Coast Eng Res. CH MR 83-10 A Numerical Model to Simulate Sediment Transport in the Vicinity of Coastal Structures ey Marc Perlin and Robert G. Dean MISCELLANEOUS REPORT NO. 83-10 _ MAY 1983 ¢, %, cy, Le p74 "anne Approved for public release; distribution unlimited. Prepared for U.S. ARMY, CORPS OF ENGINEERS COASTAL ENGINEERING RESEARCH CENTER eh lh Kingman Building 263 Fort Belvoir, Va. 22060 Me 22-10 Reprint or republication of any of this material shall give appropriate credit to the U.S. Army Coastal Engineering Research Center. United States Limited free distribution within the of single copies of this publication has been made by this Center. Additional copies are available from: National Technical Information Service ATIN: Operations Diviston 5285 Port Royal Road Springfteld, Virginta 22161 The findings in this report are not to be construed as an official Department of the Army position unless so designated by other authorized documents. win | Minti OTN SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM T. REPORT NUMBER 2. GOVT ACCESSION NO|| 3. RECIPIENT'S CATALOG NUMBER MR 83-10 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED A NUMERICAL MODEL TO SIMULATE SEDIMENT TRANSPORT IN THE VICINITY OF COASTAL STRUCTURES Miscellaneous Report 6. PERFORMING ORG. REPORT NUMBER 8. CONTRACT OR GRANT NUMBER(S) 7. AUTHOR(s) Marc Perlin and Robert G. Dean DACW72-80-C-0030 10. PROGRAM ELEMENT, PROJECT, TASK 9. PERFORMING ORGANIZATION NAME AND ADDRESS AREA & WORK UNIT NUMBERS Coastal and Offshore Engineering and Research, Inc., 70 S. Chapel Street Newark, DE 19711] 11. CONTROLLING OFFICE NAME AND ADDRESS Department of the Army Coastal Engineering Research Center (CEREN-EV) Kingman Building, Fort Belvoir, VA 22060 14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) C3155] 12, REPORT DATE May 1983 13. NUMBER OF PAGES 119 15. SECURITY CLASS. (of this report) UNCLASSIFIED 15a, DECLASSIFICATION/ DOWNGRADING SCHEDULE Approved for public release, distribution unlimited. 16. DISTRIBUTION STATEMENT (of this Report) 17. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report) 18. SUPPLEMENTARY NOTES 19. KEY WORDS (Continue on reverse side if necessary and identify by block number) Bathymetric response Numerical model Shoreline evolution Littoral barrier Sediment transport Wave transformation 20. ABSTRACT (Continue am reverse side if necessary and identify by block number) An implicit finite-difference, n-line numerical model is developed to pre- dict bathymetric changes in the vicinity of coastal structures. The wave field transformation includes refraction, shoaling, and diffraction. The model is capable of simulating one or more shore-perpendicular structures, movement of offshore disposal mounds, and beach fill evolution. The structure length and location, sediment properties, equilibrium beach profile, etc., are user specified along with the wave climate. DD , BAITS 1473 ~—s EDITION OF 1 NOV 65 tS OBSOLETE UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) nn eee SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) 2 SECURITY CLASSIFICATION OF THIS PAGE(Whe: in Data Entered) PREFACE The purpose of this report is to provide coastal engineers and researchers with a numerical model which predicts sediment transport and the resulting bathymetry in the vicinity of coastal structures. The work was carried out under the U.S. Army Coastal Engineering Research Center's (CERC) Numerical Modeling of Shoreline Response to Coastal Structures work unit, Shore Protec- tion and Restoration Program, Coastal Engineering Area of Civil Works Research and Development. This report was written by Marc Perlin and Robert G. Dean, Coastal and Offshore Engineering and Research, Inc., under Contract No. DACW/2-80-C-0030. The CERC contract monitor was Dr. F. Camfield, Chief, Coastal Design Branch, under the general supervision of Mr. N. Parker, Chief, Engineering Development Division. Technical Director of CERC was Dr. Robert W. Whalin, P.E. Comments on this publication are invited. Approved for publication in accordance with Public Law 166, 79th Congress, approved 31 July 1945, as supplemented by Public Law 172, 88th Congress, approved 7 November 1963. Huw, 6 Ad ic, oe ee ea Colonel, Corps of Engineers Commander and Director II ITI IV APPENDIX A CONTENTS CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) TNTIRODUGTNON Geer tay acl eees anes: omucite iyteoy Ne sriac nett: 1. General. .. LST AY bccn ec ALE tie ip a ee At alge? 2. Study Objectives ‘ BACKGROUND. .... UA aay Merisae Sete Bs to suitor is Ave aM coe Pa 1. Wave Refraction. is Arges aie rc EE cicsydirs she, ies one iy a 2. Crenulate Bays (LeBlond, 1972) ... 3. Crenulate Bays (Rea and Komar, 1975) 4, General One-Line Shoreline Model THE NUMERICAL MODEL ..... Binh cae setae 8 He DESCRIP T MO Mealy ci cea sneha Gale rian yaya Shee Nc sou tr tries Miecene 2RORE EAGT OlNvisarty, rer Get nate ene Oe No awe wears | ate 3. Diffraction. .. 4. Sand Transport Model SIMULATIONS AND VERIFICATION. ; 1. Simulation of Savage's Physical “Model ‘Tests. 2. Several Runs Using Shore Perpendicular Structures to Demonstrate Effects of Altering Some of the Pertinent Parameters. 3. Simulations ‘of Sediment. Transport ‘of Dredge Disposal in the Vicinity of Oregon Inlet. abs 4. Simulation of the Longshore Sand Transport Study at Channel Islands Harbor, California. are ade SUMMARY AND RECOMMENDATIONS . EITERATURE CIED.< DISCUSSION OF CONSTANTS AND SOME OF THE VARIABLES eae BNF aE MODE sae imaae: Sores Ourculs ck Mamta nde Gasly raVeter Mook eee. ee Ubayin Ge Reet gs PROGRAM LISTING . CONTOURS AND SCHEMATIC ILLUSTRATIONS. METHODOLOGY AND PROGRAM LISTING OF COMPUTER PROGRAM WHICH CONVERTS BATHYMETRIC DATA INTO MONOTONICALLY DECREASING DEPTH CONTOURS. LS HN CMA ra santa 6 Sar oeg ‘ USER DOCUMENTATION AND INPUT AND OUTPUT FOR PROGRAM VERIFICATION. : Sec omo tye : TABLES 1 Summary of results at Oregon Inlet . Zita MONIC Hail Vou VeiILUES iz Osfiase A Visa stats abl a sisocdurSeaanautetamsieesuale ae me ce gh (pune es a a ae Oo? 20 All CONTENTS--Continued FIGURES Definition sketch . Flow chart. . Definition sketch for wave diffraction. Distribution of sediment transport across the surf zone . Schematic representation of banded matrix if not stored in banded storage mode. Simulation of the physical model of Savage (1959) Equilibrium planform, case 4.2a . Equilibrium planform, case 4.2b . Equilibrium planform, case 4.2c . Equilibrium planform, case 4.2d . Stepped version of equilibrium profile used in the Oregon Inlet modeling . x Bo eee APs Initial contours used in the numerical model for all the Oregon Inlet Simulations. Liyarst eip VocdNee asta ome Monthly incremental values of Ay due to dredge disposal illustrated for the block between 7- and 11-foot contours. tert ROA yee ou ea, Initial and final 7- and 11-foot contours (no distortion) . Initial and final contours for case 2.cl. Stepped version of equilibrium profile used in the Oregon Inlet modeling, h=Ay2/°, case 4. Se eae aca Shore-perpendicular cross section of disposal mound . Incremental values of Ay due to dredge disposal Idealized numerical model representation of offshore breakwater at Channel Islands Harbor, California . . CIH simulation of shoreline contour . CIH simulation of (JMAX)th contour. Page 12 13 17 1) 26 29 31 32 39 34 36 37 39 41 42 44 45 45 47 49 50 CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) UNITS OF MEASUREMENT U.S. customary units of measurement used in this report can be converted to metric (SI) units as follows: EE Multiply by To obtain inches 25.4 - millimeters — 2254 centimeters Square inches 6.452 Square centimeters cubic inches 16.39 cubic centimeters feet 30.48 centimeters 0.3048 meters Square feet 0.0929 Square meters cubic feet 0.0283 cubic meters yards 0.9144 meters Square yards 0.836 square meters cubic yards 0.7646 cubic meters miles 1.6093 kilometers square miles 259.0 hectares knots 1.852 kilometers per hour acres 0.4047 hectares foot-pounds 1.3558 newton meters 1.0197 x 1072 kilograms per square centimeter millibars ounces 28.35 grams pounds 453.6 grams 0.4536 kilograms ton, long 1.0160 metric tons ton, short 0.9072 metric tons degrees (angle) 0.01745 radians Fahrenheit degrees 5/9 Celsius degrees or Kelvins! lt obtain Celsius (C) temperature readings from Fahrenheit (F) readings, use formula: C = (5/9) (F -32). To obtain Kelvin (K) readings, use formula: K = (5/9) (F -32) + 273.15. A NUMERICAL MODEL TO SIMULATE SEDIMENT TRANSPORT IN THE VICINITY OF COASTAL STRUCTURES Marc Perlin and Robert G. Dean I. INTRODUCTION ‘lee General. The need for reliable predictions of shoreline response to man-made or natural modifications is increasing due to environmental concerns and the rising cost of remediai measures. The capability of numerical modeling in addressing problems of shoreline response has advanced with improvements in wave climatology, programs to better understand sediment transport relationships, and improvements in numerical modeling. In-situ and remote sensing technology for the measurement of directional wave characteristics has been developed and applied, primarily within the last two decades. In addition to providing the necessary climatology, the resulting measurements have provided the basis for evaluation and refinement of directional wave prediction procedures. Studies such as the Channel Islands Harbor Longshore Sand Transport Study (Bruno, et al., 1981) and the Nearshore Sediment Transport Study (NSTS) (Gable, 1979) have yielded a better understanding of surf zone dynamics and the resulting sediment transport. The increased capacities of large computers and reduced computing costs combined with improved numerical modeling algorithms have resulted in an extremely promising potential for the numerical modeling of shoreline problems. Although it is doubtful that numerical modeling will ever replace completely the use of movable-bed physical models, the former type offers many advantages. The modeling of shoreline response is somewhat analogous to the problem of simulating storm surges in the coastal zone in which the scale effects and measurement difficulties essentially preclude physical modeling. For shorelines, the scale effects inherent in modeling sediment are well recognized and the costs of representing a substantial length of shoreline may be prohibitive. The laboratory representation of a realistic wave climate is at the forefront of technology. The investigation of shoreline response can best proceed by several approaches, with each approach selected for the particular strengths which it offers. Field programs are costly, usually because of the considerable equipment and the extensive time required, but these programs are essential for quantifying the values of constants or parameters, the forms of which may be available from laboratory measurements or theoretical considerations. Laboratory studies occupy a special niche by allowing the wave conditions and independent variables to be controlled readily, experiments to be repeated, and selected measurements to be conducted. Although, as noted before, scale effects are present in laboratory measurements of sediment transport, the physics governing the process should be the same. However, the relative magnitudes of suspended versus bedload transport in the laboratory and field may differ. Laboratory studies can also provide an excellent base for evaluating certain aspects of a numerical model, including wave refraction and diffraction and the resulting shoreline patterns due to, for example, the placement of a littoral barrier. Numerical modeling offers the capability to incorporate all the hydrodynamic wave-surf zone and sediment transport knowledge that is available from laboratory and field studies. Numerical modeling has the potential of providing accurate predictions of shoreline response to various structural and nourishment alternatives. Additionally, the possibility exists of employing numerical models and available field measurements to learn more about sediment transport mechanisms. In this latter mode, various candidate mechanisms or coefficients would be evaluated by determining the best match between measured and predicted shorelines and the bathymetry. Generally, this mode would require high-quality measurements of the forcing function (waves and nonwave-related currents) and the associated response (sediments) as well as the knowledge of appropriate conditions at the boundaries of the model. The present report documents the development and application of an n-line numerical model to investigate bathymetric response to time-varying wave conditions and shoreline modification. The model includes both longshore and onshore-offshore sediment transport. Based on laboratory results, a new distribution of longshore sediment transport across the surf zone is used. The wave climate is specified on the model boundaries which do not need to extend to deep water. Efficient algorithms are employed for representing wave refraction and diffraction. The equation of sediment continuity and transport are solved by a completely implicit algorithm which allows a large time-step. Specified sediment transport values or specified contour positions can be accommodated at the model boundaries. The model is suitable for investigating the shoreline response to a variety of modifications such as one or more groins, terminal structures, structures with variable permeability, and beach nourishment with or without terminal structures. 2. Study Objectives. The objectives of the present study include (a) the documentation of state-of-the-art models, (b) the development and documentation of an improved model which includes the capability to represent n-contour lines and (c) the application of the model to several relevant coastal engineering problems. II. BACKGROUND This discussion describes significant contributions which either address numerical modeling of shorelines directly or provide improved capability for modeling. 1. Wave Refraction (Noda, 1972). Noda developed an algorithm for solving the following steady state equation for wave refraction vxk=0 (1) in which ¥, the horizontal vector differential operator, and k, the wave number, are defined in terms of their components as Vos ie fg ee (2) k =ik +jk (3) where 7 and j are the unit vectors in the x and y directions respectively. Equation (1) can be expressed as a(ksine) _ a(kcose) in which e is the direction of the vector wave number relative to the x-axis and k denotes |k|. Noda expanded Equation (4) to the following form k cose = + sine ee = -k sine . + cose * (5) Since ok and oe are known from the angular frequency o, the water depth h, and the dispersion equation s@ = g k tanh kh (6) Equation (5) can be solved numerically, although there are problems of directional stability. The primary advantage of Equation (5) is that it allows the wave direction 6 to be determined on a specified grid, compared to unspecified locations that would be obtained by, for example, wave ray tracing. 2. Crenulate Bays (LeBlond, 1972). LeBlond attempted to model the evaluation of an initially straight shoreline between two headlands into a crenulate bay. The model constitutes a one-line (shoreline) representation. The transport equation employed related the total sediment transport to total water transport in the surf zone as predicted by the formulation provided by Longuet-Higgins (1970). The. initial shoreline patterns resemble crenulate bays in nature; however, the predictions were found to be unstable for reasonably long periods of computa- tional time and did not approach a realistic planform. 3. Crenulate Bays (Rea and Komar, 1975). Rea and Komar employed a rather ingenious system of orthogonal grid cells to provide a cell which locally is displaced perpendicular to the general shoreline orientation. A one-line representation was employed. A simple and approximate representation of wave diffraction was employed. Although the model yielded reasonable results for the examples presented, the unique coordinate system would not be suitable for a general model as the coordinate system must be "tailored" to some degree to conform to the expected shoreline configurations. 4. General One-line Shoreline Model (Price, Tomlinson, and Willis, 1Y/c). Price, Tomlinson, and Willis' formulation consists of the sediment continuity equation and the total sediment transport equation i 0.70 a (nC) sina, COSa, Fie eT OLS ST = hoe (7) A) S in which E represents the wave energy density, (nC) the group velocity, a the angle between the breaking wave front and the shoreline, y, the Specific weight of water, p the in-place sediment porosity, and S. the specific gravity of the sediment relative to the water in which it is immersed. The subscript "b" represents values at breaking. Two formulations were presented by Price, Tomlimson, and Willis (1972). In the first, Equation (7) was substituted into the continuity equation and the results cast into a finite-difference form. In the second, the two equations were employed separately. The latter formulation was selected due to its simplicity and used for the results presented. Computations were carried out for the case of beach response due to the placement of a long impermeable barrier. The total sediment transport equation by Komar (1969) was used and the planform was calculated at successive times. Refraction was apparently not accounted for in the numerical model. To verify the computations, a physical model study was carried out for the same conditions using crushed coal as the modeling material. The comparison was interpreted as good for up to 3 hours; however, for greater times, substantial differences occurred and these were inter- preted as being due to wave refraction not being represented. The crushed coal was supplied to the model at the updrift end at a rate based on the Komar equation, and the results were interpreted as substantiating this relationship. However, the updrift end of the model beach receded substan- tially both in the numerical and physical models. In the physical model, this can only be interpreted as due to the Komar equation predicitions being less than the actual transport rate, possibly due to the low specific gravity (1.35) of the crushed coal. The predicted recession of the updrift beach is puzzling, although it could be due to a problem in properly representing the updrift boundary condition. Other one-line models for shoreline changes in the vicinity of coastal structures were developed by LeMehaute and Soldate (1977) and Perlin (1978). Perlin also developed a two-line model formulation, with one-line represent- ing the shoreline and the second the offshore. Dragos (1981) developed an n-line nodel for bathymetric changes due to the presence of a littoral barrier. III. THE NUMERICAL MODEL 1. Description. There are several methods of modeling bathymetric changes due to the presence of a littoral barrier. An attempt can be made to either model the complete hydrodynamics and the resulting sediment transport or model using a combination of analytical and empirical sediment transport equations. The second method was chosen due to past relative success. At least two methods of employing sediment transport equations exist: a fixed longshore and cross-shore grid system where the depth is allowed to vary or a fixed longshore and depth system where the cross-shore distance is allowed to change. Although it may seem somewhat awkward, the latter system was chosen for the model. This method allows the modeler to think of bathymetric changes due to a littoral barrier in terms of the effect on the contours; i.e., the contour realinement due to the structure's presence is observed. One limitation of this approach, at least as it was applied here, is that each depth contour must be single-valued; it is not possible to represent bars. The next step in formulating the model was choosing the specific representation of the bathymetry. The model is an n-line representation of the surf zone in which the longshore direction x is divided into equal segments each ax in length. The bathymetry is represented by n-contour lines, each a specified depth, which change in offshore location according to the equation of continuity. There are two components of sediment transport at each of the contour lines, a longshore component, Qx, and an offshore component, Qy. Figure 1 is a definition sketch showing the beach profile representation in a series of steps and the planform profile representation and notations used. Implementation of the sediment transport equations requires knowledge of the wave field and the equilibrium offshore profile. A discussion of the refraction and diffraction schemes follows. The equilibrium profile is introduced when it is convenient. As an introduction to the logic used in the numerical model, a flow chart is presented in Figure 2. 2. Refraction. A refraction scheme compatible with variable ay's was required because of the variable distance to fixed depth contours (as opposed to the more usual fixed grid system where a grid center has a longshore and offshore coordinate with a variable depth). One of the benefits of the n-line model is the ease with which the response of the contours to a particular wave and structure condition can be visualized. A fixed grid system and an interpolation scheme could have been used to obtain the wave field; however, this would have reduced accuracy and increased computation time. The scheme developed also saves computation time because it does not use differential products terms. 