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- Permanent Link:
- http://ufdc.ufl.edu/UF00078566/00001
## Material Information- Title:
- Beach nourishment performance predictions
- Series Title:
- UFL/COEL (University of Florida. Coastal and Oceanographic Engineering Laboratory) ; 91/013
- Creator:
- Dean, Robert G.
- Place of Publication:
- Gainesville
- Publisher:
- Coastal and Oceanographic Engineering Department, University of Florida
- Publication Date:
- 1991
- Language:
- English
## Subjects- Subjects / Keywords:
- Beach nourishment
## Notes- Funding:
- This publication is being made available as part of the report series written by the faculty, staff, and students of the Coastal and Oceanographic Program of the Department of Civil and Coastal Engineering.
## Record Information- Source Institution:
- University of Florida
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- University of Florida
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- All applicable rights reserved by the source institution and holding location.
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UFL/COEL-91/013
BEACH NOURISHMENT PERFORMANCE PREDICTIONS by R. G. Dean and Chul-Hee Yoo October 1991 w1 REPORT DOCUMENTATION PAGE 1. Report No. 2. 3. Recipient's Accessioo No. 4. Title and Subtitle 5. Report Date October, 1991 BEACH NOURISHMENT PERFORMANCE PREDICTIONS 6. 7. Author(s) S. Performing Organization Report No. R. G. Dean Chul-Hee Yoo UFL/COEL-91/013 9. Performing Organization Name and Address 10. Project/TaskI/work Unit No. Coastal and Oceanographic Engineering Department University of Florida 11. Contract or Grant No. 336 Weil Hall Gainesville, FL 32611 13. Type of Report 12. Sponsoring Organization Name and Address Miscellaneous 14. 15. Supplementary Notes 16. Abstract A simple method is developed for representing wave refraction and shoaling in the vicinity of a beach nourishment project. The method applies for the case of a one-line model of shoreline evolution in which the active profile is displaced seaward or landward without change of form. The model can include the presence of shore-perpendicular structures and background erosion. Background erosion rates are formulated in terms of cross-shore and longshore transport. An underlying concept of the method is that in cases where large perturbations, such as nourishment projects, are placed in the natural system, the system erodes on two time scales with the shorter time scale associated with the planform perturbation. It is recommended that shoreline modeling be carried out by conducting an ad hoc transformation in which the pre-project contours are represented as straight and parallel. The simple method is compared to a one-line model which includes a more detailed grid-based refraction and shoaling algorithm. For all cases tested, the simple method of representing refraction and shoaling results in shoreline evolution in good correspondence with the detailed method. The models are used to illustrate the effects of several features of beach nourishment projects that are of engineering interest. For a long uninterrupted shoreline, which has been nourished with the same material as the native, and in the absence of structures and background erosion, initially symmetric nourished planforms remain nearly symmetric as they evolve, even under oblique wave attack. This is interpreted as due to the small aspect ratio (additional beach width to length) of the nourishment project. For no background erosion or retention 17. Originator's Key words 18. Availability Statement Beach nourishment Sediment transport Shoreline evolution Shoreline models 19. U. S. Security Classif. of the Report 20. U. S. Security Classif. of This Page 21. No. of Pages 22. Price Unclassified Unclassified structures, the proportion of sand remaining in the project area over a twenty year period is illustrated for projects of various lengths and various effective wave heights. The effects of shore-perpendicular structures placed at the ends of the project with and without background erosion are illustrated. For normally incident waves and no background erosion, the project with or without retention structures causes only accretion to the adjacent shorelines. However, in the presence of background erosion or oblique waves, retention structures can cause localized erosion to the adjacent shorelines. Finally, the effects of nourishing with material more and less transportable than the native are illustrated. If less transportable, the nourishment tends to "armor" the project area and oblique waves can cause localized downdrift erosion. It is concluded that the simple method can provide reliable predictions of the evolution of nourishment projects and is thus useful in conducting benefit/cost analyses. UFL/COEL-91/013 BEACH NOURISHMENT PERFORMANCE PREDICTIONS by R. G. Dean and Chul-Hee Yoo October, 1991 TABLE OF CONTENTS LIST OF FIGURES LIST OF TABLES Abstract INTRODUCTION BACKGROUND Governing Equations ................................ METHODOLOGY Procedures Common to Both Methods ........................... Grid System and Transformation of Initial Geometry ............. Background Erosion ............................. Numerical Solution of Governing Equations ................... Procedures Which Differ for the Two Methods Wave Refraction and Shoaling Simplified Wave Refraction and Shoaling ...................... Detailed Refraction and Shoaling ........................... Effective Wave Height and Period ............................. RESULTS SUMMARY AND CONCLUSIONS Appendices I.REFERENCES............ II. NOTATION .............. 2 3 5 6 7 9 10 13 13 13 15 17 17 17 19 21 22 41 45 45 47 LIST OF FIGURES FIGURE PAGE 1 Definition Sketch .............................. 11 2 Definition Sketch for Numerical Model ................... 14 3 Recommended Ad Hoc Transformation for Modelling Coastal Systems in which Large Perturbations are to be Introduced . . . . . . . 16 4 Definition Sketch for Effect of Beach Nourishment on Contours . . . 18 5 Example 1. Comparison of Beach Nourishment Evolution for Simple and Detailed Methods of Wave Refraction and Shoaling, Normal Wave Incidence, Ho = 0.6 m, T = 6.0 sec, &o = 900. No Background Erosion . 24 6 Example 2. Comparison of Planform Evolution Obtained by Two Methods for 200 Oblique Waves, Ho = 0.6 m, T = 6.0 sec, ao = 700. No Background Erosion 25 7 Example 3a. Planform Evolution by Detailed Method for Deep Water Wave Directions, ao = 70*, 80', 900. Ho = 0.6 m, T = 6.0 sec. No Background Erosion . . . . . . . . . . . . . . 27 8 Example 3b. Planform Evolution by Simple Method for Deep Water Wave Directions, ao = 70*, 800, 90'. Ho = 0.6 m, T = 6.0 sec. No Background Erosion 28 9 Example 4. Even and Odd Components of Shoreline Position After 10 Years for Deep Water Wave Directions, ao = 700 and 80*. Results Obtained by Detailed Method . . . . . . . . . . . . . 29 10 Example 5. Illustration of Wave Height Effect on Rate of Planform Evolution. Results Based on Simple Method, ao = 90', T = 6.0 sec. Results Shown for 1, 3, 5 and 10 Years. No Background Erosion . . . . 31 11 Example 6. Effects of Various Project Lengths and Wave Heights on Project Longevity 32 12 Example 7a. Effects on Planform Evolution of Two Shore-Normal Retention Structures of Length Equal to One-Half the Initial Project Width. Normal Wave Incidence, No Background Erosion. Ho = 0.6 m, T = 6.0 sec.. . .......................................... 34 3 13 Example 7b. Effects on Planform Evolution of Two Shore-Normal Retention Structures of Length Equal to One-Half the Initial Project Width. Normal Wave Incidence. Uniform Background Erosion Rate at 0.5 m/yr, Zero Background Transport at Project Centerline. Ho = 0.6 m, T = 6.0 sec. ......... ...................................... 