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1:1 BF-EAC2H.- E:7ROSUC I ON MODE. (FEDEKACH 1) UISE-R SMrA N U AL 0 0 V 0 L. Li N ElI'. T-IEUR Y ANT) E4 ACK<C]R00ND E - 121 i: >' El David Li[. Kr i ebe L 0 13 N e vi a T- k n i ver-w I 97i of Uc[ w r 11~~~~~~~~Nw d 1'.D] 91 121 1)ep a r t I wenrt C.)f C csa s -1 a [. a ri d COc ea no -ra p h I c E n g i n e er I ng 1'.] Un i ver A. i -ty o)f F or i da ii~~~~~~~~~~~G ca i fln e s- Le F- Lr :?2606 El 0~~~~~~~~I he a c: ih e s a n ci9 sh or eV R e 0~ c� u . C , e n t r-er c El F I or idca SFinaite Un Iver s Ity ci 121 BilEAC~ii.-E's AliD SHIORES 11 TECi :: 1,1CA L i-ND DIE':S GN ME'MORA)NDUM No~ 84**~-i:: COASTAL ZONE ci ci INFORMATION CENTER El1 El r.:i jn(Afl/ iRevewed 1:)y El El~~~~~~~~ Bea c: lie~s a ncd Shares. Recnaor ce Cen-mter 11 121 ~~~~~~Insti-1I I tie of Sc:i ernc:ea nc F:' 1131c: A Wfta i r-s El F [or i ia, Sta*t e Un i vers i -t y0 I'A ~~~~~~~~a ni cli ci ~~~~~~~F [or i cda Of-f I Ce of Coai-s-ta L Management 11 ci ~~~~F [or I dia Depar tmen-t of Env i ronmentia L Reg u [~a t I onEy 0 ci Funded by c ci A grant' from *the Li. S. Off i cec of Coas~tal. Zone Maviagement 121 ci Nat ional[ Ocean ic: and Atmosph-er Ic: Adm in isIr at i on 12 121 (under the Coastal. Zone Managemenl"t Act o.f i 972, as amendled) Li 121 ~~~~~~~~~~~~~thro-ugh 03 El -,.~~~~~- F [or, 1 da Of f i re of Coasta 1. Management ci El~~~~r F or i cia Dep--ar tment of En-rv i ronmen a [ keg 1 1a3 o [I ~~~~~~~~~~~~~~~ani 121 Fl[or I da Depmartment of NatIuralI Resotir(:es I21 GB ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~El .F56b L no. 84-5-2 FORE WO0RD Th is users man ual d esc ri bes the numer i ca c omputer mocle L EE4EACH which is desir-ned -to est imate the t ime-dependlent beach-cluric erosion assoc:i ated w ith severe storm such as h urr icanes or northeaster s. The users manual i Is preasente cir, two parts: voLume I conta ins a descrirption of program Log ic and -var iab les used.. volume 11I prov ides a more thorough d isc uss icon of bac:kgr'u-nd theomry and mode L verif i ca ti on. The or iginal. research Pertaining to the EDEACHV model, was conduc:ted by the author under the diirect ion of IRobert G. Dean al. part of the author- 's Master or C i v i L Eng ineer i ng dlegree at the Un ivers ity of Del~aware. 'The rese.:arch phase tral:, suppor ted by the Nati ona I Oceanic and Atmospher ic Adminiir istra tion avid -the DeLaware Sea Grant Co LFege Program.~ The present work is presented in p:art ialI. fuiffi I Lment of con-tract ual. ob iIga tions of the Federa I Coast a I Zone Management Programi (subject -to prov is ions of the Coasta I Zonie Managemfrent Improvement Act of 1972.. as amended> subject to provisions of contract CM--37 ent it Led 'Enigii neer ivng Support E nh a ric e fe n t F~r* og) r a ti ~nder pr ov isio; n of DNR c on tr ac t C0,4:37, th is work1 is a scubcontracted product o.f the B'eaches and Shores Resourc(:e Center , linst~i-t u tce of Sc ienc e and Pub iic A ffa i rs F br ida State tUn ivers ity.~ The document has been adopted as a B~eacheS and Ehores-.- Techni-tca I and Des ign Memoranidum i n accordance with provisions of Chapter 16EB-3~3. F~or ida Administrative Code. At the time of subni ss ion -for contract ura I compIi ance, James H ., 14al.sIi.Ii e was the con-tract manager and Aidm i nistr ator of t-he Anal~ysis/Research Section. Flat N.~ 13e-an was Chief of -the Bur-a u of' Coastal. Data Acq~uis iftion, Deborah E. Flack Director of the D)iv is ion of Beaches and Shores, and Dr~ El.-ton J~ G issendaniner -the Ex ecu tivye D)i recctor of -thec Flbr i da Depart menit of Natural. ResoUrceS.., Deborah E. Flack, Di rector Division of Bteaches and Shores JulIy, -1 9834 PZO~rtyof C30 Library U.SDEPARTMENT OF COMMERCE NOAA P-L- ~~ ~ ~~COASTAL SERVICES CENTER I-L 2234 SOUTH HOBSON AVENUE 01 - ~~~~CHARLESTON, SC 29406-2413 (N m TABLE OF CONTENTS Chapter Page I INTRODUCTION ..................... 1 1.1 Schematic Erosion Prediction Methods ....... 3 1.2 Proposed Method ................ 12 II EQUILIBRIUM BEACH PROFILE THEORY ........... 17 III SEDIMENTS TRANSPORT EQUATION ............ 19 IV NUMERICAL SOLUTION OF CONTINUITY EQUATION ....... 26 V INITIAL PROFILE AND SOLUTION DOMAINS ......... 31 5.1 Boundary Conditions .............. 33 5.2 Simulation of Erosion Process .......... 35 VI RESULTS-GENERAL BEHAVIOR ............... 37 6.1 Relative Effects of Water Level and Wave Height 39 6.2 Relative Effects of Sand Grain Size and Beach Slope .................... 42 VII SIMULATION OF STORM SURGE HYDROGRAPH ......... 45 VIII RESULTS-SIMULATION OF HURRICANE ELOISE ........ 55 REFERENCES ........ . ............. 71 APPENDIX A ...................... Al APPENDIX B ...................... Bi LIST OF FIGURES Figure Page- 1 Seasonal Change in Beach Profile 2 2 Bruun's Equilibrium Beach Profile 3 3 Bruun's Response of Equilibrium Profile to Sea Level Rise 4 4a Edelman's Method Applied Natural Profile 6 4b Edelman's Method Applied to Schmatic Profile 6 5 Edelman's Method Applied to Concave-Upwards Profile 7 6 Swart's Schematic Beach Profile 8 7 Dean's Method for Determining Maximum Potential Erosion Due to Storm Surge Using Schmatic Equilibrium Beach Pro'file 11 8 General Characteristics of the Beach Profile Before and After the Storm 13 9a Schematic Beach-Dune Profile with Wide Berm 15 9b Schematic Beach-Dune profile with no Berm 15 10 Particle Diameter Versus Shape Factor (A) 20 11 Particle Diameter Versus Equilibrium Energy Dissipation Based on the Smooth Average Curve Presented in Figure 10 21 12a Equilibrium Profile Subjected to Increased Water Level Showing Decreased Surf Zone Width 22 12b Equilibrium Profile Re-Established Relative to Increased Water Level 22 13a Decreased Surf Zone Width Resulting From Beach Fill 23 13b Beach Fill Distributed Over Surf Zone at Equilibrium 23 14 Effect of Varing the Sediment Transport Rate Coefficient on Cumulative Erosion During the Simulation of Saville's (1975) Laboratory Investigation of Beach Profile Evolution for a 0.2 mm Sand Size 25 15 Numerical Representation of Surf Zone Showing a Sediment Transport Over Imaginary Cell 27 Figure Page 16 Time Dependent Solution for Berm Recession Due to Steady-State Forcing Conditions 38 17 Effect of Storm Surge Level on Volumetric Erosion 40 18 Effect of Wave Height on Volumetric Erosion 41 19 Effect of Sand Grain Size on Berm Recession 43 20 Effect of Beach Face Slope on Berm Recession 44 21 Method of Normalizing Axis to Represent Storm Surge Hydrograph and Maximum Potential Erosion 47 22 Schematic of Numerical Results Showing Predicted Erosion Curve Approaching Equilibrium for Smooth Increase to Static Peak Surge Level 48 23 Schematic of Numerical Results Showing Predicted Erosion Curve Reaching a Maximum as Water Level Decreases Following Peak Surge Level 48 24 Effect of Storm Surge Duration on Volumetric Erosion, 12 Hr. Storm Surge 50 25 Effect of Storm Surge Duration on Volumetric Erosion, 24 Hr. Storm Surge 51 26 Effect of Storm Surge Duration onVolumetric Erosion, 36 Hr. Storm Surge 52 27 Comparison of the Effects of 12, 24, and 36 Hr. Storm Surge on Volume Erosion 54 28 Estimated Storm Surge Hydrograph, Hurricane Eloise Bay-Walton County Line 57 29 Representative Profile, Bay-Walton County Line 59 30a Steepening of Natural Dune Form 63 30b Uniform Recession of Schematic Dune Form 63 31 Example of Predicted Post-Storm Beach-Dune Profile, Hurricane Eloise, Bay-Walton Counties, Florida 66 32 Comparison of Predicted Post-Storm Profile to Maximum Potential Erosion Due to Static Peak Surge Level, Hurricane Eloise, Bay-Walton Counties, Florida (Chiu, 1977) 67 Figure Page 33 Contour Advance/Retreat Due to Hurricane Eloise Bay-Walton Counties, Florida (Chiu, 1977) 69 34 Volumetric Erosion Distribution and Beach Face Slope Variation, Bay-Walton Counties, Florida (Chiu, 1977) 70 LIST OF TABLES Table Page. I HSTAR Values Versus Beach Slope and Sand Grain Sizes (Based on Moore's Sand Grain Size Versus Energy Dissipation) 32 II Simulation of Schematic Profile of Bay-Walton County Hurricane Eloise 60 III Observed Erosion Characteristics for Bay-Walton Counties 61 IV Equilibrium Recession For Static Peak Surge Level Schematic Profile of Bay-Walton County 61 BEACH EROSION MODEL (EBEACH) I. INTRODUCTION At present, the mechanics of sand transport in the surf zone and on the foreshore are not known with certainty. Field measurements of beach profiles and plan forms have been widely used to identify erosion trends; however, few detailed pre-and post-storm profiles exist from which empirical relationships can be developed. Theoretical expressions have been developed to explain bed load and suspended load sediment transport for both the longshore and onshore- offshore modes. Again, however, field data from severe storms are not available to verify these theoretical equations. Laboratory experiments in small and large scale wave tanks have also been used to simulate prototype conditions, but the extension of these laboratory results, to predict erosion in nature, has not always been successful. In general, it is known that as the water level rises during a severe storm, waves attack the berm or dune and large quantities of sand are transported offshore, either suspended in the turbulent water column or concentrated as bed load. At the seaward edge of the surf zone, the first breaking of the incoming waves effectively limits offshore sediment transport. While localized currents resulting from nearshore water surface gradients surely move some sediment beyond the breaking depth, most sand is deposited near the break point in the form of a bar. If no sand is gained or lost in the longshore direction, it may be argued that the idealized beach response results in a redistribution of sand over the surf zone, as the profile adjusts to a more stable form. If no overwash occurs, then the sand mass is conserved in the surf zone such that the volume deposited offshore equals the volume eroded onshore. From this arguement, several beach erosion prediction methods have been developed. These methods are based on the assumptions that: (1) sand is conserved in the onshore-offshore direction- and (2) the beach profile may be represented in schematic form, in which both the pre-and post-storm profile shapes are known. It is well known that beaches seem to reach seasonally stable shapes. In Figure 1, the "summer" or normal beach typically has a broad berm and a smooth profile, the result of onshore sand transport during a relatively mild wave climate. In contrast, the "winter" or storm beach may be characterized by a narrow berm and offshore bars, the result of offshore transport from the beach face to the seaward edge of the surf zone by larger waves from winter storms. ._.,j/"NORMAL (SUMMER) PROFILE STORM (WINTER) PROFILE Figure 1 .Seasonal Change in Beach Profile Fenneman (1902) observed that beaches also exhibit a characteristic "profile of equilibrium", in which the entire beach profile reaches a state of equilibrium for the given sediment size, wave characteristics, and nearshore currents. If irregularities in the profile, such as troughs and bars, are smoothed, beaches may be considered to reach dynamic equilibrium, in which the -2- *profile shape, relative to the mean water level is maintained as the entire profile slowly moves landward and seaward in response to changing hydraulic conditions. This concept of a unique equilibrium profile shape for any sediment size, has resulted in several highly idealized methods for predicting beach changes due to changing water levels and wave heights. 1.1 Schematic Erosion Prediction Methods Bruun (1954) discovered that beach profiles from the coasts of California and Denmark exhibited similar shapes. From these profiles, an empirical expression relating water depth, h, to the distance offshore, x, was developed such that: h = Ax21 (1) in which A is a coefficient that relates the steepness of the profile to the given sediment characteristics. This general profile form is represented in Figure 2. *~~~~* ~~~h=1 S .