11 yzero (I) BERM OFFSHORE +h (a) Beach Profile Representation Li a +| Qx(I,y) | yil,J | pL UT pC Ts am Qx(I, J+!) +y (y+ iyth Contour Dy(L-1,J-1) epee Contour mDvDoxrwnn7n7:dsno +X i=l i=2 i=I-1 i=l i=1+l i= LMAX LONGSHORE (b) Beach Planform Representation Figure 1. Definition sketch. “queyd MOL4 “zg aunbl4 + Arqawfky7eq Teury ay. autIag *saTqetien umouyun jo sm1i9] ut UTT pta3 yora sso19e qaodsueiq a10yssuoT qeqoy aya aqepTnope) *saTqetaea umouyun jo swiaz UT 9u0z jans ayq ssoioe qzodsuelq aroyssuot aya 39NqT1ISTd *Arqowkyyeq mau 97 auTwiaqap o7 xfrI1e papueq ayy aatos ~patenba usaeq Ieqep aaem ayq jo uofaeanp oy sey ipasn useq eqep aaem oy Tre sey *ganqonaqs aya 4e uoptjtpuos Aiepunoq 71odsuely quautpas a1oys3uo0~T 942 4eS -daqs awty pue ‘sainjoni4s jo uoTzeDOT pue yzUeT *Aaqowky3eq YStTqeqsd ‘aUTT Pf13 yoea OJ VOTIPIOT ST pue qystoy r9ayxee14 quatdptout ay} aqeTNopTe9 + (aatqeiaqit) pTety aaen aya o3epTnopteo suoT}OVAJJLP/uoToeajoI pue uotjzorAjai jo sauoz \stTqeqsd aarm anduy 13 The first of the governing equations used is the conservation of waves equation +i, xk=0 (8) > > where vy is the, horizontal differential operator equal to i(a/ax) + jla/ay) in which i and j are the unit vectors in the x and y directions, respec- tively, and x is the longshore direction, with positive to the right when facing the water, y the offshore direction, with positive seaward, and z the vertical coordinate, with positive defined as upwards. For the steady-state case, equation (8) yields p) t) = # (ky) - & (k,) = 0 (9) where ky and ky are the wave number projections in the respective directions. Defining as the angle k makes with the y-axis positive in the counter- clockwise direction, the equation can be written in final form as 3 3 : = (k cos @) = 5 (k sin @) (10) where @ = a + w (in radians). Noda (1972) and others have developed numerical solutions to expanded forms of equation (10). In the present study, equation (10) was initially central-differenced in the x-direction and forward-differenced in the y-direction with Snell's law used to specify the boundary conditions on the offshore boundary and one of the sides (i.e., the side of the wave angle approach). However, a numerical problem arose. The argument of the arcsine exceeded + 1.0 for large ay/ax. To overcome this problem, a dissipative interface was used on the forward-difference term (after Abbott, 1979). The final finite-differenced form of equation (10) is n+] ral 1 = sin — ft sin e) 8. led RT tele (1-2t)(k sin e). SL tan . A atl + Tk Usain OV Fas - st ((k cose) 7 GF (k 7 had where t has been taken as 0.25. The past of j and the present of j wave angles are numerically averaged to give the 6},;. Newton's method is used to compute the wave number via the linear wave theory dispersion relation. In addition, numerical smoothing is used at the conclusion of the wave field calculation. This approximates in an ad hoc manner diffractive effects (lateral transfer of wave energy along the wave) which exist in nature but have been omitted due to use of the equation for refraction (equation 8). The smoothing routine is (12) The second governing equation used in the refraction scheme is conservation of energy. Neglecting dissipation of energy due to friction, percolation, and turbulence, this equation can be expressed as Welle ce) = 0 (13) where E is the average energy per unit surface area and Ce the group velocity of the wave train. Performing the operation indicated and replacing C by its components (Cosin e) and (C.cos e) results in the following: 3 : 3 A a? (E Co sin e) + ay (EXGnZeose)e="0 (14) Assuming linear theory, E = agit (15a) where p is the mass density of water, g the, gravitational constant, and H the wave height. Dividing the equation by o> finite-differencing and weighting the forward-differenced term as before, and solving for the wave height, results in the following: r n+1 = i 2 2 ig (C,cose). j (t)(H Cqcose) sy say + (1-27)(H Cqcose) 4) (15b) 2 aY cine ¢3 “csi ae This equation is also solved by iterative techniques and the HEU and Hi ; are averaged at the conclusion of each iteration. oJ J Ce is determined by the linear wave theory relationship AG 2kh 8 8 (ag sinh OKh) (16) where h is the water depth, k the wave number, and C the wave celerity. Wave height boundary conditions are input along the same boundaries as the wave angles using linear theory shoaling and refraction coefficients. The e's have been previously determined. In both equations (11) and (15) for a variable grid system, the points (i + 1, j) and (i - 1, j) need to be determined (i.e., because the y coordinates are not fixed, adjacent values with the same subscripts can be farther or closer to shore, therefore interpolation must be used). The actual values are found by searching the (i + 1) and (i - 1) cross-shore lines, finding the adjacent values in the positive and negative y-direction, and interpolating to determine the value. 3. Diffraction. The diffraction solution (in the lee of the structure) used in the model is based on the method of Penny and Price (1952). Assumptions used in this method include a semi-infinite breakwater, which is infinitesimally thin, linear wave theory and constant depth. A definition sketch for wave diffraction is shown in Figure 3. The quantity THETAO represents the angle of wave incidence relative to the jetty axis, ANGLE represents the angle from the jetty at the point where the diffraction coefficient is to be computed, and RAD is the radial distance. The radial distance is then cast into a dimensionless parameter, RHOND (= 2n RAD/L), where L is the wavelength. This is equivalent to multiplying the radial distance by the wave number k. ~ The diffraction coefficient AMP is expressed as the modulus of the diffracted wave AMP = (Sum 1)2 + (Sum 2)@ (17) where Sum 1 = [cos (RHOND (cos (ANGLE-THETAO) )) (5 (1.0 + C. + S))] + [sin (RHOND (cos (ANGLE-THETAO))) . (- ; (S - C.))] + [cos (RHOND (cos (ANGLE+THETAO) )) peito ace siSiilce {sin (RHOND (cos (ANGLE+THETAO) )) G - (S-Ce))] (18) Sum 2 = [cos (RHOND (cos (ANGLE-THETAO))) . (- ; (S - C.))] + [sin (RHOND (cos (ANGLE-THETAO))) . (5 (1.0 +C, +S))]+ [cos (RHOND (cos (ANGLE+THETAO))) . (- : (S- C.))] + [sin (RHOND (cos (ANGLE+THETAO))) . (5 (1.0+Cp+S))] (19) In Equations (18) and (19), Cre and S represent Fresnel integrals and are computed in the model by means of an approximation after Abramowitz and Stegun (1965). Having obtained AMP, the wave height at the location in question is simply the product of the specified partially refracted incident wave height and AMP. The angle of the wave crest is computed assuming a circular wave front along any radial; this angle is then refracted using Snell's law. Throughout the refraction and diffraction schemes, the local wave heights are limited by the value, 0.78 x depth. Incident wave ray THETAO Location where diffraction coefficient, AMP, is to be calculated Ws Semi-infinite jetty Figure 3. Definition sketch for wave diffraction. 17 4, Sand Transport Model. a. Governing Equations. Three basic equations are used to simulate the sediment transport and bathymetry changes according to the wave field. The equation of continuity aq aq EN sie te Noy EL atu ay 0 (20) requires as input, knowledge of the longshore and cross-shore components of sediment transport. The total transport alongsnore has been measured by several investigators and many equations exist; however, the distribution of the transport across the surf zone is not well known. Fulford (1982) based on laboratory data from Savage (1959), developed a distribution of longshore sediment transport across the surf zone for the case of straight and parallel contours. Fulford's use of Savages experiment was based on two assumptions: 1) the structure must be a total littoral barrier and 2) onshore-offshore sediment transport could be neglected. Test 5-57 was chosen because the two criteria were nearly met. Savage reported that the groin acted as a total littoral barrier for the first 35 hours of the test (i.e., no bypassing occurred prior to 35 hours). This does not mean that no onshore-offshore transport occurred because as the profile steepens on the updrift side, onshore-offshore transport does occur. However, it was assumed to be negligible. In addition, the initial profile had been molded to an equilibrium profile via 150 hours of waves. Thus, the two criteria required to develop an inferred longshore distribution of sediment transport were nearly satisfied. This distribution is shown as a dashline in Figure 4. The smaller "maximum" is believed to be an extraneous effect of a groin downdrift from the location in the experiment where the data were taken. Therefore, this feature was replaced by a monotonically decreasing, smooth curve as shown by the "altered" curve. To analytically represent this distribution, a function of the following form was chosen n a. (y= (8) Gam) en? (21) This type of equation is convenient because it is easily integrable, and by properly choosing the constant, B, the integral of the equation from zero to infinity can be required to equal a particular value. This too is highly desirable because, aS was done in the model, the integral is set equal to one and then multiplying by the value of the well-known longshore transport equation, the value of the transport at any location across the surf zone can be determined. Further investigation suggested a value of n = 3 to produce a curve similar to Fulford's curve. A more general form of the equation which allows more flexibility and curve fitting is Pe ia | Aaa : q(y) = Bly +a)" e CYp (22) 18 least squares analysis Sediment Transport Rate (ft3/hr) D> i= T= ae se) wo = fC) = rH © Ges Or ‘— pb per - Ss ©) Lol 6 = KEY — —— Inferred sediment transport rate —-— Altered sediment transport rate a= Breaking positioneafters| 0 hirs Nonlinear least squares approximation to altered sediment transport rate = —-—-—-—-— Breaking vosition used for nonlinear (Aenl2e On ns 6 4 2 Oo -2 -4 -6 Distance Offshore From Initial MWL (ft) Figure 4. Distribution of sediment transport across the surf zone. where yp = distance to the point of breaking a = constant to allow sediment transport above mean water line (MWL) (swash transport or transport in region of wave setup) to be represented c = a constant establishing the width of the curve (to be determined) Ba = (causes fo (y) dy = 1.0) cy, O Based on Fulford's (1982) results and considering a to be proportional to the breaking height divided by the beach slope, the constant of propor- tionality was determined to be unity; i.e., a = hp/(ah/ay). Using equation (22) and a digitized version of the curve shown in Figure 4, a nonlinear least squares regression was carried out to determine the value of c. A Taylor's series expansion of the form k+1 (c,y) = £*(c,y) fesce AG (23) f where k and k + 1 represent the number of the iteration carried out. Least Squares regression minimizes the square of the difference between observed and predicted values with respect to a change in the parameter being computed, or 2 n= where fogs represents the observed values, which in this case is ax(y)ops- Carrying out the differentiation indicated and manipulating terms, Ac can be solved in terms of known quantities. An iterative procedure was then used by updating the values of fk(c,y), af/ac, and c until an acceptably small change inc results. For the data herein, the value of c was determined to be 1.25. The final form of sediment transport of a y location in the surf zone results for a shoreline with straight and parallel contours, as 3 (1.25)5(yp)° 20 This equation, which is also presented in Figure 4, predicts the relative transport at point y. To obtain the fraction of transport between two y coordinates, the integral of equation (25), from y; to y2, must be used. Ha sie Ly. = al/(t)25 9 ue Oe aa) = quiyildy =e) % Yiae ts o/ y 1 1 3 we ~El¥g# a)/(1.25 y,)] (26) Q,[ND] is dimensionless; therefore, to compute a value in, say, cubic feet per second, it must be multipled by the total transport along a perpendicular to the shoreline obtained from the total longshore transport equation used in the model 5/2 ; 0) Hp sin (2 ay) (27) See Appendix A for a discussion of the constant C'. It is noted that the transformation of qx(y) to qx(h) can be effected by multiplying by the one-dimensional Jacobian (ay/Ah). This latter form (qx(hn)) is more useful here because the present model simulates the changes in contour position (ay) rather than changes by depth (ah). In the numerical model, Qy (I,J) (see Fig. 1) is determined using equation (26) except for the shoreline contour, J=1, and the farthest offshore contour simulated, J = JMAX. The shoreline contour longshore transport, Qy (1,1), in order to include swash transport, uses equation (16); however, the first term is set equal to 1.0. The seawardmost contour transport, Qy (1,JMAX), in order to include any longshore transport not yet accounted for, neglects the second term of equation (26) (i.e., it accounts for transport from y(I,JMAX) to infinity). The dimensionless numbers are then multiplied by Q determined from equation (27). This method is based on parallel contours which may not exist. In order to compensate for the nonparallel nature of the contours (note that refraction does account for it as far as the wave field is concerned), the term sin (2ap) of equation (27) is replaced by sin (2a) shoreward of the breakpoint, where a represents the angle between the "local" wave angle and the "local" contour. It can be argued that for a spilling breaker, the remaining surf zone at any point "sees" a total transport similar to equation (27), where ap and Hp are the local values. The problem is that the constant of proportionality was determined for the entire surf zone and for nearly straight and parallel contours. This not being the case, the equation was altered on intuitive grounds to reflect the fact that the contours are no longer straight and parallel. 2 | The second input required by the continuity equation to predict the bathymetric changes is the cross-shore sediment transport. The governing equation for onshore-offshore transport (after Bakker, 1968) is Q = ax C I St eTAV re ee PON (28) Vs OE i,j-l Tse] EQS where Co is an activity factor (inside,the surf zone = 10e2 feet per second for the OF Fototype simulation herein, 10 © feet per second for the physical model simulation) (see App. A. for a een pa We (i,j) is the positive equilibrium profile distance between y(i sneiy i,j-1), determined from the equilibrium profile used in the numerical pee ae ay2/3 (Dean, 1977). See Appendix A for discussion of the value of A. The physical interpretation of equation (28) is that as this profile steepens (flattens), sediment is transported offshore (onshore). b. Methods of Solution. Three separate finite-difference techniques were used to solve the equations: (1) Explicit longshore-continuity and explicit cross-shore continuity; (2) Implicit longshore-continuity and explicit cross-shore continuity for half a time-step then vice versa; and (3) Implicit longshore-cross-shore continuity. An explicit formulation was first developed which used the refraction scheme, the distribution of longshore sediment transport across the surf zone, and the onshore-offshore sediment transport equation. Problems in addition to the usual ones which are encountered with explicit methods (e.g., computation time and cost) were immediately realized. In the explicit method, both transport computations are based on the former values of the contour locations and are completely uncoupled. Stability of an explicit scheme requires a small time-step. In addition, the noncoupled nature of the equations, in some cases, resulted in crossing of the contours due to the transport computed. It is logical to assume that an implicit formulation of the longshore transport equation used as input to the continuity equation along with the explicit onshore-offshore transport component would help the numerical stability (on the other half time-step, the longshore component would be computed explicitly and the onshore-offshore transport equation would be solved implicitly with the continuity equation). Although this scheme would be superior to the explicit procedure, it still would be susceptible to crossing contours. It should be noted that the magnitude of the coefficient used in the onshore-offshore equation is very important to the extent that the simulation models natural phenomena. If the coefficient is very small or vanishes, sediment will not move offshore and contours will cross because of the variation in the distribution of longshore sediment transport across the surf zone. If the coefficient is too large, the onshore-offshore transport, may become large enough that on a particular time step, an offshore contour 22 would move too far shoreward, thereby crossing an inshore contour or vice versa. Once the contours cross, not only does the bathymetry become unrealistic, but mathematically, the equation which computes the longshore distribution across the surf zone changes signs at some locations and the entire model becomes physically unrealistic. To circumvent these problems, an implicit scheme that simultaneously solves the three governing equations, was developed. Utilizing equation (26), and the one-dimensional Jacobian (ay/ah) to convert to Qy(h), the total longshore transport equation (27), the following equation is obtained, 3/2 Sh203 Hie. 3/223 (hy jen Hy A (ins Niges Hy, A Pane | ler (=~) a | Loa : b. ; b. x (5? | x sin (20 - 2a,) (29) 1,J Qx(i,j) represents the sediment transport between depths h(i,j) and h(i,j-1) (see Fig. 1). The term in brackets represents the normalized distribution of longshore transport between h(i,j) and h(i,j-1); © is the averaged wave angle at the location of Qy(i,j) and ac is the local contour orientation angle. Defining everything except sin (20 - 2ac) as v(i,j) and using a superscript to denote a time step, this equation can be written gntl Save Suet ee 2aQ*t) Xe « Und) (30) The assumption has been made that the wave field (H and e) do not vary during the bathymetric changes over the time-step. Using the following trigonometric identities, Sime 2a eb)! = Sinyz2a.cos 2bs—-ncos. Za Sinseb (31a) cos 2a =H2\coselalel) (31b) sin 2a =i 2usinitanGoswa (31c) and recognizing that the following expression is an approximation 1 nt+1 n+1 n n a iye al = hye s gaik Fee WA Bou (cea ee) Srna Negima ee sealed vee iey add Pela (32) 3 (oy cs aaidlbin Ne Nae Wad) i-1,j 23 along with assuming that the change in the denominator is small for a reasonable time-step (the numerator has been averaged over the nth and n + ith time-steps), equation (30) results in n+1 n+1 nt n Oy j + (S3); 5 Vioj ra (S3); Yi-1,j os (RHS1) 5 (33) all 1 where (S3). . = (5) (v. .) cos (2e) (2 cos ao.) —————————— Usd tod : 2 2\1/2 (ox + AY n mR : 2 n n ; (RHS1); = (v; ;) (2 sin @ cos e)(cos a - ASS G AWG Aeshna al Here it has also been assumed that Cos ea does not change over the time step. Equation (33) is the final form of the longshore sediment transport equation prior to its use in conjunction with the other equations. Averaging y values on the nth and (n+1)th time-steps, equation (29) can be rewritten as oe 1 n+1 n n+1 n yg song a } (5. ie a Pies if i) where Const6(i,j) = Copfli,j). ax. This is the final form on the onshore-offshore sediment transport equation. The equation of continuity, finite-differenced for the nth and (n+1)th time-steps, can be written as ne ein Via) BO aE e ne [aati faa 1 nile sn n+1 n n+1 .,n n+1 n tated = ah gh art -0r Qh ag gna ne | MUS iad) Ateled Siete g ond ean dele (35) Defining Ry,j as 1/(2axah), inserting equations (33) and (34) into equation (35), and transferring all known quantities for the nth time-step to the right-hand side of the equation result in n+1 nt+1 nt+1 n+1 Yi jt CAtRy 51535 Vag - COtRy 5)S35 4-1, 7 CPRE GS Sinn 5% 141, nt+1 il n+1 n+1 + (atR; 5 )S3 jeonst6; |) (; [ Ty ein ea i) 1 er jon 3 1 nt+1 n+1 i ap (atR, ;Const6; 5.1) (; [ Vinj Fz Yi, j+l 1) = (AWARE); (36) 24 Equation (36) can be rewritten as (1+ U+ V+ Z1 + Z2) Tae - (u)yot? - (vjyntt i-1,J i+, — (ZFS y= (Zep sy = (AWARE), (37) where Ule= AtRs 59355 PSOE Sia Z1 = (5) R; jConsté. | Z2 = (5*)R; jConst6, 41 Equation (37) is a weighted, centered scheme in which yat is computed using a weighting of itself and its four adjacent grid "neighbors". The weighting factors (U, V, Zl, and Z2) are functions of the wave climate, the slope between contours, and the variables included in the original formula- tion. An investigation of a small gridded system demonstrated that by writing simultaneous equations, one for each yj,j, a banded matrix results. This matrix can be solved by LEQT1B, one of the available routines from the International Math and Statistics Library (IMSL). A schematic representation of the matrix A which results from the matrix equation [A][y] = [B] is presented in Figure 5. In this schematic, the large zeros represent triangular corner sections of all zeros and the 0...0 represents bands of zeros, the number of which is dependent on the number of contours simulated (the number of zero bands between either remote nonzero bands and the tridiagonal nonzero bands equals two less than the number of contours modeled (in both the upper and lower codiagonals of the matrix)). An inspection of the subscripts in equation (29) yields the reason the zero bands are required. The more j values (contours) used, the more y grids there are along any perpendicular to shore. This causes zeros to appear in the matrix between bands as the weighting factors await being used to operate on y"t!(i-1,3) and yn+l(i+1,j). For this reason, the expense of simulating an increasing number of contours is exponential. The LEQTIB routine, utilizes banded storage and saves both storage and computation time; however, the routine has no special way of handling the interior zero bands. One refinement which would save computation time would be to develop an algorithm to solve and store the matrix by taking advantage of these inner zero bands; however, it is beyond the scope of this project. Of course, the matrix requires boundary values on longshore extrem- ities and on both onshore and offshore boundaries. The longshore boundary conditions are treated by modeling a sufficient stretch of shoreline so that effects of a structure's presence are minimal. The y values along these boundaries can therefore be fixed at their initial locations. In the onshore-offshore direction, boundaries are treated quite differently. The 25 ‘ x Geer Note:. Size of matrix full storage mode [ (IMAX-2) (MAX) x (IMAX-2) (JMAX) ] Size of matrix panded storage mode [ (IMAX-2)(JMAX) x (20MAX + 1)]j Figure 5. Schematic representation of banded matrix if not stored in banded storage mode. 26 berm and beach face are assumed to move in conjunction with the shoreline position. The required sediment transport is then computed by the change in position of the shoreline. The two equations are Nitviaas He n Yi, 0 yi 0 a m Y¥; 11 (38a) Mtl ee Berm Ax n+] The offshore boundary is treated by keeping y"+](i,jmax) (the contour beyond the last simulated contour) fixed, until the angle of repose is exceeded. Then, the y"*t!