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Title: Next generation beach and dune erosion model for applications of the Bureau of Beaches and Coastal Systems
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Title: Next generation beach and dune erosion model for applications of the Bureau of Beaches and Coastal Systems
Physical Description: Book
Language: English
Creator: Dean, Robert G.
Publisher: Coastal & Oceanographic Engineering Dept. of Civil & Coastal Engineering, University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: June 28, 2004
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General Note: Submitted to: Beaches and Shores Resource Center, Innovation Park, Florida State University
General Note: UFL/COEL - 2004/008
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Bibliographic ID: UF00091388
Volume ID: VID00001
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UFL/COEL -2004/008


NEXT GENERATION BEACH AND DUNE EROSION
MODEL FOR APPLICATIONS OF THE BUREAU OF
BEACHES AND COASTAL SYSTEMS






by



Robert G. Dean


Submitted to:

Beaches and Shores Resource Center
Innovation Park
Florida State University
Tallahassee, FL 32399


June 28, 2004








NEXT GENERATION BEACH AND DUNE EROSION MODEL
FOR APPLICATIONS OF THE
BUREAU OF BEACHES AND COASTAL SYSTEMS



June 28, 2004



Prepared by:

Robert G. Dean





Submitted to:

Beaches and Shores Resource Center
Innovation Park
Florida State University
Tallahassee, FL 32399






Submitted by:

Department of Civil and Coastal Engineering
University of Florida
Gainesville, FL 32611-6590







TABLE OF CONTENTS

1.0 INTRODUCTION ......................................... 1

2.0 MODEL DEVELOPMENT................................ 1

2.1 Grid Systems ........................................... 1
2.2 Bar Representation ..................................... 2
2.3 Effects of Wave Conditions on Equilibrium Beach Profiles ..... 5
2.4 Seawalls .............................................. 8
2.5 Overwash ............................................ 9

3.0 GENERAL DESCRIPTION OF THE BEACH AND DUNE EROSION
MODEL .......... ........................................ 10

3.1 Sediment Transport .................................. 10
3.2 Continuity Equation ................................ 12
3.3 Boundary Conditions ................................. 13
3.4 Wave Setup ....................................... 13
3.5 Wave Runup .........................................14
3.6 Solution of Sediment Transport and Continuity Equations .... 14

4.0 RECOMMENDATIONS FOR APPLICATION TO BBCS
ISSUES/PROBLEMS ........................................ 14

5.0 REFERENCES ......................................... 15










LIST OF FIGURES


Figure Caption Page

1. Grid Systems Used in Previous and Present Models .................. 2
2. Illustrating Torque Applied to Water Column by Breaking Wave ....... 3
3. Schematic Showing Suspension at the Crest Phase Position and
Subsequent Settling to the Sea Floor at the Fall Velocity ............... 4
4. Example of Profile Evolution Producing a Bar .........................4
5. Variation of Sediment Scale Parameter, A, With Sediment Size, D and
Fall Velocity, w.
Note to Convert A in mi"3 to A in ft1"3, Multiply by 1.5 ................. 5
6. Effect of Fall Velocity Parameter on
Near-Equilibrium Beach Profiles ...................................... 7
7. Ratio of Profile Scale Parameter (Including Effect of Fall Velocity
Parameter) to Value in Table 1 ....................................... 8
8. Example of Profile Evolution in the Presence of a Seawall............. 9
9. Preliminary Results From Overwash Modeling .......................10
10. Profiles Calculated From Equations With Two Terms
and a Single Term ....................................................11




TABLE

Table Caption Page

1. Summary of Recommended A Values.
(Units of A Parameter are in m1/3) ..... ......................... 6








NEXT GENERATION BEACH AND DUNE EROSION MODEL
FOR APPLICATIONS OF THE
BUREAU OF BEACHES AND COASTAL SYSTEMS


1.0 INTRODUCTION

The intent of this model development is to replace the previous beach and dune (B&D)
erosion models) developed for the Bureau of Beaches and Coastal Systems (BBCS) of
the Florida Department of Environmental Protection (FDEP). The processes associated
with beach and dune erosion are complex and the associated physics are not completely
understood. This scenario results in this area being under active research. The objective
was to incorporate more appropriate and complete physics into the new model and to
represent features not present in the previous model such that the model will be suitable
for a wide range of BBCS applications.