35 14 Example 7c. Effects on Planform Evolution of Two Shore-Normal Retention Structures of Length Equal to One-Half the Initial Project Width. Normal Wave Incidence. Uniform Background Erosion Rate at 0.5 m/yr, Zero Background Transport Located 4,500 m to Left of Left Structure. Ho = 0.6 m T = 6.0 sec. 36 15 Plot of K vs D (Modified from Dean, et al., 1982) . . . . . . 38 16 Example 8a. Planform Evolution for Nourishment Sand Less Transportable than the Native (KF = 0.693, KN = 0.77). Note Centroid of Planform Migrates Updrift Variation of Surface Layer K Values with Time at Locations A, B, C, D and E are presented in Figure 18. Wave and other Project Conditions Presented in Table I. . . . . . . . . . 39 17 Example 8b. Planform Evolution for Nourishment Sand More Transportable than the Native (KF = 0.847, KN = 0.77). Note Centroid of Planform Migrates Downdrift Variation of Surface Layer K Values with Time at Locations A, B, C, D and E are presented in Figure 19. Wave and other Project Conditions Presented in Table I. . . . . . .. 40 18 Example 8c. Variation of Surface Layer Longshore Transport Coefficient K with Time at the Five Locations Shown in Figure 16. Case of KF = 0.693, KN = 0.77, Ymi. = 2.0 m. . . . . . . . . . . . 42 19 Example 8d. Variation of Surface Layer Longshore Transport Coefficient K with Time at the Five Locations Shown in Figure 17. Case of KF = 0.847, KN = 0.77, Ymxi = 2.0 m. . . . . . . . . . . . 43 4 LIST OF TABLES 1 Characteristics of Examples Presented .... ................. 23 5 TABLE PAGE BEACH NOURISHMENT PERFORMANCE PREDICTIONS R. G. Dean and Chul-Hee Yoo Abstract A simple method is developed for representing wave refraction and shoaling in the vicinity of a beach nourishment project. The method applies for the case of a one-line model of shoreline evolution in which the active profile is displaced seaward or landward without change of form. The model can include the presence of shore-perpendicular structures and background erosion. Background erosion rates are formulated in terms of cross-shore and longshore transport. An underlying concept of the method is that in cases where large perturbations, such as nourishment projects, are placed in the natural system, the system erodes on two time scales with the shorter time scale associated with the planform perturbation. It is recommended that shoreline modeling be carried out by conducting an ad hoc transformation in which the pre-project contours are represented as straight and parallel. The simple method is compared to a one-line model which includes a more detailed grid-based refraction and shoaling algorithm. For all cases tested, the simple method of representing refraction and shoaling results in shoreline evolution in good correspondence with the detailed method. The models are used to illustrate the effects of several features of beach nourishment projects that are of engineering interest. For a long uninterrupted shoreline, which has been nourished with the same material as the native, and in the absence of structures and background erosion, initially symmetric nourished planforms remain nearly symmetric as they evolve, even under oblique wave attack. This is interpreted as due to the small aspect ratio (additional beach width to length) of the nourishment project. For no background erosion or retention structures, the proportion of sand remaining in the project area over a twenty year period is illustrated for projects of various lengths and various effective wave heights. The effects of shore-perpendicular structures placed at the ends of the project with and without background erosion are illustrated. For normally incident 6 waves and no background erosion, the project with or without retention structures causes only accretion to the adjacent shorelines. However, in the presence of background erosion or oblique waves, retention structures can cause localized erosion to the adjacent shorelines. Finally, the effects of nourishing with material more and less transportable than the native are illustrated. If less transportable, the nourishment tends to "armor" the project area and oblique waves can cause localized downdrift erosion. It is concluded that the simple method can provide reliable predictions of the evolution of nourishment projects and is thus useful in conducting benefit/cost analyses. INTRODUCTION Beach nourishment, the placement of large quantities of sand in the nearshore region to advance the shoreline seaward, is being applied increasingly as a method of erosion control. Advantages over other methods include maintaining a near-natural wide beach for storm protection and recreational purposes. In some localities the wide beach also provides suitable nesting areas for several endangered species of sea turtles. Sand placed in this manner along a long, uninterrupted shoreline represents a perturbation which, over time, tends to be smoothed out by longshore sediment transport. Additionally, if placed at an initial profile that is steeper than "equilibrium", offshore transport will occur with an associated narrowing of the dry beach. Realistic prediction of the performance of beach nourishment projects includes analysis of the cross-shore and longshore transport processes and is of considerable significance to a rational evaluation of the economic benefits of the project. Unfortunately, at present there are few beach nourishment projects that have been documented adequately to allow detailed and quantitative evaluation of calculation procedures. Methods have been proposed by a number of investigators to predict the performance of nourishment projects. A beach nourishment manual has been developed based on experience in the Netherlands (Pilarczyk and Overeem, 1987) which, in addition to performance, addresses dredge equipment, manner of placement, environmental effects, etc. Krumbein and James (1965), James 7 (1974), and Dean (1974) have proposed ad hoc but quantitative methods of assessing the quality of material. Application of the James method is described in the Shore Protection Manual (1984). In general, these methods attempt to establish the required volume of borrow material which is equivalent to one unit volume of native material. However, other parameters critical to project performance are not addressed by these procedures. Pilkey and his co-workers (Pilkey and Clayton, 1989; Leonard, Clayton and Pilkey, 1990; Leonard, Dixon and Pilkey, 1990) have conducted "broad brush" analyses of the performances of many projects along the East and Gulf coasts of the United States. Attempts have been made to define project performance in terms of the actual versus design lifetimes of the projects. Also performance was plotted versus project length, sediment grain size, etc. In general, they found little or no correlation between expected and actual performances. Rather, the following equation was proposed for the East Coast of the United States to represent the required renourishment volumes, VR, conducted at intervals of n (in years) for an initial restoration volume, jt, and a project life, N (in years). -VR = -V (1) n It was recommended that n = 9 for Florida, n = 3 for New Jersey, and n = 5 for the remaining portions of the East coast. Eq. (1) simply states that it will be necessary to renourish with an amount equal to the initial restoration volume every n years. The rationale for Eq. (1) is not apparent from sediment transport considerations. The studies by Pilkey and his co-workers do not recognize