Beach erosion model (EBEACH) users manual

[From the U.S. Government Printing Office, www.gpo.gov]

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4590

no. 84-5-1

FOREWORD

This userss manual. describes the numerical. c:oreputer dmode
EBEACH wh i c:h i7 des i g.ned to est i mate the t i me-dlependent
beach-d-dune erosion associat:ed with severe stormo s:uch as
h ur r i cane.s or nor -'fealster s The u.sers mianua 1. ix present-'ed in
two rar'tS: volume I contaains a descril-ption of prograram Logic:
and variables used, vo lume II provildes a more 'horough
disctlss ion of background theory and mode l verificfiatlioni The
original. research pert'aining 'to the 'EBEACI-H model. was cronduc:tecl
by the author under the direct ion of Rober't G., Dean as part of
tlhe at th:or's iMarster or Ci i I. Engjineer ing idegree at 1the
Universil. ty of DeLaware Thle research phase was suppor'ted by
the National. Oc:eanic: anid A'tmospheric Admti i.stra t ion andi tfhe
Delaware .Sea Grantl CoLLegye P:rogram.

The present workl is pre se'nted in partial. Fui. f It ment of
co'ntractual obl.igations of the Federarl Coastal. Zone
Maniagement F'rogramn (siubject to provi.,s ioons of -the (:oasta
Zone Management Improvement Act of i97'72, as amended) subject
to prov i s i on s of corntract I M-S3'7 en t i t Ledl "Eni i neeer i n SU tpor t
E:nh ancement P:rogram .. Under provisioons .of DNR contract C0037,
thi is work i ; a subcon'trac'ted prodluc't of the Beaches and:l ,Sh'ores
Fs'esotrce Center, Institut'te of Science and F'lubl.ic Affairs,
FLor icda ,';tate Univer..::i'y 'A The doc umen't has be-en adopt'eci ds as
Beac:hes arnd Shores Technical. avid Design Menmorancdlum i n
accordance wi lh provisions of Chapter 6-B-33., l-Lor ida
Adm i n i nistrati ve C or:le.

A t t h e t i me of sb m i s i on f (:r c: on t1 r a c t r a . c omlp L i a nC: e,
James H.. Ba Lsi I Li e was ''the con'lract' manager and Administrator
of the .Anal.ys is/ReJsearch Sect i on, Hal. N, Eean was Chief of the
Bureau of Co:as.tal Data Ac:r. i s i t i onr, I)eborah E. F Lac: k 1) i re:c tlor
of the Division of Beaches and Shores, anvd Dr. lIl tcn J.,
C i ssendtlanner *the Executive Di rect or- of l' he F I-orida Dep ar t'mernt
of Natural l esources,

. .... ......... a.............

Deborah LE Fl.ack, Direc tor
Division of Beaches and Shores
I

+~~~~5~~ ~Jul y, '1984

-0

-
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TABLE OF CONTENTS

Chapter Page

I INTRODUCTION .............. ........ 1

II PROGRAM DESCRIPTION ................... 3

III VARIABLE LIST ...................... 6

3.1 Spatial Variables .................. 6
3.2 Temporal Variables ................. 8
3.3 Variables for Dynamic Solution ........... 8
3.4 Variables for Geometric Solution .......... 9

IV OCEANOGRAPHIC INPUT DATA ................. 10

4.1 Storm Surge Data .................. 10
4.2 Wave Height Data ................ 10

v INITIAL PROFILE REPRESENTATION ............. 12

VI BOUNDARY CONDITIONS ................... 17

VII DYNAMIC SOLUTION ..................... 20

7.1 Energy Dissipation and Sediment Transport ...... 20
7.2 Double Sweep Solution ................ 20

VIII GEOMETRIC SOLUTION .................... 22

IX OUTPUT ........................ 25

LIST OF FIGURES

Figure Page

1 Flow Chart for the EBEACH Simulation Model 4

Schematic Profile 14

3a Wide Berm Profile Type 15

3b No-Berm Profile Type 15

4 Discrete Stair-Step Approximation with Input Variables 16

5 Evolution of Offshore Portion of Profile Showing Change
in Slope at Breaking Depth 23

BEACH EROSION MODEL (EBEACH)

I. INTRODUCTION

This report describes the programming requirements of the EBEACH model

developed at the University of Delaware by David L. Kriebel and Dr. Robert G.