The previous model which is in current use by BBCS Staff was developed in the early to
mid-1980's and has been modified to a minor degree. The model was based on a
parameterized representation of the erosion processes. This had the advantage that the
model was stable; however, the disadvantage was that the model included a low level
representation of the physics and did not represent the following features, all of which are
present in nature and can be relevant to BBCS applications: (1) Offshore bars, (2)
Seawalls, and (3) Overwash over low lying features such as barrier islands. In order to
provide a more flexible model framework, an objective of the new model was to
incorporate a system gridded in the seaward direction rather that grids in the vertical
direction as is present in the current model. The advantage of grids in the offshore
direction is that this system is more suited for the representation of offshore bars. Other
differences in the current and new models will become apparent in the following sections.

A Users' Manual is available as a companion document to the present report and provides
details of the modal calibration and application to several types of problems.

2.0 MODEL DEVELOPMENT

2.1 Grid Systems

As discussed, the grid system in the current model in use by BBCS is one in which the
elements are increments of elevation as shown in Figure la. With the vertical grids
defined such that each one represents a different elevation, the model progresses to
update the horizontal position (y) of each element as it changes in response to wave
height and tide level. The inherent difficulty in representing bars with this type grid
system is, at the elevations intercepting a bar, there will be three horizontal y positions
for each vertical element. This complicates the bookkeeping of the three y values
considerably.








The grid system used in the present model is shown in Figure lb with the grids
represented by vertical elements and the model updating the changing elevation
associated with that element. The vertical elevation is defined in terms of depths relative
to the original vertical datum such as NGVD. Thus, elevations above the current water
level (including storm tides) are negative. As noted, the model development incorporates
grids in the cross-shore direction.





Yr-i ~ -


Yi

.a) Grid System Used in Previous Model.


a) Grid System Used in Previous Model


b) Grid System Used in Present Model


Figure 1. Grid Systems Used in Previous and Present Models.

2.2 Bar Representation

As noted, the physics associated with bar development are not well understood; however,
research over the past several decades has clarified bar features and physics substantially.
The earliest interpretations linked bar formation with deepwater wave steepness, Ho/Lo,
where Ho and L, are the deep water wave height and deep water wave length
respectively. Later theories developed other causes for the formation of offshore bars. At
least five potential causes were proposed two decades ago. It is now clear (at least to me)








that most bars in nature are so-called "break point" bars and the result of the vortex which
accompanies a breaking wave, see Figure 2. An empirical parameter which describes
whether a bar will form is the so-called "fall velocity parameter", defined as









Torque Applied to Water
Column By Breaking Wave

SBar .


Figure 2. Illustrating Torque Applied to Water Column by Breaking Wave.


HbIwT (1)

where Hb is the breaking wave height, T is the wave period and w is the sediment fall
velocity. Bars tend to be present for higher values of the fall velocity parameter. Bars are
relevant to shoreline recession since they cause wave breaking and result in less wave
action at the shoreline.

