HomeMy WebLinkAboutSUP 06-10X2A; AGUA HEDIONDA OUTER LAGOON MAINTENANCE; BEACH EQUALIBRIUM ANALYSIS OF MIDDLE AND SOUTH BEACH DISPOSAL OPTIONS FOR DREDGED SANDS FROM AGUA HEDIONDA LAGOON; 2017-07-07Beach Equilibrium Analysis of Middle and South Beach Disposal Options for
Dredged Sands from Agua Hedionda Lagoon, Carlsbad, CA
Submitted by:
Scott A. Jenkins, Ph.D.
Dr Scott A. Jenkins Consulting
14765 Kalapana St
Poway, CA 92064
Submitted to:
Sheila Henika, P.E. MBA-TM
NRG Cabrillo Power Operations Inc.
Encina Power Station
4600 Carlsbad Blvd.
Carlsbad, CA 92008
Draft: 7 July 2017
Cabl-02
Cab 1-01
sand remaining@ t = + 90 days
Middle+ South Beach= 173,053 cubic yards
North+ Middle+ South Beach= 214,994 cubic yards
Executive Summary: A detailed set of beach profile surveys at Middle and South Beach in
Carlsbad CA were provided by Cabrillo Power I LLC, delineating beach surfaces before and
after the 2104/2015 dredging of Agua Hedionda Lagoon, (AHL), which placed 229,693 yds3
between the south inlet jetty to Agua Hedionda Lagoon and the Encina Power Station (EPS)
generating building. The surveys were accurately performed by Noble Engineers using
differential GPS and known historic benchmarks. Three-dimensional CAD models were lofted
from the measured points along the three (3) Middle Beach survey range lines (Cab 1-04 -Cab 1-
06) and three (3) South Beach survey range lines (Cab 1-01 -Cab 1-03) to delineate the beach
surfaces immediately before beach dredge disposal (based on the 22 December 2014 profile
measurements) and immediately after dredge operations were completed (based on the 17 April
2017 profile measurements). When these two surfaces were lofted together in a common
reference frame, it was determined that 178,584 cubic yards of beach fill have been retained after
placing 229,693 cubic yards on Middle and South Beach between 1 January 2015 and the post-
dredge survey on 17 April 2015. This calculates to an average sand loss rate of 473 yds3/day and
projects a sand retention time of 485 days. This is significantly longer than the retention time at
the North Beach disposal site where retention time projected to only 33 days.
To understand the reasons for the contrasting sand retention characteristics of North
Beach vs. :Middle and South Beach, a baseline beach evolution study was conducted using the
Coastal Evolution Model (CEM) to hindcast the fate of beach fill placed on the three receiver
beaches. The CEM was developed at the Scripps Institution of Oceanography with a $1,000,000
grant from the Kavli Foundation, (see http://repositories.cd1ib.org/sio/techreport/58/ ), and is
based on latest thermodynamic beach equilibrium equations published in the Journal of
Geophysical Research. Inputs to the CEM baseline study were based on measured shoaling wave
data, grain size data for the dredged sands, and daily beach fill volumes were derived from the
dredge monitoring reports to the Regional Water Quality Control Board (Cabrillo, 2015) and
from Cabrillo dredging data bases.
Between 1998 and 2015 there have been eight (8) different events when Agua Hedionda
dredged sands have been disposed concurrently on all three receiver beaches (North, Middle and
South Beach). Historic dredged sand volumes placed on North Beach ranged from 62,030 yds3 to
161,525 yds3, while beach fill placed on Middle and South Beach ranged from 163,996 yds3 to
281,195 yds3 . The CEM beach evolution simulations of these events determined that the
minimum sand loss rate occurs when beach fill volumes on the three receiver beaches are equal
to the critical mass, which was found to be Vc,u == 79,471 yds3 for North Beach, (see companion
North Beach report, Jenkins, 2017), and Vc,it == 200,890 yds3 for Middle and South Beach. The
critical mass is the theoretical maximum carrying capacity of a beach fill site for supporting a
beach profile in equilibrium. The carrying capacity of a beach is limited by the width of the
wave-cut platform in the bedrock on which beach sands have accumulated over geologic time
scales. The wave-cut platform at North Beach is only 550 ft. to 600 ft. in width and 600 ft. to 650
ft. at Middle and South Beach. Many of the beaches throughout north San Diego County are
perched on narrow wave-cut platforms. The platforms are narrow because they were carved by
wave action into erosion resistant bedrock formations during the present high-stand in sea level,
and these narrow wave-cut platforms physically cannot hold large quantities of beach sand; and
often become fully denuded during periods of high-energy winter waves.
Another contributing factoi to the limited cairying capacity of the three receiver beaches
is that they are exposed to a prevailing negative divergence of drift caused by the way the
bathymetry surrounding the Carlsbad Submarine Canyon produces variable wave shoaling along
the length of these beaches. The presence of the Carlsbad Submarine Canyon creates a bright
spot in the shoaling wave pattern that diminishes in intensity with increasing distances toward
the north. For example, wave heights are locally higher at the inlet jetties than further to the
North around Maple Avenue. The prevailing littoral drift transports beach sand southward
throughout the entire Oceanside Littoral Cell; but the alongshore imbalance in shoaling wave
height causes higher southerly longshore transport rates of sand at the southern end of each of the
receiver beaches than at the northern ends. Consequently more sand exits each receiver beach at
its southern end due to longshore transport, than enters at the northern ends from sand sources
further updrift. This inequality in sand transport rates between the north and south ends of the
receiver beaches is referred to as divergence of drift, and when the sand transport rates are higher
at the down-drift end of the receiver beaches, it becomes a constant loss system referred to as
negative divergence of drift. So, when beach fill volumes exceed the critical mass of the receiver
beaches, the excess sand cannot be supported in equilibrium on its narrow wave-cut platform and
is quickly lost to the negative divergence of drift. However, this effect is somewhat muted at
Middle and South Beach because the AHL inlet jetties and the EPS discharge jetties produce a
groin field which impedes the longshore transport at both the updrift and downdrift ends of the
Middle/South Beach complex, and provides extra storage capacity for sand on the wave cut
platform. Consequently retention times for beach fill on Middle and South Beach is longer than
at North Beach.
Historically, the CEM baseline study finds that when a standard 1: 10 (rise over run)
beach fill template on North Beach is filled to critical mass, the theoretical minimum sand loss
rate to negative divergence of drift is 1,495 yds3/day, and the sand retention time is 53 days (see
the companion North Beach report, Jenkins, 2107). By contrast, the standard 1: 10 beach fill
template at Middle and South Beach historically achieved minimum sand loss rates of 431
yds3/day, and the sand retention times of 466 days. But, when beach fill sand volumes at Middle
and South Beach were increased by 41 % over critical mass ( as occurred during the 2000/2001
dredge event when 281,195 yds3 were placed on Middle and South Beach), the retention time is
only increased by 7 % from T0 = 466 days to T0 = 497 days. In contrast, over-filling the North
Beach receiver site produces an even worse return on beach fill investment. During the
2002/2003 dredge event, 161,525 yds3 were placed on North Beach, (103% increase over critical
mass), but the retention time increased by only 26 % from T0 = 53 days to T0 = 67 days, while
the sand loss rate increased by 61 % to 2,411 yds3/day. This is an increase in sand loss rates at
North Beach of916 yds3/day. Unfortunately, such increases in sand loss rates at North Beach
correlate with proportional increases of sand influx rates into Agua Hedionda Lagoon.
The 2010/2011 survey data show that AHL sand loss rates also increase when the fill
volumes are less than the critical mass. Sand influx rates in 2010/2011 were 519 yds3 /day when
only 163,996 cubic yards were placed on Middle and South Beach (36,894 yds3 below critical
mass requirements). Bear in mind that the critical mass is the minimum volume of sand required
to establish an equilibrium beach profile on a wave-cut platform; and a beach is in its most stable
state with an equilibrium profile. But with a prevailing negative divergence of drift along Middle
and South Beach, equilibrium cannot be achieved due when there is insufficient sand volume,
and consequently sand loss rates increase with a destabilized, non-equilibrium profile.
Following CEM beach evolution analysis of the Middle and South Beach historic
baseline, attention was given to finding a more effective beach fill template that could increase
sand retention using beach fill from Agua Hedionda Lagoon dredging. Beach fill has typically
been placed on Carlsbad beaches using a standard beach fill template with a flat backshore
platform and a 1: 10 (rise over run) seaward facing beach slope extending down to O ft. MLL W.