Dean. The background theory, numerical scheme, and model results are presented

in Volume II. This volume includes definitions of program variables, a description

of input requirements, and a summary of the programming considerations for each

subsection of the model.

As a brief review, the objective of this model is to determine time dependent

beach-dune erosion resulting from a severe storm surge. The model includes a

steady state solution of the equation of continuity for conservation of sand in

the onshore-offshore direction:

ax Qs
(1)

This equation is solved simultaneously with a sediment transport relationship

based on Dean's (1977) theory that the equilibrium form of the beach profile,

defined by the curve:

h = Ax2/ (2)

is the result of uniform wave energy dissipation per unit volume in the surf

zone. This transport equation:

Qs = K (D-Deq) (3)

explains the total sediment transport flux, Qs, in terms of a disequilibrium

between the actual energy dissipation, D, and the energy dissipation at

equilibrium, Deq. The transport parameter, K, relates the sediment transport

volume to the excess energy dissipation in the surf zone.

As input, the model requires a schematic beach profile representation that

includes a linearly sloping beach face, a uniform berm height, a variable berm

width, a linearly sloping dune face, and uniform dune height. With this initial

profile approximation, an actual or predicted storm surge hydrograph is read into

the model and the storm surge level at each time step governs the energy dissipation

per unit volume, the sediment transport flux, and the rate of profile change.

Also, an estimate of the breaking wave height as required to limit the width of

the active surf zone.

The output of the model consists of the time-history of:

(1) The horizontal retreat/advance of any contour elevation.

(2) The beach profile cross section.

(3) The total eroded volume per linear foot of shoreline.

These may be compared to the storm surge hydrograph to determine the relative ti-me

scales of profile changes and the water level variations.. In Volume II, the

discussion of the general model results show that, qualitatively, the model captures

the essential characteristics of the time dependent beach-dune erosion. Also, in

a comparison of model results to field measurements of dune-erosion during hurricane

Eloise, it appears that the numerical model gives good quantitative agreement with

observed erosion characteristics.

-2-

1I. PROGRAM DESCRIPTION

The flow chart for the EBEACH simulation model is shown in Figure 1.

As mentioned, the user must initially input the schematic beach-dune profile

characteristics, from which the discrete form of the initial profile is established.

The main program consists of a DO loop in which each iteration represents one

time step, simulating one-half hour of real time. At the beginning of each

iteration, the water level and wave height for the time step are entered into

the program from a data file. From the updated water level and wave height, the

surf zone and active onshore profile limits are established according to the

wave breaking depth and the-location of the new water level relative to the

onshore portion of the profile.

With the boundary conditions defined, the active profile is divided into

two regions:

(1) The dynamic region below the water line, in which profile changes

are governed by the calculated energy dissipation and continuity.

(2) The geometric region, above the water line, in which profile

changes are governed by continuity only.

In the main program, several DO loops are included to calculate profile

changes in the dynamic region. First, the energy dissipation per unit volume

and the sediment transport function are determined. Then, to solve the implicit

finite-difference form of the governing equation, a recursion relationship is

introduced, and a double-sweep solution is employed. Starting from an onshore

boundary condition for sediment transport, an offshore sweep is used to determine

coefficients in the governing equation; then, starting from an offshore boundary

condition, defining the limiting depth of profile change, an onshore sweep is used

to determine the horizontal change for each point in the profile up to near the

current water level.

-3-

Initial
Profile
Representation
l
Establ ish
Initial Discrete Display Time
Profile, Dependent
Elevation, and Contour Change,
Distance Volume Erosion,
Profile Cross
Section
Begin Simulation
ITime Steps I

Read Storm No Sulatio
Surge Level Completed

Update Water
Depth for /ecord Contour
Each Contour Change or
E Volume Erosion

T
etermine Update Contour
/ouetermi ne Positions for
Boundaryons Time Step

Apply Boundary
Calculate Conditions to
Energy Find Change
Dissipation, in Contour on
Sediment Beach Face
Transport l
4r--3
'V ~~~~~~~~~~~~~~~~~~~~~~~~~~~-J
Double Sweep Check Offshore
� Solution for - Slope, Seaward
Change in ]of Breakpoint
Contour Position

Figure 1 Flow Chart for the EBEACH Simulation Model

-4-

From this point landward, the profile is governed by continuity according to

changes in the offshore profile and the limits of onshore sediment motion. Similar

to storm surge-flooding models that "flood" land by opening and closing discrete

cells, this model activates erosion in portions of the onshore profile by opening

and closing sections of the comlposite beach-dune face, This procedure is

based on the elevation of the water line and the profile shape, principally depending

on the berm width. Offshore, at the breaking depth, a loop is included to geometrically

maintain a smooth transition between the active surf zone and the adjacent offshore

profile.

