977), Up is not a true eddy viscosity for turbulent diffusion. It is really a combined transport- dispersion coefficient since much fluid is advected shoreward in the upper layers by the reference celerity for breaking waves. An excellent summary of all models before 1978 can be found in Gourlay (1978). . The trend since 1978 is to use separate models and coefficients within the surf zone and beyond the breaker line (Skovgaard, Jonsson, and Olsen, 1978; Kraus and Sasaki, 1979). Completely different closure coef- ficients result in each region (see Table 8). It will be extremely bene- ficial in the future to use the newly available field data on longshore currents (NSTS) to estimate separate Vr, and closure coefficients by fitting the data. It will also be appropriate to consider new ways to analyze the available velocity time histories across the surf zone. One possibility is to use auto- and cross-correlation techniques to estimate Lagrangian length scales from which Eulerian length scales and local eddy viscosities 190 Table 9. Estimates of eddy viscosity across the surf zone from laboratory and field experiments (after Bowen and Inman, 1974). Date Exp. x hy gh A N me GR) Or GA Inman, Tait and Nordstrom (1970) Scripps 129 83 0.80 Zou 5.9 0.064 109 ~=80 0.60 2.4 2.8 0.037 112 68 0.60 2.4 2.0 0.030 107 70 0.66 765} 2.0 0.027 El Moreno 55 55) 0.45 2.1 -30 0.065 97 4.7 0.45 2.1 12 0.030 95 4.8 0.40 2.0 eal 0.026 104 4.8 0.35 1.8 Ld) 0.034 97 3.8 0.30 Ty / -08 0.039 Harris et al. (1962) field I 18 1.4 S\57/ toil 0.041 2 14 haat 3.3 0.24 0.013 3 12 0.9 3.0 0.26 0.018 4 16 Wa? 3.4 0.42 0.020 5 8 0.6 2.4 0.28 0.036 laboratory - 0.70 0.068 0.82 0.0051 0.022 - 0.80 0.072 .0.85 0.0054 0.020 - 0.80 0.072 0.85 0.0050 0.018 - 0.50 0.043 0.66 0.0051 0.038 - 0.60 0.056 0.75 0.0028 0.016 - 0.50 0.045 0.67 0.0023 0.018 - 0.50 0.041 0.64 0.0019 0.015 - 0.30 0.027 0.52 0.0022 0.036 — 0.50 0.043 0.65 0.0012 0.010 - 0.35 0.032 0.57 0.0007 0.009 Thornton (1970a) lab (Galvin, ieee - 0.52 0.043 0.65 0.0029 0.021 field (Ingle, 1966) - 70 VS) 3.8 1.4 0.013 Table 10. Comparison of four surf zone theories for mean eddy viscosity across the surf zone (after Nielsen, 1977). Empirical Closure Relative Remarks Constant Coefficient Scatter Yeh. . Ky = 0-4N Longuet-Higgins (0.4N) |x, | (1970). Inman, Tait, (KH x /T © i 6 kK, = Coeff. by Inman and Nordstrom (1970). 2 = Jonsson, (K,) Tog, cos Oy, O Ky Coeff. by Jonsson Skovgaard, and Jacobsen (1974). 4 2) aay ‘Ia Battjes (1975). (K,) (tanB-tanm) Do i N= nf] a Ix, | ¥en, 191 can be deduced (Baldwin, and Johnson 1972)? Much additional work remains to determine which model for \); is best and to pin down the appropriate closure coefficient associated with it. All the above discussion is for onshore-offshore mixing where the characteristics length scale is the surf zone width. A model for turbulent mixing in the alongshore direction (longshore dispersion) has been developed by Lin and Horikawa (1978). It assumes the surf zone as an open channel with triangular cross section and that the lateral variation of longshore current as the primary mechanism for longshore dispersion. Rip current effects are ignored. The theory when compared with observed field and laboratory measurements is found to agree reasonably well. The model follows concepts originally developed by Fischer (1967) 8 for longitudinal dispersion in natural water courses. Results are of interest for use in two-dimensional numerical models where different cross-shore and longshore dispersion processes are possible. Rip current effects must be included in future efforts in this regard. De Surf Zone Empiricism The final closure coefficient needed in all theories for comparisons with measured data is the wave height to total water depth ratio, y. It is often used to estimate both the breaker height and the energy decay in the surf zone. An extensive discussion is presented in Chapter 3; however, this section concentrates on surf zone energy dissipation and those attempts to compare actual measurements with theory. All the longshore current theories listed in Table 3 and reviewed in this chapter assume a constant y ratio across the surf zone. Is this correct for all breaker types, beach slopes, wave steepnesses, etc.? Collins and Weir (1969) summarized the results of four experimental inves-— tigations on plane beaches as shown in Figure 72. With the coordinates shown, a 45° line would give constant y in the surf zone. In general, as beach profile decreased the y ratio departed further and further from being constant across the nearshore area. On flat dissipative-type beaches (spilling breakers as found in the field) the y ratio changed continuously as the rate of energy dissipation decreased with distance from the breaker line. Distance from the breaker line is also important for determining wave heights in the surf zone. This result was confirmed by laboratory 67BALDWIN, L.V., and JOHNSON, G.R., "The Estimation of Turbulent Diffusivi- ties from Anemometer Measurements," submitted to Journal of Fluid Mechanics, Jan. 1972 (no record of JFM publication) (not in bibliography). 68FTSCHER, H.B., "The Mechanics of Dispersion in Natural Streams," Journal of Hydraultes Division, Vol. 93, No. HY6, 1967, pp. 187-216 (not in bibliography). 192 NAKAMURA STREET HORI KAWA DIVOK Y H/Hp (@) 0.2 0.4 0.6 0.8 1.0 Figure 72. Experimentally determined wave height decay on plane beaches of different slopes (after Collins and Weir, 1969). 193 measurements of Sawaragi and Iwata (1974) for regular wave dissipation across a horizontal bed. Their results are repeated here as Figure 73(a). Recent field measurements by Suhayda and Pettigrew (1977) presented some confirmation of the fact that the y ratio varies with distance from the breakpoint and ranged from 2.0 to 0.6 (Fig. 73,b). The theory of Sawaragi and Iwata (1974) is also shown in Figure 73(b). It is based on a finite- difference solution of a set of surf zone equations of motion and contin- uity that includes excessive amounts of numerical information loss (numerical viscosity). For this reason it has been omitted from this report. Finally, the data of van Dorn (1976) are shown in Figure 74. This was obtained as part of his wave setup results previously discussed. Steep slopes (8 = 0.1) as employed in most laboratory studies of longshore current profiles produced y 2 constant whereas it is again clearly shown that y is not constant for flatter profiles. Use of constant y ratio in longshore current profile models introduces a high sensitivity to bottom profile variations that is unrealistic (Battjes, 1978). As seen in Figures 72, 73 and 74, a constant y is not applicable for flat slopes, nor is it physically justifiable for bar-trough profiles. Consequently, Battjes and Janssen (1978) developed an irregular wave model for surf zone wave height decay as described in Chapter 3 and shown in Figure 41 for examples with plane and bar-trough type beaches. It is based on the conservation of energy equation (44) and not some semiempirical y variation. A series of laboratory experiments was then conducted to check the theory in a basin 45 meters long, 0.8 meters wide using SWL depths about 0.7 meter. The free-surface fluctuations were measured by parallel- wire resistance gages and data indicated by x on Figure 40. The theory was found to give excellent results on a plane beach (tan 8 = 0.05) with large wave steepness (Fig. 40,b) than the lower steepness (Fig. 40,a) where only fair results are indicated. For bar-trough profiles, the theory was considered quite good. It followed the visual observation of essentially no wave breaking in the trough region (no energy dissipation) and also produced a high dissipation rate just inside the bar. The theory shown in Figure 40 is based on only two closure parameters. One is K in equation (124) which was expected to be about unity and K = 1 was in fact used in the theoretical calculation. The second is yp taken here as 0.8 for initial wave breaking only. The theory was found to qualitatively and quantitatively predict wave height decay and MWL changes on plane and bar-trough beach profiles. Use of the theory in combination with longshore current calculations has yet to be attempted. Battjes and Jansen (1978) attribute the disparity between theory and measurement for low steepness waves (2 0.01) on plane beaches (Fig. 41l,a) to an enhancement effect. In other words, waves with low steepness have time to shoal up to heights exceeding the deepwater wave height, before the decay due to wave breaking sets in. 194 [o_| 0.035 jepilling breaker] Oram ee | | © | 0.056 [plunging breaker | ——— | on [_— aa _—" 4 NSS es Xv/L Xa/L —>._ *X/L (a) Laboratory 4 Cs) E <= ~ <= 0 .2 4 6 8 MOebel Qtek -el6ietal-8-%. 2.0 X/VT (b) Field Figure 73. Experimental studies of wave height decay in surf zones as function of distance from breakpoint: (a) laboratory (after Sawaragi and Iwata, 1974); (b) field (after Suhayda and Pettigrew, 1977). 195 G an =aacnlal alee Cae oe T ae T Period, T-sec 5 S=.022 3S o-8 °o 1.65 ye ap e 2.37 DOG FOB oh Oa SS = 04 0.2 10 0.8 2 0.6 x ~ xr 04 0.2 1.0 0.8 20.6 xr > = 0.4 0.2 046 #06 O8 (7) +) /(Ap+Dp) Figure 74. Wave height versus total water depth and surf zone width from laboratory experiments (after van Dorn, 1976). The model developed by Goda (1975) is similar to that by Battjes (1975) and Battjes and Jansen (1978) except that wave breaking is assumed to occur over a range of wave heights with varying probability. The choice of the range was arbitrary but found to agree with both laboratory and field data. This is said to ". . . represent the inherent variability of breaker heights and partly compensate [for] the inaccuracy of using a single wave period in the estimation of breaker height" (Goda, 1975). A semiempirical nonlinear theory of wave shoaling (Shuto, 1974) 69 is also used in Goda's theory. His theoretical results for various beach slopes and wave steepnesses versus laboratory data from an irregular wave flume are presented in Figure 75. Agreement could be classed as better for low steepness waves which again shoal up considerably on steep beaches before breaking. In fatt, Battjes and Jansen (1978) indicated that their simpler model was a first effort to be later refined by incorporation of Goda's smoother cutoff criteria. The most recent experimental versus theoretical attempt at surf zone modeling has been reported by Mizuguchi (1980). It also employed the wave 69SsHUTO, N., "Nonlinear Long Waves in a Channel of Variable Section," Coasta’ Engineering in Japan, Vol. 17, 1974, pp. 1-12 (not in bibliography). 