(i,jmaxt+1) is reset (at the conclusion of the n + 1 time-step) to : position such that the slope equals the angle of repose. Note that yntl (1,0) is represented in the program by YZERO;. There are also no-flow boundary conditions required at each of the structures being modeled. These are imposed on the adjacent y-grid points which are located downdrift (i.e., in the shadow zone) of the structure and shoreward of the structures' seaward extremities. They are imposed by setting S3; j of equation (33) and DISTR; ,j (the term in square brackets in equation (29) equal to zero, thereby causing Qx(i,j) to be zero (i.e., the no-sediment flow condition). This boundary condition is imposed automatically for every shore-perpendicular structure. It was found that even with the implicit formulation, high frequency oscillations occurred in the y values immediately updrift and downdrift of the structure. The solution did not "blow up"; however, on larger time-steps "sloshing" (oscillating) did occur. Part of this problem was due to the boundary condition at the structure which had been such that either no sand was allowed along a contour line or the sand determined by the equations was allowed to be transported. Because of the very large angle which existed around the tip of the structure when a contour first exceeded the length of the structure, very large amounts of sediment transport were predicted. In the nature where analog sand transport rather than digitized transport occurs, this does not happen. Therefore, the boundary condition was altered to constantly allow sand transport around the end of the structure in proportion to that part of the contour representation which exceeded the structure (i.e., the transport was calculated for the location at tip of the structure as if the structure was not there and then a proportion of this value was allowed to bypass). Although the transport around the tip of the structure is based on the values from the past time-step, it more closely simulated the natural phenomenon. Additionally, a dissipative interface is used on the y values as follows: Yev= 2) yeas Ces alae siga (t) y (39) ieee) itl ,j where Tt was again taken as 0.25. It is noted that only high frequency oscillations in y are affected by the use of equation (39); the total sum of y values is not affected. Also, in all the dissipative interface (ATE schemes used, if a boundary point is being computed, either a forward- difference or a backward-difference of equation (39) is used (after Abbott, 1979): Backward: oy. . = (t)y,_) ay (1 - t)y (40a) lad i,j Forward: y =} (Ge Ne + (1 -t)y (40b) i,j itd isJ IV. SIMULATIONS AND VERIFICATION Several simulations were run; two were attempts at verifying the numerical model, the others were run to gain insight. Because a complete data set does not exist, only the available data are compared. The first modeling effort was to simulate the physical model tests of Savage (1959). A second set of cases was run for shore-perpendicular structures. Next, an effort was made to model sediment transport in the vicinity of a hypothetical dredge disposal site in the 1]1- to 14-foot depths off Oregon Inlet. Finally, the Channel Islands Harbor Longshore Transport Study (Bruno, et al., 1981) was modeled. Bathymetric changes were closely monitored during this study; however, the wave climate (H, e, T) used was determined from the Littoral Environmental Observation (LEO) data and uncertainties exist as to the accuracy of the data. 1. Simulation of Savage's Physical Model Tests. The numerical model was used to simulate one of the physical model tests of Savage (1959). Test 5-57 was simulated numerically for a 10-hour period. In this physical model, the mean sediment size was 0.22 millimeters, the wave height averaged 0.25 feet, the wave period was 1.5 second, the wave angle was 30° (at a depth of 2.3 feet), and the groin was approximately 9.5 feet from still water to its seaward limit. Corr was held constant at 10-4 feet per second throughout the profile for this simulation. The offshore profile is presented in Savage (1959). Figure 6 represents three of the eight contours simulated. Note that the initial 0.3- and 0.5-foot-depth contours, in the numerical representation are too far seaward by approximately 2 feet. This is due to the h = Ay2/3 equation as compared to the equilibrium physical model profile. Realizing this, it is the shape of the contour which must be used as an indication of the numerical model predictions. The general trend of the contours is similar, although the numerical model contours are displaced farther seaward as expected. The major differences are in the diffraction zone. 2. Several Runs Using Shore Perpendicular Structures to Demonstrate Effects of Altering Some of the Pertinent Parameters. In the following simulations, the models were run until their near-equilibrium values were achieved. Coefficient Corr was not a function of depth (beyond the surf zone) but was held constant throughout the simulated area. Important variables are as shown in the figures. Only one wave condition (Ho = 3 feet, T = 7 seconds, and a deepwater wave angle ag 28 "(6G61L) abeAeS JO Lapow LedLsAyd By JO UOLZeLNULS “9 |uNbL4 (44 vr'v = XG) SuUOLze907 Plug au0ysbuo7 ve ce Of 82 92 ve od O02 8! 14 O'o=uaded { 14 €°0=47daq 34 5° 0=47daq SUN0JUOD [apow LedLuawnu sYy-O| SUN0}UOD [apoWw LedLuowNu [eLZLUT S4N0JUOD |HeAeS AY-Q| PaZLJaUdSLq S4N0}UOD aHeAeS [eLILUT L[apow [eLzLuL pue eraiies aHbeaes [eLZLUL UdaMzZaq ADURAdeUDSLG :920}\ (a4) A SauoysssG GouezSLG 29 of 60°) was used as input for all four cases. Case 4.2a used an equilibrium shape factor A of 0.0899 and one groin. Case 4.2b was similar to 4.2a with the only modification being, that the A value was changed to 0.1486. In this way, a direct comparison was made based only on the shape of the equilibrium profile. Cases 4.2c and 4.2d used A-values of 0.0899 and 0.1486, respectively, but this time three shore-perpendicular, evenly spaced structures were simulated. a. Comparison of Cases 4.2a and 4.2b. The most obvious difference between Figures 7 and 8 is the volume of sand impounded updrift and eroded downdrift. This is due to blockage of more of the active transport zone in the second case (i.e€., a shorter groin is required for an equivalent performance on a steeper beach). The next obvious difference is the size of the perturbation which exists in the offshore contours. Clearly, case 4.2b is more perturbed and this is expected because larger offshore transports occur due to the steepening on the updrift side. Conversely, this means less sediment is initially bypassed (and along with the downdrift requirement for larger volumes of sand) causes larger erosional features in case 4.2b. Another interesting feature is the downdrift fillet which occurs in the third, fourth, and fifth contours. The fillet is due to the shape of the sixth contour which occurs because of the inability of the wave to transport more sediment (due to the reduction in wave height and angle in the diffraction shadow zone). The remaining difference is also due to the volume of sediment being impounded; i.e., the distance and extent of change the presence of the groin causes upcoast and downcoast. b. Comparison of Cases 4.2c and 4.2d. The variations between cases 4.2c and 4.2d are very similar to the differences between cases 4.2a and 4.2b as would be expected with a groin field (here, three groins) as compared with a single groin (see Figs. 9 and 10). There is, however, one additional feature which can be attributed to the additional groins. Note that in the direction of littoral drift, the size of the fillet is decreasing. This is due to the updrift beach having an uninterrupted supply of sediment while the downdrift groin compartments are supplied sand at a rate determined by the bypassing. Part of this feature may also be due to the system not having attained complete equilibrium. The effects of the fixed boundary conditions are evident on all cases run. In these example cases, the boundaries are clearly too close to the structure to provide a proper representation of the fillet contours. Sic Simulations of Sediment Transport of Dredge Disposal in the Vicinity of Oregon Inlet. Hypothetical dredge disposal movement in the nearshore but beyond what is normally the surf zone at Oregon Inlet's adjacent beach to the south was modeled. In order to do these simulations, the program was altered such that for every nth iteration (time periods), the contours were shifted seaward to simulate the addition of dredged sediment disposal. The program presented in Appendix B does require slight modification to simulate this situation. In general, the fifth and sixth contours were shifted seaward on a monthly basis to simulate the disposal of 121,000 cubic yards of sediment. 30 OOSt | 4 4nojuod | eul4 "ez'p ased SuluosuR{d WNLUgL|LNbdJ (44) xX S99ue}ZSLq a4oysbuo] OOOv OOSE O00€E O0O0S2Z 0002 | I | ! | ANOQUOD | eULBLUO “7 aundl4 OOoS| oool J ! 4 8 = XWWe 24 0°00L = Xd MaO0eT= Alas 44 One. = Wide s 009° LZ = 113d GO°O = adds 4680 °0=V7 UMOUS 10U SUNOJUOD B puke /=f :910N 00S | TOOI- - 00! —002 -—OO€ (44) A SauoysssQ aoueIsig 31 "Gz'y ased ‘wuojuetd wntugiiinbz °g aunbLy (44) X Sa0uRqstg auoysbuo07 OOSb OOO” OOSE OOO€ 00G2 0002 OOS! OOO] OOS \ I | | ANOJUOD | PULDLUQ sues 8 = XWWP 24 0°00L = Xa 9-2 2 AL MUS = MNES AL 28S IE SL =ol § 09°12 = 1130 0°. = MIS 2 Oe si 98b1 0 2°) 9Se) O?1=4 OO -—00]1 -002 —OO0€ (ag) A Se4ouss19 ooueystG 312 "O?'p 9SeD SWUOJURLG WNLUGL{LNDJ *G BunbL4 (44) X Sa9ue7S1q auousbuo07 OO0Sb OO00F oos¢e ooo0€E O0O0SZ 0002 oos|] 000! OOS O | | | | \ | | | | =a OO —0OO0O|] — 002 Ano JUOD | eUL4 —OO€ ANOZUOD | CULDLUQ — OOD eee 8 = XWWe 14 OOL = Xa 509 = 2 ae 14 OO€SO0ESOUE = ALLICS 44 0°€ = Wad Sf eel 009‘ Lz = 173d GOs0r= ddvdSS 9s" 026- = aH 66800 = W - Wp ase) —~90S —009 i £ Ou UMOYS }OU S4NOJUOD g puke /=f :910}; (44) A SauoyusysQ Bour SIG 33 "DZ'p aseod Suuosuetd wntuqL{inby “OL aunbl4 (44) xX ‘aduezstg suoysbuo7 OOSb OO00F OOSE OO00E O00SZ2 0002 oos! OOOl OOS O Se | TN gto / fo Ms es Nd 7) ae Se © ZEON E/E OO NEY SRS = oO, EN LI OVI LDN ES 001 ee ee OS NEN OL =4 00v 06 =u 8 = X¥WfT 34 0°00L = Xa 009 = 44 OO€*00E:00E = ALLACS 44 O°€ = WYAD Cee les 00S s 009.12 = 113d = $0"0_=_40W4S 14 0°€= H 89pL'O = VY - Pep ases K *adOUSf4Q 9OUPISILG (34) 34 In all these simulations, the following variables were held constant: (a) a time-step of 3 hours, (b) a shoreline length of 10,000 feet, (c) a longshore space-step of 200 feet, (d) an A value of 0.15 foot!/3 for the equilibrium profile (see Fig. 11), (e) a berm height of 5.3 feet with a beach face slope of 0.05, and (f) a duration of 1 year. The wave climate was provided by the U.S. Army Engineer Waterways Experiment Station Wave Information Study (WIS) 1975 data and was initiated at different times of the year as indi- cated in the specific cases below. All simulations, prior to any addition of sediment, used the bathymetry shown in Figure 12. The shoreline (relative to mean low water, MLW) was scaled from a bathymetry-topography survey provided by the U. S. Army Engineer District, Wilmington. The initial offshore bathymetry was computed according to the equilibrium profile and the 0-foot contour; i.e., the profile was shifted seaward or landward, accordingly, (see App. C.) The boundary profiles were fixed throughout the simulations. The variation of COFF outside the surf zone was used because of the importance of the time rate of change in this simulation. Table l presents the percentage of sediment which moves out of the control volume (i.e., imaginary boundaries around the area where sediment was added) directly onshore and the percentage of sediment remaining in the control volume at the conclusion of the simulation for each of the cases. In addition, a seventh (case 3) and eighth (case 4) were modeled. In Case 3, the only difference was that sediment was placed at the 11- and 14-foot contours. Case 4, however, was quite different and will be described in detail later. It has a 20,000-foot shoreline, a longshore space-step of 400 feet, and sediment was added on a weekly basis. Also, the resolution in the profile was better. diem opecitic Cases. (1) Case 2.a. In order to provide insight for the interpretation of the other modeling efforts, a simulation of the shoreline evolution using the January to December WIS time series, with no addition of sediment, was carried out. As expected, the contours almost attain an equilibrium planform shape (i.e., straight and parallel between the fixed end profiles; they do not, however, become aligned parallel to the base line because of the end conditions). Because of the scales involved, alongshore versus onshore-offshore, plotting the contours without distortion does not yield much information. Appendix C provides a listing of the final contours for all the cases modeled. (2) Case 2.b. The only difference between cases 2.a and 2.b is the suppression of the WIS wave angle which was set equal to zero (i.e., wave crest approach is shore-parallel at the offshore boundary of the model). This does not cause the longshore sediment transport to vanish completely. There are still local gradients in the contours which cause refraction and relative angles between wave crest and contour, thereby driving the longshore sediment transport (even if refraction was not considered, the local angle between the wave crest and contour would cause sediment transport). Note the larger onshore transport (Table 1) for this case compared with Case 2.a. This is due to the reduction in longshore transport caused by the wave angle of 0°. The model still tries to smooth the contour lines; however, more of the smoothing for the present case must be done by onshore-offshore transport. 39 “(e 13883 STO = VW) ¢ zAW =U ‘BuLlapow qajuy uobeug ayy ul pasn a,tjoud wntuqi{iLnba Jo uolsuaa paddays OOO€ 0002 ooo! (44) A SauoysssQ BdOURqSILG “TT aunbLy Og O02 Ol (24) 4 SMIW MOLaq yydaq 36 ("uMOYsS aLed9S AY BdLMZ SEM 7 BSED AOJ BL PIS aul) “SuoLze_NWLs qaluy uobavQ a4yz [Le AO¥ Lapow LedLreunu a4z UL pesn SUNnOJUOD [eLLZLUT "2T aunbl4 Gil Table 1. Summary of results at Oregon Inlet. Case Pct Onshore out of Pct Remaining No. Description control volume in control volume * After 17 weeks, the addition of sand caused contours to cross. WIS waves, Jan. No sediment added, WIS waves Jan. - Dec. No sediment added, WIS waves {a =0-) Jan. - Dec. 121,000 yd3 added monthly, WIS waves Jan - Dec. 121,000 yd3 added monthly, WIS waves Apr. - Mar. 121,000 yd3 added monthly, WIS waves July - June. 121,000 yd3 added monthly, WIS waves Oct. - Sept. 121,000 yd3 added monthly at the 11- and 14-foot contours 27,923 yd3 added weekly on the 7- 8-, 9-, and 10-foot contours, WIS waves Jan. - Dec. sediment added was 363,000 yds. not rerun. - Dec. Onshore Movement (992 yd3) Onshore Movement (1624 yd3) Bila: (460,264 yd3) 32.1 (466,160 yd3) 28.6 (415,784 yd3) OD (395,556 yd3) 8.9 * (32,164 yd3) 19.0 (275,796 yd3) Increase (14,148 yd3) Increase (9,356 yd3) 38.6 (559,984 yd3) 36.9 (535,392 yd3) 47.0 (682,088 yd3) 46.8 (670,848 yd3) 78.0 (283,016 yd3) 47.4 (687,525 yd3) Prior Problem was rectified; however, case was 38 (3) Case 2.cl. In this simulation, sediment is added to the system each month. It was simulated by advancing the 7- and 11-foot contours on a monthly basis to represent 121,000 cubic yards per month. Specifically, the sand volumes were "tapered" starting at the center of the nourished area over a distance of + 2,700 feet from the center. Table 2 presents the monthly ay values for the blocks between the 7- to 11-foot contours and the 11- to-14 foot contours. Figure 13 shows the planform ay values added monthly. WIS waves were used with the sequence being the normal calendar year, January through December. eames ep = 4.0 ft 5 increment 200 ft= Width of grid OFFSHORE DIRECTION 145.8 ft I = 14 16 18 50 22 ZZ as mr 32 34 36 38 Figure 13. Monthly incremental values of Ay due to dredge disposal illustrated for the block between 7- and 11-foot contours. The initial and final fifth and sixth contours have been plotted in Figures 14 and 15. The first figure has no distortion; the second is distorted 10 to 1. The simulation predicts that 31.7 percent of the dredge disposal will move shoreward out of the control volume. An additional 29.7 percent efflux occurs in the offshore and longshore directions, leaving only 38.6 percent of the total amount of sediment added remaining in the control volume. It is not clear what quantity of the sediment leaving in the longshore direction would reach shore. It is conceivable that most of this sediment would eventually reach the surf zone. The rate at which this material would move ashore would be expected to be slower than the rate at which the large mounds would move ashore because the deviation of the profile from equilibrium is much less. (4) Cases 2.c2, 2.c3, and 2.c4. The next three simulations were the same as 2.cl except the time series of wave events has been seasonally altered. Cases 2.c2, 2.c3 and 2.c4 use the 1975 wave climate from April through March, July through June, and October through September, respectively. The maximum variation is about 5 percent for the sediment volume moving onshore, and about 10 percent for the volume remaining. The variation in the 39 Table 2. Monthly values of Ay for the steps located between the 7- to 10-foot contours and the 11- to 14-foot contours. Value of I Monthly ay value (ft) for steps between 7- and 11-foot contours 11- and 14-foot contours 26 Zd5Cl/ 24,28 23,29 22,30 21531 Z20R32 19,33 18, 34 17535 16,36 152537 14, 38 13,39 All Others 40 -(UOLIAOJSLP OU) S4NOJUOD ZOOJ-| {| pue -/ [ePULJ PUR LeLILUT “HL aunbt4 1 3SVv8 | F | | visiur (9*1) 4 | I | | \| l | | ' a om) (one. 1 jours (St1)A | plyeey i etme ee eR TI Ei Ee EEE Eg Ft le a pn tle eet pete el tieay yee ta te eee i eh le ee ee ei ee eee ely ce i ee oe at eee ae ee Wee eet (ae eal (Saale ae] 4\ = U0!I}JO}SIG a1D9S :ajon “L(9°I)A pue (S°*1)A]_ [9°Z ases 4O0f SUNOJUOD [eUL} pue LeLiLUT "GL aunbL4 X “UOLJD9ULq au0ysbu07 OS Ov O¢€ 02 Ol =] I : U y I T 009 IOUT 5 87K oS Cc. O02 006 Ae 0001 IUNUT ery 4 OOl! }ouly (9°I)A 002! OO€! > Sy OS Woy aby Sire Hit Nice Oia) 42 quantity moving onshore could be caused by waves that first tend to move more sediment longshore; then, the waves that transport more sediment onshore have a less out-of-equilibrium profile to cause movement upon. The variation in percentage remaining is due to the variation of the time series of the wave climate, with the last month in the simulation being especially important. (5) Case 3. Instead of extending the 7- and 11-foot contours monthly to simulate the disposal of dredged sediments, the 11- and 14-foot contours were extended (194.4 feet each at the center of the disposal area). This case was modeled because the larger available dredge could not dump in more shallow water. The reduction and increase in the percent of onshore volume and the percent volume remaining (8.9 percent and 78.0 percent, respectively) demonstrate the sensitivity of the depths investigated. Qualitatively, these depths are the depths to which offshore bars occur along the Atlantic U.S. coast. (6) Case 4. Further investigation of the disposal process demonstrated the need for an 11,000-foot shore-parallel disposal length with the sediment placed at the 11-foot contour building to about 7 feet. It was decided to model this physical situation also. The total shoreline length was changed to 20,000 feet, and the space step to 400 feet; the length of the disposal area in the longshore direction was increased to 10,800 feet. The resolution in the vicinity of the depths of the dump was improved by adding the additional contours and the profile is shown in Figure 16. As in the other seven cases, 1,452,000 cubic yards was added annually to the system; however, the addition was accomplished on a weekly basis (27,923 cubic yards per week). Sediment was still added by extending the contours seaward, but rather than placing one-fourth of the sediment at each of the four contours, the volumes were determined based on the trapezoidal cross section shown in Figure 17. This cross section more closely resembles the disposal mound formed by hopper dredging. The incremental values Figure 18 show, in planform, the extension of the contours to simulate the weekly sediment addition at the 8-foot contour. A schematic illustration of the sediment transported from the nourished region is presented in Appendix C. Nineteen percent of the sediment added moved directly onshore out of the control volume. b. Conclusions for the Movement of Disposed Sediment in the Vicinity of Oregon Inlet. The computer simulations, tempered with engineering judgment, demonstrate that between 15 and 35 percent of the material added to the 7- and 11-foot contours,or to the 7- 8- 9-, and 10-foot contours would be transported into the nearshore transport system during the first year. If the disposal process was continued, the system would approach steady state in terms of the volume of deposited material residing offshore. For the case of sediment addition at the 11- and 14-foot contours, the computer simulations, tempered with engineering judgment, show that between 5 and 25 percent of the material added would be transported into the nearshore transport system during the first year. 43 "Jaay-OL puke “6 ‘8 *Z Ze UOLANLOSaU ayy BION “fp BSED *(¢,,9994 GT°0=V) cv zAW=4 SbuLLapow zalLul uobaug ay} UL pasn a,tjJoud wntuqi{inba 40 uOLSuaA paddazys “g9T aunbL4 O¢€ Ss (a>) TS} c+ = ion O¢z. 8 (eo) = = = = =e 0! = an O OO0O0€ 0002 OOO! 0 (94) A SauoyssfO JOUe SLA 44 28.05 pct. 9.0 ft 10.0 ft VOL:35.16 pct. 11.0 ft Elevation Figure 17. Shore-perpendicular cross section of disposal mound. The volumes represent the volume per- centage of the trapezoidal section between contours and therefore, the quantity of sediment added to the 7-, 8-, 9-, and 10- foot contours. Contour Depth =| ft Increment 400 ft = width of grid OFFSHORE DIRECTION he 3 Figure 18. Incremental values of Ay due to dredge disposal, illustrated for the block between 8- and 9-foot contours (case 4). 