At the end of the main loop, the final profile changes over the time step

resulting from the dynamic and geometric solutions, are recorded as the incremental

retreat or advance of each contour line. This updated profile is then used as

the initial condition for the next time step and this process is repeated for the

duration of the storm surge.

III. VARIABLE LIST

3.1 Spatial Variables

N = Counter to identify the number of each grid point.

HI(N) = Initial contour elevation at each grid point in feet above mean

sea level (MSL); - if above MSL, + if below MSL.

Xl(N) = Initial horizontal distance from baseline to each grid point in

feet; positive when seaward of baseline.

H(N) = Updated contour elevation in feet, relative to current water level.

X(N) = Updated horizontal distance in feet.

DELX(N) = Incremental change in horizontal position of each grid point over

time step in feet; positive seaward.

WSEL(LTIME) = Storm surge elevation for current time step in feet above mean

sea level; stored in data file initially. Note: WSEL(LTIME)

may be stored either as exact elevation or rounded to next larger

0.5' increment. In program, each value is rounded to next larger

0.5' increment for use in erosion simulation.

WSELEV = Original WSEL(LTIME) at each time step.

WAVE(LTIME) = Breaking wave height at current time step in feet; stored in data

file initially.

WH = Constant wave height in feet; may be used in place of WAVE(LTIME).

DH = Vertical grid spacing, equal to 0.5'.

BDPT = Breaking depth in feet; equal to 1.3 times wave height.

HBERM = Height of berm or elevation of a break in slope from beach face to

dune face in feet; input as positive value, in data file.

HDUNET = Height of dune in feet above MSL; input as positive value in data

file.

-6-

XDUNET = Initial horizontal distance from baseline to crest of dune in

feet.

XBERM = Initial horizontal distance from baseline to crest of berm or

change in slope from beach face to dune face in feet.

XMB = Beach face slope; input as positive slope in decimal form.

XMD = Dune face slope; input as positive slope in decimal form.

NBERM = Grid point at crest of berm; remains constant for each simulation.

NSTOP = Grid point boundary condition where Qs = 0 on beach or dune;

always equals either NBERM or 1 depending on water level and berm

width.

NBREAK = Grid point boundary condition defining breaking depth offshore.

NFLAG = Counter to indicate width of berm; NFLAG : 0 if wide berm has eroded-

or-if no berm is present initially.

HSTARI = Initial depth of intersection between beach face slope and concave-

upwards equilibrium profile. From Table I in Volume II.

HSTAR = Depth of intersection at each time step; initially equals HSTARI,

changes to 0 when water level is on dune face and no berm is present,

changes to H(NBERM) when water level is on dune face and berm is

still present.

XCRIT
: Critical horizontal spacing used to check berm width, offshore slope.
XHCRIT

XDUNE = Change is horizontal position for contour elevation of top of dune.

X20

X15 Change in horizontal position of contour elevations at 20', 15',

X1i 10', and 5' above MSL, respectively.

XMSL = Updated horizontal position for contour elevation of original mean

sea level.

-7-

VOL = Change in volume for each vertical cell from initial position to

new position at current time step in ft. 3/ft.

SUMVOL(N) = Cumulative net change in volume from NMAX, shoreward to grid point

N. SUMVOL(N) must equal zero at top of dune for continuity to be

satisfied in profile.

VOLCHK = SUMVOL(i) or SUMVOL at top of dune; used as check on continuity,

must equal zero.

VOLERO = Volume eroded in yd.3/ft.

3.2 Temporal Variables

T = Real time counter in hours, increased in increments of � hour.

DT = Real time increment in seconds determined by stability criteria.

LTIME = Time counter in main program loop; increased in increments of I for

each � hour.

JTIME = Time counter determined by stability criteria such that MAX real

time simulated at JTMAX equals � hour.

Example - If DT = 1800 seconds, then JTMAX = i and JTIME counter

equals real time increment of � hour.

