MR: TT -10 U.S-ArmY Caast-Eng. Res Ctr. MR 77-10 CAD-A04T7 641) Mathematical Modeling of Shoreline Evolution by Bernard Le Mehaute and Mills Soldate MISCELLANEOUS REPORT NO. 77-10 ~ OCTOBER 1977 ‘ DOCUMENT \ COLLECTION - oe ae Prepared for U.S. ARMY, CORPS OF ENGINEERS COASTAL ENGINEERING RESEARCH CENTER Kingman Building Fort Belvoir, Va. 22060 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: Operations Division 5285 Port Royal Road Springfield, Virginia 22151 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 MH authorized documents. i iy TT UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) REPORT DOCUMENTATION PAGE READ INSTRUCTIONS BEFORE COMPLETING FORM 1. REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER MR 77-10 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED {ATHEMATICAL MODELING OF SHORELINE EVOLUTION Miscellaneous Report 6. PERFORMING ORG. REPORT NUMBER Report No. TC-831 7. AUTHOR(S) 8. CONTRACT OR GRANT NUMBER(s) Bernard Le Mehaute Mills Soldate DACW72-7T-C-0002 9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK 11. DD , 5 On", 1473 Evition oF 1 Nov 65 1s OBSOLETE AREA & WORK UNIT NUMBERS Tetra Tech, Inc. 630 North Rosemead Boulevard FSIS 51 Pasadena, California 91107 CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE Department of the Army October 1977 Coastal Engineering Research Center (CEREN) Os eee aia Kingman Building, Fort Belvoir, Virginia 22060 4 FZ MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) 15. SECURITY CLASS. (of this report) UNCLASSIFIED 15a. DECL ASSIFICATION/ DOWNGRADING SCHEDULE DISTRIBUTION STATEMENT (of this Report) Approved for public release; distribution unlimited. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report) SUPPLEMENTARY NOTES KEY WORDS (Continue on reverse side if necessary and identify by block number) Beach slope Shoreline evolution Coastal engineering Waves Mathematical modeling ABSTRACT (Continue on reverse side if necesaary and identify by block number) A critical literature survey on mathematical modeling of shoreline evolution is presented. The emphasis is on long-term evolution rather than seasonal or evolution taking place during a storm. The one-line theory of Pelnard-Considere (1956) is developed along with a number of applications. Refinements to the theory are introduced by considering changes of beach slope, wave diffraction effects, wave variation, and variation of sea level. The case of hooked bays is also reviewed. inued UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE(When Data Entored) It’ is concluded that a finite-difference mathematical scheme could be For the large developed for engineering purposes for a small wave angle. wave angle, shoreline instability does not permit use of a reliable mathematical model at this time. 2 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) PREFACE This report is published to provide coastal engineers with a litera- ture survey on mathematical modeling of shoreline evolution, which it is hoped will lead the way in establishing a flexible and practical numerical method suitable for predicting shoreline evolution resulting from the construction of navigation and shore structures. The work was carried out under the coastal structures program of the Coastal Engineering Research Center (CERG)): The report was prepared by Bernard Le Mehaute, senior vice president, and Mills pon AEE Tetra Tech, Inc., Pasadena, California, under CERC Contract No. DACW72-7T-C-0002. Funds for the preparation of this litera- ture review pare of the contract were provided by the,U. -S. Army Engineer Division, North Central, Chicago, Illinois. The authors acknowledge the assistance of Dr. J.R. Weggel, CERC, and Mr. C. Johnson, U.S. Army Engineer District, Chicago, in providing a list of papers on the subject matter, along with pertinent comments relevant to the situation in the Great Lakes. Dr. Weggel was the CERC contract monitor for the report, under the general supervision of G.M. Watts, Chief, Engineering Development Division. Comments on this publication 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. OHN H. COUSINS Colonel, Corps of Engineers Commander and Director VI1 CONTENTS CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI). SYMBOLS AND DEFINITIONS. INTRODUCTION . THE FIRST MODEL (PELNARD-CONSIDERE) 1. Refinement and Extensions of the Peimerde Coniideie Model 2. Example of Shoreline Evolution . THE TWO-LINE THEORY OF BAKKER THE EFFECT OF WAVE DIFFRACTION . SPIRAL BEACHES . PROTOTYPE APPLICATIONS CONCLUSIONS. LITERATURE CITED . TABLES u versus ¢ (u). Summary of mathematical models for shoreline evolution FIGURES Beach depth definition Successive beach profiles updrift of a long groin before bypassing . pala. Successive beach profiles updrift of a groin after after bypassing Matching transition between solutions 1 and 2 Sand bypassing long groin as a function of time Comparison between experimental and theoretical shore- line evolution . Comparison between experimental and theoretical sand bypassing discharge. Page 7 50 14 14 19 ILE) 21 DS 23 ih WW NS 14 15 16 IY 18 19 20 Zi CONTENTS FIGURES-- continued Spreading of sand along a shoreline due to instantaneous dumping at a point . Sand