196 20 T i: See ee = i Tr 5 2 2 T T T T T Bottom Slope: 1/50 Bottom Slope: 1/10 : » : Theory (Hii00/Ho ) Ho/Lo= Theory (Hvioo/Ho') _~~-—_—___. = 6 Ho/Lo= 002 14 ; . ; =>] 1.4 Sr fears oars ¢ I a Theory H ZL =002. Theory LZ Ho/Lo*002 / Legend | (Ho/Lo) +GDi (0.021) “GK (0037) * GS2 (0.043) (He/Le) (He/Le) » S3(Q020) = D2(0042) “w(Q02!) * S2(0.044) 1 0:(0,026) “ D3(0.0S4) 4 K(0.041) °S:(0.056) : ; 5 on A ; 5 i i © O8 TOUS" 20n25 sO eS “Oo Oo OS 10 1S Bo 25) so) 3S 2O h/Ho h/Ho Figure 75. Irregular wave height variation across the surf zone on plane beaches: theory versus laboratory data (after Goda, 1975). energy conservation equation (44) but then relied on entirely different energy dissipation relations (than Battjes) as briefly described in Chapter 3. Figure 76 shows four separate cases of theory versus experiment. Here, the wave amplitude to amplitude at breaking (a/ap) is plotted for (a) horizontal, (b) plan 1:10 slope, (c) and (d) stepped beach profiles. Also shown are wave setup results (n/d,) not of interest here. The theory (solid line) underestimates the wave height decay in all cases. Better results were obtained (dashline) by using E = 1/6pga’ for the wave energy per unit area rather than the correct value of E = loga-. The factor 1/6 was found empirically to give the correct wave setup for step-type profiles. The necessary empiricism with this theory makes it less attractive than the model of Battjes and Jansen (1978) modified by the smoother, breaking cutoff criteria devised by Goda (1975). 197 Wave height and set-up in surf zone of uniform Uniformally sloping beach depth 0.5 2 Bi a td ee Step—type beach Field data on step-type beach Figure 76. Irregular wave height variation across the surf zone for four different beach profiles: theory versus data (after Mizuguchi, 1980). III. NEARSHORE CIRCULATIONS I Rip Currents. To date, all theory has concentrated on rip current spacing as dis- cussed in Chapter 3. Mechanisms for rip current generation are categorized in Table 1 and reviewed extensively in Chapter 2. Because of the large number of possible mechanisms that can cause rips to form on beach profiles ranging from the reflective-type (steep) to the dissipative-type (flat, broad), no single theory has emerged for comparison with laboratory and 198 field spacing measurements. In fact, much of the knowledge in this area must still come from empirical correlations of the available data. One such effort was made by Sasaki (1977) (see Sasaki and Horikawa, 1975a, 1975b). Field data from the United States, South Africa, and Japan were analyzed in terms of the rip current spacing Y,; (L; in eq. 111), surf zone width Xp», beach slope tan 8, and wave steepness. The latter two variables can be combined into the surf similarity parameter —& (Battjes, 1974a, 1975, eq. 93) or here defined as the Irribaren Number, I,- Note that the deepwater wave height H, is used in contrast to Battjes' similarity parameter & which employed H, the height at the "toe" of the slope. Gourlay defined I, if H, (at breaker) is involved. eeanwgs fe HME a From the data sources shown in Figure 77 for the correlation between ie and X,, Sasaki plotted the normalized rip current spacing Y,-/X, against I,- Considerable scatter was found to exist, as shown in Figure 78, where rip spacing ranged from about one to eight times the surf zone width. From an extensive analysis of the data in which breaker type, beach slope, reflec- tion coefficient, and other factors were considered, Sasaki (1977) hypothe- sized the three domains for rip current generation mechanisms shown in Figure 78. These were an edge wave, instability, and infragravity domain. A detailed description of each domain for nearshore circulation systems is given in Table 11 (after Sasaki, 1977). The demarcation between each domain was given as 1 O25 eand) sie as shown in Table 11. The reasons for this choice were never explicitly stated by Sasaki (1977). The infragravity domain (I; < 0.23) was hypothe- sized by Sasaki (197%) to explain why flat beaches with wide surf zones (spilling-type breakers) could reflect relatively low waves (H, < 0.3 m) with longer periods of the order 30 to 120 seconds. These waves are known as infragravity waves and are produced by irregular wind waves and also called surf beat. The existence of these low mode edge waves within the surf zone on flat beaches has subsequently been measured by Huntley (1976a) and Sasaki, Horikawa, and Hotta (1976) in the field and Bowen and Guza (1978) in the laboratory. The data of Figure 78 for each domain give the ratio Y,/Xp as (1) infragravity domain (I, < 0.23) OPCS Sy Li’ (167) 70 SASAKI, T., HORIKAWA, K., and HOTTA, S., "Nearshore Current on a Gently Silopine Beach," Proceedings, 15th Coastal Engineering Conference, 1976, pp. 626-644 Gar in bibliography). ieNe) * r Nondimensional spacing of rip current, Y 1000 ¢ 8 Y, (m) Spacing of rip current, Figure 77. Sasaki infragravity wave o Ajigaura @ Shoonan © Katsuura ® Kashiwazaki © Kujyuukuri e@ Kashima © Seagrove, Fila 6 Silver Strand,Calit 9 Scripps, Calif O@ El Moreno, Mexico © Virginia, South Africa Japan 10 100 1000 Width of surf zone, Xp (m) Correlation between rip current spacing Y; and surf zone width, Xp (after Sasaki, 1977). (1974 ) Hino (1973) instability Bowen-Inman (1969) edge wave upper limit of data © Hino (1973) . (>) ~ Sasaki(1974) 9 Ws (e} -) On n=l &) OS 0) Se fo) ) Os (Yo) -O @ PHS oO ~ n=2 QO Bowen-Inman(1969) ' o- YF? ; surge [ties | plunging Folens 4 6 8 2 4 6 8 2 4 6 8 0.1 1 10 Irribaren Number I, Figure 78. Dimensionless rip current spacing related to Irribaran number and three domains of rip current generation mechanisms (after Sasaki, 1977). 200 Table 11. (after Sasaki, 1977). Domain Infragravity Instability Author Sasaki (1974) Hino (1973) Range of I ik 2 at 2) (oe 3) é ve i Breaker type Spilling Spilling plunging Surf zone Exists or does not exist Always exist Three domains of the nearshore current system Edge wave Bowen and Inman (1969) Plunging Ysurging/collapsing Does not always exist Number of waves in the surf zone More than 3 waves 1v3 waves Reflection coefficient Incident wave character- istics Wind waves & swell Less than 1 wave =% Te 0) Arbitrary Micro- topography Longshore bar Crescentic bar Classifica- tion by Guza and Inman (1975) Dissipative system Incident wave with large wave steep- ness Gentle bottom slope] tanB=1/2011/40 tanB<1/50 Remarks 201 Beach cusp Reflective system Incident waves with small wave steepness Steep bottom slope tanB>1/10 (2) instability domain (0.23 < ue 53 11) Y/X, 24 (168) and (3) edge wave domain (1. = il) ap ° yy Sih YI = | on (169) 3G n= 2 Equation (167) was obtained empirically by Sasaki (1977) from the data shown whereas the other relations are theoretical. The edge wave theory of Bowen and Inman (1969) with modal number n appears to agree well with data from the steep beach (tan 8 # 0.15) at El Moreno, Mexico (eq. 169). In the instability regime, the theory of Hino (1974) falls within the variation of data shown. The theory of Dalrymple and Lozano (1978), given by equation (112), and the theory by Miller and Barcilon (1978), given by equation (114), result in WealPGe e 2 and Y,/Xp 2 10, respectively, for typical values of tan 8, y, and Cr. Thus, their more sophisticated eigenvalue analyses appear to give results farther outside the range of observed rip current spacings in Figure 78. More data are needed to sub- stantiate the various theories. For example, use of the LEO data gave Bruno and Dalrymple (1978) 71 encouragement for the applicability of equa- tion (112) in the instability regime. Rip current generation mechanisms can also be categorized as either the free type (instability, eigenvalue theory) or the forced typed (edge wave, structural interaction, etc.). Mizuguchi and Horikawa (1976), based on their experiments, argue that observed rips in the field should all be interpreted as the forced type where irregular bottom bathymetry controls the motion. Much research remains to be done on rip currents and the development of two-dimensional numerical models such as by Vreugdenhil (1980) to study migrating rip currents will provide needed insight in the future. No theory exists for rip current velocity, width or size, and little field data are available to verify the numerical simulations in this regard. Sasaki (1977) does provide some data and correlation for the offshore extent of rip currents beyond the breaker line (x. aia) Jdeifers VIKA) S 71BRUNO, R.O., and DALRYMPLE, R.A., "Held Observations of Rip Currents," University of Delaware Report, 1978 (not in bibliography). 202 De Nearshore Circulations. The data base for experimental measurements of mean water surface variation and two-dimensional currents, i.e. nearshore circulations, is extremely limited. Only a few field and laboratory studies have been made, and the results are not readily accessible for use. While significant strides have been made in development of numerical models to simulate near- shore hydrodynamics, data to calibrate, verify, and test these models are practically nonexistent. The Japanese recognized this problem in the early 1970's and devel- oped their BACS (ballon-borne camera systems) with drogues. It was modi- fied into the STEREO-BACS system by Sasaki, Horikawa, and Hotta (1976) 70 to permit wave height estimates to be included. Two cameras are suspended by ballons that permit stereographic projections. Problems with this method were discussed in Chapter 2. The results are not generally available for use by other researchers. In the United States, the problem is currently being addressed by General Government agencies such as the National Sea Grant Office of the National Oceanic and Atmospheric Administration (NOAA) in their NSTS project and the CERC in their Atlantic Remote Sensing Land Ocean Experi- ment project. Data from the NSTS experiments are being made available on magnetic tape. The data could provide the information necessary to substantiate the nearshore circulation models in the near future. Figure 79 provides an analytic solution, based on Bowen's (1969) linear theory for the same wave characteristics and bathymetry as the field measurements of currents with drogues and current meters by Sonu (1972) shown previously in Figure 2. Comparing the two figures showed general qualitative agreement but, the symmetry in the theory was not found, no quantitative comparison of results was attempted. Also, it was observed that the circulations pulsated but the analytic solution was for the steady state. The undulating surf zone bed was postulated as the mechanism trigger- ing rip current formation. It was concluded that the theory substantiated the observations in this regard. The simplified numerical model NCSS developed by Sasaki (1977) was exercised to simulate two different cases as measured in the field by the BACS system. In one example (CASE-2) drogues were scattered outside the breakers to facilitate comparison and a meandering longshore current-rip current pattern was present. Figure 80(a) is the nearshore current velocity field as deduced from the camera photos, and Figure 80(b) shows the numerically computed velocities under corresponding wave and bottom contour conditions. Sasaki (1977) stated that to the left and near the pier ". . . the direction of the predicted current is different from that of the observations [which] has a predominant offshore component." In fact some of the strongest measured currents (> 30 centimeters per second) appear precisely where the model indicates a large-scale eddy with zero motion. In terms of transport stream-function contours the overall flow 7OSASAKI, HORIKAWA, and HOTTA, op. cit. 203 aGen Breaker line —— Treaspeort velocity 0.0 a *fooc/a ——— Tranopert streamline —— __ Bettom contour e Correat motor S Weve gevge 4 areed 40 2 30 mY 20 10 SEAGROVE, FLA. JULY 31, 1968 meTeRs Distribution of transport velocities and streamlines in the circulation cell. Streamline separation is 0.4 m*/sec, H» = 0.395 meter, 7» = 5.0 sec, and a = 0. 