Equilibrium Beach Profiles: Characteristics and Applications Robert G

JournalofCoastal Research I 7 I 1 1 53-841 FortLauderdale, FloridaIWinter 1991

Equilibrium Beach Profiles: Characteristics and Applications Robert G. Dean

Coastal and Oceanographic Engineering Department University of Florida 336 Weil Hall Gainesville, FL 32611, USA ABSTRACT

DEAN, R.G., 1990. Equilibrium beach profiles: characteristics and applications. Journal of Coastal Research, 7(1), 53-84. Fort Lauderdale (Florida), ISSN 0749-0208. ?? ?? An understanding of equilibrium beach profiles can be useful in a number of types of coastal engineering projects. Empirical correlations between a scale parameter and the sediment size or fall velocity allow computation of equilibrium beach profiles. The most often used form is h(y) = Ay2/3 in which h is the water depth at a distance y from the shoreline and A is the sediment-dependent scale parameter. Expressions for shoreline position change are presented for arbitrary water levels and wave heights. Application of equilibrium beach profile concepts to profile changes seaward of a seawall include effects of sea level change and arbitrary wave heights. For fixed wave heights and increasing water level, the additional depth adjacent to the seawall first increases, then decreases to zero for a wave height just breaking at the seawall. Shoreline recession and implications due to increased sea level and wave heights are examined. It is shown, for the equilibrium profile form examined, that the effect of wave set-up on recession is small compared to expected storm tides during storms. Profile evolution from a uniform slope is shown to result in five different profile types, depending on initial slope, sediment characteristics, berm height and depth of active sediment redistribution. The reduction in required sand volumes through perching of a nourished beach by an offshore sill is examined for arbitrary sediment and sill combinations. When beaches are nourished with a sediment of arbitrary but uniform size, it is found that three types of profiles can result: (1) submerged profiles in which the placed sediment is of smaller diameter than the native and all of the sediment equilibrates underwater with no widening of the dry beach, (2) non-intersecting profiles in which the seaward portion of the placed material lies above the original profile at that location, and (3) intersecting profiles with the placed sand coarser than the native and resulting in the placed profile intersecting with the original profile. Equations and graphs are presented portraying the additional dry beach width for differing volumes of sand of varying sizes relative to the native. The offshore volumetric redistribution of material due to sea level rise as a function of water depth is of interest in interpreting the cause of shoreline recession. If only offshore transport occurs and the surveys extend over the active profile, the net volumetric change is zero. It is shown that the maximum volume change due to cross-shore sediment redistribution is only a fraction of the product of the active vertical profile dimension and shoreline recession. The paper presents several other applications of equilibrium beach profiles to problems of coastal engineering interest.

ADDITIONAL INDEX WORDS: Beach erosion, nourishment, sea level rise, seawalls.

INTRODUCTION In a broad sense, it is obvious that sand particles are acted upon by a complex system of A of the quantitative understanding charac- constructive and destructive generic "forces" teristics of beach is equilibrium profiles central with the constructive forces acting to displace to rational design of many coastal engineering the sediment particle landward and vice versa. projects and to the interpretation of nearshore Constructive forces include landward directed processes. Several features of equilibrium bottom shear stresses due to the nonlinear beach profiles are well-known: (1) they tend to character of shallow water waves, landward be concave upwards, (2) smaller and larger sand directed "streaming" velocities in the bottom diameters are associated with milder and boundary layer (BAGNOLD, 1946; PHILLIPS, steeper slopes, respectively, (3) the beach face 1966), the phasing associated with intermittent is and waves approximately planar, (4) steep suspended sediment motion, etc. The most result in milder and a for bar slopes tendency obvious destructive force is that of gravity cou- formation. pled with the destabilizing effects of turbulence 90048 received26 February 1990; accepted in revision 8 June 1990. induced by wave-breaking; others include the 54 Dean

effect of seaward directed bottom undertow cur- iments, and ranges of wave heights and periods rents and forces due to wave set-up within the and initial slopes of planar beaches. Three surf zone (e.g. SVENDSEN, 1984; STIVE and beach profile types were established in labora- WIND, 1986). Indeed, the above represents only tory experiments including one erosional and a partial listing of the complex force system act- two accretional types. SUH and DALRYMPLE ing on sediment particles and serves to illus- (1988) applied concepts of equilibrium beach trate the difficulty of a rational physics-based profiles to address the same problem as Suna- prediction of equilibrium beach profiles. mura and Horikawa and identified one ero- Several approaches have been pursued in an sional profile type and one accretional type. attempt to characterize equilibrium beach pro- Comparison of laboratory data demonstrated files. KEULEGAN and KRUMBEIN (1949) good agreement with their criteria and predic- investigated the characteristics of a mild bot- tions of profile changes. tom slope such that the waves never break but Numerous investigations have been carried rather are continually dissipated by energy out to develop appropriate scale modeling cri- losses due to bottom friction. BRUUN (1954) teria including DALRYMPLE and THOMPSON analyzed beach profiles from the Danish North (1976), NODA (1972), HUGHES (1983) and Sea coast and Mission Bay, California, and VAN HIJUM (1975). HAYDEN et al. (1975) found that they followed the simple relation- apparently were the first to apply the concept of ship empirical orthogonal functions (EOF) to extract the principal modes of change from a set of = h(y) Ay2/3 (1) beach profile data. Numerous later studies have in which h is the water depth at a seaward dis- used this approach to investigate the character tance, y, and A is a scale parameter which of the dominant modes of profile change, e.g. depends primarily on sediment characteristics. WINANT et al. (1975), WEISHAR and WOOD EAGLESON, GLENNE and DRACUP (1963) (1983), AUBREY et al. (1977), and AUBREY developed a complex characterization of the (1979). The EOF is a purely descriptive method wave and gravity forces acting on a particle and does not address the causes or processes of located outside the zone of"appreciable breaker profile change. influence" and developed expressions for the HAYDEN et al. (1975) assembled a data set seaward limit of motion and for the beach slope comprising 504 beach profiles along the Atlan- for which a sand particle would be in equilib- tic and Gulf coasts of the United States. DEAN rium. (1977) analyzed these profiles and used a least SWART (1974) carried out a series of wave squares procedure to fit an equation of the form tank tests and developed empirical expressions h = Ay" (3) relating profile geometry and transport characteristics to the wave and sediment conditions. to the data and found a central value of n = 2/3as The active beach profile was divided into four BRUUN had earlier. It was shown that Eq. (3) zones and empirical expressions were developed with n = 2/3 is consistent with uniform wave for each zone. VELLINGA (1983) investigated energy dissipation per unit volume, ., within dune erosion using wave tank tests and devel- the breaking zone, i.e., oped the following "erosion profile" which included the effect of water 1 0 deep significant Z/.= - (ECG) (4) wave height, H,,, and sediment fall velocity, w, h ay

1.28 where E and CG are the wave 67.6 0.56 energy density i7 h = 0.47i y+)18 and group velocity, respectively. It can be 2.0)( shown from linear shallow water wave theory (2) that D. and A are related by 2/3 in which the values of all variables are in the S24~.(D) A = 24(D) metric system. It can be shown that Eq. (2) is in 5pg/22 (5) reasonably good agreement with Eq. (1). SUN- AMURA and HORIKAWA (1974) examined and in which E is the wave energy density, CG is the characterized beach profiles for two sizes of sed- group velocity, p is the water mass density, g is

Journal of Coastal Research, Vol. 7, No. 1, 1991 Equilibrium Beach Profiles 55 gravity, D is sediment particle diameter and K log plot) as presented by the dashed line in Fig- is a constant relating wave height to water ure 1. depth within the surf zone. The interpretation KRIEBEL (1982), KRIEBEL and DEAN of Eq. (3) is that a particle of given size is char- (1984, 1985) and KRIEBEL (1986) have consid- acterized by an associated stability and that the ered profiles out of equilibrium by hypothesiz- wave breaking process results in the transfor- ing that the offshore transport is proportional mation of organized wave motion into chaotic to the difference between the actual and equi- turbulence fluctuations; these fluctuations are librium wave energy dissipation per unit vol- destructive forces and, if too great, cause mobi- ume, i.e. lization of the sediment particle with resulting = K( - .) (6) offshore displacement and a milder beach slope, Q •0 which reduces the wave energy dissipation per Eq. (6) and a sand conservation relationship unit volume eventually resulting in an equilib- have been incorporated into a numerical model rium profile. Later MOORE (1982) collected and of sediment transport with generally good con- analyzed a number of published beach profiles firmation between laboratory profiles (Figure 5) and developed the relationship between A and and field results (Figure 6). LARSON (1988) D as shown by the solid line in Figure 1. As and LARSON and KRAUS (1989) have consid- expected, the larger the sediment size, the ered the active region of sediment transport in greater the A parameter and the steeper the four zones and have developed empirical coun- beach slope. Figures 2, 3 and 4 present several terparts to Eq. (6) for each zone, thus allowing profiles employed by Moore in establishing the solution of the transient beach profile problem relationship presented in Figure 1. The profile including a capability for generating longshore in Figure 2 is of particular interest as the sed- bars. iment particle size ranges from 15 cm to 30 cm, approximately the size of bowling balls! DEAN MODIFIED EQUILIBRIUM BEACH (1987a) has shown that when the relationship PROFILE presented in Figure 1 is transformed to A(w) rather than A(D), where w is the fall velocity, An unrealistic property of the form of the the relationship is surprisingly linear (on a log- equilibrium beach profile represented by Eq. (1)