explicitly the effects of background erosion, sediment quality or project length on project performance. Dean (1983) reviewed available methods of predicting beach nourishment performance and showed that based on the Pelnard Considere (1956) solution and in the absence of background erosion, the time, tp, required for a project to lose p percent of the material placed is 122 t, a 1f/ (2) b in which i is the project length and Hb is the height of the breaking waves which mobilize the sediment causing the spreading out beyond the project limits. Recently Dean (1988b) has shown 8 that the proportionality factors for 50% "loss" are approximately, for the units as shown t50% = K 12 (3) j5/2 in which t50% is in years and K" = 0.172 for i in km and Hb in meters, and K" = 8.7 for i in miles and H in feet. It is emphasized that the material "lost" from the project area is transported alongshore to project adjacent areas and continues to provide benefits there (Dean, 1988a). BACKGROUND The bases for predicting the performance of beach nourishment projects are the equations of continuity and transport. In general, these may be used to develop a one-line model in which only one contour (usually the mean water line) is used to represent shoreline changes (e.g. Bakker, 1968; LeMehaute and Soldate, 1977; Perlin, 1978; Walton and Chiu, 1979; Perlin and Dean, 1985; Hanson, 1989; Hanson and Kraus, 1989). Bakker (1968) has developed a two-line model which, for example, allows for profile steepening and flattening updrift and downdrift of a structure, respectively. Hanson and Kraus (1987) have compared the results of one-line numerical models to analytical solutions applicable for several beach nourishment initial planforms. An n-line model has been developed by Perlin and Dean (1985) in which an arbitrary number (n) of contour lines is used to represent the beach profile. A restriction of the multi-line models developed to date is that the profiles represented must be monotonic. An alternative formulation method would be to represent the topography by a grid system as is commonly done in hydrodynamic modeling thereby eliminating the monotonic requirement. For models which represent the profile by more than one contour, it is necessary to specify a relationship for cross-shore sediment transport. Such models have been developed by Kriebel (1982) and Kriebel and Dean (1985) for profiles which must vary monotonically and Larson (1988) and Larson and Kraus (1989) for models not requiring monotonocity. 9 Governing Equations The equation of sediment conservation can be expressed for three dimensions as Oh - Oq, aq( -= V-q=a + a- (4) in which h is the water depth relative to a fixed datum, t is time, V is the horizontal vector differential operator (V () =i + 3 q is the horizontal sediment transport vector (q=7 q,+ j q.) as presented in Figure 1, and i and j are the unit vectors in the x and y directions, respectively. Integrating Eq. (4) in the cross-shore direction from a landward location, yi, where the cross-shore transport (qy) is zero to a seaward location, y2, where qy is similarly zero, yields a j hdy a j qdy = 0 (5) The first term is recognized as minus the time rate of change of volume of sand,-V and the second integral is the total longshore sediment transport, Q. Making these substitutions &V aQ -- + -- = 0 (6) In one-dimensional model formulations, it is usually assumed that accretion or erosion of a profile is associated with a seaward or landward displacement respectively of the profile without change of form. The vertical extent of this change is from some depth, h., of limiting sediment motion up to the berm elevation, B. Thus the change, Ay, in any contour associated with a change in volume, AV, is AY AN' (7) h. + B which when substituted in Eq. (6) yields the one-dimensional equation for conservation of sand, ay 1 Q t+ -x=0 (8) Ot +(h +B) Ox The one-dimensional equation of sediment transport can be expressed as (Komar and Inman, 1970) I = KP, (9) 10 Depth Contours -y \y 0 Wave Crest x B h* (x,y) Depth of Limiting Motion Figure 1. Definition Sketch. 11 in which I is the immersed weight sediment transport rate, Pt, is the longshore energy flux factor and K is a non-dimensional sediment transport proportionality factor. These two quantities can be expanded I = Qpg(s 1)(1 p) (10) Pt = EbCGb sin Ob cos Ob (11) in which s is the ratio of mass densities of sediment to water (= 2.60), g = gravitational acceleration, p = in-place sediment porosity (taken here as 0.35), E = wave energy density, CG = wave group velocity, 9 = angle between the wave crests and the bottom contours, and the subscript "b" denotes that the subscripted variable is to be evaluated at the breaker position. Based on small-amplitude shallow water wave theory and assuming that the breaking wave height Hb is proportional to the breaking water depth, hb (H1 = Khb) H2 Eb = P9 (12) CGb = b = \f = g Hb/ (13) in which r is a proportionality constant (~ 0.78) and Cb = wave celerity at breaking. Substituting Eqs. (10), (11), (12) and (13) into Eq. (9), Q = _____ _____ sin Ob cos Ob (14) 8 (s 1)(1 p) Eqs. (8) and (14) form the basis for a one-dimensional numerical model. Pelnard-Considere (1956) has shown that by linearizing Eq. (14) with respect to perturbations in the predominant shoreline alignment and combining the result with Eq. (8), the one-line model is transformed into the heat conduction equation = G a (15) T9t OX2 in which G is the "longshore diffusivity", defined as K H 5/2 Vr G =( H512( F (16) 8(s 1)(1 -p)(h + B) 12 In the finite difference solution of Eq. (15), it can be shown that a critical time increment, At,, exists which, if exceeded, will cause the numerical solution to become unstable, 1 AX2 (17) At=2 G in which Ax is the alongshore grid spacing. By inspecting Eqs. (16) and (17) it can be seen that the smaller the grid spacing and the larger the wave height, the smaller the allowable time increment. For multi-line models, it is necessary to specify relationships for the distributed longshore (q,) and cross-shore (qy) sediment transport. Kriebel (1982) and Kriebel and Dean (1985) have proposed the following for the cross-shore transport rate qy = K'(V D,,) (18) in which D and V. are the actual and equilibrium values of wave energy dissipation per unit water volume and K' is a proportionality factor. METHODOLOGY Only one-line models are discussed here. Two models using quite different methodologies for representing wave refraction and shoaling will be presented, applied and the results compared. In the first model, refraction and shoaling will be represented by a very simple one-step procedure whereas in the other a detailed grid-based solution is used. Procedures Common to Both Methods Grid System and Transformation of Initial Geometry The continuity and transport equations (Eqs. (8) and (14)) were solved using an explicit method with the grid system shown in Figure 2. The shoreline displacements are maintained fixed while the transport is computed and in the second part of the same time step the transport is held constant while the shoreline displacements are computed. Thus although the primitive equations are employed, the limiting time step specified by Eq. (17) is still valid. In areas of special interest where greater resolution is required, smaller grid elements, (Ax), can be used in the grid system presented in Figure 2. 13 T Ax Ae --1 q/ Note: -- Ob= 1-ab Yi yi+1 x Figure 2. Definition Sketch for Numerical Model. 