This convention dates back to the Regional Beach Sand Project, (AMEC, 2002). However,
stable beach profiles in Nature have a much more gradual, curving profile with slopes that range
between 1 :50 to 3: 100. Formulations of equilibrium beach profiles are found in the U.S. Army
Corps of Engineers Shore Protection Manual and later the Coastal Engineering Manual; and the
latest most advanced formulation is known as the elliptic cycloid. The elliptic cycloid
formulation can account for continuous variations in the equilibrium beach profile due to
variability in wave height, period and direction when occurring in combination with variations in
beach sediment grain size and beach sand volume. Therefore, a new beach fill template has been
proposed here for Middle and South Beach referred to as the cycloid-dune template (see Figures
ES 1-6). The shape of the template is based on the extremal elliptic cycloid which is the
equilibrium profile for the highest wave in the period of record. But the extremal elliptic cycloid
extends below the MLL W tide line and earth moving equipment which spread out the beach fill
cannot work below MLL W. So, the template truncates the extremal elliptic cycloid at MLL W
and places the residual volume of critical mass (totaling 49,680 yds3) in a back-beach dune that
stretches 3,680 ft. from the south inlet jetty to Agua Hedionda Lagoon to the north end of the
EPS generating building.
While an elliptic cycloid is an equilibrium beach surface, it does not produce a state of
zero sand loss in the presence of a negative divergence of drift, which is the persistent littoral
drift state along Middle and South Beach. When the divergence of drift is negative, the
equilibrium cycloidal beach profile will progressively shift landward as it loses sand to negative
divergence of drift, eventually intersecting the basement surface of the critical mass envelope.
Once this happens, then the cycloidal shape of the profile is disrupted, and the equilibrium state
of the profile is lost. The concept behind the cycloid-dune template is that, as the cycloid begins
to approach an intersection with the basement surface of the critical mass envelope, (under the
erosional effects of continued negative divergence of drift), it also intersects the base of the dune
and receives additional sediment cover as the dune erodes and spreads out downslope across the
still intact cycloidal surface. Thus; the dune acts as a restoring mechanism that re-supplies the
cycloid with sand lost to negative divergence of drift.
The construction method envisioned for the cycloid-dune template begins with building
the back-beach dune portion first, starting at the south inlet jetty and adding sections to the
dredge pipeline until the build-out of the dune extends beyond the South Groin abeam of the
north end of the EPS generating building. Building the dune first creates a "safe" reservoir of
sand before the template can be fully constructed, and sand from this reservoir is only released to
the lower eroded basement surface during periods of the highest tides and waves. After the
buildout of the dune to the southern end of South Beach, the cycloid portion of the template is
laid out beginning from the toe of the dune and spreading the material down slope to MLLW,
and working back across Middle Beach to the south inlet jetty, removing pipeline sections as the
cycloids are completed
CEM beach evolution simulations of the Middle and South Beach cycloid-dune templates
were run for future conditions with the South Groin removed, and show significant
improvements in sand loss rate and retention time relative to the historic baseline. Again, the
most efficient use of Agua Hedionda dredged sands occurs when the cycloid-dune template is
filled to no more than critical mass (200,890 yds3 for Middle and South Beach), which reduces
average sand loss rates on Middle and South Beach to an absoiute minimum of 306 yds3/day,
while extending retention time to 656 days. This is a 35% improvement in sand retention time
over historical dredge disposal practices at Middle and South Beach. If the cycloid-dune template
is filled to more than critical mass by adding more sand to the back-beach dune, then Middle and
South Beach retention times will increase beyond 656 days. If the reserve sand volume in the
dune on Middle and South Beach were increased by a factor of 2.6 to 129,985 yds3 (producing a
cycloid dune equivalent to the historic maximum placement volume of V0 = 281,195 yds3) then
retention time could be extended to a maximum of 693 days. But, again, this is not a good return
on doubling the investment in reserve beach fill placed in the back-beach dune because retention
time is only increased by 5 weeks while the sand loss rate on Middle and South Beach would
increase by 33% to 406 yds3/day, (an additional 100 yds3/day of sand loss). This inefficiency
occurs because the enlarged dune encroaches further seaward into the middle bar-berm portion
of the profile that is subject to more frequent wave attack, and the groin field formed by the inlet
and discharge jetties at Middle and South Beach is already filled to carrying capacity at the
critical mass of vcril = 200,890 yds3•
On the other hand, under-filling the cycloid-dune template, (by building a reduced dune),
leads to accelerated sand loss rates and reduced retention times. If the Middle and South Beach
cycloid dune templates were filled with the historic minimum beach fill of V0 = 163,996 yds3,
(by under-building the back-beach dune with only 12,786 yds3) then sand loss rates would
increase to 375 yds3/day and retention times would be reduced by to T0 = 437 days, a 33%
reduction in sand retention time relative to the ideal build using beach fill equal to critical mass.
The prevailing negative divergence of drift across Middle and South Beach causes the initial
cycloid profile in the lower portion of the template to shift landward, and once intersection with
the basement surface of the critical mass envelope occurs, there are insufficient sand reserves in
the reduced dune to resupply the cycloid in the presence of continued negative divergence of
drift. Once the reserve sand supply in the dune is exhausted, the cycloidal shape of the profile is
disrupted, and the equilibrium state of the profile is lost. Even so, if the cycloid-dune template on
Middle and South Beach were filled to a volume equivalent to the 2104/2015 disposal event ( V0
= 229,693 yds3) by using a dune containing only 78,483 yds3, then sand retention times are still
significantly better than what was achieved using the standard 1: 10 (rise over run) template.
With this over-built dune in combination with the cycloid, retention times following the
2014/2015 dredge cycle could have been T0 = 676 days with sand loss rates reduced to 340
yds3/day, an improvement of 39% over what was achieved using standard Middle and South
Beach disposal practices.
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22 December 2014 (most eroded historic profile)
Cycloid Beach Fill Template with Dune
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100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500
Distance offshore from benchmark (ft.)
Figure ES-1: Proposed beach fill template for South Beach range Cab 1-01 , based on the
extremal equilibrium profile truncated at 0 ft. MLL W with a back beach dune to hold-and-
release residual critical mass as the profile adjusts to changing wave climate.
Coordinates of Bench Mark:
Northing (ft): 1994408.5
Easting (ft): 6228847.4
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22 December 2014 (most eroded historic profile)
Cycloid Beach Fill Template with Dune
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Distance offshore from benchmark (ft.)
Figure ES-2: Proposed beach fill template for South Beach range Cab 1-02, based on the
extremal equilibrium profile truncated at 0 ft. MLL W with a back beach dune to hold-and-
release residual critical mass as the profile adjusts to changing wave climate.
Coordinates of Bench Mark:
Northing (ft): 1995102.9
Easting (ft): 6228562.5
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22 December 2014 (most eroded historic profile)
Cycloid Beach Fill Template with Dune
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Figure ES-3: Proposed beach fill template for South Beach range Cab 1-03, based on the
extremal equilibrium profile truncated at O ft. MLL W with a back beach dune to hold-and-
release residual critical mass as the profile adjusts to changing wave climate.
Coordinates of Bench Mark:
Northing (ft): 1995576.9
Easting (ft): 6228365
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22 December 2014 (most eroded historic profile)
Cycloid Beach Fill Template with Dune
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Figure ES-4: Proposed beach fill template for Middle Beach range Cab 1-04, based on the
extremal equilibrium profile truncated at 0 ft. MLL W with a back beach dune to hold-and-
release residual critical mass as the profile adjusts to changing wave climate.
Coordinates of Bench Mark:
Northing (ft): 1996164.9
Easting (ft): 6228090.5
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23
22
21
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19
18
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15
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Middle Beach Profiles, Cab 1-05
22 December 2014 (most eroded historic profile)
Cycloid Beach Fill Template with Dune
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Figure ES-5: Proposed beach fill template for Middle Beach range Cab 1-05, based on the
extremal equilibrium profile truncated at O ft. MLL W with a back beach dune to hold-and-
release residual critical mass as the profile adjusts to changing wave climate.
Coordinates of Bench Mark:
Northing (ft): 1996778.5
Easting (ft): 6227826.9
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22 December 2014 (most eroded historic profile)
Cycloid Beach Fill Template with Dune
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Distance offshore from benchmark (ft.)
Figure ES-6: Proposed beach fill template for Middle Beach range Cab I-06, based on the
extremal equilibrium profile truncated at 0 ft. MLL W with a back beach dune to hold-and-
release residual critical mass as the profile adjusts to changing wave climate.
Coordinates of Bench Mark:
Northing (ft): 1997015.5
Easting (ft): 6227720.2
•
Beach Equilibrium Analysis of Middle and South Beach Disposal Options for
Dredged Sands from Agua Hedionda Lagoon, Carlsbad, CA
by: Scott A. Jenkins, Ph.D.