40 Breaker line + W=0 merTers ‘Shoreline Computed transport streamlines of a circulation system for normal-wave incidence and undulatory surf zone: H, = 0.395 meter, 7» = 5.0 sor, a, = O, tan B = 0.03. Figure 79. Comparisons of field observations of nearshore circulations and analytical theory of Bowen (after Sonu, 1972). 204 BACS-731214-2 Wind direction : SSW-WSW T =8.0sec 160 bee Sy iar 2.5 ~2.6m/sec Arrows showing float 15 velocities(cm/sec) 140 oe 5 «=: Above 30cm/sec | aT) 2 < : Below 30cm/sec — 15 —: Bottom contours 120 Saas Aes — Lom __: Significant EK gljom ae a es breaker : ars 9 00} 2 ou SS Ne By ey 2 4 ; 1.0m as Law ers 54 yee? 1.0m Pr74 9 2 3 NN eo on il a <_ = Sa “= a Ba ue <— > 51 FLO IE) 2 os = Of 6 3525 Sa - RECEIVER antenna 22S 2005 \ sive access roa LL BASE cap Figure 81. Plan view of experiment site and horizontal coordinate system (after Allender, et al., 1978). Model input included local bathymetry, offshore measured wave charac- teristics, wind histories, and local water level changes. Comparisons between model and observed ¥V and Hymsg for six different sled transits (sled position fixed) are shown in Figure 82. In general, for all examples, agreement is far from satisfactory for both currents and rms wave heights. For example, in Figure 82(b), the peak measured currents are less than 30 percent of the peak currents predicted. And, in most ases, the model energy levels in the surf zone are too high (e.g., Fig. 82,d). The peaked- ness of the model profile could be due to these wave height discrepancies 206 DISTANCE OFFSHORE (m) (a) DISTANCE OFFSHORE (m) 0 00 (b) Figure 82. MODEL MODEL OOON --—-— OION DATA WAVE STAFF a 05m( ABOVE DATA o e@ 10m)5 BOTTOM 70 + a 60+ 50+ a 40 30 a 20+ OF [Birra Aa ae 0 00 05 10 LONGSHORE CURRENT (m/s) RMS WAVE HEIGHT (m) Comparison of model and observed current profiles (a) and wave heights (b). Model results for adjacent Computational Rows are shown in (a). i, MODEL CASE A ———— CASE 8B | DATA OATA H 2 05m ABOVE 405m, AS0VE e 10m 5 BOTTOM e 10m \ BOTTOM 0 05 10 15 00 ck) 19 LONGSHORE CURRENT (m/s) LONGSHORE CURRENT (m/s) Comparison of model and observed current profiles for different wave climates at the beginning (Case A) and end (Case B) of a sled transit (a) and for a less severe wave climate (b). Comparisons of field observations of nearshore currents and numerical model of Birkemeier and Dalrymple (1976) (from Allender, et al., 1978). 207 MODEL DATA a 05m] ABOVE MODEL WAVE STAFF DATA @ e@ 10m) BOTTOM & Ss & 6 OISTANCE OFFSHORE (m) 3 nh o a | Ss 3S T 00 05 10 ie) 00 05 10 LONGSHORE CURRENT (m/s) RMS WAVE HEIGHT(m) (c) Comparison of model and observed current profiles (a) and wave heights (b). Model underestimates differences between wave heights near breaking and heights inshore of breaker zone (b) resulting in sharply peaked model current profiles (a). MODEL MODEL | DATA | WAVE STAFF | a» 10m ABOVE DATA B BOTTOM @ eae a w e § rall 2 2 s z 30 ° 3 [ 3 20 a bn eee : O 00 05 10 15 00 05 10 LONGSHORE CURRENT (m/s) RMS WAVE HEIGHT (m) (d) Sharply peaked model current profiles (a) result from underestimation of wave height decay (b) across the surf zone. Figure 82. Comparisons of field observations of nearshore currents and numerical model of Birkemeier and Dalrymple (1976) (from Allender, et al., 1978) .--Continued 208 MODEL MODEL DATA 4 05m) ABOVE © 10m \ BOTTOM WAVE STAFF DATA B DISTANCE OFFSHORE (m) 0 —l 00 05 10 15 LONGSHORE CURRENT (m/s) RMS WAVE HEIGHT (m) - (e) Additional comparisons of model and observed current profiles (a) and wave heights (b) suggest that more energy is extracted from the wave field inshore of the breaker zone than the model predicts. Figure 82. Comparisons of field observations of nearshore currents and numerical model of Birkemeier and Dalrymple (1976) (from Allender, et al., 1978) .--Continued plus the lack of lateral mixing stress term. Allender, et al. (1978) cited the surf zone model by Battjes and Janssen (1978) to handle bar and trough bathymetry and the addition of lateral mixing stress terms as needed improve- ments. The latest version (Ebersole and Dalrymple, 1979) includes the convec- tive acceleration and lateral mixing terms but retains the wave breaking ratio y to specify the surf zone energy dissipation. Published comparisons of numerical versus experimental results are unknown. As previously dis- cussed (see Fig. 35), the present version contains sufficient numerical viscosity to negate the quantitative reliability of the results for some cases. All the above were finite-difference numerical models. An example of a finite-element solution for the circulation eddy in the lee of a harbor and some physical model data was previously presented in Figure 36. No quantitative numbers were given to make a comparison. This is also true for the finite-element model by Liu and Lennon (1978). Finally, it should be noted here that the complete experimental results of Gourlay (1978) from his laboratory experiments with an offshore break- water (Fig. 8) are included in his Appendix 4. Sufficient current, wave height, and MWL information is presented to serve as acheckon any of the numerical models cited. Whereas confined laboratory basins posed problems 209 when measuring uniform profiles on infinite beaches, they provide the ideal controlled environment for the much-needed data to check two-dimensional, nearshore circulation models. This is true for both the models based upon radiation stress theory discussed above and the newly emerging Boussinesq theory numerical models. IV. BOUSSINESQ THEORY Limitations in the range of application of Boussinesq theory compared with cnoidal and stream-function wave theories were shown in Figure 44. Here, those few examples of numerical solutions of the Boussinesq equations versus experimental results that have been published are presented. Also, the fact that a numerical theory carries with it inherent accuracy limitations based upon the numerical integration methods, grid sizes, and time steps used in the calculation is emphasized. Differences between theory and experiment may also be due to the poor choice of a numerical method with large truncation errors. ike Wave Shoaling. One-dimensional wave shoaling studies (solitary and pistarlo ete) were conducted first. Early pioneering efforts iy Peregrine (1967)22 » Camfield and Street (1969) 2, Madsen and Mei £1969)" » Madsen, Mei, and Savage (1970)72, and Chan and Street (1970) concentrated on the physics of soli- tary and periodic wave propagation (nonpermanent form, soliton development) rather than accuracy of numerical versus experimental data. For example, Figure 83(a) shows the calculated shoaling curves for a solitary wave (no/d = 0.1) on three different beach slopes by Madsen and Mei (1969)? conpared with some experiments by Camfield and Street (1969) 59 for °2PEREGRINE, op. cit. S9CAMFIELD and STREET, op. cit. “*OMADSEN and MEI, op. cit. 72MADSEN, 0.S., MEI, C.C., and SAVAGE, R.P., "The Evolution of Time-Periodic Long Waves of Finite Amplitude," Journal of Fluid Mechanics, Vol. 44, Pt. 1, 1970, pp. 195-205. 60CHAN and STREET, op. cit. to pads Sa ipide 210 1:0 hlhg Amplitude variation with depth for initial amplitude to depth ratio, 79/hy = 0-1. Caleutated results for slopes, &: ----, 0065; - , 0-05; —--—-, 0-023. O, experiments by Camfield & Strect for a = 0-02. 1 ao : [ loin, 4 1/20 71 No 3:5 t (sec) Figure 83. Comparisons of Boussinesa EROoEY ,ginte experiments for wave shoal- ing (after Madsen and Mei , 1969). tan 8 = 0.02. The data are seen to agree well with the theory in the present report but they omitted data at large slopes because of too much scatter. Figure 83(b) displays even greater disparity when a second crest appears and this is attributed to the neglect of viscous damping. The Boussinesq 49 : MADSEN and MEI, op. cvt. 21 theory correctly predicted the disintegration of the solitary wave and phase of its second peak which was of main interest. Madsen and Mei numer- ically solved the characteristic form of the equations by the method of characteristics. Abbott, Petersen, and Skovgaard (1978) have published the most exten- sive engineering test results for the Boussinesq theory to date. Figure 84 presents their comparison of the theory (System 21 Mark 8, diamond) against experiments made by the CERC in their large, outdoor wave tank (Madsen, Mei, and Savage 1970)./2 The tank is 194 meters long, 6.1 meters deep and 4.6 meters wide. Four second periodic waves with initial height of 0.2 meter were observed shoaling on a 1:15 slope. Agreement was said to be within 5 percent in elevation, over the whole test range. Cnn CERC Experiments Crest oO -0 | Trot giiee System 21, Wave Height | is Wave Height ® CERC, Wave Height -, J | System 21 | ats : L Mark 8 @ mie Theo Figure 84. Comparison of numerical computation of shoaling waves obtained using Boussinesq theory and CERC experimental results (after Abbott, et al., 1978). 