Mathematical modeling of the fate of beach fill

CoastalEngineering, 16 ( 1991 ) 83-! 14 Elsevier Science Publishers B.V., Amsterdam

83

Mathematical modeling of the fate of beach fill Magnus Larson a a n d Nicholas C. K raus b aDepartment of Water Resources Engineering, Institute of Science and Tec!;nology, University of Lund, Box 118, Lurid, S-221 00, Sweden bCoastal Engineering Research Center, U.S. Army Engineer Waterways Experiment Station, 3909 Halls Ferry Road, Vicksburg, MS 39180-6199, USA (Received 1 May 1989; accepted after revision 8 April 1990)

ABSTRACT Larson, M. and Kraus, N.C., 1991. Mathematical modeling of the fate of beach fill. In: J. van de Graaff, H.D. Niemeyer and J. van Overeem (Editors), Artificial Beach Nourishments. Coastal Eng, 16:83-114. This paper describes mathematical approaches for calculating the fate of beach fill under wave action. General properties of the spatial and temporal behavior of fills under average longshore sand transport conditions are first reviewed through use of simple closed-form solutions of the sediment continuity equation. This well-known solution procedure is then extended to describe the collective movement of sediment or "'longshore sand waves". Because of the increasing use of beach fills as storm-protection harriers, the paper focuses on prediction of storm-induced beach erosion. An empirically based model is described and tested with a high-quality field data set of storm-induced beach erosion. One synthetic tropical storm and one synthetic extratropical storm representing typical midAtlantic storms with approxi~ate 2-5-yea¢ return period are then used to simulate the erosion of two hypothetical beach fill configurations and the subsequent post-storm recovery process. Eroded volume and contour movement are evaluated as a function of storm type, fill cross section, grain size, and time. Quantitative and qualitative features of storm erosion are reasonably well reproduced, and calculated changes in beach fills impacted by the synthetic storms indicate that little erosion protection benefit is gained for moderate storms (2-5-year return period) by placing fills with grain sizes greater than 0.4 mm. Simulation of post-storm beac'l recovery is presently limited to qualitative reproduction of observed trends.

INTRODUCTION

Beach nourishment is becoming a preferred shore protection measure because it is often the least expensive alternative, augments existing shore protection structures with great flexibility, and provides a natural and enjoyable coastal environment. Prediction of the performance of the fill, for example, its longevity and protective functioning against storms, is required to evaluate the benefits of a nourishment project in relation to other shore protection alternatives. Quantitative description of coastal sediment transport processes and beach change involves numerous hydrodynamic, sediment, and geomet0378-3839/91/$03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

84

M, LARSONAND N.C. KRAUS

ric factors that vary in space and time. Mathematical modeling of beach profile and fill change provides a general and quantitative approach for representing the simultaneous effects of these factors. Successful numerical simulations are limited only by our ingenuity in developing the underlying theory and the availability of data to refine and verify the models. Major engineering needs for predictive models of beach change include: ( 1 ) design of protective beach and dune complexes against erosion and inundation; (2) prediction of the effects of coastal structures and engineering activities on the beach; and ~3 ) prediction of seasonal changes in the profile shape and shoreline position. Kraus (1989) examined various types of numerical models developed to address these problems, classifying them by temporal and spatial ranges of applicability, and concluded that only two types of models, shoreline change models and beach profile change models, were sufficiently reliable and powerful for widespread engineering use at present. Shoreline change models involve longshore sand transport, whereas beach profile change models involve cross-shore sand transport. At the present state of practice, our capability is in a transition period in which longshore and cross-shore transport must be calculated independently. It is clearly a goal of the 1990"s to combine these components into one model that will be more general and reliable than the individual models. Figure 1 gives a sketch of the nearshore region containing a beach fill and establishes notation and axes convention used in discussion. The fill is exposed to persistent longshore currents induced by waves and wind, which may deplete it if no updrift sand source is available and no maintenance occurs; the fill is acted on by cross-shore wave forces to mold it to a dynamic equilibrium shape, which varies according to season; and the fill is also impacted by

Approximate Vertical 11 Exaggeration 20 =-

~ -

As-Constructed

Berm

i

Pre-f

Profile J ' - -

Nl~x'x'x--~ "~ lx~N....,_~---~x~n,/" Bar - - /

Design (Adjusted) Profile

' ~ 'hm -~m - ~. - ~~- ~ - - - _

Distance f r o m Bosellne Fig. I. Definition sketch categorizing a beach fill.

MATHEMATICALMODELING OF THE FATE OF BEACH FILL

85

episodic and potentially catastrophic storm events. Over the course of a few weeks to months, the as-construcled berm ~ill adjust to aporoximate the design profile expected to form under ordinary wave action. The design profile defines a fill cross section or template of a specified elevation and width on the subaerial beach. Different templates will provide different degrees of erosion and inundation protection, and an important objective of numerical modeling is evaluation of these protective properties. This article concerns mathematical prediction or modeling of the fate of beach fill under wave action. Principal inputs to the models are data readily available or reasonably estimated in engineering studies, namely: time series of a statistically representative wave height, direction, and period, and water level (at approximately 1- to 6-h intervals); estimates of the initial beach plan shape and profile; and average fill and native grain sizes. SHORELINE CHANGE MODELING

In this section we review a simple analytical model whereby changes in shoreline position are calculated from spatial and temporal differences in the longshore sand transport rate. The material follows the presentation of Larson et al. ( 1987 ), which may be consulted for derivatic,s and references to the literature. The treatment is supplemented by new material describing the collective movement of sand alongshore. Mathematical modeling of shoreline change was originated by PelnardConsidere ( 1956 ) and, in recent years, such modeling has become a standard engineering technique for predicting the long-term (order of months and years) evolution of the beach plan shape (see, e.g., Hanson and Kraus ( 1989 ) for a description of an advanced shoreline change numerical model). Three basic assumptions underlie shoreline change modeling: ( 1 ) permanency of the beach profile shape; (2) existence of a depth of closure of longshore sediment movement; and (3) dependence of the longshore sand transport rate on wave direction. If the further assumptions are made that the incident waves are constant in height and direction through time and that they arrive at a small angle to the trend of the coast, a simple equation governing shoreline change can be derived (Pelnard-Considere, 1956) for which closed-form or analytical solutions can be found (e.g., Larson et al., 1987 ): O:'Y OY Ox 2 _ &

( 1)

in which the shoreline position Y ( x , t ) is a function of the distance x along the shore and time t, and ~ is a semi-empirical coefficient related to the particular predictive longshore sand transport rate formula used. If the transport rate Q is given by the "CERC" formula (Shore Protection Manual, 1984), then:

86

M. LARSON AND N.C. KRAUS

Q= Qosin20b

(2)

in which 0b is the angle of the breaking wave crests to the trend of the local shoreline. The quantity Qo is called the amplitude of the longshore sand transport rate (Larson et al., !987) and is given by:

1£1 )p(HZCg)b

Qo-16(S_ 1

(3}

in which K~=, empirical coefficient (approximately 0.8 if root-mean-square wave height is used), H = w a v e height, C~=wave group speed, b = subscript denoting wave breaking condition, S = ratio of sediment density to water density (2.65 for quartz sand ), and p = sediment porosity of bed material (0.6). With this choice of the longshore sand transpe~ rate formula, e becomes: E=

2Qo h,

(4)

in which h, is the depth of closure. Equation 1 is identical to the one-dimensional heat conduction or diffusion equation, and it is rich in closed-form solutions derived according to the initial and boundary conditions imposed. The parameter ~, having the dimensions of length squared over time, is interpreted as a diffusion coefficient (in the present context, "longshore diffusion") expressing the time scale of shoreline change as controlled by the strength of wave action.