If DT = 300 seconds, then JTMAX = 6 and JTIME counter

equals real time increment of 1/12 hour.

3.3 Variables for Dynamic Solution

A = "A" parameter or shape factor in Ax2/3 profile shape. Determined

from mean sand grain size. Note: Due to discrete profile approximation,

profile shape is governed by finite difference form of energy

dissipation equation using equilibrium energy dissipation value.

Therefore, A value is input to determine DISSE but is not actually

used in calculations.

DISSE : Equilibrium energy dissipation per unit volume associated with sand

grain size and A parameter. From Figure 11 in Volume II, or equal

-8-

to 46.3 A 1.

KQ = Transport coefficient defining sediment transport volume in terms

of excess energy dissipation per unit volume. Determined by

Moore (1982) as 0.001144 ft. 4/lb.

KD = Energy dissipation coefficient equal to yK2/ or 55.24.

DISS(N) = Energy dissipation per unit volume occuring in cell between grid

points N and N-1.

QS(N) = Sediment transport flux between grid points N and N-1; initially

unsmoothed, smoothed for use in dynamic solution.

QI(N) = Working variable used in smoothing function.

QQ = Smoothed sediment transport flux at HSTAR; sediment transport curve

extended uniformly from QQ to zero at NSTOP in increments of QQQ.

NQQ = Grid point at HSTAR.

BDT

AN
Double sweep coefficients defined by DISS, QS, DT, DH, and
BN =
horizontal grid spacing.
CN

E(N) Double sweep coefficients defined by recursion formula; used to

F(N) calculate DELX.

K = Ratio of breaking wave height to breaking depth,

3.4 Variables for Geometric Solution

NSTAR2 = Grid point at HSTAR.

DELSUM = Change in cumulative volume required to satisfy continuity in

profile at end of time step NSTAR2 and N = 1,

HAVSUM = Change in cumulative volume at beginning of time step between

NSTAR2 and N p 1.

-9-

IV. OCEANOGRAPHIC INPUT DATA

4.1 Storm Surge Data

Two types of storm surge input may be used:

1) The actual measured storm surge hydrograph.

2) The calculated storm surge hydrograph resulting from a numerical storm surge

model.

For standard model application, the numerical storm surge predictions of

Dean and Chiu (1981) are input directly into the model. However, by following

the specified formats any storm surge hydrograph may be utilized. It should

be noted, however, that model does not allow dune overtopping. Therefore,

the model should only be used in cases where dune crest is higher than peak

storm surge elevation.

The water surface elevation must be listed at one-half hour intervals for

the duration of the storm surge. In the main DO loop, the time counter, LTIME,

is increased by 1 for each time step, to a maximum, LTMAX, which is determined

from the time history of the surge hydrograph. The real time counter, T, increases

by 0.5 hours eact time step. To supply the storm surge data, the water surface

elevation at each time step, WSEL(LTIME), is listed according to the format

specifications. Initially, when LTIME = 1, T = 0.5, and WSEL(1) = 0.0. Since

the model uses discrete vertical cell widths of 0.5 feet, the storm surge at any

time is approximated to the next greatest multiple of 0.5 feet. The current water

level is then compared to the previous water level to determine the incremental

change in water level, DELWS, at the beginning of the time step.

4.2 Wave Height Data

Two methods may be used to input the wave height:

1) The observed, hindcast, or forecast wave height can be read at each time

step from a data file.

2) A constant representative wave height can be used as a simple approximation.

-10-

As noted in Volume II, the exact determination of the wave height is of

secondary importance in the numerical solution. From Figures 17 and 18 and

Table 11 changes in wave height have less effect on the storm erosion than

changes in the water level . For storms of short duration, a constant wave

height of greater than 7 to 10 feet may often be used to obtain a quick estimate

of beach-dune changes.

Note: Wave heights of less than about 5 feet probably should not be used

in any simulation. For smaller wave heights, end effects associated with the

smoothing function applied to the sediment transport curve can significantly

change beach recession rates. For simulation of storm conditions where wave

heights are greater than 5 feet, these effects seem to be negligible.

V. INITIAL PROFILE REPRESENTATION

The schematic beach profile characteristics are depicted in Figure 2.