14 In conducting numerical modeling of generally straight shorelines where a large perturbation has been introduced, it is recommended, in general, that the original shoreline and offshore bathymetry be represented by straight and parallel contours. This is equivalent to the ad hoc transformation shown in Figure 3. The rationale for the transformation is that, with the exception of the long-term background erosion that will be discussed subsequently, the nearshore system has approached a near-equilibrium, the details of which present modeling techniques cannot represent adequately. The equilibrium may depend on rather subtle and perhaps nonlinear three-dimensional wave transformation (refraction, diffraction and shoaling) over minor bathymetric features. Usually, if a model is applied to a shoreline situation such as Figure 3a, it is found that the shoreline and offshore contours will approach unrealistically straight alignments and/or the changes will occur over time scales that are much shorter than actual. Thus numerical modeling is much more effective in cases where substantial perturbations (human-induced or natural) have placed the system out of balance. The transformation presented in Figure 3 recognizes that the changes caused by the perturbation introduced will be of a much shorter time scale than those associated with the pre-existing non-equilibrium causative features. Subsequent to the modeling, the results can be transformed back to the actual system. Background Erosion Long-term background shoreline changes must be accounted for appropriately if the nourishment evolution is to be predicted with good accuracy. This is especially the case if stabilizing structures are to be installed. Long-term background changes could be caused by divergences in longshore or cross-shore transport (cf Eq. (4)). Thus, the general treatment considers the long-term background changes composed of a longshore transport component, YB,L, and a cross-shore transport component, yB,C 8YB 9YBL + 09YB,C (19) at at at and the background change, 'y, is expressed as background longshore transport, according to Eq. (8), QB,L(X) = QB,L(XR) (h. + B)] OYBlL dx (20) J at 15 .. a) In a Shoreline Contours Shoreline Contour. Ad itial Actual Shoreline nd Contours b) Initial Shoreline and Contours to be Modeled. Figure 3. Recommended Ad Hoc Transformation for Modelling Coastal Systems in which Large Perturbations are to be Introduced. H 0's Hoc Transformation in which QB,L(XR) denotes the background transport at some reference location, XR, to be based on local knowledge or other calculation procedures not discussed here. Numerical Solution of Governing Equations The azimuth, P;, of the shoreline normal at the nth time level, is established to represent the value at the grid line associated with Qg (see Figure 2) = p + -tan-' (4 ) (21) 2 \Xi+1 Xi ) The background transport, QB,LS (Eq. (20)), is added to the longshore transport resulting from the planform anomaly to yield the total transport, Qn, Q Q + QB,L, (22) Finally, the shoreline position is updated from the nth to the (n + 1)th time level At W -' ) (23) Ax(h + B) +,) at Procedures Which Differ for the Two Methods Wave Refraction and Shoaling Simplified Wave Refraction and Shoaling We start by showing several results from linear wave theory using the wave and contour directions shown in Figure 4. The difference AP, between the nourished contour orientation, #,(h < h.), and the deep water contour orientation, Po, (h > h.), can be considered small since the ratio of nourished beach width, Y, to length, f, is generally on the order of 0.02 at most. Conservation of wave energy flux from deep water to a water depth, h*+I is EoCGO cos(flo ao) = E*+CG*+ cos(#lo a*+) = E.CG. cos(Po a.) (24) in which the subscripts "+" and "" indicate conditions just seaward and landward of the depth transition respectively (cf. Fig. 4). The energy flux just landward of the transition may be equated to that at breaking by E.CG, cos(#0 a,,) = EbCG, cos(#o ab) (25) Since 1,(x) = Po + Ap(x) EoCGo cos(3o ao) EbCGb sin(o, ab) sin(AO) (26) EbCGb cos(3 ) = (cos(AP) 17 North Shoreline A Region Influenced by Beach Nourishment x y h=h* h > h. Contours N a*(x) Waves N Deep Water Contour N a0 A a) Planform Showing Perturbed Contours to Depth h, Original Profile Nourished Profile b) Profile Through A-A Figure 4. Definition Sketch for Effect of Beach Nourishment on Contours. 18 1 W - Utilizing shallow water linear wave theory and the wave height breaking proportionality factor, neglecting terms modified by sin(A#) sin(#, ab) and approximating cos(A#) by unity (A# is small), Cb 9 g2HCG cos(#o ao) 0.2(27) I 2 cos(#, ab) I which will be useful later. Applying Snell's Law in a similar manner across the transition and to breaking, sin(p, a*+) sin(#, a.) sin(#, ab) (28) C*+ C. Co 28 and a*+ can be determined by applying Snell's law from deep water to h*+. Eqs. (26), (27) and (28) will be used to express the transport relationship (Eq. (14)) primarily in terms of deep water wave conditions. With a limited amount of algebra, we obtain K IH4^Chig0^ cos1.2( Po -a0). Q = KH24sm OS. P e)i(P, a.) (29) 8(s 1)(1 p)CK04 Si( in which terms modified by sin(Af) sin(#, ab) have been neglected and cos(Afl) has been approximated by unity. It is interesting to note that if Eq. (29) is linearized in the form of Eq. (15), the appropriate value of longshore diffusivity, G, in terms of deep water wave conditions can be expressed as G =(HO2 CG'2 9C0.4 cosi.2(po ao) cos 2(po a.) (30) 8(s 1)(1 p)Cn04(h, + B) cos(#o a.) Eq. (29) is the relationship for the longshore sediment transport rate that completes the simple model development. Detailed Refraction and Shoaling The detailed method of solution differs from the simple method as refraction and shoaling are carried out by the detailed procedure. This method as first presented by Noda (1972), and employed by Perlin and Dean (1983) and Dalrymple (1988), is based on the irrotationality of wave number and conservation of wave energy. The reader is referred to the papers noted above for further detail. V x k=0 (31) 19 in which k is vector wave number. Eq. (31) can be expanded to a(k sin ) = a(k cos ) (32) In the present model, a two-dimensional grid is established and Eq. (32) is expressed in finite difference form as r(k cos )i-1,-1 + (1 2r)(k Cos 1)i,j+l 1 = cos- +r(k cos O)i+1i+1 (33) ((k sin O)i,j+i (k sin 9)ijj) in which r is a smoothing factor taken as 0.25 in application here and all k and 0 values on the right hand side are understood to be at the (n + 1) time step. The above equation is solved by considering the wave direction, 0, at the offshore boundary as known and the wave directions along the lateral boundaries given by Snell's Law. These conditions along the lateral boundaries should not affect the interior solution if the lateral boundaries are sufficiently distant from the area of interest. With the refraction solution for a given time step, the conservation of wave energy equation is solved to determine the wave height field, Hij. The equation for conservation of wave energy (Eq. (24)), is expressed in finite difference form as r(H2CG sin 0)i-1,I+l + (1 2r)(H2CG sin )i,+- 1/2 ("+= sn) +,r(H2CG sin 9)i+1,j+1 (34) iJ (CG sin 0)ijj [-A ((H2CG cos 0),+,1 (H2CG cos 0),_1,j) J As with wave refraction, in solving Eq. (34), the wave height along the outer boundary is considered as known and along the lateral boundaries, it is assumed that a = 0. Eq. (34) is iterated until ax convergence of the solution is obtained to within an arbitrary pre-defined limit. The wave height calculations are carried landward to a location where the inequality H',j > W,j occurs. The transport, Q', is then calculated by transforming the wave conditions at i,j' -1 to the breaking location much as was done for the simple method in transforming wave conditions from deep water to the breaking point. 