1) Beach Profile Surveys and Dredge Disposal:
Six beach profile survey range lines were monitored on South Beach and Middle Beach
before and after the 2014/15 Agua Hedionda Lagoon dredging event. These surveys are plotted
in Figures 1-6, and labeled from south to north as: Cab I-01, Cab 1-02, and Cab 1-03 on South
Beach; and Cab I-04, Cab I-05 and Cab I-06 on Middle Beach. The captions in Figures 1-6 also
give the California planar coordinates of the bench mark for each range line.
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-6
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-8
-9
-10
-11
100 200
South Beach Profiles, Cab 1-01
----17 April 2015
----22 December 2014
---6 November2014
---5 September 2013
300 400 500 600
Distance offshore from benchmark (ft.)
Coordinates of Bench Mark:
Northing (ft): 1994408.5
Easting (ft): 6228847.4
700 800
Figure 1: Measured beach profiles at survey range Cab I-01 on South Beach, before and after
the most recent Agua Hedionda Lagoon dredging, which was begun on 31 December 2014 and
completed on 15 April 2015. See Figure 7 for bench mark locations.
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~ 3
~ 2
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-11
0 100
Coordinates of Bench Mark:
Northing (ft): 1995102.9
Easting (ft): 6228562.5
South Beach Profiles, Cab 1-02
----17April2015
----22 December 2014
----6 November 2014
----5 September 2013
200 300 400 500
Distance offshore from benchmark (ft.)
600 700
Figure 2: Measured beach profiles at survey range Cab 1-02 on South Beach, before and after
the most recent Agua Hedionda Lagoon dredging, which was begun on 31 December 2014 and
completed on 15 April 2015. See Figure 7 for bench mark locations.
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8
~ 7
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C 2 8 1 n, > 0 Q)
[jj -1
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-6
-7
-8
-9
-10
-11
0 100
Coordinates of Bench Mark:
Northing (ft): 1995576.9
Easting (ft): 6228365
South Beach Profiles, Cab 1-03
----17 April 2015
----22 December 2014
----6 November 2014
----5September2013
200 300 400 500
Distance offshore from benchmark (ft.)
600 700
Figure 3: Measured beach profiles at survey range Cab 1-03 on South Beach, before and after
the most recent Agua Hedionda Lagoon dredging, which was begun on 31 December 2014 and
completed on 15 April 2015. See Figure 7 for bench mark locations.
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9
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~ 7
...J 6
...J 5 ~ 4 ~ 3
C: 2 0 .::. 1 ro > 0 Q) w -1
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-5
-6
-7
-8
-9
-10
-11
0 100
Coordinates of Bench Mark:
Northing (ft): 1996164.9
Easting (ft): 6228090.5
Middle Beach Profiles, Cab 1-04
----17April2015
----22 December 2014
----6 November 2014
----5 September 2013
200 300 400 500
Distance offshore from benchmark (ft.)
600 700
Figure 4: Measured beach profiles at survey range Cab 1-04 on Middle Beach, before and after
the most recent Agua Hedionda Lagoon dredging, which was begun on 31 December 2014 and
completed on 15 April 2015. See Figure 7 for bench mark locations.
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~ 7
...I 6
...I 5 ~ 4 E, 3
C: 2 0 ~ 1 (1) > 0 Q)
ui -1
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-8
-9
-10
-11
0 100
Coordinates of Bench Mark:
Northing (ft): 1996778.5
Easting (ft): 6227826.9
Middle Beach Profiles, Cab 1-05
----17 April 2015
----22 December 2014
----6 November 2014
----5 September 2013
200 300 400 500
Distance offshore from benchmark (ft.)
600 700
Figure 5: Measured beach profiles at survey range Cab 1-05 on Middle Beach, before and after
the most recent Agua Hedionda Lagoon dredging, which was begun on 31 December 2014 and
completed on 15 April 2015. See Figure 7 for bench mark locations.
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9
8 -7 ~ 6 ...J ...J 5 :'E 4 ~ 3
C: 2 0 :; 1 <ll > 0 Q)
[j -1
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-5
-6
-7
-8
-9
-10
-11
0 100
Coordinates of Bench Mark:
Northing (ft): 1997015.5
Easting (ft): 6227720.2
Middle Beach Profiles, Cab 1-05
----17 April 2015
----22 December 2014
----6 November 2014
----5 September 2013
200 300 400 500
Distance offshore from benchmark (ft.)
600 700
Figure 6: Measured beach profiles at survey range Cab 1-06 on Middle Beach, before and after
the most recent Agua Hedionda Lagoon dredging, which was begun on 31 December 2014 and
completed on 15 April 2015. See Figure 7 for bench mark locations.
Each range line was surveyed four times, three times prior to the most recent Agua
Hedionda beach disposal event, (5 September 2013, 6 November 2014 and 22 December 2014);
and once immediately upon completion of beach disposal on 17 April 2015. The dredge logs
indicate that no dredging or beach disposal was begun until 31 December 2014. Since the 22
December 2014 beach surveys were performed just 9 days prior to onset of lagoon dredging,
these surveys captured the receiver beaches close their most denuded condition. In fact there had
been no placement of new sands on these beaches since April 2011; and inspection of the profile
measurements in Figures 1-6 indicates little profile change between the 6 November 2014 and 22
December 2014 surveys because much of the beach surface was exposed hard-bottom substrate
(bedrock, cobbles and basal conglomerate, see Figure 9).
Middle Beach was the first to receive beach fill from the 2014/15 dredge cycle beginning
on 31 December 2014, and received a total of 156,056 cubic yards; followed by South Beach that
began receiving 73,637 cubic yards of beach fill beginning on 21 February 2015. Unlike
previous dredge events, North Beach was the last to receive beach fill, and 64,968 cubic yards
were placed there between 23 March 2015 and 15 April 2015, (cf. Appendix-A). Therefore, the
17 April 2015 beach profile surveys represent the receiver beaches in their most built-out state,
because these surveys were performed only two days after the beach disposal of dredged sands
from Agua Hedionda was completed.
Three-dimensional CAD models were lofted from the measured points along the 1 o
beach survey range lines (Cab 1-01 -Cab 1-10) to delineate the beach surfaces immediately
before dredge disposal (based on the 22 December 2014 profile measurements) and immediately
after dredge disposal (based on the 17 April 2017 profile measurements). When these two
surfaces were lofted together in a common reference frame (as determined by the bench marks of
each survey range line), places where the 17 April 2015 surface intersected the 22 December
2014 surface identified areas along the receiver beach where sand was not retained. Conversely,
places where these two beach surfaces remained separated identified areas where dredged sands
were being retained to at least some degree. This is shown in Figure 7 where the fully built-out
beach surface (post-dredge disposal, 17 April 2015) is lofted in brown, and the severely eroded
beach surface (pre dredge disposal, 22 December 2014) is lofted in silver. It is clear from the
large areas of silver in this composite CAD model, that beach fill has been poorly retained on
North Beach and the northern end of Middle Beach despite the fact that this was the last beach to
receive beach fill. The eroded silver areas between Cab 1-06 and Cab 1-10 in Figure 7 are
especially prominent in the seaward half of the bar-berm section of the beach profile, but also in
the upper portions of the profile on the berm itself.
The mass properties tool of the SolidWorks 3-d CAD software was invoked to calculate
the rlifference in beach volmne between the surfaces defined by the 22 December 2014 and the
17 April 2015 surveys, and determined that 13,780 cubic yards of beach fill have been retained
after placing 64,968 cubic yards on North Beach between 23 March 2015 and 15 April 2015.
This means that only 21 % of the beach fill had been retained over a 3 ½ week period that ended
just two days after completion of pumping beach fill to North Beach! Clearly sand was being lost
at a high rate (1,969 yds3/day, or about 1/3 to 1/2 daily pumping rates) during placement of sand
in the beach fill template. A further concern, is that the beach fill placed on North Beach is up-
drift of the lagoon inlet in the prevailing southerly littoral drift, particularly during the winter
season when the large waves that erode the beach are from the northwest. Consequently, when
beach fill is not retained it is immediately ingested by Agua Hedionda Lagoon, where it provides
no useful function in protecting the shoreline against erosion and wave overtopping, and where it
restricts the tidal prism of the lagoon and degrades the lagoon water quality. On the other hand
sand retention was much better on Middle Beach and South Beach. After placing 229,693 cubic
yards on these beaches between 1 January 2015 and 23 March 2015, the SolidWorks mass
properties tool calculates that 178,584 cubic yards of beach fill have been retained at the time of
the post-dredge survey on 17 April 2015. Therefore, the sand loss during this time was only
51,109 cubic yards, or a loss rate of only 473 yds3/day, about I/4th the rate of loss of sand on
North Beach.