45 4. Simulation of the Longshore Sand Transport Study at Channel Islands Harbor, California. The CIH Longshore Sand Transport Study (Bruno, et al., 1981) was the only field study found suitable for verification purposes. Wave data collected included the LEO data and a two pressure-sensor gage array. Although the pressure gages were not in operation throughout the study, it was expected that the data they produced would be superior to that of the LEO data. However, these data were not available in a reduced form, so the LEO data were used. An adjustment of 11° was made to the breaker angle to orient . the angle with respect to the base line, rather than to the local shoreline orientation angle. Observations had been taken twice daily at three locations; the middle location was used (observer No. 5714). Waves which approached the shoreline at angles too large to have originated in a depth of 10 meters, according to Snell's law, were set equal to 90° at that depth (crest of wave perpendicular to the baseline). The 10-meter depth was chosen because it is the approximate depth at the tip of the offshore breakwater (for this reason, it was also chosen as the depth of the step beyond the y(I, JMAX + 2)th contour). It was assumed that each of the two daily observations occurred for 12 hours and using a time-step of 6 hours, this meant two time-steps per wave. In cases where parts of the wave data (Hp, ab, or T) were missed by the observer or were equal to zero, the data were ignored (no computations were made), but the time was included. Because the time rate of change is important for this simulation, the variation of Corr outside the break point was used. The period chosen to model was 20 April through 1 December 1976. The initial survey was taken after dredging of the sediment trap and for this reason was known to be out of equilibrium. The bathymetric surveys were con- ducted using several methods, the most advanced being a Lighter Amphibious Resupply Cargo vessel (LARC) proceeding along shore-perpendicular lines (approximately in the vicinity of each survey station) taking fathometer readings every 10 seconds, with positioning systems trilaterating the vessel's position concurrently. These data were recorded on tape. The beach-face data were taken using standard surveying methods. Because the data fluctuated randomly about the stations, depending on the speed of the craft, the (x, y) coordinate positions had to be altered to fixed changes in x and y. This was accomplished using an interpolation routine. The x values were made to coincide with the stations used in the surveys, and the y values were determined at 100-foot intervals beginning from the base line. Stations 100+00 and 118+00 were located at the north jetty and termination of the detached breakwater, respectively (these correspond to I values of 16.5 and 34.5 in the model). See Figure 19. Monotonic profiles of the form h = Aly - yde1)2/3 were fit to the data along each station line. "ydel" represents the zero location of the fitted shoreline, the value of which was unknown. Because dredging had been done in the lee of the breakwater, there was no reason to expect the A value to correspond to the value upcoast where the influence of the structure and the dredging was negligible. For this reason, the profiles of Stations 122+00 through 134+00 were evaluated separately to determine an A value for the equilibrium profile to be used in the numerical model. For the detailed method used (LaGrange Multipliers and Newton-Raphson Method for nonlinear 46 es [ea fe) 800 ft Offshore Breakwater Jetty Approximate Shoreline After Dredging PROTOTYPE Station 100+00 118 + 00 North ‘spur’ Jetty Approximate Shore li-ne After Dredging -% ? > eee ete t ee IDEALIZED Figure 19. Idealized numerical model representation of offshore breakwater at Channel Islands Harbor, California. 47 equations) and the computer programs see Appendix D. The two values obtained for the surveys of 20 April and 1 December 1976 were averaged to obtain the value used in the model, A = 0.2606. Stations 101+00 through 121+00 were treated separately for the purpose of obtaining values with which to initial- ize those parts of the contours in the model and for comparison of the model predictions with the prototype values. Note that although the breakwater extends only to about Station 118+00, the influence of the structure and dredging extends beyond that location and so, although arbitrary, the 121+00 station was chosen as the dividing line. The initial and final values of the scaling parameter A for the profiles were 0.3233 and 0.3528, respectively. Because the initial shoreline is so irregular, a discontinuity between 121+00 and 122+00 is not evident. One further idealization was made. The jetty-breakwater system was idealized as shown in Figure 19. This was required to simplify the physical situation, and although waves, currents, and sediment do pass through the opening in the prototype, it is hoped that they are of secondary importance. The results of the numerical modeling of Channel Islands Harbor are presented in Figures 20 and 21. The first figure presents the shoreline contour (depth = 0); the second figure presents the farthest offshore, modeled contour. In both cases, the initial shoreline represents the model and prototype (after fitting of the profiles). The initial shoreline contour is further offshore along the section of beach beyond the end of the breakwater, while in the lee of the breakwater, as would be expected after dredging, the shoreline is closer to the base line. The final prototype contour has undergone erosion along the reach beyond the tip of the structure, and accretion in the ‘ee. The model's shoreline contour has undergone similar changes, and on the average, represents the final prototype contour quite well. The JMAXth contour has been displaced quite similarly to the shoreline contour with shoreward movement (erosion) along the reach beyond the tip of the breakwater and seaward movement (accretion) within. It appears that the final model's shoreline has predicted too much erosion and not enough accretion. Several parameters could be incorrect, with the onshore-offshore sediment transport rate coefficient, Corff, perhaps the most likely. Overall, the model seemed to predict reasonable values of the contours. V. SUMMARY AND RECOMMENDATIONS Some of the parameters that the model does not include are important and should be mentioned. As stated previously, the model does not include bar formation. This is precluded by an n-line system. There are no provisions for water level fluctuations or currents. Improvement to the model could also be facilitated with better longshore and cross-shore sediment transport relationships. A more reliable equation for distribution of sediment transport across the surf zone would also be helpful (or further testing and calibration of the equation proposed herein). Finally, combining refraction and diffraction using equations to predict their combined effect would improve the wave field. The program was constructed such that improvement 48 OQ ss YD ot Mw —- cI OS Oar a) a) S&S) MK 1600 1400 1200 1000 Final (Model) 800 Initial (Model and Prototype) 200 p Final (Prototype) I= 18 20 25 30 35 40 45 50 Longshore Direction, x Figure 20. CIH simulation of shoreline contour, 20 April - 1 December 1976 (from LEO data). 49 MEO SVects Nea Mus OBS: Os—he— ho. «< 1600 1400 Final Initial Oo (Prototype) (Mode! and Prototype) 800 Final Final (Model) (Prototype) Initial 200 17 20 25 30 35 40 45 50 Lonashore Direction, x Figure 21. CIH simulation of (JMAX)t" contour, 20 April - 1 December 1976 (from LEO data). 50 could be accomplished with minimum effort. Therefore, if a more suitable equation becomes available, the change of a subroutine should be sufficient for implementation of the equation. Although the model is limited by the omission of the aforementioned parameters, it is reasonably correct. The ability to simulate various physical situations (shore-perpendicular structures, beach fills, breakwater and shore-perpendicular structures) has been demonstrated. In the CIH simulation where the data were first transformed to monotonically decreasing contours and LEO wave data were used, the model still predicts the prototype shoreline changes in a reasonable fashion. Further research and model development should include exercising the model in a number of different situations. Several theoretical cases should be simulated, which if analyzed properly, would provide a tool for the coastal engineer. Combined refraction and diffraction should be included, if possible, along with any of the aforementioned parameters which have been omitted and for which relationships exist. Perhaps the most difficult prob- lem to researchers working on modeling sediment transport in the vicinity of structures is the availability of field data. High-quality concurrent wave and bathymetric change data in the vicinity of coastal structures do not exist. One suggested field experiment is to monitor changes both updrift and downdrift of a jettied inlet which has a bypassing plant. Monitoring should begin immediately after bypassing, when the profiles are out of equilibrium. The recorded bathymetric and wave data would then provide data with which to calibrate, verify, and evaluate the existing models. 5| LITERATURE CITED ABBOTT, M.B., Computational Hydraultes, Pitman Publishing Ltd., London, 1979. ABRAMOWITZ, M., and STEGUN, I., eds., Handbook of Mathematical Functions, Dover Press, 1965. BAKKER, W.T., "The Dynamics of a Coast with A Groyne System," Proceedings of the 11th Conference on Coastal Engineering, American Society of Civil Engineers, 1968, pp. 492-517. BRUNO, R.O., et al., “Longshore Sand Transport Study at Channel Islands Harbor, California," TP 81-2, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., Apr. 1981. DEAN, R.G., "Equilibrium Beach Profiles: U.S. Atlantic and Gulf Coasts," Ocean Engineering Report No. 12, University of Delaware Press, Newark, Del., 1977. DRAGOS, P.A., "A Three Dimensional Numerical Model of Sediment Transport in the Vicinity of Littoral Barriers," M.S. Thesis, University of Delaware, Newark, Del., 1981. FULFORD, E., "Sediment Transport Distribution Across the Surf Zone, M.S. Thesis, University of Delaware, Newark, Del., 1982. GABLE, C.G., "Report on Data from the Nearshore Sediment Transport Study Experiment at Torrey Pines Beach, California, Nov.-Dec. 1978," Institute of Marine Resources IMR No. 79-8, Dec. 1979. KOMAR, P.D., The Longshore Transport of Sand on Beaches, Ph.D. Dissertation, University of California, San Diego, Calif., 1969. KOMAR, P.D., and INMAN, D.L., “Longshore Sand Transport on Beaches," Journal of Geophystcal Research, Vol. 75, 1970, pp. 5914-5927. LeBLOND, P.H., "On the Formation of Spiral Beaches," Proceedings of the 15th Conference on Coastal Engineering, American Society of Civil Engineers, 1972, pp. 1331-1345. LeMEHAUTE, B., and SOLDATE, M., "Mathematical Modeling of Shoreline Evolution," MR 77-10, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., Oct. 1977. LONGUET-HIGGINS, M.S., "Longshore Currents Generated by Obliquely Incident Sea Waves, I, II," Journal of Geophysical Research, Vol. 75, 1970, pp. 6778-6801. MOORE, B., "Beach Profile Evolution in Response to Changes in Water Level and Wave Height," M.S. Thesis, University of Delaware, Newark, Del., 1982. NODA, E.K., "Wave Induced Circulation and Longshore Current Patterns in the Coastal Zone," Tetra-Tech No. TC-149-3, Sept. 1972. 52 PENNY, W.G., and PRICE, A.T., “The Diffraction Theory of Sea Waves and the Shelter Afforded by Breakwaters," Philosophical Transactions of the Royal Soctety, Series A, 244, Mar. 1952, pp. 236-253. PERLIN, M., "A Numerical Model to Predict Beach Planforms in the Vicinity of Littoral Barriers, M.S. Thesis, University of Delaware, Newark, Del., 1978. PRICE, W.A., TOMLINSON, K.W., and WILLIS, D.H., "Predicting Changes in the Plan Shape of Beaches," Proceedings of the 13th Conference on Coastal Engtneering, American Society of Civil Engineers, 1972, pp. 1321-1329. REA, C.C., and KOMAR, P.D., "Computer Simulation Models of a Hooked Beach Shoreline Configuration," Journal of Sedimentary Petrology, Vol. 45, No. 4, Dec. 1975, pp. 866-872. SAVAGE, R.P., “Laboratory Study of the Effect of Groins on the Rate of Lit- toral Transport: Equipment Development and Initial Tests," TM 114, U.S. Army, Corps of Engineers, Beach Erosion Board, Washington, D.C., June 1959. 53 ee ia APPENDIX A DISCUSSION OF CONSTANTS AND SOME OF THE VARIABLES REQUIRED BY THE MODEL Establishing the grid-contour system requires several variables. IMAX represents the number of cross-shore grid lines desired and JMAX the number of contours simulated. DX represents the spacing between the IMAX grid lines and DY the spacing between the contours. DX is a value which must be chosen along with IMAX and JMAX such that sufficient detail is obtained where necessary (e.g., in the shadow zone, if diffraction effects are believed to be very important, DX must be assigned a sufficiently small value so that at least some points lie within the shadow zone for the larger wave angles). DY is not a constant, but a dimensional array which is computed by the model according to the contour location. Once the depths of contours to be modeled are chosen, the initialization of DY and the y values are computed with the following equation after Dean, 1977 h=A y2/3 GAS) where h is the depth, y is the offshore distance and A is the scaling parameter Dean gives values for A for several diameter sediments; however, if long-term beach profiles are available for the site being modeled, the modeler may want to choose a slightly different A value to more closely match the site-specific beach profile. Figure A-1 presents values of A versus diameter (after Moore, 1982). The model is programmed to input the h({I,J) values (depths as shown in Figure 1, called DEEP (I,J) in the program) read in the value of A (called ADEAN in program) and it then computes the y values. Also shown in Figure 1 is the height of the berm (BERM) and this value, along with the beach-face slope (SFACE), is required as program input and can be obtained from beach profile site data. Because the model does not include water level fluctuations such as tides, all values are to be referenced to a chosen datum. Other geometrical constants depending on the site include SJETTY (the length of the jetty), MMAX (the number of structures to be input), and IJET (M), M = 1,2,...MMAX (the smaller I value adjacent to the mth structure's location). If no structure is required, as in a beach fill, the value of SJETTY must be entered as 0.0, with MMAX and IJET (M) entered as 1 and (IMAX/2), respectively. As set up presently, the groin locations must be equally spaced. One constant used throughout the program is the breaking wave criteria (CAPPA in the program) equal to 0.78. It is required in several different computations and always governs the maximum wave height allowed according to the depth. Another group of variables assigned values within the program is the sediment and fluid properties. These include fluid mass density, sediment mass density, porosity, and the angle of repose (e.g., RHO = 1.99, RHOS = 5.14, POROS = 0.40, and REPOSE = 32°, respectively). The values can easily be changed to reflect site conditions. SIS) 0-00! "(2861 ‘a400W 49zJe) WeJOWeLP ZUBWLpasS SNSuar Y (wu) UazoWeL] PUSWLPaS O'Ol O'l “T-y aunbey 56 Another very important set of constants is the constant chosen for the longshore and cross-shore components of sediment transport. Equation (27), the total longshore transport equation, contains the constant C' equal to ant yl/2 Gu oy (A-2) ib. 2a) Se) Ge 2 K Ps p where K = 0.77 (Komar and Inman, 1970) g is the acceleration of gravity (32.17 ft/sec2) pg and p are the mass densities of the sediment and the seawater (5.14 and 1.99 slugs per cubic feet, respectively p is the porosity (0.40), and K 7s taken as 0.78. Using these values to compute C' (TKSI in the program), a value of 0.325 is obtained. It is stressed that if any of these values are different for the site to be modeled, they should be changed and the program will compute another value for C’'. The parameter Corf is an “activity factor" which, based on earlier work primarily within the surf zone, was found to be eyed Corr = 10s sht/se h hy (A-9) 1 Cao° 3 paageieneiel f H x 10° (A-10) OFF 5r 3/2 2,3/2, \sinh kh GivtaicsAnaan in which fT is a parameter relating the efficiency with which breaking wave energy (which occurs primarily near the water surface) mobilizes the sediment bottom (0 < r <1). Herein, © is taken to be one. Figure A-2 presents an example of the variation of the activity coefficient versus relative depth for a particular wave period and deep water wave height. It is seen that the activity coefficient reduces rapidly with increasing depth. The value of Corr for the physical modeling of Savage's (1959) data was taken as 10-4 feet per second. Perlin (1978) presents some rationale for choosing a value of Corr; however, very little testing has been done and none is based on actual field measurement. 58 “SUOLZLPUOD BAPM UE{NILJued UOJ ‘u Syqdap uazem snsuan 449 *quarsijya0o ATLALZI yO aldwexy °Z-y aunbl4 (44)4 “U4daq 4910M S| Ol S O gl x#4°9 @U0Z J4NS apis}noO z2'0 >bO HES) 10°0 ='9 9°0 s@g@=lL 4) © =H 8°0 O'l auoZ yins 59 Finally, wave data are read into the program and the simulation begins. (For information regarding "Read Formats" for the various constants and variables, see Appendix E). Wave data required are wave height, wave period, wave angle relative to the x-axis of the model at a depth, WDEPTH and the duration of the wave climate (HS, T, ALPWIS, and a combination of NTIMES x DELT, respectively, in the model). As is always the case with numerical models, the time step and space steps are very important to both stability and accuracy. Time steps on the order of 3 to 6 hours (10,800 to 21,690 seconds) or less are recommended. However, the complexity of the bathymetry, variation and time series of the wave data, constants used (especially Corr) along with several other factors, greatly influences the stability and accuracy of the results. Table A-1 lists several of the important variables in the computer program. Table A-1. List of important variables in the program ABAND The input banded matrix which stores the values from equation (37) ADEAN The value of the scaling parameter in the equilibrium beach profile ALPHAS The angle a contour makes with the x-direction base line (counter-clockwise is positive) ALPWIS The angle (-90° to +90°) the wave crest makes with the x-direction (counter-clockwise is positive) AMP The amplitude of the diffracted wave in the shadow zone ANGGEN The wave angle at a depth, WDEPTH ANGLOC The local contour orientation angle AWARE See equations (36) and (37) BERM The height of the berm above water level BMATRX The matrix which, upon solution of the banded matrix problem yields the new y values C The wave celerity CAPPA The breaking wave index cc Constant which establishes the width of the distribution of sediment transport across the surf zone CG The group velocity throughout the wave field CGEN The linear wave theory celerity at a depth, WDEPTH 60 CGGEN co COFF CONST CONST6 DEEP DEEPB DEEPBI DELT DIAM DISTR DX DY HBQ The linear wave theory group velocity at a depth, WDEPTH The deepwater, linear wave theory wave celerity The onshore-offshore transport rate coefficient within the surf zone The constant in the longshore sediment transport relationship (Osu) The space step, DX, multiplied by the activity coefficient The water depth at any grid location The initial breaking depth along each profile (between adjacent profiles) The initial breaking depth along each profile (at each profile, rather than between them) The time-step in seconds (DELT x NTIMES = wave condition duration) The mean diameter of the sediment particles See equations (36) and (37) The alongshore space-step in the x-direction (distance between I values) The onshore-offshore space-step in the y-direction as defined by the stepped profile The deepwater, linear wave theory wavelength The wavelength at the tip of the structure The change in the wave number which is acceptably small The acceleration of gravity (32.17 feet/second2) The specific weight of seawater The wave height throughout the wave field The maximum wave height which could exist throughout the wave field (where H = 0.78 * h) The initial breaking wave height along any profile at the y values rather than between them The initial breaking wave height along any profile, between adjacent profiles 61 HGEN Average wave height at a depth, WDEPTH HS The significant wave height input I The longshore grid location IBREAK The leeward side of the initial breaker location J value IJET Represents the lesser I value adjacent to the structure (these must be evenly spaced alongshore) IMAX The total number of grid points in the x-direction (alongshore) J The offshore contour location JMAX The value of the seawardmost contour simulated JUSE (JMAX + 2) the seawardmost contour at which the wave field is calculated Jl Landward contour of refraction zone J2 Seaward contour of refraction zone J1REF Landward J values of boundary of refraction zone J2REF Seaward J values of boundary of refraction zone MMAX The number of shore-perpendicular structures to be simulated (present maximum of 16) NITER The counterindex in the refraction routine NTIME The counterindex in the time simulation "DO" loop NTIMES The number of iterations of time-step, DELT, for which a particular wave is simulated NUNIV The total number of time-steps simulated at any time PI The value of 7 = 3.141592654 POROS The porosity of the sediment QX The longshore sediment transport rate at a specific location QXTOT The total alongshore sediment transport rate due to the height and angle of the initial breaking wave QY The onshore-offshore sediment transport rate at a specific location R See equations (36) and (37) 62 REPOSE The angle of repose of the sediment RHO The mass density of seawater RHOND The dimensionless distance from the tip of structure where diffraction is initiated RHOS The mass density of sediment RK The wave number $3 See equations (36) and (37) SFACE The slope of the shoreface SJETTY ie ies of the shore-perpendicular structure (from the base ine SIGMA The wave radian frequency Tf The wave period TAU The dissipative interface parameter THETA The wave angle throughout the wave field THEATO The wave angle at the tip of the structure TKSI The longshore sediment transport rate coefficient TWOP I Twice the value of U See equations (36) and (37) UCRIT The critical velocity required to move the sediment according to the Sheid's diagram V See equations (36) and (37) WDEPTH The depth of water in meters to which the input wave conditions are to be transformed WEQ The equilibrium profile distance between contours as defined by the stepped profile XCOOR The x-coordinate where the wave field is to be calculated. Together with YCOOR, they determine whether the position is within or beyond the diffraction shadow zone XDISTN The location of the structure along the shoreline in feet Y The distance offshore to the contours 63 YCOOR The y-coordinate where the wave field is to be calculated. Together with XCOOR, they determine whether the position is within or beyond the diffraction shadow zone YDISS The value of y after the use of the dissipative interface YOLD The previous value of y YZERO The berm contour location Z1 See equation (37) Z2 See equation (37) 64 APPENDIX B PROGRAM LISTING 65 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000 5100 5200 5300 5400 5500 5600 5700 5800 5900 6000 6100 6200 6300 6400 6500 6600 6700 6800 6900 7000 7100 7200 C* *** KK KKH PROGRAM IMPLICIT SEDTRAN C*THIS PROGRAM IS SET-UP TO HANDLE MULTIPLE GROINS(M<=10). COMMON/A/ C(60,20),RK(60,20),Y(60, 20) ,DEEP(60, 20) ,ALPHAS(6O, 20) COMMON/AA/YZERO(60) COMMON/BB/WEQ(6O0, 20) COMMON/B/ THETA(6O,20),QXTOT(60), OLDANG(60,20), DY(60,20) COMMON/C/ H(60,20),CG(60, 20) ,HOLD(60, 20) ,HB(60, 20) , YB(60) COMMON/N USED/JUSE,T,CO,CGEN, CGGEN, ANGGEN, DX ,BERM, THETAO( 10) , MMAX COMMON/D/SIGMA,G,ELO, UMAX, IMAX,PI, TWOPI ,PIO2,HGEN, IJET(10),SUJETTY COMMON/F/ADEAN, REPOSE,DIAM COMMON/AAA/DELT ,NTIMES COMMON/COUNT /NUNIV COMMON/EXPL/QYEXP(60, 20), YIMP(60, 20) DIMENSION CHANGE(20),HC(10),TC(10) DIMENSION YORIG(60, 20), YZEROO(60) ,SANGLE(20) NUNIV=O JMAX=8 JUSE=JMAX+2 IMAX=50 PI=3.141592654 TWOPI=PI1*2. PIO2=PI/2.0 REPOSE=32.