~~~~~~~~ Figre Bru' Eqilibriu B e c Prfl e ~ ~ ~ ~ -3 Bruun (1962) proposed that, for the two-dimensional case in which longshore sediment transport gradients do not exist, beach erosion due to a slow sea level rise is predictable since the profile geometry is always known relative to the still water line. As illustrated in Figure 3 , for an increase in the water level, S, the profile must be shifted upward by an equal amount, S,to maintain the same relative vertical position. Since this net upward motion requires a considerable addition of sand in the offshore region out to the depth of closure (about 30 feet) the beach must recede by a distance, R,to provide the necessary sand volume. By equating the volume of sand eroded onshore with the volume of sand deposited offshore, the beach recession may be predicted for any given water level rise. Bruun estimated that for a 6 mm/yr rise in water level in southeast Florida, a 2.0 to 2.5 ft/yr recession is required to maintain equilibrium. This is in general agreement with the 1.0 to 3.0 ft/yr average recession observed on most of the East coast of Florida. EQUILIBRIUM PROFILE AFTER WATER LEVEL RISE INITIAL ~ -.. EQUILIBRU --- PROFILE Figure 3 Bruun's Response of Equilibrium Profile to Sea Level Rise -4- Edelman (1968) observed many beach changes during storms in the Netherlands and suggested that: (1) the post-storm beach profile shape is mainly the result of wave transport of sand perpendicular to the coast, and (2) the short-term, ie. storm duration, transfer of sand occurs from the dunes offshore to the edge of the surf zone, where it is limited by the incoming shore-breaking waves. By using an idealized profile representations in Figures 4a and 4b, Edelman predicted maximum dune erosion due to specific storm conditions by balancing eroded and accreted volumes to satisfy continuity. Edelman (1972) subsequently modified his earlier work to incorporate: (1) time dependent erosion, and (2) concave-upwards beach profiles as sugguested by Brunn and as observed in nature. With a profile shape, given in Figure 5, which the curve AB can be determined empiricallyEdelman theorized that the erosion process could be represented by shifting the profile horizontally, with velocity u, and vertically, with velocity v. By assuming that the profile "keeps up" with or moves at the same speed as the rising water level , the recession occuring during any time, At, is determined by balancing volumes. The results of this analysis yielded predicted recessions that were much larger than those observed during actual storms; Edelman concluded that erosion does not occur as fast as the water level rise. Swart (1974) analyzed numerous small and large-scale laboratory experiments and combined the concept of a concave-upward equilibrium profile shape with a simple onshore-offshore sediment transport theory,, in the first schematic erosion prediction method to include a time-dependent mechanism of sediment transport. In Figure 6, Swart defined three regions of sediment transport in the beach profile: (1) the backshore, above the wave run-up limit, in which only wind-blown transport occurs, (2) the transition area, seaward of the offshore limit of the surf zone, in which only bed load transport occurs, and (3) the developing or D-profile, including the surf zone and the swash zone, in which bed load and -5- . DUNE EROSION STORM SURGE LEVEL ._ _ _ ,_jii ! i ,- - MEAN SEA LEVEL ; '...~ ~"~ ~ EQUILIBRIUM h �4 . �.-a~ ~PROFILE B SLOPE DEPOSITION Figure 4a Edelman's Method Applied to Natural Profile * EROSION STORM SURGE LEVEL . ~':"< .. MEAN SEA LEVEL . ~- ~~ ~ EQUILIBRIUM h �*.. - <PROFILE ": "~.'.. ~. '--...SLOPE DEPOSITION Figure 4b Edelman's Method Applied to Schmatic Profile -6- x + x 14HIGHEST STORM SURGE LEVEL NOT MEASURED, AFTE Figure 5 Edelman's Method Applied to Concave-Upwards Profile -7- t ONSHORE BACKSHORE LIMIT l _WAVE SETUP SWL : L 1 -_, ,-- _ - I SWL i I------ k---; -- - -/- /! 'L/ - //-./://0 D PROFILE L2 OFFSHORE TRANSITION Figure 6 Swart's Schematic Beach Profile -8- suspended load transport due to breaking waves occurs. By assuming that a stable, equilibrium profile is eventually attained, Swart developed a simple equation for sediment transport in the D-profile: S sy(Rw - Rt) (2) in which Sy is a time dependent sediment transport flux in the D-profile, Sy is a transport rate coefficient dependent on specific boundary conditions, Rt is a time dependent parameter related to the sediment transport flux, and Rw is the equilibrium value of Rt. Note that in this expression Swart relates the total sediment transport flux to the disequilibrium of certain surf zone parameters. Swart then proposed that equation 2 could be defined in the D-profile of Figure 6, such that a schematic length (L2 - L1)t and an equilibrium value of (L2 - L1), or W, represented the sediment transport mechanism, thus: Sy S y [W - (L2 - LI)t] (3) Swart (1974), (1976), developed lengthy, complex empirical relationships to define the D-profile, sy, and W, in terms of deep water wave characteristics, sediment size, and berm height. Dean (1977) developed theoretical expressions for possible mechanisms causing equilibrium beach profile shapes. Three possible causes were considered: (1) longshore shear stress in the surf zone; (2) uniform wave energy dissipation over the surf zone plan area; (3) uniform wave energy dissipation per unit volume of water in the surf zone. Using linear wave theory, Dean found a theoretical form of the beach profile to be: h = Axm (4) in which A is a shape factor dependent on sediment size, and m is an exponent -9- for the profile curvature, such that m equals 0.4 for causes (1) and (2) given previously, and 0.67 for (3). Upon analysis of 502 beach profiles along the U.S. Atlantic and Gulf coasts, Dean determined the best-fit empirical exponent, m, to be 0.6 to 0.7. This seemed to confirm Bruun's original profile shape in Figure I and also identified uniform wave energy dissipation per unit volume as the best explana- tion of the profile shape. In a geometric analysis similar to that used by Edelman (1972), but for a simple square-berm profile, Dean (1976) analytically integrated the eroded and accreted volumes, and found the maximum "potential"1 berm recession as a function of wave height, berm height, water level rise, and the A coefficient as in Figure 7. Dean plotted the dimensionless water level rise, S' = SIB, versus the dimensionless breaking depth, h~ 'b/B' and developed curves for the dimensionless berm recession, R' = R/W b$ in which Wb is the surf zone width defined by the breaking depth and the A coefficient. Dean's method allows fast prediction of the maximum berm recession for a highly idealized profile geometry but does not include time dependence. For storm surge-erosion prediction, Chiu (1977), has found that both Dean's method and Edlman's method over-predicted the actual erosion occuring during Hurricane Eloise, by and much as a factor of five. It seems clear that the rise and fall of water level occurs too quickly for the maximum erosion protential of the peak surge height to be realized. Since most predictions are based on a steady-state maximum storm surge is not surprising that the predicted maximum erosion is often much greater than observed in nature. Of the schematic beach erosion methods described, none is well verified or completely adequate for general practical application. As mentioned, Edelman and Dean do not include realistic time dependence of the erosion process. Dean -10- ', .--- EROSION :;1~~~~~R .~~~~~~~~~W *:1B~~~~ i- Eoin Du.oSorug Uin Sceaic EqulLibrIu B \ i. -- * ; DEPO~~~~EPSITION I R' = S'---,2h' [1- C-R')2/3] ht =8 B! R hB WB hB]3/2 Figure 7 Dean's Method for Determining Maximum Potential Erosion Due to Storm Surge Using Schematic Equilibrium Beach Profile and Swart do not include realistic beach face representations and only 'Edelman' s method may be applied to analyze dune erosion. While Swart makes significant advances with the inclusion of a time dependent sediment transport mechanism, the complex, interrelated empirical parameters used in the solution prohibit easy application. In the state of Florida, another prediction method is also available. Chiu (1972) has developed a modified form of Edelman's (1968) scheme, in which the actual pre-storm profile is approximated by a linear offshore profile, a sloping beach face, and a vertical dune face, illustrated in Figure 8 . To find the beach recession or eroded volume, the post-storm profile is represented by linear bottom slope that intersects in peak storm surge elevation. This solution is carried out by computer and has been found to be a good approximation for some storm conditions. 1.2 Proposed Method This report summarizes the development of an alternative schematic method of predicting beach and dune erosion due to changes in water level and wave height associated with severe storms. This method differs from previous methods in that it is based on: (1) more realistic representations of the beach profiler, including sloping beach and dune faces; (2) a general mechanism for onshore-offshore sand transport in the surf zone; (3) the time history of the storm surge, such that the time-dependent shoreline response is predicted. From preliminary results, thi~s model agrees both qualitatively with observed erosion from Hurricane Eloise. In general, the model uses a steady state solution of simplified finite difference equations that describe equilibrium beach profile evolution. From -12- dune eroded wave breaking storm tide level material mean sea level beach profile before erosion deposited material beach profile after erosion Figure 8 General Characteristics of the Beach Profile Before and.After the Storm Dean's theory, where wave energy dissipation per unit volume in the surf zone governs profile form, a sediment transport equation maybe proposed of the form: Qs = K (D - Deq) (5) where Qs is the volumetric sediment transport flux in the onshore-offshore direction, K is a transport rate parameter, and (D - Deq) represents the excess energy dissipation per unit volume during storm conditions. This equation is solved simultaneously with the equation of continuity for sand transport normal to the shoreline: ax Qs (6) This ensures that: (1) sand is conserved, i.e. the eroded volume equals the deposited volume and (2) time dependence is included. Three general types of input are required. First, the actual pre-storm profile is represented in schematic form, as illustrated in Figures 9a and 9b. The initial dune profile is approximated by a constant dune height, hD, and uniform dune face slope, MD. Foreshore geometry is expressed as a uniform slope, M, and a berm height, h To accomodate a variety of profile forms, the berm width, WB, may be varied to simulate a wide berm or to simulate a concave profile form with a break in slope between the beach and dune faces. Offshore, the profile is approximated by the equilibrium profile shape h = Ax213 (7) which is defined by Dean (1977). Second, for specific applications the storm surge hydrograph is required, since the water level at each time step governs profile change. As a third input requirement, wave height values may be input either at each time step or in the form of a constant design wave height. In terms of the Figure 9a Schematic Beach-Dune Profile with Wide Berm WB=O :\ MD hhe3 Fi Schmatic Bach-Due Profle wit no B.r Figure 9b Schematic Beach-.rin Figure 9b Schematic Beach-Dune Profile with no Berm numerical solution, the total water level at each time step may be thought of as the driving force inducing profile change with the wave height serving as a boundary condition limiting the active surf zone width. -16- II. EQUILIBRIUM BEACH PROFILE THEORY As a wave breaks over a sloping bottom, energy is dissipated in the form of turbulence and bottom friction. In general terms, this net energy loss in some control volume of unit width, h Ax, may be expressed by the change in energy flux over the volume. The energy dissipation per unit volume, D, may be expressed as: D = F(x + Ax),- F(x) (8) h Ax where F is the energy flux at the volume boundary and is-defined as positive in onshore direction,,.and F is the average depth over-the distance; Ax. In differential form, the energy dissipation may be given as: D =~ ! (9) ax From linear wave theory, the energy flux is equal to the total wave energy, E, times the speed at which the wave energy is transmitted, the group velocity, Cg. Thus, F = ECg (10) In the surf zone, the energy flux may be rewritten in terms of the water depth only. Employing both the shallow water and spilling breaker assumptions, Equation (10) is found to be: I 2 g1/2h5/2 F YK� (11) where K is taken as 0.78 and y is the specific weight of sea water. By substituting this last expression for the energy flux into the differential form of the energy dissipation to be dependent on the water depth and bottom slope: 0 =5 2 1/2h1/2 ah (12) -17- From this expression Dean determined the theoretical beach profile shape resulting from uniform wave energy dissipation per unit volume in the surf zone as: h : Ax2.3 (13a) in which A - (13b) 5 �2 g [D]j1/12/3 Dean's results seem to confirm the equilibrium profile shape suggested by Bruun where A is a shape parameter dependent on the sediment size. Dean (1977) performed a least-squares analysis of 502 beach profiles from the East and Gulf coasts and determined the best-fit A parameter for each profile. In similar analysis, Hughes (1978) and Moore (1982) extended the empirical results to include a wide variety of sand grain sizes. In Figure 10, Moore (1982) presents a summary of these results in which the shape factor, A, is plotted versus grain size. In Figure 11 the equilibrium energy dissipation per unit volume is plotted versus grain size based on the A values in Figure 10 and Equation 13b. It is also possible to perform a least squares or similar analysis to determine A from a recorded offshore profile. 