72MADSEN, MEI, and SAVAGE, op. cit. 212 Recently, new laboratory experiments on wave shoaling have been made by Svendsen and Buhr-Hansen (1977) /3 and Flick (1978)/4. Some experimental results are shown in Figure 85 where it is again apparent that linear wave shoaling theory is far from correct. Most existing radiation stress models use linear shoaling theory to estimate wave height fields. The errors resulting could be excessive. Svendsen and Hansen utilized laboratory equipment with a special flap-type wave generator that eliminated free second harmonies in the wave profile. Test of the Boussinesq theory up to the breaking limit is an active research area and the boundary shear stresses must be included in the theory to affect a fair comparison. In this regard, Svendsen and Buhr—Hansen (1978)75 have shown that (1) Both wave height and profile show good agreement near break- ing even though the H/h ratio is large; (2) wave setdown is adequately described; and (3) the horizontal velocity ment even at the breaking point shows remarkably good agree- .'' with measured values. The theoretical results were obtained analytically and not by numerical methods. Field shoaling data for irregular waves are also needed to test the Boussinesq theory. The nonlinear process resulting in transfer of energy from the dominant frequency to harmonics via strong interactions is further discussed by Guza and Thornton (1978) where some field results are presented. Di Two-Dimensional Tests. Abbott, Petersen, and Skovgaard (1978) were primarily interested in testing the accuracy of their numerical model in two dimensions. For this purpose they duplicated a series of well-documented physical model experi- ments of the Danish harbor at Hanstholm. Because of the uncertainties associated with periodic wave tests, irregular waves based upon field- measured, time series of water surface elevations at the harbor entrance were employed. This illustrated the fact that numerical models can be 73SVENDSEN, I.A., and BUHR-HANSEN, J., "The Wave Height Variation for Regular Waves in Shoaling Water," Coastal Engineering, Vol. 1, 1977, pp. 261-284 (not in bibliography). 74PLICK, R.E., "Study of Shoaling Waves in the Laboratory," Ph.D. Dissertation, S.1.0., University of California, San Diego, 1978 (not in bibliography). 7SSVENDSEN, I.A., and BUHR-HANSEN, J., "Prebreaking Behaviour of Waves on a Beach," Euromech 102, University of Bristol, July 1978 (not in bibilography). 28) so a 0: RR |H,/Lp = 0.0357, 1Vg7h =4.35 cn sin IH/h =0.104 H,/L, = 0.0099, 97h = 8.70 H/h= 0.106 20) 10} a, ROKR E Thrbt Cait 4 al eR oz 00% 006 008 rn mma 0 c NY uy \ —-— SINUSOIDAL ——— CNOIDAL BH MOVING AVE. EXPER. 50 PEAK-TROUGH HEIGHT 40 Ls = us 30) Hy/Lo = 0.0039 ; WaZh =13.05 " T=1.93sec “4 _+ 0.0103 H/h= 0.111 S 20) 10 Hae SNM Mee re NN ere YT eau oe ee Suv le © DATA 18DEC75-1-2 O07 002 0.03 004 005 H/o THIRD ORDER 145 F - STOKES THEORY = = = = = 7 a a = —--— LINEAR THEORY “. Variation of wave height for three different wave steepnesges (a) BREAK POINT (b) Figure 85. Some results of recent wave shoaling experiments: (a) after Svendsen and Hansen (1977) 42> (b) after Flick (1977)24 7SSVENDSEN and BUHR-HANSEN, op. cit. 74PLICK, op. cit. 214 operated with any type of input boundary data. Their results are presented in Figure 86 (a, b, c). The surface elevation contours (a) result from H = 1.5 meters (T = 10.5 seconds) at the harbor entrance and generate the wave amplification factors K shown in (c) for 10 different positions along the quay (b). The difference between rms wave heights in the physical and numerical models was small enough to provide an acceptable level of confidence in the accuracy of the models generated by the Boussinesq theory. Figure 46 is a perspective plot of waves in the outer harbor shown in Figure 86(a). This is the only published comparison of the two-dimensional numer- ical results with physical experiments found in the literature. Comparisons of the two-dimensional computations with other analytical methods are also available. Figure 87 demonstrates a comparison for pure diffraction theory with the classical analytical results of Sommerfield for linear, light waves (Abbott, Petersen, and Skovgaard, 1978). The analy- tical wave orthogonals and fronts shown as dashlines are found to agree with the numerical methods as long as waves with very small amplitude (linearized wave theory) are tested.. Hebenstreit and Reid (1978) 7° tested their numerical model against some experimental results for solitary wave reflection from vertical barriers and linear theory (Snell's Law) refraction over a plane beach. The finite- difference algorithm devised by Street, Chan, and Fromm (1970) was employed. Figure 88 demonstrates the numerical results (solid line) for wave reflection against the ripple tank measurements of Perroud (1957)78. The Mach-Stem effect was observed in the numerical work for incident angles between 20° and 45°, as expected. Numerical accuracy was considered quite good since the calculations neglected wall friction, the measured values were quite small, and it was not certain if steady state had been reached in Perroud's values. The wave refraction studies were very revealing. Significant differ- ences in wave crest bending (refraction) and wave shoaling were observed in the model as compared with that predicted by linear wave theory and Snell's Law for refraction. The Boussinesq theory simulations for solitary 7€HEBENSTREIT, G.T., and REID, R.O., "Reflection and Refraction of Solitary Waves--A Numerical Investigation," Report 78-/-T, Oceanography Department, Texas ASM University, July 1978 (mot in bibliography). 77STREET, R.L., CHAN, R.K.C., and FROMM, J.E., "Two Methods for the Computa- tion of the Motion of Long Water Waves--A Review and Applications," Stanford University, Department of Civil Engineering, Technical Report No. 136, 1970 (not in bibliography). 78PERROUD, P.H., "The Solitary Wave Reflection Along a Straight Vertical Wall at Oblique Incidence," University of California, Berkeley, Technical Department, Series 99, Issue 3, Berkeley, Calif. (not in bibliography). Zio INNER HARBOUR —_ a. b. WHARF NO. 13 WHARF NO. 17 Ki Ff ama SS Se = (SSS 1 Ka 057 o5 | | 04 | +04 OS} a | 0.3 Of 0.2 i POSITION NUMBER PHYSICAL MODEL ( T = 10.5 sec.) ——— MATHEMATICAL MODEL KG = Pala entrance Figure 86. Comparison of wave amplication factors by physical and numerical models in Hantsholm Harbor, Denmark (after Abbott,et al., 1978). 216 rs --- WAVE ORTHOGONAILS (0) ¢* AND FRONTS (F) ACCORDING E HEIGHT !S ONE METER 8 wav se classical diffraction theory for linear, small amplitude waves Comparison of pure wave diffraction by numerical model and (after Abbott, Petersen, and Skovgaard, 1978). Figure 87. 217 (n = as) Angle of incidence Figure 88. Comparison of pure wave reflection by numerical model and experiments of Perroud (after Hebenstreit and Reid, 1978). 76 waves consistently produced a smaller degree of wave turning than predic- ted by Snell's Law. The reason was primarily in the phase speed which was far less in the numerical simulation than Ved. Wave heights due to 7€4UERENSTREIT and REID, op. cit. 218 shoaling were always greater than given by linear theory which was consis-— tent with the one-dimensional tests above (Fig. 85). Refraction coefficients for the a = 30° and 45° tests are shown in Figure 89. Unprimed values are based on angles calculated by Snell's Law. The numerical model showed less departure from linear theory at 30° than at 45°. It was concluded that wave refraction techniques based on linear theory (Snell's Law) do not accurately predict the behavior of nonlinear dispersive waves in shoal- ing and refraction. Regarding wave refraction, it was also concluded that "The lack of similar laboratory and field work inhibits the range of conclusions that can be drawn from these results (Hebenstreit and Reid, 1978, p. 95). 76 The limited comparisons between Snell's Law and laboratory measurements shown by Wiegel (1964) 22 exhibit systematic differences and wide scatter in some instances. It would now appear that these measurements made in the 1950's must be supplemented by more sophisticated experiments to verify the new numerical models of the 1980's. (2) o Energy refraction coefficient, ve) oo 40 30 Depth (m) Wn oO Figure 89. Comparison of pure wave refraction coefficients by numerical model and linear wave theory (Snell's Law) (after Hebenstreit and Reid, 1978). 76HEBENSTREIT and REID, op. cit. 22WIEGEL, op. ctt. 219 V. SUMMARY OF STATE-OF-THE-ART Live Physical Description. A review of more than 60 years of observations and measurements of longshore current, nearshore circulations and rip currents worldwide reveals that longshore currents are primarily caused by oblique wave inci- dence and breaking at the coast which creates an excess longshore momentum flux to drive the current. In nature, the longshore current varies across the surf zone, along the coast, with depth and in time at any location. Typical values are less than 1 meter per second with maximum 3.5 meters per second or more. The use of continuous recording current meters at fixed points arrayed across the surf zone and down the coast in recent experi- ments has permitted researchers a look at far more details of the current structure. Literally millions of instantaneous values of these currents are now available from the two NSTS experiments and most have yet to be fully analyzed. The point velocity measurement in the longshore direction continuously varies in time due to the unsteadiness of nature and because many forces are at work together. Wind waves, surface winds, tides, and turbulent eddies from many sources all contribute to the signals recorded. The influence of surface winds has yet to be examined in detail. By definition, the longshore current is a time-averaged current. But, as of today, an appropriate temporal averaging time is not known. This fact plus the ability of these meters to record storm-induced currents has resulted in even larger longshore current values being reported. Far more is known about horizontal circulations around vertical axes and resulting rip currents than is known about the vertical velocity distributions and circulations about horizontal axes in the surf zone. Many plausible mechanisms have been advanced for triggering rips to form and resulting neashore circulation cells to develop. Whatever the origi- nal mechanism, strong bottom currents in rips can create troughs in offshore bars to fix rip positions. The local bathymetry then controls the future flow patterns observed. Beach slope, profile, and sediment characteristics all play important roles in coastal hydrodynamics. A complete understanding of steep (reflective) and flat (dissipative) beaches has yet to be achieved. The existence and role of secondary currents has yet to be firmly established. New instruments are being studied to measure surf zone currents. Even though problems exist with dynamic calibrations, the EM current meter remains the most proven and rugged type available for multiphase (liquids, sediment, air bubbles) flows in the surf zone, and the new slope array device should prove to be a valuable tool to examine the theory of long- shore currents in both the field and the laboratory. Because of the unsteadiness of nature and all the uncontrollable forces present simultaneously, controlled laboratory experiments have been conducted. Problems with the boundaries are being overcome so that infinite beach theories can be examined at the scale of the laboratory 220 tests. Additional laboratory experiments are needed to verify new two- dimensional numerical models applied to situations with irregular bathy- metry, coastal structures (breakwaters, groins, etc.), and irregular wave fields. A large number of wave, wind, geometry, fluid, and sediment factors interact to create the currents observed. Considerable simplification and idealization is necessary to formulate the theory and equations needed to obtain a solution. 