The offshore profile shape is governed by two parameters, the shape factor,

A, and the equilibrium energy dissipation per unit volume, DISSE. For a given

sand grain size, these values are determined from Figure 11 in Volume II or

may be determined by fitting the x, curve to the actual profile. Onshore,

the composite beach-dune profile is defined by the height and horizontal

location of each break in slope. Generally, a representative beach face slope,

XMB, and dune face slope, XMD, are determined for the profile of interest. Then,

the dune height, HDUNET, the berm height, HBERM, and the depth of intersection,

HSTARI, are specified. Note: Depths below sea level are defined as positive and

elevations above sea level are defined as negative. Unless otherwise specified

all elevations and distances must be given in feet, rounded to the nearest one-half

foot. Refer to Figure 2 for physical interpretation of each variable.

To establish initial horizontal locations of various points, the profile

must be classified into one of two types. In Figure 3a, if the profile has a

wide berm, two horizontal distances from the references baseline are required:

(1) the distance to the top of the dune, XDUNET; (2) the distance to the

top of the berm, XBERM. In Figure 3b, where no broad berm is present, only

the distance to the top of the dune, XDUNET, is needed. XBERM is set to equal 0.0.

To completely define the profile shape, it is necessary to consider a stair

step approximation of the profile, as in Figure 4. To provide a reasonable level

of detail in the erosion simulation, a uniform vertical spacing, DH, of 0.5 feet

is used. For the finite difference solution, the grid points, N, are considered

to be at the center of each vertical cell. Each cell is then identified by its

grid number; such that at the top of the dune, NOUNET, corresponds to N = 1.

-12-

Offshore beyond the limit of sediment motion, N = NMAX, Note that the dimension

size of the dimensioned arrays must be greater than NMAX. Several important

grid points, the top of the berm, NBERM, and the foot of the dune, NDUNEF, must

be specified; NDUNEF must always equal NBERM - 1.

To establish the intial profile shape, the elevation of each grid point

is first established. From the initial elevation, HI(N), the initial location

of each grid point, XI(N) is also determined according to the profile specifi-

cations above. While Hi(N) and XI(N) values do not change throughout the

program, two working arrays H(N) and X(N) are established to update the elevations

and horizontal positions for each time step.

Note: To simulate a wide berm profile, a GO TO control defines the

location of the grid point NBERM from the input value of XBERM.

To simulate a profile with no berm, XBERM is entered as 0.0.

This establishes the berm location at the change in slope from the

dune to the beach face.

It should also be noted that because of the finite difference solution

for energy dissipation, the offshore profile shape

h = Ax2/3 (4)

must also be approximated in a discrete fashion. To do this, the energy

dissipation equation must be used to determine the equilibrium horizontal

grid spacing, (Xn - Xn+l) based on the equilibrium energy dissipation, DISSE.

This approximation results in a slightly shallower profile than the contin-

uous Ax2/3 profile.

-13-

XMD

,\--Widt -W
XDUNET 1 I

XBERM XB

HSTAR I

Figure 2 Schematic Profile

XDUNET

XBERM

Figure 3a Wide Berm Profile Type

XDUNET

XBERMt = 0.0

Figure 3b No-Berm Profile Type

-15-

X =0
NDUNET=1
XDUNET

FDUINET

NDUNEF
N NIERIM
XBERIM

HBERM

HSTARI

NDUNET =
XDUNET

HDUNET Note: XBERM = 0.0

NOUNEF
NBERM

HSTARI

Figure 4 Discrete Stair-Step Approximation with Input Variables

-16-

VI. BOUNDARY CONDITIONS

As noted in Volume II, boundary conditions are required to define the

limits of sediment transport in the profile. Offshore, the breaking depth

defines the width of the surf zone and establishes a limit for offshore sediment

transport; thus, the principal function of the wave height in the model is

to control the offshore limit of the active profile. Onshore, the water

surface elevation relative to the berm height and berm width, from NDUNEF to

NBERM, define the shoreward limit of wave induced sediment transport. If a

wide berm is present in the initial profile, the model requires that the berm

be eroded back to the base of the dune before dune erosion begins. If the berm

has eroded back to the dune -or- if no berm is present in the initial profile,

the entire beach-dune face erodes uniformly for any water level rise. To

regulate the "opening" of the dune to erosion; two parameters are determined

at each time step, HSTAR and NSTOP.