111. SEDIMENT TRANSPORT EQUATION It was noted that Swart calculated the time-dependent profile change using a sediment transport relationship in which the sediment transport flux is dependent on the disequilibrium of the surf zone system. With Dean's theory that the equilibrium form of beach profiles results from uniform wave energy dissipation per unit volume, a similar sediment transport equation based on the disequilibrium of the energy dissipation may be proposed as: Qs= K(D -0De) (14) In this expression, Qd is the volumetric flux of sand, K is a transport rate parameter, D is the actual energy dissipation, and Deq is the equilibrium energy dissipation obtained from Figure 11. With Equation (14 ), if 0 is greater than Deq then Qs is positive and offshore. For example, in Figures 12a and 12b, consider a profile initially in equilibrium with a constant breaking wave height, H B and constant breaking depth hB If the water level is increased, then waves break closer to shore and the surf zone widt~h is decreased, concentrating wave-energy into a smaller region. This causes the energy dissipation per unit volume to increase. Likewise, note in Equation ( 13a) that the energy dissipation is slope dependent. With a water level rise the local slope for any water level, h, is greater than the equilibrium slope, thus the calculated energy dissipation must be greater than equilibrium. Clearly, the original profile is not In equilibrium relative to the increased water level ; thus, to regain an equilibrium position, the system requires offshore transport, such that sand is moved from the beach face to the offshore limit of the surf zone. Dean (197Q) disscusses many similar qualitative examples of the usefulness of the energy dissipation theory in predicting beach profile changes, for a variety of changes in water level and wave height. Equation (14 )can b e used to explain most of the profile changes, including onshore sediment transport -19-. 11111 i 1 III 11 I 1 1111111 1 I I11111 Avoro3o of 40 Beach Profiles Hughas' Results -t--+ Analysis of Swoarts Piofllu -.. Smooth Avorage 1.0 - A ( M) . - = ' Figre10 arice DamterVesu ,. _ (A I *C o. I.o 0o0.0 .oo0. PAIrIIt. DIAMI[aR IMM) Figure 10 Particle Diameter Versus Shape Factor (A) 'I~~~~~~~~~~~~~~ z ~~~~~~~~~~~~~~rsne nFgr 100.- 0 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~PRIL IMI2(M - BsedonSmooth Average CrePeetdi iue1 '~ ~~~~~~hB Figure 12a Equilibrium Profile Subjected to Increased Water Level Showing Decreased Surf Zone Width ~\ t~~ < ~ m ~ ___ _ __ ~_ _I _____A__ I ~ I~ B thB Figure 12b Equilibrium Profile Re-Establish ed Relative to Increased Water Level -22- * .. ". * * M *. ~ DESIGN BEACH FILL *: �* * * � x ~~~. ~~~ .~ ~MSL ORIGINAL : EQUILIBRIUM .* ' PROFILE 5 hB Figure 13a Decreased Surf Zone Width Resulting From Beach Fill MSL FINAL EQUILIBRIUM FORM OF BEACH FILL Figure 13b Beach Fill Distributed Over Surf Zone at Equilibrium -23- if the water level drops relative to the equilibrium profile, and evolution of a linear beach fill to an equilibrium shape, as in Figures 13a and 13b. Assuming that the empirical determination of thq A parameter and the equilibrium energy dissipation Deq are valid, then only the transport rate parameter, K, must be determined to fully quantify the transport equation. It is evident in Equation ( 14 ) that K relates the excess energy dissipation to the combined suspended and bed load transport resulting from the destructive forces in the surf zone. Swart (1974) found the onshore offshore transport as a complex empirical function of both wave and sediment characteristics. Given the form of Equation ( 14 ), it appears that, since energy dissipation is a function of sediment size, water depth, and the breaking wave type, spilling breaker, the transport parameter must be either a constant or a function of other wave characteristics, i.e. period or steepness. Unfortunately, theoretical relationships or empirical results from pre-and post-storm profiles to determine K are beyond the present state-of-the-art. Moore (1982) has determined empirical values of K that give good agreement with erosion rates found in the laboratory and in nature. Using a numerical model for onshore-offshore transport based on Equation (14 ), but that included a detailed simulation of breaking waves and bar formation, Moore simulated the large scale laboratory tests of beach profile evolution conducted by Saville (1957). -6i4 f4 These tests resulted in a constant value of K, 2.2 x 10- 6 or 0.001144 f for both 0.2 mm and 0.4 mm sand, seemingly verifying that K is independent of sediment size of Figure 14. Likewise, Moore simulated profile changes occuring over one year at Santa Barbara, California, using daily average wave characteristics.-. These results seem to further indicate that K is independent of wave characteristics. -24- 30 - 20- . A. _X 10 // , -;kZ' o K .- X to JJ/~~~~~~~~Bs ' Fit #2 / Oo. w . . . , , ,. . , , / .,....!...... O t- 1 2 :) / -./, ..... ~TK= 2.4 x IOUS) Figure 14 Effect of Varing the Sediment Transport Rate Coefficient on Cumulative Erosion During the Simulation of Saville's (1975) laboratory Investigation of Beach Profile Evolution for a 0.2 mm Sand Size. *~. 0 S I0-- IS 20 ( S ~~~~~~~~~~~~~~~~~~~~~30,r Ln ~~ ~ ~~~~~~~~~~~ IM ()IUIS Fiur 14Efct fVrnteSdmntl Tasort Rate Coffcintn ~~~CumuaieEoinDrn h iuaino ail' 17)lbrtr IV. NUMERICAL SOLUTION OF CONTINUITY EQUATION In addition to the transport equation, schematic beach profile changes are governed by the continuity equation for conservation of sand in the surf zone. In general, the change in depth of any point in the surf zone over time is the result of gradients in the flux of sediment which, for the onshore-offshore problem considered here may be given as: ax As (15) at ah where the minus sign is consistent with the definition of the sign of the depth, h. For numerical application, the surf zone may be represented by a series of cells in which the incremental change in depth, Ah, is uniform, as in Figure 15. With this definition, hn is the water level, referenced to mean sea level, at the grid point, n. The storm surge, or departure from mean sea level, is represented by n and the total water depth at the grid point, h' '~~~~~~ n' equals hn + n . The distance from the shoreline datum to the grid point, Xn, is determined initially from the equilibrium profile form. In finite difference form, Equation 15 can be expressed as: .AX, QS QS( n sn. n+1 n16) where AXn is the change in position of the elevation contour as the water level remains constant over the time step. In this model, wave setup in the surf zone and wave runup on the beach face are not considered. Therefore, n is uniform throughout the surf zone and Equation 16 reduces to: AXn Qsn Qs at Ah: (17) From this expression, it is clear that if Qs > s there is a net n 5n+1' teei e addition of sand to the cell and the contour advance offshore, a AXn, iS positive. -26- XmsI xmS1I v SURGE LEVEL N hi I ,, . I 1 IMSL I hn+I Xn-1 I-. -xn , h, ' =~A(xn -x '2l)/3 Xn+.....I Figure 15 Numerical Representation of Surf Zone Showing a Sediment Transport Over Imaginary Cell If Qsn< Qsn+ there is a net loss of sand from the cell and the Xn contour recedes. As Qsn = Qsn+1 = 0, the system reaches equilibrium and wave energy is effectively dissipated over the cell volume without a net gain or loss of sediment. The solution for beach profile response may be obtained from the implicit solution of the two governing equations: AXn n - Sn+l (18) At Ah and Qsn = K(Dn - Deq) (19) Substituting Equation 19 into Equation 18, the contour retreat or advance can be expressed as: AX KAt ~0 D 1 (20) AXn = Ah [Dn - Dn+l] (20) While this form appears quite simple, the implicit numerical solution for the energy dissipation per unit volume is rather detailed. First, it is convenient to rewrite the finite difference form of the energy dissipation per unit volume occuring in some cell: F F Fn+1 _- n Dn+1 - (hn+1 + hn) (Xn+l - Xn) (21) From Equation 11 for the energy flux, this energy dissipation can then be expressed in terms of water depth and x distance only as: 5/2 5/2 hn+1 - hn n+1 lKd 1 n (22a) (hn+: + hn) (Xn+l -X) where Kd = 2 y g /2 (22b) -28- Substituting Equation 22a into Equation 20, the solution for AXh is now a function of both the position and depth of adjacent grid points: 5/2 5/2 5/2 5/2 KAt Kd h n - hn_1 hn+l - hn AXn Ah (h n+ h) (X Xn Kd(hn+1 + hn) (Xn+1 Xn (23) It is important to note that in the solution scheme, water depths, hn+1, hn and hn_1 are constant over the time step in the steady state solution. However, horizontal positions, Xn+1, Xn and Xn.1 vary as time progresses. Preliminary numerical solutions using constant X positions proved to be highly unstable; therefore, an average value of X over the time step AX Xn = Xn + 2 (24) was introduced in an effort to achieve unconditional stability. In Appendix A, Equation 24 is substituted into Equation 23 and it is shown that the resulting solution is of the form: An Xn_1 AX + Cn AXn+1 Zn (25a) where D An = (Xn X (25b) Dn n+l Bn= :1 + - Xn _ Xn.l Xn (25c) Dn+l n - n+1 Xn (25d) Zn K2 (sn+1 Qsn) (25e) = K:t (25f) This solution represents the matrix multiplication of the variables, AXn_ls AXn, and AXn_1 with known coefficients, An, B,, Cn, and Zn. While all -29- coefficients are dependent on the energy dissipation per unit volume, Zn acts as the driving term in the equation and represents the net sediment flux over the cell. Thus, any perturbation in the system that causes gradients in the sediment transport also causes a change in the profile position. To solve Equation 25a, a recursion formula must be introduced: AXh-i = En OXn + Fn (26) in which one unknown AXn_1 is given in terms of another unknown, AXn. Substituting this form into Equation. 