This fall velocity parameter as a determinant for bar formation was developed by Dean
(1973) with the following simple conceptual model. Consider a sediment particle to be
suspended under the wave crest phase position and falling with the sediment fall velocity.
If the sediment falls in less than one-half of the wave period, it can be shown that the
particle will experience a net shoreward motion whereas if the fall time is greater than
one half the wave period, the net sediment transport will be seaward. For the latter case, it
was hypothesized that bars would be formed. A graphical interpretation of the process is
shown in Figure 3 where the suspension height is assumed to be approximately
proportional to the local breaking wave height. For locations within the inner portions of
the surf zone, the suspension is relatively small (since both the local water depth and
local breaking wave height are small), thus the net sediment particle displacement is
landward. For larger and larger breaking waves, the suspension height is larger and larger
and for some breaking wave height, the net sediment particle is seaward and a bar is
formed. To correlate the model, the only "free" parameter was the elevation to which the
particle was suspended. For this purpose, Dean used a combination of laboratory and
field data which resulted in the following expression








H/WT i<0.85,No Bars
0 w[> 0.85, Bars Form J


Kraus, et al (1991) later examined field data and found the constant to be approximately
3.2.

With the above description as background, it is desirable to incorporate, in a rational
manner, the effects of the fall velocity parameter into the transport equation. A
substantial effort was directed toward this objective; however, no physics based approach
(ie, an approach based on the concepts illustrated in Figure 3) was found. Rather, the
effects were incorporated by altering the wave energy dissipation per unit volume, D*, in
accordance with the fall velocity parameter. An example of the profile yielded by
incorporating these effects into the transport equation is presented as Figure 4.


Figure 3. Schematic Showing Suspension at the Crest Phase Position
and Subsequent Settling to the Sea Floor at the Fall Velocity.


0 100 200 300 400 500 600
Horizontal Distance (ft)


Figure 4. Example of Profile Evolution Producing a Bar.











2.3 Effects of Wave Conditions on Equilibrium Beach Profiles

Prior to this project, the considerable work that had been conducted to quantify the
profile scale parameter, A in the EBP relationship


h(y) = Ay23 (3)
had resulted in a relationship between this parameter and the median sediment size, D.
This recommended relationship was first developed by Moore (1982) and was later
refined with more data added resulting in the graphical relationship presented in Figure 5
and summarized in tabular form the more common beach sand sizes (0.1 mm < D < 1.09
mm) in Table 1. However, it is well known that, in addition to sediment size, wave
characteristics play a role in the beach profile with large steepness waves (storm
conditions) causing milder profiles. A portion of this project effort was dedicated to
quantifying this effect of wave characteristics.


SEDIMENT FALL VELOCITY, w (cmls)
0.01 0.1 1.0 10.0 100.0


E.

En
11




in

rL:
0


1.0 Suggested Empirical
Relationship A vs. D _
(Moore) -O

From Hughes r A 0.067 w0-4
rom Individual Field Field Resullts t ,
rofales Where a Ring
I Sand Sizes was Glv n
0.10
.0r l Bsed L don rnsfonning
F O A vs. D Cu ve Using
SFll Veloci Rellatonship

From Swart '
Laboraory Results
0.01


0.01 0.1 1.0 10.0 100.0
SEDIMENT SIZE, D (mm)

Figure 5. Variation of Sediment Scale Parameter, A, With Sediment
Size, D and Fall Velocity, w. Note to Convert A in m1/3 to A in ft'3,
Multiply by 1.5.













Table 1. Summary of Recommended A Values.
(Units of A Parameter are in m1"3)