HSTAR is defined as a transition depth between the dynamic and geometric

solutions. When the water level is below the berm elevation or the break in slope

elevation, HSTAR is always equal to HSTARI. Therefore, the beach face slope, XMB,

is always steeper than the steepest portion of the concave offshore profile. When

the water level is above the berm of break in slope elevation, HSTAR is determined

according to the width of the berm. If the berm is wide, HSTAR is temporarily

set equal to the water depth at the berm crest. This allows the profile

seaward of the berm to erode quickly and assume a concave shape. If the berm

has eroded back to the dune -or- if no berm is initially present, HSTAR must

be determined at the point of tangency between the dune slope, XMD, and the

concave profile. Since dune slopes are typically steep, 1:2 to 1:4, this

transition depth is usually at the water line; thus, HSTAR is set to equal

-17-

to zero. This results in a final profile shape in which the steep dune face

changes to the shallow concave profile at the still water line.

Whil-e HSTAR defines the transition between the dynamic and geometric

region, or between the linear and the concave profiles, NSTOP governs the

onshore limit of sediment transport. In particular, the sediment transport

curve is always extended linearly from the calculated value of Qs at HSTAR,

to zero at NSTOP. For cases in which the water level is below the berm height,

and when a wide berm exists, NSTOP is set equal to NBERM. This means that

Qs equals zero at the top of the berm. Similarly, if no berm is present and

the dune face intersects the beach face, NSTOP is set equal to I and the entire

beach-dune face erodes uniformly.

For the case in which the water level is above the berm height, the

location of NSTOP again depends on the berm width. If a wide berm exists,

NSTOP equals NBERM, allowing rapid berm erosion. Once the berm erodes, or for

cases in which no berm is present, NSTOP is set equal to I and the dune erodes

as a unit.

When the water level begins to decrease after the peak surge, the beach

continues to erode for a short time, reaching the maximum erosion some time

after the peak surge. Since, it is impossible to predetermine this time lag,

the program must allow the beach to either erode or accrete at each step.

In this case, HSTAR and NSTOP must be allowed to vary depending not only on the

water level, and berm width, but also on whether the profile is eroding or

accreting. In general, when the water level decreases, the same criteria

are used to assign values of HSTAR and NSTOP. However, it should be noted that

if: (1) a wide berm had eroded, or (2) if no berm is present initially, a

flag, NFLAG, is activated which sets NSTOP equal to 1 at each time step. Therefore,

-18-

the solution assumes that the entire profile will continue to erode as the

water level decreases.

As the water level continues to fall, erosion slows and eventually the

profile begins to rebuild. It is clear that if the numerical solution results

in dune rebuilding or accretion, this does not correctly model nature, where the

er'oded dune is not rebuil t by wave action. Therefore, if. DELX(1), at the top

of the berm, is ever found to be positive, indicating that the dune is rebuilding,

a GO TO statement transfers control back to the beginning of the dynamic solution

where HSTAR and NBERM and modified to "close" the dune from profile recovery.

Thus, the model simulates recovery of the berm only while the dune remains in

an eroded condition.

-19-

VII. DYNAMIC SOLUTION

7.1 Energy Dissipation and Sediment Transport
Calculation of the energy dissipation per unit volume and the sediment

transport flux is fairly straightforward and follows the governing equations

identified in Volume II. For cells seaward of NBREAK or landward of HSTAR,

the energy dissipation and sediment transport are set equal to zero. DISS(N)

and QS(N) are determined in the active surf zone only. Then, the simple

smoothing function is applied to the sediment transport curve, such that most

of the discontinuities are eliminated from QS and DISS. Also, the dynamic

solution is extended to two cells beyond the breaking depth to provide a

smoother transition to zero transport and zero energy dissipation outside the

surf zone. Onshore, energy dissipation is not defined from HSTAR landward;

however, the sediment transport curve is extended linearly from HSTAR to zero

at NSTOP. This establishes a smooth transition in the sediment transport

function at the depth, HSTAR,and determines the rate of sediment transport from

the beach face into the dynamic portion of the profile.

7.2 Double Sweep Solution
With the smoothed sediment transport function and the associated energy

dissipation per unit volume, the double sweep coefficients--.AN, BN, CN, ZN, E,

and F are easily determined. At N = 1, the onshore boundary condition that E = 0

and F = ZN. is applied. This in turn establishes E = 0 and F = ZN for all cells

landward of HSTAR. Offshore beyond HSTAR, E,and F are determined according to

equations given in Chapter III of Volume II. At the offshore limit, NMAX, the

boundary condition DELX(NMAX) = 0 is applied and DELX(N - 1) values are determined

according to the recursion formula. This onshore sweep proceeds up to the top

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of the dune and the governing equation:

An A Xn 1 + Bn A Xn + Cn A Xn+1 n (5)

is satified every where in the active profile.