Cab 1-01,
... -----..------11-,0
2000 1000
Northing, ft
D
Figure 7: Three-dimensional composite CAD model of two overlaid beach surfaces on North, Middle and South
Beach, 1) immediately before dredge disposal (as delineated in silver from the 22 December 2014 profile
measurements) and 2) immediately after dredge disposal (as delineated in brown by the 17 April 2015 profile
measurements). CAD model shown with 10 to 1 vertical exaggeration.
2) Sand Retention Issues:
The poor sand retention characteristics of the receiver beach of back-passing are
due to several factors including: the timing of placement of back-passing sands, non-
equilibrium distribution of those sands, unfavorable geomorphology, and placing more
sand than the geomorphology can support in equilibrium. Beginning with timing, the
largest fraction of sand that is lost from the beach fill placed on the receiver beaches
occurs in the first few months after placement. Because the least tern nesting season
restricts Agua Hedionda Lagoon dredging to the winter season, placement of the beach
fill typically occurs in the midst of the onslaught of the largest winter waves. However,
during the period when sands from the 2014/15 dredge event were being placed on the
receiver beaches, (1 January 2015 -15 April 2015) the winter waves were not unusually
intense, with the highest waves reach Hmax = 1.94 m while average significant wave
heights were <H> = 1.1 m, with an average wave period of <T> = 13.8 sec and a typical
northwesterly wave direction averaging< <X > = 279°, (cf Figure 8b). But even these
rather ordinary winter waves can exert significant erosional effects due to the way waves
refract and shoal at along the receiver beaches, and particularly at North Beach. The
presence of the Carlsbad Submarine Canyon immediately south of Agua Hedionda
Lagoon creates a bright spot in the shoaling wave pattern immediately north of the inlet
jetties, where wave heights are locally higher than further to the north around Maple Ave.
This alongshore imbalance in wave energy leads to a negative divergence of dr(ft in the
longshore transport rates, which in turn causes higher southerly longshore transport rates
of sand exiting North Beach at the inlet jetties than enters North Beach at Maple Ave,
(see Section 5 for more detail). These same refraction effects from the bathymetry along
the north rim of the Carlsbad Canyon also propagate along Middle and South Beach,
although their effect on divergence of drift is weaker due to the obstruction provided by
the inlet and discharge jetties, (the so-called groin.field effect).
So we turn our attention to how the dredged beach fill has been placed and what
quantities. Engineered beach fill has typically been placed on Carlsbad beaches with a
1: 10 (rise over run) slope. This convention dates back to the Regional Beach Sand
Project, (AMEC, 2002). However, stable beach profiles in Nature have a much more
gradual, curving profile with slopes that range between 1 :50 or 3: 100, (Inman et al.,
1993). The theory on equilibrium beach profiles began with Dean (1977) and Bowen
(1980) who developed formulations for an equilibrium profile having the form, h=Ax'",
where h is the local water depth, x is the horizontal distance offshore, ,n = 2/3, and
A is an empirical factor. Later Dean ( 1991) developed analytic approximations for the
empirical factor, A, and that formulation was incorporated into the U.S. Army Corps of
Engineers Shore Protection Manual and later the Coastal Engineering Manual. However,
recently Jenkins and Inman, (2006), proved that the Dean (1977 and 1991) solutions are
not unique, and represent only one of a family of equilibrium beach profiles known as an
elliptic cycloids. The elliptic cycloid formulation can account for continuous variations in
the equilibrium beach profile due to variability in wave height, period and direction when
occurring in combination with variations in beach sediment grain size and beach sand
volume. Equilibrium beach profiles obey the maximum entropy production formulation
of the second law of thermodynamics, and are the most efficient shape for a beach profile
because it adjusts itself to dissipate all of the available wave energy. When the waves
encounter an inefficient non-equilibrium beach shape, such a steeply sloping beach fill
4
3 E
E
C> ·a; :i:2
Q) > t'O
~1
25
20
~ (/)
'815
·c:
if.
G> 10 > ~
5
CDIP Station #043-Camp Pendelton Nearshore, 33.2198 N, -117.4394 W
2.3-year mean= 0.97 m ~ Hm•x = 3.8 m <H> =0.95m
\
2.3-year mean = 13.8 sec <T> = 13.8 sec
~
0 __.___.___.___.___.___.___.___.___.___.___.___.___.___.__.__.__.__.__.__._ ............................ __.____.__...__....__
c: 280 0 .:= (.) ~ iS 240
! ~ 200
2.3-year mean = 276°
1/1/13 1/1/14 1/1/15
Figure 8: Shoaled significant wave heights, periods and directions at Carlsbad State
Beach based on back refraction of wave monitoring data from CDIP Station 043 at Camp
Pendleton during the beach survey period. Data shown in black occurred during the
2014/15 Agua Hedionda Lagoon dredging event.
templates, then the wave energy is not fully dissipated and the excess wave energy
begins eroding and moving that beach fill around until an equilibrium profile is finally
achieved. While this is occurring, the beach fill can be highly mobile, particularly in large
winter waves; and at Carlsbad, the net southward flowing longshore currents (particularly
in winter) will rapidly transport the eroded beach fill from North Beach toward the south
and the inlet to Agua Hedionda Lagoon; although some of the mobile North Beach sands
are transported onto Middle Beach along the by-passing bar at the lagoon inlet during
ebbing tide.
There are also geomorphic factors that contribute to poor retention of Agua
Hedionda dredge sands at the North Beach disposal site, while limiting retention times at
Middle and South Beach. A sandy beach cannot be supported in equilibrium against wave
forces without a wave-cut platform in the bed rock to provide a foundation. Wave cut
platforms are notches that have been eroded in the bed rock during protracted still-stands
in sea level. Once formed, sediment collects in these notches forming a beach which is
subsequently molded into an equilibrium shape by wave action. Figure 9a shows an
annotated seismic reflection profile measured by USGS across the continental shelf off
Carlsbad CA on range 223X of the 1991 Kolpack surveys, (Kolpack, 1991). It shows a
series of wave-cut platforms that were formed at present and earlier still-stands of sea
level, and subsequently covered with sediment. The most striking feature in Figure 9a is
how much more pronounced the paleo wave-cut platforms are than the modern platform;
and how very thin the sediment cover is over the modem wave-cut platform, as compared
with the thickness of Holocene sediment over the paleo platforms. Although the paleo
platforms have been subjected to longer periods of sedimentation, the geometric
constraints imposed by small wave-cut platform prevent them from retaining thick layers
of sediment cover.
Many of the beaches throughout north San Diego County are perched on narrow
wave-cut platforms. The platforms are narrow because they were carved by wave action
from erosion resistant Del Mar formation during the present high-stand in sea level, and
these narrow wave-cut platforms physically cannot hold large quantities of beach sand;
and often become fully denuded during periods of high-energy winter waves, as shown in
Figure 9b. Sub-bottom surveys by Elwany, et al., (1999) discovered narrow wave cut
platforms and exposed hard bottom substrate in the surfzone and nearshore at North
Beach and South Beach (Figure 10) while 1884 railroad surveys reveal beach cobble
ridges before the influence of Mankind at Agua Hedionda, ( e.g. HWY 101 and the deep
water lagoon). When beach cobble ridges and hard-bottom features are found this close to
shore, it indicates that these beaches (particularly, North Beach and South Beach) are not
geomorphically well suited to retain large volumes of sand. In the particular case of
North Beach, attempts to back-pass and place more sand there than its carrying capacity
will simply result in low retention time and increased sand influx of into Agua Hedionda
Lagoon. The remainder of this report focuses on determination of the carrying capacity at
North Beach and the optimal distribution of that carrying capacity within a beach fill
template in order to maximize sand retention time of dredged sands.
3) Critical Mass and Middle/South Beach Equilibrium Profiles:
The criticai mass is the minimum voiume of sediment cover required to maintain
equilibrium beach profiles and represents the nominal carrying capacity of a particular
beach. When a long term collection of beach profiles are plotted together over a broad
range of wave heights, a well-defined envelope of variability becomes apparent, (Figure
1 la). This envelope of profile variability is referred to as the critical mass envelope, and
the volume of sand within critical mass envelope, Ve , increases with increasing wave
height and period but decreases with increasing beach grain size, as shown in Figure 11 b.