25a results in: -Cn Zn - AnFn (7 AXn :[ Bn +AnEn ] AX+1 + [ n +AnEn ] (27) Comparing Equation 27 to the recursion relationship, it is clear that -C n En+1 =Bn + AnEn (28a) and Z AF F .n - n n (28b) Fn+l : Bn + AnEn Since all An, Bn, Cn, and Zn coefficients are known, a double sweep solution may be employed as follows: (1) From and onshore boundary condition, specify E1 and F1. In sweep number 1, all En+1 and Fn+1 coefficients are calculated according to Equations 28a and 28b. (2) From an offshore boundary condition, specify that some AXn+1 offshore equals zero. In sweep number all AXn values may be determined according to the recursion relationship. -30- V. INITIAL PROFILE AND SOLUTION DOMAINS As mentioned, this model allows schematic representations of a complex berm-dune system onshore and a concave-upwards profile shape offshore. In Figures 9a and 9b, the dune is approximated with a constant dune height, hD. and a uniform dune slope, M. The berm with, WVB may be varied to simulate a variety of initial profile configurations. Likewise, the constant berm height, h., and a uniform beach face slope, MB are required. To achieve continuity in the profile, the linear beach face slope is assumed to intersect the equilibrium profile at some depth, h*, such that there is a continuous decrease in slope in the offshore direction. Table I summarizes this depth of intersection for a variety of sand grain sizes and possible beach slopes. With this basic profile definition, the double sweep solution is valid from h* seaward to beyond the breaking depth. This region is governed by the energy dissipation and is termed the dynamic solution domain. From h* onshore, any number of simulation schemes may be employed. In this model, this region is defined as the geometric solution domain and is governed by the continuity equation only, since the energy dissipation per unit volume is not defined above water line according to the governing equations. In a more advanced model, it is anticipated that a more complete swash zone transport relationship and a variable upper limit of wave induced transport may be introduced. For now, beach slope is kept constant as the beach face recedes uniformly. Similarly, for more severe storms, the dune slope is kept constant as the dune face recedes uniformly. This representation does not always represent nature, where the eroding profile steepens; however, it seems to give good agreement for eroded volume and an order magnitude agreement for contour erosion. It is anticipated that statistical or empirical corrections can be developed to adjust the uniform slope recession to better represent natural steepening. -31- Table I HSTAR Values Versus Beach Slope and Sand Grain Sizes (Based on Moore's Sand Grain Size Versus Energy Dissipation) Sand Grain Beach Slope Size DIA(mm1 1 1 1 1 1 1 1 DIA (mm) I I I I 0.2 0.0 0.0 0.0 0.0 0.0 0.5 0.5 1.5 0.3 0.0 0.0 0.5 0.5 1.0 1.5 4.0 --- 0.4 0.0 0.5 0.5 1.0 1.5 3.0 --- 0.5 0.5 0.5 1.0 1.5 2.5 --- --- 0.6 0.5 1.0 1.5 2.0 0.7 0.5 1.0 1.5 2.5 --- --- -32- 5.1 Boundary Condtions Two types of boundary conditions are required for the double sweep solution. In nature, onshore-offshore sediment transport is zero at the top of the berm (in the absence of overwash), is maximum somewhere between the swash zone and the break point, and is zero again at some point seaward of the breaking depth. Therefore, Q. = 0 is the formal boundary condition for both onshore and offshore limits of the profile in the numerical model. As mentioned, the dynamic solution extends from h* offshore to the breaking depth. One feature of the finite-difference solution is that large discontinuities may exist in the sediment transport function at h*, at the breaking depth, and near the peak of the Q curve. To eliminate these sharp jumps, a smoothing function is applied to the transport curve. This serves two purposes: (1) to spread, the sediment transport over adjacent cells, broadening the curve and reducing the peak; (2) to extend the transport curve beyond the breaking depth, helping to ensure a smooth transition in this region. Onshore, uniform recession of the beach/dune face is ensured by extending the Qs curve linearly from h* to zero at the top of the berm/dune. This also serves two purposes: (1) providing a smooth transition in the Q. curve at h*; (2) ensuring that all Zn on the beach/dune face are equal. In terms of the numerical model, the onshore and offshore boundary conditions may now be specified. First, in Sweep #1, the E and F coefficients are defined down to h* as: En = 0 (29) Fn = Zn (30) -33- From this condition, E.+ and F.+1 may be determined by the dynamic cal culations out to beyond the breaking depth. Offshore, at the point where Qs = 0, there is no wave energy dissipation and no sediment transport. In Sweep #2, all AXn- for this point seaward are defined equal to zero. Then according to the recursion relationship, AXn values are determined in the landward sweep. At the end of each time step continuity requires that the total eroded volume must equal the total deposited volume. During the onshore sweep, the net change in volume from offshore up to the point h* + Ah determines the volume that must be eroded to achieve continuity at the top of the berm/dune. This volume is distributed equally among the grid points from h* landward, ensuring that the beach/dune face recedes uniformly. This procedure is required because the discrete nature of the sediment transport function can cause the grid point at h* to erode beyond adjacent, grid points, resulting in an unstable, and unreasonable, numerical solution.. As with most numerical solutions, a stability criteria can be found for selecting a stable time step. In Equation -23 , the coefficients are dependent on the relationship of the time step to the horizontal grid spacing. Through extensive testing of the model it has been found that: KAt < 0.25 (31) 'n - )-i is required to avoid numerical instability. In particular, with a linearly sloping beach/dune face, the minimum grid spacing in this region governs the stability. It might-be noted, that while the implicit solution scheme was used to avoid instability, a Taylor series approximation used in the derivation in (Appendix A) seems to introduce the instability at large time steps. In program application, time steps must be reduced when simulating steep dune slopes. -34- 5.2 Simulation of Erosion Process With the complex berm-dune configuration in Figures 9a and 9b, several erosion scenarios may be simulated: (1) erosion of berm only, for cases with wide berm; (2) erosion of the berm which then exposes the dune to subsequent erosion; (3) erosion of entire beach face-dune face, for cases in which no berm is present. Therefore, in the model two conditions are monitored: (1) the surge height, S, relative to the top of the berm elevation, h B; (2) the berm width, WB. Consider the case in which the berm width is wide and a distinct berm is present. In this case, any water level rise up to but not exceeding the top of the berm is assumed to cause only the beach face to recede, thus Qsis set equal1 to zero at the top of the berm. If erosion proceeds until the berm crest reaches the base of the dune then it is assumed that the entire beach-dune face erodes uniformly as waves would now run up the face of the dune. Thus, Qs is extended to zero at the top of the dune. In nature, waves may overtop the berm and erode the base of the dune before the berm recedes completely. In the model, the geometric solution approximates this dune erosion only after the berm has receded; however, it is believed that the critical nature of the berm and the dune, as reservoirs of sand, are included in a fairly rational fashion. Now consider the case in which water level quickly rises above the berm height. First, if this occurs before the berm is eroded to the dune, then the numerical model will calculate the energy dissipation per unit volume -35- between the dune foot and the top of the berm to be less than equilibrium, implying onshore transport and accretion of the dune face. In nature, any water level above the berm would allow waves to reach dune foot in a turbulent bore. From field observations, this results in rapid smoothing of the berm crest while the dune erodes and steepens. In the model, this process must be approximated to avoid unrealistic dune accretion; therefore, the berm is again required to erode completely before dune erosion is initiated. Continuity is satisfied initially by setting Qs 0 at the dune foot such that the berm erodes quickly, then by setting Qs 0 at the top of the dune to allow uniform dune recession. If the original profile has no distinct berm width, then the entire beach-dune face is allowed to erode uniformly by setting Qs 0 at the top of the dune. Once again, exact processes are highly approximated; however, the essential characteristics of the berm-dune system seem to be represented. As a final note, the recovery process is governed by the same numerical solution for energy dissipation and sediment transport. Whil~e berm recovery and rebuilding are permitted, dune erosion is considered permanent. Although the time dependent erosion-recovery process;.may be represented by the proposed transport equation little attention has been paid to the recovery process. This model is intended only for the prediction of beach recession due to severe storms and is not intended for simulation of beach recovery. -36- VI. RESULTS-GENERAL BEHAVIOR To observe the general behavior of the numerical solution, consider a profile in which there is an instantaneous increase in the water level which is then maintained with a constant incoming wave height. Since the berm recession is dependent on the disequilibrium of the system, the time-dependent berm recession in Figure 16 shows that the recession is largest initally, then decreases as the system approaches equilibrium. With this behavior, the actual recession approaches the maximum potential recession, R ., asymptotically. This behavior has been verified in the laboratory by Saville (1957), Swart (1974), and Hughes and Chiu (1982). Although this ideal behavior is quite simple, it is significant for several reasons. Because the transport parameter, K, is small, i.e. 0.001144 ft4/lb, the actual berm recession requires considerable time to reach equilibrium. Many coastal scientists have observed that erosion occurs at a slower rate than the change in water level; however, little field data from severe storms is available to quantify erosion rates. As noted in Chapter I, the beach erosion prediction methods of Edelman and Dean yield only the maximum potential recession, here defined as R... It is clear that without accounting for the time dependence indicated in Figure 16, these methods over predict the actual recession. Because the numerical solution accounts for the time-dependent recession toward the maximum potential recession, it should allow more realistic representations of natural beach response. -37- z 0 (n v) i,! / LLI LLI u I0 100 200 300 TIME (hrs) Figure 16 Time Dependent Solution for Berm Recession Due to Steady-State Forcing Conditions -38- 6.1 Relative Effects of Water Level and Wave Height Recalling the finite difference expression for the energy dissipation per unit volume in some cell: h512 .h5/2 D n-i- Kd n+1 fl n(32) D n+l: Kd(In+, + hn) (n+l - Xn) it is clear that each incremental increase in the total water depth, h; profoundly effects the energy dissipation, thus, the magnitude of the sediment transport in the cell. In Figure 17, a plot of the recession curves for several steady-state water levels with a given breaking wave height reveals that berm recession varies almost linearly with the water level. In-fact, for a given wave height, the'recession at any time is nearly proportional to the change in water level. This result is substantiated by the laboratory model tests of Hughes and Chiu (1982). Now consider the same initial profile with a given steady-state water level but with several different breaking wave heights. The primary effect of increasing the wave height is to change the width of the surf zone and the offshore limit of the active profile. Since the onshore portion of the Qs curve is not effected by a change in wave height, the berm recession in Figure 18 is initially the same for each wave height. Only after some length of time, does the increased surf zone width increase the berm recession, as the volume of sand required to reach equilibrium is greater for wider surf zones. From these results, it is noted that water level behaves as the forcing function in the numerical solution while wave height acts essentially as a boundary condition controlling the surf zone width and the total volume of sand that must be moved to achieve equilibrium. This result agrees well with lab tests of Hughes and Chiu (1982) where water level was found to be the -39- 125 D.50 0O.3 mm 24mj~ E z w > - 5 00 ~~50. 100 150 200 TIME(hrs) Figure 17 Effect of Storm Surge Level on Volumetric Erosion 24 505Q-0.3 mm H- B m 30m M13 1:10 . S = 1.5m Is- 12-. 01 0 25 50 75 100 TIME Whs) Figure 18 Effect of Wave Height on Volumetric Erosion single most important parameter in the storm-surge erosion process. For the numerical simulation, this result indicates that exact wave height determination is not critical for data input. 