dumping along a finite stretch of beach Equilibrium profile between two headlands Two theoretical forms of shoreline equilibrium of river deltas Differences on shoreline configuration due to onshore- offshore transport near a groin Notation for the two-line theory Evolution of shoreline and offshore beach limit near a groin Effect of wave diffraction Hooked beaches Indentation ratio for a range of wave obliquity . Orthogonal arrays for numerical scheme of hooked bay Orthogonal arrays for numerical scheme of hooked bay Semilogarithmic profiles Relationship between shoreline retreat and change in mean water level . Page 25 25 27 29 52 34 38 41 43 43 45 46 47 48 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: Multiply by To obtain inches 25.4 millimeters 2.54 centimeters Square inches 6.452 square centimeters cubic inches NOs SY 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 Lo BSDZ kilometers per hour acres 0.4047 hectares foot-pounds 1.3558 newton meters millibars 10197 = 107 ® kilograms per square centimeter ounces 28.35 grams pounds 453.6 grams 0.4536 ki lograms ton, long 1.0160 metric tons ton, short 0.9072 metric tons degrees (angle) 0.1745 radians Fahrenheit degrees 5/9 Celsius degrees or Kelvins? = ——— Ito obtain Celsius use formula: == = (C) temperature readings from Fahrenheit (F) readings, Ge G/9) E +82) To obtain Kelvin (K) readings, use formula: 3 (G/2) CF 252) = 27/5015. Ox OY E(u) SYMBOLS AND DEFINITIONS time horizontal axis at S WL parallel to the (initial) beach profile horizontal axis at S W L perpendicular to the (initial) beach profile beach depth (depth beyond which sediment transport is negligible) wave angle with beach profile wave angle with beach profile at infinity longshore transport (littoral drift) discharge constant = Ep ~ DD de Qa O parameter u = ~(4Kt)® ie 2 : 2 -u- Fresnel integral = E(u) = = e du length of groin time for the beach profile to reach the end of the groin transform time t, = 0.62t, 1 sinusoidal beach amplitude (at time t = 0) parameter related to beach wavelength [|:, = (72) K parametric value of x defining volume of beach dumping parametric value of y defining volume of beach dumping parameter used to define hypocycloid beach profile between headlands COSinriELEMcES WSOGl sin Tlie WaliceOrail Chemie scoramullal co characterize the effect of wave angle breaking wave height water depth at inception of wave breaking group velocity Intictconeil GCheisee Comsteume 6.42 > Ome distance of shoreline from a horizontal axis parallel to the initial beach profile distance of the offshore beach limit from a horizontal axis parallel to the initial beach profile equilibrium distance y, - yy onshore-offshore transport per unit length of beach onshore-offshore transport parameter (dimension iT) longshore sand transport discharge in shallow water longshore sand transport in deeper water eer? 2 distance of the beach profile to a spiral center angle parameter in mathematical description of hooked bays spiral angle in mathematical description of hooked bays depth of hooked bays distance between headlands MATHEMATICAL MODELING OF SHORELINE EVOLUTION by Bernard Le Mehaute and Mills Soldate I. INTRODUCTION This interim report presents a critical literature survey on the subject of mathematical modeling of shoreline evolution. Hopefully, this review will lead the way in establishing a flexible and practical numerical method suitable to predict shoreline evolution, resulting from the construction of navigation and shore protection structures in the Great Lakes. To focus attention on the most pertinent literature, the subject under consideration is limited to long-term shoreline evolution as defined below. Three time scales of shoreline evolution can be distinguished: (a) Geological evolution taking place over centuries; (b) long-term evolution from year-to-year or decade; and (c) short-term or seasonal evolution and evolution taking place during a major storm. Associated with these time scales are distances or ranges of influ- ence over which changes occur. The geological time scale deals, for instance, with the entire area of the Great Lakes. The long-term evolution deals with a more limited stretch of shoreline and range of influence; e.g., between two headlands or between two harbor entrances. The short-term evolution deals with the intricacies of the surf zone circulation; e.g., summer profile-winter profile, bar, rhythmic beach Paeeerisp meee. For the problem under consideration, long-term evolution is of pri- mary importance, the short-term evolution appearing as a superimposed per- turbation on the general beach profile. Evolution of the coastline is characterized by low monotone variations or trends on which are super- imposed short bursts of rapid development associated with storms. The primary cause of long-term evolution is water waves or wave- generated currents. Three phenomena intervene in the action which waves have on shoreline evolution: (a) Erosion of beach material by short period seas versus accretion by longer period swells; (b) Pp etkect om) CMake) level changes toniverosion yan (c) effect of breakwaters, groins, and other structures. Even though mathematical modeling of shoreline evolution has