20 This concludes description of the simple and detailed procedures for calculating shoreline evolution following a nourishment project. While the required computer times are not large for either procedure, the times for the simple procedure were approximately 1/250 of those for the more detailed procedure. Effective Wave Height and Period Although the wave height and period change continuously with time, it is evident that in the absence of littoral barriers, there is an effective constant wave height and period that will produce the same spreading out of the beach nourishment material as the actual time-varying values. In the following development, a Rayleigh distribution f(H) for wave heights will be assumed f(H) = e (35) in which Hm, is the root-mean-square wave height. Referring to the sediment transport equation (Eq. (29)) expressed in terms of deep water conditions, for a given sea state characterized by the significant deep water wave height H,, the effective deep water wave height, Ho.,f, is given by Hoeff = [j H2.4p(H)dH] 24 (36) Evaluating Eq. (36) numerically and using the approximate Rayleigh distribution relationship between significant and root-mean-square wave heights (H, ~ v/2Hm,), it can be shown that Ho.,, = KrmHrm, = KH, (37) where Krm, = 1.04 and K., = 0.735. Thus the long-term effective wave height Hoeff at a particular location is 1 N )24 . Ho.ff = [-- (K KH)2.] (38) .n=1 in which HI,, is the significant wave height of the nth record in a series of N records encompassing the time period of interest. Examining Eq. (29), it is clear that a somewhat more appropriate but more cumbersome value of effective wave height, H1 is given by f = 1(39) N n= C [c N 21 and the effective value of C/C. (or equivalently, wave period) to be used in Eq. (29) is the denominator of Eq. (35) raised to the 2.4 power. RESULTS Several examples were investigated to demonstrate the effects of various design options and to establish, by comparison with the detailed procedure, the relative validity of the simple procedure. In the examples the background erosion will be taken as zero unless otherwise stated. For those examples including effects of background erosion, only longshore transport contributions will be represented. Unless noted otherwise, the grid spacing, (Ax), and project length for all examples were 150 m and 6,000 m, respectively. At the lateral boundary conditions, the shoreline position changed according to the background erosion rate and the domain was selected to be sufficiently large that the perturbation effects at the ends of the domain were insignificant. Other characteristics of these examples are presented in Table I. Example 1 Initially Rectangular Planform, i = 6000 m, Y = 30 m, Ho = 0.60 m, T = 6.0s, ao = 0o = 90', h, = 5.5 m, B = 2.5 m. The results of this example comparing the two methods is presented in Figure 5 for 1, 3, 5 and 10 years after nourishment. The simple and detailed procedures yield very similar results which is somewhat surprising in view of the extremely simple algorithm employed to represent wave refraction and shoaling. Again the interpretation is that the aspect ratio of the nourishment project (Y/l) is quite small. Example 2 Initially Rectangular Planform, f = 6000 m, Y = 30 m, Ho = 0.6 m, T = 6s, ao =70', o = 90', h. = 5.5 m, B = 2.5 m. Conditions for this example are the same as for Example 1, except the deep water wave direction is 200 oblique to the shoreline. Reference to Figure 6 demonstrates that the two methods are in quite good agreement with surprisingly little planform asymmetry due to the oblique wave direction. This asymmetry is examined in greater detail in the next two examples. 22 TABLE I. CHARACTERISTICS OF EXAMPLES PRESENTED* Wave Characteristics Results Special Example Ho T ao in Characteristics Purpose of Example (m) (sec) (0) Figure(s) 0.6 0.6 0.6 0.6 0.4,0.8 Variable 0.6 0.6 6 6 6 6 6 6 6 6 900 700 700,800,900 700,800 900 900 900 700 * Unless Stated Otherwise, the Nourished Length, t, of Shoreline is 6,000 m. 23 5 6 7,8 9 10 11 12,13,14 16,17 None None None None None Various Wave Heights and Project Lengths Two Project Retention Structures Present, With and Without Background Erosion Nourishment with Sands of Different Transport Characteristics Compare Two (Simple and Detailed) Methods for Normal Wave Incidence Compare Two Methods for Oblique Waves Compare Two Methods for Range of Wave Directions Detailed Examination of Antisymmetric Component of Response Effect of Wave Height Effects of Wave Heights and Project Lengths on Project Longevity Illustrate Effects of Retention Structures for Various Background Transport Conditions Effect on Evolution Patterns 50.0 - SIMPLE METHOD DETAILED METHOD Initial Planform 30.0 --1 year 3 years -' 5 years 10 years 10.0 -10.0 ' 0. 1800. 3600. 50o0. 7200. 9000. 10800. 12600. 14400. 16200. 18000. X (M) Figure 5. Example 1. Comparison of Beach Nourishment Evolution for Simple and Detailed Methods of Wave Refraction and Shoaling, Normal Wave Incidence, Ho = 0.6 m, T = 6.0 sec, ao = 90*. No Background Erosion. I 5 0 0I I I I 1 r - I I I I I I I I I I I -------- SIMPLE METHOD -- DETAILED METHOD Initial Planform 30. 0 year 3 year - 3 yearsy (m) 5 years 10 years 10.0 -10.0 0. 1800. 3600. 5400. 7200. 9000. 10800. 12600. 14400. 16200. 18000. x (M) Figure 6. Example 2. Comparison of Planform Evolution Obtained by Two Methods for 20* Oblique Waves, Ho = 0.6 m, T = 6.0 sec, ao = 70*. No Background Erosion. 50.0 Example 3 Same As Example 2, Except Response to Different Wave Directions Figures 7 and 8 present shoreline planforms at 1, 3, 5 and 10 years after nourishment for wave directions of 70*, 800 and 900, i.e. 20*, 100 and 00 obliquity to the general shoreline alignment. It is seen that the results obtained by the two methods are quite similar and that the effects of wave direction are relatively small. Although slight, the major effect seems to be that for the more oblique angles, there is less wave energy flux toward the shoreline resulting in less longshore sediment transport. A small asymmetry is evident. Example 4 Same as Example 2, Except Detailed Examination of Planform Asymmetry for Oblique Wave Directions The purpose of this example is to examine the asymmetry resulting from an oblique wave acting on a nourished planform which is initially symmetric. Figure 9 presents the results ten years after nourishment for deep water wave directions, ao, of 70* and 800, i.e. 200 and 100 obliquity to the pre-nourished shoreline alignment. In order to quantify the antisymmetry, the planforms were separated into even and odd components, ye(X') and yo,(') where x' is the longshore coordinate with origin at the project centerline. Denoting yT(x') as the total shoreline displacement, it can be shown that the even and odd components are determined as 1 ye(x') = [yr(X') + YT(-X')] (40) 2 y0(X') = 1 [YT(X') YT(-X')] (41) The maximum displacement associated with the odd component is 1 m. The reason that the odd components are so small is that the aspect ration (Y/t) of the nourished planform is so small, in this case 30 m/6000 m = 1:200. Thus the asymmetries are small and are primarily due to the nonlinearities in the sin 20 term of the transport equation. 26 50 .0 I I I I I I I -------- ao= 90* --a0 80* - 700 Initial Planform 30.0 --1 year - 3 years y () 5 years 10 years 10.0 -10.0 0. 1800. 3600. 5100. 7200. 9000. 10800. 12600. 4qqO0. 16200. 18000. x (M) Figure 7. Example 3a. Planform Evolution by Detailed Method for Deep Water Wave Directions, ao = 70', 800, 900. Ho = 0.6 m, T = 6.0 sec. No Background Erosion. 