*TWOPI/360. WRITE(6,732) 732 FORMAT ( TORO OR ORO IO OO OF I OI OI IOI OR a ok rk tk doko 1) WRITE(6,733) 733 FORMAT(2X,’TO WHAT DEPTH ARE THE WAVES TO BE TRANSFORMED’ ) C*WDEPTH MUST BE A DEPTH BEYOND THE END OF THE STRUCT, PREFERABLY AT C**DEEP(JUMAX) OR GREATER(OR ELSE SNELL’S LAW OR SHOAL COULD BLOWUP IN C***DEEPER WATER. IT’S IN METERS HERE! READ(5,770) WDEPTH 770 FORMAT(10X,F 10.3) WDEPTH=WDEPTH*3. 28084 WRITE(6,762) WDEPTH 762 FORMAT(2X,"THE DEPTH (IN FT) WAVES TRANSFORMED TO, WDEPTH= ". * F10.3) WRITE(6,732) WRITE(6,777) 777 FORMAT(2X,"ITS TIME FOR SJETTY, BERM, SFACE, AND DIAM",/) C*SJETTY MUST BE MUCH LESS THAN Y(I,JUMAX). READ(5,776) SUETTY,BERM, SFACE,DIAM 776 FORMAT(2F10.3,F10.4,F10.3) WRITE(6,761) SUETTY 761 FORMAT(2X,’THE LENGTH OF THE STRUCTURE, SJETTY= ’,F10.3) WRITE(6,740) BERM 740 FORMAT(2X,’THE HEIGHT OF THE BERM, BERM= ‘,F10.3) WRITE(6,739)SFACE 739 FORMAT(2X,’THE SLOPE OF THE BEACH FACE, SFACE= ’,F10.4) WRITE(6,738) DIAM 738 FORMAT(2X,’THE SEDIMENT DIAMETER, DIAM= ’,F10.3) WRITE(6,732) 780 FORMAT(2X,’SUPPLY MMAX( THE NO. OF GROINS) AND THEIR I-LOC’,/) UCRIT=16.3*SQRT (DIAM*O.00328) C*THE NO. OF MULTIPLE GROINS,MMAX MUST BE GIVEN THEIR X LOCATIONS. READ(5,779) MMA X 779 FORMAT(I3) DO 760 M=1,MMAX C*IJET REPS LESSER I-VALUE ADJACENT TO STRUCTURE. 760 READ(5,779) IJET(M) WRITE(6,759) (M,IJET(M),M=1,MMAX) 759 FORMAT(2X,’THE NUMBER’,I5,’ GROIN IS LOCATED AT GRID’,I5) WRITE(6,732) C*CONVERT TO RADIANS. C*FIRST MUST GIVE Y COORS POSITIONS AND DEPTHS. C*FIRST, MUST SET UP ALL OF THE DEEP-VALUES. WRITE(6,773) 773 FORMAT(2X,"NOW ENTER THE VALUE OF ADEAN") READ(5,774)ADEAN 774 FORMAT(F10.4) WRITE(6,749) ADEAN 749 FORMAT(2X,’THE VALUE OF ADEAN= ’,F10.4,’ IN THE EQ. H=AY**2/3’) WRITE(6,732) 66 7300 WRITE(6,772) 7400 772 FORMAT(2X,"READ IN THE SPACE STEP,TIMESTEP",/) 7500 READ(5,775) DX ,DELT 7600 775 FORMAT(2(F10.3)) 7700 WRITE(6,737) DX 7800 737 FORMAT(2X,’THE VALUE OF THE LONGSHORE SPACE-STEP, DX= ’,F10.3) 7900 WRITE(6,736) DELT 8000 736 FORMAT(2X,’THE TIME-STEP IN SECONDS, DELT= ’,F10.3) 8100 DATWARCHANGE/Hm 2 Se Seite dis 6 1405 4725) 39" BOs) 1O*xO.0/, 8200 DO 220 J=1,UJUMAX+2 8300 DO 220 I=1,IMAX 8400 220 DEEP(I,JU)=CHANGE (J) 8500 DATA(HC(1),1=1,8)/1.87,0.5,0.35, .25, .21,.20,.19,.19/ 8600 DATAGICCID IM 8))),2os 4p Ga Olin ioeued|4ee/) 8700 DO 200 J=1,JUMAX+2 8800 DO 200 I=1, IMAX 8900 200 Y(1I,U+1)=(0.5*(DEEP(I,JU+1)+DEEP(I,U))/ADEAN) **1.5+Y(I,1) 9000 WRITE(6,732) 9100 Co Fe tee OK KOK 9200 C*WE WILL ALWAYS REQUIRE Y(I,JMAX+2) TO COMPUTE DY AND YBAR. 9300 C*WE WILL ALWAYS REQUIRE DEEP(I,JMAX+2) TO COMP SEDIMENT TRANSPORT. 9400 C % Ke ee eK ee eK OK 9500 WRITE(6,734) 9600 734 FORMAT(2X,’THE BOUNDARY Y-VALUES, I=1,IMAX ARE AS FOLLOWS’ ,/) 9700 WRITE(6,801) (Y¥(1,JU),J=1, UMAX+2) 9800 WRITE(6,801) (Y CIMAX,JU),J=1, UMAX+2) 9900 WRITE(6,732) 10000 WRITE(6,735) 10100 735 FORMAT(/,2X,’THE DEPTHS BETWEEN CONTOURS ARE AS FOLLOWS’ ,/) 10200 WRITE(6,801) (DEEP(1,JU),J=1, UMAX+2) 10300 WRITE(6,732) 10400 801 FORMAT(2X,10(F8.2)) 10500 DO 2 I=1,IMAX 10600 2 YZERO(1I)=Y(I,1)-(BERM/SFACE) 10700 C*WILL COMPUTE THE EQUIL WIDTH BETWEEN CONTOURS, HERE. 10800 DO 1 I=1,IMAX 10900 WEQ(I,1)=Y(1,1)-YZERO(I) 11000 DO 1 J=1,JMAX 11100 IF(JU.NE.1) GO TO 32 11200 YTEMP1=0.0 11300 GO TO 33 11400 32 YTEMP1=((0.5*(DEEP(I,JU-1)+DEEP(I,U)))/ADEAN)**1.5 11500 33 YTEMP2=((0.5*(DEEP(1I,J)+DEEP(1I,U+1)))/ADEAN) **1.5 11600 WEQ(1I,JU+1)=YTEMP2-YTEMP 1 11700 4 CONTINUE 11800 C*LET’S STORE THE ORIG VALUES TO COMPUTE VOL CHANGES OVER CONTOURS,LATER 11900 DO 796 I=1,IMAX+1 12000 YZEROO(I)=YZERO(I) 12100 DO 796 J=1,JMAX+2 12200 796 YORIG(I,J)=Y(I,u) 12300 CEO OR III I I IOI II III ICICI I aC ICR kok eke kee ae ak a 12400 C*READ THE DISK FILE AND TRANSFORM PARAMETERS INTO MY UNITS. 12500 Co ¥ RO RK KR KK | ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 1 ! ! ! ! ! ! ! ! | eo KK EK KK KK 12600 C*ALL ADJUSTMENTS TO WAVE ANGLE,HEIGHT,CELERITY,GROUP VEL, WILL BE MADE 12700 C**HERE, AND THRUOUT THE REST OF THE PROG, THEY WILL BE AS IF OCCURRED 12800 C***AT WDEPTH! 12900 798 READ(5,799, END=1000) HS,T,ALPWIS 13000 799 FORMAT(10X,3F6.1) 13100 NTIMES=1 13200 NCHECK=NUNIV+NTIMES 13300 HGEN=O. 707107 *HS 13400 SIGMA=TWOPI/T 13500 G=32.17 13600 CO=G*T/TWOPI 13700 ELO=CO*T 13800 IF(T.LE.2.0) GO TO 797 43900 HCC=0.23 14000 DO 444 I=2,7 14100 T2=TC(I) 14200 IF(T.GT.T2) GO TO 444 14300 T1=TC(1-1) 14400 DELTAT=T2-T1 67 14500 14600 14700 14800 14900 15000 15100 15200 15300 15400 15500 15600 15700 15800 15900 16000 16100 16200 16300 16400 16500 16600 16700 16800 16900 17000 17100 17200 17300 17400 17500 17600 17700 17800 17900 18000 18100 18200 18300 18400 18500 18600 18700 18800 18900 19000 19100 19200 19300 19400 19500 19600 19700 19800 19900 20000 20100 20200 20300 20400 20500 20600 20700 20800 20900 21000 21100 21200 21300 21400 21500 21600 DT=(T-T1)/DELTAT DTT=(T2-T)/DELT HCC=HC(1)*DT+HC( 1-1) *DTT GO TO 446 444 CONTINUE 446 CONTINUE IF(HGEN.LT.HCC) GO TO 797 ANGGEN=ALPWIS*TWOPI/360. CC FO I I II I OI I OR 2 IK aK oe ok CALL WVNUM(WDEPTH, T, DUMKK) C*ANGGEN,HGEN,CGEN,CGGEN REPRESENT THE WAVE ANGLE,HEIGHT,CELERITY AND C**GROUP VEL(RESPECT.) OF THE SPECIFIED WAVE INPUT AT A DEPTH, WDEPTH CALL WVNUM(11.0,T,DUMKKK) C11=TWOPI/(T*DUMKKK ) CG11=0.5*C11*(1.+(2. *DUMKKK*11.0/SINH(2.*DUMKKK*11.0))) CGEN=TWOPI/(T*DUMKK) CGGEN=0.5*CGEN*(1.+(2.*DUMKK*WDEPTH/SINH(2.*DUMKK*WDEPTH) ) ) CALL TRANS 797 IF(NCHECK.NE.NUNIV) NUNIV=NCHECK 709 GO TO 798 1000 CONTINUE STOP END Co KR ee Eo oo oo a oa oo eo oo I oo KK 2 KK oR RK eK ok Kk OK SUBROUTINE TRANS C*THIS SUBROUTINE WILL COMPUTE SEDIMENT TRANSPORT COMMON/A/ C(60,2C),RK(60,20),Y(60, 20) ,DEEP(60, 20) , ALPHAS(60, 20) COMMON/AA/YZERO(60) COMMON/BB/WEQ(60, 20) COMMON/B/ THETA(60,20),QXTOT(60), OLDANG(60,20), DY(60,20) COMMON/C/ H(60,20),CG(60, 20) ,HOLD(60, 20) ,HB(60, 20), YB(60) COMMON/N USED/JUSE,T,CO,CGEN, CGGEN, ANGGEN, DX ,BERM, THETAO( 10) , MMAX COMMON/D/SIGMA,G,ELO, JMAX, IMAX,PI, TWOPI ,PIO2,HGEN, IJET( 10), SUETTY COMMON/E/RHO,RHOS,POROS,CONST,TKSI COMMON/F/ADEAN, REPOSE,DIAM COMMON/G/IBREAK(60) , HNONBR( 20) COMMON/P/HBQ(60) , DEEPB(60) COMMON/ZZZ/NTIME COMMON/AAA/DELT,NTIMES COMMON/COUNT/NUNIV DIMENSION YOLD(60, 20),R(60,20),S(60, 20) ,HC(60, 20) ,QY(60, 20), YDISS( * 60,20) DIMENSION RHS1(60, 20) ,S3(60, 20), THETAB(60, 20) , ANGLOC(60, 20) DIMENSION DISTR(60, 20) ,AWARE(60, 20 CE OK eo ko I a I I Ra IC ko ke ake ke ok ok FIC ok ok oo oko akc ok ke ake ok ok ke ke ok ok ke ke a ke ake ake ok ake ake ke ok ake Co ee eo ke ko i ok ke oko ok 2 oo ie ke ke kk kok oko EE oe ke keke ke ake ke ke ok ke ok oe ok oe oe ok oe ok ok ote ote ote ote oe oe ake ake ake ake ke ke ke CR eee ee ee ee NO ITE: : SIZE OF ABAND AND XL HAVE TO BE CHANGED CRE OR ERE eee Le ACCORDING TO JMAX+1+JMAX AND JMAX+1,RESPECT. CORSE EAE AE RE SEICOE AE REAR BERS CHANGE REQ’D AT 7040 AND 18650 CK I ok OK I 2K a aK ook koko ook ok ok ok oka oko kok oe oe ok ok ok oe ok ok ok ok ke kok ke ke oe ke * ), BMATRX (432), ABAND(432, 19) ,QX(60, 20) ,XL(432, 10) , CONST6(60, 20) COMMON/MP/ RKB(60),HBI(60) ,DEEPBI(60) COMMON/EXPL/QYEXP(60, 20), YIMP(60, 20) DIMENSION SANGLE(20) C*LET’S ZERO-OUT ALL OF THE DIMENSIONED MATRICES. DO 1000 J=1,UMAX+2 SANGLE(JU)=0.0 DO 1000 I=1,IMAX+2 YOLD(I,JU)=0.0 R(I,JU)=0.0 S(I,JU)=0.0 HC(1I,JU)=0.0 QY(I,JU)=0.0 YDISS(I,U)=0 RHS1(1I,U)=0. S$3(1,JU)=0.0 THETAB(I,U) ANGLOC(I,Ju) DISTR(I, u)= AWARE(I,J)= QX(I,JU)=0.0 CONST6(I,JU)=0.0 O. oO. 68 21700 21800 21900 22000 22100 22200 22300 22400 22500 22600 22700 22800 22900 23000 23100 23200 23300 23400 23500 23600 23700 23800 23900 24000 24100 24200 24300 24400 24500 24600 24700 24800 24900 25000 25100 25200 25300 25400 25500 25600 25700 25800 25900 26000 26100 26200 26300 26400 26500 26600 26700 26800 26900 27000 27100 27200 27300 27400 27500 27600 27700 27800 27900 28000 28100 28200 28300 28400 28500 28600 28700 28800 28900 4000 CONTINUE RHO=1.99 RHOS=5. 14 POROS=0.40 CONST=0.77 CAPPA=0.78 TAU=0.25 TKSI=(CONST*RHO*SQRT(G) )/((RHOS-RHO)*(1.O-POROS)*16.0*SQRT(CAPPA) C* QX(I,JU) IS THE TRANSPORT BETWEEN THE (1I,JU+1) AND (I,J) CONTOURS. C*THE ‘DO 1 LOOP’ SIMULATES TIME---TIME=DELT*NTIMES. COFF=0.00001 GAMMA=RHO*G DO 1 NTIME=1,NTIMES NUNIV=NUNIV+1 C*THE MATRICES ABAND AND BMATRX MUST BE "ZEROED OUT" K=O DO 26 I=2,IMAX-1 DO 26 J=i,JUMAX K=K+1 BMATRX(K)=0.0 DO 26 L=1,JUMAX+1+JMAX 26 ABAND(K,L)=0.0 XNT IME=1.0*(NTIME) CALL PREDIF C*SMOOTHING OF THE WAVE ANGLE,THETA, IS RE’D TO ACCT FOR DIFF EFFECTS. CALL SMOOTH( THETA, IMAX, UMAX, IJET, SUETTY ,MMAX,Y) CALL QTRAN C*FIRST THE LONGSHORE SEDIMENT TRANSPORT WILL BE DISTRIBUTED C****ACROSS THE SURF ZONE.... CC=Hh25 C***QX(I,U) WILL BE DETERMINED BY SUBTRACTING FROM THE INTEGRAL C**OF QX FROM DEEP(I,J-1) TO INFINITY, THE INTEGRAL OF QX FROM DEEP(I,u) C***TO INFINITY. IN THIS WAY THE SEDIMENT TRANS FROM JMAX OUT GETS C***INCLUDED IN QX(I,JMAX). TO INCLUDE THE SWASH TRANS, WHEN J=1 C*WE WILL SUBTRACT FROM 2 TO INFINITY FROM 1.0 C*LOOP FOR VALUES WHICH ARE HELD CONST AND STORED. THETAB(1,1)=0.5*(THETA(1,1)+0.0) R(1,1)=0.5/(DX*(DEEP(1,1)+BERM/2.)) DO 290 I=2,IMAX R(1,1)=0.5/(DX*(DEEP(1,1)+BERM/2.)) c* THETAB(I,1)=0.25*(THETA(I,1)+THETA(I-1,1)+0.+0. ) THETAB(1I,1)=0.5*(THETA(I,1)+THETA(I-1,1)) C*NO NEED TO COMPUTE PROP ANGLE AT STRUCTS BECAUSE QX =0.0O AT IJET(M)+1 ANGLOC(I,1)=ATAN((Y(1,1)-Y(1I-1,1))/DX) C*HBQ(IJET(M)+1) IS PROPERLY SET IN THE SUBROUTINE QTRAN. DISTR(I,1)=1.0-EXP(-((DEEP(I, 1)**1.5+HBQ(1I)*ADEAN**1.5)/ * (CC*DEEPB(1)**1.5))**3) DISTR(I,1)=DISTR(I,1)*TKSI*HBOQ(I)**2.5 DO 280 J=2,JUMAX R(1,JU)=0.5/(DX*(DEEP(1I,U)-DEEP(I,U-1))) THETAB(I,JU)=O.5*(THETA(I,JU)+THETA(I-1,uU)) ANGLOC(I,J)=ATAN((Y(1I,JU)-Y(1I-1,U))/DX) DISTR(I,JU)=EXP(-((DEEP(I,JU-1)**1.5+HBQ(1I) *ADEAN**1.5)/(CC*DEEPB(1) * **4.5))**3)-EXP(-((DEEP(1I,JU)**1.5+HBQ(I) *ADEAN**1.5)/(CC* * DEEPB(I)**1.5))**3) DISTR(I,JU)=DISTR(I,JU)*TKSI*HBQ(1)**2.5 290 CONTINUE DO 301 J=1,JMAX DO 301 I=2,IMAX AWARE(I,JU)=DELT*R(I,JU)*(QX(I,JU)-QX(I+1,JU)+QY(1I,JU)-QY(I,U+1))+Y(I,U * ) S1=2.*SIN(THETAB(I,JU))*COS(THETAB(I,JU))*(-1.+2.*(COS( * ANGLOC(I,uJ)))**2) S2=COS(2.*THETAB(I,U))*COS(ANGLOC(1I,U))/(SQRT(DX**2+ * (Y(I,U)-Y(1-1,J))**2)) $3(1,JU)=S2*DISTR(I,u) IF(SUJETTY.EQ.0.0) GO TO 302 DO 325 M=1,MMAX IF(I.NE.IJET(M)+1) GO TO 325 IF(THETAO(M).GE.O.0) ISIDE=IJET(M) IF(THETAO(M).LT.O.0) ISIDE=IJET(M)+1 YSEA=0.5*(Y(ISIDE,J)+Y(ISIDE,uU+1)) YSHORE=0.5*(Y(ISIDE,JU)+Y(ISIDE,uU-1)) 69 29000 29100 29200 29300 29400 29500 29600 29700 29800 29900 30000 30100 30200 30300 30400 30500 30600 30700 30800 30900 31000 31100 31200 31300 31400 31500 31600 31700 31800 31900 32000 32100 32200 32300 32400 32500 32600 32700 32800 32900 33000 33100 33200 33300 33400 33500 33600 33700 33800 33900 34000 34100 34200 34300 34400 34500 34600 34700 34800 34300 35000 35100 35200 35300 35400 35500 35600 35700 35800 359300 36000 36100 36200 IF(YSEA.GT.SJETTY.AND.YSHORE.GT.SJETTY) GO TO 302 IF(YSEA.GT.SJETTY.AND.YSHORE.LE.SJETTY) GO TO 298 C*BECAUSE A NO FLOW B.C. IS USED ALONG THE STRUCT, NO ATTN WAS PAID C**TO GETTING PROPER VALUES OF ANGLOC, THETAB,DISTR,ETC. S$3(1,JU)=0.0 DISTR(I,JU)=0.0 GO TO 302 325 CONTINUE GO TO 302 C***ABOVE, ALL PARAMETERS(I.E.,S1,S2,S3, THETAB,DISTR,ANGLOC) C***ARE COMPUTED AS IF THE STRUCT IS NOT THERE. THE B.C. AT THE C***STRUCT TIP ASSUMES QX COMPUTED AS IF NO STRUCT PRESENT AND THEN C***BYPASSES ACCORDING TO "RATIO". 298 RATIO=(YSEA-SJETTY)/(YSEA-YSHORE ) S$3(1,JU)=S3(1I,JU)*RATIO DISTR(1I,JU)=DISTR(I,U)*RATIO 302 RHS1(I,U)=DISTR(I,JU)*S1-S3(1,U)*(Y(1,U)-Y(1-1,u)) 301 CONTINUE CALL BREAK(IMAX, JUMAX) C*TO DETERMINE DECAY OF CONST6(I,JU),NEED WAVE NO. AT BREAKING. DO 754 I=1,IMAX+1 754 CALL WVNUM(DEEPBI(I),T,RKB(I)) C*USING SHIELD’S DIAG,Y AXIS=0.05 & (TAUO=RHO*C*U**2),GET UCRIT(FT/SEC) UCRIT=16.3*SQRT(DIAM* .00328 ) DO 750 I=1, IMAX+14 CONST6(I, 1)=COFF*DX DO 750 J=2,UMAX+2 C*CONST6(1I,U) GOES W/ QY(I,JU) WHICH IS ASSOC W/ DEEP(I,uU-1) IF(DEEP(I,J-1).LE.DEEPBI(I)) GO TO 751 C*HERE, MUST CAUSE COFF TO DECAY (WE’RE BEYOND SURF ZONE) UMAXB=HBI (1 )*G*T*RKB(1I)/(2.*TWOPI*COSH(RKB(1I)*DEEPBI(I))) UMAX=H( I, JU-1)*G*T*RK(I,J-1)/(2. *TWOPI*COSH(RK(I,U-1)*DEEP(I,U-1))) IF(UCRIT.LT.UMAX.AND.UCRIT.LT.UMAXB) GO TO 749 CONST6(I,U)=0.0 GO TO 750 749 TOP=0.01*H(1I,JU-1)**3*SIGMA**3/(SINH(RK(I,J-1)*DEEP(I,JU-1))**3) BOT=DEEP(I,U-1)*(0.625*TWOPI*G**1.5*O.78**2*ADEAN**1.5+ *(0.01*HBI(1)**3*SIGMA**3/(DEEPBI(1I)*(SINH(RKB(I)*DEEPBI(1)))**3))) CONST6(1,JU)=DX*COFF*TOP/BOT GO TO 750 751 CONST6(1I,U)=COFF*DX 750 CONTINUE K=O C**PUT INTO BANDED FORM USING THE ALGORITHM A(M,N)->B(M,NN) WHERE C***NN=KB+1-M+N(KB IS THE NUMBER OF LOWER CODIAGONALS(=UMAX,HERE)). DO 304 I=2,IMAX-1 DO 304 J=1,UMAX K=K+1 AWARE(1,U)=AWARE(I,JU)+DELT*RHS1(1,JU)*R(I,U)-DELT*R(I,JU)*RHS1(I+1,J as )+DELT*R(I,U)*CONST6(1I,JU)*WEQ(1,JU)-DELT*R(I,JU)*CONST6(I,JU+1)* * WEQ(I,U+1) YDUM=YZERO(1) IF(JU.NE. 1) YDUM=Y(I,J-1) AWARE(I,U)=AWARE(1I,JU)+DELT*R(I,JU)*CONST6(I,JU)*O0.5*(YDUM-Y(I,U)) * -DELT*R(I,U) *CONST6(I,JU+1)*O0.5*(Y(1I,JU)-Y(1I,U+1)) U=DELT*R(I,JU)*S3(I,u) V=DELT*R(I,JU)*S3(I+1,U) Z1=DELT*R(I,JU)*CONST6(I,JU)*0.5 Z2=DELT*R(I,JU)*CONST6(I,U+1)*O.5 C*NOW WILL SET UP THE MATRICES ABAND AND BMATRX. ABAND(K, UMAX+1)=1.O0+U+V+Z1+Z2 IF(I.NE.2) GO TO 305 AWARE(1I,U)=AWARE(I,JU)+U*Y(I-1,U) GO TO 310 305 ABAND(K,1)=-U 310 IF(I.NE.IMAX-1) GO TO 306 AWARE(1,JU)=AWARE(I,JU)+V*Y(IMAX,J) GO TO 311 306 ABAND(K, JMAX+1+JMAX )=-V 311 IF(JU.NE.1) GO TO 307 ABAND(K, JMAX+1)=ABAND(K, UMAX+1)-Z1 AWARE(1,1)=AWARE(I,1)+Z1*(YZERO(I)-Y(I,1)) GO TO 312 70 36300 307 ABAND(K, JUMAX)=-Z1 36400 312 IF(U.NE.UMAX) GO TO 308 36500 AWARE(1,JU)=AWARE(1I,JU)+Z2*Y(I, UMAX+1) 36600 GO TO 309 36700 308 ABAND(K,JMAX+2)=-Z2 36800 309 BMATRX(K)=AWARE(I,J) 36900 304 CONTINUE 37000 KMAX=K 37100 C**CALL IMSL ROUTINE LEQT1B TO SOLVE THE BANDED MATRIX. 37200 CALL LEQT1B(ABAND,KMAX, JUMAX , UMAX,432,BMATRX,1,432,0,XL, IER) 37300 C*NOW, GIVE Y’S THEIR NEW VALUES STORING OLD VALUES IN YOLD. 37400 K=O 37500 DO 315 I=2,IMAX-1 37600 YOLD(1I, UMAX+1)=Y(1I, UMAX+1) 37700 DO 315 J= 1,JMAX 37800 K=K+1 37300 YOLD(I,J)=Y(I,U) 38000 Y(1I,JU)=BMATRX(K) 38100 315 CONTINUE 38200 DO 320 J=1,UMAX+3 38300 YOLD(1,U)=Y(1,u) 38400 320 YOLD( IMAX,JU)=Y(IMAX,J) 38500 C*WILL USE ABBOTT’S DISSIPATIVE INTERFACE TO RID HIGH FREQ OSCILLATIONS 38600 DO 650 J=1,JUMAX 38700 DO 650 I=2,IMAX-1 38800 YDISS(1I,JU)=TAU*Y(I-1,JU)+(1.-2.*TAU)*Y(I,JU)+TAU*Y(I+1,U) 38900 IF(SJETTY.EQ.0.0) GO TO 650 39000 DO 649 M=1,MMAX 39100 IF(I.NE.IJET(M).AND.I.NE.IJET(M)+1) GO TO 649 39200 IF(Y(IJVET(M),JU).GT.SJUETTY.OR.Y(IJUET(M)+1,U).GT.SJETTY)GO TO 649 39300 IF(I.EQ.IJET(M) )YDISS(I,JU)=TAU*Y(I-1,U)+(1.-TAU)*Y(I,J) 39400 IF(1I.EQ. (IUET(M)+1))YDISS(1I,U)=TAU*Y(I+1,JU)+(1.-TAU)*Y(I,U) 39500 649 CONTINUE 39600 650 CONTINUE 39700 DO 651 J=1,JUMAX 39800 DO 651 I=2,IMAX-14 39900 651 Y(I,JU)=YDISS(I,u) 40000 C*THIS LOOP WILL STORE THE IMPLICIT Y VALUES REQ’D TO COMP QY&QX 40100 DO 40 I=1,1IMAX+1 40200 DO 40 J=1,JUMAX+3 40300 40 YIMP(I,J)=Y(I,U) 40400 C*THIS LOOP WILL EXPLICITLY MOVE CONTOURS SEAWARD IF REPOSE EXCEEDED. 40500 KOUNT =O 40600 SLOPEM=TAN(O.9*REPOSE ) 40700 DO 48 I=1,IMAX 40800 43 KOUNT=KOUNT+1 40300 IF (KOUNT . GT .50000) GO TO 41 41000 C*LET US COMPUTE ALL THE SLOPES(PSLOP) FOR EACH CHANGE IN DEPTH. 41100 DO 47 J=1,JMAX+1 41200 DUM=-BERM/2.0 41300 IF(JU.NE.1) DUM=DEEP(I,U-1) 41400 DELH=0.5*(DEEP(I,JU+1)+DEEP(I,JU))-0.5*(DEEP(I,U)+DUM) 41500 PSLOP=DELH/(Y(I,JU+1)-Y(1I,U)) 41600 47 SANGLE(J)=ATAN(PSLOP ) 41700 C*FIND THE MIN NEG SLOPE ANGLE OR THEN THE POS SLOPE>REPOSE OR FORGET IT 41800 ASLOPM=-1.0E50 413900 ASLOPP=0.0 42000 DO 46 J=1,JMAX+1 42100 IF(SANGLE(JU).GT.0.0) GO TO 45 42200 IF(SANGLE(JU).GT.ASLOPM)ASLOPM=SANGLE (J) 42300 TF (ASLOPM.EQ.SANGLE(u)) JM=J 42400 GO TO 46 42500 45 IF(SANGLE(J).GT.REPOSE.AND.SANGLE(J).GT.ASLOPP )ASLOPP=SANGLE (J) 42600 IF(ASLOPP.EQ.SANGLE(Uu)) JP=J 42700 46 CONTINUE 42800 IF(ASLOPM.EQ.-1.0£50.AND.ASLOPP.EQ.0.0) GO TO 42 42900 IF(ASLOPM.EQ.-1.0E50) GO TO 44 43000 DUM=-BERM/2. 43100 IF(JUM.NE. 1) DUM=DEEP(I,UM-1) 43200 ALTER=((0.5/SLOPEM*(DEEP(1I,JM+1)-DUM) )-(Y(I,JUM+1)-Y(I,UM)))/ 43300 * (1.0+((DEEP(I,JM+1)-DEEP(I,UM))/(DEEP(I,JUM)-DUM) ) ) 43400 Y(1I,JUM+1)=Y(1,UM+1)+ALTER ll 43500 43600 43700 43800 43900 44000 44100 44200 44300 44400 44500 44600 44700 44800 44900 45000 45100 45200 45300 45400 45500 45600 45700 45800 45900 46000 46100 46200 46300 46400 46500 46600 46700 46800 46900 47000 47100 47200 47300 47400 47500 47600 47700 47800 47900 48000 48100 48200 48300 48400 48500 48600 48700 48800 48900 49000 49100 49200 49300 49400 49500 49600 49700 49800 49900 50000 50100 50200 50300 50400 50500 50600 50700 44 42 48 Y(1,JUM)=Y(1,JUM)-(ALTER*(DEEP(I,JUM+1)-DEEP(1I,JUM))/(DEEP(I,UM)-DUM) ) QYEXP(1,JM+1)=QYEXP(I,JUM+1)+DX/DELT*ALTER*(DEEP(I , UM+1)-DEEP(I,UM) * ) GO TO 43 CONTINUE DUM=-BERM/2. IF(JUP.NE.1) DUM=DEEP(I,UP-1) ALTER=((0.5/SLOPEM*(DEEP(I,JUP+1)-DUM) )-(Y(I,JUP+1)-Y(I,UP)))/ * (1.0+((DEEP(1I,JP+1)-DEEP(1,JUP))/(DEEP(1,UP)-DUM) )) Y(I,JUP+1)=Y(1,JUP+1)+ALTER j Y(I,UP)=Y(1I,JUP)-(ALTER*(DEEP(1I,JP+1)-DEEP(I,UP))/(DEEP(1I,JUP)-DUM) ) QYEXP(1I,UP+1)=QYEXP(1I,JUP+1)+DX/DELT*ALTER*(DEEP(I,UP+1)-DEEP(1,UP) * ) GO TO 43 WEQ(I, UMAX+1)=Y (I, JMAX+1)-Y(1I, UMAX) CONTINUE C*IF WE GET SENT HERE, LOOP 444 WILL CATCH THE CROSSED CONTOURS. 319 318 323 CONT INUE WE CAN COMPUTE QX’S AND QY’S! DO 318 I=2,IMAX IMPLIC AND EXPLIC MOVEMENT OF YZERO WILL BE TAKEN CARE OF HERE QY(1, 1)=-BERM*DX*(Y(1I,1)-YOLD(I,1))/DELT YZERO(I)=YZERO(1I)+(Y(1,1)-YOLD(I,1)) DO 318 J=1,JUMAX QX(1,U)=RHS1(1,U)-S3(1,JU)*YIMP(I,JU)+S3(1I,U)*YIMP(I-1,u) QY(1,U+1)=CONST6(I,JU+1)*(0.5*(YIMP(I,JU)+YOLD(I,JU)-YIMP(I,J+1) * -YOLD(I,uU+1))+WEQ(I,uU+1)) DO 323 J=1,JUMAX QX(1,U)=QX(2,U) QX (IMAX+1,JU)=QX(IMAX, J) C*TOTAL QYS WILL BE COMP FROM IMPLIC AND EXPLIC VALUES.THEN ZERO QYEXP 39 DO 39 I=1,IMAX+1 DO 39 J=1,UMAX+3 QY(I,JU)=OY(1I,JU)+QYEXP(I,J) QYEXP(I,JU)=0.0 C*THIS CHECK WILL BOMB THINGS OUT IF CONTOURS HAVE CROSSED. DO 444 II=1, IMAX DO 444 JJ=1,JUMAX C*IF CONTOURS CROSS AT ANY TIME WANT PROGRAM TO STOP! 150 151 152 103 19 444 C*THE C*LET 926 800 Cc* c*900 Cc* Cc*903 Cc* C*906 c* Cx755 801 107 15 IF(Y(II,JUJU).LT.Y(II,JJU+1)) GO TO 444 WRITE(6, 103) WRITE(6, */) NUNIV DO 150 J=1,JUMAX WRITE(6, 100) (QX(I,JU),1=1, IMAX) DO 151 J=1,JUMAX WRITE(6, 101) (QY(I,U),1=1, IMAX) DO 152 J=1,JMAX WRITE(6, 100) (Y(1,JU),1=1, IMAX) FORMAT(2X,’THE CONTOURS HAVE CROSSED AND SOMETHING IS WRONG’ ,/) DO 19 J=1,UMAX WRITE(6,100) (YOLD(I,JU),1=1,IMAX) GO TO 445 CONT INUE WRITE(6,*/) NUNIV FOLLOWING STATEMENT DETERMINES AT WHAT FREQ EVERYTHING IS WRITTEN! IF (MOD(NUNIV, 10) .NE.O) GO TO 1 ’S WRITE ALL OF IT OUT. WRITE(6,926) NUNIV FORMAT(2X,’THE TOTAL ELAPSED NUMBER OF TIME-STEPS. NUNIV= ’,15,/) FORMAT(2X,14(F8.4)) DO 900 I=1,IMAX WRITE(6,800) (THETA(I,JU),J=1, UMAX) DO 903 J=1,UMAX+1 WRITE(6,801) DEEP(1,uU) DO 906 I=1,IMAX WRITE(6,800) (H(I,JU),J=1,JUMAX) DO 755 J=1,JUMAX WRITE(6,800) (CONST6(I,U),1I=1, IMAX) FORMAT(2X,14(F8.2)) WRITE(6, 107) FORMAT(/,2X,’THE LONGSHORE TRANSPORTS,QX, FOLLOW’ ) DO 15 J=1,JMAX WRITE(6, 100) (QX(1,JU),1=1, IMAX) UZ 50800 50900 51000 51100 51200 51300 51400 51500 51600 51700 51800 51900 52000 52100 52200 52300 52400 52500 52600 52700 52800 52900 53000 53100 53200 53300 53400 53500 53600 53700 53800 53900 54000 54100 54200 54300 54400 54500 54600 54700 54800 54900 55000 55100 55200 55300 55400 55500 55600 55700 55800 55900 56000 56100 56200 56300 56400 56500 56600 56700 56800 56900 57000 57100 57200 57300 57400 57500 57600 57700 57800 57900 58000 WRITE(6, 108) 108 FORMAT(/,2X,’THE ON-OFFSHORE TRANSPORTS, QY, FOLLOW’ ) DO 17 J=1,JUMAX 17 WRITE(6, 101) (QY(I,JU),1=1, IMAX) WRITE(6, 109) 109 FORMAT(/,2X,’THE NEW CONTOUR VALUES, Y, FOLLOW’) DO 18 J=1,JMAX 18 WRITE(6, 100) (Y(1I,JU),1=1, IMAX) 100 FORMAT(2X,13(F9.3)) 101 FORMAT(2X,13(F9.4)) 1 CONT INUE RETURN GO TO 446 445 STOP 446 CONTINUE END CK ee oo eee kk ok OK ka KK KK kK ok ak aK ok kk a ok ie akc ok ok ke ok G SUBROUTINE QTRAN C*THIS SUBROUTINE CALCS THE BREAKER HEIGHT FOR EACH C*OF THE I GRID LINES. METHOD--FINDS Y-LOCATIONS BEFORE AND AFTER C*BREAKING HAS OCCURRED BY ‘’REFRAC’, THEN USES SHOALING TO GET THE C*HBQ.SNELL’S LAW IS USED FOR REFRACTION OVER THE SHORT DIST TO BREAKING C* QX(I,Ju) IS THE TRANS BETWEEN(I-1,JU) AND (I,J) AT THE BLOCKCENT COMMON/A/ C(60,20),RK(60,20),Y(60, 20) ,DEEP(60, 20) , ALPHAS(60, 20) COMMON/AA/YZERO(60) COMMON/B/ THETA(60,20),QXTOT(60), OLDANG(60,20), DY(60, 20) COMMON/C/ H(60,20),CG(60, 20) ,HOLD(60, 20) ,HB(60, 20), YB(60) COMMON/N USED/JUSE,T,CO,CGEN,CGGEN, ANGGEN,DX,BERM, THETAO( 10) , MMAX COMMON/D/SIGMA,G,ELO, JUMAX, IMAX,PI, TWOPI,PIO2,HGEN, IJET( 10), SUETTY COMMON/G/ IBREAK(60) , HNONBR( 20) COMMON/E/RHO, RHOS,POROS, CONST, TKSI COMMON/P/HBQ(60) ,DEEPB(60) CAPPA=0.78 DO 1 I=2,IMAX DO 2 JJ=1,JMAX JU=JMAX-Ju+1 HDUM=(H(1I,JU)+H(I-1,JU))*0.5 HBDUM=(HB(1I,JU)+HB(I-1,JU))*0.5 C*CAN ONLY USE COND ON ONE SIDE OF STRUCT. CAN’T AVG HERE! IF(SJETTY.EQ.0.0) GO TO 4 DO 4 M=1,MMAX IF(I.NE.IVET(M)+1) GO TO 4 IF(THETAO(M).GE.0.0) ISIDE=IJET(M) IF(THETAO(M) .LT.O.0) ISIDE=IJET(M)+1 C***B.C. AT STRUCT TIP ASSUMES GX COMP AS IF NO STRUCT IS PRESENT. YSEA=0.5*(Y(ISIDE,JU)+Y(ISIDE,uJ+1)) IF(YSEA.GT.SJETTY) GO TO 3 HDUM=H(ISIDE,u) HBDUM=HB(ISIDE,uU) GO TO 3 4 CONTINUE 3. IF(HDUM.