SEDIMENTFALL VELOCITY,w (cm/s) S 0.01 0.1 1.0 10.0 100.0 E 1.0 . SuggestedEmpirical RelationshipAvs. D CE (Moore) / / From W Hughes' 44 FieldResults , A=0.067 FromIndividual FieldI -. Profileswhere a Range of SandSizes was Given\ C. W 0.10. Basedon Transforming • AAh - / A vs D Curveusing : FallVelocity Relationship

F•romSwart's S/ LaboratoryResults O 0.01 cr 0.01 0.1 1.0 10.0 100.0 SEDIMENT SIZE, D (mm)

Figure 1. Beach profile factor, A, vs sediment diameter, D, and Fall velocity, w, in relationship h = Ax"2' (Dean, 1987a; modified from Moore, 1982).

Journal of Coastal Research, Vol. 7, No. 1, 1991 56 Dean

DISTANCE OFFSHORE (m) 0.00 25.00 50.00 0.00 E -- Least Squares Fit - ActualProfile

o 5.00

10.00-

Figure 2. Profile P4 from Zenkovich (1967). A boulder coast in Eastern Kamchatka. Sand diameter: 150mm-300 mm. Least squares value of A = 0.82 m113 (from Moore, 1982).

DISTANCE OFFSHORE (m) 0.00 60.00 120.00 0.00 E ---- Least Squares Fit --- ActualProfile I- 0 3.00

I- 00

6.00

Figure 3. Profile P10 from Zenkovich (1967). Near the end of a spit in western Black Sea. Whole and broken shells. A = 0.25 m11/3(from Moore, 1982).

DISTANCEOFFSHORE (m) 0.00 25.00 50.00 75.00 100.00 0.00 E Least Squares Fit ------Actual Profile I-

1.00 -

-2.00

2.00

Figure 4. Profile from Zenkovich (1967). Eastern Kamchatka. Mean sand diameter: 0.25 mm. Least squares value of A = 0.07 m1/3 (from Moore, 1982).

Journal of Coastal Research, Vol. 7, No. 1, 1991 Equilibrium Beach Profiles 57

10.00 Erosion Model Calibration vs Saville's (1957) Large-Scale Laboratory Test 5.00 Inital Slope 1:15 5.00ItialSlope1:15 H = 5.5 ft \ = 11.3 sec. 0 HT

0.00 z Predicted at 40 brs W -5.00 - Observed uJ at 40 hrs

-10.00-

-15.00 I I I I 0.00 40.00 80.00 120.00 160.00 200.00 240.00 280.00 320.00 DISTANCE(ft)

Figure 5. Comparison of calibrated profile response model with large wave tank data by Saville (1957) (from Kriebel, 1986).

30 30 ...... "...... ".-...... ??-?;??;???i? ??l. ??l?

...... ????...... I ...... 1 1 - 1 - 1 I.- I I I...... 20 _ _.. _. ..T .....

b s e rv e d ...... ?? ???;???;???;I???;???;???; ???; ; ??;. ??;?-?;... ?? I??i;0 I??-; .....t ~???i 10 ...... ; ;...... - ...... -.' . - ...... i...... i...... i--.-...... :...... -- - ...... i...... - i...... : re d .,ic te d ...-: : : : : : : : : ....I..i ...... ?...... ???I?..:.. .. . : : : : : : ::-.:...P ...... ,7...... -" ...... "-... .'"'...... '" . ..-, ...... ,....;...... 10 ...._ ' - +...... I...,...._ . . - I. 0 ' , ...... -250 -200 -150 -100 -50 0 50 100 150 200 250

MONUMENT(ft)

Figure 6. Comparison of calibrated profile response model with field profile affected by Hurricane Eloise as reported by Chiu (1977) (from Kriebel, 1986). is the predicted infinite slope at the shoreline. (1). A slight modification to include gravita- Large slopes induce correspondingly large tional effects is gravity forces which are not represented in Eq.

Journal of Coastal Research, Vol. 7, No. 1, 1991 58 Dean

DISTANCE OFFSHORE (m) W. 0o 00 200 300

------S 1 N N T-- ". -Gravitational Effects Included NNI ------

X2 N. 0.-.Ea

Only Wave Dissipation Included--/ Figure 8. Definition sketch for profile response due to sea 41 level rise. Figure 7. Comparison of equilibrium beach profile with and without gravitational effects included. A = 0.1 m1/3 corresponding to a sand size of 0.2 mm. APPLICATIONS OF EQUILIBRIUM BEACH PROFILES OF THE FORM: h = Ay2" 0 .ah 1 - - + (EC) =. (7) may h By The utility of Eq. (1) will be illustrated by several A definition sketch of the Gravity Effect Turbulence Effect examples. system of interest is presented as Figure 8. In in which the two terms on the left hand size rep- results presented here, it will be assumed that resent destabilizing forces due to gravity and within the surf zone the wave height is propor- turbulent fluctuations due to wave energy dis- tional to the local water depth with the propor- sipation, m is the beach face slope, and as before tionality factor, K, i.e. H = Kh(K - 0.78). In par- (. represents the stability characteristics of ticular, the breaking wave height, Hb, and the sediment particle but now the interpreta- breaking depth, h., are related by Hb = Kh,. tion of 0. is broadened beyond equilibrium unit volume to energy dissipation per include Effect of Sediment Size on Beach Profile gravity as an additional destabilizing force. Eq. (7) can be to: integrated Figure 9 presents two examples of the effect h 1 of sediment size on beach profile. The scale ;pY+ h (8) = --m Ah3/232A3/2 parameter A is determined for various sediment sizes from Figure 1 and the profiles com- where, as before, A is related to A. by Eq. (5). puted from Eq. (1). In shallow water, the first term in Eq. (8) dominates, simplifying to Beach Response to Altered Water Level and Waves h = my (9) The effects of elevated and lowered water lev- i.e. the beach face is of uniform slope, m, con- els will be treated sistent with measurements in nature. In deeper separately. An elevated water with wave and water, the second term in Eq. (8) dominates level, S, sediment conditions such that the is with the following simplification profile reconfigured out to a depth, h., is assumed to h = Ay213 (10) result in equilibrium with the final state being the same profile form as before, but relative to as presented earlier. the elevated water level. This situation could 7 a of Figure presents comparison Eq. (8), pertain to a storm tide of long duration or to sea which includes the planar portion near the level rise. water line and (1) which has an infinite Eq. Referring to Figure 8, the sand volume at the water line. A form similar to (8) slope Eq. eroded, VE, is equal to the volume deposited, V, was adopted by LARSON (1988) and LARSON and KRAUS (1989). VE = VD(11)

Journal of Coastal Research, Vol. 7, No. 1, 1991 Equilibrium Beach Profiles 59

Ay = -S (15) (h. +B)

first proposed by BRUUN (1962) and now E CE referred to as the "Bruun Rule". E , For the case of lowered water levels, there w will be an excess of sand in the active system 0 I and a resulting advancement of the shoreline. Cl) LL The profile equilibration depth, h., will occur at 0 LL8 w a distance, W2, from the original shoreline. This L.) E Z I ? E equilibrium depth is considered to extend as a

Cl) horizontal terrace from the distance W2 noted above to a landward location consistent with the equilibrium profile and the shoreline advance, Ay. For this case, shown in Figure 11, the non-dimensional shoreline advancement can be shown to be

(wJ) Hld3G = 2 (1 - - 1 S'B')512 (16) y' - 5 (B' - S'B' + 1) Figure 9. Equilibrium beach profiles for sand sizes of 0.2 mm and 0.6 mm A(D = 0.2 mm) = 0.1 m1/3, A(D = 0.6 mm) = 0.20 1/3 where it is emphasized that for this case, S' < 0. Additionally,