6.2 Relative Effects of Sand Grain Size and Beach Slope As further support that the numerical scheme correctly simulates natural profiles, at least qualitatively, consider two beach profiles, both subjected to the same storm surge level and wave height, In Figure 19, the berm recession may be compared for the case in which the two profiles have different mean grain sizes, but the same beach face slope, For this condition, the steeper profile CD0 4 mm) reaches equilibrium quickly with a fairly small recession indicating that beaches with larger sand grain sizes are relatively insensitive to peak surge duration. Conversely, it is evident that the milder profile (D 50 =0.2 mm) takes langer to reach equilibrium and that the equilibrium recession is much greater due to the wider surf zone. Thus, the model indicates that beaches with small sand grain size may experience significant variation in the magnitude of the erosion depending on the storm duration. Now, in Figure 20, compare the berm recession for the two profiles with the same sand grain size but with different beach face slopes. This situation can occur over relatively short coastal segments where localized changes in bathymetry, shoreline orientation, or wave characteristics, can cause local changes in the beach face slope. For this case, the numerical results indicate that steeper beach faces erode both faster and farther than milder beach face slopes. This agrees with nature where steep beach faces often'represent a more unstable profile formation with high erosion potential. This was confirmed by Chiu (1977) from field observations where steeper pre-storm profile experienced more erosion than milder pre-storm profiles, -42- M61: 15 12 - S =0.3m Hb 3.Om D 00.2mm 0~~~~~~~~~~5 W 6- Li 0,~~~~~~50=0.4mm 0 100~I( 200 300 400 TIME (Iys) Figure 19 Effect of Sand Grain Size on Berm Recession 12 D05Q 0.2 mm E8~ z CI) 46- 0 ~50 100 150 200 250 300 TIME (hrs) Figure 20 Effect of Beach Face Slope on Berm Recession VII. SIMULATION OF STORM SURGE HYDROGRAPH Prior discussions have considered only the beach response due to a simple instantaneous increase in the still water level; however, the time-dependent recession resulting from the complete storm surge hydrograph may also be simulated. Conceptually, the continuous surge hydrograph may be approximated by a series of discrete still water levels in a stair-step approximation. With this procedure, the solution at each time step this based on the total still water level during that time increment. As the incremental time step is decreased, the discrete representation more closely approximates a continous function. In the numerical model, the solution procedure is essentially unchanged. The energy dissipation and sediment transport flux are determined from the total water depth at each time step. Likewise, the breaking depth, hB, and the transition depth, h*, are determined relative to the new water level; therefore, these depths move onshore and offshore with each new water level and wave height. Referring to Equation 22a, it is clear that as the water level increases with a new time step, the energy dissipation and sediment transport values also increase. Since the dynamic system does not respond at the same rate as the change in water level,.Qs and D are maximum at the time of the peak surge. Thus, the beach erosion rate is also maximum at this time. Intuitively, since erosion occurs at a slow rate relative to the water level rise, the peak surge level represents the stage of the greatest disequilibrium in the system. From the continuity equation: ax aqs (33) at~ an DQs it is clear that since the slope of the transport curve, h, is maximum every- axisas where in the profile at the peak surge level, the erosion rate, at is also maximum. -45- Suppose for a moment that erosion occurs instantly such that the maximum potential erosion for any water level is realized immediately. The maximum potential erosion curve resulting from an idealized surge hydrograph may be represented as in Figure 21, where the temporal surge profile may define both the water level and the associated maximum potential erosion at each time step. With this representation, the time-dependent erosion may be plotted on the same graph for comparison to both the time-dependent storm surge and the maximum potential erosion associated with the peak surge level. Consider the case, in Figure 22, in which the peak surge level is maintained indefinitely, In this case, the numerical results show that the actual erosion is much less than the maximum potential erosion for' the duration of the water level rise. Only after the peak surge level is maintained for several hundred hours of simulation time does the profile "catch up" to the maximum potential value and attain complete equilibrium for the peak water level. Now, in Figure 23, consider the case in which the water level decreases back to zero such that the peak surge level is maintained for a short time. For the duration of the water level rise, the recession characteristics are exactly equal to those noted above and the rate of recession is maximum at the time of the peak surge., However, as the water level decreases, the energy dissipation and the sediment-transport flux also decrease as the system is brought back toward equilibrium. As the slope of the transpo rt curve is decreased, the recession rate also decreases. As some point, the instantaneous water level drops to a level at which the energy dissipation is nearly in equilibrium and the sediment transport curve is equal to zero in the nearshore region. At this time, I's is equal to zero, therefore, the recession rate is equal to zero and the peak recession is attained. As the water level continues to drop, energy dissipation per unit volume decreases below the equilibrium value; hence, onshore sediment transport is initiated and -46- 4- EQUILIBRIUM EROSION' ASSOCIATED WITH STATIC - PEAK SURGE HEIGHT CURVE REPRESENTS / STORM SURGE AND / \ MAX. POTENTIAL/ i > w\ o EROSION / \ z o TIME (hrs) Figure 21 Method of Normalizing Axis to Represent Storm Surge Hydrograph and Maximum Potential Erosion WATER LEVEL AND MAX POTENTIAL EROSION PREDICTED EROSION I-~~ ~ o w~~~~~~ 0 LU W~~~ cc~~~~~~~~~~~~~~~~~~~~~~~~L TIME Figure 22 Schematic of Numerical Results Showing Predicted Erosion Curve Approaching Equilibrium for Smooth Increase to Static-Peak Surge Level / A~ WATER LEVEL AND MAX POTENTIAL / \ * ~~EROSION / Z Uj 1-41 I ~ ~ ~ ~ ~ ~ ~ PREDICTED EROSION TIME Figure 23 Schematic of Numerical Results Showing Predicted Erosion Curve Reachi-ng a Maximum as Water Level Decreases Following Peak Surge Level -48- =~~~ !~~~~~~~~~ w~~~ =~~~ ( / J 'TIME Figure 23 Schematic of Numerical Results Showing Predicted Erosion Curve ReaChi~ng a Maximum as Water Level Decreases Following Peak Surge Level -48- the profile begins rebuilding to a new equilibrium form. Hughes and Mhu (1981), have successfully modeled this behavior in small scale laboratory tests; concluding that the actual recession at the time of the peak surge is 60 to 90 percent of the peak recession. In Figures 24 thorough 26, the dynamic erosion response of a beach-dune profile is illustrated for various storm surge durations., In these examples, the storm surge hydrograph is idealized by the relationship: n = 4 sin2 a t (34) where the peak surge level is 4 feet and a equals it divided by the storm surge duration. Together with a breaking wave height of 10 feet, these ideal storm surge hydrographs are applied to a representative beach profile from the Bay-Walton county area of Florida. In order to compare the response of the profile to the various storm surges, the equilibrium erosion conditions are first determined for a static water level of 4 feet and a constant wave height of 10 feet. From this analysis, the maximum potential volumetric erosion for an infinite storm duration, i.e. over 300 hours, is found to be about 28 cu yds/ft. Thus, the right hand axis of each figure is scaled to display this maximum potential erosion for the peak storm surge level. .It may be noted that Hughes and Chiu (1981) suggest that the maximum potential erosion resulting from a gradual water level increase is greater than that obtained from an instantaneous increase to a static water level . This may be attributed to increased deposition that occurs in the offshore region as the breaking depth migrates shoreward as the water level rises slowly. This effect is also evident in the numerical model results; however, for small increases in water level and for storm of short duration, the difference in volumes is insignificant. -49- Cl IDEAL RESPNSE OF -I! REPRESENTATIVE PROFILE DAY-UALTON COUNTIEGFLA o H = 100 FT -F RB~~ STORM MORE AND MAXIJVH POTENTIAL VOLUMETRIC EROBION I.- Lii PREDICTED VOLUMETRTC EROSION %.bw 3'.o 6'8.00 DO ; i.oo lb.0o ib.C 2At.0o 2~4.00 2'.00 3b .00 Bb.00 9 o TIME - HRS Figure 24 Effect of Storm Surge Duration on Volumetric Erosion, 12 Hr. Storm Surge IDEAL RESPONSE OF REPRESENTATIVE PROFILE BAY-WALTON COUNTIES.FLA H L 10.0 FT STORM $UROE AND MAXIVH POTENTIAL.. B f VOLUKETRIC EROSION g q LL I-~~~~~~~~~~~~~~~~~n LO Cl- w w a ~~~~~~~" ~~~~~~~~~~~~4 PREDICTED VOLUIETRIC EROSION d-~~~~~~~~~~~~~~~~~ - -- 9).Co K'oo 8!.00 So00 1.�.0O lb.00 o 8.00 A .00 i4 .00 *X 2'. b Sb.00 8Ob XO Sd.OO TIME - M-RS Figure 25 Effect of Storm Surge Duration onVolumetric Erosion, 24 Hr. Storm Surge IDEAL RESPONSE OF STORM SUR(5 AND AAXItVN POTEOIML REPRESENTATIVE PROFILE VOLUMETRIC EROSION BAY--WALTON COUNTIESFLA ~� He H 10.0 FT F .flj 80 B~~~~~~~~~~~~~~~~~~~~~3 DIIR PREDICTED VOLUMiETRIC EROSION 9i.00 s'.oo 8'.00 3%m �!zxo :b.00 i�b.00 2t.( 24.O 2'r.00 Bb.Oo SA.00 SJ.Oba TIME - FIRS Figure 26 Effect of Storm Surge Duration on Vol'umetric Erosion, 36 Hr. Storm Surge In Figures 24 thorough 26, the storm surge hydrograph and the maximum potential erosion are represented along with the predicted time-dependent volumetric erosion, for storms of 12, 24, and 36 hour durations respectively. In Figure 27, the results of the three simulations are compared directly to graphically depict the dynamic erosion response. It is important to note that the predicted maximum erosion is only 14.2 to 28.5 percent of the maximum potential erosion. From these examples, the numerical analysis indicates that the time dependence of the storm surge erosion process is of fundamental importance in the prediction of storm related erosion. Specifically, it is evident that: (1) the actual maximum recession lags the peak storm surge; (2) the actual maximum recession is function of the storm duration as well as the peak surge height; (3) the erosion rate is dependent on the rate of increase of the water level; (4) the maximum potential recession predicted by Edelman and Dean is seldom attained during a typical storm surge duration. -53- 8~~~ Vi V.