€T.HBDUM) GO TO 2 DEEPB(I)=((0.5*(H(I,U+1)+H(I-1,U+1)))*((0O.5*(DEEP(1I,u+1) * +DEEP(I-1,JU+1)))**O.25)/CAPPA)**O.8 HBQ(I)=CAPPA*DEEPB(I) i C*HBQ(I) AND DEEPB(I) WILL BE COMPUTED ACCORDING TO THE WAVE DIR. C** AT THE STRUCTURE TIP,THETAO. IF(SUETTY.EQ.0.0) GO TO 1 DO 6 M=1,MMAX IF(I.NE.IJET(M)+1) GO TO 6 C**THE TRANSPORTING WAVES WILL BE COMPUTED USING THE WAVE TO PROP SIDE. IF(THETAO(M).GE.0O.0) GO TO 11 DEEPB(1)=(H(IJET(M)+1,uU+1)*DEEP(IJET(M)+1,U+1)**O.25/CAPPA)**0.8 IBREAK(I)=IBREAK(IJET(M)+1) GO TO 12 41 DEEPB(I)=(H(IVET(M) ,J+1)*DEEP(IJET(M),J+1)**O.25/CAPPA)**0.8 IBREAK(1I)=IBREAK(IJET(M) ) 12 HBQ(1)=DEEPB(I)*CAPPA GO TO 1 6 CONTINUE GO TO 1 CONT INUE 1 CONT INUE “3 58100 58200 58300 58400 58500 58600 58700 58800 58900 59000 59100 59200 59300 59400 59500 59600 59700 59800 59900 60000 60100 60200 60300 60400 60500 60600 60700 60800 60900 6 1000 61100 61200 61300 61400 61500 61600 61700 61800 61900 62000 62100 62200 62300 62400 62500 62600 62700 62800 62900 63000 63100 63200 63300 63400 63500 63600 63700 63800 63900 64000 64100 64200 64300 64400 64500 64600 64700 64800 64900 65000 65100 65200 C*IF THE OFFSHORE WAVE HT IS ZERO, NEVER GET TO HERE. C*HOWEVER IF THE H IS SUCH THAT IT WOULD BREAK INSHORE OF Y(I,2) C*DEEPB(I) WOULD STILL BE ZERO AND DISTR(I,JU) WOULD BLOW-UP. DO 20 I=1,IMAX IF (DEEPB(I).GT.0O.0O) GO TO 20 DEEPB(I)=(H(1I,1)*DEEP(1,1)**0.25/CAPPA)**0.8 HBQ( I )=CAPPA*DEEPB(1) 20 CONTINUE HBQ(1)=HBQ(2) HBQ( IMAX+1)=HBQ( IMAX) DEEPB(1)=DEEPB(2) DEEPB( IMAX+1)=DEEPB(IMAX) RETURN END CF kt tee ee eK keke te te eke kk a ke ok ke eke kek ke eo oe kk ao ee ek SUBROUTINE BREAK(IMAX,UJMAX) C*ROUTINE WILL DETERMINE HB AND DEEPB ON THE GRID LINES RATHER C* THAN BETWEEN THEM. REQ’D FOR COFF BEYOND SURF ZONE. COMMON/A/ C(60,20),RK(60,20),Y(60, 20) ,DEEP(60, 20) , ALPHAS(60, 20) COMMON/C/ H(60,20),CG(60, 20) ,HOLD(60, 20) ,HB(60, 20), YB(60) COMMON/MP/ RKB(60),HBI(60) ,DEEPBI(60) CAPPA=0.78 DO 1 I=2,IMAX DO 2 JJ=1,UJUMAX J=JMAX-Ju+1 IF(H(1I,J).LT.HB(I,U)) GO TO 2 DEEPBI(1I)=((H(1,U+1)*DEEP(I,JU+1)**O.25)/CAPPA)**0.8 HBI(I)=CAPPA*DEEPBI(1) C***ONCE THE HEIGHT & DEPTH AT BREAKING ARE FOUND, GO TO NEXT GRID-LINE. GO TO 1 2 CONTINUE 1 CONTINUE DO 20 I=1,IMAX IF(DEEPBI(I).GT.0.0) GO TO 20 DEEPBI(1)=(H(1I,1)*DEEP(1,1)**O.25/CAPPA)**0.8 HBI(1I)=CAPPA*DEEPBI(I) 20 CONTINUE DEEPBI(1)=DEEPBI(2) DEEPBI( IMAX+1)=DEEPBI(IMAX) HBI(1)=HBI(2) HBI ( IMAX+1)=HBI(IMAX) RETURN END CORO OR ORO OR III OR I I IO IOI ICI ICC IIE IIE A IIR Ik a kk ae ake SUBROUTINE REFRAC(JBEGIN, JEND,NPTS, IBEGIN, IEND, ISTART,M) COMMON/A/ C(60,20),RK(60,20),Y(60, 20) ,DEEP(60, 20) , ALPHAS(60, 20) COMMON/AA/YZERO(60) COMMON/B/ THETA(60,20),QXTOT(60), OLDANG(60,20), DY(60,20) COMMON/C/ H(60,20),CG(60, 20) ,HOLD(60, 20) ,HB(60, 20) , YB(60) COMMON/N USED/JUSE,T,CO,CGEN, CGGEN, ANGGEN, DX, BERM, THETAO( 10) , MMAX COMMON/D/SIGMA,G,ELO, JUMAX, IMAX,PI, TWOPI ,PI02,HGEN, IVET(10),SJUETTY COMMON/G/ IBREAK(60) , HNONBR( 20) COMMON/ZZZ/NTIME DIMENSION UJBEGIN(60), JEND(60) CRORE Ee Ae EEE ee THIS SUBROUTINE WILL DETERMINE H AND CR ek ESE ERE Ce THETA AT THE MID PT OF Y VALUES. C***TAU IS THE FACTOR WHICH RECOUPLES THE REFRACTION EQS.SEE ABBOTT TAU=0.25 C*MUST PRESCRIBE THE WAVE ANGLE AT THE OUTERMOSTCONTOUR BOX C*SNELL’S LAW WILL BE USED TO START THINGS OFF. C*THETA(I,JU) WILL BE AT AREA’S CENTER AND WILL USE Y(I,JU) IN NEG Y-DIR C*WILL INITIALIZE ALL THETA’S USING SNELL‘S LAW. DO 206 I=IBEGIN, IEND C*INITIALIZE TWO J-VALUES BEYOND UJUMAX,IF IN REGION 1. IF (JEND(I).EQ.UMAX) JINIT=2 IF (UEND(1I).NE.UJUMAX) JINIT=O DO 206 J=JBEGIN(I),JEND(I)+JINIT C*MUST CORRECT FOR THE CONTOUR ORIENTATION, ALPHAS. IF(I.NE.IBEGIN) GO TO 960 ALPHAS(1,U)=ATAN((0.5*(Y(1+1,JU)+Y¥(14+1,U+1))-O.5*(Y(1, J) * +Y(I,U+1)))/DX) GO TO 962 74 65300 960 IF(I.NE.IEND) GO TO 961 65400 ALPHAS(I,JU)=ATAN((0.5*(Y(1I,JU)+Y(1,JU+1))-O.5*(Y(1I-1,U) 65500 * +Y(I-1,U+1)))/DX) 65600 GO TO 962 65700 961 ALPHAS(I,JU)=ATAN((0.5*(Y(1I+1,U)+Y¥(I+1,U+1))-0.5* 65800 * (Y¥(I-1,U)+Y(I-1,U+1)))/(2.*DX)) 65900 962 DALPHA=ANGGEN-ALPHAS(TI,uJ) 66000 THETA(I,JU )=ARSIN((C(I,JU )/CGEN)*SIN(DALPHA) ) 66100 C*MUST GET THETA WRT THE X-AXIS. 66200 THETA(I,JU)=THETA(I,J)+ALPHAS(I,U) 66300 206 CONTINUE 66400 C*NOW, WE MUST COMP THE BOUN WAVE HTS SO THE HTS CAN BE COMPUTED. 66500 CW HEE AUS Exit EE: Q)e sree As DEL DOT (E*CG)=0.0 66600 C*NOW WE WILL CORRECT THE HT FOR SHOALING AND REFRACTION TO THE B.C. 66700 C*WILL ALSO INITIALIZE H’S WITH THESE EQUATIONS FOR ENTIRE ARRAY. 66800 DO 500 I=IBEGIN, IEND 66900 C*INITIALIZE TWO JU-VALUES BEYOND UMAX IF IN REGION 1. 67000 IF (JEND(I) .EQ.UMAX) JINIT=2 67100 IF (JEND(I).NE.UMAX) JINIT=O 67200 DO 500 J=JBEGIN(I1),JEND(I)+JUINIT 67300 H(1I,JU) =HGEN*SQRT(CGGEN/CG(I,JU) )*SQRT(COS(ANGGEN) /COS(THETA(1, 67400 * J))) 67500 IF(HB(I,JU).LT.H(I,u)) H(1I,JU)=HB(I,uU) 67600 500 CONTINUE 67700 (OE oe ane ae eo oe ee ee a re Oe ST et aD le me iin ee 67800 CII I I IIR RO III III IOI III I FR FO 67900 C*LET’S FILL THE DY ARRAY. 68000 C*DY WILL BE INDEXED AS THE THETA TO WHICH WE ARE GOING. 68100 DO 209 I=IBEGIN, IEND 68200 DO 209 J=JBEGIN(I )+1,JEND(I) 68300 DY(I,JU-1)=0.5*(Y(1I,JU-1)+Y(1I,J))-O.5*(Y(I,JU)+Y¥(1,U+1)) 68400 209 CONTINUE 68500 NITERS=100 68600 DO 100 NITER=1,NITERS 68700 SUMANG=0.O 68800 C*DO "6O LOOP" GOES FROM 2 TO IMAX IF ISTART =IBEGIN 68900 c*DO "60 LOOP" GOES FROM IMAX-1 TO 1 IF ISTART=IEND 69000 DO 6O II=IBEGIN, IEND 69100 C*MUST HAVE IT SET UP SO THAT THE KNOWN BOUNDARIES ANGLES AREN’T RECOMP 69200 IF(ISTART.EQ. IBEGIN) I=I1 69300 IF(ISTART.EQ.IBEGIN .AND. I1.E£Q.IBEGIN) GO TO 60 69400 IF(ISTART.EQ.IEND) I=IEND-II+IBEGIN 69500 IF(ISTART.EQ.IEND .AND. I1.EQ.IEND) GO TO 60 69600 C*ADX EQUALS ACTUAL DELTA X ACROSS SPACE STEP. 69700 C*ONLY ON BOUNDARIES WHERE FORWARD OR BACKWARD DIFFERENCING. 69800 IF(I.NE.IBEGIN) GO TO 6 69900 ADX=DX 70000 IP=I+1 70100 IM=I 70200 GO TO 12 70300 6 IF(I.NE.IEND) GO TO 10 70400 ADX=DX 70500 IP=I 70600 IM=I-1 70700 GO TO 12 70800 10 ADX=2 .O*DX 70900 IP=I+1 71000 IM=I-1 71100 12 CONTINUE 71200 DO 40 J=JBEGIN(I),JEND(I)-1 71300 C*WILL GO FROM (JMAX-1) TO 1 BECAUSE THAT’S THE DIR WAVE COMES IN FROM. 71400 JJ=JEND(I )-1-JU+JUBEGIN(T) 71500 OLDANG(I,JUJU)=THETA(I, Ju) 71600 C*LOCATE MIDPOINT BETWEEN TWO ADJACENT BLOCK CENTERS 71700 C*BECAUSE THETA’S JJ-VALUE IS THE SAME AS THE FIRST SHOREWARD Y VALUE 71800 C*MUST. USE JJ, UJ+1, AND JJ+2 TO COMPUTE YBAR. 71900 YBAR=0. 25*(Y(I,Ju)+2.0*Y(1I,JJU+1)+Y(1, UU+2) ) 72000 C*LOCATE APPROPRIATE INDICES ON IP AND IM GRID LINES. 72100 IMINUS=- 1 72200 IPLUS=+1 72300 CALL LOC(IM,JJ,JOIM, JSIM, YBAR, IMINUS) 72400 CALL LOC(IP,UJ,JOIP,JSIP,YBAR,IPLUS) 72500 C*NOW USE THE CONSERVATION OF WAVES EQUATION.............. 75 72600 72700 72800 72900 73000 73100 73200 73300 73400 73500 73600 73700 73800 73900 74000 74100 74200 74300 74400 74500 74600 74700 74800 74900 75000 75100 75200 75300 75400 75500 75600 75700 75800 75900 76000 76100 76200 76300 76400 76500 76600 76700 76800 76900 77000 77100 77200 77300 77400 77500 77600 77700 77800 77300 78000 78100 78200 78300 78400 78500 78600 78700 78800 78900 79000 79190 79200 79300 79400 79500 79600 79700 79800 PART 1C=RK(I, JU+1)*SIN(THETA(I, JU+1)) PART2=-DY(I,JUJ)/ADX C*WILL LINEARLY INTERPOLATE TO DETERMINE RK*COS(THETA) AT I+1 AND I-1. C*IF NO ADJ SHOREWARD PT EXISTS, PUT IN ZERO FOR TERMS IN GOV. EQ. IF(USIM.NE.O) GO TO 301 PART3B=0.0 GO TO 302 301 TOPIM=RK( IM, JOIM- 1)*COS(THETA(IM, JOIM-1)) BOTIM=RK( IM, JSIM) *COS(THETA(1IM,USIM) ) TOTALB=0.5*(Y( IM, JOIM)+Y( IM, JOIM-1))-0.5*(Y( IM, USIM+1)+Y(IM,JSIM) ) DUMB=0.5*(Y( IM, JOIM)+Y(IM, JOIM-1))-YBAR PART3B=((TOTALB-DUMB ) *(TOPIM-BOTIM)/TOTALB)+BOTIM 302 IF(USIP.NE.O) GO TO 303 PART3A=0.0 GO TO 304 303 TOPIP=RK(IP,UOIP-1)*COS(THETA( IP, JOIP-1)) BOTIP=RK(IP,JUSIP)*COS(THETA(IP,USIP) ) TOTALA=0.5*(Y(IP,JOIP)+Y(IP,JOIP-1))-0.5*(Y(IP,USIP+1)+Y(IP,USIP) ) DUMA=0.5*(Y(IP,JOIP)+Y(IP,JOIP-1))-YBAR PART3A=(( TOTALA-DUMA ) *(TOPIP-BOTIP)/TOTALA)+BOTIP 304 PART3=PART3A-PART3B C*NOW MUST FIND RK*SIN(THETA) FOR I+1 AND I-14 AT J+1 YBARP=O. 25*(Y(1I,JJU+1)+2.*Y(1,JJU+2)+Y(I,JU+3)) CALL LOC(IM, JJ+1,JPOIM, JUPSIM, YBARP, IMINUS) CALL LOC(IP,JJU+1,JUPOIP,UPSIP, YBARP,IPLUS) IF(UPSIM.NE.O) GO TO 305 PART 1B=0.0 GO TO 306 305 TOPM=RK( IM, JPOIM-1)*SIN( THETA( IM, JUPOIM-1)) BOTM=RK(IM, JPSIM)*SIN( THETA( IM, JUPSIM) ) TOTB=0.5*(Y( IM, JPOIM)+Y(IM, JPOIM-1))-0.5*(Y( IM, JPSIM+1)+ * Y( IM, JPSIM) ) DUMPB=0.5*(Y (IM, JPOIM)+Y(IM, JPOIM-1))-YBARP PART 1B=(( TOTB-DUMPB ) * (TOPM-BOTM)/TOTB)+BOTM 306 IF(UPSIP.NE.O) GO TO 307 PART 1A=0.0 GO TO 308 307 TOPP=RK(IP,JUPOIP-1)*SIN(THETA(IP,JPOIP-1)) BOTP=RK(IP,JUPSIP)*SIN(THETA(IP,UPSIP) ) TOTA=0.5*(Y(IP,JUPOIP)+Y(IP,JUPOIP-1))-0.5*(Y(IP,UPSIP+1)+Y(IP,JUPSIP ce )) DUMPA=0.5*(Y(IP,UPOIP)+Y(IP,JPOIP-1))-YBARP PART 1A=(( TOTA-DUMPA ) *( TOPP-BOTP )/TOTA)+BOTP 308 PART 1=TAU*PART1B+(1.-2.*TAU)*PART1C+TAU*PART1A IF(UPSIM.EQ.0O)PART1=(1.-TAU) *PART1C+TAU*PARTI1A IF(UPSIP.EQ.0O)PART 1=TAU*PART 1B+(1.-TAU) *PART1C ARG=( (PART 1+PART2*PART3)/RK(I, Ju) ) C*IF THE ROUTINE IS TO BLOWUP,USE SNELLS LAW. IF(ABS(ARG).LE.1.0) GO TO 41 ARG=(C(1I,JUJU)/C(1I, Ju+1))*SIN(THETA(I,JuU+1)) IF(ARG.GT.1.0) ARG=1.0 THETA(1I,JUJU)=ARSIN(ARG) GO TO 42 41 THETA(I,JJ)=ARSIN( ARG) 42 THETA(I,JJU)=0.5*(THETA(I,JJ)+OLDANG(I, Ju) ) SUMANG=SUMANG+(ABS(THETA(I, JJ) -OLDANG(I,uUU))) 40 CONTINUE 60 CONT INUE . C*MUST EJECT IF WE HAVE REACHED AN ACCEPTABLE ITERATION ERROR C*IF THE SUM OF THE ABSOLUTE VALUE OF ANGLE CHANGES DURING AN ITERATION c* AVERAGES LESS THAN 0.02 DEGREES PER GRID ITS CLOSE ENOUGH. IF (SUMANG.LT. (NPTS*O.0035) ) GO TO 215 IF(NITER.GE.50) GO TO 215 100 CONTINUE WRITE(6, 803) 215 CONTINUE C*ITERATION LOOP FOR THE WAVE HEIGHT. DO 501 NITER=1,NITERS SUMH=0.0 DO 510 II=IBEGIN,IEND C*MUST HAVE IT SET UP SO THAT THE KNOWN BOUNDARIES HTS. AREN’T RECOMP IF(ISTART.EQ.IBEGIN) I=II IF(ISTART.EQ.IBEGIN .AND. I1.EQ.IBEGIN) GO TO 510 76 79900 80000 80100 80200 80300 80400 80500 80600 80700 80800 80900 81000 81100 81200 81300 81400 81500 81600 81700 81800 81900 82000 82100 82200 82300 82400 82500 82600 82700 82800 82900 83000 83100 83200 83300 83400 83500 83600 83700 83800 83900 84000 84100 84200 84300 84400 84500 84600 84700 84800 84900 85000 85100 85200 85300 85400 85500 85600 85700 85800 85900 86000 86100 86200 86300 86400 86500 86600 86700 86800 86900 87000 IF (ISTART.EQ. TEND) I=IEND-II+IBEGIN IF(ISTART.EQ.IEND .AND. I.EQ.IEND) GO TO 510 C*ADX EQUALS ACTUAL DELTA X ACROSS SPACE STEP. C*ONLY ON BOUNDARIES WHERE FORWARD OR BACKWARD DIFFERENCING. 503 504 505 312 313 315 316 317 318 IF(I.NE.IBEGIN) GO TO 503 ADX=DX IP=I+1 IM=I GO TO 505 IF(I.NE.IEND) GO TO 504 ADX=DX IP=I IM=I-14 GO TO 505 ADX=2.0*DX IP=I+1 IM=I-1 CONT INUE DO 502 J=JBEGIN(I),JEND(I)-1 JJ=JEND(I)- 1-JU+UBEGIN(I) HOLD(I,JUJU)=H(I,uu) YBAR=O.25*(Y(1I,JU)+2.0*Y(1, JU+1)+Y(1,JJ+2) ) CALL LOC(IM, JJ,JOIM,JUSIM, YBAR, IMINUS) CALL LOC(IP,UJU,JOIP,USIP, YBAR, IPLUS) PART13=(H(I,JJU+1)**2.)*CG(I,JJ+1)*COS(THETA(I, JuU+1)) PART2=DY(I,JUJU)/ADX IF(JUSIM.NE.O) GO TO 311 PART4B=0.0 GO TO 312 TOPIMH=(H( IM, JOIM-1)**2. )*CG( IM, JOIM-1)*(SIN(THETA( IM, JOIM-1))) BOTIMH=(H( IM, JSIM) **2. )*CG( IM, JSIM) *SIN( THETA( IM, JSIM) ) TOTALB=0.5*(Y( IM, JOIM)+Y (IM, JOIM-1))-0.5*(Y( IM, JSIM+1)+Y(IM,JUSIM) ) DUMB=0.5*(Y( IM, JOIM)+Y(IM, JOIM-1))-YBAR PART4B=((TOTALB-DUMB ) * (TOP IMH-BOTIMH)/TOTALB )+BOTIMH IF(JUSIP.NE.O) GO TO 313 PART4A=0.0 GO TO 314 TOPIPH=(H(IP,JOIP-1)**2.)*CG(IP, JOIP-1)*SIN(THETA(IP,JOIP-1)) BOTIPH=(H(IP, JSIP)}**2.)*CG(IP,JSIP)*SIN(THETA(IP,USIP)) TOTALA=0.5*(Y(IP,JOIP)+Y(IP,JOIP-1))-0.5*(Y(IP,JUSIP+1)+Y(IP,USIP) ) DUMA=0.5*(Y(IP,JOIP)+Y(IP,JOIP-1))-YBAR PART 4A=((TOTALA-DUMA ) *(TOPIPH-BOTIPH)/TOTALA )+BOTIPH PART4=PART4A-PART4B YBARP=O0.25*(Y(I,JU+1)+2.*Y(1,JUJU+2)+Y(1, JUU+3)) CALL LOC(IM, JJ+1,JPOIM, UPSIM, YBARP, IMINUS) CALL LOC(IP,JJ+1,JPOIP,UPSIP,YBARP, IPLUS) IF(UPSIM.NE.O) GO TO 315 PART12=0.0 GO TO 316 TOPMH=(H( IM, JPOIM-1)**2)*CG( IM, JPOIM-1)*COS(THETA(IM, JPOIM-1) ) BOTMH=(H( IM, UPSIM) **2)*CG( IM, UPSIM) *COS(THETA( IM, JUPSIM) ) TOTB=.5*(Y( IM, JPOIM)+Y(IM, JPOIM-1))-.5*(Y( IM, JPSIM+1)+Y(IM,JPSIM) ) DUMPB=0.5*(Y(IM, JPOIM)+Y (IM, JPOIM-1))-YBARP PART 12=((TOTB-DUMPB ) * (TOPMH-BOTMH) /TOTB )+BOTMH IF(UPSIP.NE.O) GO TO 317 PART11=0.0 GO TO 318 TOPPH=(H( IP, UPOIP-1)**2)*CG(IP, JUPOIP-1)*COS(THETA(IP,UPOIP-1)) BOTPH=(H( IP, UPSIP)**2)*CG(IP,UPSIP)*COS(THETA(IP,UPSIP) ) TOTA=.5*(Y(IP, UPOIP)+Y(IP, UPOIP-1))-.5*(Y(IP,JUPSIP+1)+Y(IP,UPSIP) ) DUMPA=0.5*(Y(IP,JUPOIP)+Y(IP,JUPOIP-1))-YBARP PART 11=((TOTA-DUMPA ) *(TOPPH-BOTPH) /TOTA)+BOTPH PART 1H=TAU*PART 12+(1.-2.*TAU) *PART13+TAU*PART 11 IF(UPSIM.EQ.O)PART1H=(1.-TAU) *PART13+TAU*PART 11 IF(JUPSIP.EQ.0)PART 1H=TAU*PART 12+(1.-TAU)*PART13 ARG=( (PART 1H+PART2*PART4)/(CG(I,JUJU)*COS(THETA(I,uJ)))) C*IF THERE IS TO BE AN INVALID SQRT,USE LINEAR SHOALING. IF(ARG.GE.O.) GO TO 44 ARG=(CG(I, JU+1)*COS(THETA(I,JU+1)))/(CG(1, JU) *COS(THETA(I, JJ) )) IF(ARG.LT.0O.0) ARG=0.0 H( 1, UJ) =H(1I,JJ+1)*SQRT (ARG) GO TO 45 OM, 87100 87200 87300 87400 87500 87600 87700 87800 87900 88000 88100 88200 88300 88400 88500 88600 88700 88800 88900 89000 89100 89200 89300 89400 89500 89600 89700 89800 89900 90000 90100 90200 90300 90400 90500 90600 90700 90800 90300 91000 91100 91200 91300 91400 91500 91600 91700 91800 91900 92000 92100 92200 92300 92400 92500 92600 92700 92800 92300 93000 93100 93200 93300 93400 93500 93600 93700 93800 93900 94000 94100 94200 94300 44 45 H(1I,JU)=SQRT (ARG) H(1I,JJU)=0.5*(H(1,JU)+HOLD(I, UU) ) HNONBR( JJ) =H(I, JJ) C*IBREAK(I)=JJU, THEREFORE JJ WILL BE LEEWARD SIDE OF GRID AT INIT BREAK 501 507 802 803 804 805 806 * IF(HB(I,JU) .LT. H(1,JJ) .AND. HB(1I,JJU+1).GE.HNONBR(JuU+1) ) IBREAK(I)=Ju IF(HB(I,JJU).LT.H(I,uu)) H(1,JUJU)=HB(1, JJ) SUMH=SUMH+ABS(H(I,JJU)-HOLD(I, Ju) ) CONT INUE CONTINUE IBREAK( IEND)=IBREAK(IEND-1) IBREAK(IBEGIN) =IBREAK( IBEGIN+1) IF(SUMH.LT. (NPTS*O.01)) GO TO 507 IF(NITER.GE.50) GO TO 507 CONTINUE WRITE(6,803) CONT INUE FORMAT(2X,4(F15.5),////) FORMAT(2X, "AFTER NITERS ITERATIONS, CONVERGENCE WAS NOT REACHED") FORMAT(2X,"THE WAVE HT. ROUTINE CONVERGED IN, NITER= ",15,//) FORMAT(2X, "THIS IS MY CHECKING WRITE STATEMENT") FORMAT(2X,"THE WAVE ANGLE ROUTINE CONVERGED IN, NITER= ",15,//) RETURN END CI RI OR II I RO I III ICI III I II II II ICI II I IOI I a tO Ok oe SUBROUTINE DIFF(RHOND, THETAO, ANGLE, AMP) C****DIFFRACTION ABOUT SEMI INFINITE BREAKWATER (PENNEY-PRICE ) PI=3.14159265 ABSS=SIN(O.5*(ANGLE-THETAO) ) ABSP=SIN(O.5*(ANGLE+THETAO) ) ABC=COS(ANGLE-THETAOQ) ABC 1=COS(ANGLE+THETAO) XX =RHOND* ABC XXC=COS(XX) XXS=SIN(XX) XX 1=RHOND*ABC 1 XXC1=COS(XX1) XXS1=SIN(XX1) AL=SQRT(RHOND/PI ) SIG=2.0*AL*ABSS SIGP=-2.0*AL*ABSP CALL FRES(SIG,C,S,FR,FI) CALL FRES(SIGP,CP,SP,FRP,FIP) SUM1=XXC*FR+XXS*FI+XXC1*FRP+XXS1*F IP SUM2=XXC*FI-XXS*FR+XXC1*FIP-XXS1*FRP AMP=SQRT (SUM1**2+SUM2**2) RETURN END COO OI IOI III II II III II I ICICI II II A Ca a ak Ott i A 9 SUBROUTINE FRES(A,C,S,FR,FI) C*FRESNEL INTEGRAL SUBROUTINE****AFTER ABROMOWITZ AND STEGUN. 50 Z=ABS(A) PO2=1.5707963 FZ=(1.0+0.926*Z)/(2.0+1.792*Z+3. 104*Z*Z) GZ=1.0/(2.0+4. 142*2+3.492*Z*Z2+6 .670*Z*Z*Z) XX =PO2*Z*Z CZ=COS(XX) SZ=SIN(XX) C=0.5-GZ*CZ+FZ*SZ S=O0.5-FZ*CZ-GZ*SZ IF(A.GT.O.0) GO TO 50 C=-C S=-S FR=0.5*(1.0+C+S) FI=-0.5*(S-C) RETURN END CO OO OO ROO IOI II II III III III II ICICI IOI I a I I I IC I IA FOI A SUBROUTINE PREDIF COMMON/A/ C(60,20),RK(60,20),Y(60, 20) ,DEEP(60, 20) , ALPHAS(60, 20) COMMON/AA/YZERO(60) COMMON/B/ THETA(60,20),QXTOT(60), OLDANG(60,20), DY(60,20) COMMON/C/ H(60,20),CG(60, 20) ,HOLD(60, 20) ,HB(60, 20), YB(60) 78 94400 94500 94600 94700 94800 94900 95000 95100 95200 95300 95400 95500 95600 95700 95800 95900 96000 96100 96200 96300 96400 96500 96600 96700 936800 96900 97000 97100 97200 97300 97400 97500 97600 97700 97800 97900 98000 98100 98200 98300 98400 98500 98600 98700 98800 98900 93000 99100 99200 99300 99400 99500 99600 939700 993800 99900 100000 100100 100200 100300 100400 100500 100600 100700 100800 100900 101000 101100 101200 101300 101400 101500 101600 COMMON/N USED/JUSE,T,CO,CGEN,CGGEN, ANGGEN, DX’, BERM, THETAO( 10) ,MMAX COMMON/D/SIGMA,G,ELO,JUMAX, IMAX,PI, TWOPI ,PI02,HGEN, IJET(10),SUETTY COMMON/G/IBREAK(60) , HNONBR( 20) DIMENSION J1(60),U2(60),J1REF(60) ,J3REF(60) C*THIS SUB CALCS WHERE DIFFRACTION GOVERNS AND WHERE REFRACT GOVERNS. C*IT WILL CALL REFRAC FOR OFFSHORE AREA(OFF TIP OF STRUCTURE). C*THEN IT WILL DO THE SHADOW ZONE USING DIFF(IF THETAO .NE.O.O) C* IT WILL THEN FINISH THE OTHERS USING REFRAC AGAIN. C*LET’S ZERO-OUT THE DIMENSIONED ARRAYS. DO 1000 I=1,IMAX+2 Ji(1)=0.0 J2(1)=0.0 J1REF(1I)=0.0 1000 JU3REF(1I)=0.0 C*NOW, LETS FIND C,CG,RK,HB, AND WVNUM. DO 202 I=1,IMAX DO 202 J=1,JUMAX+2 DEPTH=DEEP(I,u) CALL WVNUM(DEPTH, T,DUMK) RK( I,J) =DUMK C(1I,JU)=CO*TANH(RK(1I,JU)*DEEP(I,U)) EN=0.5*(71.0+((2.*RK(1I,JU)*DEEP(I,U))/SINH(2.*RK(I,JU)*DEEP(I,J)))) CG(I,JU)=EN*C(I,U) HB(I,J)=0.78*DEEP(I,uJ) 202 CONTINUE C*WILL ATTRIB AN EQUAL REACH TO EACH SIDE OF EACH M-GROIN. DO 200 M=1,MMAX IDUML=1 IF(M.NE.1) IDUML=(IJET(M)+IJET(M-1))/2 IDUMR=IMAX IF(M.NE.MMAX) IDUMR=(IJET(M)+IJET(M+1))/2 NPTS=O DO 1 I=IDUML,IDUMR DO 2 J=1,JMAX IF(Y(1I,J).LT.SJETTY) GO TO 14 Ji(1I)=J J2(1)=JMAX GO TO 15 144 CONTINUE 2 CONTINUE 145 CONTINUE C*IF NO STRUCT IS PRESENT(SJETTY=0.0), DO REFRAC THRUOUT GRID SYSTEM IF(SJETTY.EQ.0.0) J1i(I)=1 4 CONTINUE DO 16 I=IDUML,IDUMR C* ‘REFRAC’ STARTS ON THE NEXT TO LAST JU-CONTOUR,NOT THE LAST! DO 16 J=J1(1),J2(1)-1 16 NPTS=NPTS+14 C*WILL NOW DO THE REFRACT FOR’THE REGION 1 AREA. C*ISTART REPRESENTS THE DIRECTION THE SWEEPS WILL BEGIN FROM. C*WILL USE DUMMY IMAX,IJUET,IJET+1 IN CALL STTS SO IBEGIN,IEND, AND C***ISTART WON’T CHANGE THEM.MUST RESET AFTER EACH CALL REFRAC. IMAXT=IDUMR IJETT=IJET(M) IJETP1=IJET(M)+1 IDUMLL=IDUML IF(ANGGEN.GE.O.0) CALL REFRAC(JU1,J2,NPTS, IDUMLL, IMAXT, IDUMLL,M) IF(ANGGEN.LT.O.0) CALL REFRAC(Ji1,J2,NPTS, IDUMLL, IMAXT, IMAXT,M) IMAXT=IDUMR IJUETT=IJET(M) IJETP1=IJET(M)+1 IDUMLL=IDUML JDUMN=J4(IJET(M) ) JDUMS=JU1(IJET(M)+1) XDISTN=(IJUET(M)-1.0)*DX+DX/2. ELTIP=T*0.5*(C(IJET(M) , JDUMN)+C(IJUET(M)+1,JDUMS) ) C*NOW MUST CHECK THE ANGLE AT THE STRUCTURE’S TIP TO SEE WHERE SHAD ZONE C*IF NO STRUCT PRESENT(SJETTY=0.0), FUTHER REFRAC/DIFF UNNECESSARY. IF(SUJETTY.EQ.0.0) GO TO 13 THETAO(M)=0.5*(THETA(IJET(M), JDUMN)+THETA( IJVET(M)+1,JUDUMS) ) HINC=0.5*(H(IJET(M) , JDUMN)+H( IJET(M)+1,JUDUMS) ) IF(THETAO(M))10,11,12 C*THIS SECTION HANDLES REFRAC/DIFF IF THETAO0.0 112300 12 CONTINUE 112400 C*FIRST, REGION 2- ALL REFRACTION. 112500 NPTS=O 112600 DO 110 I=IDUML,IJET(M) 112700 J2(1)=J1(1) 112800 110 Ji(1)=1 112900 DO 111 I=IDUML,IJET(M) 113000 DO 111 J=J1(I),J2(1)-1 113100 111 NPTS=NPTS+1 113200 IMAXT=IDUMR 113300 IDUMLL=IDUML 113400 IJETT=IJET(M) 113500 IJETP1=IJET(M)+1 113600 CALL REFRAC(JU1,JU2,NPTS, IDUMLL, IJETT, IDUMLL,M) 113700 IMAXT=IDUMR 113800 IJETT=IJET(M) 113900 IJETP1=IJET(M)+14 114000 IDUMLL =IDUML 114100 C*NOW WILL DO REGION 3 OF THE POS THETAO CASE. 114200 DO 112 I=IJET(M)+i,IDUMR 114300 J2(1I)=Ji(1) 114400 112 vi(1)=1 114500 DO 113 I=IJET(M)+1,IDUMR 1414600 JIREF(1)=1 114700 C*WILL GO ONE PT. BEYOND JU2(I) TO MAKE SURE OUTOF DIFF ZONE. 114800 DO 114 J=Ji(I),J2(1)+14 114900 XCOOR=(I-1.0)*DX 115000 YCOOR=0.5*(Y(1,JU)+Y(1,uU+1)) 115100 ANGLE =ATAN( (XCOOR-XDISTN)/(SJETTY-YCOOR) ) 115200 IF(YCOOR.GT.SUJETTY) ANGLE =PI+ANGLE 115300 C*IF LEAST J-VALUE IS OUT OF SHAD ZONE,SO ARE OTHER JU’S.(FOR EACH I) 415400 IF (ANGLE .GT.ABS(THETAO(M) ) ) GO TO 115 115500 RAD=SQRT( ( XCOOR-XDISTN) **2+(SJETTY-YCOOR) **2) 415600 RHOND=RAD*TWOPI/ELTIP 415700 THE=THETAO(M) 115800 CALL DIFF(RHOND, THE,ANGLE, AMP) 115900 ANGRAD=ANGLE 116000 C*WILL NOW REFRACT DIFFRACTED WAVES IN SHAD ZONE USING SNELL"S. 116100 CTIP=ELTIP/T 116200 ALPHAS(1,JU)=ATAN((0.5*(Y(I+1,JU)+Y¥(1I+1,JU+1))-0.5* 81 116300 116400 116500 116600 116700 116800 116900 117000 117100 117200 117300 117400 117500 117600 117700 117800 117900 118000 118100 118200 118300 118400 118500 118600 118700 118800 118900 119000 119100 119200 119300 119400 119500 119600 119700 119800 119900 120000 120100 120200 120300 120400 120500 120600 120700 120800 120900 121000 121100 121200 121300 121400 121500 121600 121700 121800 121900 122000 122100 122200 122300 122400 122500 122600 122700 122800 122900 123000 123100 123200 123300 123400 123500 = AGES o RAK =). 5 eea)) ) )/ C2.) ) IF(I.EQ. IUJET(M)+1)ALPHAS(I,JU)=ATAN((0.5*(Y(I+1,JU)+Y(1+1,U+1))-O.5* * (Y(I,JU)+Y¥(1I,J+1)))/DX) DALPHA=ANGRAD-ALPHAS(I,U) THETA(I,JU)=ARSIN((C(I,JU)/CTIP)*SIN(DALPHA) ) THETA(I,JU)=THETA(I,JU)+ALPHAS(I,uU) H(1,J)=HINC*AMP C*MUST CHECK TO SEE IF WAVE WOULD HAVE BROKEN. IF(HB(I,JU).LE.H(1,JU).AND.HB(1I,JU+1).GT.H(1I,JU+1))IBREAK(I)=u IF(HB(I,JU).LT.H(I,U)) H(I,J)=HB(I,J) 114 CONTINUE GO TO 113 115 JIREF(I)=J 413 CONTINUE C*NOW MUST DO REFRAC FOR REGION 4. NPTS=O DO 116 I=IJET(M)+1,IDUMR DO 116 J=J1REF(I),J2(1)-1 116 NPTS=NPTS+14 IMAXT=IDUMR IDUMLL=IDUML IJETT=IJET(M) IJETP1=IJET(M)+1 CALL REFRAC(J1REF,JU2,NPTS, IJETP1, IMAXT, IMAXT,M) IMAXT=IDUMR IJETT=IJET(M) IJVETP1=IJET(M)+1 IDUMLL=IDUML 143 CONTINUE 200 CONTINUE RETURN END CE ee te OK ee ke fe ek 2 ee ke oe ok ok kok a ok kK ok ok oe ke ok kK ek kok 2K kk ok ek 2k ok ok oe ok SUBROUTINE LOC(IM,JJ,JOIM,JSIM, YBAR, IDUM) COMMON/A/ C(60,20),RK(60,20),Y(60, 20) ,DEEP(60, 20) , ALPHAS(60, 20) COMMON/AA/YZERO(60) COMMON/B/ THETA(60,20),QXTOT(60), OLDANG(60,20), DY(60,20) COMMON/C/ H(60,20),CG(60, 20) ,HOLD(60, 20) ,HB(60, 20), YB(60) COMMON/N USED/JUSE,T,CO,CGEN, CGGEN, ANGGEN, DX ,BERM, THETAO( 10) , MMAX COMMON/D/SIGMA,G,ELO, JMAX, IMAX,PI, TWOPI,PI02,HGEN, IJET(10),SUETTY C*SUBROUTINE LOC FINDS J-VALUES WHICH ARE GREATER AND LESS THAN YBAR. JOIM=2 2 AA=0.5*(Y( IM, JOIM)+Y (IM, JOIM-1) ) IF(AA.GT.YBAR) GO TO 4 JOIM=JOIM+ 1 C*THE FOLLOWING IS REQ’D SO THAT DY/DX>0.5 C*WILL DTERMINE K SIN THETA ON IM-LINE AT A DIST YBAR. C*WILL CALL THIS POINT JUSE+1 IF(JOIM.LE.JUSE) GO TO 2 JOIM=JUSE+1 Y( IM, JOIM)=YBAR C* DEPTH AT THIS POINT WILL BE COMP ASSUMING CONST BEACH SLOPE ON I=IM DEL=.5*(Y( IM, JOIM-1)+Y( IM, JOIM-2))-.5*(Y(IM, JOIM-2)+Y(IM,JOIM-3) ) BSLOPE=(DEEP(IM, JOIM-2)-DEEP(IM, JOIM-3))/DEL DEEP( IM, JOIM-1)=DEEP(IM, JOIM-2)+BSLOPE*(Y(IM, JOIM)-Y(IM, JOIM-1)) DEPTH=DEEP(IM, JOIM- 1) CALL WVNUM(DEPTH,T,DUMK) RK( IM, JOIM- 1) =DUMK C( IM, JOIM-1)=CO*TANH(RK( IM, JOIM-1)*DEEP( IM, JOIM-1)) EN=0.5*(1.0+((2.0*RK( IM, JOIM-1)*DEEP( IM, JOIM-1))/SINH( * 2.