When Eqs. (1) and (11) are combined the follow- W'2 (1 - S'B')3/2 (17) ing implicit equation for the shoreline change, W. Ay, is obtained and the landward location of the terrace, W", is 5/3 3 +Ay hW.T W1 W' 1 + y' (18) =W = 5 B \B (12) The other non-dimensional parameters are as defined by Eqs. (14). in which W. is the seaward limit of the active Equilibrium Profile and Recession W. = profile , B is the berm height Including the Effect of Wave Set-Up and for this case of shoreline recession Ay < 0. The equilibrium beach profile, h = Ay2/3 (12) can be in non-dimensional Eq. expressed interpreted as resulting from uniform wave form as energy dissipation per unit water volume does 3 not include the effect of wave set-up. As a pre- Ay' - - (1+ Ay')513]+ S' = 0 (13) 5B B[1 cursor to developing recession predictions due to increased water levels and wave set-up, it is in which the non-dimensional variables are useful to first establish the equilibrium beach profile, including the effect of wave set-up, ~r. IAyy' We start with the well-known solution for wave across the surf zone et B set-up (BOWEN al., B' (14) 1968) = h. S K(Y)= 1b + J[hb - h(y)] (19) B = in which hb is the breaking depth (hb h. - 9b Eq. (13) is plotted in Figure 10. For small val- - S), 'qb is the set-up (actually negative) at ues of Ay', Eq. (13) can be approximated by breaking, and

Journal of Coastal Research, Vol. 7, No. 1, 1991 60 Dean

1.0

MNSONES / / S0.8D- - / I-o " i z~ /" / -/ _. ,' -. c

) 0.2

o 0.0'l l" 0.1 0.2 0.5 1.0 2.0 5.0 10.0

DIMENSIONLESSSTORM BREAKINGDEPTH, h 1 B -B Figure 10. Isolines of dimensionless shoreline change, Ay', vs dimensionless storm breaking depth, h:, and dimensionless storm tide, S'.

I4-W2 W,

B OriginalWater Level ? --'3 .. ? . ,..

Original h. Profile Tr

Horizontal Terrace

Figure 11. Profile geometry and notation for shoreline advancement due to lowering of water level.

3K2/8 heights. The equilibrium profile based on uni- J (20) =+3/81+3K2/8 form wave energy dissipation per unit volume 1 kH commences from 8 sinh (21) == 2kh. (22) (h + S + --) ) ay, (EC,) where k is the wave number. Since - is positive over most of the surf zone, it is reasonable that in which y, is directed landward, and as before it contributes much like a tide in causing reces- E is the wave energy density, and CG the group sion, especially for the larger breaking wave velocity. Assuming shallow water,

Journal of Coastal Research, Vol. 7, No. 1, 1991 Equilibrium Beach Profiles 61

The question of the relative roles of breaking ECG +) waves and storm surges can be addressed by =-K2(h+S+8 I)2Vg(h+S simplifying Eq. (25) for the case of relatively If the algebra is carried through, the not-too- small non-dimensional beach recessions, surprising result is obtained IAy'l< <1), which upon adoption of K = 0.78 and expressing in dimensional form yields h+S+ + = Ay2/3 (23) where the is now the location where H S y-origin 0.068 + - h+S = 0. 12 a of the B B + Figure presents plot (26) non-dimensional wave and the associ- Hb set-up W.Ay_ 1 + 1.28- ated equilibrium beach profile. B In the following development, the effects of a uniform storm tide, S, and the set-up which var- It is necessary to exercise care in interpreting ies across the surf zone (Eq. (23)) will be con- Eq. (26), as the surf zone width, W. includes the sidered as presented in Figure 13. Following effect of the breaking wave height, procedures similar to those used for determin- 3/2 Hb ing recession with a water level which is uni- W*= (27) form across the surf zone, volumes are equated =\KA/ as As shown the dimensionless beach 0 + Ay by Eq. (26), f S fW - Ay)2dy is much more related to [B - (y)]dy + AA(y change (Ay/W.) strongly S storm surge than wave height with the storm W + = W +Ay I Ay2/3dy+ [S + (y)]dy (24) surge being approximately 16 times as effective AY. in causing the dimensionless beach recession. which after considerable algebra yields However, during storms the breaking wave 35 height may be two to three times as great as the Ay' + + Ay']5 storm tide and the larger breaking waves may •-B[1 much than the storm tides. 1 (3/5 - J) persist longer peak (25) The reason that the storm tide plays a much B' (1 - J) -S'- 'b greater role than that due to breaking wave set- in which Eqs. (14) have been used for non- up is evident from Figure 12 where it is seen dimensionalization and = 'rb/B. that the wave set-up (actually the set-down) 9rb

Break Point - 02

0.2 0.8 1.0 Y/w.

0.2- h/h

0.6

0.8

1.04 1.0-

Figure 12. Non-dimensional wave set-up and equilibrium beach profile.

Journal of Coastal Research, Vol. 7, No. 1, 1991 62 Dean

do.

Ay<0

h(y)

Figure 13. Beach recession due to waves and increased water level, including the effect of wave set-up.

acts to reduce the mean water level over a sub- equilibrium with the elevated water level. stantial portion of the surf zone. Applying equilibrium profile concepts, it is pos- For the case in which storm surge is not sible to calculate only the second component. important and the ratio of breaking wave The system of interest is presented in Figure height to berm height is large, Eq. (26) can be 14. The profile is considered to be in equilib- simplified to represent only the effects of waves rium with virtual origin = 0. For a water yl and wave-induced set-up level elevated by an amount, S, the equilibrium profile will not be different and will have a vir- = -0.053 tual origin at y2 0. We denote the distances W.Ay = from these virtual origins to the wall as Yw1and or Yw2 for the original and elevated water levels, As in the 3/2 respectively. previous cases, approach is to establish the origin, Yw2, (now virtual) such Ay = -0.053(a) (28) that the sand volumes seaward of the seawall and associated with the equilibrium profiles are before and after the increase in water Thus for a doubled wave height, the recession equal level. In the all (h values) are induced by wave set-up increases by a factor of following, depths 2.8. referenced to the original water level except h. which, as described previously, is a reference depth related to the breaking wave Profile Adjustment Adjacent to a Seawall height. volumes as before Due to Altered Water Level and Waves Equating W2W hi(y)dy,-= I2h2y2)dy2 We first consider the case of profile lowering wi W2 adjacent to a seawall due to an elevated water which can be = level. integrated using hi(y) Ay/3 It is well-known that during storms a scour and h2(y) + S = Ay'/3 and simplified to yield trough will occur adjacent to a sea wall. For 5/2 h'w2+S' /2 5/2 here it is to consider this - 3/2 purposes appropriate h' h'w2+S')3/2h'f scour as a profile lowering due to two components: (1) the localized and probably dominant effect due to the interaction of the seawall, h'( waves, and tides, and (2) the effect due to sed- h'. 2 iment transport offshore to form a profile in 3 h'f h'*

Journal of Coastal Research, Vol. 7, No. 1, 1991 Equilibrium Beach Profiles 63

Yw2 Yw1 Sea l 7-7' Sea Level - -- w"Sa ------Y2--- i-Oncreased Irs Virtual Origin, wallh S f,,-Orlignal Sea Levei Increased A Y Sea Level hW2 h Virtual Origin, Ahw h2 hi Originial Sea Level

Figure 14. Definition sketch. Profile erosion due to sea level increase and influence of seawall.

in which the primes represent non-dimensional level rises further, with the same total break- quantities defined as ing depth, the active surf zone width decreases, such that less sand must be transported sea- hw ward to satisfy the equilibrium profile. With increasing storm tide, the surf zone width h. approaches zero at the limit h'. - (30) hwl S + = h. S hw1 S' - or hw1 Eq. (29) is implicit in h4 and must be solved by S' = h'. - 1 iteration. Defining the change in depth at the wall, Ahw, as Which corresponds to the upper line in Figure 15. It is emphasized that the increased depth at = Ah, hw2- h,1 the seawall predicted here does not include the and in non-dimensional form scour interaction effect of the seawall and waves. Ah, Ah'w =- We next consider the case of lowered water hw,, level (S < 0) adjacent to a seawall as shown in The quantity Ah4is now a function of the fol- Figure 17. In this case sediment will move land- lowing two non-dimensional variables: S' and ward due to the disequilibrium caused by the lowered water level. The notation is the same h'. The relationship Ah(h'., S') is presented in Figure 15 where it is seen that for a fixed h' and as in the previous case. Equating volumes eroded and is as increasing S', the non-dimensional scour, Ahh deposited expressed first increases and then decreases to zero. Fig- WA -YW2 ure 16 presents a specific example for h' = 6. The interpretation of this form is that as S' wI +YW1hi(y)dyI increases, the profile is no longer in equilibrium + - and sand is seaward to the =f h2(2)dy2 h.(W' W.) (31) transported develop w2 equilibrium profile and the water depth adjacent to the seawall increases. However, as sea which can be integrated and simplified to yield

Journal of Coastal Research, Vol. 7, No. 1, 1991 64 Dean

20.0 S I I T 1 _ S-h hw, - o 10.0o h hb 8.0 hw, f LU - 6.0- ALw/Ahw

) uJ 2.02.5 S 1.0 - - - - 0.8 z 1.5 - U 0.6 o 0.4 0.2 0.1 0.5 0 0.41 %I 1 I 0 " I I 1 2 4 6 810, 20 , 40 6080100 DIMENSIONLESSSTORM BREAKING DEPTH, h', Figure 15. Isolines of dimensionless seawall toe scour, Ah/•vs dimensionless storm tide, S' and dimensionless breaking depth h'.