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C Ilk~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-r I- SC TIME~~~~36R - R Fgr27CmaioothEfetof1,4ad3HrStrSreoVoueEoi a 4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~, 12 MIfS , , a~~~~a - O~~~~~~-. - . , 8 - - - -~ 8. c~00 3.oo 8.00 2.00 �a.O i.oo �t~a b.O 2ii~x i4.0 2~.0 .oo ior sb.oo as3Eo TftlE - fIRS Figure 27 Comparison of the Effects of 12, 24, and 36 Hr. Storm Surge on Volume Erosion VIII. RESULTS-SIMULATION OF HURRICANE ELOISE As a preliminary verification of the numerical model', beach-dune erosion in Bay-Walton County area resulting from Hurricane Eloise is simulated. Hurrica'ne Eloise crossed the Florida Panhandle on September 23, 1975 and was one of the most severe storms ever to strike the Bay-Walton County area. While no reliable tide gage records exist for this area, it appears that a peak storm surge of 8-12 feet occurred over much of the two-county area. Using numerical storm surge models, the National Weather Service has suggested the peak open coast storm surge to be about 10.5 feet above mean sea level at the Bay-Walton County Line. From high water mark, data, the Mobile District of the U.S. Army Corps of Engineers estimated peak surge elevations in the area to be 12-16 feet. While high water mark data does not correspond to the elevation of the peak surge, it seems clear that a peak surge of 10-12 feet occured near the Bay-Walton County Line. Chiu (1977) summarized beach erosion data from Bay and Walton counties, collected under the Florida Coastal Construction Setback Line Program. Two years prior to the storm, beach-dune profiles were taken at approximately 1,000 feet intervals. Also, aerial photographs at a scale of I" = 100' were taken in the 1973 survey. This data provided the pre-storm profiles and, in the low energy environment of the Gulf of Mexico, were assumed to be- representative of the actual pre-storm profiles. After the storm, similar data were collected; however, beach profiles were taken up to 3-4 weeks after the storm. By this time some recovery of the beach face had occurred. While dune erosion characteristics should be recorded accurately, Chiu notes that both the eroded volume and the contour'advance/retreat of the contours near mean sea level reflect some beach recovery. To simulate these beach-dune changes, three types of data are required: (1) the storm surge hydrograph; (2) the breaking wave height; (3) the pre-storm profile characteristics. The storm surge hydrograph was estimated with a Bathystrophic Storm Surge Model written by the author. To simulate the storm surge on the open coast near the Bay-Walton County line, the characteristics of Hurricane Eloise at the time of landfall were supplied as input to the Bathystrophic model. Since Eloise made landfall almost normal to the coast, the peak storm surge was determined at the point of maximum wind, 20 to 22 miles from the storm center crossing. At this point, just east of the Bay-Walton County line, a peak surge height of 7.5 feet was calculated. Burdin (1977) shows a recorded storm surge record for Destin, Florida. While this tide gage is located in a bay and does not reflect the open coast surge directly, an'initial water level rise of from 1.5 to 3 feet is evident up to 24 hours before the peak surge. By adding an initial rise of 1.5 to 3 feet, the predicted peak surge elevation agrees well with other predictions. The predicted storm surge hydrograph is shown in Figure 28. The breaking wave height is extremely difficult to predict with any reliability. Because of this uncertainty, and because wave height is not of primary importance in this model, a constant breaking wave height is used for the duration of the model simulation. Numerical cases were run with breaking heights 7.5 and 15 feet. Beach profiles for Bay-Walton County vary significantly in terms of dune height, beach slopes, and dune configurations. Chiu (1977) and Hughes and Chiu (1981] give several schematic beach-dune profiles from which a representative or, -56- 12 HURRICANE ELOISE BAY-WALTON CO. LINE 9 - 6. -3 -4 I 0 4 8' 12 16 20 24 28 32 36' TIME,. HRS Figure 28 Estimated Storm Surge Hydrograph, Hurricane Eloise Bay-Walton County Line perhaps, an average profile form can be determined. From these sources, the initial profile was assumed to have an average dune height of 18 feet above mean sea level and a break in slope at the vegetation line elevation, approximately 7 feet. Average dune face slopes appear to be between 1:4 and 1:2; beach face slopes average 1:15 to 1:10. This schematic profile is shown in Figure 29. After viewing actual pre-storm profiles from Walton County, this seems to be a very good approximation for several actual profiles. Offshore, the equilibrium profile shape was modeled with A = 0.2055 ft 1/3 {0.13 m1/3} which is empirically determined for 0.3 mm sand. Hughes and Chiu (1982) found the representative A parameter for this area to be 0.19 ft 1/3 In this preliminary verification, 20 test cases are simulated: (1) to determine the quantitative validity of the proposed model relative to observed erosion data; (2) in an effort to display some of the effects of the basic parameters on the erosion characteristics. Since exact values of the peak surge, wave height-, and beach slopes are not known, various combinations of 3 beach slopes, 2 dune slopes, 2 wave heights,and 2 peak surge elevations are tested. In Table II, the results from each test case are summarized, such that the maximum erosion, in terms of contour change and eroded volume, is shown. For comparison, average erosion statistics presented by Chiu (1979) are shown in Table III. Upon inspection, the model results show good agreement with observed average erosion characteristics. First, consider the beach-dune change as measured by the volumetric erosion. From the 20 test cases, volumetric erosion varies from 8.3 to 15.3 cu yds/ft; this compares to average values of 7.3 and 8.14 for Bay and Walton counties. respectively-and an average of nearly 10 cu yds/ft near the point -58- . DUNE SLOPE 12 TO 1:4 4- E * 3- o : 2 XB BEACH SLOPE 1:10 TO 1:15 w I- iwt o -\ MSL EQULIBRIUM PROFILE A= 0. 13 m1/3 ' . Figure 29 Representative Profile, Bay-Walton County Line -59- Table II Simulation of Schematic Profile of Bay-Walton County Hurricane Eloise Beach Dune Wave Peak Contour Advance/Retreat Volume Slope Slope Heights Surge 15' 10' 5' 0' Eroded (ft) (ft) (ft) (ft) (ft) (ft) (yd3/ft) 1:10 1:2 7.5 10.5 -22.4 -22.1 -12.7 +22.9 10.8 1:10 1:2 15.0 10.5 -23.0 -22.6 -15.9 +13.8 12.0 1:10 1:4 7.5 10.5 -20.1 -20.8 -15.2 +22.2 10.6 1:10 1:4 15.0 10.5 -20.6 -21.4 -18.4 +13.0 11.9 1:13 1:2 7.5 10.5 -19.1 -18.9 -10.3 +13.5 9.2 1:13 1:2 15.0 10.5 -19.6 -19.5 -12.9 +.2.9 10.4 1:13 1:4 7.5 10.5 -17.0 -17.7 -12.6 +13.0 9.1 1:13 1:4 15.0 10.5 -17.5 -18.2 -15.1 + 3.0 10.3 1:15 1:2 7.5 10.5 -17.5 -17.3 - 9.6 + 7.5 8.3 1:15 1:2 15.0 10.5 -18.1 -17.8 -11.8 - 1.9 9.6 1:15 1:4 7.5 10.5 -15.6 -16.2 -11.7 + 5.8 8.3 1:15 1:4 15.0 10.5 -16.1 -16.7 -13.9 - 3.1 9.6 1:10 1:2 7.5 11.5 -29.9 -28.2 -13.1 +29.7 13.3 1:10 1:2 15.0 11.5 -31.0 -29.4 -18.9 +16.0 15.3 1:10 1:4 7.5 11.5 -26.6 -27.7 -16.1 +29.2 13.1 1:10 1:4 15.0 11.5 -27.6 -28.8 -21.9 +14.8 15.1 1:15 1:2 7.5 11.5 -24.1 -22.2 - 8.4 +12.1 10.4 1:15 1:2 15.0 11.5 -25.1 -23.2 -13.1 + 1.0 12.3 1:15 1:4 7.5 11.5 -21.1 -21.9 -11.1 +10.8 10.3 1:15 1:4 15.0 11.5 -22.2 -22.9 -15.8 - 1.7 12.4 -60- Table III Observed Erosion Characteristics For Bay-Walton Counties Avg. Contour Advance/Retreat Volume Location 15' 10' 5' 0' Erqded (ft) (ft) (ft) (ft) (yd /ft) Bay County - 8.9 -23.5 - 5.9 +26.5 7.30 Walton County -128 -35.0 -11.3 +22.5 8.14 Bay-Walton County Line -20.0 -42.0 - 5.0 +42.0 10.00 (20-22 miles from landfall) Table IV Equilibrium Recession For Static Peak Surge Level Schematic Profile of Bay-Walton County Beach Dune Wave Peak Contour Advance/Retreat Volume Slope Slope Height Surge 15' 10' 5' 0' Eroded (ft) (ft) (ft) (ft) (ft) (ft) (yd3/ft) 1:10 1:2 7.5 10.5 -115 -110 + 20 +165 45 1:10 1:2 15.0 10.5 -225 -215 -110 + 70 105 1:10 1:2 7.5 11.5 -130 -110 + 45 +160 48 1:10 1:2 15.0 11.5 -250 -230 -100 + 95 115 -61- of the peak surge. While the predicted values are larger than observed 'eroded volumes, it should be remembered that the post-storm profiles reflect a partial rebuilding of the profile near the mean sea level line. Post storm profiles from Walton County show a well defined ridge of sand on the foreshore between * the 0' and 3' elevations. Chiu notes the presense of this ridge and suggests that approximately 240,000 cu yds of sand were returned to the beach face in Walton County at the time of post-storm survey. Without this additional sand volume, the volumetric erosion at the time of the maximum erosion would be greater than the eroded vol-umes present in Table1~II; Chiu has estimated that perhaps an additional 2 cu yds/ft may have been eroded at the time of maximum erosion. From this discussion, it appears that the model correctly includes time-dependent erosion such that the magnitude of the eroded volume is essentially preserved. To further illustrate this, the maximum potential erosion resulting from static water level and wave height conditions is presented in Table IV, for a 1:10 beach slope and a 1:2 dune slope. Comparing these results to the observed erosion statistics in Table 1I, it is clear that the maximum potential volumetric erosion is 5 - 10 times the actual observed values. In contrast, when the time dependent storm surge levels are considered, predicted erosion is, to the first order, in agreement with observed values. Thus, it seems that erosion rates are simulated accurately such that the eroded volume is conserved in the time-dependent simulation. Now, consider the profile response as measured by the contour advance or retreat. First, it is well known that in nature, an eroding dune face steepens considerably, sometimes approaching a vertical slope. In Figure 30a, it is clear that for a water Ilevel of about 10 feet the 10' contour woul'd erode further than the 15' contour, as the latter is subjected to less direct wave -62- 15' CONTOUR RETREAT . -'' X . *~~* , x i<PRE-STORM DUNE PROFILE '.'"\ w ~10' CONTOUR RETREAT /.*.-! -X POST STORM .* . DUNE PROFILE Figure 30a Steepening of Natural Dune Form ..� , . � . . .X15' CONTOUR RETREAT * * - - PRE-STORM DUNE.PROFILE *~.'-~~ \10t CONTOUR RETREAT POST STORM DUNE PROFILE Figure 30b Uniform Recession of Schematic Dune Form -63- attack. This behavior is quite evident in the observed erosion data from Hurricane Eloise, both in Table III and in Figure 33 from Chiu (1977). In the numerical model, it has been noted that the dune face is approximated by a linear slope which is maintained as the dune erodes. Therefore, in Figure 30b, the numerical solution results in equal recession of the 10 and 15 foot contours for a water level rise of up to 10 feet. For a slightly larger water level rise, as in the simulated storm surge hydrograph in Figure 28, the dynamic solution begins to affect the 10' contour as the concave profile shape is established below the still water level. Thus depending on the initial dune slope and the peak storm surge duration, the 10' contour recession may be greater than or less than the 15' contour recession by a small amount. In the present model, without a rational method of including dune steepening, the numerical results for dune recession cannot be compared directly to the observed dune erosion. However, in analyzing the predicted results in Table II with respect to the observed dune recession in Table III, it may be argued that since the eroded volume is conserved, then the parallelogram approximation of dune erosion represents an average retreat of the dune face. In other words, since the volume eroded from the dune face in the model is essentially equal to the volume eroded from the dune face in nature, then the predicted dune recession should equal the average dune recession observed in nature. If the recession of the 10' and 15' contours in Table III are averaged, then the "average" dune retreat is 16.9 feet in Bay County, 23.9 feet in Walton County, and 31 feet in the region of the peak surge. Clearly this simple averaging has little physical significance, however, it does suggest that the model results are of the correct order-of-magnitude; numerical results range from 15.6 to 31 feet. Again, it is emphasized that the proposed numerical solution accounts for time-dependent erosion effects in a rational fashion. In Table IV, the maximum -64- potential dune recession due to static peak surge conditions varies from- 120 to 250 feet and requires more than 500 to 1,000 hours to reach equilibrium. In contrast, use of the storm surge hydrograph to determine the erosion for the duration of the storm surge, results in predicted values of dune recession that are 5-10 times less than the maximum potential recession. In Figure 31, the characteristics of the beach-dune erosion predicted for Hurricane Eloise are illustrated. Graphically, the difference between thi.s predicted erosion and the maximum potential erosion for the static peak surge level is depicted in Figure 32. In Table II, the results of the numerical simulation of erosion due to Hurricane Eloise reinforce earlier conclusions that: (1) the time-dependence of the storm surge-erosion process is critical for valid predictions of hurricane related beach erosion; (2) steeper beach/dune slopes erode farther and faster than mild slopes; (3) the' change in water level is the dominant forcing function in the hurricane-related erosion process, while changes in wave height have less effect on storm induced dune erosion. In the previous discussion, it has been noted that the inclusion of the time-dependent change in the water level represents a significant improvement in the ability to predict dune erosion associated with the hurricane storm surge. While the results in Table II represent the solution for idealized test cases., the close agreement between these results and observed erosion characteristics in Table III are encouraging. Likewise, the results in Table IV clearly suggest that use of the maximum potential erosion may greatly overpredict storm-induced erosion, especially for a fast moving storm such as Hurricane Eloise. Aside from the storm surge duration, the magnitude of the storm surge has a significant effect on the storm related erosion. In Table II, storm durations -65- Ha= 15.0 FT 16 - \ ~~~~14 ~ \ \ MD 1:2 14 --D 12 - \(4~~~~ \ ~~~~~~PEAK SURGE LEVEL i10 -\\ l-a I 0 * 6 PREDICTED POST ~~6 STORM PROFILE VOLUME ERODED 1:10 4 - EQUALS 12.0 CU YDS/FT B 2- 0 l I I I I i l 0 20 40 60 80 100 120 140 DISTANCE, FT Figure 31 Example of Predicted Post-Storm Beach-Dune Profile, Hurricane Eloise, Bay-Walton County, Florida 18 X \ z HB= 15.0 FT PREDICTEDM POST STORM \ PROFILE \ 9 INITIAL'PROFILE 12 CU YDS/FT 12 - PEAK SURGE LEVEL UILIBRIUM >. 