Collective sand movement Equation 1 describes the change in shoreline position produced by the movement of individual sand grains moved alongshore by wave action. Kraus et al. ( 1982 ) estimated from their tracer experiments that the longshore advection speed of sand grains was on the order of 1/ 100th of the mean speed of the longshore current. Assuming an annual mean longshore current speed of 10 cm s-~, a sand grain that remains active in the surf zone is expected to move on the order of 30 km in a year. In recent years, with increased experience from beach fill monitoring, interest has heightened in what Sonu ( 1968 ) first called the "collective" ~lovement of sand in the surf zone, in which large bodies having the dimensions on the order of the width of the surf zone move along the shore in the direction of the predominant sand drift with speeds on the order of 0.5 to 4 km/year (see, e.g., Inman, 1987). This collective movement can often be associated with intermittency in sand supply, such as occurs during placement of a small beach fill ("small" meaning short in length compared to the particular stretch of sandy beach). The movement of large shoreline forms can be incorporated in Eq. 1 by introducing the advective term - ~Y/Ox to give:

MATHEMATICAL MODELING OF THE FATE OF BEACH FILL

02Y

ifiY 0Y

%-~x2 -

Ox- ot

87

(5)

in which V is the speed of propagation of the sand wave moving alongshore (parallel to the x-axis), to be contrasted with the speed of individual sand particle movement alongshore, which is much more rapid. No predictive expressions are available to specify V~ although Sonu (1968) has estimated that it is proportional to the longshore extent of the feature to the minus 4/5th power; that is, large features move more slowly. The speed Vis most conveniently taken as a constant for obta;ning simple analytic solutions of Eq. 5. Let the shoreline position Y(x,t) be reexpressed by the arguments ~ = x - Vt and ~= t, which describes a transformation to a coordinate system moving in the positive x direction with speed F. By the chain rule of differentiation, Eq. 5 then reduces to the same form as Eq. 1. Therefore, if there are no disturbances at the boundaries, analytical solutions of Eq. 1 are also solutions of Eq. 5, but with the arguments ~ and ~. Figure 2 shows the spreading and attenuation calculated by Eq. i of an initially symmetric triangular-shaped beach fhl piaced on an initially straight shoreline to which waves are normally incident. Spreading of the fill is plotted as a function of dimensionless time t ' = ~t/a 2, in which a is half the length of the fill alongshore at t=O. The ordinate in Fig. 1 plots the width of the fill across the beach made dimensionless by the initial maximum width Yo. The initial condition is:

0.8

0.6 -- _ 0.15-""'" \ \

o. o

o

0.,-

o.o

I

0.0

0.5

I

1

1.5

Distonce Alongshore (x/o) Fig. 2. Longshore spreading o f a triangular-shaped beach fill.

"

I

2

88

M. LARSONAND N.C. KRAUS

yo a - x

O<_x
yoa+x a

-a
a

Y(x,t) =

0

(6)

Ixl>a

and the solution for the shoreline position as a function of time is:

Y(x,t)=~a~(a-x)er Yo ~ "a-x ~ (a+ x) erf(_____~ a+x _ 2xerfl/_____ x "] \2v/~t] \2x/~t/ \2v/~t] (7) N ~k for ¢> O and - ~ < :~< or, in which erf is the error function. By the previously discussed coordinate tran~;°~lnation, Eq. 7 describes spreading of a triangular-shaped beach segment that is also moving as a unit with speed V.

Simple properties of beach fill evolution alongshore Two fundamental properties of beach fill behavior emerge from solutions such as Eq. 7, which are expected to hold in more sophisticated modeling, under the assumption that material is not moved offshore. First, from Eq. 7 the rate of spreading of a finite-length beach fill is governed by the longshore diffusion coefficient c. From Eqs. 3 and 4, the rate goes as H 5/2, because Cg b'~" H l/2. Thus the material spreads much more rapidly than linearly with wave height as the wave height increases. The second property was pointed out by Deanj (1984) for the case of a rectangular beach fill, but applies zs well as to the ~Lriangular-shaped fill solution given by Eq. 7. By noting thzt the same dimensionless volume of fill spreads in the same dimensionless time at differenlt projects (fills of different width or fills at different sites), one finds that the time lp2 for a certain percentage to be lost from the original fill compared with the corresponding time tpl for another fill is:

\ a l / e2 For the same site, e i = e2; therefore, movement of material across the lateral borders of the original project size goes as the square of the length of the fill. Figure 3 illustrates the percentage volume of a triangular-shaped fill that remains in the original project area for given dimensionless elapsed time. The

89

MATIJEMATI('ALMODELING OFTItE FATE()F BEACH FILL

!00,

R e m a i n i n g Fill Volume (%)

75

50

25

0

I

0

1

_ _

l

____t_

2

I

3

4

Dimensionless Time, t' F,u. 3. P e r c e m ol r e m a i n i n g fill v o l u m e .

rate of spr,.ading is initially high, with 50% of the fill volume moving out of the original site in time tr,=a2/E (because t;, = 1 ), slowing to 75% moving out in time tp= 5a2/~. The time evolution of other fill shapes, for example, rectangular fills, is quantitatively similar (Larson et al., 1987 ). Relatively speaking, short fills spread more rapidly to adjacent beaches than long fills. From such considerations, it might be argued that a beach is best protected by filling the entire littoral cell or subcell. In any case, no matter at what rate the fill material moves alongshore, it is not "'lost", but remains in the littoral zone to provide benefits to neighboring beaches. Delft Hydraulics ( 1987 ) and Dean ( 1988 ) discuss economic benefits of fill to adjacent beaches. STORM-EROSION MODELING

Short-term cross-shore transport processes associated with storms are of major engineering concern because of catastrophic erosion and inundation, and shore p~'otection planning and beach fill design must consider potential damage produced by storms. Major factors controlling cross-shore sediment transport and storm-induced beach profile changes are: ( 1 ) offshore bathymetry and 13rofile shape prior to the storm; (2) grain size distributions of the native beach and fill; (3) surge plus tide hydrograph; (4) waves (wave height, period, setup, and rump); and (5) fill and beach cross section. All models of beach profile erosion developed for engineering use have relied on some assumption about the shape of the profile in its transition from one state to another over the course of wave action and water level change. Early models were effectively time independent in predicting profile change from an initial (pre-storm) to final (post-storm) shape, both shapes speci-

90

M.I.ARSON m, :~ N.C. KRAUS

fled a priori (Edelw ~, 1972; Vallianos, 1974; Swart, 1976). On the basis of extensive tests of dune erosion in both small and large wave tanks, Vellinga ( 1983, 1986) developed a simple mathematical mode! for estimating catastrophic erosion of steep protective dunes impacted by extreme storm events as might occur along the Dutch coast. Pre~:~ctions from the aforementioned models are effectively independent of the temporal details of the forcing functions, implying potentially large error if used outside their range of calibration. For example, it is known that dune erosion is sensitive to the duration and shape of the surge hydrograph; for coasts with a substantial tidal range, coincidence of the high tide and peak surge might brin~ severe damage, whereas arrival of the peak storm surge during low tide migbI r~sult in minor or no damage and beach loss. Wave properties change with time during a storm, and beaches begin to recover (accrete) under the long-period swell at storm's end (e.g., Kriebel, 1987). Therefore, more reliable estimates of storm erosion will result which include the major forcing parameters in a time-dependent formulation. Realistic time-dependent simulation of beach and dune erosion was first accomplished by Kriebel (1982, 1986) and Kriebel and Dean (1985), who extended the static equilibrium profile concept of Bruun (1954) and Dean ( 1977 ). Their model a!lows calculation of a profile of monotonically increasing depth with distance offshore. This shape is given by the function:

h=A(y- y)2/3

(9)

in which h is the water depth and y is the distance offshore [y measured from an arbitrary baseline, and h referred to mean sea level (MSL) ], and A, called the shape parameter, is an empirical coefficiem found to be an increasing function of grain size (Moore, 1982) or sediment fall speed (Dean, 1987; Kriebel et al., 1991 ). Dean (1977) showed that Eq. 9 was consistent with a cross-shore sediment transport rate proportional to wave energy dissipation per unit volume of broken waves. In the model of Kriebel and Dean, the transport rate at any point in the surf zone is thus given as: q=Kc(D-Dcq)