NON-DIMENSIONAL CHANGE WIA= + Y'w2- Y'w1 (33) IN DEPTH AT SEAWALL,Ah'W 1 --) =r o r" o o The non-dimensional depth change (decrease) I* at the wall is

z = - 1 (34) 0. Ah'w h'w, z which can be established by solving Eq. (32). z0 o of z Figure 18 presents a plot Ah'h., S'). cn In Eq. (32), the term h, + S' represents the z W non-dimensional total depth at the seawall on 0 the equilibrium profile for the lowered water -4 level. A limiting case for the above formulation is for this water depth to be zero, i.e. h4 + S' mcn c = Ur 0, yielding C) m 2 (32 - 5(h'- S')5/2 + h'-S'- + h'1/2 = 0 (35) 5 55 5 5

which is plotted as the upper dashed line in Fig- 16. Non-dimensional change in water depth at wall Figure ure 18. Ah/as a function of non-dimensional breaking depth h" = 6.0.

Response from Initial Uniform Slope [3 21 hpW2+S [(h'w2+S') h':J -:(h S')512 For simplicity, many wave tank tests commence with an beach + h'I.-S'3) +2h'1/2 = 0 initially planar slope, mi. S 5/ 5 5 It is of interest to examine the relationship of (32) the equilibrium and initial profiles. As presented in Figure 19, there are five types of equi- The non-dimensional distance WA( WA/IW.) is librium profiles that can form depending on the given by initial slope, mi, and the sediment and wave

Journal of Coastal Research, Vol. 7, No. 1, 1991 Equilibrium Beach Profiles 65

YWI

---o . WA Wi

-777 Seawall (h• S

Ahw Virtual Origins

Horizontal Terrace

Figure 17. Definition sketch, profile response adjacent to a seawall for case of lowered sea level.

-1.0

-0.5 - 11i

. Case wuJS - Limiting 0 W -0.2- - S' = .-J -0.2 h'2+ W I" Q -0.1 - S-0.4 =A hw -0.3 Z -0.05 - 0 -0.2 z w -0.1 -0.02-

0 zI I tutl I ll I I 1 I IIuu I I 1 2 5 10 20 ! 50 100

NON-DIMENSIONALBREAKING DEPTH, = h', h*/hwl

Figure 18. Non-dimensional shallowing adjacent to a seawall due to lowering of water level. Variation with non-dimensional breaking depth, h' and non-dimensional water level, S'.

Journal of Coastal Research, Vol. 7, No. 1, 1991 66 Dean

Initial Type 1 Profile -Shoreline (Recession, No Terrace)

-- Initial Profile

Ay< 0 Type 2 Profile (Recession, No Terrace) -.. InitialProfile h.

Ay> 0

Type 3 Profile (Recession, Terrace) Initial Profile

Profileih

Type 4 Profile (Advancement, No Terrace) initial Profile h*

Ay> 0

Type 5 Profile (Advancement, Terrace) initial Profile h. h.

Figure 19. Illustration of five equilibrium profile types commencing from an initially uniform slope. characteristics. Wave tank tests by SUNA- teristic is that a scarp is formed at the shoreline MURA and HORIKAWA (1974) identified pro- and no berm is deposited. The wave tank profile file Types 1, 2 and 5. Shoreline responses for presented in Figure 5 is an example of a Type 1 Types 1 and 3 have been investigated by SUH profile. Type 2 profile, also a case of shoreline and DALRYMPLE (1988) using equilibrium recession, occurs for a somewhat milder rela- profile concepts and good agreement with wave tive slope (initial to equilibrium), sediment tank data was demonstrated. transport occurs in both the onshore and off- Referring to Figure 19, for the Type 1 profile shore directions and a berm is formed at the the initial slope is much steeper than that for shoreline. With still milder relative slopes, sed- the equilibrium profile and only seaward sedi- iment transport occurs only shoreward result- ment transport occurs. An additional charac- ing in a Type 3 profile characterized by shore-

Journal of Coastal Research, Vol. 7, No. 1, 1991 Equilibrium Beach Profiles 67

line recession with all of the sediment trans- in which ported deposited as a berm feature. A terrace or "bench" (here assumed horizontal) is formed at (38) the seaward end of the equilibrium profile. This y' type profile is probably the least likely to occur due to the unrealistically high berm elevation For this type profile, SUH and DALRYMPLE required. Type 4 profile is one of shoreline (1988) have denoted h./W. as the equilibrium advancement, occurring for still milder initial slope, me, and have shown a correlation slopes and is characterized by sediment trans- between the ratio of initial to equilibrium port in both the landward and seaward direc- slopes and the resulting profile changes. tions. Finally, Type 5 profile is one of shoreline advancement with only landward sediment Type 2 and Type 4 Profiles transport and leaving a horizontal terrace or bench at the seaward end of the equilibrium For these profile types, the non-dimensional profile. The following paragraphs quantify the change, Ay', depends on and non-dimensional profile characteristics and shoreline changes h" berm height, B', for each of these five types. As described previously, shoreline recession and advancement 6 will be denoted by negative and positive Ay, ' = -(B'+1)+ (39) Ay' = -(B'+1)+ 2B'-+5 . (39) respectively. It will be shown that the non- 5?7 dimensional shoreline change, Ay'( Ay/W.) is a function of the non-dimensional depth of lim- For type 2 and Type 4 profiles, the values of Ay' will be iting profile change, h.(= h./W.mi) and non- negative (recession) and positive dimensional berm height, B'(- B/W.mi). The (advancement), respectively. developments associated with these profile The equations for non-dimensional volumes types will not be presented in detail. Methods transported seaward past any location, y', for are similar to those applied earlier in this Type 2 and Type 4 profiles are report for example for the case of shoreline recession due to an elevated water level. Figure i '2 20 presents Ay'(h',B') and the associated 2B'-1[B12_y121 (40a) of regions occurrence for the five profiles types. - (y'+B') , -B'

Journal of Coastal Research, Vol. 7, No. 1, 1991 68 Dean

2.0\ 0 O

000" -06/-/0 -0.7 Type - '.00 J-0. -0.3 X 0.2 / _70/ I/ .000e-0.1 • cA 0/T00 /, /T .4 / y'= .1+0.2 +0.3 au- +061 m? 0.5 I++.40. ,, z - "2/\ / -- +0.7 0

,f yp 2 Type5 cn ~ . ye2A+0.8 1.0 2.0 3.0 1.0 wI m.06 +0.9 Typ~e I ype4\/ 0 1.0 2.0 3.0 NON-DIMENSIONALSEAWARD DEPTH OF ACTIVEMOTION, h NON-DIMENSIONAL SEAWARD DEPTH OF ACTIVE MOTION, hl*-hh'-

Figure 20. Regimes of equilibrium profile types commencing from an initially planar profile showing five types of equilibrium profiles and non-dimensional profile advancement, Ay'.