in- spired some research, it has received only limited attention from practicing engineers. The present methodology is based mainly on (a) the local experience of engineers who have a deep knowledge of their sectors, understand littoral process, and have an inherent intuition of what should happen; and (b) movable-bed scale models that require extensive field data for their calibration. In the past, theorists have been dealing with idealized situations, rarely encountered in engineering practice. It seems that mathematical modelers have long been discouraged by the inherent complexity of the phenomena encountered in coastal morphology. The lack of well-accepted laws of sediment transport, offshore-onshore movement, and poor wave climate statistics have made the task of calibrating mathematical models very difficult. Considering, on one hand, the importance of the subject of deter- mining the effect of construction of long groins and navigation structures and on the other, the progress which has been made in determining wave climate and littoral drift, it now appears that a mathematical approach could be useful. The complexity of beach phenomena could, to a large extent, be taken into account by means of numerical mathematical scheme, (instead of in closed-form solutions), dividing space and time intervals into small elements, in which the inherent complexity of the morphology could be taken into account. Furthermore, better knowledge of the wave climate, a necessary in- put, will allow a better calibration of coastal constants such as found in the littoral drift formula. This study emphasizes the relative importance of various reports and reviews the most important ones. Conclusions based on this review are presented, pointing out the deficiencies of the state-of-the-art. (Sub- sequent investigators should attempt to bridge the remaining gaps. ) The reports are presented individually, primarily in chronological order. Two milestone developments from this survey are reports by Pelnard-Considere (1956) and by Bakker (1968b). Others are extensions and refinements, experimental verifications, support papers, numerical procedures, and side issues, including the latest developments on "hooked beaches" or crenulate-shaped bays. Il. THE FIRST MODEL (PELNARD-CONSIDERE) The idea of mathematically formulating shoreline evolution is attri- buted by Bakker (1968a) to Bossen, but no reference to Bossen is given. The first report which appears in the literature, on mathematical model- ing of shoreline evolution, is by Pelnard-Considere (1956). His theoretical developments were substantiated by laboratory experiments made at Sogreah (Grenoble), France. The experimental results fit the theoretical results very well. It is surprising that such relatively simple theory has not been more frequently applied to prototype cases by the profession (at least as it would appear from the open literature), a fact which may be attributed to the lack of knowledge of wave climates. Pelnard-Considere assumed that: (a) The beach profile remains similar and determined by the equilibrium profile. Therefore, all contour lines are parallel. This assumption permits him to consider the problem to be solved for one contour line only. (b) The wave direction is constant and makes a small angle with the shoreline (<20°). (©) Mae lomgsinore tremsporct, ©@ , 28 linearlky wellaced co ene EAMGSME Ore IS emeile Gre wmeiclencSs @ C(@ | se(@), s(@)) = ein w). (dd) the beach has ‘a fixed Gili=derined)) depth, DD) (Fussy le D is a factor relating erosion retreat to volume removed from profile, which could be defined by the threshold velocity of sand under wave action. A practical method of determination Ot D IS wiyeEm wm Sectiom WIL, Despite the crudeness of these approximations, the Pelnard-Considere model can be considered as a milestone in demonstrating the feasibility of mathematical modeling of long-term shoreline evolution. For this GeAasSOlpeetiasEsmyudced use tol descrmbel in=someudetally hiss theoretical development. Consider an axis, ox , parallel to the main coastal direction and an axis, Oy , perpendicular seawards (Fig. 2). The angle the deepwater wave makes with the axis, ox , is a. The angle of the wave with the shoreline a at any location is asSumed to be small; therefore, S A = chy dy dy a = a tan ae . (1) xe (0) ox fe) OX {le ge I (o} Le} ie) | e i} I (y = £(x,t) gives the form of the shoreline as function of time t). The littoral drift Q is a function of angle incidence a and can be put into a Taylor series: Figure 1. Beach depth definition. DBS SWAB MWw AAAS Figure 2. Successive beach profiles updrift of a long groin before bypassing (from LeMehaute and Brebner, 1961). 