The critical mass envelope is always limited by the breadth of the wave cut platform,
a)
MSL 0
present 20
wave-cut 40 shelf platform sediment 60
paleo country rock 80
wave-cut platform vertical exaggeration 23x
7 6 5 4 3 2 1 0
Distance, km
Figure 9: Wave-cut platforms in north San Diego County: a) Annotation ofUSGS
Geopulse sub-bottom seismic profile along range line 223-X in the inner shelf off
Carlsbad showing present and ancient wave cut platforms (after Kolpack, 1991); b)
exposed wave cut platform in Solana Beach during the 1983 El-Nino winter.
E
..c: -a. Q)
0
362000---
361000-
360000
358000--
357000
356000
355000
354000--
353000--
3520oo---;
Pacific Ocean
D 10-30% hard substrate
D 30-60"-'> hard substrate
D 60-100% hard substrate
1660000 1661000 1662000 ' 1663000
I
1664000
. -,.,
·-----...... ...
C A I.
1665000 1666000 1667000
l
' _..-.-·· ··-· --· ---·---::------~,~ ---;
Figure 10: Nearshore survey (left) showing exposed rocky reefs outcrops and other hard bottom substrate; and (right) 1884 railroad
survey map showing beach cobble ridge, both indicating minimal sediment cover on the beaches around Agua Hedionda Lagoon.
a)
-----------MSL ---
__ .,.. ... -
V.E. 33x
E 2500 --..,
E_ 2000
":;:.O
l 1500
~
~ ~ 1000
(.)
0
Q)
E :::,
500
1600 1200 800
Distance, m
b} --D, = 12011m, 0, = 80 µm (Rockport, TX)
--01 = 200 µm, 0, = 100 µm (Torrey Pines, CA)
--D1 = 200 11m, 0, = 200 11m (Scripps Beacll, CA)
--o, = 400 µm, 0, = 150 µm (Duck, NC)
400 0
~ 0 i,c;..._.,___ ........ _..J...._.....,__---1. _ __,L....-_,L..__...L..._...J...._.....J
3
E
" UJ'
f 2 C: ti :2 I-
m rn
Cll 1 ~
~ ~ (.)
0
c)
1600
2 3 4 5
rms Incident Wave Height, H,., , m
--D1 = 120 llffl, 0, = 80 µm (Rockport, TX)
--D1=20011m, 0, = 100 µm (TOIT8y Pines, CA)
--o, = 200 µm, D, = 200 µm (Salpps Beach, CA)
--D1 = 400 µm, D, = 150 µm (Duck, NC)
1200 800 400 0
Distance, m
10
5
0 E
£
5 5r 0
10
15
Figure 11: Features of the critical mass of sand: a) critical mass envelope for waves ranging
from 1 m to 5m in height; b) volume of critical mass as a function of wave height and sediment
grain size; c) variation in the thickness of the critical mass as a function of distance offshore.
which forms a hard-bottom boundary condition on the critical mass envelope. The best way to
calculate the critical mass is to find the volume between the wave cut platform ( or its layer of
basal conglomerate) and the elliptic cycloid equilibrium profile that corresponds to the native
beach grain size in combination with the wave height and period of the extreme event wave in
the period ofrecord. The volume integral between the surfaces of the wave cut platform and the
extremal event elliptic cycloid then give the critical mass volume. At Middle and South Beach,
the sub-bottom reflection data is too spotty to resolve the complete surface of the wave-cut
platform along the 3,680 ft. length of the Middle/South Beach disposal site (between south inlet
jetty and the north end of the Encina generating building). Therefore we will use the surface
given by the 22 December 2014 beach surveys as a surrogate bottom of the critical mass
envelope. This is a reasonable approximation because there has been no placement of new sands
on Middle and South Beach since the 2010/11 dredge event, which only placed 163,996 cubic
yards on Middle and South Beach in April 2011 (cf. Appendix-A). Inspection of the profile
measurements in Figures 1-6 indicates little profile change between the 6 November 2014 and 22
December 2014 surveys because much of the beach surface had partially exposed hard-bottom
substrate (bedrock, cobbles and basal conglomerate).
The extremal elliptic cycloid equilibrium profile is a curve that is traced by a point on the
circumference of a rolling ellipse, see Figure 12b. It is calculated from Jenkins and Inman (2006)
using the following:
h = 7r& X ( 1-COS 0) + Z I
2J?l 0-sin 0 (1)
Note this has the same basic formulation of the original Dean (1977 and 1991) solutions in the
U.S. Army Corps of Engineers Coastal Engineering Manual Here Z1 is the elevation of the berm
crest (cf. Figure 12a) given by Hunt's Formula [Hunt, 1959; Guza and Thornton, 1985;
Raubenheimer and Guza, 1996]:
(2)
In equation (2), r is the runup factor taken herein as r = 0. 76, and H b is the breaking wave
height. The cycloid in (1) is based on the elliptic integral of the second kind that has an analytic
approximation, I !2l = ~ (2 -e2 ) / 2 , where e is the eccentricity of the ellipse given by
e = ~ 1-b2 I a 2 , with, semi-major and semi-minor axes are a, b, (cf. Figure 12b). The wave
parameter,€, in equation (1) is given by:
( )
l/2 4/S ( J2/5 Hb O' H00 S=O' -"" ----y g -2"s gy
(3)
here a-= 2n/period is radian frequency, H 00 is incident wave height, g is the acceleration of
gravity, and yis the wave breaking criteria taken as y= 0.8. The rolling angle of the ellipse is:
[
( 1-l \a]
0 = arccos 1 -2l ~ j _A y hc_
(4)
a.
-5
0 E
~ 5
£ ~ 10 0
15
20
Et :::1:
Shorerlse
Xc2
Xe
b. type-a cycloid
X
C.
\
\
\ 8=7t t
' ' ' ....
a) profiles: eccentricity and shear stress linearity
----e= 0.845: n= 3
----e=0.798;n=2
----e=0.707:n=1
----e=0.447;n=O
't
----e = 0 ; n = -0.33 ; brachistochrone solution
700 600 500 400 300 200
Cross-Shore Distance x2 ,m
Bar-Berm
-1
X1 ~A X1
h
0
2
4 ..I "' 6 ~
E -
8 ~""-
10
:g_
Cl) 0
12
14
100 0
Figure 12. Equilibrium beach profile theory: a) nomenclature, b) mathematical basis for an
elliptic cycloid, c) Typical range of elliptic cycloids on a 700 m wide wave-cut platform.
where A is the shoaling factor relating breaker height to incident wave height, A= Hool Hb,
(
.2 / )
115
which for shoaling Airy waves, becomes A= 2215 H~5 a/ g y . The closure depth, he in
equation (4) is grain size and wave period dependent and is given by:
(5)
where k = a I~ ghc is the shallow water form of the wave number, Ke and 1// ~ 2.0 are non-
dimensional empirical parameters, set at Ke= 2.0 and 1// ~ 0.33; D50 is the median grain size;
and Do is a reference grain size taken as D0 = 100 µ m. Equation (5) is transcendental and is
solved numerically within the CEM.
Calculation of the extremal elliptic cycloid equilibrium profile at Middle and South
Beach requires knowledge of the characteristic median grain size, D50 , of the dredged sediments
to be placed there. Recent sediment grain size analyses by Merkel, (2008) based on three
sampling locations on the flood tide bar in the West Basin of Agua Hedionda Lagoon (Samples
L1 -L3) were compared against native sediments on the three receiver beaches (RB 1-RB3).
These grain size distribution are plotted in Figure 13. Note Middle and South Beach is
represented by samples RB 1. Grain sizes at the lagoon sample sites and beach sites were similar
with median grain sizes of 0.32 millimeters (320 microns) on the food tide bar in the West Basin
of Agua Hedionda, while residual sediments that still remained on Middle and South Beach prior
to disposal of material from the 2008/09 dredging averaged 0.374 millimeters (374 microns).
To determine the highest waves to reach to effect Middle and South Beach disposal, the
waves measured at ½ hour sampling intervals at CDIP Station 043 were back refracted into deep
water from the monitoring location off Camp Pendleton, and then forward refracted into Middle
and South Beach. An example of this procedure is shown in Figure 14 for a wave occurring 8
January 2002. This effort produced a continuous wave record throughout the historic period
when Middle and South Beach disposal of Agua Hedionda dredged sands was practiced, (1998-
2015), see Figure 15 .. The highest energy wave (extremal) event occurred in January 2007, when
a Gulf of Alaska storm brought 4.8 m high waves approaching Carlsbad at 276 ° with a 15
second periods. This extreme event wave was used to calculate the extremal elliptic cycloids on
Middle and South Beach.