D(mm) 0.00 ,0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.1 0.063 0.0672 0.0714 00756 0.0798 0.084 0.0872 0.0904 0.0936 0.0968
0.2 0.100 0.103 0.106 0.109 0.112 0.115 0.117 0.119 0.121 0.123
0.3 0.125 0.127 0.129 0.131 0.133 0.135 0.137 0.139 0.141 0.143
0.4 0.145 0.1466 0.1482 0.1498 0.1514 0.153 0.1546 0.1562 0.1578 0.1594
0.5 0.161 0.1622 0.1634 0.1646 0.1658 0.167 0.1682 0.1694 0.1706 0.1718
0.6 0.173 0.1742 0.1754 0.1766 0.1778 0.179 0.1802 0.1814 0.1826 0.1838
0.7 0.185 0.1859 0.1868 0.1877 0.1886 0.1895 0.1904 0.1913 0.1922 0.1931
0.8 0.194 0.1948 0.1956 0.1964 0.1972 0.198 0.1988 0.1996 0.2004 0.2012
0.9 0.202 0.2028 0.2036 0.2044 0.2052 0.206 0.2068 0.2076 0.2084 0.2092
1.0 0.210 0.2108 0.2116 0.2124 0.2132 0.2140 0.2148 0.2156 0.2164 0.2172
Notes:
(1) The A values above, some to four places are not intended to suggest that they are jmiwn to that
accuracy, but rather are presented for consistency and sensitivity tests of the effects of variation in
grain size.
(2) As an example of use of the values in the table, the A value for a median sand size of 0.24 nun
is: A = 0.112 m". To convert A values to a3, multiply by 1.5.

Figure 6 presents three near-equilibrium profiles from Saville's (1957) large wave tank
tests. These profiles commenced with a uniform beach slope of 1:15 and were run for
various times to near equilibrium with the final run time of approximately 40 hours.. The
fall velocity parameters are shown on each of the three profiles and range from 4.94 to
13.51. The role of the fall velocity parameter on the bar formation is immediately obvious
with the bar accentuation increasing with increasing fall velocity parameter. Less
apparent is the effect of this parameter on the profile landward of the trough. This aspect
was investigated by fitting a least squares profile of the form of Eq. (3) to each of the
profiles over the range from the mean water position out to the apparent most landward
location of the bar/trough influence. This resulted in a factor which depends on the fall
velocity parameter as shown in Figure 7. These results demonstrate that higher FVP will
have smaller A parameters and thus will have milder slopes and thus the beach will be
more eroded compared with a wave of the same height but with a longer period. These
results are in accord with observations in nature where longer period waves tend to
transport sand shoreward and widen the dry beach and short period high waves transport
sand offshore and form a bar. The results are also consistent with seasonal shoreline
beach changes.




















Case 300 H/wT 4.94


2

I0

1-2

-4

_,


-40 -20 0 20 40
distance [m]
Case 400 H/wT 9.64


2



-2

5-4


-40 -20 0 20 40
distance [m]
Case 500 H/wT= 13.51


-40 -20 0 20 40 60 80
distance m]

Figure 6. Effect of Fall Velocity Parameter on Near-Equilibrium Beach Profiles.


60 80


60 80


-.. .
S......... ... .. .. ..... ........ '........ ........

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


....... ........ . ... .... ... . ....... .......
. . ; ...... ", ...... .. ...

.... . . .. ..... ....... ... . . .
; 5 ,,,
o.. . .... ... ....0 .... ,. .. 0
.S. ;


!~ ~. ............... ................ ... ......

...... . . .

.........~...................'


RI


AI


IIIII


A



























0 2 4 6 8 10 12
Hb&T



Figure 7. Ratio of Profile Scale Parameter (Including Effect of Fall Velocity Parameter}
to Value in Table 1.




2.4 Seawalls

As noted, an objective of the model development was to include the capability to
represent seawalls which limit the extent of landward recession. This issue is of interest
to BBCS as many of the beaches in Florida are backed by seawalls and may be activated
during storms. During storms, seawalls are known to result in a phenomenon termed "toe
scour" which results in a locally deepened area in the immediate vicinity of the seawall.

Figure 8 presents an example of a computer simulation of profile evolution in the
presence of a seawall.
















ii i i .. .... 40 Hours



--.......-........



.1 ........... ........I.: ......-.. -........... -7-.--- .........
'Ti



-20
100 200 300 400 500 600
Horizontal Distance (ft)

Figure 8. Example of Profile Evolution in the Presence of a Seawall.