(10)

in which q is the cross-shore sediment transport rate per unit width of beach (units: m 3 s-1 m-~), D is the wave energy dis,qpation per unit volume, and Ocqis the energy dissipation occurring when the profile has achieved an equilibrium shape given by Eq. 9. Kc is a dimensional empirical coefficient found to have a (revised) value of 8- 10 -6 m 4 N - I for the Kriebel and Dean model (Kriebel, 1986). The energy dissipation per unit volume is caiculated as D = ( d F / d y ) / h , in which F is the energy flux specified in the model by linear wave theory under the assumption of spilling breaking waves (constant wave height to water depth ratio). According to Eq. I 0, if D > D~q then the beach will erode, whereas if D < D~q the beach will accrete. This is sensible, as high

MATHEMATICAL MODELING OF THE FATE OF BEACH FILl

91

waves are expected to erode the beach and low waves to promote recovery, other factors being equal. As the beach adjusts under wave action and D approaches Dcq everywhere on the profile, the transport rate given by Eq. 2 approaches zero and the profile reaches an equilibrium shape. Kriebel and Dean (1985 ) demonstrated that their model produced reasonable results through comparisons with the limited field data on dune erosion. They also performed a number of hypothetical simulations, showing, for example, that the predicted maximum rate of erosion lags the peak surge. The Kriebel and Dean numerical model is simple to program and operate, and it produces the correct order of magnitude of eroded beach volume. It is mainly driven by change in water level, which is known to be the leading variable controlling dune erosion (Hughes and Chiu, 1981 ). Principal limitations of the model are: ( 1 ) weak dependence of predictions on the magnitude of wave height mid wave period; (2) somewhat unrealistic profile shape, including absence of bars; and (3) limited capability to simulate accretion and berm buildup on the foreshore. Bars are a natural defensive response of the beach to limit sediment movement to the offshore, also serving as sediment reservoirs, and it is considered important to reproduce these formations. Similarly, beaches begin to recover at the waning stage~ of a storm, and typically accrete in between storms and during "'summer" wave conditions. Accurate simulation of long-term profile response and impacts of multiple storms therefore requires representation of accretionary processes as well as erosional processes. SBEACH

The authors have attempted to remove the aforementioned limitations through development of a numerica~ model called "SBEACH", an acronym for Storm-Induced BEAch CHange model (Larson and Kraus, 1989b, Larson el al., 1990). A description of the model and selected results are given in Larson (1988), Lar~en et al. (1988), and Larson and Kraus (1989a,b). In the following, the model will be briefly described, tested with a high-quality storm erosion data set, ther, applied to examine the short-term response of beach fill to storms.

Direction of transport A criterion expressed in terms of the deepwater wave steepness Ho/Lo and dimensionless fall speed Ho/wTwas developed to predict the net direction of transport. These parameters best predicted transport direction and frequently emerged in empirical correlations of dynamic and morphologic properties of the profile. Ho and Lo are the deepwater wave height and wavelength, respec-

92

M. LARSON A N D N.C. KRAUS

(a)

~i

+

LWT

cE:c.,~P, t

CEErosion

i

II

0.1000

.J

~[

i

022ram

~

~1 -,,- cEE,o,,oo: o:4omm !

Data

ACCRETION

O CEAccretion, 0 2 2 m m -~'~'CEAccretion, O.4Omm - i l l - CRIEPi Erosion. 0.47 mm a CRIEPIErosion, 0 2 T r a m

~

. : . :

i i :

i !

:<~_ : . / - l -

t/ m CRIEPi Accretion, TL ~ ~ _ _ ~

i < i

0.27 mm i :

~ :

~ ~,

; : i : ~I--

i : ,

,

~

:LI -~-cn,EP, Accre~oo.o.47mm I

~

l :

-

.

'

II

•

-:-

; : ! "

• •

II

. . . .

~

~ :

!

.j

/"

t i

i :~

:

i ~il :

:

O

i

T °

.....

°°°1° i

: : ~

o.s

~°s'°N

.~

~ ~ •

t.o

lO.O

20.0

Ho/WT

(b)

Field

Data,

Mean

0.1000

Wave

Height

::

-

"~

! :

ACCRETION

o.oolo

....... 0.1

+

:

~

0.0001

/

-~........ 4-

~0

D~"

:°

-~ - - I / - ~

i

:

,<.

l t ~ -

,~:

:

:

: EROSION

:

:

:

:

LI 7:i . . . . +--__+-__' ~_' ~__+_+ +..+¢- . . . . . . . . . 1.0

10.0

20.0

Fig, 4. Criterion for predicting beach erosion and accretion. ( a ) Monochromatic waves, ( b ) Field conditions.

laboratory

Ho/WT

MATHEMATICAL MODELING OF THE FATE OF BEACH FILL

9.3

tively, ,, is the sediment fall speed, and T is the wave period. The criterion is given by: Lo--

(11)

in which M = 0.00070 is an empirically determined dimensionless coefficient. If the left side of Eq. 11 is less (greater) than the right side, the profile is predicted to erode (accrete). Figure 4 shows the separation of erosional and accretionary events obtained for ta,~,k experin~ents witla heights and periods of large monochromatic waves in two independent tests performed by the U.S. Army Corps of Engineers (CE) (Saviile, 1957; Kraus and Larson, 1988~ ) and the Central Research Institute of Electric Power Industry (CRIEPI) in Japan (Kajima et al., 1982), and Fig. 4b shows a classification of field data. In order to use Eq. 11 with M=0.00070 for the field situation, which involves random waves, the wave height should be taken as the mean wave height and the period as the peak spectral or significant period (Larson and Kraus, 1989b; Kraus et al.. 1991 ). In Eq. 11, the wave steepness accounts for the wave asymmetry, which is known to control the direction of net sediment transport, whereas the wave height and period appearing in the fall speed parameter account for the absolute magnitudes of those quantities. Saville (1957) showed that the magnitude of the wave height controls erosion and accretion in addition to the wave steepness. It is interesting to note that a value of about H o / w T = 2 separates the erosional and accretionary cases in the field data ( Fig. 4b), which mainly pertains to beaches with fine-medium sand (grains size of about 0.2 mm ), but this is not substantiated by the large-scale tank data which included coarser sands (grains size of about 0.4 mm ). Kraus et al. ( 1991 ) discuss the predictive capability of Eq. 11 and similar predictors in detail. Net transport rate In the model, the transport rate is calculated in four distinct zones as depicted in Fig. 5, corresponding to analogous zones of different wave properties identified in hydrodynamic studies of the surf zone (e.g., Svendsen et al., 1978; Basco, 1985). Empirical transport rate formulas for each zone were established by integrating the mass conservation equation of sediment between profile surveys made in the large wave tank experiments, supplemented by physical reasoning and field data. Zone I (prebreaking region) The net transport rate was found to decay with distance seaward from the break point as:

94

M. LARSON AND N.('. KRAUS pp

SWASH ZONE

BP

BREAKER TRANSITION {]ROKE N WA vg ZONE

ZONE

PRE 8R£A K tNG ZO~/£

Fig. 5. Regions of cross-shore sand transport.

q=qbe ~'~''-'~'~

yb
(12)

in which qb is the transport rate at the break point calculated from Eq. 13, and Yb is the horizontal location of the wave break point from the base line. The empirical spatial decay coefficient 2~ had different values for erosional and accretionary cases in the large wave tanks. For erosional cases, regression analysis gave 2 j = lO.4(Dso/Hb) "47, expressed in units of m-~, in which Dso is the median grain diameter. The mean value of 2~ was 0.18 m-~ for the erosional cases. For the accretionary cases, no, dependence on grain size or wave properties was found, and the mean value was 2 ~= 0. ! ! m - ~.

Zone H (immediately postbreaking region) In this relatively short region defined by the break point and the plunge point 3'o, the net transport rate is given by:

q = qpe>:(''''p)

3'~,< y < .Vb

( 13 )

in which qr, is the transport rate at the plunge point as calculated from Eq. 14, and 22=0.20 2t.