To calculate the critical mass of Middle and South Beach, we combine the extremal
waves with theD50 grain size values from Figure 11 to solve equations (1)-(5) for the extremal
elliptic cycloid profile. These extremal cycloid profiles are plotted on the Middle and South
Beach Range survey range lines (Cab-1-07 -Cab 1-10) in Figures 16-21. These profiles
represent the beach shape that can be sustained in an equilibrium state during the most severe
wave events of the 1998-2015 Middle and South Beach disposal period. These profiles form the
top of the critical mass envelope, while the most eroded profile (from the 22 December 2014
c,11~ui::n1c., \ f-n h~uP Ar-r-11...,..~rl 1n that ciamP t'\Pt-1Arl Af rPf'nrrl rlPtin,=.c thP hnttnm nf r-ritir~ 1 m!:l~~ IJWJ. YVJiJJ \.V J.J.""""' VVVUl..&.VY J.J.J. l,J.J.U-1, ..,1,,.4,J.J.J.V yvJ.J.Vllo.A v.a. ........ V\J.1.-...................... ..., ............. ...,.....,. ................ '-'& _ _...._ .. ._ __ ......... -.... ....
envelope. When lofted in the 3-d CAD SolidWorks software, the SolidWorks volume tool
calculates the critical mass envelope to hold of 280,361 cubic yards along the entire 3,680 ft. of
• Dredge Area L 1
• Dredge Area L2
• Dredge Area L3 • • C:.:• • Reciever Beach RB 1 • • Reciever Beach RB2
• Reciever Beach RB3 • •• •
• • •• • • • •
• •• • • •
• •• •
• •
10 1 Grain Size (mm) 0.1
100
90
80
&
70 ! ~ CD
60 ~ .c en i 50.;.; .c en 'i
40 i
i i
30 'S E
20
10
0
0.01
::::, u
Figure 13: Grain size distributions form Agua Hedionda Lagoon (Samples L1 -L3) and from
the receiver beaches (RB1-RB3). Note Middle and South Beach is represented by samples RBI,
(from Merkel, 2008).
33.200
33.175
33.150
33.125
33.100
33.075
117.450 117.425 117.400
0
117.375
Longitude
2 3
B.Jena \Asta Lagoon
117.350 117.325
4 5
Incident Wave Height at CDIP-043 = 1 .66 m, Period = 18 sec, Direction = 263 deg
117.300
Figure 14: Regional wave shoaling during 8 January 2002 from back-refraction of wave
monitoring data at CDIP Station# 043 in 20 m local water depth offshore of Cam Pendleton.
Carlsbad Waves Derived from CDIP Station #043-Camp Pendelton, 33.2198° N, -117.4394° W
5
4
25
20
~ °8 15
·c: Q)
D..
Q) 10 > ~
5
360
340
Q> 320 ~ 300
c 280 0
ts 260 ~ o 240
~ 220 (1'
3: 200
180
1998
Hmax = 4.8 m ---..,.,.:l-
long term mean = 0.95 m
2000 2002 2004 2006 2008 2010
1998 2000 2002 2004 2006 2008 2010
long-term mean = 272°
2012 2014 2016
2012 2014 2016
160 --....---.---.----.---.----r-..-----r-..----.--r--.---,..---.---,--,--~--.---,
1998 2000 2002 2004 2006 2008 2010 2012 2014 2016
Figure 15: Shoaled significant wave heights, periods and directions at Carlsbad State Beach
based on back refraction of wave monitoring data from CDIP Station 043 at Camp Pendleton for
the period of record of Middle and South Beach disposal, 1998-2016.
12
11
10
9
8
7
6
5
4
~ 3 ...J ...J 2 :::ii
s, 1
C: 0
0 -1 .:;
«l > -2 (I) w -3
-4
-5
-6
-7
-8
-9
-10
-11
100 200
South Beach Profiles. Cab-01
----22 December 2014
----Extremal Cycloid (H"'"' = 4.8 m)
Critical Mass, Ve
300 400 500 600
Distance offshore from benchmark (ft.)
700 800
Figure 16: Critical mass envelope at range line Cab 1-01 on South Beach based on the extremal
elliptic cycloid solution using a 4.8 m high design wave height with 15 second wave period.
12
11
10
9
8
7
6
5
4
:i: 3 ..J ..J 2 :;:
~ 1
C: 0
0 -1 :,= ro > -2 Q) w -3
-4
-5
-6
-7
-8
-9
-10
-11
0 100
South Beach Profiles, Cab-02
----22 December 2014
----Extremal Cycloid (H,..,., = 4.B m)
Critical Mass, V c
200 300 400 500
Distance offshore from benchmark (ft.)
600 700
Figure 17: Critical mass envelope at range line Cab 1-02 on South Beach based on the extremal
elliptic cycloid solution using a 4.8 m high design wave height with 15 second wave period.
17
16
15
14
13
12
11
10
9
8
s:: 7
..J 6
..J 5 ~ 4 = -3
C: 2 0 ~ 1 ~ 0 Q)
[i -1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
0 100
South Beach Profiles, Cab-03
----22 December 2014
----Extremal Cycloid (Hm., = 4.8 m)
Critical Mass, V c
200 300 400 500
Distance offshore from benchmark (ft.)
600 700
Figure i8: Critical mass envelope at range line Cab I-03 on South Beach based on the extremal
elliptic cycloid solution using a 4.8 m high design wave height with 15 second wave period.
17
16 Middle Beach Profiles, Cab-04
15 22 December 2014
14 Extremal Cycloid (H.,,,, = 4.8 m)
13
12
11
10
9
8 -7 ~ 6 ..J ..J 5 ::::iE = 4 Critical Mass, Ve -3
C: 2 0 ~ 1 n, > 0 G.l jjj -1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
0 100 200 300 400 500 600 700
Distance offshore from benchmark (ft.)
Figure 19: Critical mass envelope at range line Cab 1-04 on Middle Beach based on the extremal
elliptic cycloid solution using a 4.8 m high design wave height with 15 second wave period.
17
16
15
14
13
12
11
10
9
8
~ 7
6 ::I 5 ~ 4 ~ 3
C: 2 0 .:; 1 co > 0 Q)
iii -1
-2
.3
-4
-5
-6
.7
-8
-9
-10
-11
0 100
Middle Beach Profiles, Cab-05
----22 December 2014
----Extremal Cycloid (H"'"' = 4.8 m)
Critical Mass, Ve
200 300 400 500
Distance offshore from benchmark (ft.)
600 700
Figure 20: Critical mass envelope at range line Cab 1-05 on Middle Beach based on the extremal
elliptic cycloid solution using a 4.8 m high design wave height with 15 second wave period.
17
16
15
14
13
12
11
10
9
8
~ 7
...J 6
...J 5 ~ 4 ~ 3
C: 2 0 .:. 1 n, > 0 Q) w -1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
0 100
Middle Beach Profiles, Cab-06
----22 December 2014
----Extremal Cycloid (H.,., = 4.8 m)
Critical Mass, Ve
200 300 400 500
Distance offshore from benchmark (ft.)
600 700
Figure 21: Critical mass envelope at range line Cab I-06 on Middle Beach based on the extremal
elliptic cycloid solution using a 4.8 m high design wave height with 15 second wave period.
the Middle and South Beach disposal site. This volume represents the optimal carrying capacity
of the Middle and South Beach disposal site. Lesser amounts of beach fill will not be able to
sustain an equilibrium profile during the highest energy wave events; and without an equilibrium
profile, the beach will not dissipate all the incident wave energy, and the excess wave energy will
erode the beach. If Middle and South Beach is over-nourished with more than the critical mass of
sand, then two processes will intervene: a) the excess sand will spill off the wave cut platform
(which is 600 ft. to 650 ft. wide) and be re-suspended over the rocky outcrops and hard bottom
substrate at Middle and South Beach (cf. Figure 10); and/orb) the excess sand will be swept
away by the net longshore transport (littoral drift) which flows from north to south throughout
the Oceanside Littoral Cell ( cf Figure 22).
111roo'
,,,,.-----,
' ' ,,
\ '-,
i \
km
I I
\ -\ -· .. ' ' \ \
I
,,-...)
I
------USA ________ _
iio1co
33•30'
• WOO'
32'45'
---
Figure 22: Oceanside Littoral Cell showing net north-to-south littoral drift, (from Inman and
Brush, 1970)
4) Cycloid Beach-Fill Template Design:
Ideally the optimal beach fill templates for Middle and South Beach would duplicate the
critical mass envelope in Figures 16-21, as these would prescribe an adequate amount of sand to
support an equilibrium profile in the presence of extreme event waves without exceeding the
carrying capacity of the otherwise limited wave cut platform that exists there. However, Figures
16-21 indicate that the critical mass envelope extends well below mean lower low water
(MLL W) to depths ranging from -4 ft. MLL W to -8 ft. MLL W. With present beach fill
construction methods, it is not possible to build a template that extends below the waterline.