1 V' = The limit for no shoreline change, Ay' - 0, 2B-(B'2 -y'2) (42a) depends on whether or not a terrace is present. - (y' +B'), - B' 0, Shoreline Recession ?= 2h'. +h'. < Shoreline Advancement 5B ' 1 2 (42c) , 0, • (45) 1 1 h'. , 1 + Ay'

Limits of Profiles Types 4 = 0, No Shoreline Change B'2 - h'2 + -h'. > 0, Shoreline Recession The criterion for berm formation is 5 < 0, Shoreline Advancement 3 < No Berm Formed (46) B 0, *2 > 0, Berm Formed (43) 1h Figures 21 and 22 present examples of profile response and associated volumes transported Here B' is as the non-dimensional interpreted for profile Types 1 and 4. berm height that would be formed if the berm height exceeds the scarp height cut by the Perched Beach shoreline recession into the uniform slope profile. For berm > formation, Ay -B/mi. The offshore extent of a beach is termi- The limit for terrace formation is perched nated by a shore parallel structure which pre- < vents the sand from seaward. In + 0, No Terrace moving conjunc- (B' Terrace h'.)2_-2B, 6h.5i-f> 0, tion with a beach nourishment project, it is pos-

Journal of Coastal Research, Vol. 7, No. 1, 1991 Equilibrium Beach Profiles 69

//0.23 \

/ 0.1- -0.2 0 0.2 0.4 0.6 0.8 1.0 W*

VolumetricTransport Past Any Location

0.2 0.2 0.4 0.6 0.8 1.0 , Y 0.2 Initial Profile

h .4 , ho Profile 0.6 -Equilibrated

0.8 Profile Response - 1.0-

Figure 21. Example of Type 1 (recession) profile response from an initially uniform slope and associated volumetric transport. Note only positive (seaward) transport. sible in principle to obtain a much wider beach profiles (c.f. Figure 24a and b). For simplicity, for the same volume of added sediment. Denoting only the results for non-intersecting profiles the "native" and "fill" sediment scale parameters (without the sill) will be presented here. The as AN and A, respectively (c.f. Eq. (3)). Referring AV . fractional reduction in volume, A~, is given by to Figure 23 for terminology, the required vol- V, ume, V, for the case of the sand just even with the 3/2 top of the submerged breakwater is 3/2 3/2- AV '5B?' A, h = 3/2 5/3 3/2 - E(; -- A hAN,! FA (47) Ay'+ 5/2 5/2- F, + [AN ) - A F ) (49) in which = the volume that would be and the added beach width, Ay, is V, to advance the shoreline seaward 3/2 3/2 required by h an amount, Ay, without the sill and y' = y2/W., A \A,(48) = hjh., = = and B' = hA, hi h2 h2/h., Ay' Ay/W., B/h.. Referring to Figure 23, it is possible to calcu- As an example of the application of Eq. (49), late the reduction in required volume through a consider the following parameters: AN = A, = perched beach design. The results can be devel- 0.15 m1/3, h. = 6 m, hi = 4 m, h2 = 3 m, B = oped for intersecting and non-intersecting type 2.0 m, and Ay = 48.3 m (From Eq. (48)).

Journal of Coastal Research, Vol. 7, No. 1, 1991 70 Dean

-V 0.2.IBWo 0.1

_ __ _I I - y -0.2 0.2 0.4 -- 0.6 0.8 1.0 W

-0.2 - VolumetricTransport Past Any Location

-0.2 0.2 0.4 0.6 0.8 1.0 Y wo 0.2- Profile - h 0.4Initial Equilibrated Profile h* 0.6-0. Profile Response 0.8- 1.0 - Figure 22. Example of Type 4 (advanced) profile response from an initially uniform slope and associated volumetric transport. Note both positive (seaward) and negative (landward) transport.

- Yi

Perched Beach

o h2

Toe Structure Figure 23. Perched beach, demonstration of nourishment volumes saved.

The width of the surf zone without the sill, Profile to Beach Nourishment = = = Response W., is W. (h./AN)3/2 253.0 m and Y2 (h2/ AF)3/2 = 89.4 m. The fractional reduction in volume is AV/V, = Intersecting, Non-Intersecting and 0.342 i.e., there is a 34% reduction in sand vol- Submerged Profiles. Beach nourishment is umes with the perched beach design. considered with sediment of arbitrary but uni-

Journal of Coastal Research, Vol. 7, No. 1, 1991 Equilibrium Beach Profiles 71

Ay.

" B 3 V B t Added Sand h,

a) Intersecting Profile AF>ANI

W..-

b) Non-intersecting Profille

' Virtual Origin of Nourished Profile ..

Added Sand c) Submerged Profile AF

form diameter. As indicated in Figure 24, nour- must be finer than the native. Figure 25 illus- ished beach profiles can be designated as "inter- trates the effect of placing the same volume of secting," "non-intersecting," and "submerged" four different sized sands. In Figure 25a, sand profiles. A necessary but insufficient require- coarser than the native is used, intersecting ment for profiles to intersect is that the placed profiles result and a relatively wide beach Ay is material be coarser than the native. Similarly, obtained. In Figure 25b, the same volume of non-intersecting or submerged profiles will sand of the same size as the native is used, non- always occur if the placed sediment is the same intersecting profiles result and the dry beach size as or finer than the native. However, non- width gained is less. More of the same volume intersecting profiles can occur if the placed sed- is required to fill out the milder sloped under- iment is coarser than the native. For "sub- water profile. In Figure 25c, the placed sand is merged" profiles to occur, the placed material finer than the native and much of the sand is

Journal of Coastal Research, Vol. 7, No. 1, 1991 72 Dean

92.4m B 1.= s5m

h.=6m

Intersecting Profiles, AN= O.1mi'AF = O.14m13 45.3m

{h. 6m

Non-Intersecting Profiles = 0.1m1/3 AN= AF

h.= 6m

/3 AN = 0.1m M113A = 0.09ml I-- 710

h = 6m > 5. h5 W<" r LimitingCaseof Nourishment Advancement Non-Intersecting Profiles, AN= 0.1m/3 ,AF= 0.085m113

L I I I I ! 0 100 200 300 400 500 600 OFFSHORE DISTANCE (m)

Figure 25. Effect of nourishment material scale parameter, AF, on width of resulting dry beach. Four examples of decreasing AF with same added volume per unit beach length.

used in satisfying the milder sloped underwater the seaward limit of the placed profile moves profile requirements. In a limiting case, shown seaward. in Figure 25d, no dry beach is yielded with all We can quantify the results presented in Fig- the sand being used to satisfy the underwater ures 24, 25 and 26 by utilizing equilibrium pro- requirements. Figures 26a through 26d illus- file concepts. It is necessary to distinguish the trate the effects of nourishing with greater and three cases noted in Figure 24. The first is with greater quantities of a sand which is consider- intersecting profiles such as indicated in Figure ably finer than the native. Figure 26d is the 24a and requires A, > A,. For this case, the vol- case of formation of an incipient dry beach, i.e. ume placed per unit shoreline length, V, asso- the same as in Figure 25d. With increasing vol- ciated with a shoreline advancement, Ay, is pre- umes, the landward intersection of the native sented in non-dimensional form as and placed profiles occurs closer to shore and

Journal of Coastal Research, Vol. 7, No. 1, 1991 Equilibrium Beach Profiles 73

It can be shown that the critical value of (Ay') for intersection/non-intersection of is OFFSHORE DISTANCE (m) profiles 0 100 200 300 400 500 given by - - I I iI I I

0 o

< 0, Intersecting Profiles 0. AddedVolume120 m3Im > 0, Non-IntersectingProfiles The critical volume associated with intersecting/non-intersecting profiles is =1 ? 1 ( (54) +B(k5B'!L AN]AI and applies only for (A,/A,) > 1. Also of inter- SAdded Volume = 900 m m a, est, the critical volume of sand that will just yield a finite shoreline displacement for non- > intersecting profiles (A,/A, < 1), is

( - 1 5B=,A (55)

Figure 27 presents these two critical volumes Added Volume = 1660 m3 m versus the scale parameter ratio A,/A, for the Case of Incipient Dry Beach special case h./B = 4.0, i.e. B' = 0.25. The results from Eqs. (50), (52) and (53) are Figure 26. Effect of increasing volume of sand added on in form in 28 and = presented graphical Figures resulting beach profile. AF 0.1 m"/3, AN = 0.2 m11/3, h = 6 29 for cases of = 2 'and 4. Plotted is the m, B = 1.5 m. (h./B) non-dimensional shoreline advancement (Ay') versus the ratio of fill to native sediment scale parameters, A,/AN, for various isolines of V' = (50) Ay'+ (Ay')'/3 - 3/22/3 dimensionless fill volume per [1 (AN)/2 V' (= unit length of beach. It is interesting that the shoreline advancement increases only for in which V'(- is the non-dimensional slightly V1/BW.) A,/A, > 1.2; for smaller values the additional volume, B is the berm height, W. is a reference shoreline width, Ay, decreases rapidly. For offshore distance associated with the breaking A,F/ values slightly smaller than plotted, there depth, h., on the original (unnourished) profile, AN is no shoreline advancement, i.e. as in Figure i.e. 25d. 3/2 Referring to Figure 24c for submerged pro- (51) files, it can be shown that (N) N3 and the breaking depth, h. and breaking wave Ay - (56) Y, height, Hb are related by h. = Hb/K with K(= 0.78), the spilling breaking wave proportional- where Ay < 0 and the non-dimensional volume ity factor. of added sediment can be expressed as For non-intersecting but emergent profiles 3/2 5/3 (Figure 24b), the corresponding volume V2 in non-dimensional form is 5B'(AN) + Ay] 3 = ' + • '2 y (-Ay')/3 AN?3/2 5-73/2 5/3 3/2 (52) 3/2 2F/3- -(57) [(~) -1]