2.5 Overwash

During periods of large storm surges acting on low topography, the process of overwash
can occur with sand being transported landward and deposited as a layer or plaque termed
a"washover deposit". In some unusual cases, these deposits can be up to 6 feet thick and
in Hurricane Opal (1995) thicknesses of 4 feet were reported. This landward transport is
sand extracted from the nearshore system and results in an additional recession of the
shoreline.

As of this date, the overwash portion of the program has not been verified. Figure 9
presents an example of the overwash results. At hour 1, landward transport had caused a
deposit that advanced the shoreline seaward. However, for the remainder of the times
plotted, the initial deposit had eroded and the shoreline had reverted back to near its
initial landward position. The final profile reflects net landward transport at the landward
limits of the active profile.


































Cross-shore Distance (ft)

Figure 9. Preliminary Results From Overwash Modeling.

3.0 GENERAL DESCRIPTION OF THE BEACH AND DUNE
EROSION MODEL

This section describes the conceptual elements of the beach and dune erosion model.
Information relating to application of the model are presented in a Users Manual. In
general, the beach and dune erosion model consists of the schematization of the following
two processes in nature: a sediment transport equation and a continuity equation which
keeps track of or accounts for the differences of sediment transport into and out of an
element. In the model, these two equations are solved simultaneously. The equations
governing these two processes are described and presented below.

3.1 Sediment Transport

Cross-shore sediment transport, Q, is represented by


Q=KID-D, (M-) (D-D.)


In which Q, the cross-shore sediment transport, is positive seaward, K is a sediment
transport coefficient, D is the wave energy dissipation per unit volume, M is an exponent


10


0 Hours
i1 Hour
- .. 5 Hours
--- 10 Hours
--- 20 Hours
.- 40 Hours


Level = 7 feet


Storm Water





.......:... .......... ..i~ .................:

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


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


i i i i


........ ...


1







in the transport equation and governs, in part, the rate of profile evolution, and D. is the
equilibrium value of the wave energy dissipation per unit water volume. In the present
report, M is included as a user defined variable. In past applications, D. has been taken
as a constant; however, here D* will be varied across the profile to represent bar
formation and toe scour. The quantity D is expressed by


5 h3/2 D. ah
D= pg1.5 2a D+ (5)
24 ay m ay

Inserting Eq. (5) into Eq.(4) and setting the transport equal to zero results in the following
equilibrium beach profile (EBP)

= 5 pg.5Z h3/2 + 1h (6)
24 D. m

For small water depths (and on the beach face), the second term dominates, resulting in

h = my (7)

where m is the beach face slope. Eq. (7) is consistent with the observed uniformly sloping
beach face. For deep water, the second term dominates resulting in


h(y) = [524D ] 2/3 = Ay2/3 (8)


consistent with Eq. (3) and earlier findings on EBPs.

Figure 10 presents an example of a profile calculated from Eq. (6) with A = 0.19 ft1/3 and
m = 1:30 (solid line) and a profile calculated from Eq. (8) with an A value of 0.14 ft1/3
(dotted line).
















. -5

C
0

I
-10






-15


0 100 200


300 400 500 600 700 800 900 1000


Cross-shore Distance (ft)

Figure 10. Profiles Calculated From Equations With Two Terms and a Single Term.

3.2 Continuity Equation

The continuity or "conservation of sand" equation is expressed in differential form as


Ah a3Q
at y


And in finite difference form as


h+l' = h, + [Q+in1/2 Q+1/2]
Ay


(10)


Where the sediment transport value, Q, apply at the grid lines and the depth, h, value
apply at the grid element centers. This equation essentially carries out the bookeeping of
differences of sediment transport into and out of an element.