14 (@=@_ j} & os 5 (2) oO Ci = @ oO in which Q denotes the transport, Q , when the angle of the wave incidence is ao: Substituting equation (1) into equation (2) yields: #3 3Q oy Ue Qos E a= a OX : (3) fo) During the interval of time, dt , the shoreline recedes (or accretes) by a quantity dy . Therefore, the volume of sand which is removed (or deposited) over a length of beach, dx , is D dx dy . The quantity jus equal to the difference of longshore transport during time, dt , between x @imGl x 4 GS Moo, ONderancd ands (GON a8 ax ) oe 3 io@e's 9Q ane dt Therefore, dQ oy Il @Q) D = — pe eS dx dy ae Gbxalig =, © ve Dox - (4) Substituting the expression for Q , a being small, and defining , _ L dQ <= 0 do ©) a= a (o) yield: 2 Oo _ oy cas TENE (6) OX which is the well-known diffusion or heat-flow equation. K is approximately constant at a given site. Bakker (1968a) found K equal to 0.4 x 10° cubic meters per meter depth per year, at an exposed site along the coast of the Netherlands. Equation (6) demonstrates that the rate of accretion or (erosion), 2, is linearly related to the curvature of the coast, the derivative of the longshore transport rate with respect to the angle of the wave incidence, a , and inversely propor- tional to the beach depth, D . The above equation will be recognized as the well-known diffusion equation. A number of classical solutions of mathematical physics are applicable to the diffusion equation when boundary conditions are specified. Pelnard-Considere (1956) applied his theory to the case of a littoral barrier or long groin. This case is reviewed below: The longshore transport rate along a straight, long beach is sudden- ly stopped by the construction of a long groin built perpendicular to the beach (see Fig. 2). The boundary conditions are: (a) y= oO or alll x Wine c= © Wine Cheracceirizes em initial straight shoreline. o which is (b) At the groin, the longshore transport rate Q = 1.e€., when realized when the waves approach the shore normally; @ 927 i (Ww) 1S the Birssnel inteerrelll ,, E Z te (u) = = e du (8) u Values of E (u) or more frequently, @¢ (u) = 1 - E (u) , can be found in tabulated form as given in Table 1: Table 1. u versus ¢ (u). ~¢(@) @.112 0.225 0.528 0.428 0.520 ©.067 O.799 O-9iO O.995 i Fig. 2 illustrates the shoreline evolution as defined by equation (8). It 1s interesting that these curves are homothetic with respect to the OUAIVIO HO wBEHER Sar oA = oB = oC ssieie — GieSo os oA~ oB~ oC > The horizontal lengths grow with t , and in particular, tan e oy = ———— 2 Ke 2 Vie A tangent to the shoreline at the groin intersects the initial shoreline defined by y=o0 at a point a distance of 2 VkKt/t updrift from the groin. The ratio of the area of sand accumulation, such as is in OYX, 5 to the area of sand contained in the triangular fillet, oyx , is 1.56 and the distance OX, = 2.7 ox . This ratio permits rapid assessment of the total amount of sand accumulated updrift from a single measurement of the angle as > and determination of D as shown in Section IV. The end of the groin of length, oy = 2 , is reached when 2 e =) = (10) 4K tan oa 1e) When t ae , the boundary conditions must be modified since the groin no longer traps all the sand but bypasses some of it. If the same theory is applied to the beach downdrift of the groin and if assumed that the wave diffraction effects are negligible, the beach is eroded in a form symmetric with the updrift accretion. When t = t,, the end of the groin is reached by the shoreline and sand begins to be bypassed around the groin. The boundary condition at the groin becomes oy = & (constant) for anor The solution then becomes (Fig. 3): WV. 1B x ° (11) V4Kt The curves representing the shoreline become homothetic with respect to ie. AUS OY B 1.80% The area between the shoreline and the ox axis (oy x) is given by: 5 18 5 Vikt The area of triangular fillet, OY 6X Hence, OV 26° = Sh eS Ee vee ee oe (12) OY 9X 1 1 . Vikt and Ox = 2x O The shoreline as described by equation (7) at time t = ty is slightly different from the shoreline defined by equation (11) at iene | AS shown in Fig. 4. The volume of sand defined by both curves is equal when the time ty of in equation (11) in such a way , equation (7) is replaced by the time ty that 2 ee 16 cot [—) 5 LESS Les US) Onze : (13) ct (SH Figure 3. Successive beach profiles updrift of a groin after sand bypassing (from Le Mehaute and Brebner, 1961). Eq.11(t’, = 0.62t, ) Eg: 7, (t=t,)) Figure 4. Matching transition between solutions 1 and 2. Therefore, the shoreline evolves initially as represented by equation (then whent its — t , the shoreline keeps evolving as given by equation (11) as at the time were t = 0.38t, . Then’, the sediment dis- charge, Q , bypassing the groin is equal to the incoming. discharge Q minus the volume of sand which accumulates per unit of time. KD2 Oe) =) Qin ; (4) a [pKce-9- 584) | M2 i.e., Oey) = @ Ge : (15) S tana [ rK(t-0.38t,) | ye O il or again 0.638 QE is NO Tt 12 (16) [‘e/ep Z 0-38 | In dimensionless terms, the following values are obtained for equation (lo), (See Rie, 5) 2 DO Re Re eB ion S- S|) So SS © WG oO ~S Bs j=) Orv OV ial It takes a long time before the value of Q approaches initial dis- charge, Qo , downdrift of the groin. 