Zone III (smf zone) An energy dissipation type formula similar to Eq. 10 is used, modified by a bottom slope-dependent term: Q dh'~ Q dh

Kc(D-Ocq + K~,d~)' q=

D> D e q - K c d y Q dh

O,

(14)

D < Dcq - K c d y

for the region y~,< y < Yo, in which y, is the location of the shoreward end of

MATHEMATICALMODELING OF THE FATEOF BEACH FILL

95

the surf zone, and % ~ 0.001 m 2 s - i is an empirical coefficient controlling the slope- or gravity-dependent transport term.

Zone IV (swash zone) The limited data available on the transport rate in the swash zone and on the beach face support a linear decrease in the rate from the seaward limit of the surf zone to the limit of wave run up:

q=qs y-y------&~

yr
(15)

Ys - - Y r '

in which qs is the transport rate at ys as given by Eq. 14, and Yr is the limit of wave run up. Expressions for predicting the position of the plunge point (generalized to include spilling breakers), the limit of wave rump, and other needed parameters are given in Larson and Kraus (1989b). In use of Eqs. 12-15, the transport rate is first calculated by Eq. 14, for which values at the boundaries of Zone III are obtained to calculate in Zones II and IV. The numerical model of Dally et al. ( 1985 ) is used to provide the wave height needed to calculate the location of the break point, energy dissipation, set up, and other waverelated quantities at each point on the profile change model grid. Although the mean wave height is used as the statistically representative wave height to define the net direction of transport by Eq. 11, experience has shown that the location of the average break point and the magnitude of the net transport rate are best specified by use of the significant wave height. These inferences based on model simulations of field profile change are supported by laboratory results (Mimura et al., 1986 ). An explicit finite-difference solution scheme is used v9 solve the mass conservation equation: ~Jh 0q at -Oy

(16)

The seaward limit of the model calculation is generally set at a location where Eq. 12 gives negligible transport over the course of the simulation, and the shoreward limit is given as the limit of the wave rump for the case of a sandy beach face, or q = 0 if a seawall is present. An avalanching routine is also included to limit the steepness of the beach slope. In typical situations involving storm waves of height of 5 to 10 m, and cross-shore extent of 500 to 1000 m, the time step of the model is typically 5 to 20 min and the grid cell length is 1-5 m. Input to the model are time series of wave height, wave per~cd, water level, and the initial profile shape and representative grain size. The water level and offshore wave conditions are input at each time step, and the wave height across shore and profile change are then calculated.

96

M. LARSON AND N('. KRAUS

SAMPLE OF MODEL CALIBRATION

Any type of model must be calibrated prior to use in design, and the present empirically based numerical model is no exception, since values of the main parameters (K~, %, 2) were established for a limited range of wave heights and periods, and sand grain sizes. Here an example is given of a calibration using a high-quality data set on storm-induced beach erosion in the United States. The site is Point Pleasant and Manasquan Beaches, located, respectively, on the south and north sides of the Manasquan River jetties, and facing to the Atlantic Ocean on the coast of New Jersey. Point Pleasant Beach is a relatively wide sandy beach with no active coastal structures present except the 678-m long jetty to the north, Manasquan is a relatively narrow sandy beach with short groins located at approximately 200- to 300-m intervals. A location map and details on the data and site are given in Larson et al. (1990). The profile was surveyed to wading depth along eight transects at approximate 200- to 300-m longshore spacing at Point Pleasant and on nine transects midway between groins at Manasquan on March 27-28, 1984. A strong extratropical storm (northeaster) arrived on March 29, halting scheduled fathometer surveys to extend the measurements to deeper water. The storm caused significant erosion and, after it had passed, profile surveys were made along the transects. The post-storm profiles taken on April 2-3, 1989 include an estimated 3 to 4 days of recovery waves. Water level changes produced by the storm surge and tide were recorded at l-h intervals at a tide gage located at Manasquan Inlet, and statistics of tile wave height and period were determined at 1- to 6-h intervals, depending on the magnitude of the wave height, from recordings of sea surface elevation by a wave buoy located on the 15-m contour directly offthe inlet. The water level between profile surveys is shown in Fig. 6 and the wave height and period in Fig. 7. For the model calibration, two profiles near the jetties were eliminated from the analysis since they showed comparatively small eroded volume (probably a longshore effect), and a mean initial (pre-storm) profile was developed using a December 28, 1983 survey for the subaqueous profile joined to the mean measured subaerial profile from 27-28 April. A mean profile was similarly developed for the post-storm condition using the survey data. The prestorm profiles at the two beaches differed substantially, with the Point Pleasant profiles increasing monotonically in depth with distance offshore and the profiles a' vlanasquan having a subaqueous terrace located near the end of the groit.J. Since grain size information is lacking for the date of the storm, a composite grain size was established from samples taken in the 1950s and in April 1989, with 0.5 mm (0.3 ram) assigned to the portion of the profile above (below) MSL at each beach. Further work on determining the most representative grain size should allow more accurate modeling of the ob-

MATHEM-~TI('ALMODELINGOF ] H E FATE OF BEACHFILL

97

Water Level (m)

1.5

Manasquan, 1984

I I

0.5

NJ

A

-O,Sf -

1 26

x 27

l

28

L

29

L

t

30 31 March / April

J. . . . . . . .

1

• . . . . . .

2

• ........

3

Fig, 6. Water level mea:~ured at Manasquan Inlet during the March 1984 storm.

served change. In the calibration, a value of Kc=2.0 10 6 m 4 N - ~~as determined by least-squares minimization, with all other model parameters kept at average values found for the large wave tank tests. Because of the small height and long period of the post-storm waves (Fig. 7 ) that arrived after March 30, partial recovery undoubtedly occurred prior to post-storm surveying, obscuring the true erosion that took pi,,ce. Measured volume change is therefore contaminated by accretion during the recovery stage (which is likely the case for all field data on storm erosion ). Tabie! lists Wave Height and Period 14

Manasquan, NJ 12

10 8 6 4 2

0 26

27

28

29

30 31 March / April

1

2

3

Fig. 7. Wave height and period measured at Manasquan Inlet during the March 1984 storm.

08

M. L A R S O N A N D N.C. K R A U S

Elevation (m)

(a) Point Pleasant Beach, NJ 1984

0 -2 -4

---

Initial 27 Mar

--

Poat-Storm2Apr

~ ' . . .

--

Calc. Eroslon3OMar

~

,

,.,

~ , . .

Calc. Recovery 2 Apr -6

I

0

50

" i

I

__

100 150 Dist~=~nce Offshore (m)

Elevation (m)

-.

i

200

250

(b) Manasquan Beach, NJ 1984

4

2 o

-2

--

Initial 26 Merr

Post-Storm 3 Apr -4[

"'" -, ~

.

.... t a l c . Erosion 30 Mar Calc, Recovery 3 Apr

0

L

i

L

!

50

100

150

200

250

D i s t a n c e O f f s h o r e (m)

Fig. 8. Storm-induced profile change at Manasquan Inlet, New Jersey. (a) At Point Pleasant Beach. (b) At Manasqu~m Beach.

the measured and calculated volumes of erosion for the two beaches, and Fig. 8 allows visual comparisons of the nearshore profile shapes. Averages for the measurements are given for all~profiles and for the subset selected for use in comparisons with model predictions. The minimum and maximum measured erosion shown in Table 1 shows that considerable vari-