12


Witf Tm Terms
......... OnrOnemTerm











3.3 Boundary Conditions


With the governing equations specified, the formulation is completed by the specification
of boundary conditions. The seaward boundary condition is that water related transport is
zero seaward of the breaking depth, hb, defined in terms of the breaking wave height, Hb,
as

hb = H /K = 1.28Hb (11)

The landward boundary condition is that water related transport is zero landward of the
wave runup. In addition to the water related transport (ie, governed by the transport
equation Eq. (4)), the possibility exists of a slope occurring which is greater than realistic
at either the seaward or landward ends of the computational domain. Basically, this
process of relaxing an oversteepened profile is treated by specifying maximum values of
seaward and landward slopes, which if exceeded, will result in "avalanching" such that
the limiting slopes are restored. The process of this restoration is geometric rather than
based on physics. Thus, after each time step, the slopes are checked across the profile
region to determine whether they exceed the maximum limits, and if exceeded locally,
avalanching occurs such that the slopes are restored to the limiting values.

3.4 Wave Setup

Wave setup is the increase in water level across the surf zone due to the transfer of
momentum from the waves to the water column. Conceptually, this transfer is the same
as when wind transfers momentum in the form of a wind stress to the water column
resulting in a "tilt" to the water surface which, when integrated across the continental
shelf, results in the storm surge. The transfer of wave related momentum to the water
column also results in a tilt to the water column. Basic simplifying assumptions establish
that the wave setup i(h) at a particular water depth, h, is given by

5 3 2)
(h) 16 128 2 3 8 (12)
1+321/8 1+3K2/8

If the usual value K = 0.78 is applied,

7](h)= 0.148hb -0.198h (13)

In most BBCS applications, for example the 100 year surge, the wave setup is included in
these surge values and thus addition of wave setup would constitute "double" counting
the setup estimates and thus setup should not be added to the storm surge values.
Therefore in the program, no provision is provided for calculating wave setup.









3.5 Wave Runup


Wave runup occurs on the beach face and here is considered to extend the landward limit
of water related transport. Wave runup, R, was calculated according to the Hunt (1959)
equation


R=FH mef (14)
R Hb I L(

In which meff is the effective slope for wave runup calculations here and is defined as the
average slope between the breaking depth and the runup limit, FR is a runup coefficient
taken as unity in the program, Hb is the breaking wave height and Lo is the deep water
wave length.

3.6 Solution of Sediment Transport and Continuity Equations

The solution of the two equations is detailed and for purposes here, only the generalities
associated with the solution are discussed. The two equations are solved simultaneously
by a so-called "Double Sweep" method. The equations are solved for a specified time
increment, At, which updates the profile form and then the equations are solved
repetitively. If the total time is sufficiently long, the resulting profile should approach an
equilibrium. Figure 4 presents the evolution from an idealized initial profile toward
equilibrium. Various experiments with the run times have established that time
increments of 30 seconds are appropriate for most cases.

4.0 RECOMMENDATIONS FOR APPLICATION TO BBCS
ISSUES/PROBLEMS

In applications for BBCS applications/problems, the following values are recommended:

Sediment Transport Coefficient, K = 0.005

Exponent in Transport Equation, M = 2.0

Onshore Limiting Slope:, Slopeon = 3.0

Offshore Limiting Slope, Slopeoff = 0.15

Beach Face Slope (associated with erosion) = 0.5

Time Increment, At = 30 seconds








5.0 REFERENCES


Dean, R. G. (1973) "Heuristic Models of Sand Transport in the Surf Zone", Proceedings,
Engineering Dynamics in the Surf Zone, Sydney, pp. 208 214.

Kraus, N.C., M. Larson and D. L. Kriebel (1991) Evaluation of Beach Erosion and
Accretion Predictors", Proceedings, Coastal Sediments '91, American Society of Civil
Engineers, Seattle, pp. 572 587.

Moore, B. L. (1982) "Beach Profile Evolution in Response to Changes in Water Level
and Wave Height", MCE Thesis, Department of Civil Engineering, University of
Delaware, 162 pages.

Saville, T. (1957) "Scale Effects in Two- Dimensional Beach Studies", Proceedings, 7th
General Meeting, International Association of Hydraulic Research, pp. A3-1 A3-10.




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