20 "(1961 ‘Louqerg pue syNeYyey, oT wWorz) OWT} FO UOTIOUNF e& Se UTOIS BuO, BsuLsseddq pues v € : 2 NOISNAdSNS NI LYOdSNVUL HLIM 3AYND WOlLoWud——“~ ee JAYND 1WW9ILIYOSHL *¢g oinstTy y3.1.VMDIvaHE |} Ad GaddOLS—> GNVS T1V 00/0 2\ The shoreline may be deduced at any time, t , by a homothetic trans- formation about the oy axis from the knowledge of the shoreline at a given time, ts > and also by applying the simple relationship (see Fig.3): AD i AC ie a 1/2 Z 0.38t, | [*2 f 0.58, | The theory of Pelnard-Considere has been verified in laboratory ex- periments with fairly good accuracy. The steady-state littoral drift, Q, , was obtained experimentally from preliminary calibration over a straight shoreline. The results of these experiments are shown in Figs. 6 and 7. However, the shoreline predicted by theory is not expected to be valid downdrift of the groin because of the influence of wave diffraction around the groin tip. Some sand begins to bypass the groin by suspension before t =t (see Fig. 5). Also, different boundary conditions apply to different contour lines since the deeper contour lines reach the end of the groin before the contour lines which are near the shoreline, which implies the one-dimensional theory is no longer entirely satisfactory. (17) Subsequently, Lepetit (1972) also conducted laboratory experiments which verify the results of a numerical scheme based on the theory of Pelnard-Considere. He used the law, Q=Gsimk oP VEOS a) >.) Mepetaiit ls exe periments were carried out with a very small angle between wave crest and shoreline. 1. Refinement and Extensions of the Pelnard-Considere Model. After Pelnard-Considere's contribution, the mathematical formula- tion of shoreline evolution has proceeded at a slow pace. The first refinements came in improving the longshore transport rate (littoral drift) formula, in particular, modifying the expression relating sedi- ment transport to incident wave angle. Based on results from laboratory experiments performed by Sauvage : : 3 er) and Vincent (1954), Larras (1957) introduced the function f(a) = sin rie also used by Le Mehaute and Brebner (1961). New theoretical forms of shoreline evolution are determined as solutions of the diffusion equa- : : : , ail, (ON ts tion. Introduction of the relationship f(a) = sin = instead of tana, allows obtention of solutions valid for larger wave angles. Of particular interest are the cases of shoreline undulations, since assuming linear superposition, any form of shoreline may be approximated by a Fourier series. The solution of the diffusion equation is then of the form: ae “(QS61 ‘eLepTsUuoj-pxreUuteg WorZ) edLeyOstp BSutsseddq pues [eoTJeLO09Yy} pue [ejUoUTIodxe usemzeq uostszedwoj) °*/ oansTy L yi “(9S61 ‘SLOpTsuoD-preuyted WoIZ) UOT NTOAS QUTTOELOYS [TBOTELO9Y} pue [eJUoWTLedx9 useMIEeq UOSTIeduOD A¥YO3HL LNSWIYSdxXZ ——-— — 20/0 "9 oinsTy 31VOSs (23) shin eae y< y =ube cos K éae x5) (18) which indicates that shoreline undulations tend to decay exponentially and disappear with time. B defines the beach undulation amplitude at eune, tc = © , zinc YN WS sellencecl co whe welyellemeenl, bf Or dais winewila= tion through the relationship: 2n \2 19) Ls 2) K ; ( 16 Shoreline evolution due to the sudden dumping of material at a given point may be represented by: rs 4K Viel Equation (20) gives the spreading of the sand along the shoreline since the integration J ydx, which expresses the conservation of sedi- MEME Win tne SySeOm,) 1S) 2 Comswehre' (SES Rie, 5 Wnts solution was also mentioned by Pelnard-Considere. It is interesting that much later, Noda (personal communication, 1974) investigated the same problem by taking an initial condition for sand dumping. Y = constant when |x| x as shown on Fig. 9. Using the functional relationship now commonly accepted, f(a) = sin 2a , Noda found that the solution: to the diffusion equation to be: - erf a ° (21) 24 Figure 8. Spreading of sand along a shoreline due to instantaneous dumping at a point. Figure 9. Sand dumping along a finite stretch of beach (initial condition). 25 Even though the initial condition is different from the previous one, the solutions tend to be similar as time increases and are, therefore, both applicable to the problem of shoreline sand dumping. Also of interest is the solution, proposed by Larras (1957), of a beach equilibrium shape between two headlands or groins described by the equation: where s is the distance along the shoreline. This indicates no sand transport along shoreline configuration and, therefore, yields an equilibrium to obtain: ds = L cos ue da (where L is a proportionality constant), which gives 4 5 4 (22) YS ols [eos um aR gOS =| Equation (22) defines a hypocycloidal form as might be found between two headlands (see Fig. 10). R is a parameter which is related to the relative curvature of the shoreline. When R-+o , a straight shoreline solution is obtained. Another family of solutions was given by Grijm (1960, 1964). In these two publications, Grijm used the most commonly accepted expression for dependence of longshore transport on angle, f(a) = sin 2a , and applied the theory to cases where the angle of incidence, a , is not necessarily small. Subsequently, he established the kind of shoreline which can exist mathematically under steady-state conditions. Even though the theoretical approach obeys the same physical assump- tion as the previous theory (except for the allowable range for the angle of incidence), his mathematical formulation is not as simple. The shoreline is defined with respect to a polar coordinate axis. The con- tinuity equation is solved in parametric form, which is integrated either by computer or by graphical methods. Details of Grijm's compu- tations are not available. 