2 Theory and Experimental Verification. The time-averaged equations of horizontal motion and continuity, including radiation stress terms, have become the accepted basis for a theory of coastal hydrodynamics in the 1970's. Short-period wind waves when time-averaged create additional momentum flux terms, i.e. the radia- tion stress gradients, which become the primary driving forces in the theory. Solution of these three equations determines the three dependent variables of interest: the MWL change nj, the longshore current V, and the cross-shore current u. Wave heights must be specified a priori to affect a solution because the local wave energy appears in the radiation stress terms. This requires use of wave shoaling, refraction, diffraction, reflection etc. theory outside the breakers, a wave breaking criterion, and some model (general, empirical) of surf zone energy dissipation. Additional stress gradient terms appear in the motion equations to account for surface wind and bottom friction shears and lateral, internal, turbu- lent mixing stresses. All require semiempirical turbulence stress models with appropriate closure coefficients determined experimentally. The wave breaking and surf zone empirism along with general knowledge of the required closure coefficients remain the weak links in the theory. The original theory makes extensive use of linear (Airy) wave theory. It forms the basis for calculation of the radiation stress components both outside and within the highly nonlinear surf zone and is used in wave shoaling, refraction, diffraction, ete. calculations. Applications to simplified geometrics with appropriate boundary conditions have tradition- ally been in the following categories: (1) Mean water level changes (setdown and setup), (2) uniform longshore current profiles, and (3) nearshore circulation systems with rip currents. The findings in these three areas are summarized below. Major modi- fications to the original theory have taken place in (1) Bed shear-stress formulations, (2) extensions to irregular waves, 221 (3) surf zone energy dissipation models, (4) different mixing models inside the surf zone and beyond breakers, and (5) by use of two-dimensional numerical models to compute solutions. In general, the experimental data from both field and laboratory measurements substantiate the qualitative aspects and trends of the radiation stress theory. However, sufficient data under widely varying conditions are lacking to quantitatively verify all aspects of the theory, to select the best submodels, and to determine the closure coefficients. In some cases (e.g., rip current formulations and spacing) mechanisms are still being studied as part of the theory so that controlled labora- tory investigations still play a major part in the research effort. Extensive new field data collection efforts (NSTS, ARSLOE) will signi- ficantly enhance efforts to verify the radiation stress theory and estab- lish the needed coefficients in the near future. This background provides a beginning in understanding the current state-of-the-art, and the radiation stress approach will continue to be researched. Further refinements will be made by use of nonlinear wave theories in both the radiation stress and wave field computations, use of better wave breaking and surf zone simulation models that include breaker type and beach profile features, determination of rational procedures to select closure coefficients, and incorporation of wave-current interactions in all aspects of the computation. Accurate two-dimensional numerical models will be constructed which include all the physically important terms and minimize the numerical inaccuracies. However, signi- ficant improvements in knowledge will always be limited by the time- averaged nature of the radiation stress approach. In effect, a plateau of advancement has been reached in which a somewhat rational framework is now available for time-averaged coastal hydrodynamic theory. This approach will benefit from further research but only moderate improvements in fundamental, physical knowledge are anticipated because the time-averaging omits many local, time-varying details. Waves inthe nearshore and surf zones are highly nonlinear. Solitons form and harmonics appear in nature with energy levels on the same order as the dominant period. Unsteady currents appear due to many forces including wave breaking rollers, eddies, and turbulence. The longshore current is by definition a time-averaged current yet a univer- sally accepted averaging time is not known. Efforts to verify the radia- tion stress theory by new field measurements will be hampered by the need to average the time-series obtained over some time interval yet to be determined. The Boussinesq theory offers the possibility to eventually raise the fundamental knowledge of coastal hydrodynamics to a higher level. No time-averaging is involved. Nonlinear wave propagation and resulting 222 wave height variations are all automatically produced as part of the calcu- lation procedure. The unsteady asymetrical currents and instantaneous water surface variations as solutions to the governing equations are only obtain- able with the aid of large high-speed computers. Solution techniques and applications are in their infancy. Wave breaking and surf zone simulations have yet to be implemented. Nonbreaking solutions have indicated signifi- cant differences with linear theories of wave height variation (Snell's Law, diffraction) to require additional experimental data for analysis. The state-of-the-art summary of all the theories and experiments is presented below. a. Mean Water Level Changes. Generally, the results are for normal wave incidence which in itself greatly limits their practicality. The original theory for wave setdown on plane beaches, based upon linear theory radiation stress, sometimes referred to as first-order theory, cannot be substantiated by the experiments. Consequently, equation (3) and its variations (eqs. 31 and 32) are incorrect. The nonlinear theory (Svendsen and Hansen, 1976) using first-order cnoidal wave theory and given by equation (155), has been verified right up to the breaking limit. It is repeated here A computer-based solution would facilitate and should be no deterrent to its use. The irregular wave models of Battjes (1974a) or Goda (1975) also derived setdown values close to measured data. The original theory for wave setup as given by equation (35) can only be verified for relatively steep plane beach slopes (tan 8 £ 0.1). It makes use of a constant breaker height-to-depth ratio y in the surf zone which also has only been verified under steep beach conditions. Wave steepness is also a factor. Consequently, use of equation (35) and varia- tions therefrom (eqs. 68 and 69) should not be generally applied to long- shore current theory as a correction for wave setup effects. This result is contrary to the popular belief that the theory applies to spilling breakers on dissipative-type (relatively flat) beaches. In fact, it has only been verified on steep laboratory beaches with plunging-type breakers. The nonlinear setuptheory of James (1973, 1974a) requires further experimental verification. No cnoidal theory in the surf zone has been attempted. The irregular wave setup theory of Battjes (1974a) could not be con- firmed by laboratory experiments although it gave reasonable agreement with limited field data. However, later efforts of a similar nature by Battjes and Janssen (1976) in which a more sophisticated, hydraulic jump model of surf zone dissipation was employed (eqs. 44 and 124) resulted in good to 223 excellent agreement with new laboratory data. This was for both H,p. and N variations across plane and bar-trough profiles. Improvement in the theory to match the data for low steepness waves is expected by incorpora- tion of a wave breaking criteria over a range of wave heights with varying probability (following Goda, 1975). Wave height variationacross the surf zone and wave-induced setup are obviously related. Ali experimental evidence indicates that as beach slope decreases, the y ratio departs further from a constant value across the surf zone. Distance from the breaker line is an important variable. A constant y ratio in the surf zone is incorrect (eq. 33) for all beach profiles. The surf zone energy dissipation model based upon equation (44) with equation (124), i.e. dF x — ae + Wasi 0 with as developed by Battjes and Janssen (1976) is the most acceptable for all beach profiles at this time. Local wave heights in the surf zone are found by integrations so they logically depend upon all preceding sea- ward water depths plus the local depth. It is recommended that Battjes' model of the surf zone be studied further for regular wave setup and wave height decay. Additional comparisons with experimental data plus some minor modifications to the theory are also warranted. Wave setdown and setup theoretical equations for oblique wave inci- dence (eqs. 39, 40, and 41) are not recommended. They.are based on linear radiation stress theory, constant y in the surf zone, and neglect the feedback of the longshore current induced on the wave motion. No comparisons with experimental data have been attempted. bye Longshore Currents. If a rough estimate of the average long- shore current is needed it is recommended that the expression wv = K/gh, sin (20, ) K : 0.3-0.6 (170) be employed. Komar and Inman (1970) empirically obtained K =z 0.6 for the midsurf velocity v1 from results of sand transport studies. Dette (1974) obtained K 2 0.3 from measurements in the North Sea and an analysis of all formulas including those based upon radiation stresses for the average current. Beach slope and bottom roughness do not appear in the empirically determined equation (169) since it has been found theoretically that the midsurf velocity is relatively insensitive to changes in these independent variables (Kraus and Sasaki, 1979). 