50.0 1 1 1 ------- ao = 900 -- o= 80* - o = 70* 3-- Initial Planform 7'.0 1 year 3 years y (m) 5 years 10 years 10.0 -10.0 0. 1800. 3600. 5400. 7200. 9000. 10800. 12600. 14400. 16200. 18000. x(M) Figure 8. Example 3b. Planform Evolution by Simple Method for Deep Water Wave Directions, ao = 70*, 80*, 90*. HO = 0.6 m, T = 6.0 sec. No Background Erosion. 50.0 1 - ao=70* ao=80' 30.0 ---- ----------------------------------- Initial Planform Even Components N)M %.0 10.0 Odd Components 10.0' -10 .0 i i i I 0. 1800. 3600. 5400. 7200. 9000. 10800. 12600. q44OO. 16200. 18000. x (M) Figure 9. Example 4. Even and Odd Components of Shoreline Position After 10 Years for Deep Water Wave Directions, ao = 70 and 80'. Results Obtained by Detailed Method. Example 5 Effect of Wave Height Figure 10 presents the planforms calculated by the simple method at 1, 3, 5 and 10 years after nourishment and for 0.4, and 0.8 m wave heights and normal wave incidence. Of particular interest is the major role of wave height in causing spreading out of the nourished planform. This effect is also evident from Eq. (2) which indicates that the spreading out is proportional to Hb. Example 6 Effects of Various Wave Heights and Project Lengths Earlier examples have demonstrated (for homogeneous sediment conditions) the relative insignificance of wave direction. The simple model was exercised to demonstrate the effect on the fraction of sand remaining in the project area for a wide range of wave heights and project lengths. These results are presented in Figure 11 where the proportion remaining is shown over a 20 year time period for nine combinations of wave heights and project lengths. For a small wave height and long project (Ho = 0.30 m, t = 24,000 m), it is seen that at the end of a 20 year period, over 95% of the material remains. By contrast, for a four-fold larger wave height and a length one-sixteenth of the former, the sand remaining after one year is less than 40% of that placed. This illustrates the significance of project length and wave height. It has been noted earlier that the longevity of a project varies with the square of the project length and inversely with the wave height to the 2.5 power. It is emphasized that the results presented in Figure 11 do not include effects of background erosion which can be of considerable magnitude, especially over long time periods. Example 7 Effects of Retention Structures Several examples will be presented illustrating the effects of retention structures, with and without background erosion. The algorithm for the transport boundary condition at a structure is fairly complex and will not be described fully here. In general the algorithm is consistent with Eqs. (21), (22), (23) and (29) with transport determination based on relative positions of structure tip and the shorelines on the two adjacent grid cells and the background erosion transport rate components. Diffraction was not included in the examples presented here. 30 50 .0 I I I I I I I HO = O.4' - H 0.87 Initial Planform 30.0 f / ~ 10.0 -10.0 0. 3000. 6000. 9000. 12000. 15000. 18000. 21000. 24000. 27000. 30000. x(M) Figure 10. Example 5. Illustration of Wave Height Effect on Rate of Planform Evolution. Results Based on Simple Method, ao = 90', T = 6.0 sec. Results Shown for 1, 3, 5 and 10 Years. No Background Erosion. 1.0 1 0-- -- - .-. .....- ...H-. m, -= 24- m H0=0.3 m, 2= 24000 m f.t ..... .- H 0-=1.2 m, 2 000 M Z 0.8 \4 __ 0=1.2 M,224000 m H0=0.3 m, 2=6000 m O L 0.6 1H0=0.6 m, 2=6000 m -O 0.4 1.=H=1.2 m, =6000 m 0. c cf f . 0 ZO Oo- H=0.9m, 2= 1500 n 0H 0.2 0=0.6m, 2=1500 M 0 CL 1-0=1.2m, =1500 m 0 2 4 6 8 10 12 14 16 18 20 YEARS AFTER NOURISHMENT Figure 11. Example 6. Effects of Various Project Lengths and Wave Heights on Project Longevity. Example 7a and Figure 12 present results for no background erosion and normal wave incidence. Structures one-half the length of the initial project width (30 m) are present at the two ends of the project. Here, the structure length should be interpreted as the "effective" structure length. Consistent with intuition, it is seen that the shorelines immediately adjacent to the project advance as sand from the project area "spills" around the structures; however as the project recedes approaching the structure length, less and less sand is transported to the project adjacent areas and the planform evolves toward one of shoreline segments straight and parallel to the incoming waves with the beach width within the project area equal to the lengths of the stabilizing structures. Example 7b and Figure 13 present the results for the same conditions as Example 7a, except there is a uniform background erosion of 0.5 m/year and the reference background erosion is taken as zero at the project centerline. The transient results are qualitatively similar to those of Example 7a. However, because of the background erosion, an equilibrium planform exists only within the confines of the stabilization structures; within the project area, the equilibrium planform is concave outward and symmetric about the project centerline. For equilibrium conditions within the project area, the background erosion transport is exactly balanced locally by the planform orientation transport due to the waves. This accounts for the character of the planform in Figure 13. At a great distance from the project area, the shoreline retreats at the background rate of 0.5 m/year; however, immediately adjacent to the retention structures, the erosion rate would be greater to compensate for the effect of the reduced erosion (at later times) within the project area. The final example (7c) that will be presented illustrating the effects of structures is the same as example 7b, except now the zero reference background transport is located 4,500 m to the left of the left structure in Figure 14. It is seen that updrift of the left structure, transport is toward the structure and the shoreline accretes there. Inside the two structures, sand is transported initially in both directions, but somewhat later the positive transport prevails and sand is carried past only the right hand structure. Eventually, as before, the planform within the structures will be aligned for equilibrium and both the downdrift and updrift shorelines will erode. For zero reference 33 50.0 I Initial Planform 30.0 ---- ---1 year 3 years 5 years 10 years 10.0 Structure Structure -10.0 ' 0. 3000. 6000. 9000. 12000. 15000. 18000. 21000. 24000. 27000. 30000. x(M) Figure 12. Example 7a. Effects on Planform Evolution of Two Shore-Normal Retention Structures of Length Equal to One-Half the Initial Project Width. Normal Wave Incidence, No Background Erosion. Ho = 0.6 m, T = 6.0 sec. 50.0 Initial Planform 30.0 ----------------- ---1 year 3 years y(m) -ya 5 years ----- -10 years 10.0 Structure Structure :L -10.0 0. 3000. 6000. 9000. 12000. 15000. 18000. 21000. 24000. 27000. 30000. x(m) Figure 13. Example 7b. Effects on Planform Evolution of Two Shore-Normal Retention Structures of Length Equal to One-Half the Initial Project Width. Normal Wave Incidence. Uniform Background Erosion Rate at 0.5 m/yr, Zero Background Transport at Project Centerline. Ho = 0.6 m, T = 6.0 sec. 50.0 30.0 Initial Planform - - -- - - - --A " 1 year 3 years 5 years - 10 years Structure Struct ------ ----- ------ -- 11 --- - --- - -- - - - I I I I I I I III I I I I I I I I I i I 0. 3000. 6000. 9000. 12000. 15000. 18000. 21000. 2q000. 27000. 