6 w6- EUPROFILE 105 CU YDS/FT 0I I. I I I 0 60 120 180 240 300 360 420 DISTANCE, FT Figure 32 Comparison of Predicted Post-Storm Profile to Maximum Potential Erosion Due to Static Peak Surge Level, Hurricane Eloise, Bay-Walton County,.'Florida are equal for both 10.5 and 11.5 ft. peak surge levels; however, it is -clear that both volumetric erosion and contour recession are greater for the larger storm surge. This result is significant when compared to the erosion changes that result from a doubling of the wave height from 7.5 to 15 feet. While the equilibrium or maximum potential erosion is greatly affected by the wave height, for representative storm surge durations, wave height has a small impact on erosion. Finally, the numerical results in Table II show that the beach face slope is a major parameter in determining the extent of the profil~e change. For any given water level and wave height, the steepest profile, a 1:10 beach slope and a 1:2 dune slope, exhibits the greatest dune recession and volumetric erosion. Likewise, the steepest profile shows the greatest advance of the 0' contour, as the contour must build seaward a greater distance to reach a stable position. Chiu compared volumetric erosion directly to the beach slope and found a general trend toward increased erosion in areas of the greatest beach face slope. While local variations exist, possibly due to the presence of structures due to the variability in dune heights, it is clear in Figure 34 that volumetric erosion exhibits a distinct variation with beach slope. Likewise, comparing Figure 34 to Figure 33, it is evident that contour advance or retreat is greatest in areas of steepest beach slopes. BAY COUNTY *i t -2'- d- ................... __. _ 15--..SHELL .... - I V .................... i...... ...... ...... 16 98X20 22 it-6~ 2$S - i do.3 E1.) I�- MSL >.2 0 ISLAND _ 2d, ~ 4d f - . DISTANCE FROM LANDFALL-POINT IN MILES OKALOOSA COUNTY WALTON COUNTY 4d-~~~~~~~~~~~~~~~I- ,P - i o' -' I ..... . ...a.. ......... OE~ ~I DISTANCE FROM LANDFALL-POINTS IN MILES-. --. Figure 33 Contour Advance/Retreat Due to Hurricane Eloise Bay-Walton Counties (Chiu, 1977) WALTON COUNTY XB~~~~~~~~~~~d~~~EAC SLOPE P~~~ -' - EROSION-1 I J * A w� 1''0 B 6 4 2 0 4 6 i i24 16 DISTANJCE FROM LANOFALL-POINTIN MILES 0 20 ' B 6AY COUNTY c x~~~ a - 0 ~~~~~~~~~~~~~~~~~~~~~~0 & -- E~OSION/)-- ~8 wm C- - 16 to 20 22 24 26 28 30 32 34 0 DISTANCE FROM LANVFALL-POINT IN MILES Figure 34 Volumetric Erosion Distribution and Beach Face Slope Variation, Bay-Walton Counties (Chiu, 1977) REFERENCES 1. Bruun, P., "Coast Erosion and the Deveolpment of Beach Profiles," U.S. Army Corps of Engineers, Beach Erosion Board, Technical Memorandum No. 44, 1954. 2. Bruun, P., "Sea Level Rise as a Cause of Shore Erosion," Journal of Waterways and Harbors Division, ASCE, Vol. 88, WW1, February 1962. 3. Burdin, W.W., "Surge Effects from Hurricane Eloise," Shore and Beach, Vol.45, No. 2, April 1977. 4. Chiu, T.Y., "Beach and Dune Response to Hurricane Eloise of September 1975," Coastal Sediments '77, ASCE, 1977. 5. Chiu, T.Y., "Dune Erosion During Storm Conditions," Unpublished, Dept. of Coastal and Oceanographic Engineering, University of Florida, 1972. 6. Chiu, T.Y. and Dean, R.G., "Combined Total Storm Tide Frequency Analysis for Dade County, Florida," Dept. of Coastal and Oceanographic Engineering, University of Florida, 1981. 7. Dean, R.G., "Beach Erosion: Causes, Processes, and Remedial Measure," CRC Reviews in Environmental Control, CREC Press, Inc., Vol. 6, Issue 3, 1976. 8. Dean, R.G., "Equilibrium Beach Profiles: U.S. Atlantic and Gulf Coasts, "Ocean Engineering Report No. 12, Dept. of Civil Engineering, University of Delaware, Jan. 1977. 9. Edelman, T., "Dune Erosion During Storm Conditions," Proc. 11th Conf. on Coastal Engineering, London, 1968. 10. Edelman, T., "Dune Erosion During Storm Conditions," Proc. 13th Conf. on Coastal Engineering, Vancouver, 1972. 11. Fenneman, N.M., "Development of the Profile of Equilibrium of the Subaqueous Shore Terrance," Journal of Geology, Vol. X, 1902. 12. Hughes, S., "The Variation of Beach Profiles when Approximated by a Theoretical Curve, " M.S. Thesis, University of Florida, 1978. 13. Hughes, S.A. and Chiu, T.Y., "Beach and Dune Erosion During Severe Storms," UFL/COEL-TR/043, Dept. of Coastal and Oceanographic Engineering, University of Florida, 1981. 14. Moore, B., "Beach Profile Evolution in Response to Changes in Water Level and Wave Height," M.S. Thesis, University of Delaware, 1982. 15. Saville, T., "Scale Effects in Two-Dimensional Beach Studies," Trans. 7th Meeting of Intl. Assoc. of Hydraulic Research, Lisbon, 1957. 16. Swart, D.H., "Offshore Sediment Transport and Equilibrium Beach Profiles," Publication No. 131, Delft Hydraulics Lab., Delft University of Technology, 1974. -71- 17. Swart, D.H., "A Schematization of Onshore-Offshore Transport," Proc. 14th Conf. on Coastal Engineering, Copenhagen, 1974. 18. Swart, D.H., "Predictive Equations Regarding Coastal Transports," Proc. 15th Conf. on Coastal Engineering, Honolulu, 1976. -72- APPENDIX A DERIVATION OF IMPLICIT EQUATION OF CONTINUITY GOVERNING ONSHORE-OFFSHORE BEACH PROFILE EVOLUTION In Chapter III, an implicit finite-difference formulation for beach profile evolution was developed from a consideration of: (1) the offshore sediment transport, based on the excess energy dissipation per unit volume in the surf zone, and (2) the onshore-offshore continuity equation for conservation of sand in the surf zone. This solution involves a space staggered numerical scheme where the change in position of an elevation contour, or horizontal cell of height Ah, over time At, is given as: Kd(h'5/2 _h' 5/2) Kd(h5+/2 521 (A.1) K~}~ n -1 Kah+l-h x - LCh-+h1) -(hn+i+hA) .n4i-Xn) j Also, it is recognized that for a static water level, the position of each contour, n, varies over the time step, such that the average position, Xn, is given as: Ax + 2 (A.2) n n 2 Substituting Equation. (A.2) into Equation (A.1) results in the rather lengthy expression: Kd ht 5/1 h, / KZAt n n_ L (X+ + 11 2 -n-l 2 5/1 '52 - 4d+1 -n (A.3) (h' + h' ) (xnf - -. nh~+1 r>(,~+ - n 2 where the interdependence of adjacent contours is evident. h-1 t n I I ~ S.: L I -Yn+l1 FIGURE A.1DEFINITION SKETCH OF ENERGY DISSIPATION AT VERTICAL CELL BOUNDARY -A2- From Figure (A.1) , the change in position of the nth contour over a time step is defined by the excess energy dissipation in the water column associated with a vertical cell containing the nth contour. In Equaticn *(A.3), the energy dissipation terms at the beginning of the time step may be identified as: 31 h 5/2 - .. D =-( (A.4) n ih+hl)(x - xnhl) Therefore, the implicit form of the continuity equation may be expressed as: AX = K 1t n rd+i (A.5) 1 + 2 (xnxnl ) +| [ n2 (Xna1 X) where it is clear that 6Xn-� , 6Xn, and bXnL are the only unknowns. With these unknowns in the denominator of the right handd:side of Equation (A.5), an analytical solution is difficult, if not impossible, to obtain. However, if a Taylor series expansion is introduced, namely: 1 = 1 2 3 4 (A.6) a-+a a +a l+ca -A3- then, to the first order:' 1 Axc - Axs n 2(.x n-- (A. 7) Arn Ax ~n-1 2 (xn x n-1 n n- i2( Xn'~i-)n n- Now, Equation (A-5) may be rewritten-as: A~x A x A A- Ax KAt F D X n-X1 D + D I n +D I h21L 'n+1 rdij where the unknowns are in the numerator. This form may be rearranged,. such that: K3tI D� 3�nXh-l Din n A~x = Ah L ~+lU-1) (-x) A~ x D + -e Dn+iAxn + n+iC-,,1+ (A.9) 2(~~Xn+~) (n�-) and a constant coefficient may be defined as: KA t 2Ji (A.10) By grouping like terms, with the unknowns on the left hand side, the final form of the continuity equation is: -A4- L xj ~j x~1+ L+ + +n- - n- Xn (fl -- DN+I) A.N ~~~x~ 1D n, A. l1 which may be more clearly given as: Anx n-I + 3it An+ Cn n+l Zn where A -n- (A. 12a) n Xn X-1 =1+ 8D +~n+ (A.12b) n n-IL n+IL n OD - I (A.12c) n+1 = - DD J (A. 12d) n n r,-AS As a final note, the expression, Deq-Deq=O, may be added to both sides of Equation (A.12); thus, the driving term, Zn, may be defined as: Zn 2 5[ D es+D) (A.13) or :, -- Qs n+(A.14) n K QS1 Qsn (A.14) where the sediment transport flux across the vertical cell boundaries is now included for convenience. -A6- APPENDIX B LISTING OF COMPUTER PROGRAM C C EBEACH PROGRAM C DAVID KI(IEBEL--ULNIV:EIRSITY OFI- DE..AWARE(--F:P'RIL 1 932 C C THIS PROGRAM CALCULATES THE TIME-DEPENDENT BEACH-DUNE EROSION C USING THE EQUlILIBRIUM BEACH PROFlII.E THI-IEORY OF' l:ROBIERT G. DEAN. C ACCORDING TO THIS THEORY, THE EQUILIBRIUM PROFILE SHAPE RESULTING C FROM THE UNIFORM WAVE ENERGY DISSIPATION PER UNIT VOLUME IN THE C SURF ZONE IS OF THE FORM: C H = A*(X**(2/3)) C WHERE HI- IS THE WATER DEF'1-PTH AT SOME DISTANCE X i',OM THE SHORELINE C AND A IS A PARAMETER DESCRIBING THE PROFILE STEE:NNESS. A SEDIMENT C TRANSPORT RELATIONSHIP MAY BE PROPOSED SUCH T-lHAT TIHE VOLUMETRIC C FLUX OF THE ONSHORE-OFFSH-IORE SEDIMENT TRANSPORT I1S EXPRESSED IN C TERMS OF THE EXCESS ENERGY DISSIPATION IN THE SURF ZONE; THUJS: C QS = K*( D-DISSE ) C WHERE K IS AN EMPIRICAL TRANSPORT PARAMETER, D IS THE ENERGY C DISSIPATION PER UNIT VOLUME, AND DISSE IS THE EQUILIBRIUM VALUE C OF D. THIS TRANSPORT EQUATION IS SOLVED SIMULTANEOUSLY WITH THE C ONSHORE-OFF:SHORE EQUATION OF CONTINUITY FOR THE CONSERVATION OF C SAND IN THE SURF ZONE: C (DX/DT) = (DQS/DH) C C�__.__------- ------------- C C REQUIRED INPUT: C (1) SCHEMATIC REPRESENTATION OF" INITIAL BEACH PROFILE, WITH: C A) CONSTANT DUNE HEIGHT C B) UNIFORM LINEAR DUNE FACE SLOPE C C) INITIAL BERM HEIGHT -OR- ELEVATION OF CHANGE IN SLOPE C FROM DUNE FACE TO BEACH FACE ( UJSUALLY THE VEGETATION C LINE ELEVATION) C D) UNIFORM LINEAR BEACH FACE SLOPE C E) OFFSHORE PROFILE OF THE FORM A*(X**(2/3)) WHICH C INTERSECTS THE LINEAR BEACH FACE AT SOME DEPTH, HSTARI, C SJUCH THAT THE PROFILE SLOPE CONTINUOUSLY DECREASES IN C THE OFFSHORE DIRECTION. C C (2) STORM SURGE HYDROGRAPH: C A) SURGE LEVEL FROM MSL RECORDED AT ONE-HALF HOUR INTERVALS C C (3) BREAKING WAVE HEIGHT: C A) CONSTANT ESTIMATE OF:' DESIGN WAVE HEIGHT, -OR- C B) OBSERVED OlR ESTIMATED WAVE HEIGHT AT ONE-HALF HOUR C INTERVALS'. C C GENERATIED OUTPUJT: C (i) BEACH F'ROFILE CROSS-SECTION AT TIME OF MAXIMUM EROSION, C (2) CONTOUR ADVANCI. OR RETRIEAT FOR SPECIFIC ELEVATIONS, C (3) VOLUMETRIC EROSION IN CUBIC YARDS PER LINEAR FOOT. C COORDINATE SYSTEM: C (1) DISTANCES (X DIRECTION) ARE POSITIVE SEAWARID OF SOME C ARBITRARY BASELINE DATUM C (2) ELEVATIONS (Y DIRECTION) ARE PF:'OSITIVE BELOW THE WATER LINE C (3) A DISCRETE PROFILE REPRESENTATION IS USED WITH UNIFORM C VERTICAL CELLS OF WII)TII, DH - 0.5 FEET. MIDPOINT OF EACH C CELL IS DEFINED BY THE GRID NUMBER, N, SUCH TIHAT N=t AT C THE TOP OF THE DUNE, N=NMAX AT LAST POINT OFFSHORE. C C INPUT VARIABLES: C DISSE = EQUILIBRIUM ENERGY DISSIPATION PER UNIT VOLUME; C OBTAINED FROM THE "A" PARAMETER FROM THE F:ROFILE C SHAPE. C C HDUNET = AVERAGE INITIAL ELEVATION OF TOP OF DUNE IN FEET C RELATIVE TO MSL; NOTE: MUST HAVE NEGATIVE SIGN. C C HBERM = AVERAGE INITIAL ELEVATION OF TOP OF BERM -OR- C BREAK IN SLOPE BETWEEN DUNE AND BEACH SLOPES (USUALLY C THE ELEVATION OF THE VEGETATION LINE); NOTE: MUST HAVE C NEGATIVE SIGN C C XDUNET = INITIAL HORIZONTAL DISTANCE FROM BASELINE DATUM TO C CREST OF DUNE IN FEET; NOTE: MUST BE P'OSITIVE NUMBER. C C XBERM = INITIAL HORIZONTAL DISTANCE FROM BASELINE DATUM TO C CREST OF- BEFERM -OR- BREAK IN SLOPE BETWEEN DUNE AND C BEACIH FACE, IN FEET C C NBERM - GRID NUMBER OF BERM CREST -OR.