224 The original model theory for uniform longshore current profile on a plane beach includes many assumptions and is based upon the idealized environment in Table 2. It has laid the foundation for all subsequent modified theories by providing both a qualitative and an order-of-magni- tude agreement with the data. Consequently, equations (61), (66), and (67) plus those associated with them,must now be replaced by more general and accurate formulations. Longuet-Higgins (1972b) recognized these defi- ciencies and made some recommendations for improvements that were incor- porated in the subsequent modified models. The major limitations of the original model theory were (a) Bed stress model for weak currents and small angles, (b) wave setup effects neglected, (c) excessive lateral mixing stresses outside the surf zone, and (d) results only applicable to plane beach slopes. Table 3 identifies the modified theories. All include some wave setup effects based upon a constant y ratio in the surf zone. For the reasons just discussed, it is concluded that further refinements are possible in the near future. With these limitations, two modified theories have emerged recently that have shown good comparisons with experimental data. The theory of Skovgaard, Jonsson, and Olsen (1978) is more generally applicable (i.e., monotonically decreasing profiles) but requires numerical solution procedures. Its key strength is a different lateral mixing form- ulation outside the surf zone that matched some limited laboratory data for currents and also observations of mixing intensities in nature. Its major weakness is the strong current small-angle formulation for bed shear stress. This assumption is inconsistent with the fundamental physical fact that strong longshore currents appear when the angle of wave attack is large. Small incidence angles produce nearshore circulation cells, rip currents, and two-dimensional flow patterns. The best available analytic theory at this writing is that recently developed by Kraus and Sasaki (1979), but holds only for plane beach slopes. It includes wave refraction effects, wave setup, different lateral mixing formulations within and beyond the breakers, and a weak current large-angle bottom stress formulation. For the latter assumption, the ratio of v,/up, is found to be relatively small in the field to justify the weak current large-angle model employed. The results compared favor- ably with new laboratory data (Fig. 66). The theoretical curves are fitted to the data by requiring the theoretical location of the maximum velocity Vm to match the experimental results. The dimensionless longshore current profile given by equation (91) and Appendix D (see also Fig. 30) with key dimensionless parameter P* (eq. 92) should be employed. It should be tes- ted against other laboratory and field data from which Cr and I values can be determined. For bar-trough profiles, a numerical solution procedure is necessary. More exact bottom shear-stress and surf zone dissipation models can then be employed in the numerical procedures. 225 The nonlinear current profile theory of James (1973, 1974a) requires further experimental verification. No application of cnoidal theory has been attempted. The limited comparisons with James' data were all favor- able suggesting that further work needs to be performed. Longuet-Higgins (1972) stated the need to use nonlinear theory in Sxy- Irregular wave theories of Battjes (1974a) and Goda (1975) need experimental verification. Battjes' (1974a) model should be modified to. incorporate the Goda (1975) breaking criteria, the new Battjes and Janssen (1976) surf zone model, and include lateral mixing stresses. The available data should be curve fit to the theory at vin to permit extraction of closure coefficients as by Kraus and Sasaki (1979). It was surprising to discover that additional theoretical effort with irregular waves has not been pursued since the mid-1970's. The proven existence of harmonics of the incident wave period in the surf zone and the detailed field test results from the NSTS experiments are two good reasons for further theoret- ical work with irregular waves. Go Closure Coefficients. Modified bottom shear-stress and lateral mixing stress models were discussed in detail in Chapter 3 (See Tables 3 and 4). The more general expressions for TBy (e.g., eq. 80 by Bijker and v.d. Graaff, 1978) 26 for strong currents and large angles require numerical solution techniques. Solutions for arbitrary variations of bottom profile and roughness require numerical methods. The new surf zone energy dissipation models require numerical integration. Also, different models for lateral mixing stresses across the surf zone further complicate the analysis. In short, all factors indicate that oversimpli- fication to obtain analytic solutions is being abandoned in favor of more general numerical solution techniques. Nonlinear and irregular wave theories require such methods. The more general bed shear-stress formulations of Jonsson, Skovgaard, and Jacobsen (1974) (eq. 76); Liu and Dalrymple (1978) (eq. 78 and others not included); and Bijker and v.d. Graaff (1978) 26 (eq. 80) should be considered further. In some cases, the closure coefficients for Tpy are in themselves calculated from knowledge of the local bed roughness and wave characteristics. This procedure is used in the Jonsson, Skovgaard, and Jacobsen, and Bijker and v.d. Graaff models.- It is preferred since it utilizes more fundamental fluid mechanics stress coefficients based upon uniform ‘open channel flow and oscillatory water tunnel experiments. The Bijker model especially warrants further study since it explicitly defines the location above the bed where the velocity components are vectorially combined. The more general lateral mixing stress models of Skovgaard, Jonsson, and Olsen (1978) and Kraus and Sasaki (1979) (see Table 4) are preferred since they separate the reduced mixing outside the surf zone. Both are variations of earlier efforts and warrant further study. 26BITJKER and GRAAFF, op. cit. 226 Closure coefficients required in these models can be obtained by fitting the theory for longshore current profile to available data. Some results are given in Table 9. They must all be used with caution since they only apply to the theory and limited range of data involved in their determination. No state-of-the-art values will be given here for this reason. The wide range of time-averaged bottom friction coefficients in the literature is partly due to the many assumptions in the time-averaged bed shear-stress models employed. Use of the maxi- mum longshore current and its location to fit the data is better than use of mean values which are relatively insensitive to bed slope and bottom roughness. Both friction and mixing coefficients are calculated from the data by this method. An alternate method would be to specify the bed roughness, calculate the friction coefficients from fundamental theory, and use the longshore current profile data to estimate mixing coefficients. These closure coefficients would be based on the overall longshore current profile and can be obtained from field or laboratory experiments. Local closure coefficients in the surf zone can also be obtained directly from equation (42), local velocity measurements in the surf zone and independent measurements of S,,. This was done by Huntley (1976) and Thornton (1980) who both neglected the lateral mixing stress gradient in the analyses. It may be possible to independently obtain estimates of Vy, from the velocity time histories using auto- and cross-correlation techniques. Then the full equation (42) could be uti- lized to calculate friction coefficients. Only detailed and extensive two-component field data are appropriate for this purpose. The empirical wave breaking ratio, y};, and surf zone energy dissi- pation model complete the list of required coefficients to theoretically estimate current profiles. As stated by Battjes (1978a), use of a con- stant y ratio imparts an excessive sensitivity of the currents to bottom profile variations. Comments above for wave setup and setdown are equally appropriate for longshore current. It is anticipated that some fundamental new concepts in wave breaking criteria will result from the present numeri- cal modeling simulations of the wave breaking process. d. Nearshore Circulation Systems. All available models are based on vertically integrated or depth-averaged flows. This implies that the dominant motions are horizontal and relatively uniform with only weak secondary currents in the vertical taking place. Little data are available to substantiate (or refute) this assumption. The two-dimensional equations of motion written in conservation form (eqs. 107, 108, and 109) are preferred over the Eulerian form since the discharges per unit width include wave-induced mass transport. Many terms were discarded in the earlyanalytic theory. For general application, all must be included which means numerical solution methods are now neces-— sary. Additional interaction stresses resulting from mean flow, oscilla- tory motion, and turbulence interactions are present but for lack of suffi- cient detailed data, they will continue to be lumped with boundary shear and lateral shear terms. 227 All the presently available numerical models have serious limitations and none are recommended for engineering purposes. The earlier versions omitted important terms in the momentum balance equations, and the latest models simply lack accuracy due to excessive truncation errors in the algorithm employed. The lateral (eddy viscosity) mixing stress terms physically smooth the current profiles, circulation patterns, and rip current jets. The present models display evidence of excessive numerical viscosity (Fig. 35) that translates into numerical inaccuracy. Calibration of such models requires use of physically unrealistic closure coefficients that could be a disaster for use as predictive models. The evidence to support this conclusion is vividly displayed in the generally poor comparisons between model and experiment shown in Figures 80, 81, and 83. As outlined by Vreugdenhil (1980), it is clear that the Delft Hydrau- lic Laboratory has begun a concerted effort to develop a comprehensive numerical model. It is by far the best effort discussed to date because it includes all the physically important terms and proper numerical methods to ensure their accuracy in the simulation. The report by Vreugdenhil (1980) discusses the model requirements, equations, and numerical procedures. Calibrations, tests, and other results will be reported when completed. It is clear that additional comprehensive model- ing projects are desirable as both an alternative and in support of physi- cal model tests and expensive field experiments. Previous efforts were hampered by lack of data to calibrate, verify, and test the two-dimensional models. Some additional