30000. x (m) Figure 14. Example 7c. Effects on Planform Evolution of Two Shore-Normal Retention Structures of Length Equal to One-Half the Initial Project Width. Normal Wave Incidence. Uniform Background Erosion Rate at 0.5 m/yr, Zero Background Transport Located 4,500 m to Left of Left Structure. Ho = 0.6 m, T = 6.0 sec. ure y(m) LUJ M. 10.0 -10.0 transport located much farther to the left than for the example shown here, sand would bypass the updrift structure, flow past the intrastructure segment and onto the downdrift shoreline. However, both the updrift and downdrift shorelines at great distances from the project would continue to erode at the background rate and only the updrift shoreline immediately adjacent to the project area would experience a net accretion. Example 8 Effect of Different Transport Characteristics for Native and Nourishment Sands Dean, et al. (1982) have examined the sediment transport factors, K (Eq. (9)), determined from various field programs and have proposed the dependency of K on sand diameter, D, as shown in Figure 15. Intuition suggests that the effects of nourishing with KF : KN could have a significant effect on the adjacent shorelines. The subscripts "F" and "N" denote fill and native, respectively. This effect will be greater for the case of oblique waves. Figures 16 and 17 present the case of waves at a 200 obliquity acting on nourishment projects with KF/KN = 0.9 and 1.1, respectively. It is seen from Figure 16 that for the case in which the nourishment material is less transportable than the native, the nourishment planform acts as an erodible barrier with an associated accretion and erosion on the updrift and downdrift sides of the barrier, respectively. For the case in which the sand is more transportable than the native, Figure 17, the pattern is qualitatively a mirror image of that noted. As the project evolves, the calculation procedure requires determination of the degree to which the sand exposed to the waves is of nourishment and native character. This was accomplished by the following algorithm. A mixed layer of minimum thickness, Ymix, was assumed. If, during a time increment, erosion occurred such that the mixed layer was less than Ymix thick, the remaining material within the mixed layer was mixed with the underlying sand to re-establish a thickness of Ymix. The character (i.e. K) of the mixed layer was calculated. If deposition occurred, the mixed layer thickness and character were determined based on the thickness and character at the previous time step and the character of the material in the littoral stream. Obviously, it was 37 2.0 c 0 1.0 . LL ~LL. DIAMETER, D (mm) Figure 15. Plot of K vs D (Modified from Dean, et al., 1982). 38 50 0I III I AB C DE 30.0 Initial Planform 1 year y (m) 3 years \ 1- 5 years 10.0 10 years -10.0 0. 3000. 6000. 9000. 12000. 15000. 18000. 21000. 2q000. 27000. 30000. x (m) Figure 16. Example 8a. Planform Evolution for Nourishment Sand Less Transportable than the Native (KF = 0.693, KN = 0.77). Note Centroid of Planform Migrates Updrift. Variation of Surface Layer K Values with Time at Locations A, B, C, D and E are presented in Figure 18. Wave and other Project Conditions Presented in Table I. I 50.0 1 I I I I I I I I I I I AB C DE Initial Planform 1 year 3 years 5 years ~ \* 10 years I I I I I I I I I I I I I I I I I I I 0. 3000. 6000. 9000. 12000. 15000. 18000. 21000. 24000. 27000. 30000. Figure 17. Example 8b. Planform Evolution for Nourishment Sand More Transportable than the Native (Kp = 0.847, KN = 0.77). Note Centroid of Planform Migrates Downdrift. Variation of Surface Layer K Values with Time at Locations A, B, C, D and E are presented in Figure 19. Wave and other Project Conditions Presented in Table I. 50.0 30.0 Y (m) 0 10.0 -10.0 I I I I I I I I necessary to calculate the character of the sand flowing into and out of a cell; this was accomplished by starting with the updrift grid line as a boundary condition and, on cells where erosion occurred, modifying the magnitude and character of the material in the littoral stream. In the nourished area, the K value was set equal to KN when the nourished thickness reached zero. Figures 18 and 19 show the composite surface layer K values at the five locations indicated in Figures 16 and 17, respectively. It is noted that the centroids of the planform anomalies migrate updrift and downdrift for KF < KN and KF > KN, respectively and that this profile migrational signature could possibly be used in conjunction with a field monitoring program to establish the relative sediment transport coefficients, KF and KN. SUMMARY AND CONCLUSIONS Two numerical methods have been presented for calculating shoreline evolution subsequent to a beach nourishment project. These methods fall within the class of one-line shoreline models in which the active vertical portion of the profile is represented by only one contour line. Both models are capable of representing shore-perpendicular structures such as groins that might be used to stabilize beach nourishment projects. One model is quite simple, with refraction and shoaling from deep water to breaking conditions occurring in a closed form. In the second method, refraction and wave shoaling are carried out on a two- dimensional grid using conditions of wave number irrotationality and conservation of wave energy, respectively. The simple model requires approximately 1/250 of the computer time of the more detailed model. In applying the two methods, it is recommended that the actual pre-project bathymetry be replaced by contour lines which are straight and parallel with an alignment of the general shoreline. This recognizes implicitly the shorter time scales associated with the project evolution as compared to that of the original bathymetric and shoreline disequilibrium. Pre-project background erosion is interpreted as due to cross-shore and longshore transport components and is assumed to continue unchanged after project construction. 41 0.80 -T-i 7 -------.------------7------------ ----- ---- -r -~ -0.75 - -- ---A ---- ATI -4--TE0.6 0 C) u 0. 0 .. 2.. ..6. 7 .. 1 . to --- - - ---------A-Y0r AfTe N8rsmn 7) - A S0.70 / Years~~- Afe Nuismn Figure 18. Example 8c. Variation of Surface Layer Longshore Transport Coefficient K with Time at the Five Locations Shown in Figure 16. Case of Kp = 0.693, KN = 0.77, Yi = 2.0 m. 0 9 U) .4-D 0 o 0.85 U) ~ 0. 80 - ~ - -- - w 0) C -- --- --- --- -- --- --- --- --- -- --- --- --- --- -0 ------ AT A 0.75 T-- TB C13 --- AT C- AT Ca, - AT E C/) 0.70 ' 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Years After Nourishment Figure 19. Example 8d. Variation of Surface Layer Longshore Transport Coefficient K with Time at the Five Locations Shown in Figure 17. Case of KF = 0.847, KN = 0.77, Yij = 2.0 m. Comparison of results of applying the two models has established that they yield essentially the same evolution. In the absence of littoral barriers, the planform asymmetry due to oblique waves is shown to be small and is interpreted to be a result of the small aspect ratio (width to length) of the nourished project. The interaction of nourishment retention structures with background transport is examined. Such structures can cause a wide range of effects within and adjacent to the nourished area depending on the characteristics of the background transport. The effects of nourishing with sediment of transport characteristics different than the native are investigated. If the nourishment material is transported less readily, the nourishment project acts as an erodible barrier causing accretion and erosion on the updrift and downdrift sides of the project, respectively. For cases in which the nourishment material is transported more readily than the native, the shorelines updrift and downdrift of the project both accrete with the greater accretion occurring on the downdrift side. For nourishment materials which are less and more transportable than the native, the centroids of the planform anomalies migrate updrift and downdrift respectively with time. The results presented herein predict much greater longevity than those by Pilkey and coworkers although it is cautioned that most of examples do not include background erosion which can be a significant site-specific factor. It is concluded that for the range of cases and conditions examined, the simple model yields results as valid as the more detailed model which requires much greater computer time. Finally, models such as those developed herein can be effective in rational benefit/cost analyses of potential beach nourishment projects. 