99

MATHEMATICAL MODELING OFTHE FATE OF BEACH FILL TABLE 1

Measured and calculated volumes of profile erosion (m3/m) Contours

Measured Average of all

Calculated Average of selected

Min

Max

Max erosion

With recovery

Point Pleasant Beach 0 48.9 l-m 45.8

55.8 53.8

9.6 8.5

66.0 63.1

86.4 54.6

73.9 54.2

Manasquan Beach 0 36.3 l-m 32,4

39.3 34.7

14.7 13.4

62.6 56.2

51.5 37.2

44.2 37.2

ability in erosion occurred, about a factor of 6 difference for Point Pleasant and a factor of 4 for Manasquan. Point Pleasant experienced more erosion, suggesting that the groins along Manasquan Beach may have had a stabilizing effect through the subaqueous sediment reservoir (terrace) and, possibly, suppression of longshore transport during the storm. Volume of erosion was calculated at the end of the erosional phase (March 30) to estimate the maximum erosion and at the time of the April surveys to include post-storm recovery volumes. Calculated maximum eroded volumes are greater than measured volumes; this tendency is expected since the measured erosion is an underestimation because it includes recovery. The calculated volume of erosion including recovery was overestimated for Point Pleasant, mainly because the model could not reproduce the large berm which formed above MSL (Fig. 8a). Since the profile at Manasquan did not experience large berm growth (Fig. 8b) as at Point Pleasant, the calculated eroded volumes are in better (good) agreement with the measurements. It is interesting to note the nearshore multiple bars which appeared in the calculation for Point Pleasant are much more subdued for Manasquan; the model thus tends to produce the proper accretionary feature for the existing initial profile shape. Performance of the model is considered to be good in replicating change in the profile shape and in predicting volume of erosion. Simulation of beach recovery has qualitatively correct tendencies, but is quantitatively inadequate. BEACH FILL TEMPLATE DESIGN

A verified model enables examination of the shore protection functioning of alternative fill cross sections or fill templates at a beach nourishment project under impact of storms. Such calculations must be made to perform costbenefit analysis and to esti~a'e the minimum fill volume needed to provide

100

M. LARSON AND N.C. KRAUS

a specified level of storm protection against erosion and/or inundation. In the overall evaluation, longshore variability of the protective beach complex and storms must also be considered (Van de Graaff, 1983; Savage and Birkemeier, 1987 ). The remaining material in this section is adopted from Larson and Kraus (1989a). Two approaches can be taken to estimate storm impact on a beach fill complex; one may be called the design-storm approach and the other the stormensemble approach. The design storm is either a hypothetical or a historical event which produces a specified storm surge hydrograph and wave condition at the project. Surge is a water level rise caused by wind stress and atmospheric pressure variation; waves also produce a rise in mean water level at the shore called setup. The time average, on the order of an hour, of surge, wave setup, and tide is called the stage. In stage-frequency analysis, the design storm may have a certain frequency of occurrence, for example, a 100-year storm. The design storm approach is problematic for use in dune erosion modeling because beach change is sensitive to storm duration, surge shape, and wave height and period, in addition to peak stage. The maximum water level associated with the surge of a design storm may produce less erosion than a storm of lower surge but longer duration, or than a storm of lower surge but higher waves. The solution to the problem of the many-to-one relation between beach erosion and stage frequency is to use the storm-ensemble approach, i.e., to calculate erosion for a large number of storms, and key the erosion to the frequency of storm occurrence (Scheffner, 1988, 1989). This yields an erosion- or recession-frequency of occurrence curve. In this process, at present, tropical and extratropical storms must be treated independently because they have different physical characteristics. Tropical storms are infrequent events of short duration and high intensity, whereas extratropical storms are more frequent and usually of longer duration and lower intensity. The storm-ensemble approach is recommended for project design, although it requires a storm data base and is much more computationally intensive than the design storm approach. Here, the response of nourished beach profiles to a representative tropical storm (hurricane) and an extratropical storm (northeaster) is calculated to examine predictions of the model to storms of differing waves and water levels (surge and duration). The two storms were synthesized to produce erosion resulting from a 2-5-year storm-surge event for the mid-Atlantic Ocean coast of the United States. The amount of eroded volume for such events is on the order of 20-30 m3/m (Savage and Birkemeier, 1987 ).

Example representative storms Surge hydrographs for these hypothetical storms are shown in Fig. 9. The hurricane surge has a duration of approximately 12 h, with a peak surge of 2

MATHEMATICAL M O D E L I N G OF THE FATE OF BEACH FILL

I 01

Surge (m) Synthetic Storms ~ , Hurricane

/

1.5

\

\

0.5

0 I 150

155

160

165

170

175

180

185

190

Time (hr)

Fig. 9. Surge hydrographs for the synthetic storms.

m, and a duration above half the peak surge ( I m) of 6 h. The shape of the hurricane surge was generated from an inverse hyperbolic cosine squared. The surge of the northeaster has a duration of 36 h, with a peak surge of 1 m, and a duration above half the peak surge (0.5 m) of 18 h. The shape of the northeaster surge was generated by a cosine squared function. The peak surge of the hurricane is higher because the wind speeds in hurricanes are, on the average, greater than in northeasters. The time history of the wave height and period assigned to the hurricane and northeaster are shown in Fig. 10a and b, respectively. Both have peak wave heights of 5 m, which occur during the time when the respective surges are greater than half the maximum. The duration of high waves for the northeaster is thus three times that of the hurricane. Since the radius of a northeaster is typically several times greater than that of a hurricane, the fetch is longer, resulting in longer wave periods assigned to the northeaster. Wave height and period of I m and 7 s were applied for 6.5 days before start of the storms to mold the profiles into a realistic shape under typical waves at the site. Following the storm, the wave height and period were char'.ged to 0.5 m and 10 s to simulate long-period recovery swell wave conditions. A sinusoidal tide was applied with a 12-h period and 0.5-m amplitude, and a peak in the tide occ• :red during the peak surge of each storm.

Beach profile shape (fill templates) Following the procedure of Kraus and Larson (1988b), two different beach fill cross sections or templates were designed for exposure to storms. One, an artificial herm, h~d . . . . . .mc,~t . . . . . .,,t" . th,~ fill v--,,-,-~"l°"'~'~on the beach av,d above mean sea

M. LARSONAND N.C. KRAUS

i U~

(a)

Height / Period

1°I

/ Period

(sec)

Synthetic Hurricane Waves !Height {m)

0

I

0

50

I

t

I

I

I

I

I

100 150 200 250 300 350 400 450 500 Time {hr)

Height :' Period 14

I

J

(b) r-~~Period(see)

12

10 Synthetic

8

Northeaster Waves 6

lHeight (m)

4

2 0 0

50

100 150 200 250 300 350 400 450 500 Time (hr)

Fig. 10. Time history of wave height and period for the synthetic storms. (a) Hurricane. (b) Northeaster.

level (MSL). The as-constructed berm extended horizontally for a distance of 16 m at an elevation of 3 m and then tapered with a 1 : 20 slope to join the original beach profile at a depth of i.4 m (Figs. 11 and 13 ). This is a c o m m o n beach fill design in the United States. The other fill template is termed profile noztrishment (Bruun, 1988 ). In the example, the mate,:ial was placed over the profile from an elevation of + i to - 2 m (Figs. 12 and 14) in an approximation of the existing profile. The a m o u n t of fill was the same for each template, 140 m3/m. Figures 11-16 pertain to a 0.20-mm sand beach, for which both the fills and the beach had the same grain size. Runs were also m a d e for fill grain sizes

103

M A T H E M A T I C A L M O D E L I N G O F T H E FATE O F BEACH FILL

Elevation(m) A r t i f i Cila,Berm: Hurriesn~

. Profilewithoutfill - - Profile with fill

4~ 3-

\

2-

....

Pre-storm

~

Poet-storm

1-

O -1 -2 -3

i

0

40

=

q

I

80 120 160 DistanceOffshore (m)

200

Fig. 11. Response of artificial berm to hurricane impact.

Elevation (m) 4 ~

ProfileNourishment: Hurricane

3

- -

Profile without

- -

Profile

....

2 1

~ ~ , ~ ~= ,~ . ~

~ ~'~"..~"~.~ _~

with

fill

fill

Pre-etorm Post-storm Poet-storm recovery

o -1 -2

-3

i

0

40

i

i--

~

80 120 160 DistanceOffshore (m)

---

i

200

Fig. 12. Responseof nourished profile to hurricane impact. in increments from 0.2 m m to 1.0 mm. In these cases, the grain size was specified as the fill size over the portion of the profile originally occupied by the fill, and 0.2 m m elsewhere. A water temperature of 20 °C was specified in the model for computation of the sand fall speed.