controlled labora- tory data are now available (Gourlay, 1978; Visser, 1980; Mizuguchi, Oshima, and Horikawa 1978) and can be obtained for this purpose. Also, the extensive NSTS field data tapes can be used. Some fundamental problems remain with the time-averaged simulation models. The wave height fields must be specified by other means. For this purpose, the model developed by Noda, et al. (1974) for wave shoal- ing and refraction that includes current interactions remains a popular choice. Wave diffraction, reflection, transmission (breakwaters), etc. transformations must also be prescribed. The numerical programs for wave height transformation may require considerable development effort, by themselves, if not already available. In addition, it was seen earlier in this chapter how direct wave refraction calculations by Boussinesq theory produce significant differences with Snell's Law. This indicates that nonlinear refraction and other wave transformation theories should be employed. Finally, the correct numerical simulation of circulations, eddies, and subgrid-scale turbulence is still an active research area in computational hydraulics. ‘ A verified theory to predict rip current spacing for all possible conditions does not exist. The best engineering estimate available is the semiempirical hypothesis of Sasaki (1977) summarized in Figure 78 and equations (167), (168), and (169). No theory exists for rip current ©7MIZUGUCHI, OSHIMA, and HORIKAWA, op. cit. 228 flow rate, width or size, and surprisingly there are little field data to compare with numerical simulations. Finally, for near-normal wave incidence, vertical gradients in the velocity fields can create vertical radiation stress gradients to drive circulations about the horizontal axis. No attempts were uncovered to apply radiation stress theory to this case for estimates of circulation currents schematized in Figure 22. e. Boussinesgq Theory. Research has’essentially just begun. Out- side the surf zone, numerical shoaling results compare very favorably with experiments. More research and comparisons are necessary near the breaking limit. Only one comparison of two-dimensional wave transforma- tions against physical laboratory measurements has been published in the literature (Fig. 86). Much more evidence is desired. The correct form of the Boussinesq equations for variable bathy- metry has yet to be determined. No wave breaking, surf zone, wave setup, or longshore current simulations have been reported in the literature. Great care is required in the numerical methods to ensure accuracy in wave amplitude and phase propagation. Large and high-speed computers are necssary. For these reasons, future developments will remain for research purposes rather than engineering applications. The analogy.with time-averaged tidal hydraulics analyses of the 1950's was cited earlier. The digital computer permitted calculations within each tidal period to enhance knowledge. Guza and Thornton (1978) in their article on time variability of longshore currents make a similar comment. In the 1950's, large-scale ocean circulations were thought to be driven by average winds to produce generally weak and horizontally smooth currents. The relative importance of drag and eddy diffusivity terms was discussed, as we have summarized in this report. Nonlinear terms were then needed to explain the jetlike Gulf Stream; as their importance in rip current dynamics is now recognized. Still later, instan- taneous measurements showed large temporal and spatial fluctuations, in contrast to observations with floats that matched the theory for yearly means. Large cooperative experiments were then conducted in an effort to determine the importance of the shorter scale fluctuations on the longer scale flows. These final questions remain unresolved. Considering the gross nonlinearity of the surf zone and the strong analogy cited with oceanic research, Guza and Thornton (1978) concluded that it is overly optimistic to expect simple solutions to mean nearshore flows. The Boussinesq theory does offer some possibilities in this regard by going beyond the time-averaged mean to look within each wave period at the physics taking place. 229 CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS The following conclusions and recommendations are based on the entries in the Surf Zone Currents, Annotated Bibliography," (Vol. II) and the review and summary of physical processes, theory, and experimental verification described in this report. I. CONCLUSIONS il General. A general time-averaged coastal hydrodynamic theory now exists based upon sound conservation laws of physics and radiation stress principles. It has been proven to qualitatively describe the mean water surface varia- tions and currents generated by short-period surface gravity waves as shown by laboratory and field data comparisons. However, to improve the quantita- tive accuracy, further research and development is required. Specific con- clusions on various aspects of the time-averaged theory are presented below. Two inherent shortcomings of the method are: (a) Wave height fields must be specified by independent methods. Are these methods accurate? (b) Closure coefficients must be determined from field data fitted to the theory. The proper averaging time for the field data is not known. Spatial and temporal variability of field data cannot be explained nor studied by the method. The Boussinesq theory offers an alternative approach. It is founded in the same conservation laws of physics but no time-averaging takes place. Instantaneous water surface and current variations are considered. Conse- quently, the two shortcomings listed are not present in this method. To date, only conditions up to breaking have been simulated by the theory. It is concluded that research efforts to extend the method into the surf zone should be conducted. Specific conclusions on the Boussinesq theory are given below. The theory will primarily serve as a research tool to further understanding of the physical processes taking place in the surf zone within each wave. Consequently, it will also serve to improve the time-averaged method. Both theories rely on wave breaking and surf zone empirism that requires fundamental research for improvement. ie Time-Averaged Radiation Stress Theory. ale General. 230 (1) Nonlinear wave theory (e.g., cnoidal waves) should be studied further. Cnoidal theory fits the data best for wave setdown near breaking. Its use for wave field specification and in radiation stress calculation should be considered in a general higher order current theory. (2) Irregular wave theory (e.g., Battjes and Goda theory) should be studied further. Surf zone probability models could be devised to in- clude the wave harmonics observed in nature. Lateral mixing terms must also be included. (3) Analytic theories require too many assumptions and thus limit generality and accuracy. Computer solutions are now necessary for both longshore current (one-dimensional) and nearshore circulation (two-dimensional) computations. bie Mean Water Level Change. (1) First-order (linear) theory gives incorrect results for wave setdown. Cnoidal (nonlinear) theory has been verified to near the breaking limit. Use first-order cnoidal wave theory for normal incidence setdown cal- culations. (2) First-order (linear) theory is only verified for wave setup on steep plane beaches (tan 8 * 0.1) where constant y ratio is also observed. Use of a constant y ratio across relatively flat (tan B * 0.01) and bar-trough profiles in incorrect. New surf zone energy dissipation models (Battjes and Janssen) used to compute wave setup show promise but need further research. Use of wave setup theory based upon constant y ratio to modify longshore cur- rent formulas gives incorrect emphasis on bottom profile. Nonlinear wave theories for wave setup need further research. (3) Special formulas for setdown and setup under oblique wave incidence should be avoided. Solution must be based on coupled two-dimensional equations where wave-current interaction effects are included. @5 Longshore Currents. Gb) “Was WSK Yeu, sin 2a, (K = 0.3-0.6) to roughly estimate the average longshore current. (2) The original model by Longuet-Higgins (1970), which gives qualitative results, paved the way for all subsequent versions, but is now relatively incorrect. (3) The analytic model of Kraus and Sasaki (1979) should be used for plane beach computations on relatively steep beaches. (4) No verified analytical or numerical model exists to compute currents on relatively flat (tan8# 0.01) or bar-trough profiles. (5) Existing nonlinear and irregular wave current theories require some modification and extensive comparisons with laboratory or field data sets in order to be useful. (6) The model of Kraus and Sasaki (1979) should be developed further to include flat and bar-trough profiles plus more general bed-stress formulations. Numerical solution methods will then be required. (7) Complete strong current large-angle bed-stress formulations should be considered in future models where numerical methods are employed. The model devised by Bijker and v.d. Graff is recommended for further study since fundamental closure coefficient and boundary layer principles are ap- plied. 231 (8) Different lateral mixing stress models should be employed within and outside the breakers. Further research is needed to determine which are most accurate. (9) No recommendations for state-of-the-art closure coefficients are made. Continued indirect and direct determination from field data is required. Bed friction factors must be correlated with relative boundary j roughness. Eddy viscosity closure coefficients should be related to surf zone, Reynolds and Iribarren numbers. (10) Wind-induced longshore currents can be created in some numerical models but have yet to be studied in detail. d. Nearshore Circulations. (1) All theory is based on the assumption (not verified) that the two horizontal motions dominate the flows present nearshore. No three- dimensional theory or models have been attempted. (2) The two-dimensional, conservation form equations of motion including all acceleration, préssure and stress gradient terms should be employed. Additional interaction terms discussed by Harris and Bodine (1978) are incorporated indirectly by the closure coefficients. (3) None of the existing two-dimensional models are recommended for engineering use. All exhibit serious limitations due to neglect of im- portant terms or excessive numerical inaccuracies. Additional comprehensive two-dimensional numerical modeling efforts are desirable. (4) Specification of wave height fields by linear wave theory (shoaling, Snell's Law, diffraction theory) is questionable. Existing numerical schemes for this purpose should be revised to incorporate nonlinear (cnoidal) theory. (5) Numerical simulation of circulations, gyres, eddies, etc. in free-surface flows is still an active research area in computational hy- draulics. The eddy viscosity model is only one of many new turbulence closure models being investigated. (6) No single theory is valid and verified to predict rip current spacing. Semiempirical engineering estimates are available. While many plausible mechanisms exist to trigger rip currents, bathymetry usually con- trols their location. (7) Radiation stress principles can be employed to study vertical circulations about a horizontal axis. No such studies were found in the literature. Shs Boussinesq Theory. a. General. (1) The complete and correct form of the equations of motion for variable bathymetry applications have yet to be finalized. Limited appli- cations to date have been for gradually varying bathymetry. (2) Higher order accuracy numerical methods are required for a solution and are still in the development stage. (3) Considerable research and development remains for applica- tion to surf zone hydrodynamics. 