26 WAVE acer HYPOCYCLOID HEADLAND 34° HEADLAND Figure 10. Equilibrium profile between two headlands. 27 The main interest of the report lies in the results. When the long- shore transport rate reaches a maximum value (a = 45°) , the shoreline tends! £0. commun cusped: ec jedmCapema Si SNOW mlm kslsern lulls Also of interest is Grijm's (1964) mathematical formulation for different forms of river deltas for which he finds two possible solu- tions, one with an angle of wave incidence everywhere less than 45°, and another with the angle of incidence greater than 45°. The shoreline curvature also depends upon the angle a as shown on Fig. 11. The problem remains indefinite since it is unknown which solution is valid. The formulation of Grijm does not lend conveniently to numerical adaptation. Bakkere and Edelman (1964) also studied the form of river deltas, but instead of,using f(0) = sin 20 , as Grijm, they used the linear approxi. mation as given by Pelnard-Considere; i.e., f(a) = k, tana for o The onshore-offshore transport is defined by: OA OR yest MON C3 (25) where a, is a proportionality constant (dimension We). When Y, - Y5 + w) iS positive, the transport is seaward; when negative, it is shoreward. q. has been found by Bakker for a part of the Dutch coast equal to 1 to 10 meters per year for a depth Dy = 3 meters. Leen 7 = 4 = 5 Ene, q = Gy, Oa = TD. Now, following Pelnard-Considere; i.e., developing the expression for the longshore transport rate Q ina Taylor series in terms of a O21) Ae See. (26) which gives in linear approximations: S dQ dy Gs Ge E =| ; eh 33 Figure 13. Notation for the two-line theory. 34 Defining d aa [32 | ] (28) Co = @& ’ ) then, ea) Vis Qs gre ae This equation is now applied to both lines, y, Co and y,(): oy : 2 etl (29) US Ron Shy tee oy » a ; (30) = Wan = Sp aE The equation of continuity, CO ye aN , ew) is modified by the term a due to onshore-offshore transport so that 3Q, yy See Corer Wi oe (2) 3Q oy D sf 2 “ pabaacrs Re =) Bre () Substituting equations (1), (2), and (3) for Q> Q,5 oF gives: 2 ay dy 1 B 1 34 SO is, Os 3) il Be oP 35 2 : fn Go ie yh 2D 22 ‘ (35) CD ORD ge een ay Om le CE ox Adding both equations yields: 2 DD Z MCLE eee Raley eghnes n(n (nena ae 2 DB p? ah ae ax° of in which Gla ar Gl 1 UPR YE Ae wes 37 eee ne eee mp ee, Pale (S71) leh ha, 2 49 For simplicity, Bakker (1968b) assumes De ee ee which implies that 1 2 derivatives of the littoral drift transport with respect to @8 are proportional to depth D . Then, dividing a = a O equation (6) by Dy and D, respectively, and subtracting, yield : oO.) q_D JOY.) ES Dep OM a) aaeen (38) aR eee jew) jes) < oo where y, = ee wo Oe which is the equation for the offshore-onshore transport Ys . It is interesting that the offshore-onshore transport is independent of the longshore transport. Using the auxiliary variable, FS ye exg0 ao , (39) 36 the diffusion equation is still obtained: ony 8Y 6 (40) 2 dt Bakker has applied his theory to a number of idealized cases, in- cluding the behavior of a sand beach near a groin, assuming DBD. = D 2 Tenant (41) The boundary conditions are: gle Ihaicngnll Comebiietom (ic = ©))e Yau Dore Or O< 45° , the cvastline becomes unstable as previously mentioned). The two-line theory has been verified experimentally (Hulsbergen, Van Bochove, and Bakker, 1976), and shows a trend identical to the ex- perimental results. There are some differences at a small scale due to secondary currents, breaking wave type, changes of wave height due to small changes in morphology, etc. These, however,*%are short-term rather than long-term evolution phenomena. IV. THE EFFECT OF WAVE DIFFRACTION The effect of wave diffraction was subsequently taken into account by Bakker (1970). Initially, this was done for the one-line theory of Pelnard-Considere and later for Bakker's two-line theory. Pelnard-Considere's equations, Oy a“ Q=9W-ae 2 , aw =—% (42) a = a fo) and oy . _ Geo) 29) (43) at D dX Still apply. OR and oq) vare now, tuncerons som soy Sumce) both the incident wave height and angle of approach vary along the shore with x , because of wave diffraction. Inserting the expression for Q in the continuity equation, yields: tO (44) Shs) It is assumed that the longshore transport rate, Qo , 1S proportional to the angle of wave incidence, (a, - ax » and the square of the rela- tive wave height. The variation of wave height with x is given by the diffraction theory of Putnam and Arthur (1948). The modification of wave diffraction by wave refraction is neglected. A similar approach has been proposed by Price, Tomlinson, and Willis (1972), who assume that Q = Wee. E sin 20 , where E is the trans—- S Mikecedvenernay, whach ass alsionalstunctelon Ole xq asmas) » oun (anid ie 1S cle submerged density of the beach material). Price, Tomlinson, and Willis then obtain the one-line theory equation: ORS : dE da Ca Y sin 2a ae 2E cos 2a ae ap ID) aa ae (45) which is solved numerically with MOVE (46) Laboratory experiments were performed with crushed coal by Price, Tomlinson, and, Wallis ((1972))> * theytheory, eivange, thevette cel ots waive diffraction was verified by the experiments at the beginning of the test. After a 3-hour test which may correspond to a prototype storm duration, it is stated that the wave refraction pattern invalidates the input wave data and a complex boundary condition developed at the up- drift end of the wave basin. Bakker's (1970) consideration of wave diffraction has been included in his two-line theory where, oy) i Tacorle dani rosa Gn oy 5 Bo = Boo Glo. ie ) Neither the deepwater line, defined by y.