Beach fill is pumped to Middle and South Beach via a hydraulic dredge pipeline and initially
deposited as a slurry. After the slurry dewaters, the sands are spread out across the beach using
conventional earth-moving equipment, which cannot effectively operate in anything deeper than
ankle deep water. Therefore, we must pose a beach-fill template that adapts to this construction
constraint.
We begin by examining the percentage of time that a dry beach is available for
construction operations at the lower end of the beach profile during in the months of September
to mid-April, (the months during which dredge disposal is permitted in order to avoid impacting
the least tern nesting season). Figure 23 plots the relationship between ocean water level and
percent time a given elevation remains dry, (referred to as the hydroperiodfunction), based on
ocean water levels measured at the nearby Scripps Pier tide gage (NOAA# 9410230). Figure 23
indicates that the beach fill construction operations can proceed down to elevations as low as 0
ft. MLLW at least 7% of the time, or during about 50 hours in a given month. These times are
clustered during the spring tides that occur twice each month. If the beach fill template is filled
from the top down (ie, spreading sand at the highest elevations of the template first, and then
proceeding downslope towards the lowest end), then 50 hours should be adequate to allow filling
the lowest portion of a template that terminates at 0 ft. MLL W.
A significant fraction of the critical mass envelope in Figures 16-21 lies below the 0 ft.
MLL W water level, and if the extremal cycloid profile is terminated at that elevation in the
beach fill template design, then additional sand must be added elsewhere to the template in order
to achieve the critical mass volume along the entire 3,680 ft. reach of the Middle and South
Beach disposal site. The additional sand is provided by combining a back-beach dune with the
elliptic cycloid that has been truncated at Oft. MLLW. The back-beach dune placement strategy
was first implemented in Carpinteria by Bailard and Jenkins (1980 and 1983) and later during the
replacement of seawalls at Mission Beach Sea (Jenkins 2014) and Del Mar (cf Figure 24a.).
These previous implementations of the back-beach dune strategy involved very popular beach
sites, yet no adverse recreational incidences were encountered. The back beach dune is a
conservation/storage mechanism that prevents rapid sand loss from over-builds of the intertidal
portion of the beach profile, yet still allows the fill site to receive its full allocation of critical
mass, and provides gradual re-nourishment as the dune erodes during brief periods of spring
tides and/or high waves. The dune proposed for the Middle and South Beach disposal site is
shown in Figure 24b, and is roughly 9 ft high and 55 ft. wide, with a reserve storage capacity of
13.5 cubic yards per running ft. of beach. When built along the 3,680 ft. length of the Middle and
South Beach disposal site, this dune will provide 43,200 yds3 sand perched in the upper portion
of the truncated equilibrium cycloid profile. The beach fill templates that result from this strategy
are shown in Figures 25-30 for Middle and South survey range lines Cab I-01 -Cab I-06. When
lofted over the entire 3,680 ft of the Middle and South Beach disposal site using the SolidWorks
3-d CAD software, ( cf. Figure 31 ), we calculate that these templates will provide 200,593 cubic
yards of disposal volume, which compares almost exactly with the required critical mass of
200,890 cubic yards, (the optimal carrying capacity of the Middle and South Beach disposal
sites.)
~ ....J ....J
~
~
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~ Q)
....J ....
2 ~
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Q)
()
0
8
7
6
5
4
3
2
1
0
-1
-2
-3
0
La Jolla, Scripps Pier Tide Gage
NOAA # 9410230
EHW = +7.47 ft.
MHHWz: +5.13 ft.
• +2.11 ft.
I
Beach Dry @ ' MLW 17% of the time
---------r --------------
l
Beach Dry@ iMLLW 7% of the time --r ---------r --------------
1 1 % Lowest Water Level
r : ~ =-1.2ft.MLLvV ~,
I
~I if..1 1_1
1 ~I ~ I ~
I I
EL:W = -3.06 ft MLLW
10 20 30 40 50 60 70
Percent Time Dry
80 90 100
Figure 23: Hydroperiod function of ocean water levels during the months of September-April,
based on the Scripps Pier tide gage (NOAA # 9410230; based on the 1983-2001 tidal epoch,
(from Jenkins and Taylor 2015; and Jenkins and Wasyl, 2011)
a)
back-beach dune
b)
10 -----.----.-----.----.--,--,.--,--,---,---,--,---.--....-~-.-~~~~~~~~~---.----.-----.----.----.---,
9 -------1---1-----'f--l--l-+--L
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E 5 -----1-----1-----1-----1-----1-----1-----1-~·
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0 10 20 30 40 50
Dune Width (ft.)
Figure 24: The back-beach dune beach fill placement strategy, a) as implemented in Del Mar
during seawall replacement; and b) as proposed at Middle and South beach in combination with
a truncated elliptic cycloid beach profile.
60
25
24
23
22
21
20
19
18
17
16
15
i14
...J 13
...J 12 ~ 11
~10
C: 9
.Q 8 I ~
5
4
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-1
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------------------------------
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l"
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\,,
'
:\
\
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\
)
\
\
'
X_-'-''llkl',-'-
....
r,i..,._
South Beach Profiles, Cab 1-01
22 December 2014 (most eroded historic profile)
Cycloid Beach Fill Template with Dune
\ 1, -......... -.... ....... -..... I'-,. --NI:. ---......... I ---..... ....... ~ ...
' I ,1 '
I
' '
100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500
Distance offshore from benchmark (ft.)
Figure 25: Proposed beach fill template for South Beach range Cab 1-01, based on the extremal
equilibrium profile truncated at 0 ft. MLL W with a back beach dune to hold-and-release residual
critical mass as the profile adjusts to changing wave climate.
Coordinates of Bench Mark:
Northing (ft): 1994408.5
Easting (ft): 6228847.4
25
24
23
22
21
20
19
18
17
16
15 i14
..J 13
..J 12 ~ 11
~10
C: 9 .2 8 ~ 7 w 6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-----------j -I -Ir --'( -~r -,,
-------------
. ,,
I
\ I
',\
I
I
b\ .,~ '' ·, l
\ --. -,~
\ ......
South Beach Profiles, Cab 1-02
22 December 2014 (most eroded historic profile)
Cycloid Beach Fill Template with Dune
~-'I)
:--., f .......... ,1· ~ ... .....
I\. I . ----._. I 1-------........ ~-_,,. ~ ~---'
o ~ ~ oo oo 100 1~ 1~ 100 100 ~mm~~~ m ~ ~ ~ ~
Distance offshore from benchmark (ft.)
Figure 26: Proposed beach fill template for South Beach range Cab I-02, based on the extremal
equilibrium profile truncated at O ft. MLL W with a back beach dune to hold-and-release residual
critical mass as the profile adjusts to changing wave climate.
Coordinates of Bench Mark:
Northing (ft): 1995102.9
Easting (ft): 6228562.5
25
24
23
22
21
20
19
18
17
16
15
i14
..J 13
..J 12 ~ 11
~'10
C: 9
,Q 8 i 7
iii 6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-------------------------------
I\
I \
\ I I
j
/·.
\
\ ...
" 'l
1 \
' \.
'I
"Ii
I'.. "'Nil '
South Beach Profiles, Cab 1-03
22 December 2014 (most eroded historic profile)
Cycloid Beach Fill Template with Dune
.,....,
....
1 " ...... ..... .... .... -.... ....... ,....._
r-.... .· ...........
" -....... ~--........ -----'"' ,.,,. 111
o m ~ ~ oo 100 1m 1~ 1~ 100 ~ ~ ~ B ~ ~ ~ ~ ~ ~ ~
Distance offshore from benchmark (ft.)
Figure 27: Proposed beach fill template for South Beach range Cab 1-03, based on the extremal
equilibrium profile truncated at O ft. MLL W with a back beach dune to hold-and-release residual
critical mass as the profile adjusts to changing wave climate.
Coordinates of Bench Mark:
Northing (ft): 1995576.9
Easting (ft): 6228365
25
24
23
22
21
20
19
18
17
16
15
i14
...1 13
...1 12 ~ 11
~10
C: 9
.Q 8 ~ 7 ~ 6
5
4
3
2
1
0
-1
-2
-3
-4
-5
----------= ------. ----. ---. . ---
" ·,
.I
I
' I
]. .. -~ J: --
I
1:l
.\
·\..
J " -Oil/'l
I\. I ~
' '-\.
'll ,
" ...
Middle Beach Profiles, Cab 1-04
22 December 2014 (most eroded historic profile)
Cycloid Beach Fill Template with Dune
'"-.
' ".",
I\. .....
