Citation
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
Creator:
Dean, Robert G.
Place of Publication:
Gainesville, Fla.
Publisher:
Coastal & Oceanographic Engineering Dept. of Civil & Coastal Engineering, University of Florida
Publication Date:
Language:
English

Notes

General Note:
Submitted to: Beaches and Shores Resource Center, Innovation Park, Florida State University
General Note:
UFL/COEL - 2004/008

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University of Florida
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University of Florida
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All applicable rights reserved by the source institution and holding location.

Full Text
UFL/COEL -2004/008

NEXT GENERATION BEACH AND DUNE EROSION MODEL FOR APPLICATIONS OF THLE 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 in 13 to A in ft"13, Mvultiply 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 aSingle Ten.. ............................................................... 11
TABLE
Table Caption Pg
1. Summary of Recommended A Values.
(Units of A Parameter are* in in"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 model(s) 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.
Yi-I hi-

Yi
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 I, 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
Ba r .
Figure 2. Illustrating Torque Applied to Water Column by Breaking Wave.
HblwT (1)
where Hbis 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'w {4 < 0. 85, No Bars
[/f 0.85, Bars Form

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 (if)
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) = Ay2/3 (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

0.
El
.,
E o1
in

1.0 Suggested EmpIrcal "
Relationship Avs. D (Moore) -O
From Hugher -A 0.067 wo-4
From Individual Field F ie.ld Results
roles Whare a Rasng
31 Sand Sizes was Glv n
0.10 1ndll'lll .... BIdollioln
I 1 8asedoan 1renstonning f A vs. D CUl we Using
Faill Veloci V Rela*tnhp
0.01 Labortory_____

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 in1/3 to A in ft1"3, Multiply by 1.5.




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

D(mm) 0.00 0.01 0.02 0.03 00 0.05 0.06 0.07 0.08 0.09.
0.1 0.063 0.0672 0.0714 0.0756 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 0.1.21 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 0.2094 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) Mre A values above, some to four places are not intended to suggest that they are kgnon to that
accuracy, but rather are presented for consistency and sensitivity tests of the effects of variation in
pai size.
(2) As an example of use of the values in the table, the A value for a medtian sand size of 024.u
is: A = 0.112 mII. To convert A values to W5,3 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 13.5 1. 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
0
j-2
-.4

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

2
-2
6-4

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

-40 -20 0 20 40 60 80 dFftance Ne]
Figure 6. Effect of Fall Velocity Parameter on Near-Equilibrium Beach Profiles.

60 80

60 80

......... ... ..... ........ ......... ........ ........
.. ...... .. ... ,,.

........ . . . ... . . .. .. .. .. .. .. ...
. . ; ...... ", ...... .. ...
. ...... ........ .....
* ; 5 ..,
... . .... ... ....0 .... . .. .. 0
.S. ;

.... .. ... . . .... .. .. ... .. i.. .. . . .. .. ...
... .. .. ..
....... .......... ............... ..... .... ..

RI I I .

AI

IIIII

A




0 2 4 6 a 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.




0~~~N ---'--- -- --
.. .. .. ... -------I10 ......... . . . . . . . . u r
-201
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.




10Storm Water Level =7 feet
. . . . . . .. . . . . . . .
0
- - - - --- .. . . . . . . ; . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . .
-10
-20
0 100 200 300 400 500 600
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 = K ID_-D. M-, (D -D.) (4)
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




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
D=_. 9pg1.5lK2ah_. D+ Ah (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)
Y= 5 pg.5 K2 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) = [52 K ]2/3 = 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 ft"13 and m = 1:30 (solid line) and a profile calculated from Eq. (8) with an A value of 0.14 ft1/3 (dotted line).




0
i i : :: : :: i :: :: : :: : IW ith Two Terms I :
W -5o
0
-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 h Q (9)
at a
And in finite difference form as h I At [Qn+1 /2 Q+112i (10)
hi+'= ,,A+ y ,, + -i ](0 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




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 =Hb / K =.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 ij(h) at a particular water depth, h, is given by
5-3 2)2
16 128 '2h 3K__/8
11()h=bK T- (12)
1+3K 2/ 8 1+3K 2/ 8
If the usual value K = 0.78 is applied,
il (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 =FRHb mff (4
R Hb/ILo (4
In which mff 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 L, 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.