Profile change Figures 11-14 illustrate the impacts of the two storms on the two nourishment projects. The bold line labelled "profile without fill" gives a hypothetical dune, beach, and subaqueous equilibrium profile for reference. The solid

104

M. LARSON A N D N . C KRAUS

Elevation (m) 4 -~

Artificial Berm: Northeaster Profile without fill

2-

~ k

~ \ ,~

.... Pre-storm ~ Pos~-storm - - Post-storm recovery

0

r 40

0

~ ----'TI 80 120 160 Distance Offshore (rn)

i 200

Fig. 13. Responseof artificial berm to northeaster impact. Elevation (m) i

~

Profile Nourishment: N o r t h e a s t e r Profile without fill - - Profile with fill .... Pre-etorm Foet-ltorm Poet-storm recovery Ir~. : - ~ .

1 o_ -1

0

40

80 120 160 Distance Offshore (m)

200

Fig. 14. Responseof nourished profile to northeaster impact. line labelled "profile with fill" shows the fill configuration prior to storm action (at the completion of construction ). The dashed "pre-storm'" line shows the profile after 6.5 days of ordinary waves. The line with a marker represents the post-storm profile configuration, prior to the start of the recovery wave period. The line labelled "post-storm recovery" shows the profile after experiencing approximately two weeks of recovery waves. Pre-storm

The pre-storm profiles ('°design profiles" of Fig. 1 ) of the berm and nourished profile differ significantly in the inner surf zone. A steep step is pro-

MATHEMATICAL MODELING OF THE FATE OF BEACH FILL

105

duced in the berm, whereas the nourished profile experiences gentler changes since it was placed in a near-equilibrium configuration. For both cross sections, a small breakpoint bar formed at about the 210-220-m mark (measured from an arbitrary baseline). For all cases, material was removed from the inner surf zone and distributed along the profile beyond the 2-m depth. Thus, regardless of the initial fill configuration, the model predicts fill material will be transported considerably offshore to the point of incipient storm wave breaking in the process of molding the surf zone profile to an equilibrium shape. Post-storm Subaqueous sections of the post-storm profiles are very similar, being reworked by strong breaking wave action to the same equilibrium shape independent of initial profile configuration and type of storm. The shoreline position (0-depth contour, MSL) actually advanced seaward of its pre-storm location, with the material supplied from the ordinarily subaerial portion of the profile that was inundated during the storm surge and high waves. A small bar formed at approximately 5-m depth under the high storm waves, but is not shown here to better display changes near the beach. An important outcome of the predictions is that, under the action of the particular hurricane and northeaster used, resultant profile change was very nearly the same. This demonstrates that use of one storm descriptor, for example, the maximum stage, to estimate shoreline recession or volume of eroded material can produce misleading results. Post-storm recovery In all cases, a substantial berm was created which is connected to the offshore by a broad trough. The upper foreshore of ~he nourished profile experienced more accretion than the artificial berm cases. These results are consistent with the concept that the beach profile in a natural shape can best respond to changes in the incident waves. Volume eroded and contour change In this discussion, the 0-depth contour and the l-m contour (above MSL) are taken as reference datums (defined with respect to MSL). It is proposed that both the 0- and l-m contours be used as data future studies in reporting results of storm-induced beach erosion. (Here, the 0.5-m contour was used as a substitute for the 1-m contour for the profile nourishment example because of the low relief of the fill in this particul,2r case. ) The 0-depth contour defines the lower boundary of the subaerial beach and is a commonly used datum to define eroded volume and beach recession. However, the shoreline position often acts as a pivot point through which

106

M. LARSON AND N.C. KRAUS

sand is transported; in fact, shoreline position referenced to the 0-depth datum can advance seaward during a storm (Birkemeier et al., 1988). Thus, another datum is needed. Although this second datum is arbitrary, the authors suggest the 1-m contour (above MSL). The advantages of reporting eroded volumes and beach recession with respect to the 1-m contour are ( 1 ) very small storms will not significantly impact this contour, so that "noise" is eliminated from the analysis, and (2) post-storm recovery will be limited at the l-m contour, thereby avoiding a possible underestimation of eroded volume and recession. Scheffner (1988, 1989) developed dune-erosion-frequency of occurrence curves by using the maximum recession of any contour on the profile between MSL and the dune crest. Maximum recession is a good physical measure of beach erosion, but it may not be convenient for issuance of permits and in planning assessments.

Volume eroded Figure 15 plots the time evolution of eroded volume above the 0- and 1-m (and 0.5-m) contours. The volume eroded above the 0-depth contour increases rapidly at the beginning of the pre-storm ("ordinary") wave action, describing the behavior of the fill material during initial profile adjustment. In contrast, the eroded volume above the 1-m contour shows a much less rapid increase. The profile nourishment case experiences greater initial erosion during the early stage of wave action, but also greater recovery in the poststorm period. It should be emphasized that longshore processes are omitted from the present discussion; bean nourishment places a greater amount of material into the active littoral zone, thereby increasing its potential for transport alongshore and out of the project reach. The volume of eroded material above the 0-depth contour does not show significant increase during the storms, changing only from about 22 to 27 I l l 3 / m in the case of berm erosion during the nortl:, aster. Eroded volume above the l-m contour abruptly increases at the start of the storms, going from about 3 to 17 m3/m in the case of the artificial berm impacted by the northeaster. The reason why the volume of erosion above the 0-depth contour is relatively unchanged is that these moderate storms primarily remove material from the upper portion of the profile and reoistribute it over the beach face, not transporting it far offshore. The hurricane and northeaster produce nearly the same amount of erosion, 25-30 m3/m above the 0-depth contour and 13-16 m3/m above the l-m contour. Eroded volumes above the 0-depth contour are comparable to those associated with 2-5-year return period storms impacting the mid-Atlantic coast (Savage and Birkemeier, 1987 ). The longer surge duration of the northeaster was, therefore, approximately equivalent in erosion capacity to the higher surge of the shorter duration hurricane. Time evolution of the eroded vol-

107

MATHEMATICAL MODELING OFTHE FATE OF BEACH FILL Volume (ma/~)

Eroded

35 30

25

(a)

/ C o n t o u r (m)

Case

Berm

0

--

Berm

1

....

Nourllh 0

Hurricar,~ ". . . . . . . . . . . . . . . .

,..-'"

.-

20 15 10 5 0 0

Eroded

i

i

t

1

50

100

150

200

Volume (maim)

30-

/ Contour Berm 0 Berm

25 -

....

i

400

7 - - - -

450

soo

Northeaster

•........

. ~" - -

NourlShr~

Nourish 0

-r~

a50

(b)

(m)

1

I

300 Time (hr)

35Case

I

250

.,-" ~

............... "

""

-

-

.

-

~

............

20 /

.

"'-.

_

10 5 0

i

0

50

100

150

i

200 2so 300 Time (hr)

r

~

r

3so

4oo

450

5oo

Fig. 15. E r o d e d v o l u m e a b o v e specific ¢onlours. ( a ) H u r r i c a n e . ( b ) N o r l h e a s t e r .

umes above the shoreline shows an approximate exponential approach to an equilibrium value. These trends are in general agreement with those obtained by Kriebel and Dean ( 1985 ), who numerically examined eroded volume and berm recession as a function of wave height, surge level, and other parameters. It is interesting that there is a tendency in Fig. 15 for the hurricane-impacted beach to recover more rapidly than the northeaster-impacted beach. This is in agreement with qualitative observations of post-storm recover), on the Florida coast (A. Hobbs, pers. commun., April, 1989). An explanation

108

M. LARSON AND N.C. KRAUS

lies in the nature of the storm hydrograph; material is not removed as far offshore during hurricanes as in the case of the longer duration and lower surge extratropical storms. This material is deposited closer to shore and can be moved back on the beach more quickly under recovery wave action. Contours Figure 16 plots the time evolution of the 0- and 1-m contours. Decrease in magnitude of contour position indicates recession of the beach at that conContour Position (m)

(a) Hurricane

140 [ 120 t

""'""

"""

"

100 80 60

Berm

O

--

Berm

1

....