(x,t), nor qd, and Qo2 5 us affected by diffraction.) Filg. 15 presients typical results lobjtained trom this theory for the case of beach evolution near a groin and between two groins. 40 Wave incidence Offshore Contours 16At 16At sAt Shoreline Contours Accretion and erosion near a groin, numerical solution with diffraction (two-line theory) Wave incidence Offshore Contours BAt 16At Shoreline Contours 16At 4At 8At 8At 16At 4At Behavior of beach and inshore between two groins (two-line theory) Figure 15. Effect of wave diffraction (from Bakker, 1970). 4 V. SPIRAL BEACHES Hooklike beaches (Fig. 16) are common along exposed coasts and are formed by the long-term combined effects of refraction and diffraction around headlands. Yasso (1965) discovered that the planimetric shape of many of these beaches could be fitted very closely by a segment of log- arithmic spiral; the distance, r , from the beach to the center of the Spiral increasing with the angle 0 according to Eiryts Exp E cot 6 | (49) in which gB is the spiral angle. Bremmer (1970) has shown the logarithmic spiral to give an excellent fit for the profile of a recessed beach between two headlands. The evolution of spiral beaches belongs to the geographical time- scale domain (Sylvester and Ho, 1972). However, similar evolution has also been observed over smaller time scales in consonance with the definition of long-term shoreline evolution adopted in this study. So far, only empirical rather than theoretical mathematical repre- sentations of spiral beaches are available. The empirical approach has been fruitful in providing the spiral coefficients 8 as function of wave angle, a , with the headland alinement (Fig. 16) (Sylvester and Ho, 1972). The "indentation ratio" (depth of the bay to width of open- ing) also depends upon a and, in most cases, varies between 0.3 and 0.5 (Fig. 17). There have been many attempts to explain this peculiar beach forma- tion (Leblond, 1972; Rea and Komar, 1975). Leblond assumed that the rate of sediment transport is proportional to the longshore currents as given by the theory of Longuet-Higgins (1975). He also assumed that the beach profile is not modified by erosion or accretion so that the con- tinuity equation from the one-line theory can be used in a two-dimen- sional coordinate system. Thus, the variation in longshore current iE ens tty with wave angle Will yalelid he alte) or erosionwor accretion: Difficulties arise in expressing this variation of longshore current in areas subjected to wave diffraction. Leblond (1972) points out that classical wave diffraction theories are too complicated to be used in his theoretical scheme. Another difficulty arises from the fact that the barrier (headland) is not thin as it is assumed in the theory of diffraction of Putnam and Arthur (1948). To account for this effect, Leblond introduces an empirical correction coefficient to the theory of Putnam and Arthur over a two-dimensional network. The results of 42 HEADLAND Figure 16. Hooked beaches. Figure 17. CRENULATE- SHAPED BAYS LEGEND e@ Vichetpan Experiment O Ho Experiment X prototype bays ® typical bay A Bedock Singapore Indentation ratio for a range of wave obliquity (from Sylvester and Ho, 1972). 43 such a complex scheme, which is plagued with numerical instabilities, are shown in Fig. 18. Even though the results show how oblique waves initiate an erosion pattern that might eventually lead to the formation of hooklike beaches, they do not show that the beaches represent a good fit to segments of a logarithmic spiral. Rea and Komar (1975) developed an approach to overcome the numeri- cal instability encountered by Leblond. They combined two orthogonal, one-dimensional arrays as shown on Fig. 19. In this way, deformation of the beach can proceed in two directions without the necessity of a two- dimensional array. The wave configuration in the shadow zone was described by various simple empirical functions which resulted in beach configurations fairly approximated by a logarithmic, spiral. 5 -* The main interest in the work of Rea and Komar (1975) is that they show the lack of sensitivity of the shoreline evolution in the shadow zone to the actual pattern of incident waves used. Also, the sensitivity of the beach shape to the energy distribution seems to be small. VI. PROTOTYPE APPLICATIONS The application of mathematical models of shoreline evolution to pro- totype conditions is not very well documented in the literature. It is certain that, at least in its simplified form such as given by Pelnard- Considere, the method has been used by practicing engineers and designers. It has been reported in unpublished reports but very little has appeared in the open literature. Weggel (1976) has formulated a numerical approach to coastal process- es which is particularly adapted to prototype