232 be Specific. (1) The theory for plane slopes has been verified near the break- ing limit. Further research at the breaking limit is needed, including wave setdown investigations. (2) Limited evidence available suggests the theory is applicable to two-dimensional nonbreaking wave transformation studies as presently con- ducted with physical hydraulic models. (3) No wave breaking nor surf zone simulation results have been published in the open literature. Such efforts have been recommended and are currently being pursued by European researchers in coastal hydrodynamics. (4) Large high-speed computers are required for accurate simula- tions. 4. Data Base and Measurement. a. Data. (1) The overall data base to verify the theories is inadequate. However, new laboratory experiments (Visser, 1980; Gourlay, 1978; Mizuguchi, Oshima, and Horikawa, 1978) and field data (NSTS) are becoming available for this purpose. (2) Results of the recent extensive NSTS field experiments need further detailed study. The results are readily available for this purpose. (3) Additional controlled laboratory tests are needed to study and verify aspects of both theories. A spiral wave maker can be readily designed for this purpose. Tests with relatively flat beach slopes and bar- trough profiles are critically needed to verify new surf zone energy dissi- pation models. (4) New, extensive field investigations should be delayed until analyses of the NSTS results are completed. (5) Field investigations of coastal hydrodynamics have yet to be made on Gulf of Mexico beaches. Experiments on this relatively flat low- energy coast are needed. (6) No data on the vertical current profiles over the water column exist except for some rip current profiles. Also, long-term current time histories are needed for statistical purposes at selected locations. (7) Further extensive use of the slope array device (Seymour, et al.) for direct measurement of Ss and a is recommended. bre Measurement. (1) The use of extensive arrays of fixed current meters in the surf zone is the best way to obtain sufficient and useful field data for verification of theory. (2) Current meters with flat gains beyond 2 hertz are needed to obtain additional turbulence information in the surf zone. (3) The STEREO-BACS system could be significantly improved to overcome problems with camera drift and surfboarding drogues but will be limited to observations of nonstorm events. 233 II. RECOMMENDATIONS It is recommended that fundamental, long-range research be conducted to extend the Boussinesq theory into the surf zone. For practical engineering investigations of coastal hydraulics, knowledge from Boussinesq simulations will be most useful to improve the time-averaged radiation stress approach and its eventual coupling to sediment and pollution transport models. Therefore continued refinement of radiation stress numerical models is also recommended to include nonlinear and irregular wave effects and improved wave breaking and wave energy dissipation models in the surf zone. 234 Pecan paq9a3ap ssautpeazsu 201°0 2210°0 2PLM 002-00 4eg das/44 -G°l h Edin ~patue, s0°0 -10°0 ost ~ mia ie a : L96L} LLouss ‘auoo LUO LL eI ‘ Saul 4 LS6L uutnd ‘uewuy KassO}| OG6| | uewuy *paedays uuadu09 2149 JO eLusoyt [29 paedays uoLzdauLG usay nos e7ed plat Jot Ae uMOUY S91 [4eg N — eLusOsL Le apLsuesc9Q Syuny Sweuznd 9u0Z | a40ysidoud- WINE Xe syueuiay Pounseoy quauun) OLzep eA uo} eAUaSqO auoyshuo7 uray e7eq oAeN SLNJYUND JYOHSONOT - SNOILYAYASEO d1S14 VY XIGN3ddV 235 e es a SUOLIRLULCA awit}? UoL}eunp ayNuLW GL paAuasqo sdiu dtu-9as/946"¢ HoLzepnouly B4OYSUPaN [eee apy cpa hes aa Squauung dry uoLze{noaLy| a4OYSueay x< Gi auo7 jasoysoud- |}, guns} Buoj}4aq ahkq |2e014 U paunseay uoLzeLUeA 40} pouzay uoLzeAUaSqQ €°05S°1 ° [=) N +1 “sn, pnang pue| bug ydeag uozunes Pas YI4ON Se ESS e1peugsny yoeag seuung eplso|4 aaoubeas uebLydiw puel [oH epl4ol4 OOLXaW “JELED “OS SOdU0W *yaueqgoW * ,eYOURW uaMmog *Xa,quny Jayoquyn4 ‘91100 oueglys *ekiyons| 236 pyeses *emMey LUO} staeg ‘x04 ounug das /ul WINE x BY| quauung auoysbuo7] uray e7eq aAeM SINAYYND AYOHSONOT - SNOILWAYSSdO A114 sag uoL}e907 : . uebtitW uds piled : -4e e ‘uo P yrdag a ULI {Z Japua| Ly (Q-€) pau ey 70°L *o0L “SSPH| 8/61 | AeAawyuLuuarg -Seau suo} “angst es} > 2/61 -eLueA Yyzdaq L9qtas eLusose ed 9L6L “edSaueW 12 Uday} 40N uebiystw}] 2/61 Kel cx L°0> oSt | 2°b-S"e axel) 9/61 sone olé-Sl apiszsey SL6L ; eLpul Lueydapese, *yoeag G/6L Sekkeuaap aynbue| ey *AQ4NW uebLyotw aye] 35209 ULSUODSLM SZ6L SLLtsied £9 PLusOJL | 29 nBnw “4d aie X 20 je face 082-S S-S°2 | 9°0-40°0 e7190 OLIN ef | E Ee EB a> ~ oO fy 0b x X F£°0 9°0 vl-8 lwemcunl uedep vL6L fyeses 12 nPAeOgs4Ns,, . eunebi ly s}e0|4 Ess 9u07] a4u0y sy4euiay | suns} bu07q sqgQ} uotze07] ON ON paunseay quauung sof quaaun) UOLZeLUeAL - poyzey auoysbuo07 ueay e7eG ancy uO LZ eAUaSGO SINJYYND JYOHSONOT - SNOTLVAYSSHO C1414 237 Ean CY) Oo Aauhes PLUdOFL| OD puedays et lorey BLusos L129 euequeg Ou *ezny satpnys Lapou get osty PeyxL4eH yoeag Luseueby] yoeuwoueuey Lyeses SUO}7ELURA yydaq “peue) zans ueay pue|od (spa) ua_pLaz ysuep lassew Lyeses *sneuy suoty -eAUasqgo 071 Aa|ur4 * | Ppawiwny uojUsOYL Sezng 3U07} au0us| des /w das /w Blac: 44ns]} 6uo7 1 ural win Lx ey) uoLze907 paunseay quading sos quauung uOLzeL eA poyiay auoysbuo7] uray uoLzeAUasqg 38 2 uedee OAWOL $0 “ALU N N i=) Lo-vek OL’O — S*L-0°L}60°0-L0°0 yueman aueme lag 40 °ALUQ eLLeugsny pue|suaand JO “ALUQ s}uasund wu404LuN-UON Aejunoy epLsoly BX) AYLSABALUN 4a eu! aaem [eutds eed Say_dwhuleg Peg pextd Sa Ka ayxsag “SELED oN uLaquinuy 3 uedep 8961 tyeses a, qeaow © eme> LAOH S°0 60L°0] 982 - L G* 1-60] 90°0-LO"O| BE | e5pruquied S a epeue) 2s x ss°0 L0°0} SE - Z | EL°L-8°0} Z0°0-€0°0 AL SABALUN) sinydwey pue c0‘0 $s ,uaanh * sauqaug eLusose [e9 210 L°L- L*b| S*1-£°0}20°0-10°0 Aa ax4ag OLLbAeS £ aL qeaow ge] “Wd eLusOsL Le Se°0 092°0|S°ZL-8L"€] 2°0-2°0]} pL°0-v0°O} LE | AeLex42¢ dol Ae - 990°0 gel “Wed yun, ‘uewgnd ang das /u das /w ay aXq }2e014 uray UWNWLX BY ado|s POLuad quawun) 40} pdyje quauung yorag uoizeAsasqg auoysbuo07 uray peunseay, UOLPELALA ezeg sAeM SLNJYYND JYOHSINO] - SNOTLWAYASAO AYOLYYORV aA VIANMIIIYW 259 auoysueay sa, tsoud auoyusbuo7 WU04LUf) Syseuay auo7 | a40ys jung} 6Buo7 pouns Pay UOLZeLAeA 4aya) bide 3uUadAN) 4O} poyray) uoLyeAUaSGg quauung auoysbuo7 uray YOHSINOT - SNOTLVAYSSHO AYOLVYOSY BM2> | 4OH PUI-WOH] ouewLys 4assiA APPENDIX C LONGSHORE CURRENT FORMULAS ier piste Ma One : | eae | sin2a b [6.978 A tan Be? 29 Putnam-Munk- Traylor (1949) Energy Conservation, Solitary waves Momentum Conservation, Asymmetric-periodic waves sinfsina ae 1/2 Eagleson (1965) (3 gxH, N|> [a + 4 Pee = ij Putnam-Munk- Momentum Conservation, Traylor (1949) Solitary waves A = 20.88 B cosa, H, Galvin-Eagleson (1965) Mass Conservation Inman-Bagnold (1963) Mass Conservation, Rip currents included Bruun (1963) Mass Conservation k£tanBcosa Ty Mass Conservation, Bruun (1963) Rip currents included 2.31 b 1/2 Gs + 2.28gH, sina, ) - = Inman-Quinn (1951) Empirical-based on momentum analysis tanBH, cosa, A = 108.3 T 4 2/3 8.0 afield 2 {sin 1.65a fo) Brebner-Kamphius Empirical-based on (1963) momentum analysis pl/3 + 0.1 sin 3.300, ] 3/4 1/2 hyo mvp 14.0 sin [sin 1.65 a Brebner-Kamphius (1963) Empirical-based on energy analysis + 0.1 sin 3.30 a, ] 0.241 H, + 0.0318 T + 0.0374 oy Empirical-least Square analysis Harrison (1968 ). + 0.0309 tan8 - 0.170 241 APPENDIX D LONGSHORE CURRENT PROFILE THEORY EQUATIONS The equations for the longshore current profile theory of Kraus and Sasaki (1979) are found by expanding v in a power series and determining the unknown coefficients by the boundary conditions. Namely, v and dv/dx must be continuous at the breaker line at v finite at the limits X+0 and infinity. Introducing dimensionless variables V and X defined by V=- kes = “5 where be 5 tan 8B sina, = ven (D-1) b 16 (143/y28) b Ce and essentially equivalent to the modified reference velocity with correction for wave setup without the cosa, term (eq. 72). Expressed in dimensionless variables the general solution of equation (42) is foe) He CAG XceBEOXe)) cnn OM With the definitions, SP Are eS = 2 ose esa = 2 ioe (D-4) S' = Y (mtl1)A 5 SY Ss 8 (ptn)B 5 Sse & @an)s n=0 oh P n=0 =0 from continuity at the breaker line, B Ss SY — SYS )D/GS) = Sas¥) ie) q q Pq Pq (D-5) GC Ss Gs! ows Gis, |] S.8¥) .e) ( D Pp Pq Pq where in the above Ps ts ee [8 = ti tan 8 (D-6) £ 1+3y2/8 f and p=- 2+ V@/16)+C72) g=- - VG/64)+0/@ 243 129 TADS /—COyaou iwTgcn* €07OL OstOAS SLAs) OOM £((°S°N) tequeg yoieesey ButTiseuTSuq Te}seoj) 3iodei1 snosueTTeosTW :SeTIeS “AI ‘*SutiseuTsuy TTATO jo quemjzzedeq *AQTSISATUN WV sexe], ‘III *(°S*N) 1aque9 yoIessoy ButTaeuTSuq TeqseoD “II “STITL “I *squedind 9uoz ging *9 ‘*squeaaind dty *¢ ‘sTepom TeotTiowNN *y *suOTQ -P[NOATS aloysieeN *€ “*SqJUeTAND as1o0YyssuoT *Z ‘*sotmeuAporpAH °T *popnyToUuT aie eJep JetTTIee 3nq /96[ BedUTS sqoedse TedTJeIOSBYA [Te uo sf uotTeAQUs.U0D “*SjUeTAIND dtT1i pue ‘suofAeTNoOATO szoysiesu *squeiind sioyssu0—T :sjusuodwod utem 9814} sqT pue sotTweudpo1rpky yTeqseod uo yoieaseail jo Arewwns jie 9yj-jJo-3}e9S e saptAoad ya1odoy *AQISISATUN W9V Sexe, ‘3upTiseuTsuq TTAT) Jo Juewmjz1edaq Aq paaedeig *(€000-0-08-ZZMOVA 39e139U09) (TA £/-ZQ cou £ JaqUaDQ YOIeesoy BuTIveUu -TS8uq Teqseo) *S*n--jiodsei snosueTTeosTW)--°wo gz f "TTT : *d [€47] “7861 *SIIN Worz atTqetTtTeae : ‘eA ‘pTetys3utads ‘1equea) yoieasoy SutiaseuTsuq Teqaseo9 ‘siseutsuq jo sdiop ‘Away *S*n : “eA SATOATAaG 10q--* ODSeg “ua praeq Aq / a8patmouy Jo a3e9S “*T aWNTOA *squUaTAIND duUOZ Fans ""Y praed ‘ooseg Le9 Tea