*RK( IM, JOIM-1)*DEEP(IM, JOIM-1)))) CG( IM, JOIM-1)=C( IM, JOIM-1)*EN C*WILL USE SNELL’S LAW TO DETERMINE THE WAVE ANGLE HERE C*ANGLE OF CONTOUR WILL BE ASSUME TO BE THE SAME AS THE JMAX+1 CONTOUR IF(IDUM.EQ. 1)ALPH=ATAN((Y( IM, JOIM-1)-Y(IM-1,JOIM-1))/DX) IF (IDUM.EQ.-1)ALPH=ATAN((Y(IM+1,JOIM-1)-Y(IM, JOIM-1))/DX) DALPHA=ANGGEN-ALPH THETA( IM, JOIM-1)=ARSIN((C( IM, JOIM-1)/CGEN) *SIN(DALPHA) ) THETA( IM, JOIM-1)=THETA( IM, JOIM-1)+ALPH 4 JSIM=JMAX- 4 6 AA=0.5*(Y(IM, JSIM)+(Y(IM, USIM+1))) IF(AA.LT.YBAR) GO TO 8 JSIM=JSIM-1 82 123600 123700 123800 123900 124000 124100 124200 124300 124400 124500 124600 124700 124800 124900 125000 125100 125200 125300 125400 125500 125600 125700 125800 125900 126000 126100 126200 126300 126400 126500 126600 126700 126800 126900 127000 127100 127200 127300 127400 127500 127600 127700 127800 127900 128000 428100 128200 128300 128400 128500 128600 128700 128800 128900 129000 129100 129200 129300 129400 129500 129600 129700 129800 129900 130000 130100 130200 130300 130400 130500 130600 130700 130800 C*IF JSIM=0,THERE IS NO ADJ PT, SUB REFRAC CAN HANDLE IT. IF(JUSIM.EQ.O) GO TO 8 GO TO 6 8 RETURN END CK KK Fe oe ee kee te keke oo ke oe kek ke kk ok oko kk oo ko kok aK ok I a ok oo aK a aK ok ok ok ok ok SUBROUTINE WVNUM(DEPTH,T,RK) G=32.17 EPS=0.001 TWOPI=6.283185307 SIGMA=TWOPI/T RK=TWOPI/(T*SQRT(G*DEPTH) ) DO 100 IT=1,20 ARG=RK*DEPTH EK=(G*RK*TANH(ARG) )-(SIGMA**2) EKPR=G* (ARG*((SECH( ARG) )**2)+TANH(ARG) ) RKNEW=RK-EK/EKPR IF (ABS(RKNEW-RK) .LE.ABS(EPS*RKNEW) ) GO TO 120 RK=RKNEW 100 CONTINUE WRITE(6, 1000) IT,DEPTH, RK é 1000 FORMAT(///,10X,"ITERATION FOR K FAILED TO CONVERGE AFTER" * ,3X,13,"ITERATION",/, "OUTPUT: DEPTH, RK",3X,2F 13.5) CALL EXIT 120 RK=RKNEW IF(RK.GT.O.O) GO TO 140 WRITE(6, 1020) DEPTH,RK 1020 FORMAT(///,10X," RK IS NEG",/," OUTPUT DEPTH,RK",3X,2F13.5) CALL EXIT 140 RETURN END Ce OR ee ee oe ee eo oe oo ok kk kak kk kk ok eo ok ok ke ok ko oo ok ok SUBROUTINE SMOOTH( THETA, IMAX,JMAX,IJET,SJETTY ,MMAX,Y) C*THIS WILL SMOOTH THE WAVE ANGLE FIELD TO ACCT FOR DIFF(ARTIFICIALLY) DIMENSION TEMP(60, 20),Y(60, 20), THETA(60, 20), IJET( 10) C*(MMAX+1) IS REQ’D BECAUSE M-GROINS HAVE M+1. REACHES OF SHORELINE. DO 10 M=1,MMAX+14 IF(M.NE.1) GO TO 3 ILEFT=2 IRIGHT=IJET(1) GO TO 5 3 IF(M.NE.MMAX+1) GO TO 4 ILEFT=IJET(MMAX )+14 IRIGHT=IMAX-1 GO TO 5 4 ILEFT=IJET(M-1)+14 IRIGHT=IJET(M) 5 CONTINUE DO 1 J=1,JMAX-1 DO 1 I=ILEFT,IRIGHT IF(I.NE.ILEFT.AND.I.NE.IRIGHT) GO TO 15 C*TO GET HERE, MUST BE ON BOUN OR ADJ TO A STRUCTURE. IF(I.EQ.2.0R.1.EQ. IMAX-1) GO TO 15 C*TO GET HERE,ADJ TO A STRUCT AND CAN BE ILEFT OR IRIGHT. IF(Y(1,JU).GE.SJETTY) GO TO 15 C*IF HERE, WITHIN JETTY AND ADJ TO EITHER SIDE. IF(I.EQ.ILEFT)TEMP(1I,JU)=0.5*(THETA(I,JU)+THETA(I+1,U)) IF(I.EQ.IRIGHT)TEMP(I,JU)=0.5*(THETA(I,JU)+THETA(I-1,U)) GO TO 1 145 TEMP(I,J)=0.25*THETA(I-1,JU)+O.50*THETA(I,J)+O.25*THETA(I+1,U) 4 CONTINUE 10 CONTINUE DO 2 J=1,JUMAX-1 DO 2 I=2,IMAX-1 2 THETA(I,J)=TEMP(I,uJ) RETURN END CE OI IOI FO IO IO I III III I II I EI IC Oa ae ak ae FUNCTION SECH(A) SECH=1.0/COSH(A) RETURN END C****HERE IS WHERE THE IMSL ROUTINES MUST GO! 83 APPENDIX C CONTOURS AND SCHEMATIC ILLUSTRATIONS This appendix presents tables of the original contours at Oregon Inlet and the final contours for the eight numerical simulations (Tables C-1 to C-9). Also included are schematic illustrations of sediment volumes transported from the nourished region (Figs. C-1 to C-8). 84 t=ce 8zz°zagl Szz°zE9l ez2°zeagl Bzz°zeBgt Bcz°cBst 8ZzZ°zB9l Bzz°zB9l Bzz°zegl BzZe*cacdt BZz°ZELI BZzZ°atct @zz°ZOLI Bzz°ZOLI Bzz°zE9l B2z°cggt BZzZ°ZSOI @zz°zs9l sBzz°zZsgl Bez" zeal QZZ°Zb9I Bze°cv9l Bzz*Zv9l Bzz2°cBgl Bce"coLt QZZ2°ZILI Bzz°zoOL9R9 e2°689 990°069 969°069 0915°169 6Sb°T69 LoS 169 TeS°T69 f2n°le9 Ont*feo LmZ°069 992°069 Lte%ou9 e2t°eed 205°989 9297209 92°L89 209°999 Sbht°oe9 $99°Se89 0s2°sea 2£06°re9 6£9°n99 ghh’ned IEE°hA9 £e2°nAd 162°RA9 n9E°Hed 2LH°hes9 g909°RG9 QSL°Hge? py ee yg — 8290 72m ANE % een Ena*22n E00°s2n B0S*S2h O20°Hem gos’henh gnl*sen noL°Sen 22n°92%n 2h °Len 209 L2h 619° een 80S 62m HSF*OSh Bot"lih 920°2En e2e°2hh OL6°SEn eLe°ninh Ole’nih ean’ sth 906° Stn eLe°9%n 669°9Eh 2S6 9fh 9eb Lk 2S2°LSh 20E°LEn eBe°Lin G22e°ushm L2l°Leh ETO°LED wee°oth eeoL°9EM 222°9h £99° OSH 969° 9ih OLL°9S Flo 9fh o2h*sto Ten°seh oge°ein 282°ehn ShL°eth I2E°ofh tse osm 129°0hm tein e20°2nn : : fh 682 Ohm 692 Blh’oa2 BLS°oRe SEL*oR2 MO°bBe L91°062 22n*062 COL"0G2 ETO'Voe ESE tE2 £24) tee 025% 22 bnG' 262 6L6 262 62n°ko2 LBO°S6S 9FE°Mo2 HAL*noe 022°S62 O0N9°S62 2n0°962 Een°ro2 1$02°962 Ott Loe MEH" Loz OFL*LO2 L00°GE2 B92*ae2 BIS*BO2 LSL°G62 666°R62 oH2*H62 LIS*662 B0Ro62 FEI°00E 2on°00E 969°00E obf°TO£ SsQg tog nth 208 L20°SOf M69°EOE Tin’noe Szi°Sde 066°SOe LIg’90E 89°L0E O6S°gue Chh°o0k : : £29° te2 Is9° 182 969°IE2 F9L°UES? O9E°TE2 Glortee ofl°2t2 e2e2°2c2 LOW 2k2 929°2k82 606°2E2 sot'see foprste the°ec2 9SO' Oke HAL*ne2? V2uz°nk2 g90°ss2 VtH*se2 ase’sse solose wzeM°9s2 L9z°9S2 ognrLEZ2 Q0n 4€2 wed Lee Of0 BF2 SGF*efe EL9"ete2 Gee ak2 CLE°Of2 29°62 GH0°One OHM’ One LSe°One Lig Ihe eel °tne Lte°2e2 veec°2ve son fv2 Ost *one 8n8° tne 68S°Sh2 OLL°9Me LEN LHe SEO°Qn2 606°@n2 20e°en2 yol ose F29°tse = Bees ee CORO ££0 002 140 002 _ $2b°002 set °oo2 se2°002 £9€°002 Ogn°002 919°002 €224°002 v56°002 out toe Log toe L09 102 Ase*toz szt°z02 90n°202 S69°202 266°202 22°C “ges'eb2 ToOe'k02 Loe "wos aIS°noe 60°02 Lot ’so2 fin°so2 60@ S02 eSt°902 225°902 {€0e°902 GOE*L02 B24°L02 Ogt*gdz 099°g02 225°402 ot2°o02 LOC OT2 2H OTS GaStite Teet2te CEO°Ete VSta°st2 S2o°nmte Oen’ste ene°ol2e oe2 ute ani ate 20%%t2 oVv0°oz2 “(UOLZLppe JUawWLpas snjd syaam /T) € aSeD *SunoqUOD |eULy -g-) aI qR] 92 PED-OS2T Sh6°0S23 Qim°ts2t Sue°2sc2l cE5°252) OSC°EGEI FwS°ES2E FOI Sel G2o°nSel 2£1°SS2I 0n9°Sset mos*gg28 2n9°9S2h mek es2s $29°LS2T 200° HS23 945° BS25- G70° 6520 90526525 -096°6S25 -LUM 0921 Shel 09ZS-SL2°-5921-So9- 8925 - GOb°2923 S$05°2923 S6a°2921 mL2°E920 mH9°F925 SIO°H9OZ2E YGF HG2l DUL MO! Fn0°S92E QLE°S9e) SuzL°S9eZt 1EU°992T ESE°992T 229°9928 066°992) 90E°L9ZS 029°7928 FEH°L9Z1 SM2°y92d 9SS° HI? 99G°HOZE 9Lb°692) QyR° 692} For oged SOL ULSI Hin°ogel : 921°096 £16°L96 b6b°SLE FRECHE SOG OYH- -bNG7966- CEH’? TNYE GLe°OTUS EEO ONUE Lemp Ceub gez°o2z05 2Ka°SEOR AVL UMOL VLE°LMOL GLZL°2508 Ther eguld 262°2908 292° L996 Sim EGE CrboGeOd Vin aLEl end sul CE LBOL Osyg°ugud 0PL°SQOE 696°SB01 975°SG01 Goh ngdt 99Qg°2Q01 EAzZ°OBOE 190% ALU TNUTGIOL HLS TLEL GEG L9SL 99L°F90UT USM OSUT Nog’? MSOs COFSOSOL SOS*SHOT OLO° SLOT FES HES BAO B20I- 2o2e-E2uUl—2om2LIOV--LESSEIOL-SiH2GUGl SS¥°o4n— ULI £06 —140°99b—S2l-0y0 — — - 299°S@L 265°008 {£SS°ST@ 155°0fH 909°Sng SEL°09H Y¥Eo°SLe B2° tou S2S°9CeE Ihe° leo 260°Lfo G95°2S56 900°L96 «9th 386 292°S6H «SAE BOOK 229°0201 F2e°TEOl ANB THY aG°OSOL Ome LSU 26g°f90F Oze°gout gzt°hzot TGS°2L0B- OG£°SLOL FOLTOLOF S05°L90} -GEg*2q05--£2p°9SUI—SS26n0 J—LNG20pEd-929°.66 04-0226 101-166", 0U5-2gn-S66—BLb* 2g0-—- 090°696 pfn°SS6 o79°lne6 Sle°L26 noe°st6 922°0U06 185°989 nlo°228 ei °oSs Gau’Sra@ fSn°efe VSY°elg 299°S0y V20°n0L bob ote Lok hse HE9°OML 92o°H9L ME2°0GL NMIL°SoL Klecbha 292°928 VEE°ena 999°LS8 ——. L62°51e0-—-1ES°ORG— 250°F06 - L26°LS6 - SOQGL1E6-—-6162DN6—-960°.156—SIL° G26 ENG LLb —-£92°986—S20°$ 66050-8661 B2-100I— -.. €59°2008 L5b°2001 $29°H66 21L°S66 186°6R6 609286 668°FL6 YLO°L9L 945°2S6 NGI*In6 GI9°B2H . UNS*STo T60°2U6 92€°ae@ BIS°nLe@ £85°09g 959°9ng 22u°2ce 956°aI9 222°S0g P9OS*toL SLo°LdLL Gnn°nge Sme°OSL LEP “EL 20° nee we ee en V8 °929—. F012 SE9 EO °SH9— 2262089. 119°599_ LOL 649 2 ¥92209..-192°209__fgo°10L vee bbg-..cnorbeg—.- atsctee Cnt’ Une SmO°tse L4£S°09L StL°69L 99n°BLL 699°9RL bel*HHL ££6°008 FSe°9CyY HSS*Iby oM2°>Sta 952°L1e 9p0°6i9 Lo0°ota T2o°Lte 659°SIe mL0°25g MSS5°L09 OVI°20g vea’SeL NOR’ gaL s4tE°teL gee°fde $00°S9L foH°9SL ——LOL°LolL—BVoo° aed bi2°0k1.- 226° ter__nag°?2te 9ff 2 n0L 20y°S69 G1S°199 262° 619 _O010°19 159°299 969°nS9_ 125° 909 £SS°ESS £94°@SS 226°S9S 283°69S Tof’nLS S6S*6LS 06L°nBS 116°9S 925°S6S 2H2°009 L0E°S09 962°039 bet°SI9 On6°6t9 Of5°29 Dbe°g2o £50°f89 196°959 LIm°0n9 @55°f09 Se2°909 095°g09 956°0S9 059°5S9 ——~.-420°269---£99°2S59__€26°259 — 6692159 . .SIp20S9 6142909 S09°9n9_ ELI °nn9d S92°109_ ¢9329f9_ Loe? bf9_ 082°3£9_ 95H" 229-_.._ BL6°S29 «BeS°6t9 «25S5°SI9 gef°IS9 «222°709 260°F09 92—°H6S 139°RES Sot°M0S CQl°SgS B96°19S O094°LLS £59°ELS 920°22h g0n’Seh neL°een 2st°2efn 10S°Sih Itne°otn Ont’2nn t2n°Sne O0S59°eub S2g°sSn ef6°uSo ——-SLo° LS 226°09nh 992L°S9b_. 06n°99b -S20°69n_-SOS° Thm 09u° Ech 228°SLN_ nLO°LLh.. 6622 6Lb._ 99° 0en_SIAQ-tean_Sa9o°2go_ e92°canp 129°feh 929°fen Lon°feh SnO°fgn 95f°2un 2nn len LIC Vu" o66°ALn 90S°LLm LSa°SLo Of0°DLd Sot eed ent Osh Sv0°e9n S99°S9n 229°f9n net°loh nae’eSe 909°9Gn L02°hSn LeL* Sh nGk°onn St6°9nn 2L0°non g20°2ne Se enim ae ----. -. 3p0°Q92. Onn°OL2. nog°2z2_-2n]2°Sre_. €29°4L2_ 0G626L2.. yOSS2yu2__109°naed__.£S9° 982 -_9SD° be __NOZ* Soe B02°C4e 10L°S62 SE2°Lo2 640°662 S2g°00E m9 26E Bee'LOe BEL°SHOL 459°90F QuL°LOE TeL°QGk 994° 60k Lee°UTE 2be°Ote LOTTE Soe°TIE OSn°RIE GOEEIE OFI HIE F9L°OTE NA2°OTE HF9° SOE FoR BOE JO" YOE FOE°LOE B90°90E BSo°90E OPL°FOL--12S°20¢ - Q12° 308. 699° 662-1902 G62 9602 262_. 92525622602 n62__ 266 °262—_1 00° 562_-F9S°692— 1 b0°gge2__ 0u0°002 102°202 LOH *n02 auS°S02 U9L°Q02 UIH°OI2 Ve0'Eh2 O21°Ste Hyl°ele eet ose bes°see TWhHo’s2e CLe°mee 285°922 on2°a22 L2e°oz2 HOE°IE2 BAd°2E2 LSe°EE2 OFF°SES ENE gfe BMU'Lte PM2Q°Lte Lw°ate 20o°OLS -B5£°Of2.. 109° HES. BIL O82. £69°6F2—60S*oS2— ng2°ok2---006°Gh2_.-Shp°afe2_f29°Lh2__9f 12 L£2_J 96° 982 b0S*Sf2_. ELS°wE2 GLS°kES Lis®*2e2° gOn°te2 ame°or2 050°o22 Lin°se2 Sss°g22 O12°S22 L90°S22 259°222 gek* tee 000°022 mOWW4S “A “SZINIVA BNOANOD MAN BHI *p ased *‘Sunojuod [eUuLy *6-) alge, =)S) 17-ft level 240,352 ya? 237,284 ya? a —— 3 3 530,160 yd 522,784 yd SP 4-ft level _ _ Ke — — Ke eee a aaa ae eee eee 7T Case 2.a. Period Considered: Twelve months, January through December, using 1975 WIS wave hindcasts Sediment Budget Summary : Amount of sediment added: None c Op aA gy Amount of sediment transported shoreward from nourished region: 992 yd Amount of sediment transported seaward from nourished region: 96 vale 3 Net amount of sediment transported alongshore from nourished regioml9,444 yd Total amount of sediment transported from nourished region: 29,356 yd Figure C-1. Schematic illustration of sediment volumes transported from region, case 2.a. 94 Sf eleven) mien 14-ft level 4,936 wae — 1ll-ft level 12,020 ya? Se Ke — - —- —~ —- — — -— b ——— 7-ft level —--4J E 1,624 ya? - ee aes tae ak acc FE nerf ai ee Snel SiN Lae eee | ka ee 4-ft level ea Case No. 2b. Period considered: Twelve months, January through December, using 1975 WIS wave hindcasts, but wave angle always set equal to. O°. Sediment Budget Summary : Amount of sediment added: None Amount of sediment transported shoreward from nourished region: 1,624 ae Amount of sediment transported seaward from nourished region: 132 ve 3} Net amount of sediment transported alongshore from nourished region-15,904 yd Total amount of sediment transported from nourished region: -14,148 sail Figure C-2. Schematic illustration of sediment volumes transported from region, case 2.b. 95 -- Cer err erm rm rw ans os —sCd' T= Ft level 27,808 ya? 14-f+ level 3 3 143,192 yd 327,888 yd ——— LTC TEP LVAD ELEY OT VLG LG LEY ALE IA, _— Soa SSS — ll-ft level 2) 3 380,412 yd 599,660 yd ao ELT LILI EE aE LO LP LT LT LD OL RIT LO > Case 2.cl. SIS es er ot — — — —~A— ~—~SY—i«*7-ft level 460,264 ya? -_—-—- — _—_-— — CO - 4-ft level Period considered: Twelve months, January through December, using 1975 WIS wave hindcasts. Sediment Budget Summary: Amount of sediment added: 1,452,000 Rae (on 7- and 11-ft contours) Amount of sediment transported shoreward from nourished region: Amount of sediment transported seaward from nourished region: Net amount of sediment transported alongshore from nourished region: 403,944 yd Total amount of sediment transported from nourished region: Figure C-3. 460,264 yd°(31.7pc¢h 27,808 sale (1. 9pct) So7sepet! 892,016 yd*(61.4pctl Schematic illustration of sediment volumes transported from nourished region, case 2.cl. 96 ae a ie fteleve lio 28,920 ee 14-ft level 5} 138,864 yao 336,876 yd —_ —- are eS tiotia S ae ftalevel sad 3 637,584 yd 414,168 yd ¢ eae ——p SS --—-3- 466,260 yd Case 2.c2. Period considered: Twelve months, April through March, using 1975 WIS wave hindcasts. Sediment Budget Summary: Amount of sediment added: 1,452, 000 i (on 7- and 11-ft contours) Amount of sediment transported shoreward from nourished region: 466,260 vai (32.1pct) 3 Amount of, sediment transported seaward from nourished region: 28,920 yd (2.C pct) Figure C-4 3 Net amount of sediment transported alongshore from nourished region: 421,428 yd (29.Opct) 3 Total amount of sediment transported from nourished region: 916,608 yd (63.lpct) Schematic illustration of sediment volumes transported from nourished region, case 2.c2. Sif —-— 17-ft level 23,728 vdeo 14-ft level 3 147,984 yd 298,056 wae — ae = = — — = = Sse ; 3 598,400 yd 418,072 yd —$»~ —_ Son ui eulantatasia an siete ct 7-ft level =o" 415,784 ya? a Case 2.c3. Period considered: Twelve months, July through June, using 1975 WIS wave hindcasts. Sediment Budget Summary: Amount of sediment added: 1,452,000 vale (on 7= and 1l-ft contour) Amount of sediment transported shoreward from nourished region: 415,784 sek (28.6 pet) Amount of sediment transported seaward from nourished region: 23,728 vale (1.6 pet) 5 2 3 Net amount of sediment transported alongshore from nourished region330,400 yd™ (22.8pcr) Total amount of sediment transported from nourished region: 769,912 ya? (53.0pct) Figure C-5. Schematic illustration of sediment volumes transported from nourished region, case 2.c3. 98 17-ft level _. — SS SSS OO SI 28,452 yd> 14-ft level ——— 3 151,592 yd° TIGERS Sel _—_—— 3 613,548 yd SECS. SC) aoe — > Ia 395,556 yd Case 2.c4 Period considered: Twelve months, October through September, using 1975 WIS wave hindcasts. Sediment Budget Summary: Amount of sediment added: 1,452,000 way (on 7- and 11-ft contours). 3 Amount of sediment transported shoreward form nourished region: 395,556 yd (27.2 pct) Amount of sediment transported seaward from nourished region: 28,452 vide (2.0 pet) Net amount of sediment transported alongshore from nourished region: 348,144 ya? (24.0 pet) Total amount of sediment transported from nourished region: M12), Loe ya? (53.2 pet) Figure C-6. Schematic illustration of sediment volumes transported from nourished region, case 2.c4. 99 4,112 ya? — oe ee eee ee ee — ss LJ ft) level! 3 3 11,240 ya LE ELT TLE TEE ANS Sho eo ZZ eer ae ee E 14-f£t level Be 3 PE SAA 3 54,528 yd DID ap ALT DAT ELE CLAP LTT UTALLCTE TEE PUT TEFL TET TET Ss ee —_— =— =— -— LI=f£t level 32,164 ya? Case 3. Period considered: Four months, January through April, using 1975 WIS wave hindcasts. Sediment Budget Summary: Amount of sediment added 363,000 de (on ll- and 14-ft contours). Amount of sediment transported shoreward from nourished region: 32,164 vde (8.9pct) Amount of sediment transported seaward from nourished region: 4,112 ee (1.1 pet) Net amount of sediment transported alongshore from nourished region: 43,708 vada (12.0 pct) Total amount of sediment transported from nourished region: 79,984 ya> (22.0 pct) Figure C-7. Schematic illustration of sediment volumes transported from nourished region, case 3. 100 85,375 ya Pe 96,950 yd° eS iswadgiyde ——e 134,680 vai ——— —_—_—— —- ie ee — Ss 392,000 ye At=.ftalevel eo = 3 119,813 yd LI LEY ELLY ELIT UIT UO IF EY LY UO CY BD LEP LPL LS, ETL EDITS LLIOF LELOES LD PPL NN EEE Ses he Se on O-ft revel: _- a5; 350nyde LL YH LY LI LEP LES CLT UT LA (EAD UY EY LEG (TLE EDL EP ED ED ALA AT UF (BF TIBI a ne ae ee a) Re ee oe ott level =i 139,391 ee LEFT EG IG EIGHT LET LOL LEG (ES. EDD LE LP LD ETT UTS LP ODPL AED EP LT L3\ S SSS Ss a m BPO ES as _-_— 134,373 yd? © ALIE YY OTTO UTED LY ITER EET FLED (LT ELTON T Ut FRSC Case 4. 275,796 yd? Period considered: Twelve months, January through December, using 1975 WIS wave hindcasts. Sediment Budget Summary: Figure C-8. Schematic illustration of sediment volumes transported Amount of sediment added: 1,452,000 sale (on 7-, 8-, 9-, and 10-ft contours). Amount of sediment transported shoreward from nourished region: 275,796 ya? (19.Opct) Amount of sediment transported seaward from nourished region: 392,000 yaa (27.C pet) Net amount of sediment transported alongshore from nourished region: 96,679 vain ( 6.7pcet) : 3 3 Total amount of sediment transported from nourished region: 764,475 Y¢ (52.6 pct) from nourished region, case 4. 101 APPENDIX D METHODOLOGY AND PROGRAM LISTING OF COMPUTER PROGRAM WHICH CONVERTS BATHYMETRIC DATA INTO MONOTONICALLY DECREASING DEPTH CONTOURS In order to simulate prototype shorelines (and in this case to help verify the numerical model via Channel Islands Harbor data), the (x, y, z) data points must be transformed into a form suitable for use in the model (i.e., bars can not be present). First, the bathymetric data have to be put into a form with fixed longshore and offshore spacings (i.e., ax and ay equal constants). This can be accomplished using one of the many available canned programs which do the interpolation. The problem is then one of finding the most suitable value of the constant, A, in the equation h = Ay2/3, However, as iS usually the case, the exact location of the shoreline (h = 0) is unknown. In addition, one requires the added constraint is required that the volumes of sediment (or conversely, the water above the profiles) balance. The problem is solved using LaGrange Multipliers and the Newton Raphson technique for non linear equations. The equation to be minimized is IMAX IMAX 2 F(A,ydel,, ydel,, ... ydel Ses anh - jh ) 3 1°? ’ (D-1) 2 IMAX fail) ail meas; 3 pred; ; where A is the scale parameter in the equilibrium beach profile, ydel. are the locations of the shoreline for the IMAX profiles, h is the interpdlated depth from the survey, and Nored is the depth predicted by the equation 2/3 - ydel,) (D-2) g(A,ydel,, AAA ydel ray) = V oe Brine of i Faty - yaet ,)2/° dy P i=1 ydel, IMAX a 3 5/ 3am = 2 R Ax Alye - ydel, ) = Vane (D-3) where Vpreq is the predicted volume of water above the profile to the reference datum, Vmeas is the measured volume computed from the survey, Ax is the longshore distance between onshore-offshore profiles, and yf is the distance offshore to the last point on each of the measured profiles (it was a constant after the interpolation routine was used). 102 LaGrange Multipliers procedure says to form the quantify F* as eg eye (D-4) take the total differential of equation (D-4) S _ (dF dF dF* = dF - x dg = € dA + dtydet, J d(ydel , ) aa O56 dtydeTs aay aC) d d d ce) (4 dA + atyder 7 d(ydel, ) ieiaetie\(s cee sy) Rearranging Se eee dF d Ole 5 ve i) a ea mA tye, AT Se oe (D-6) It is clear that the terms in brackets in equation (D-6) must individually equal zero, however this leaves (IMAX + 2) unknown (udel i = to IMAX, A, and a4) and only (IMAX = 1) Equations. The (IMAX + 2)th equation jis taken as equation (D-3). The following system of equation then results: IMAX JMAX ede dam 2/3 is 2/3 CRaecivat dat fr” oe eee ec con ee Nye ly del ys My ven er Vel jel Vs IMAX SNE zB Ax (Y~ - ydel rae (D-7-1) 103 wen Ona aydel,) ren atyde1,)= 2( Peas, j . AY > Bb) 2/3) -1/3 2/3 *(2/3 Aly, ar ydel,) +r ax A (ye - ydel,) (D-7-2) JMAX 278 dF = » Fost - Aly .-ydel ) d{ydeliyay) — dlydeT yay =1 [zen meastuay. j IMAX, j IMAX “1/35 Als * (2/3 A(Y qx 57-¥92) ray) + ax A (Ye - ydel ray) (D-7-(IMA¥+1)) IMAX V See 5/3 meas ~ , (3/5 ax Aly e- ydel,) ) 1 (D-7-(IMAX+2) ) Because Equations (D-7) is a system of nonlinear equations, it can not be written in matrix form as a [D] [x] = [E] system of equations (the brackets denote matrices). To solve the equations, a Newton-Raphson Iteration technique for nonlinear equations was used. This is done by differentiating each of the (IMAX + 2) equations with respect to each of the unknowns, the resulting equations are then linear in terms of Aa, Aydelj, . . Aydeltmax, 4X - The resulting matrix is inverted to obtain the a(unknown) and the quantities are added to the original estimates to produce a better estimate. This iterative procedure is continued until the changes become acceptably small. The solution converged rapidly. Benenaillye the first row of the matrix to be inverted is (aj] represents the kth row and the 1th column of the matrix). IMAX JMAX aq) % cS 2(y. ydel ye i=l jel JMAX . 4 -1/3 f ‘ 2/3 oa? as 3 (Y. f ydel,) (reas ij any 5 ydel) ) 104 a =a (y . - ydel (h 1, IMAX+1 3 ‘7 IMAX, j IMAX meaSimay j jel 2A(Y ray TLD crepes) IMAX ay IMAX+2 =. 