- BREAK IN SLOPE IF C NO BEIRM IS PRESENT IN THE INITIAL PROF-IL.E NOTE: MUST C BE AN INTEGER; N=i AT TOP OF DIUNE AND INCREASES BY i FO C EACH 0.5' DECREASE IN ELEVATION. C C EXAMPLE: IF HDUNET--i8,0' AND HBERM---7,0', THEN C NBERM=1 +INTEGER (( 8-7)/)05)=23 C C XMD = UNIFORM LINEAR DUNE FACE SLOPE IN DECIMAL FORM (FT/FT) C C XMB = UNIFORM LINEAR BEACH FACE SLOPE IN DECIMAL FORM (FT/FT) C C HSTARI= DEPTH OF INTERSECTION OF LINEAR BEACH FACE SLOPE AND C EQUILIBRIUM PROFILE; ESTABLISHED AT THE POINT OF C TANGENCY SUCH THAT SLOPE DECREASES IN OFFSHORE DIREC- C TION; USED TO I)EFINE THEr TRANSITION BETWEEN DYNAMIC C AND GEOMETRIC SOLUTION REGIONS. NOTE: MUST BE ROUNDED C TO NEXT GREATER DEPTH IN 0.5' INCREMENT. C C LTMAX = MAXIMUM VAL. UE OF TIHE TIME COUNTER, ILTIME, USED TO C IDENTIFY TIME DEPENDENT STORM SURGE ELEVATION. C LTIME=0 AT T=O HOLIRS,LTIME=LTMAX FOR FINAL WATER LEVEL C DATA POINT. L...TIME INCREASES BY I FOR EACH ONE-HALF C HOUR INCREASE IN REAL TIME. C C WSEL(LTIME)=-S0ORM SURGE LEVEL. RELATIVE TO MSL ATl TIME STEP LTIME; C NOTE: POSITIVE F-OR WATER LEVEL. INCREASE C B2 C WH = DESIGN BREAKING WAVE HEIGHT IN FEET; CONSTANT VALUE. C C WAVE(LTIME)= BREAKING WAVE HEIGHT AT TIME STEP LTIME C C DIMENSION Hi(0:250),H(O:250),Xl(0:250),X(0:250),DELX(0:250) DIMENSION DISS(0:250),QS(0:250),Qi(0:250),E(0:250),F(0:250) DIMENSION SUMVOL(0:250),WSEL(0:250),WAVE(0:250) REAL KQ REAL KD CHARACTER*8 PROF C----FORMAT STATEMENTS FOR INPUT/OUTPUT�---------------------- C 920 FORMAT(iX, 'ERODED VOLUME EGIRIALS',FI:7.2, CUBIC YARDS/FT',//) 9211 FORMAT(MX, 'TIME = ',F6.2,/) 925 FORMAT(MX,'GRID',iX, 'INITIAL',IX, 'INITIAL',iX, 'UPDATED', I 2X,'UPDATED',2X,'ENERGY',3X,'SED.',4X,'DELTA',4X,'CUM.',/, 2 'POINT', 3X, 'ELEV',4X, 'DIST', 4X, 'ELEV',4X,'DIST', 3 4X, 'DISS',4X, 'TRANS', 6X, 'X' ,VX '''/) 930 FORMAT(15,SF8 2,F8.4,2F8.2) 935 FORMAT(2X, 'TIME',3X, 'SURGE',3X, 'WAVE' ,21X, i 'CONTOUR AI)VANCE/RETREAT',21'X, 'ERODED',4X,'CH1ECK',/., 2 9X, 'HEIGHT' ,IX, 'HEIGHT' ,5X, 'DUNE' ,5X, '20 FT', 3 5X, 'i5 FT' ,5X, '10 FT' ,6X, '5 FT' ,6X, 'MSL' ,6X, 4 'VOLUME' ,3X, 'CONTINUITY',/) 940 FORMAT(F6.1,F8.2,F7.1,7F10.2,Fl0.1) 945 FORMAT(1X,/////) 950 FORMAT(7X,F7.2) 960 FORMAT(F6.2) 975 FORMAT('PROFILE', 5X, 'I)UNE' ,iOX, 'BERM' ,7X, 'BEACH',4X, I 'DUNE' ,/,9X, 'ELEV' ,3X, 'DIST' ,3X, 'ELEV' ,3X, 'DIST', 2 3X, 'SLOPE' ,3X, 'SLOPE',/) 980 F0RMAT(A8,F8.1,3F7.I,2F8.3,//) 990 FORMART(A8;2F6.O,Fr6.,F6.3,F6.i,2F .0,F6.i,F6.3,F6. 3) C C----INPUT DATA ----� C C EQUJILIBRIUM BEACH PROFILE SPECIFICATIONS C 707 CONTINUE C READ (10,990, END=777) PROF, XMSL , XBERM, HBERM, XMEI, HIDLJNEF, XDUNEF, I XXDUNET, HDUNET, XMD, SMULT WRITE(6, 990) PROF, XMSL, XBERM HbERM, XMB, HDUNEF, XDUNEI:, If XXJDUNET, H--IDUNET, XMD, SMULT REWIND(I) WRITE(3,975) WRITE(4,975) WRITE(4, 980) PROF, HDUNET, X)UNET, H--IBERM, XBERM, XMB, XMD WRITE(3, 980) PROF, HIDUNE-T, XDUNET, FIBERM, XBERM, XMFI, XM) WRITE(4,935) C B 3 DISSE=46.32*(A**i.5) C BERM AND DUNE SPECIFICATIONS C HDUNET=-HDUNET HBERM=-HBERM NMAX=5i +IFIX((-HDUNET--30. )/0.5) NBERM=i+IFIX(--(HDl.NET-HBERM)/0.5) NDUNEF=NBERM-i C COEFFICIENTS FOR ENERGY DISSIPATION AND SEDIMENT TRANSPORT EQNS. KD=55.24 KQ=0.0i1i44 C TIME CHARACTERISTICS C T=0.0 LTMAX=25 C C INITIAL VALUES C NSTOP=NBERM NFLAG=O MTWE=O MFIF=O NGOTO= I IF(XBERM.LT.1.0) NGOTO=2 C C SELECT INPUT FORMAT FOR WAVE HEIGHT C IF IWAVE=i, USE CONSTANT WAVE HEIGHT, SET WH VALUE BELOW C IF IWAVE=2, READ WAVE HEIGHT DATA AT ONE-HALF HOUR INTERVAL IWAVE=i C IF(IWAVE.EQ.E) WH=15.0 C C----ESTABLISH INITIAL PROFILE�-----------------------�-------�--- C DO 110 N=I,NMAX HI(N)=P4DUNET+DH*(N-i) WRITE(6, 1 120)N, HI (N) , Hi (N-I ) , HDUNET 1120 FORMAT(15,F8.2,F8.2,F8.2) IF(N.LE.NDONEF) GO TO 103 IF(Hi (N) .NE. HBERM) GO rO 199 C IF PROFILE HAS WIDE BERM, USE 102 IN NEXT GO TO C IF PROFILE DOES NOT HAVE BERM, USE 103 IN NEXT GO TO C GO TO(iO2,103) NGOTO C 99 IF(HI(N).LE.HSTARI) GO TO 104 HA=Hi (N)**2.5--Hi (N-i )-*-2,5 HP=Hi (0J).+-Hi (N-i ) XH=(KDI-FIA)/(HB*DISSE) B4 Xi (N)=XI (N-I )+XH GO TO 1 09 102 Xi(N)=XBERM GO TO 109 103 Xi (N)=XDUNET+(HI-l (N)-IHDUNET)/XMD GO TO 109 104 Xi(N)=XI(NBERM)+(HI (N)-H-IERM)/XMI GO TO 109 109 X(N)=XI(N) H(N)=HI(N) IF(HI(N).EQ.0.0) MSL=N IF(HI(N).EQ.-5.0) Ml:IV=N IF(HI(N).EQ,.-i0.0) MTEN=N IF(HI(N).EQ.-i5.0) MFIF=N IF(I-li(N).EQ.-20.0) MTWE=N 110 CONT:INUE C C RECORD INITIAL PROFILE C IF(T.EQ,.0.0) GO TO 999 997 CONTINUE C C C CALCULATE CHANGE IN PROFILE FOR GIVEN WATER LEVEL. AND) WAVE HEIGHT C C C----BEGIN MAIN TIME LOOP IN INCREMENTS OF ONE-HALF HOUR---------�---- C DO 1 LTIME= ,LTMAX C C READ STORM SURGE LEVEL AT EACH TIME STEP C READ(i,950) WSEL(LTIME) WRITE(6,950) WSEL(LTIME) WSEL(LLTIME)=WSELLTIME)*SMLJLT C C READ WAVE HEIGHT AT EACH TIME STEP UNLESS CONSTANT C FORMAT OPTION IS USED SUCH THIAT WI--=CONSTANT C IF(IWAVE.EQ,1) GO TO 400 READ(2,960) WAVE(LTIME) WH=WAVE(LTIME) 400 CONTINUE C UPDATE REAL TIME IN INCREMENTS OF 0.5 HOURS C T=T-0.5 C C APPROXIMATE THE STORM SURGE TO NEAREST 0.5 FEET AND C DETERMINE INCREMENTAL CHANGE IN WATER LEVEL, DELI4S C NOTE: FOR OUTPUT, WSELEV CONTAINS ACTUAL STORM SURGE LEVEL, C WSEL(LTIME) IS ALTER:ED TO CONTAIN APPROX. SURGE LEVEL C WS=0.0 WSEL(O)=0.0 F'S WSELEV=WSEL (LTIME) C DO 200 K=i,50 IF(WSELEV.L.E.WS) GO TO 201 WS=WS+0. 5 200 CONTINUE 20 WSEL(L TIME)=WS C DELWS=WSEL(LTIME)-WSEL(LTIME- ) C C UPDATE DEPTH AT EACH TIME STEP C DO 300 N=1,NMAX H(N) =1-(N) +DEL.WS 300 CONT'INUE C C---- SET SECONDARY TIME STEP TO AVOID NUMERICAL INSTABILITY------------ C IF(H(NBERM).GE.0.0) GO TO 111 DT=600..0 JTMAX=3 GO TO 1 2 111 CONTINUE DT=I 00.0 JTMAX=1 8 112 CONTINUE C C BEGIN SECONDARY TIME LOOP USING DT IN SECONDS C NOTE: TOTAL REAL TIME SIMUL ATED IN FOLLOWING LOOP MUST C EQUAL ONE-HALF HOUR DO 2 JTIME=i ,JTMAX C C---- DETERMINE BOUNDARY CONDITIONS LIMITING WIDTH OF ACTIVE PROFILE---- C C ESTABLISH OFFSHORE _IMIT 1'0 SURF ZONE AT BREAKING DEPTH C DEFINED BY THE SPILLING BREAKER ASSUMPTION C BDPT=t .30*WiH C DO 113 N=i,NMAX IF((H(N).GE.BDF'T) .AND.(H(N-1),LT.EBDPT)) NBREAK=N--1 113 CONTINUE C C ESTABLISH ONSHORE LIMIT TO SEDIMENT TRANSPORT AND ESTABLISH C THE T'RANSITION DEPTH, HSTAR, BETWEEN DYNAMIC AND GEOMETRIC C SOLUTIONS C IF(H(NBERM).LE.O.0) GO TO 114 IF(H(NBERM).GT.0.0) GO TO 115 C IF WATER LEVEL. IS ON BEACH FACE, SET HSTAR EQUAL TO ORIGINAL C TRANSITION DEPTH, HSTARI C IF WIDE BERM IS PRESENT, SET NSTOP EQUAL TO NBERM C IF BERM ERODES -OR- IF NO BERM IS PRESENT, NSTOP EQUALS I AT TOP B6 C OF DUNE C 11 4 HSTAR=HSTARI NSTOP=NBERM IF((X(NBERM)-X(NBERM-i)).LE.((0.5/XMD)+0.2)) NFLAG=i IF(NFLAG.EQ.i) NSTOP=1 GO TO 118 C C IF WATER LEVEL IS ON DUNE FACE, TRANSITION DEPTH VARIES WITH C THE BERM WIDTH: C IF WIDE BERM IS PRESENT, HSTAR EQUALS WATER DEPTH AT NBERM, AND C NSTOP IS EQUAL TO NBERM-- THI-S ALLOWS BERM TO ERODE QUICKLY. C IF BERM ERODES -OR- IF NO BERM IS PRESENT, HSTAR EQUALS ZERO C AND NSTOP IS EQUAL TO i AT TOP OF DUNE C 115 HSTAR=H(NDERM) NSTOP=NBERM XHCRIT=((H(NBERM)**2.5-H(NBERM-i)*-)2.5)*K-D)/ I((14(NBERM)+H(NBERM-i))*DISSE) IF((X(NDERM)-X(NBERM-I)).LE.(XHCRIT+0.2)) NFLAG~:z IF(NFLAGEQi) GO TO 116 GO TO 118 116 HSTAR=0.,0 NSTOP=1 GO TO 118 C C IF SOLUTION RESULTS IN DUNE ACCRETION, RESET HSTAR AND NSTOP C SUCH THAT ONLY BERM REBUILI)S, THUS DUNE EROSION IS P-'ERMANANT C 117 CONTINUE IF(H(NBERM) .GT.0.0) HSTAR=H(NBERM) NSTOP=NBE ERM 118 CONTINUE C C-------------------------------------------------------- C BEGIN DYNAMIC SOLUTION C---- ----------- C CALCULATE ENERGY DISSIPATION PER UNIT VOLUME AND C SEDIMENT TRANSPORT FLUX IN SURF ZONE C DO 121 N-:::,NMAX IF(H(N).LE.HSTAR) GO TO 119 IF(H(N).GT.H(NBREAK)) GO TO 119 HA=H(N)**2,5-H(N-1 )**2.5 I-IB=H (N) Hl-I (N-I) HC=X(N)-X(N-1) D ISSE(N) = (K D*HA) / (HB-11,14C) QS(N)=KQ*(DISS(N)-DISSE) GO TO 120 119 DISS(N)=0.0 Q (N)=O.O 120 Ql(N)=QS(N) 121 CONTINUE C APPLY SMOOTHING FUNCTION TO SEDIMENT TRANSPORT CURVE B 7 C FROM TRANSITION DEPTH, HSTAR, TO BEYOND BREAKING DEPTH C DO 137 N=2,NMAX IF(H(N).LE.HSTAR) GO TO 138 B(N) =0 .B1*(~1N-2) +0,i5*CQ1( N- ) +0 25*cQ1(N)+ I 15*QI i(N+1 )+0 1i*Q (N+2) TF(H(N).GT.H(NBREAl<+2)) GO TO 138 DISS(N)=DISSE+QS(N)/l<Q IF((QS(N).NE.0.0).AND,(QS(N-ILE4.0.0)) GO To 139 GO TO 137 138 QS(N)=Q1 (N) DISS(N)=0.0 GO TO 137 139 QQ=Qs(N) 137 CONTINUE C C EXTEND SEDIMENT TRANSPORT CURVE LINEARLY FROM TRANSITION DEPTH C TO ZERO AT NSTOP TO SATISFY CONTINUITY BETWEEN IYNAMIC C AND GEOMETRIC SOLUTION REGIONS. C qQGQ=QQ/ ( NQC-NSTOP) QS(NSTOP)=0.0 NA=NSTOPi + NB=NQQ-1 C DO 136 N=NANB QS(N)=QS(N-i)+QQQ 136 CONTINUE C C----CALCULATE COEFFICIENTS FOR DOUBLE SWEE:: SOLIJTION--- ----------------- C OFFSHORE SWEEP NUMBER I C C BDT=DT/ (21. 0*Dl--l) NB=NMAX-I DO 130 N=I,NB IF(NMEQJ.) GO TO 128 XA=X(N)-X(N-i) XB=X(N+1 )-X(N) 127 AN=-BDT*KQ*DISS(N)/XA BN=l,0FB T*KQ (DISS(N)/XA+D SS(N+1)/XB) CN=-BDT*KQ*DISS(N+l)/XB GO TO 129 128 E(1)=O.O F(i)=-(BDT/0.5)*(QS(2)-QS(i)) XB=X(N+i )-X(N) XA=XFI GO TO 127 129 E(N+1 )=-CN/(BN+AN*E(N)) F(N+i)=(ZN-AN*F(N) )/(BN,+-AN)*E(N)) 130 CONTINUE C C----CALCULATE CHANGE IN POSITION OF EACH GRID POINT ----------------- BO C ONSHORE SWEEP NUMBER 2 NB=NMAX-I DELX(NMAX)=O.O C DO 140 M=1,NB N=NMAX--(M-I) DEL.X(N-1)=E(N)*DELX(N)+F(N) 1 40 CONTINUE C C - -------------------------------------- C DYNAMIC SOLUTION IS COMPLETED, NOW APPLY GEOMETRIC CRITERIA C TO MAINTAIN SMOOTH PROFILE AND CONTINUITY C--------------------�----------�--------------------------- C� C ESTABLISH OFFSHORE SLOPE TO LIMIT CELL GROWTH AT BREAKING DEPTH C XCRIT=0.33/XMB NA=NBREAK-3 NB=NBREAK+i C DO 144 NN=i,5 1OLINT=0 DO 143 N=NA,NB XFOR=X(N-1- ) +DELX(N-i) XBAC=X(N)+DELX(N) XCI--IECI<=XFOR--XBAC IF(XCHECK.LT..(0.6-XCRIT)) GO TO 142 GO TO 143 142 KOUNT=1 XD:EFF=XCRIT-XCI-IECK DELX(N)=DELX(N)--XDIFF/2.0 DELX(N+1)=DELX(N+i)+XDIFF/2.0 143 CONTINUE IF(KOUNT.GT.0) GO TO 144 GO TO 145 144 CONTINUE 145 CONTINUE C C ESTABLISH UNIFORM BEACH FACE SLOPE AND DUNE FACE SLOPE C SUCH THAT CONTINUITY IS SATISFIED AND ERODED VOLUME EQUALS C DEPOSITED VOLUME C NOTE: FROM THIS POINT ON, WE ARE ONLY CONCERNED WITH FINDING C PROFILE CHANGE IN BEACH-DUNE REGION C IF(DELWS.LT.0.0) GO TO 151 IF( (H(NB�1~1�IERM) iGTS0.) SAN. (,NSTOP I ) GO TO 193 IFC(H(NBERM).GT,0.O0LAND.(DELWS.GEO)) GO TO 150 GO TO 151 150 IF(HSTAREQ. H(NBERM)) HSTAR=H(NBERM-i) 151 CONTINUE C DETERMINE VOLUME REQUIRED TO SATISFY CONTINUITY C BY FINDING NET CHANGE IN VOLUME BETWEEN ORIGINAL C PROFILE AND CURRENT PROFILE IN DYNAMIC SOLUTION REGION B9 C BETWEEN OFFSHORE POINT AND TRANSITION DEPTH, HSTAR. C SUMVOL(NMAX+i)=O.O C DO 160 M=1,NMAX N=NMAX--(M-lI) VOL=DH* (X(N) +DELX(N)-X1 (N) SUMVOL.(N):=SUMVOL(N+1)+VOL IF(H(N).EQJHSTAR) GO TO 161 160 CONTINUE 161 NSTAR2=N DELSUM=-SUMVOL(NSTAR2+1) C C DETERMINE VOLUME ERODED IN PREVIOUS TIME STEPS IN ONSHORE C ONSHORE PORTION OF TH E PROFCL E C HAVSUM=0.0 DO 170 N=1,NSTAR2 HAV=DH*(X(N)-Xi(N)) HAVSUM =lHA VSUM + 1- -AV 170 CONTINUE DSUM=DELSUM-HAVSUM C C )ETERMINE INCREMENTAL CHANGE IN VOLUME FOR CELLS ABOVE WATER LINE C TO SATISFY (i) UNIFORM SLOPE REQUIREMENT, AND (2) CONTINUITY C DELX(NSTAR2)=DSUM/((NSTAR2-NSTOP'+-)*DH) C DO 180 M=I,NSTAR2 N=NSTAR2-(M-1) DELX(N)=DELXCNSTAR2) IFCN.LT.NSTOP) DELXMN)=0.0 VOL=DH*(X(N)+DELX(N)--Xl(N)) SUM VOL (N) =:StJMVOL (N41 ) +VOL 180 CONTINUE C C C IF SOLUTION RESULTS IN DUNE ACCRETION, REPEAT CALCULATIONS C WITH ACCRETION LIMITED TO BERM AND BEACH FACE C 193 IF(DELX(I).GT.0.0) GO TO 117 * ~C C UPDATE POSITIONS OF EACH GRID POINT IN ENTIRE PROFILE C DO 666 N=1,NMAX XCN)=X(N)+DELX(N) 666 CONTINUE C OUTPUT TO RECORD EROSION STATISTICS C 999 CONTINUE Bio C FIND CONTOUR LOCATION C X20=0.0 X15=0.0 XDUNE=X(i)-XI(1) IF(MTWE.NE.0) X20=X(MTWE:-XI(MTWE) IF(MFIF.NE.) X15=X(MFIF)--Xi(MFIF) Xi0=X(MTEN)-Xl('MTEN) X5=X(MFIV)-XI(MFIV) XMSL=X(MSL)-Xi(MSL) VOLCHK=SUMVOL i) C C FIN) ERO)ED VOLUME NVOL = DO 500 N=i,NMAX IF(SUM VOLL(N+l ).LT..SUJVOL (N)) MVOL=N IFCSELMVOL(MVOL).GTESUMVOL(NVOL)) NVOL=MVOL 500 CONTINUE V0LERO=SUMVOL(NVOL)/27. C C IF((T.NE.26.0).AND.(T.NE.30.0)) GO TO 990 IF(JTIME.LT.JTMAX) GO TO 998 C C -----WRITE STATEMENTS----I-BEACFI PROFILE DATA----------------------------- C WRITE(3,921)T WRITE(3,920) VOLERO WRITE(3,925) WF 3ITE( 3,9P30 ) (N,HI�II (N) , Xi (N) ,NI(N ) , X(N ) ,DJ:Iss( N) ,cs ( N)> i DELX(N),SUMVOL(N),N=i,75) WRITE(3,945) C 998 CONTINUE C C 2 CONTINUE C, C----WRITE STATEMENT----TIME HISTORY OF STORM SURGE AND RECESSION------- C WRITE(4,940) T,WSELEV,WHXD[JNE,X20,Xi5,XiO,X!5,XMSI,,VOL.ERO,VOLCHI< C IF(T.EQ.0.0) GO TO 997 CONTINUE GO TO 707 C 777 CONTINUE C C ST0FP END Dii