Journal of Coastal Research, Vol. 7, No. 1, 1991 74 Dean

15 3

wz / /(2) z w Q as i w

z zU- 0 0 0 1 2 3 AF /AN Figure 27. (1) Volumetric requirement for finite shoreline advancement (Eq. 55); (2) Volumetric criterion for intersecting profiles (Eq. 54). Variation with AFIAN. Results presented for = 4.0. h,/B

Variation in Sediment Size Across the Surf ues. The two uniform cases are for A equal to Zone 0.1 m1/3 and 0.068 mi/3 corresponding to sediment diameters of 0.2 mm and 0.1 mm, respec- All cases presented earlier have considered tively (Figure 1). The profiles for these two the sand size to be of uniform size across the cases are presented as the dashed lines in Fig- surf zone. In most cases, there is some sorting ure 30b. As presented in Figure 30a, for the with the sand grading to finer sizes in the sea- remaining case, the A value varies linearly ward direction. With the relation of A(D), and from 0.1 m1/3 to 0.068 m"/3 at 800 meters sea- thus W.(D), known (c.f. Eq. (5)) it is possible to ward, in accordance with a decreasing sediment calculate equilibrium profiles for cases of a con- size with distance offshore. The profile for this tinuous variation of sand sizes across the surf case is the solid line in Figure 30b. As expected: zone and a distribution composed of piecewise (1) the two profiles for A = 0.1 mn/" and variable uniform diameter segments. A agree in shallow water since they differ little in this region, and (2) the profile for varying A Continuous Arbitrary Distribution of lies between the two profiles associated with the Sand Sizes Across the Surf Zone. The dif- limiting A values for the variable A profile. ferential equation for an equilibrium beach profile is given by Piecewise Uniform Sand Sizes Across the Surf Zone. Denoting the sand size as Ah3/2 D. = A3/2(D) (58) over the segment y, < y < y,, +, the water depth ay in this region is obtained from a slight variation of from which Eq. (1) is obtained readily. Inte- Eq. (59) grating across the surf zone yields the equilib- h3/2(y) = h3/2(y) +A3'2(D,)[y -y,] (60) rium profile which applies for y, < y < y,+ 1 h3/2(y) =LA3/2(D)dy (59) Comparison with Empirical Orthogonal If the sediment scale parameter, A, varies lin- Functions early with distance offshore, y, it is possible to develop an analytical solution of Eq. (58). Fig- It is instructive to compare results obtained ure 30 presents a comparison of equilibrium from a simple application of the equilibrium profiles for uniform and linearly varying A val- beach profile with those developed by various

Journal of Coastal Research, Vol. 7, No. 1, 1991 Equilibrium Beach Profiles 75

10.0

sit

It el - 0.2 -V'-0.01

for &y 0 0 0.010 = 0.005 , ?.1 SY'

= VBW 0.002 AFym 0. =

0.001 - Definition Sketch 0 1.0 2.0 2.8 A A' = AF/ANA.F/A' =

Figure 28. Variation of non-dimensional shoreline advancement Ay/W., with A' and V. Results shown for h./B = 2.0.

researchers (e.g. WINANT, INMAN and volumes and can be shown to be approximately NORDSTROM (1975) and WEISHAR and WOOD (1983)) in their application of Empirical = Ah' Orthogonal Function (EOF) methods to time h. (61) , series of natural beach profiles. This method = B'(1-S')+(y'-Ay')213 Ay'

Journal of Coastal Research, Vol. 7, No. 1, 1991 76 Dean

1.0 _------Non-intersecting ------, -__ - Profiles s - V= V/BW, = 5.0 -_

ntersec-ting----- 2/ 2Profiles 11

0.1-- --0.------

V 0.05 Asymptotes for Ay = 0 I

0.01 V 0. 1

- -- wF-V-o0.0051-MID-o- -

' = 0.0021

0.001 .- -- 001 - v. ?N-II.

Definition Sketch

A' = AFA /AN

Figure 29. Variation of non-dimensional shoreline advancement Ay/W., with A' and V. Results shown for h./B = 4.0.

Effects of Sea Level Rise on Beach year 2100 are extremely unlikely. While it is Nourishment Quantities clear that worldwide sea level has been rising over the past century and that the rate is likely to increase, the future rate is very poorly Recently developed future sea level scenarios known. Moreover, probably at least 20 to 40 (HOFFMAN et al., 1983) have been developed will be before our confidence based on assumed fossil fuel and years required consumption level of future sea level rise rates will other relevant factors and have led to concern improve substantially. Within this period, it will be nec- over the viability of the beach nourishment essary to assess the viability of beach restora- option for erosion control. First, in the interest tion on a project-by-project basis in recognition of objectivity, it must be stated that the most of possible future sea level scenarios. Presented extreme of the scenarios published by the Envi- below is a basis for estimating nourishment ronmental Protection Agency (EPA) amounting needs for the scenarios in which there is no to sea level increases exceeding 3.4 m by the

Journal of Coastal Research, Vol. 7, No. 1, 1991 Equilibrium Beach Profiles 77

0.1 -

DISTANCE OFFSHORE, y (i) 0 100 200 300 400 500 600 700 800

S---. h(y) for A VariationIn Panel (a)

10 b) EquilibriumBeach Profiles

Figure 30. Example comparing equilibrium beach profiles for uniform sediment scale parameters (two dashed curves, panel b) and linearly varying sediment scale parameter shown in panel a (associated profile is solid curve, panel b). landward sediment transport across the conti- tance from the shoreline to the depth, h., asso- nental shelf and there is a more-or-less well- ciated with the seaward limit of sediment defined seaward limit of sediment motion; in motion and B is the berm height. Assuming the second case the possibility of onshore sedi- that compatible sand is used for nourishment ment transport will be discussed. (i.e. A, = A,)

Case I-Nourishment for the V Quantities AYN = + B (64) Case of No Onshore Sediment Transport. h. The Bruun Rule (1962) is based on the consid- and Vis the beach nourishment volume per unit eration that there is a well-defined depth limit, length of beach. Therefore h., of sediment transport. With this assumption, the only response possible to sea level rise 1 Ay = (65) is seaward sediment the (h,.+B) transport. Considering (h. +B)[V-SW.] total shoreline change Ay, to be the superposi- The above can be in tion of recession due to sea level rise and equation expressed rates Ays by, the advancement due to beach nourishment, AyN, dy 1 - WdS = + dt (h. +B) dt AY Ays AYN (62) dV_ (66) and, from the Bruun Rule (Eq. 15) were dSldt now represents the rate of sea level rise and dVldt is the rate at which nourishment W. - S (63) material is provided. It is seen from Eq. (66) Ays = h, +B that in order to maintain the shoreline stable in which S is the sea level rise, W. is the dis- due to the effect of sea level rise the nourish-

Journal of Coastal Research, Vol. 7, No. 1, 1991 78 Dean

00Z 0

0 J 100 200 300 400

SEAWARD DISTANCE, y(m) Most Significant Elgenfunction of Profile Change (From Winant, Inman and Nordstrom, 1975)

0.3

L< 0.2

Z a

S0 0.4 0.6 0.8 ]W o

O Non-dimensionalElevation Changes Based Z On EquilibriumBeach Profiles

Figure 31. Comparison of beach profile elevation changes by equilibrium profile concepts with results from field measurements. ment rate dV/dt is related to the rate of sea level Case II-Nourishment Quantities for the rise dSldt by Case of Onshore Sediment Transport. Evi- dence is accumulating that in some locations dV dS W there is a substantial amount of onshore sedi- dt dt dt- dt(67) ment transport across the continental shelf. Of course, this equation applies only to cross- DEAN (1987b) has noted the consequences of shore mechanisms and therefore does not recog- the assumption of a "depth of limiting motion" nize any other causes of background erosion or in allowing only offshore transport as a to longshore transport losses from the project response sea level rise and proposed instead area. It is seen that W. behaves as an amplifier that if this assumption is relaxed, onshore of material required. Therefore, it is instructive transport can occur leading to a significantly to examine the nature of W. and it will be useful different profile response to sea level rise. Con- sider for this purpose to consider the equilibrium pro- that there is a range of sediment sizes in the file given by Eq. (1), active profile with the hypothesis that a 3/2 sediment particle of given hydraulic characteristics is in under certain wave con- HW- (68) equilibrium \KA/ ditions and at a particular water depth. Thus, if sea level rises our reference particle will seek i.e. W. increases with breaking wave height and equilibrium which requires landward rather with decreasing A (or sediment size). than seaward transport as required by the