Nourish 0 Nourish 0.5

40 0

1

s

I

I

50

100

150

200

Contour Position

(m)

140i,,,,........

100 ~ 80-

t

250 300 Time (hr)

350

400

450

500

3so

400

45o

500

(b) Northeaster

.~ C

a

s

e

~

Berm 0 --

Berm

1

60- .... Nourish0 Nourish 0.5 40 , ,o 50 loo ;50

2~o

250

zoo

Time (hr) Fig. 16. M o v e m e n t o f selected c o n t o u r s . ( a ) H u r r i c a n e . ( b ) N o r t h e a s t e r .

109

M A ' I H E M A T I ( ' A L M O D E L I N G O F T I l E E,VI'E O F BEACH FILL

tour. The 0-depth contours for both fills and both storms show recession during pre-storm and storm periods, but begin to advance before the end of the storms and prior to arrival of the recovery waves, as the surge subsides and the wave height decreases. The l-m contour shows no recovery for the berm because the subaerial d u n e / b e r m complex is steep, whereas the 0.5-m contour for the nourished profile does show some recovery since the gentler slope allows the post-storm wave runup to build a berm. In the model, berm formation and growth are largely controlled lzy the elevation reached by wave

(a)

Eroded Volume (mS/m)

35 /

Hurricane

Case / Contour (m) - - - Berrn 0

"'.,,..

28

--

Berm

---

~ . . _ . ' ' " "

1

Nourioh 0 Nourish 0.5

""'-.

21

I~~_

14

~ + . -

'

. . . . . . . . . . . . . . . . . . . . . . =-~

7-

0

0.2

i

I

0.3

0.4

Eroded Volume (mS/m)

35

I

I

I

0.5 0.6 0.7 Grain Size (ram)

f

0.9

(b) Northeaster

28

I

0.8

"""

Case / Contour (m) Berm 0 --

Berm

....

Nourish 0

1

Nourish

0.5

21

14

t

:1 0.2

0.3

0.4

0.5 0.6 0.7 Grain Size (ram)

0+8

0.9

Fig. 17. Effect of grain size on eroded volume. (a) Hurricane. (b) Northeaster.

I i0

M. LARSON AND N.C. KRAUS

runup and local slope (Larson, 1988; Larson and Kraus, 1989a), and berm processes are diminished on steeper slopes.

Eroded volume and grain size Figure 17 plots eroded volume at the end of the storm (prior to recovery' wave action) as a function of the grain size of the fill. As previously mentioned, the native beach grain size was set at 0.2 ram, and the area in which the fill was placed was assigned the grain size of the fill. This procedure does not allow tracking of movement of the different grain sizes. However, since surf zone sedir~ents are usually sorted with coarser material located higher on the active profiie, this simple procedure is considered to provide a reasonable first approximation of the response of a natural beach of varying grain size. For the particular synthetic storms used, Fig. 17 shows a relatively steep decrease in eroded volume as grain size increases through the range of 0.2 mm to 0.4 ram, with a gentle decrease thereafter. This behavior follows from the property of the empirically determined functional dependence of the wa,~e energy dissipation needed to generate an equilibrium profile of given grain size. The property is manifested by computed dissipation rates which .,ise steeply in the range of 0.1 to 0.4 mm, and then increase at a lower rate with increasing grain size (Moore, 1982). Since the rate of decrease in erosion is small beyond 0.4 mm, and the cost of beach fill typically increases substantially for larger size material, calculations such as those illustrated in Fig. 17 allow an evaluation to be made of initial nourishment volume and subsequent fill maintenance costs. In project planning, curves equivalent to those in Fig. 17 must be developed with consideration of the design profile and the storms characteristics of the area, using the storm-ensemble approach. SUMMARY

AND CONCLUDING

DISCUSSION

Two classes of models of beach change are currently in use, one which calculates longshore transport and shoreline change and another which calculates cross-shore transport and profile change. In the future, these two classes of models will be undoubtedly be merged to provide more general and reliable results. Also, results from models based more directly on the underlying physics of fluid motion and sediment movement (e.g., Roelvink and Stive, 1989 will be incorporated, enhancing model accuracy and generality. The present state of ~o~elir.g doe~, provide substantial quantitative information on both the long-term and short-term evolution of beach fill. Analytical models of the long-term fate of fill were used to illustrate fundamental properties, including the dependence of spreading of the fill as wave height to the 5/2 power and inverse square of the longshore extent of the fill. It was

MA'I'ItEMATICAL MODELING OF TItE FATE OF BEACH FILL

1l 1

also demonstrated that the movement of sand waves or large localized shoreline forms could be described in the shoreline change models. Main focus of this paper was on beach fill template design for storm protection. Principal results from this simulation effort with a high-quality storm erosion data set were: ( 1 ) The numerical model performed well in describing measured storminduced erosion, showing correct dependencies on the surge and wave input, as well as the initial profile shape, Qualitative features of the post-storm recovery phase were also reproduced, but further work is required to both understand and simulate recovery. Many more high-quality data sets on storminduced beach erosion are needed to refine and test simulation models. (2) The model can be used to judge the relative behavior and merits of various beach fill cross sections exposed to ordinary and extreme waves for time intervals on the order of days to weeks. (3) Storm-induced beach and dune erosion cannot be uniquely specified through a single storm-related parameter such as the maximum stage. This result demonstrates the limited usefulness of the design storm approach. (4) Fill placed on the upper beach was calculated to move offshore to relatively great depths (region of incipient wave breaking), in agreement with the generally inferred behavior of the movement of beach fill material. ( 5 ) The l-m contour (elevation above MSL) is a useful datum to which to refer storm-eroded volume and beach recession, in addition to the shoreline or 0-depth (MSL) datum. (6) The limited number of calculations performed here indicates that it may not be cost effective 1o use beach fill for storm protection with a median grain size much greater than 0.4 mm owing to the typically greatly increased cost and the declining benefit in decreased volume of eroded material. Design alternatives for specific situations (fill grain size, fill cross section, and storm climate ) could be evaluated with the model. ACKNOWLEDGEMENTS

We would like to express our appreciation to Mr. Jeffrey A. Gebert, oceanographer, U.S. Army Corps of Engineers, Philadelphia District, for providing the beach erosion data set at Point Pleasant and Manasquan, New Jersey, and assisting us with enthusiasm in understanding the data set and related conditions. This excellent data set was collected through the planning of Mr. Gebert and with the support of the Monitoring of Completed Coastal Projects program, Corps of Engineers. The contribution of N.C. Kraus was made as part of the activities of the Cross-Shore Sediment Transport Processes work unit of the Shore Protection and Evaluation Program, Coastal Engineering area of Civil Works and Development, being executed by the Coastal Engineering Research Center