44 Appendix I REFERENCES Bakker, W.T. (1968) "The Dynamics of a Coast With a Groyne System", Proceedings, Eleventh International Conference on Coastal Engineering, ASCE, pp. 492-517. Dalrymple, R.A. (1988) "Model for Refraction of Water Waves", Journal of Waterway, Port, Coastal and Ocean Engineering, Vol. 114, No. 4, July, pp. 423-435. Dean, R.G. (1974) "Compatibility of Borrow Material for Beach Fills", Chapter 77, Proceedings, Fourteenth International Conference on Coastal Engineering, Copenhagen, Denmark, pp. 1319-1333. Dean, R.G. (1983) "Principles of Beach Nourishment", In: Komar, P.D. (Ed.), CRC Handbook of Coastal Processes and Erosion, Boca Raton: CRC Press, pp. 217-232. Dean, R.G. (1988a) "Realistic Economic Benefits from Beach Nourishment", Chapter 116, Proceedings, Twenty-First International Coastal Engineering Conference, Malaga, Spain, pp. 1558-1572. Dean, R.G. (1988b) "Engineering Design Principles", Short Course on Principles and Applications of Beach Nourishment, Gainesville, FL: Florida Shore and Beach Preservation Association, 42 p. Dean, R.G., E.P. Berek, C.G. Gable and R.J. Seymour (1982) "Longshore Transport Determined by an Efficient Trap", Chapter 60, Proceedings, Eighteenth International Conference on Coastal Engineering, Cape Town, South Africa, pp. 954-968. Hanson, H. (1989) "Genesis A Generalized Shoreline Change Numerical Model", Journal of Coastal Research, Vol. 5, No. 1, pp. 1-28. Hanson, H. and N. Kraus (1987) "Comparison of Analytic and Numerical Solutions of the OneLine Model of Shoreline Change", Proceedings ASCE Specialty Conference on Coastal Sediments '87, p. 500-514. Hanson, H. and N. Kraus (1989) "Genesis: Generalized Model for Simulating Shoreline Change", Technical Report CERC-89-19, Coastal Engineering Research Center, Waterways Experiment Station, U.S. Army Corps of Engineers, Vicksburg, Mississippi. James, W.R. (1974) "Beach Fill Stability and Borrow Material Texture", Chapter 78, Proceedings, Fourteenth International Conference on Coastal Engineering, Copenhagen, Denmark, pp. 1334-1349. Komar, P.D. and D.L. Inman (1970) "Longshore Sand Transport on Beaches", Journal of Geophysical Research., Vol. 75, pp. 5914-5927. Kriebel, D.L. (1982) "Beach and Dune Response to Hurricanes", M.S. Thesis, Department of Civil Engineering, University of Delaware, Newark, DE. Kriebel, D.L. and R.G. Dean (1985) "Numerical Simulation of Time-Dependent Beach and Dune Erosion", Coastal Engineering, Vol. 9, No. 3, p. 221-245. Krumbein, W.C. and W.R. James (1965) "A Log-Normal Size Distribution Model for Estimating Stability of Beach Fill Material", Technical Memo. No. 16, U.S. Army Coastal Engineering Research Center, Washington, D.C. 45 Larson, M. (1988) "Quantification of Beach Profile Change", Report No. 1008, Department of Water Resources Engineering, Lund University, Lund, Sweden. Larson, M. and N.C. Kraus (1989) "SBEACH: Numerical Model for Simulating Storm-Induced Beach Change Report 1: Empirical Foundation and Model Development", Technical Report CERC-89-9, Coastal Engineering Research Center, Waterways Experiment Station, Vicksburg, Mississippi. Leonard, L.A., K.L. Dixon and 0.H. Pilkey (1990) "A Comparison of Beach Replenishment on the U.S. Atlantic, Pacific and Gulf Coasts", Special Issue No. 6 Artificial Beaches, Journal of Coastal Research, pp. 127-140. Leonard, L.A. T.D. Clayton and 0.H. Pilkey (1990) "An Analysis of Replenished Beach Design Parameters on U.S. East Coast Barrier Islands", Journal of Coastal Research, Vol. 6, No. 1, pp. 15-36. LeMehaute, B. and M. Soldate (1977) "Mathematical Modeling of Shoreline Evolution", U.S. Army Corps of Engineers, Coastal Engineering Research Center, Miscellaneous Report No. 77-10, Noda, E.K. (1972) "Wave Induced Circulation and Longshore Current Patterns in the Coastal Zone", Tetra-Tech No. TC-149-3. Pelnard-Considere, R. (1956) "Essai de Theorie de l'Evolution des Formes de Rivate en Plages de Sable et de Galets", 4th Journees de l'Hydraulique, Les Engergies de la Mar, Question III, Rapport No. 1. Perlin, M. (1978) "A Numerical Model to Predict Beach Planforms in the Vicinity of Littoral Barriers", Thesis presented to the University of Delaware, Newark, DE in partial fulfillment of the requirements of the Master of Science Degree. Perlin, M. and R.G. Dean (1983) "An Efficient Numerical Algorithm for Wave Refraction/Shoaling Problems", Proceedings, Coastal Structures '83, ASCE, Arlington, VA, pp. 988-1010. Perlin, M. and R.G. Dean (1985) "3-D Model of Bathymetric Response to Structures", Journal of Waterway, Port, Coastal and Ocean Engineering, Vol. 111, No. 2, March, pp. 153-170. Pilarczyk, K.W. and J. van Overeem (1987) "Manual on Artificial Beach Nourishment", Report 130, Center for Civil Engineering Research, Rijkswaterstaat and Delft Hydraulics, 195 pp. Pilkey, O.H. and T.D. Clayton (1989) "Summary of Beach Replenishment Experience on U.S. East Coast Barrier Islands", Journal of Coastal Research, Vol. 5, No. 1, pp. 147-159. U.S. Army Corps of Engineers (1984) "Shore Protection Manual", Coastal Engineering Research Center, Volume I, Superintendent of Documents, U.S. Government Printing Office, Washington, DC. Walton, T.L. and T.Y. Chiu (1979) "A Review of Analytical Techniques to Solve the Sand Transport Equation and Some Simplified Solutions", Proceedings of the ASCE Conference on Coastal Sediments '79, p. 809-837. 46 1 APPENDIX II. NOTATION The following symbols are used in this paper: b = subscript denoting breaking; B = beach berm height; C = wave celerity; CG = group velocity; D = characteristic sediment size; V = wave energy dissipation per unit volume; E = wave energy density; f = probability distribution; g = gravitational constant; G = longshore diffusivity; h = water depth; h. = depth of closure; hI. = depth of original profile, immediately adjacent to filled profile, see Figure 4b; H = wave height; I = immersed weight sediment transport rate; k = wave number; K = sediment transport proportionality factor; K' = proportionality factor for cross-shore sediment transport; K" = proportionality factor for project longevity; I = nourishment project length; n = renourishment interval; N = project life; p = in place sediment porosity; Ps = longshore energy flux factor; s = relative sediment density, ps/p; t = time; T = wave period; q.,y = sediment transport distribution in x, y directions; Q = total longshore sediment transport; -V = volume; x = longshore coordinate; y = cross-shore coordinate, position seaward; Y = initial nourished beach width; a = wave direction, relative to north; # = shoreline or contour orientation, relative to north; 0 = wave crest orientation, relative to bottom contours; r = breaking wave height proportionality factor; I = measure of ambient shoreline orientation; p = water mass density; p, = sediment mass density; r = smoothing factor; and V = horizontal vector differential operator. 47 |