Journal of Coastal Research, Vol. 7, No. 1, 1991 Equilibrium Beach Profiles 79

Bruun Rule. Figure 32 summarizes some of the be considered as representing the "ambient" def- elements of this hypothesis. icit rate due to cross-shore sediment transport Turning now to nourishment requirements in resulting from long-term disequilibrium of the the presence of onshore sediment transport, the profile and source and sink terms. conservation of sediment yields In attempting to apply Eq. (70) to the prediction of profile change and/or nourishment needs = under a scenario of increased sea level rise, it A- + sources - sinks (69) ay at is reasonable to assume that over the next several decades the ambient deficit rate (or sur- in which h is the water referenced to a depth plus) of sediment within the active zone will vertical datum and the sources fixed could remain constant. However, an increased rate of include natural contributions such as hydro- sea level rise will cause an augmented demand genous or biogenous components, and suspended deposition or human related contribu- which can be quantifiedas W. (dS - tions, i.e. beach nourishment. Sinks could include removal of sediment dS through suspen- in which is the reference sea level sion processes and offshore advection. Eq. (69) can be integrated seaward from a landward change rate during which time the ambient limit of no transport to any location, y demand rate is established. Thus the active zone sediment deficit rate will be = Q(y) (sources-sinks)dy fo atdy (70) New Deficit Rate - [f dV ahy1d [+WdSW (71) If only natural processes are involved and there f ady+ dt )•dS dt dt are no gradients of longshore sediment transport, the terms on the left hand side of Eq. (70) in which dV•dt represents the nourishment rate represent the net rate of increase of sediment and the subscript "0" on the bracket represents deficit as a function of offshore distance, y. For the reference period before increased sea level y values greater than the normal width, W., of rise. In order to decrease the deficit rate to zero, the zone of active motion, the left hand side can the required nourishment rate is

POSSIBLE MECHANISMOF SEDIMENTARYEQUILIBRIUM

Increased Sea Level

-.S __ OriginalSea Level ...

Sediment Particle

Subjected to a Given Statistical Wave Climate, A Sediment Particle of a Particular Diameter Is in Statistical Equilibrium When In a Given Water Depth

Thus When Sea Level Increases, Particle Moves Landward

Figure 32. Possible mechanism of sedimentary equilibrium (After Dean, 1987b).

Journal of Coastal Research, Vol. 7, No. 1, 1991 80 Dean

dV_ wah dS VA = Aydh, -B

+ S) (2/3 . -( The above integration results can be cast into

we AY<0

U 0 0.05 .10 y Bw, II'

. •0.5

1.0

DefinitionSketch. - Non-DimensionalVolume - Redistributedvs. Non-Dimensional Depth.

Figure 33. Definition sketch and non-dimensional volume redistributed as a function of non-dimensional depth due to sea level rise, S. Case of = 0.25, S/B = 0.5. B/h,

Journal of Coastal Research, Vol. 7, No. 1, 1991 Equilibrium Beach Profiles 81

non-dimensional form with the following non- that could occur due to the surveys not extend- dimensional parameters Ay' = Ay/W., h' = h,/ ing to a sufficient depth is approximately 25% h., h' = h,/h., A' = A/h., S' = SI/B, B' = B/h., of the volume deficit associated with the sea and V' = VI/WB. level rise. The non-dimensional volumes are It is also possible to develop approximate expressions describing the volumetric redistribution. we the non-dimen- V'A = AyI(1+) , -B'

V'c= Ay'(1+ h78 2 which for our example yields Ay' = - 0.100 vs + [(h')5/2 - (h' + (79) 5B' S'B')5/2, the complete equation result of -0.103. The 0

V'D = 1 + + 2(h,)5/2 (81) Ay ( h' \S'B' -- -h',(3 + 1 (80) 5B' ' B'

Journal of Coastal Research, Vol. 7, No. 1, 1991 82 Dean ditions under which Eq. (15) is most valid and/ tive motion. It is shown that the maximum or of the ratio W./(h. + B). error is only a fraction of the sediment "demand". SUMMARY AND CONCLUSIONS The effects of sea level rise on nourishment needs are evaluated for cases with and without Equilibrium beach profile concepts provide a onshore sediment transport across the conti- useful basis for application to a number of nental shelf. It is shown that the sediment vol- coastal engineering projects. In addition to umes required to maintain a shoreline position addressing conditions at equilibrium, these vary directly with wave height and sea level concepts establish a foundation for considering rise rates and inversely with profile slopes. the response of profiles out of equilibrium. It is hoped that the results presented herein Based on analysis of numerous profiles rep- will provide guidance for coastal engineering resenting laboratory and field scales, a reason- projects and serve as a framework for interpre- able approximate and useful form of a mono- tation of project performance and behavior of tonic beach profile appears to be h(y) = Ay2/3 in natural beach systems. which h is the water depth at a distance, y, offshore and A is a scale parameter depending on ACKNOWLEDGEMENTS sediment characteristics. Representations of the sediment scale parameter variation with Support provided by the Sea Grant College sediment diameter and fall velocity are pre- Program of the National Oceanic and Atmos- sented. pheric Administration under Project R/C-S-22 Methods are presented for quantifying the is hereby gratefully acknowledged. This sup- shoreline response due to elevated water levels port has enabled the author and students to and wave heights on natural and seawalled pursue a range of problems concerned with shorelines. Additionally, results are presented rational usage of the shoreline. Ms. Cynthia for calculating nourishment quantities for sand Vey and Lillean Pieter provided their usual of uniform but arbitrary diameter. Depending flawless typing and illustration services, on volumes and sizes of sediment added, three respectively. I appreciate the many discussions types of profiles can occur: intersecting, non- and collegial support provided by present and intersecting and submerged. The advantages of former students. using coarser sand are quantified and equations are presented expressing the volume of a par- LITERATURE CITED ticular sand size required to yield a desired additional beach width. Many laboratory stud- AUBREY, D.G., 1979. Seasonal Patterns of Onshore/ ies of beach profiles commence with a planar Offshore Sediment Movement. Journal of Geophys- slope which could be much steeper or coarser ical Research, 84(C10), 6347-6354. AUBREY, INMAN, and than the overall equilibrium slope consistent D.G.; D.L., NORDSTROM, C.E., 1977. Beach Profiles at Pines, CA. Pro- the in Torrey with sand size the experiments. Applying ceedings of the Fifteenth International Conferenceon equilibrium beach profile concepts, it is shown Coastal Engineering, American Society of Civil that five profile types relative to the initial pro- Engineers, pp. 1297-1311. file can occur. Three of these are erosional BAGNOLD,R.A., 1946. Motion of Waves in Shallow- types Water-Interaction Between Waves and Sand Bot- and two are accretional. Three of these profile toms. Proceedings Royal Society, 187 (Series A), 1- types have been identified in laboratory stud- 15. ies. Conditions under which a profile type will BOWEN, A.J.; INMAN, D.L., and SIMMONS, V.P., occur are quantified and all results including 1968. Wave Set-Down and Wave Set-Up. Journal of Research, 2569-2577. shoreline change are incorporated into a single Geophysical 73(8), BRUUN, P., 1954. Coast Erosion and the Develop- graphical representation. ment of Beach Profiles. Technical MemorandumNo. The volumetric redistribution of sediment 44, Beach Erosion Board. across the profile due to sea level rise is exam- BRUUN, P., 1962. Sea Level Rise as a Cause of Shore ined in detail and compared with the total sed- Erosion. Journal of Waterway, Port, Coastal and iment "demand" as a result of the sea level rise. Ocean Engineering, American Society of Civil Engi- neers, 88, 117-130. An application is the possible error if the sur- CHIU, T.Y., 1977. Beach and Dune Response to Hur- vey does not extend over the full depth of effec- ricane Eloise of September 1975. Proceedings of the