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VIII. GEOMETRIC SOLUTION

Although the governing equation is satisfied at this point, the solution

must be modified to maintain a smooth profile offshore, to recede the beach-dune

face uniformly onshore,and to help guarantee a stable solution. First, because

sediment transport decreases immediately beyond the breading depth, the grid

point, NBREAK, grows seaward rapidly, as QS into the breaking cell is much

larger than QS leaving that cell. The result of this rapid growth is that the

contour at NBREAK tends to grow much more quickly than adjacent offshore contours.

Clearly some limit must be placed on the growth of NBREAK to prevent a discontin-

uous profile slope offshore.

To permit rational growth of NBREAK and adjacent offshore cells, a loop

is included that compares the actual horizontal grid spacing against some

critical spacing. This critical spacing could be set equal to the angle of

repose or some percentage of the beach face slope for convenience. The spacing

is set equal to the critical spacing by taking one half of the difference

from the cell at N and adding it to the cell at N + 1. In this way, the

offshore region is allowed to grow as further deposition occurs yet is

maintained at a constant slope. NBREAK remains at the break in slope, as in

Figure 5.

The geometric solution onshore is rather confusing in the computer program

but is conceptually fairly simple, in general, the beach face and dune face must

erode uniformly when activated and continuity must be satisfied such that there

is no gain or loss of sand from the system. To determine geometric profile

changes, HSTAR and NSTOP, are used to limit the geometric solution region.

Also, the cumulative change in volume over the profile is monitored to determine

the volume changes required to satisfy continuity.

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STORM SURGE LEVEL

INITIAL
EQUILIBRIUM
PROFILE

Figure 5 Evolution of Offshore Portion of Profile Showing
Change in Slope at Breaking Depth

The volume change offshore at NMAX is clearly zero, as this point never

changes location. Since the geometric limits on the offshore slope are imposed

previously, the change in volume in each contour from the original profile to

the profile at the end of current time step may be determined from NMAX onshore

to HSTAR. By cumulatively adding these incremental changes in volume, the

total volume change required to satisfy continuity at the top of the berm or

dune may be found as SUMVOL(NSTAR2 + 1) or DELSUM, where NSTAR2 is the grid

point at HSTAR.

On the beach-dune face from HSTAR landward, the change in volume in each

contour from the original profile to the profile position at the beginning of

the time step is known. These incremental volume changes may be summed to give

the net change in volume occuring over the previous time steps in the geometric

region. This value is labled HAVSUM. To determine the uniform beach-dune face

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recession necessary to satisfy continuity in the current time step it is a

simple matter to distribute the necessary volume, DSUM = DELSUM-HAVSUM, as

an incremental change in volume or position, for each cell between NSTAR2

and NSTOP. At this point, the geometric solution is complete, unless the

water level is decreasing. As mentioned, if the profile is in a state of

recovery, the dune face is not permitted to rebuild. Therefore, if the

solution results in uniform dune accretion, the solution is repeated with the

dune isolated and only berm recovery permitted. This is accomplished by the

check in statement labeled 193.

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IX. OUTPUT

At the end of each time step, JTIME or LTIME, the horizontal contour change

over the time step, DELX(N), is used to update the horizontal positions of

the grid points. From these updated values, the contour locations at the end

of the time step are recorded. These values are then compared to the original

profile position to determine the retreat/advance of each contour. The output'

consists of contour changes for the dune crest, the still waterline, and-the

5, 10, 15, and 20 foot contour lines. If the dune elevation is less than 15

or 20 feet, the output will record zero change for these contours.

The maximum value of SUMVOL, represents the total eroded or deposited

volume in ft3/linear ft. Therefore, from this maximum volume change, the
eroded volume in cubic yards per linear foot may be obtained. Finally, the

entire profile characteristics at any time step may be printed or, with the

proper software interface, plotted to graphically depict the profile change.

It should be noted that in the printed output, the original and current spacial

variable, X and H are given for the time step. Also, the energy dissipation,

sediment transport flux, incremental change in position, and cumulative volume

from offshore are provided.

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