Il2

M. LARSON AND N.C. KRAUS

(CERC), U.S. Army Engineer Waterways Experiment Station. The contribution of M. Larson was partially supposed by the work unit Surf Zone Sediment Transport Processes of the same program, and funded by CERC through the U.S. Army Research, Development, and Standardization Group - UK. Permission was granted by the Chief of Engineers to publish this information. REFERENCES Basco, D.R., 1985. A qualitative description of wave breaking. J. Waterw., Port Coastal Ocean Engl, 111(2): 171-188. Birkcmeier, W.A., Savage, R.J. and Leffler, M.W., 1988. A collection of storm erosion field data. Misc. Coastal Eng. Res. Center, U.S. Army Eng, Waterw,. Exp. Station, Vicksburg, Miss.. Pap. CERC-88-9. Bruun, P., 1954. Coast erosion and the development of beach profiles. Beach Erosion Board, U.S. Army Corps of Eng., Tech. Memo. 44. Bruun, P., 1988. Profile nourishment: its background and economic advantages. J. Coastal Res., 4(2): 219-228. Dally, W.R., Dean, R.G. and Dalrymplc, R.A., 1985. Wave height variation across beaches of arbitrary profile. J. Geophys. Res., 90(C6): I 1,917-11,927. Dears, R.G., 1977. Equilibrium beach profiles: U.S. Atlantic and Gulf coasts. Dep. Civil Eng., Ocea~l Eng. Univ. of Delaware, Newark, Del., Rep. 12. Dean, R.G., 1984. Principles of beach nourishment. In: P.D. Komar (Editor), CRC Handbook of Coastal Processes and Erosion. CRC Press, Boca Raton, Fla., pp. 217-231. Dean, R.G., 1987. Coastal scdiment processes: toward engineering solutions. Proc. Coastal Sediments '87, ASCE, pp. 1-24. Dean, R.G., 1988, Realistic economic benefits from beach nourishment. Proc. 21 st Coastal Eng. Conf., ASCE, pp. 1558-672. Delft Hydraulics, 1987. Manual on artificial beach nourishment. Centre for Civil Eng. Res., Rijkswaterstaat, Gouda. Edelman, T., 1972. Dune erosion during stoma conditions. Proc. 13th Coastal Eng. Conf., ASCE, pp. 1305-1311. Hanson, H., 1989. Genesis - a generalizcd shorcline change numerical model. J. Coastal Res., 5(I ): 1-27. Hanson, H. and Kraus, N.C., 1989. GENESIS: gencralized model for simulating shoreline change; Report 1, technical reference, U.S. Army eng. Waterw. Exp. Station, Coastal Eng. Res. Cen!or, Vicksburg, Miss., Tech Rep. CERC-89-19. Hughes, S.A. and Chiu, T.Y., 1981. Beach and dune erosion during severe storms. Coastal Oceanogr. Eng. Dep., Univ. of Florida, Gainesvil!e, Fla., Rep. U FL/COEL-TR/043. Inman, D.L., 1987. Accretion and erosion waves on beaches. Shore Beach, 55 (3-4): 61-66. Kajima, R., Shimizu, T., Maruyama, K. and Saito, S., 1982. Experiments on beach profile change with a large wave flume. Proc. 18th Coastal Eng. Conf., ASCE, pp. 1385-1404. Kraus, N.C., 1989. Beach change modeling and the coastal planning process. Proc. Coastal Zone '89, ASCE, pp. 553-567. Kraus, N.C. and Larson, M., 1988a. Beach profile change measured in the tank for large waves, 1956-1957 and 1962. Coastal Eng. Res. Center, U.S. Army Eng. Waterw. Exp. Station, Vicksburg, Miss. Tech, Rep. CERC-88-6. Kraus, N.C. and Larson, M., 1988b. Prediction of initial profile adjustment of nourished beaches to wave action. Proc. Beach Technol. '88, Florida Shore a~id Beach Prese~w. Assoc., pp. 125137.

MATHEMATICAL MODELING OF THE FATE OF BEACH FILL

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Kraus, N.C., Isobe, M., Igarashi, H., Sasaki, T. and Horikawa, K., 1982. Field experiments and longshore sand transport in the surf zone. Proc. 18th Coastal Eng. Conf., ASCE, pp. 969988. Kraus, N.C, Larson, M. and Kriebel, D.L., 1991. Evaluation of beach erosion and accretion predictors. Proc. Coastal Sediments '91, ASCE, pp. 572-587. Kriebel, D.L., 1982. Beach and dune response to hurricanes. Unpubl. M.S. Thesis, Univ. of Delaware, Newark, Del. Kriebel, D.L., 1986. Verification study of a dune erosion model. Shore Beach, 54 (3): 13-21. Kriebel, D.L., 1987. Beach recovery following Hurricane Elena. Proc. Coastal Sediments '87, ASCE, pp. 990-1005. Kriebel, D.L. and Dean, R.G., 1985. Numerical simulation of time-dependent beach and dune erosion. Coastal Eng., 9:221-245. Kriebel, D.L., Kraus, N.C. and Larson, M., 1991. Engineering methods for predicting beach profile response. Proc. Coastal Sediments '91, ASCE, pp. 557-571. Larson, M., 1988. Quantification of beach profile change. Dep. Water Resour. Eng., Inst. Sci. Technol., Univ. ofLund, Rep. 1008, 293 pp. Larson, M. and Kraus, N.C., 1989a. Prediction of beach fill response to varying waves and water level, Proc. Coastal Zone '89, ASCE, pp. 607-621. Larson, M. and Kraus, N.C., 1989b. SBEACH: numerical model for simulating storm-induced beach change; Report 1, empirical foundation and model development. Coastal Evg. Res. Center, U.S. Army Eng. Waterw. Exp. Station, Vicksburg, Miss., Tech. Rep. CERC-89-9. Larson, M., Hanson, H. and Kraus, N.C., 1987. Analytical solutions of the one-line model of shoreline change. U.S. Army Eng. Waterw. Exp. Station, Coastal Eng. Res. Center, Vicksburg, Miss., Tech. Rep. CERC-87-15. Larson, M., Kraus, N.C. and Byrnes, M.R., 1990. SBEACH: numerical model for simulating storm induced beach change; Report 2, numerical formulation and model tests. Coastal Eng. Res. Center, U.S. Army Eng. Waterw. Exp. Station, Vicksburg, Miss., Tech. Rep. CERC-899. Larson, M., Kraus, N.C. and Sunamura, T., 1988. Beach profile change: morphology, transport rate, and numerical simulation. Proc. 21 st Coastal Eng. Conf., ASCE, pp. 588-601. Mimura, N., Otsuka, Y. and Watanabe, A., 1986. Laboratory study on two-dimensional beach transformation due to irregular waves. Proc. 20th Coastal Eng. Conf., ASCE, pp. 1393-1406. Moore, B.D., 1982. Beach profile evolution in response to changes in water level and wave height. Unpubl. M.S. Thesis, Univ. of Delaware, Newark, Del. Pelnard-Considere, R., 1956. Essai de theorie de l'evolution des forms de rivage de sable et de galets. 4th Journees de l'Hydraulique, Les Energies de la Mer, Question III, Rapport No. 1, pp. 289-298° Roelvink, J.A. and Stive, M.J.F., 1989. Bar-generating cross-shore flow mechanisms on a beach. J. Geophys. Res., 94(C4): 4785-4800. Savage, R.J. and Birkemeier, W.A., 1987. Storm erosion data from the United States Atlantic coast. Proc. Coastal Sediments '87, ASCE, pp. 1445-1459. Saville, T., 1957. Scale effects in two dimensional beach studies. Trans. 7th General Meeting IAHR, 1: A3-I-A3-10. Scheffner, N.W., 1988. The generation of dune erosion-frequency of occurrence relationships. Proc. Symp. on Coastal Water Resources, TPS-88-1, Am. Water Resour. Assoc., Tech. Publ. Ser. TPS-88-1: 33-47. Scheffner, N.W., 1989. Dune erosion-frequency of storm occurrence relation ships. Proc. Coastal Zone '89, ASCE, pp. 595-606. Shore Protection Manual, 1984. 4th ed., 2 Vols. Coastal Eng. Res. Center, U.S. Army Eng. Waterw. Exp. Station, UoSoGov. Print. Offi., Washington, D.C.

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Sonu, C.J., 1968. Collective movement of sediment in littoral environment. Proc. 1 l th Coastal Eng. Conf., ASCE, OP- 373-398. Svendsen, I.A., Madsen, P.A. and Buhr Hansen, J., 1978. Wave characteristics in the surf zone. Proc. 14ill Coastal Eng. Conf., ASCE, pp. 520-539. Swart, D.H., ~776. Predictive equations regarding coastal transports. Proc. 15tta Coastal Eng. Conf., ASCE, pp. 1113-1132. Vallianos, L., 1974. Beach fill planning - Brunswick County, North Carolina. Proc. 14th Coastal Eng. Conf., ASCE, pp. 1350-1369. Van de Graaff, J., 1983. Probabilistic design of dunes. Proc. Coastal Structures '83, ASCE, pp. 820-831. Vellinga, P., 1983. Predictive computational model for beach and dune erosion during storm surges. Proc. Coastal Structures '83, ASCE, pp. 806-819. Vellinga, P., 1986. Beach and dune erosion during storm surges. Delft Hydraulics Comm. No. 372, Delft Hydraulics Lab.