Journal of Coastal Research,Vol. 7, No. 1, 1991 EquilibriumBeach Profiles 83

Specialty Conference on Coastal Sediments '77, Resources Engineering, Lund University, Lund, American Society of Civil Engineers, pp. 116-134. Sweden. DALRYMPLE, R.A., and THOMPSON, W.W., 1976. LARSON, M., and KRAUS, N.C., 1989. SBEACH: Study of Equilibrium Profiles. Proceedings of the Numerical Model for Simulating Storm-Induced Fifteenth International Conference on Coastal Engi- Beach Change-Report 1: Empirical Foundation neering, American Society of Civil Engineers, 1277- and Model Development. Technical Report CERC- 1296. 89-9, Coastal Engineering Research Center, Water- DEAN, R.G., 1973. Heuristic Models of Sand Trans- ways Experiment Station, Vicksburg, Mississippi. port in the Surf Zone. Proceedings of the Conference MOORE, B.D., 1982. Beach Profile Evolution in on in the Insti- Engineering Dynamics Surf Zone, Response to Changes in Water Level and Wave tution of Civil Engineers, Australia, Sydney, pp. Height. Masters Thesis, Department of Civil Engi- 208-214. neering, University of Delaware. 1977. Beach Profiles: U.S. DEAN, R.G., Equilibrium NODA, E.K., 1972. Equilibrium Beach Profile Scale Atlantic and Gulf Coasts. of Civil Department Engi- Model Relationship. Journal Waterways, Harbors neering, Ocean Report No. 12, Univer- Engineering and Coastal Engineering Division, American Soci- sity of Delaware, Newark, Delaware. ety of Civil Engineers, pp. 511-528. DEAN, R.G., 1987a. Coastal Sediment Processes: PHILLIPS, O.M., 1966. The Dynamics of the Upper Toward Engineering Solutions. Proceedings of the Ocean. Cambridge, England: Cambridge University Specialty Conference on Coastal Sediments '87, Press, American Society of Civil Engineers, pp. 1-24. 26lp. SAVILLE, T., 1957. Scale Effects in Two-Dimensional DEAN, R.G., 1987b. Additional Sediment Input to the Beach Studies. Transactions the Seventh General Nearshore Region. Shore and Beach, 55(3-4), 76- of 81. Meeting of the International Association of Vol. DEAN, R.G., and MAURMEYER, E.M., 1983. Models Hydraulic Research, 1, p. A3-1 to A3-10. and 1986. for Beach Profile Response. In: KOMAR, P.D., (ed.), STIVE, M.J.F., WIND, H.G., Cross-Shore CRC Handbook of Coastal Processes and Erosion. Mean Flow in the Surf Zone. Coastal Engineering, Boca Raton, CRC Press, Chapter 7, pp. 151-166. 10(4), 325-340. EAGLESON, P.S.; GLENNE, B., and DRACUP, J.A., SUH, K., and DALRYMPLE, R.A., 1988. Expression 1963. Equilibrium Characteristics of Sand Beaches. for Shoreline Advancement of Initially Plane Journal ofHydraulics Division, American Society of Beach. Journal of Waterway, Port, Coastal and Civil Engineers, 89(HY1), pp. 35-57. Ocean Engineering, American Society of Civil Engi- HAYDEN, B.; FELDER, W.; FISHER, J.; RESIO, D.; neers, 114(6), 770-777. VINCENT, L., and DOLAN, R., 1975. Systematic SUNAMURA, T., and HORIKAWA, K., 1974. Two- Variations in Inshore Bathymetry. Department of Dimensional Beach Transformation Due to Waves. Environmental Sciences, Technical Report No. 10, Proceedings of the Fourteenth International Confer- University of Virginia, Charlottesville, VA. ence on Coastal Engineering, American Society of HOFFMAN, J.S.; KEYES, D., and TITUS, J.G., 1983. Civil Engineers, pp. 920-938. Projecting Sea Level Rise; Methodology, Estimates SWART, D.H., 1974. A Schematization of Onshore- to the Year 2100 and Research Needs. U.S. Environ- Offshore Transport. Proceedings of the Fourteenth mental Protection Agency, Washington, D.C. International Conference on Coastal Engineering, HUGHES, S.A., 1983. Movable Bed Modeling Law for American Society of Civil Engineers, pp. 884-900. Coastal Dune Erosion. Journal of Waterway, Port, SVENDSEN, I.A., 1984. Mass Flow and Undertow in Coastal and Ocean Engineering, 109(2), 164-179. a Surf Zone. Coastal Engineering, 8, 347-365. KEULEGAN, G.H., and KRUMBEIN, W.C., 1949. VAN HIJUM, E., 1975. Equilibrium Profiles of Coarse Stable Configuration of Bottom Slope in a Shallow Material Under Wave Attack. Proceedings of the Sea and Its Bearing on Geological Processes. Trans- Fourteenth International Conference on Coastal actions ofAmerican Geophysical Union, 30(6), 855- Engineering, American Society of Civil Engineers, 861. pp. 939-957. KRIEBEL, D.L., 1982. Beach and Dune Response to VELLINGA, P., 1983. Predictive Computational Hurricanes, M.S. Thesis, Department of Civil Engi- Model for Beach and Dune Erosion During Storm neering, University of Delaware, Newark, Dela- Surges. Proceedings of the Specialty Conference on ware. Coastal Structures '83, American Society of Civil KRIEBEL, D.L., 1986. Verification Study of a Dune Engineers, pp. 806-819. Erosion Model. Shore and Beach, 54(3), 13-21. WEISHAR, L.L., and WOOD, W.L., 1983. An Evalu- KRIEBEL, D.L., and DEAN, R.G., 1984. Beach and ation of Offshore and Beach Changes on a Tideless Dune Response to Severe Storms. Proceedings of the Coast, Journal of Sedimentary Petrology, 53(3), Nineteenth International Conference on Coastal 847-858. Engineering, American Society of Civil Engineers, WINANT, C.D.; INMAN, D.L., and NORDSTROM, pp. 1584-1599. C.E., 1975. Description of Seasonal Beach Changes KRIEBEL, D.L., and DEAN, R.G., 1985. Numerical Using Empirical Eigenfunctions. Journal of Geo- Simulation of Time-Dependent Beach and Dune physical Research, 80(15), 1979-1986. Erosion. Coastal Engineering, 9(3), 221-245. ZENKOVICH, V.P., 1967. Processes of Coastal Devel- LARSON, M., 1988. Quantification of Beach Profile opment. Editor: J.A. Steers. Edinburgh: Oliver and Change. Report No. 1008, Department of Water Boyd, 738p.

Journal of Coastal Research,Vol. 7, No. 1, 1991 84 Dean

[ R9SUME [l La comprehension des profils d'dquilibre est utile A bon nombre de projets d'ingdnierie c6tiore. A partir d'dquations empiriques entre un parambtre d'6chelle et la taille du sediment ou sa vitesse de chute, on peut calculer le profil d'6quilibre. La forme la plus utilis6e est de la forme: h(y) = A y2/3 Ot h est la profondeur A une distance y du rivage et A un parametre dependant du s6diment. Les expressions du changement de la position du rivage sont donnds pour des niveaux de la mer arbitraires et diverses hauteurs de la houle. L'application des concepts de profil d'6quilibre A des modifications du profil c6t0 mer d'une digue tient compte des effets de la modification du niveau de la mer et des hauteurs de la houle. Pour des vagues d'une hauteur donn6e et un niveau de l'eau croissant, la profondeur adjacente A la digue croit d'abord, puis d6croit jusqu'd z6ro (houle brisant sur la digue). Le recul de rivage, les cons6quences de la hausse du niveau de la mer et de la hauteur de la houle sont examinds. Ainsi, l'effet de l'4tablisse- ment de la houle sur le recul est faible, compar6 A celui des marfes de temp6te. L'6volution des profils Apartir d'une pente uniforme peut doner 5 types diffdrents qui d6pendent de la pente initiale, des caracteres du s6diment, de la hauteur de la berme et de la profondeur de la redistribution active des s6diments. La r6duction du volume de sable n6cessaire A l'engraissement de la plage par une buse placde au large est examinde pour un s6diment quelconque et toutes les combinaisons. Quand les plages sont nourries, il trois types de profils: 1) profils les sediments d'apport sont plus petits que ceux d'origine: tous les s6diments rdsulte submerges ohl s'6quilibrent sous l'eau, sans dlargissement de la plage seche; 2) profils ne se recoupant pas, oft le mat6riau d'apport est localisd sur le bas de plage et repose sur le profil originel A cet endriot; 3) profils se recoupant: le sable d'apport est plus gros que celui d'origine et recoupe le profil originel. Des dquations et des graphes donnent la figure de la largeur de la plage seche pour diffdrents volumes de sable et des tailles varides de s4diments d'origine. Les redistributions volum6triques de mat6riaux au large, dues A la mont6e relative du niveau de la mer en fonction de la profondeur sont utiles Al'interprdtation de la cause du retrait du rivage. S'il y a seulement transport vers le large, et que lon considbre aussi les profils actifs, le changement volumetrique net est nul. On montre que le changement maximum de volume df a la redistribution transversale des s6diments de la plage n'est qu'une fraction du produit de la dimension du profil vertical et du retrait du rivage. Cet article pr6sente aussi plusieurs applications des profils d'6quilibre A des probl6mes d'ingdnierie c6tibre.--Catherine Bressolier-Bousquet, Ggomorphologie EPHE, Montrouge, France.

Journal of Coastal Research, Vol. 7, No. 1, 1991