situations. In particular, it includes: a. A method for determining the water depth beyond which the onshore- offshore sediment transport is negligible. This information is particu- larly useful in determining the quantity D used in Pelnard-Considere's theory and others. It is also useful in determining the effect of a change of sea level. Beach profile data are plotted on semilog paper and the base elevation of the most seaward point varied until an approxi- Mate straight line is obtained (see Fig. 20). He found D = 70 feet at Pt. Mugu, California. b. The effect of a change in sea level, a situation pertinent to the Great Lakes, is also taken into account in a way proposed by Bruun (1962). Using the principle of similarity of shoreline profile, the shoreline recession Ay is related to the change of water level a _ by the relationship (Fig. 21): ab Oe Guat Ge 44 f WAVE DIRECTION Figure 18. Orthogonal arrays for numerical scheme of hooked bay (from Leblond, 1972). 45 HEADLAND Figure 19. Orthogonal arrays for numerical scheme of hooked bay (from Rea and Komar, 1975). 46 ELEVATION ABOVE SEAWARDMOST POINT (ft) 100 PROFILES AT PT. MUGU, CALIFORNIA (STATION 1467 + 60) 10 3rd approx. 2nd approx. 1st approx. Ne 0 1000 2000 3000 4000 5000 6000 DISTANCE OFFSHORE (feet) Figure 20. Semilogarithmic profiles (from Weggel, 1976). 47 “TOAST 103eM uvoWl UT O8UeYD pUe }e91}9I SUTTOIOYS Usemjeq dtysuoT\epoy “TZ 9an3ty 48 ec. A numerical scheme in which the effect of wave diffraction could be included. d. A statistical characterization of wave climate and longshore energy iF LU Re Examples of recent prototype analysis and prediction of shoreline evolution by mathematical modeling are Apalachicola Bay by Miller (1975) and the Oregon coastline by Komar, Lizarraga-Arciniega, and Terich (1976). Both studies are based on numerical schemes related to the Pelnard- Considere (one-line) formulation. VII. CONCLUSIONS There are two methods of approach to the problems related to littoral processes. The first one, typified by the previously discussed reports, consists of analyzing global effects. The method essentially based on establishing ''coastal constants" for a model by correlation between long-term evolution and wave statistics and subsequently, to use the model for predicting future effects. It appears that this method is the most promising for engineering purposes and could be termed the macro- scopic view. The main results are summarized in Table 2. The second approach, the microscopic view of the problem, consists in analyzing sediment transport, step-by-step, on a rational Newtonian approach, starting with wave motion, threshold velocity for sand trans- port, equilibrium profiles of beaches, etc., until the individual com- ponents can be combined into an overall model to predict shoreline evolution. The second method or scientific approach has not progressed to the point where it can be applied to engineering problems in the foreseeable future. However, much progress has been made in the last 5 years toward understanding the hydrodynamics of the surf zone through application of the "radiation-stress'" concept. In theory, establishing a reliable mathematical model of surf zone circulation should permit a determina- tion of the resulting sediment transport. Practically, however, inter- action between a movable bed and the surf zone circulation, and the inherent instability of longshore currents limit this approach to the realm of research. Among the problems that make this approach difficult are the refraction and diffraction of water waves, uncertainty in pre- dicting rip current spacing, and the effect of free turbulence generated by breaking waves on the rate of sediment suspension. Finally, the complexity of mathematical formulation, based on the radiation-stress concept, makes it difficult to use as a predictive tool when dealing with forcing functions expressed by statistical multi- directional sea spectra. This method is promising in explaining local effects (e.g., near groins), rhythmic topography, beach cusps, and short- term evolution due to unidirectional sea states. All these effects are 49 Table 2. Sediment transport Sediment transport Experimental Summary of mathematical models for shoreline evolution. Application to ideal Date Author alongshore Validity onshore-offshore Theoretical developments, verification cases 1956 Pelnard- Griencean é ay Considere i) ax Very small angle No Diffusion equation Laboratory Groins closed-form solution with pumice sin 2 asa. - 2% ° 1957 Larras 4 ° ax Small angle (<2S ) No Diffusion equation, No Groins-sudden dump closed-form solution sinusoidol undulation, equilibrium shape between groins 1960 Grijm sin 2a Sin all angles. In No Nonlinear differential No Forms of deltas case of large angle equation an inconsistency in the assumption ay small x 1961 Le Mehaute, Ja Brebner Sera Diffusion equation No Groins-sudden dump closed-form solution sinusoidol undulation, equilibrium shape between groins 1964 Grijm sin 2a implied Small and large No Cylindrical system of No Forms of deltas angle coordinates-numerical or graphical method 1964 Bakker, kK, tan Small angle No Nonlinear differential No Forms of deltas Edelman ko tan ale octana