05 Ls AX (Ye - yde1 )°/? ] (D-8) j=l The second row of the matrix is as follows: JMAX é 4 Ss) 6 1/3 ao 4 Si [3 h meee i - ydel,) ie Aly, jydel,) / ] is VJ D + AX (Ye - yde1,)°/° JMAX i 4 SAS a2 =21/3 - » (2/3) ax A ly, ~ yder,) 71/9 e23t= 0 49 IMAX+1 = 2 es 2/3 i a> IMAX+2 7 ox A (ye- ydel,) (D-9) The third row is simply these elements repeated except that the ones on the right-hand side of the first and last elements are changed to twos, and the a3 3 element is similar to the ap 9 except the ones on the right hand side become twos. The remaining Column elements (i.e., those when the k = 1) are zeroes. This process is continued to fill the array, except for the last row. The (IMAX+2)th row is as follows: 105 4IMAX+2,1 7 | Bax (Ye AIMAX+2,2 = -ax A (y~ - ydel,) AIMax+2, IMAX+1 ~ “4X A (Ve - ydelinay) 2/3 AIMAX+2, IMAX+2 ~ © (D-i0) The E matrix in the [D] [x] = [LE] system of equations is IMAX JMAX 28 2/3 Se) We Bieaeth = A(yi> 2 -ydell. im wily. aasy.del) 1 il jel meas; 4 lied 1 Is 1 IMAX SUG! OF (3) Ax (Ye - ydel )°/7] j=l JMAX : 2/3 ane -1/3) ON ha ee 2(h meas, Aly, 4° ydel,) )((3) A Whe ydel, ) ti enAxaA (Ye - yde1,)2/)) JMAX 2/3 E = - ff) 5 2(h -Aly . - ydel ) ) IMAX+1 aa meas Tmay j IMAX,j IMAX *((§) A (yy - ydel,) nie) + (ax A (Ye - ydel,)°/°)] IMAX 3 5/3 E =e pe ese Cm) MAA Gye mvdelll =) mand) i= av ] IMAX+2 72 5 f I meas (p-11) 106 The [D] [x] = [E] system of equations was then solved, as explained previously, by solving the x column vector (which represents the changes in the unknowns, AA, Aydel] ... Aydelymax. AA), adding these changes to the respective variables and iterating until a final solution is obtained. The computer program which did these calculations for the Channel Island Harbor simulation follows. A user-supplied matrix inversion routine is required (Line 37,200). 107 GRESET FREE CreceeercoveessPROGRAM CIH/BVALUE 1 FILE S(KIND=PACK, TITLE="CIH42076A" ,FILETYPE=7) FILE 6(KIND=REMOTE ) C*THIS PROGRAM USES THE INTERPOLATED PROFILES OF CIH. C*IT FINDS THE LOCATION OF THE SHORELINE, YDEL AND THE BEST C*FIT LEAST SQUARES "“B" VALUE FOR H=BY**2/3 C*USES LAGRANGE MULTIPLIERS TO CONSTRAIN THE VOLUMES(SO THEY ARE EQUAL) C*THEN IT USES NEWTON-RAPHSON ITER FOR NON-LIN EQS DIMENSION X(40) DIMENSION WKAREA(600) , AMATRX(23,23),BMATRX(23, 1) DIMENSION Y(40,20),Z(40, 20), YDEL(40), JBEGIN( 40), YDELI (40) DIMENSION DYTWO( 40, 20) ,DYONE (40, 20) ,.DYMTWO( 40, 20) ,.DYMONE (40, 20) NIMENSION DYMFOR( 40,20) ,0YFOR( 40,20). YDONE( 40,20), YOMTWO( 40, 20) DIMENSION YOMONE (40,20), YETWO( 40) , YEONE (40) , YEMONE (40) DIMENSION YEMTWO( 40), YEMFOR( 40), YEFIVE( 40) EXPON=2./3. THIRD=0. 3333333333333333 C*FIRST READ IN THE PROFILES FROM DISKPACK. DO 1 I=1,34 DO 1 J=1,15 1 READ(5, 100) X(I).Y¥(1,U),2Z(1,u) 100 FORMAT(14X,F6.0,F5.0,F5.0) C*NOW WE MUST GET A FIRST APPROX FOR YDEL C*WE WILL USE LINEAR INTERPOLATION TO DETERMINE IT. IBEGIN=1 IMAX=21 JUMAX=15 C*CHANGE PROFILE TO SPAN 1 TO IMAX(IF ALREADY DONE,WON’T HARM THINGS) ITEMP1=1 ITEMP2=IMAX-IBEGIN+1 K=-1 OO 777 I=1,I1TEMP2 K=K+1 DO 777 J=1,JUMAX Y(1I,U)#Y(IBEGIN+K,J) 777 2(1,.JU)=Z(IBEGIN+K, J) IMAX=I TEMP2 OX= 100.00 DO 2 I=1, IMAX DO 3 J=1,JUMAX IF(Z(1,JU).GE.0.0) GO TO 3 C*FIRST NEG POINT ON THE PROFILE IS SEAWARD OF 220.0 C* WE MUST ALSO REMEMBER THIS LOCATION. C*IF Z(1,1) = caannsWaa ai Wa=Paniaa Wie eco aliana al ee PaO Se Ge er on mal IEC ae rs [Se Ss Oe Tee $e ELUTED ae sa) ob) fonts a ae GH bse wa why Ssenpes 2 eles fea eeley gnitles ee Tne a1 5 an. hy noe Sawa Sa PS Ee SSS TC MT DUTT Ce ore RE a Oo Ste Walaa san a Uap [Oy 2 RED eps wen ap ys Bley wv elals ew eh, a aishians weeps es se sp. eee eo TO PH eee ashi ha wpe se wap na els oe hes ees steely ci Gee EADS 8 ED AI cE ETAT) FO SES i CEES PE SURE SUES HS J ALPWIS Sel By 5 till CARDS 7-36. empires tl" Yee we eS Ne a ad SSS 7 eee ae, eT Sy apes efi ewan efile eye alesis wis s ay se Gauls: iT ooo Rigi PUL TUCOU OC OFOOD ONO. io oNG Gegooedd | ae ile Be) eerie SIE ree] PORTER al PE Ed Ce eC S791 4 SELEB 57 38% Selo ADE ae ayak Oe 5 yn ey | PP Say PoE TSN O TIED DNA D Hon Gena gp sd it | aie Dd raNd! Digeaha) a peDeDan DIC) 22, Daud deaiiaeale 02 ghclidied: daa |2e2) Moe QU 22212) 2h2) Qnallahek okahall2) 2h2h2 railed ee 2k} Ona eae 22 SPS OF we ik Cee Ta and Is DUSTRSaLE Weer] Ut ern cet ys Sarr Dyn PUP Pa KET) CU LS eT eb YS a ED sist fog. es leg gs Sec vl oD PDISis (Gh Pee AD! tb ea Sp th UN Mae ae gee ig 2 ee \ . ae ; ROC ae nCn Enns Cr Cin Ome GUGn in Mnmnc rie (Cie nie] ici ae 4 | ; aR pete TY Poa Wi Sek lars) pone BSE) FURS ra on) RTDs Dito (EYER, OO a Cart nC CaS Da ea 73 RLS Or Ses OF He oO fb baeo cc bapa, MEO ESSES S555 55991585555 595559555 5559559990955" 2 Ooo S 3 Heim eT gt a ANE Mon P| hao alee jes Sicsctherousessakeossueessssssebesessscersssssassschossacussscassons a Hesa Daca Sorefer aie: boon wher analy edn ye Mesa meagan a2 age aefas a2 ag aa og hs oz 27 aa sg fea so SSO OMN VE eae ec | i rica 0 é | : | 6G GR TCO AEG IB i peapeteseassssasasssceseessssassssacaeaesasss oem OES Canty CCC OCCT aR He GB} LUNE AB 8 BEAD 2A) a0) DG) CRY ED SPP) Behe CF SEED Oc sp 999591998 EIS 79S 7999999 999999-9199999999 9999999599 9999939999979S I 99959909 Abeediat lhe a | | : ve fea MND Th de pcb TE Bd WGP aD dag th at ce de de CPO) CME Ee ed he al Figure E-1. Card deck input for program verification. 12 FILE DUM INPUT: 10.000 100 3.000 0.0500 0.220 300.000 200 0.1500 100.00 5900 21600.00 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 {13 1900 2000 2100 2200 2300 2400 30) 2900 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500 3600 000°0 000 O 000 O 000 O 000 O 000 O 000 0 000 O 000 0 000 °0 000 O 000° 0 000 O 000 9 000 O OVO O 000 0 000 0 000 0 000 0 000° 0 000'O 000 0 000 O 060° 0 000 O 000 0 000 O 000 0 000'°0 000 O 000 O 000 °O 000 © O00 O Goo O MO1104 “XO SLaOUSNVaL JYOHGONOD dtd ol 2AINON 'Sd31S-3NIL JO YIBWIN G3SdV19 W101 Ie “Ob > ATNON “6=AINNN “@2ALNON “L-ALNON “9=AININ “G-AINON “P=AINNN "€=AINNN “@=ATNIIN “b=AINON ee ee ee ee ee eo ie i eer $e°7€ 00° SZ oo Lt OO FI oo 4h OO 4 oo'sS oO °C oo 2 Go 4 SMOVIOS SV 34V SYNOLNOD N33M139B SiH1d30 BL PP CH COCO COOH es Cre Ser eo es Ore eee eroseeesereseooresine S@ PL9% OS 959} bh OSOK EL O9L 94 HOF BL 7SZ JL LEI 'O' 89 co 4 00°O SB PL97 OS 959} bh OSO! CL O9L SL b9P BE 7GZ tz LE; 10 89 tS FE 00 0 SMOVIOS SV JUV XVWI'E=1 “SANTVA-A AUWOMMOG 4H 000 009172 =1730 “SONUDGS NI d3LS-I3WEL HL 000 O01 =¥O 'd91S-39VdS JSCHSONOD Jt JO GA IVA Gt dILSIWI1° d3LS JOVdS 31 NI Qv3a4 TOPS CR e Ho eos reese oes meet eros eseeseessseeeseee C/TeekVeH OF BL NI OOSI'O =NVIOV JO 4MIWA 3k NVIQV 40 ANIWA [hth Y31N3 MON eerccres ecoos eecererescecos Gt al ees ercere 1V G31V907 SI NIOuD 4 AIHWON Gh cece eceese eielsrelsienere sisloistelsleleleieis(aceveis elsrss:e ova ame) SWVIO "43L3WVIGO IN3W1G9S JHL 00SO O -99V4dS "4OVI HOVIG Bt 4O 3dOIS JL 000 Ss FWH39G WHI Bt JO LNYTIN Bed 000 O0€ FALLIPS “JMNILONALS Jd 4O HIDNID Dad WVIQ CNY "2394S “WUIH “ALLGCS YOd GWE Sil ecoee eeroes Se i i a ear e08 te Sea oo ctl O Got 6c1 O (54 {kex{9) $1070 60°70 bo Oo 600 0 L000 #00 0 Woo oo ROO! ooo mo°0 00 oO cOO O coo O 0 ctOO O cOO O (¢) ZOO 0 coo Oo ZOO O coo 0 116 bib OSOt ble OSO} bly OSOb whe OSO} ble OSO| bib OSOl whe OSOt bib OSOt 97L O9L 97L OSL 97L°O9L Spl vor 9tL°O9L 97L°O9L 97L° O9L 97L'O9L 97L O9L 000° 0 000'0O 000 O 000 'O 000'O 000 'O 000° O 000 °O 000 °O 000 O 000 O 000 O 000°O 000 '°O 000 '0 ie) 000 °0O 000 O 000 O 000 0 000 O 000 O O7n Oo 000° O 000 O 000° 0 000 O OOO O 000° O 000° 0 000 O 000 O 000 O 000 O 000 O 000 O 000 O OOO O 000° O 000 O MO1104 “xO°S1LYOdSNVYL AYOHSONOT abd OE =eAINON ‘Sd31iS-3W11 JO YABWIIN G3ISdV13 IVLO1 JHE “OE=AINNN “67T=ALNNN “@Z=ALNIIN “L@=ALNAN “97=AINIIN “G@=AITNIIN “pO=AITNON “ET=ALNAN “€0=ATNNN “bO=ATNAN bly: OSO! blp OSO! bib OSOl by OSO} pIb OSOF bib OSOt FIb OSO! vir OSOt bib OSOl bir OSOI ple 'OSOt ple OSOt vib OSO! blk OSO} bie OSO! FIP OSOF vib OSO1 viv OSOt blip OSOI PIF OSOF ple OSOl ple’ OSO| ble’ OSOl bib OSO} vib OSOt vib OSO! FIP OSO! bib OSOt vib OSOl blr OGOI bib OSO) ble’ OSO! bie OSO! wih OSOl wIy OSO! bib OSO! bib OSOL vip OSOL bie OSOb Fhe OSOl bie OSOI vib OSOt QZL°O9L 9ZL'O9L 9%L'O9L 97L'O9L 97L'O9L 9%L°09L BCL O9L BEL OSPR SCI OSLER ICLEOSL 97L'O9L 97L'O9L 9%L°'O9L 9%L'O9L 9L O9L 9Z7L O9L 9ZL'O9L 9tL°O9L 97%L' O9L Y%L OL Q9TL°O9L 9%L°09L 97L'O9L 97L'O9L 9TL'O9L B7L OBL BCL O9L 8SL POP bel b9b Chl POP 9SL°p9b 19L' b9b 99L H9b TLL Ob eee 1S2 bi19 OST Lit St 89p 9Eb O9S bth 677 GEN SESE LS Sie€ 6S 069 89 €9p te Glb LTé Gee 1e 470 O- 676 O- 6700 786 7ST Lye sGt S09 4S% 6rpe SSt PvOr SSz@ 90L LEt sei 9€+ be SEI Ove OSt tet ort 971 O9L 92L O9L B%L O9L B%L O9L BEL OFL 972 091 9%L O9L BTL O9L BtL OBL BEL OL 97L O9L 97L O9L Y~L O9L Y9~L O9L YL OBL LSL’v9p QSL P9b SSL’ b9b PSL b9b ESL P9b CSL W9P OSL ¥9p Gel '9P BL b9b SHL VS tpl b9b Obl’ P9b Obl v9b GEL b9h ObL HOP Ih POP tpl bob bet v9b BbL PIP 2GL v9OV 69L b9b SLL’ p9b ELL bob PLL 9b SLL vob GlL W9P GLL HOP SHENID DEL OV OD SELL vote tLL'p9b OLL'P9P B9L° b9b LOL VIP 99L b9bP SOL 9b POL b9b E92 b9b 192 b9P OSL VOY 6SL ¥9b BGL FOV e6e'7St Sie'z7St sel z%St Gr9 TST PSS ZSt BSb e%St LSE %St Gre tTGt «web CSS) TNO CSE @Sp'1GZ BOE 1SZ BSI 1SZ 110 1G% 218 OSt LHL OSE €v9 OGZ 69S OSt SEG OSZ 65S OS¢ Ltp SSt €6E€ SSt G6IE'SSt SIZ SS% O60 GSt 156 1S¢ SOB vGt 669 £G% GOS YSZ 6SE SZ 18O0'pSt ~%S6 €St EB ESt BIL ES~ B09 EST LOS’ EST the ES2 OS9°LE} eS LE} 9ES LE} 69h LE 76E LEI pp Sl LS6' res BE FE! B69 EEF 168 CEI 8O0b' Lb) S68'Sh} 98S bhi Lb EME 12S CHI Lp6 BC+} E1L' BEL GIS BE€+ BSE BE SGtz BEI bit Bel 7ZO BEI 4pO 89 s6e 19 (27°19 @SL 9L BPS SL StO 89 400° 89 e686 19 b96 19 9€6 19 zoo 149 67L 99 o6e 99 696 S9 Grb So LSt eel €10 €L£ LSO ZL 7 a rpe9 OL oss 89 6€b 89 4Se€ 89 482 89 977 89 wel 69 ert 89 e799 4E The Fe Gre ve 991 SE 464 SE 619 4€ SIS FE oro 4e €09 +e G6s 1e€ BS7 IE Op Le 666 O€ 608 O€ €9S O€ 46b ve e668 Ce 966, C€ 466 72E ce9 Ze vel te bee be pie be t69 IE pL9 te 199 FE oso Ie 000 0 000 'O- 400 °0O- 400 O- tOO O- €00 O- 6z0 O- Oro O- 9S0 0O- 810 O- 801 O- 4Sb O- th= O- et6 O 4€8 0 s89 0 ves O 4Ov Oo b6z O wiz O 41700 s10 0 410 0 800 0 900 0 poo O £00 0 WZE ESC per €S% OSGi SZ L90 ESt CBE 2S~ ooe .e€' O6b LEI LSO LCh 968 9E1 zwOL 9EN 6c6 ici t7e OC} cIG 67h 466 L2h Wet Deh Pie tut cco IPl LGY Opt €L6 6E) 89S 6EI cre Let 818 LEI 618 LEI ESR LEW) 9OL LEI tO6 19 858 19 CCe L9 EELS LS Spo 19 608 19 pZO +9 990 C9 Océ 19 LtS 09 Sib Oe S69 69 SiH Be ) CuO 69 v96 489 61) 69 960 89 §LO 89 es0 89 110 69 Ses ic CLS VE GGG Ie CES" VE TOG IE Brz OC 6re 62 1GE 62% 4GL 82 SO 82 gb 2e Bvt Ze tor ce Bue IC tOG FE ero se 9c9 IC HESEKE 9c9 Fe C79 FE roo O- soo O- BOO O- 440 0- Sto O- vet O- 4Op O- GES O 989 O- tce O- tsi O 601 O #10 0 960 0 Ovo O TOO O 400 O 100 O 000 O 000 O MO1NI04 “A “S3NIWA YNOINOD MAN 3rd WN, 0000 °O 0000 'O 0000 'O 0000°O- O000'0- 0000'0- 0000 0- on00 Oo 0000 0- O0000'0- 0000 '0- 0000°0- 0000 0- 0000'0- 0000'0- 0000'0- 0000'0- 0000'0- 0000 O- 0000 0 0000 O- OOOO O- 0000 0- 0000 O 0000 °O 0000 'O 0000 °O 0000 'O 0000 O 0000 '0O 0000 '°O 0000'0 0000 O 0000 O 0000 O 0000 °0 0000 O 0000°O 0000 'O 0000 °O 0000 'O 0000 °O 0000 '0 0000 °O 0000 'O 0000 0 0000 O 000U O 0000°O 0000 O 0000 °O 0000°0- 0000°0- 0000'0- 0000'0- 0000'0- 0000 0- 0000 O- ©0000 0- 0000 O- O00N0 o- 0000°0- 0000°0- 0000°'0- 0000°'0- 0000'0- 0000°0- 0000°0- 0000 0- 0000 O 0000°O0- 0000 O- 0000 O- O000 O- 0000°O0- 0000°0 0000 °O 0000 °O 0000 'O 0000 'O 0000 ' 0 0000 O 0000 O 0000 ' O 0000 O 0000 O 0000 0 0000 °O 0000 '°O 0000 'O 0000 '0O 0000 '°0 0000 O 0000 'O 0000°O 0000 '0 0000 '0 0000 '°O 0000 'O GG00 oO 000 O 0000°0- 0000°'0- 0000°0- 0000'0- 0000 0- 0000 0- 0000 o- 0000 O- 4000 O- 40000 4000°O- 4000°0- 4000°0- 14000°0- 14000°0 4000'O- 4000°0- 4000 0- 4000 O- 4$000'O- 4000 0- 4000°0- 4000 O- 4000 °O- 4000°O 4000 'O 4000 °0 4000°O 4000'O 4000°0O 4000 0 4000 O 40000 4000 0 4000 0 4000 0 4000°0O 4000°O 4000°O 4000°O 0000 °O 0000 O 0000 O 0000 'O 0000 0 0000 '0 0000 'O 0000 '°0O 0000 O 0000 'O 0000 'O 0000 ° 0 0000 ' 0 0000 'O 0000 'O 0000 0 0000 0O- $000 90- 2000 O- €000 O- bOo0O'O- 9000°0- 6000°0- 7%400°0- 9400°0- 14200°0- 4200 0- peoo0 0- EProo Oo vs0O O- 4900 0- E800 O- z010 O 6410'O- 64110'0 tO10'O €800°O 4900°O b’SO00'O €vo00'o beoo Oo 4700 0 1Z00°0 9100 0 t100 0 6000 O 9000 '0 ~OOO O €000 0 zO000 O +000 '°0O 0000 O 0000°O- 0000'0- 0000 0 0000'0- 0000 0- O0000'O- OL00 O 0000 'O 4000 °O 4000 °0 7t000 0 €000'0 €000 0 bO0O O S000 O 9000 O 40000 8000 O 6000 '°0 0100 °0 7400°O b4100°O s100°0 41C0'O 0z00'0 t~OO O StOO O 4Z00 O O0LOO O beOO O coo O +00 'O }POO'O- BE00°O- EOO'O- O£00'0- L%00'0- Gz%00'0- %2000- 0200 0- 1100 0- S100 0- P100 O- ¢400 O- 0100°0- 6000°0- 8000°0- 4000'0- 9000'0- S000°0- p000'0- €£000'0- £000'0 zO000 O- 1000 O 4000°0 0000 O 0000°0- 0000 0 4000 °O 4000 '0O 4000°0 z~000 'O tOOO O €000 O OOO O bOoOO O 5000 O 9000°O 4000°O 8000 °O 0100 '0 t4100'0 b100'O 9100°0 6100 0 tZ00 O 9c00 O OEGO O ScoO O tPrOO O BrOO'O 8POO'O- ZPOO'O- SEOO'O- O£00'O- 9200'0- Z2200'0- 6100°0- 9100 0 pi00 O- f€100 O- 0100 0 8000 O- 4000°O0- 9000'O0- SO000'O0- %000 0- E000'0- £000'0- 72000 0- tOOO O- 4000 °O- 4000 O- 4000 O- 0000 O- 600 O 0000 '°O 0000 'O 0000 O 0000 O 4000°0O 4000°0 4000 O 4000 0 c0O00 O zO0O0O O tooo O €000 0 vOOO O S000 °O 9000 °O L000 '°0 6000 0 4400°0 €100 0 9100 O 6100 O tcOO O 4c00 O teO00 O L€00°0O £€00°O0- ZE00°0- Lz%00'0- %200°0- 61CO'O- 9100°0- €100 0- 4100 0- 6000'0- 4000 O0- 9000 O0- GO0O O- booO'O- €000'0- €000°0- %000'0- %000'0- 1000°0- 14000'0- 14000 0- 1:000°0 0000 0- 0000 O- 0000 O0- OLLO O 0000 °O 0000°O 0000 ° 0 OOcO O 4000 °0 4000 '0 4000 °O 4000 O 4000 0 cO00 O c000 O €000 0 €000'° 0 bO00 O S000 '°O 9000 °O 8000 '°O 6000 '0O 4400°0 €400 O $100 0 4100 O 8100 O 6100 O 0z00°O OtO0'O- 6400 0- 8100 0- L100'0- Si00°O0- €100'0- 4100 O- 6000 O- 8000 O- 9c00 O- SOOO O- booo O €000°'O- €000°'0- %000'0- 2000'0- 4000 '0- 4000°O- 4000 0- $000 '0- 4000°0 0000 0- ©O00G O- 0000 O- G0O0O0 O MOVIOd "AO “SLYOdSNVHL JYOHS4IIO-NO 3HI 000 '0 000 'O 000°O 000 0 000 O 000°0 000 0 000 O 000 0 000 O 00U O 000 °0O ooo 000°O 000 °O 000 °O 000 'O 000 'O 000 0 000 °O 000 O oco O 000 O 000 O 000 °O 000 000 °O 000°O 000 '°O 000°O 000 O 000° 0 000° 0 000 '°0O 000 O 000 O 000 O 000° 0 000° O 000°0O 000 °O 000 '0 000 °O 000 °O 000 O 000 O 000 O 000 O 000 '0 600° 0 400° 0 400'0 4000 4000 100°0 400 0 400 0 400 O 400 0 400 O 400 O 400°0 400° 400°O 400°O 400'0 400 '0 400°O 4000 400 0 400°0 400 0 100 O 100 O 400 °0 400 400°O 400°0 400°0 400 0 400'O 100'0 400 0 400 0 1CoO-O 400 0 400 O 400°0 4000 400°0 400-0 400 0 400'0 100 0 ‘00 0 400 0 400 0 100 0 400 0 1G0-0 6L€ 0 61.€'0 6.e€ 0 6l€°0 6LE 0 6L€'0 6L€ 0 6LE O BLE O 6LE O 6LE O 6Le°0 6LE 6l.e€ 0 6L€'0 61€ 0 6L€ 0 61€ 0 6l€ 0 6.€ 0 (HWHESO) OVE O Ove 0 Ost O ose oO OBe ose oO ose oO 6LE 0 6l€ 0 6l€°0 61€° 0 6LE 0 61€ 0 61E O 61€ 0 6LE O 6Le°0 6le€ 0 6LE'O 6LE€'0 6LE°0 6LE°0 6lLE O 6ie O 6l€ 0 614€ O bLE O 6/€ O 61e O 6re O Bre Oo Bre O ere oO ere Oo (HAS=) Lve oO Si ea0)) WIE= 0) Gre O Spe O sre'o Gre Spe oO See oO 9Pre 0 Lee Oo 6re Oo tSE O sse 0 O9c O G9E O vie O 11€ 0 B6E'0 LLe bic O 99€°0 [o}:] sme) ssc oO tSE O 6re€ oO Lre O 9vVE O GIte oO Ste=O Spe oOo Spe'o Sve oO Sre oO 9re'O 9bE O Lve oO UAS?O) ere O ere oO Bre O HeE © 6re O (5=(0) sto Sto sto bz Oo vzi-o cto za 1210 Oct oO gis 0 Co mre) bib O ‘oe 801 O bOI! O 001 0 S60 0 680'°0 780°0 PLO © S90 0 SGO 70 BLO O 'SO O 000°0 4SO B£0 0 sso'oO s90°0O blLO O t80'O 680 0 S60 0 oO! O POL O BOL O sbb Oo Am ee) 9140 CT mre) oy Ware) i ee) ttt oO €z1 oO rl ane) pzi Oo sti oO StL O Gti Oo Geo 440'0 410°0 410°0 410°0 110°0 0°70 10 © Woo $100 110 0 140 0 440°0 440 410°0 o10'O 010 0 o10 0 o10'O 600 0 600°0O 800 O BvCO O 3900 © oo O 000° 0 800 900 0 800 0 800 0 600 0 600°0 o10 Oo O10 O 010 O 010-0 10 0 Woo 4410°0 410°0 410°0 410°0 4400 4410 0 10°00 410 O 4100 10 O 140 0 100 110 0 z00'O tOoO0'O tO00'O t00'O tOO O tO0O O t00 O tOO O COO O zO00 O coo O tO00°O zoo t00°O t00'O too 'O 700 O zoo O z00°0 z00°0 z00 O z00°0 TOO © cOO O 000 '0O TOO too O TOO O t00'O 7000 t00'O z00 0 z00 © zO00 O €G0° UO tOO 0 cOO O tOO O tOO'O too O t00 0 tO0O0 'O z00 ~*~ tOO O zOO O ZOO O ZOO O COO 0 tTO00°O COO O 118 ple OSOl bib’ OSO} bhb'OSO! bib’ OSOl FIb OSO! PIb OSO) plb OSOb Pip OSOI PIV OSO} FIP OGOF PI OSO ple OSOl bie OSOF bib OSO! bib OSOt blb'OSO! bly‘ OSO! bib‘ OSOl bib OSO! bib OSOF bib OSOI PIP OSOb ble OSO! bib’ OSOL pip'OSOl bhp OSOF bh OSO! bib OSOF bIbe OSOI plp'OSO) php’ OSO} Fie OSO! bib OSO} viv OSO! PIy OSOL PIP osot rit OSOb pip OSOl bhb'OSOl bib OSO! bie OSOl whe OSOL bhp OSOb bib’ OSO} bib OSO! bib OSOI PIP OSOI the OSOL PIP OSOl tip OSOt QZL'O9L 9TL'O9L 9TL'O9L 9%L'O9L 9%L OBL tL O9L 9%L°09L 9%L O9L BL O9L BEL O9L YBtL OIL 9ZL°O9L 9TL'O9L 9~L'O9L 9TL'O9L 97L'O9L 97L°O9L 9%L'O9L 9%L'O9L 9tL O9L 9ZL O9L YBZL OBL BEL OSES ISLS OFE 97L'O9L 9TL O9L 9~L O9L B9~L'O9L 9TL'O9L 92L O9L 92L'O09L 9%L O9L 97L O9L 9ZL O9L B@%L O9L BEL O9L 9tL O94 9ZL°O9L 97L°O09L 97L° O9L Q9ZL'O9L B7L'O9L BEL O92 9%L'O09L 9%L'O9L 97L O9L B%L O9L YL OBL BEL OSL —9TLIOBE 8SL°b9b 9SL'b9v PSL’ h9b ISL bob G6rL H9F ORL POP PRL ror TpL p9p GEL V9p LEL t9v GEL v9OV CEL b9b FEL b9r OCL b9h G6ZL H9b BTL HOP @ZL b9b 6TL bob OCL b9b TEL P9t SEL b9b GEL ror pel bob Gbhl SV 9SL'p9b E9L'b9b 69L ¥9Pb FLL’ Pov BLL YOP 18L°b9p CBL P9b SBL Pop 9BL 19 BBL POP BB PIV BBL vor SB8l POV POL PSb TEL’ P9b OBL POP BLL 9b BLL HOP PLL p9b TLL b9b OLL POP B9L PS! S9L F9P ESL POV 49L b9v BSL POV Ze6°7St OSB ~ST ITL TST CES TST WIP ~SGe CEE @S% 9617S “4SO0 ZS E16 I1S% 99L'1G% 919 196 Z9p'1SZ 9OE° 1S G6rt 4St 66 OST THB OST L69 OS? p9S OSZ Lr OSt ESE OS% OG OSZ Y99¢ OGZ Eb OSE O6E OS2 60S'O0St Sp SSt ELS SSt 699 SS%T S69 SSt 19 SSz B09°SSZ pIS SSt LEE SSt 9% GSSZ OZ GS& E696 PSZ vie pSe LS9'pSt OS bSt Bre’ PST B6l HST 14150 HS B06 EST 69L €St €€9'ESZ 10S ESt TLE ESt Wee CSt Siti CSE CBE tTSZ SOL’ LE} S@S'LEb B9p'LEb OSE LES 72e LES tBO LES 976 9E1 «BRL SEI LvuG 9€4 SIE SEI B8rO 9EI Or. SE} pBE SE} 116 FE E6r PE} GEE CEI Bec CEI pSS TEb P69 FE 669 OL} LvS 6Zt E'S eI £89 92b O89 HCI pSt C7) Oph tSt 7hS OS) BIL Bri ZEI LPI 098 Sri Obl brh SFL EPL 958 CPI vie tbs ELv tel O76 OPI cpr Ove O€0'Or) PL9 6El 99€'6E! 660 6C} B98 BEI 399 BCl 68b BE} CE BEI ‘6b BCI p90 BEF Gr6 LE} 8BcB LES 90L LEI +rO' 89 ee6 19 €c6 19 918 19 bie 19 vee L9 ES9RL9, 69S L9 esr 19 LEECES GEV ES 186 99 [VEYE SE) sos 99 w64 99 478 S9 ele S9 t7SB P9 672 PO 686 C9 big c9 OwS 19 91€ O09 Grp6é 8S phi LS LS€ BL 6th LL OOL' SL e6r bl 99b EL t6S TL poe id bee bL 904 OL voc OL t6B 69 OBs 69 ere 69 860 69 €16 89 LSl°89 979° 89 91S 89 \%p 89 Ove 89 Olt 89 B07 89 OSt uo S6O 89 'rO 89 eto Fe poo te pes te ios 4€ zes te gos 1e€ GLb Ve Sev Ie Lee ue ace “Ne LStT NE OLb te e€90 Fe e€e6 O€ ELL O€ LLS OE 6€e€ O€ 8rO O€ S69 62 OLt of POL 8% Bt Be Lev LZ 67L 9E TZ4 9S Bb Le 71S 9E bSL'SE SLO SE O8b Pe SL6 CE sc €Ee B6I LE g06 CE 699 ZE pip ce pie ce per ze LLO°7E O66 tC ere te o98 \eé sve ve bei Ve ee. Ie Ore ve suo Ie €99 FE GUISE ez79 Ve 000 0 €00'0- L100 'O- 440 0- 910 O- CCORO= 620 O- eco O- OSO 'O- vygO O- €80 O- 901 O- Let O- 9Lb O- Lt O- t6Z O- 9LE€ O- pep O- 819 O- eel O- 8L6 O- 9 Giese cab h- bz9o I- HSL b= osi } ezo 4 bcos 9614 LL6 0 ee. oO 819 0 bap O LLe€ 0 Cte O [hata e(0) LENGO) 1e1o Lo1'o 80 0 p90'0 0so'O ge0 O 670 O tzo'O 910 0 10 0 100 0 C00 O 000 O MOQVIOd °A “SINIWA AYNOINOD MAN FHL 119 sive! te RAN Bh eeeene OT-€8 °ou awTgcn* £07201 “(°S*°N) Jequag yo1ressoy SutTiseutsuq Teqyseog) yiodaa snosueTTeostW settles “AI °*(*S*‘N) toque) yoieesey ButJseuTSsuyq Teyseop “III ‘*) 3roqoy ‘ueaq “IT *OTAUTL ‘I ‘Aetzaeq TeI0ZITI *G “UOTIJeWAOFSUeIQ |BAeM *y *R10d -sue1} JUSUTpes *—€ ‘*UOTINTOAD sUTTeIOYS *Z ‘*TepoW TeoTIsuUNN *T *uOTIOeAJJTIP pue “BuTTeoys ‘uoTOeAJeA sapnyTout uoTIeMIOFSUeIQ PTeTJ aAeM BY *SaINqoNajs Te IseOD FO AATUTOTA 9yQ UT Aajawmsyjeq Burqp[Nsea oy pue yaiodsueiq quewtTpes sqzotpead yoTyM Tepow yeoyieunu auT[—-u ‘aoueressTp-eTUTZ YTotTdwy ue saptaoad j10dey E861 AEH, *2aTIT} AaAog ‘ *“(OI-€8 ‘Ou § AeqUeD YOIeesey BuTis.8U -TBuy Teqseog / jysodei snosaueyTTeosTW)--"wo gz f *TTET : *d [6TT] “€86I ‘SIIN Worx aTqettTeae : *eA ‘pTatTyButadg ‘iequep yoieesoy ButlseuTsuyq Teqyseop ‘saveuTsuq jo sdiop ‘Amway *S*n : *eA SATOATOG qaog—--*ueaq *5) Jleqoy pue ut{ieaq DAeW Aq / sainjoniqzs Te }sSeOD jo AYLUTOTA dy} UT JlodsueijZ JUeUTpes ajeTNWTs 03 TepowW TeoTieunu y DaeW fUTTIAd OI-€8 “ou awyTgcn* €072OL *(°S°N) Jeaqueag yoiessoy SutiseuTZuq Te3seo7) yiodaa snosueTTe9sTW :seTies “AI *(°S*N) Jaque) yoieeasey ButTaseutTsuy [eqyseo) “III “9 Jrzaqoy ‘urea “IT *OTUIL ‘I ‘aetzaeq Te1033T] *G ‘*uoTJeWIOJSUeIR aAeM *y *Q10d -suei} qUaeMTpes °*€ “UOTANTOAS sUTTeAOYS *Z “*Tepow TeoTAoWNN °] *UOTIOPIFJTP pue ‘BuTTeoys ‘uoT.oOeAJaA sopnyTouT uoTJeWAOFSUeA PTETJ BAeM DUT “*SaInqoNaqs Teqseod Jo ARTUTOTA 9yQ UT Aajawkyjeq Butq[Nsez syQ pue qiodsuesq yuawtpes szoOTpead yoTYyM Tepou yeoyiounu suT[—-u ‘eoueieTjTp-eTUTsZ yroty~dmy ue saptaoid qaiodey E861 APH, *aT3T] 128A09 *(OI-€8 ‘ou { tequaQ YyoIeesey BuTis90eUu -T3uq Teqseog / q1odai snosueTTeostW)—--"wo gz f “TTF: *d [CIT] “E861 SSILN Worjy oeTqeTTeae : ‘eA ‘pTetyZutadsg ‘19qua) yoreessy ButaoeutTSZuy Te3seog ‘saveutSsuy jo sdiog ‘AwIAy *S*N : “eA SAITOATAG qdog—--*uesqg *) JAieqoy pue utTTIeq d1eW Aq / Seinjoniqzs Te }seOD jo AYLUTOTA |3y} UT Qaodsueiz JUeWTpas aje[NWTs 0} TepowW TeoTiouNnu y Dae ‘UT TIed OT-€8 “ou AwTgcn* €02OL *(°S°N) JaqueQ Yyoiesssy ButTiseutZuq Teqyseo)) 4iodaa snosueTTe9sTW :setties “AI °(*S*N) qajueg yoreasay BuTiveuzsuy TeIseoD “III “9 Areqoy Sura “II *O@TIIL ‘I ‘aetazeq Te10jIIWTI *G «“UOTIeWAOFSUeAR BAeM *Y *3210d -sueij Jualtpes *€ ‘“uoTANTOAS suTTeIOYS *Z “*Tepow TeoTowNN *T *uoTIOeAFJIP pue ‘BuptTeoys ‘uoT,IeAyei sapnypoutT uoT}eWIOFSUeA PTETJF AEM ay, *SeIN}ZONI}S TeqRseOD Jo ARTUTOTA 9yQ UT Aajzawkyjeq Bur3{[Nsea oyq pue jysodsue1} QuewWtpes sjoTpead yoTYyYM Tepow yeotieunu sut[—-u ‘aouerezztp—-aqpuTy 3pOTTdwmy ue sapfaoad qaoday w E861 APH, *eaTaT} 1aA0D *(OT-€8 ‘ou { Jaqueg YyoIeesey BuTiedU -[3uq Teqseog / qaodei snooueTTeosTW)--*wo gz ! “TTF: *d [6IT] “€861 ‘SIIN Woaz eTqetTyTeae : *eA ‘pTetssutads ‘19queQ yoiressay BupTiseuTsuq Teyseog ‘siveuTZuyq jo sdiog ‘Away *S*n : “eA ‘ITOATAG Jaog--*ueaq *9 JAeqoy pue uf{ied oaeW Aq / SeinjzoONAAS TeqseoD Jo AYTUPOTA |yQ UT ZAodsueaq Juawy{pas ajzeTNWTs 03 Tepow TeoTJeunu y DaeQ SuUTTIed OT-€8 °ou awTgsn’ €02OL *(°S°N) Jeaqueg yoieessy ZutTieeuTsuq Te}seog) Jlodei1 snoaueT[e9sStW settles “AI *(*S*N) jaqua) yoieassy ButiseuTZuq Te3seo) “III *9 Jleqoy ‘urea “II *aTATL ‘I ‘eTsizeq TeIOIITI *G ‘UOTIJeWIOJSUPIZ BALM “Hh *3Q410d -suelj JUaUTpes *€ ‘*UOTANTOAS sUTTeIOYS *Z “*Tepow TeoTiswNN *T *uoTJOeIJJIP pue ‘BuTTeoys ‘uoTOeIjZe1 saepnyTouT uoTjeMOFSUeI PTeTJ eAeM ay, *SeINnjzoNiqs TeIseoD jo AQTUFOFA |ayQ UT Aajowkyjeq But} [Nser ayj pue z1zodsue1q JueUTpes sjoTpeid yoTYyM Tepow yeotieunu autT—-u ‘aoueteysTp-a3tutTy ALOT Tdwy ue saptaoid j10day E861 APH, *aTIT} Jeo) *(OI-€8 ‘Ou § Jaquag Yyo1eesey BuyIaeU -T8ug Teqseog / qtodei snosueTTe0sTW)--*wo BZ * “TIE: ‘d [61T] “€R861 “SIIN Woajz eTqeTteae : ‘eA ‘ppTetzsutadg ‘raqueD yoireessy ButzseutTsuq Teqseog ‘siaeuTSuq Jo sdiop ‘Away *S*f : "eA “ITOATOG qaoqg—--ueaq *9 J1eqoy pue uTTIadq IAeW Aq / SeanjoONays TeIseod jo AVTUTOTA ay UT Qaodsuesq JueUTpes ajeTNWTSs 03 Tepow TeRoT1cuMU Vy DieQ SUT TIed S