·, ....... ... ,... .~ ....
I'... ) ,......._
' I T ..... -r-,;... 1i:1.~1:1 ~ .....
r--... I ......... ---=-, ....
..J.'-: ........... -............ '-r---,-! ' -~
I
o ~ ~ w ~ 100 1~ 1~ 1w 1~ ~ m ~ ~ ~ D m ~ ~ ~ ~
Distance offshore from benchmark (ft.)
Figure 28: Proposed beach fill template for Middle Beach range Cab 1-04, based on the extremal
equilibrium profile truncated at 0 ft. MLL W with a back beach dune to hold-and-release residual
critical mass as the profile adjusts to changing wave climate.
Coordinates of Bench Mark:
Northing (ft): 1996164.9
Easting (ft): 6228090.5
25
24
23
22
21
20
19
18
17
16
15
i14
...1 13 :jj 12
. 11
!!:,. 10
C 9
.Q 8 ! 7
iii 6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-------------------,_ -------------
-r-,..,.:
.......
' I\.
"
-
A /J,\ ,, \
\
' j
I '-I
'\
~
-r,...,
X
Middle Beach Profiles, Cab 1-05
22 December 2014 (most eroded historic profile)
Cycloid Beach Fill Template with Dune
'-" h
\
" \.
" ' '-'~,,-V -..
r-.... '-I',. ...... "'oi;; --...... .... ---..... --.... ---..... --
0 ~ ~ M 00 100 1~ 1~ 1M 100 D m ~ ~ ~ ~ m ~ ~ ~ ~
Distance offshore from benchmark (fl.)
Figure 29: Proposed beach fill template for Middle Beach range Cab 1-05, based on the extremal
equilibrium profile truncated at O ft. MLL W with a back beach dune to hold-and-release residual
critical mass as the profile adjusts to changing vvave climate.
Coordinates of Bench Mark:
Northing (ft): 1996778.5
Easting (ft): 6227826.9
25
24
23
22
21
20
19
18
17
16
15
i14
...1 13
...I 12 ~ 11
~10
C: 9 .Q 8 1ii 7 > 4) 6 iii 5
4
3
2
1
0
-1
-2
-3
-4
-5
--
Middle Beach Profiles, Cab 1-06
----22 December 2014 (most eroded historic profile)
----Cycloid Beach Fill Template with Dune
-A
---+--t--+--t-+-+--t--t-+-+~, ... \------------------------t----1 -----+--+--+--+-+--t---t--t-+-+---HOhl '--------------------------.... ---... -t--+--t-+-1--t--t-+-+-ttm . i ________________________ ....
-----j ' ..... . . .
----------. --
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....... -1--1---1---1---..i-;"''<!l!," ' '1-1--t--+--t--+-+-t--t---t·-+--+--t--+-+-t--t-+-+--t--+--+-+-----1 '~~: : :li~~ \ ---+--+--+-+--1--1--'<I'''' .
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" ' ' j'.'I.,. -4-+-+-+-+--1----f-+-ll--l-+--I',;:_--...., I , l ~~~ ... -+-+--+--t--+----+-t-1-+--+-+--+--t--+-+--I----II--I----,
i:, "I[;· ..... ___ _,_ ___ ,_, ___ _,_ ___ ,__,_
'"""'-'-+-+-+-11-t--+-+--t---t--+-+-1-f-l--+---, ~"""'--ij~ I '""""' -4-+-+-+-+-+-+-+-11-+-+-+-+-+-+--f-+-ll-""i-..:._,,,j ~ I I' ....... ..._:-+-+--+--t--+-+-+--1--1-+---,.-+-+--i
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-4-+-+-+-+-+-+-+-11-+--+-+-+-+-+-+-+-'l--l--+-+-P ... ~-~ . -~ ":' .. _ ........ ___ __,_-t---1 ----
o ~ ~ ~ ~ 100 1~ 1~ 1~ 1~ ~ ~ ~ ~ ~ ~ m ~ • ~ ~
Distance offshore from benchmark (ft.)
Figure 30: Proposed beach fill template for Middle Beach range Cab 1-06, based on the extremal
equilibrium profile truncated at O ft. MLL W with a back beach dune to hold-and-release residual
critical mass as the profile adjusts to changing wave climate.
Coordinates of Bench Mark:
Northing (ft): 1997015.5
Easting (ft): 6227720.2
Cabl-01
Cabl-01
-----------a4,o 2000 1000
Northing. ft
Figure 31: Three-dimensional CAD model of the proposed composite cycloid/dune beach fill template on
North, Middle and South Beach, where the silver surfaces indicate the beach surface on which the template is
built (from the 22 December 2014 profile measurements) and brown represents the beach surface resulting
from the proposed beach fill template.
5) Beach and Shoreline Evolution Analysis of Middle and South Beach Disposal Options
Sand movement and retention on the receiver beaches (North Beach, Middle Beach, and
South Beach) is not only determined by wave-induced erosion and littoral drift, but is also
influenced by the flow of sea water in and out of Agua Hedionda Lagoon. Therefore two
distinctly different hydrodynamic models are required to address the study objectives listed in
Section 1, namely: 1) a tidal transport model used to compute tidally induced transport of littoral
sediments in and out of the inlet; and 2) a littoral transport model used to determine the
equilibrium states of the receiver beaches, as well as the rates of delivery of littoral sediments to
and from those receiver beaches and across the lagoon inlet and discharge channels. These
models are coupled in a generalized Coastal Evolution Model (CEM) whose architecture is
shown in Figure 32 with computer code detailed in Jenkins and Wasyl, 2005, which is available
on-line from the Digital Library of the University of California at
(http://repositories.cdlib.org/sio/techreport/58D. In the work described herein, these models are
driven by wave measurements and Encina Power Station (EPS) flow rate measurements and
calibrated with beach profile measurements and dredge disposal volumes preceding and
following the most recent Agua Hedionda Lagoon dredging, December 2014 -April 2015.
The tidal transport model used in this study is known as TIDE _FEM originally published
in Inman & Jenkins, 1996. TIDE_FEM was built from some well-studied and proven
computational methods and numerical architecture that have done well in predicting shallow
water tidal propagation in Massachusetts Bay [Connor & Wang, 1974] and estuaries in Rhode
Island, [Wang, 1975 ], and have been reviewed in basic text books [Weiyan, 1992] and symposia
on the subject, e.g., Gallagher (1981). The TIDE_FEM model was peer reviewed by the Science
Advisory Panel (SAP) of the California Coastal Commission, and used in three certified
environmental impact reports (EIR's), including: the San Dieguito Wetland Restoration Project,
(SCE, 2000, 2005); the Interstate 5 (I-5) North Coast Corridor Project for the California
Department of Transportation (Jenkins and Wasyl, 2011), and the Otay River Estuary
Restoration Project for Poseidon Water LLC (Jenkins, 2014; Jenkins et al., 2015). A discussion
of the physics of TIDE_FEM along with its significant algorithms can be found in Appendix-B.
The littoral transport model is formulated from the longshore transport relations of
Komar & Inman (1970) and Longuet-Higgins (1970), and from the cross-shore transport and
equilibrium beach profile relations of Bowen (1980), Inman et al. (1993) and Jenkins and Inman
(2006). The littoral transport model has two computational domains, afar-field computational
cell (Figure 14) and a near.field computational cell (Figure 34) that is nested inside the farfield
and contains the details of the lagoon inlet and discharge channel and adjacent North Beach,
Middle Beach and South Beach and the South Beach Groin. (The South Beach Groin is retained
in the nearfield cell during the calibration runs, and removed to reflect future conditions for
model runs testing optimal beach fill templates). The far-field cell provides mass balance for
sediment entering and exiting the nearfield cell while the nearfield cell computes the interaction
between the lagoon tidal and discharge transport and the littoral transport. The nearfield cell also
computes the beach profile changes on North, Middle and South Beach due to sand volume
changes occurring from dredge disposal, and subsequent erosion occurring in between dredging
events.
The Coastal Evolution Model (CEM) is a process-based numerical model. It consists of a
Littoral Cell Model (LCM) and a Bedrock Cutting Model (BCM), (Figure 32), both coupled and
operating in varying time and space domains determined by sea level and the coastal boundaries
of the littoral cell at that particular sea level and time. Over the time scales of this study, the
LCM is the relevant module. At any given sea level and time, it accounts for transport of mobile
sediment along the coast by waves and currents. The BCM accounts for the cutting of bedrock
once the sediment cover is denuded by wave erosion. However bedrock cutting, and notching of
the bedrock to form a wave cut platform is a process that occurs over decadal to millennial time
scales.
~ ~
~ ~ "' :::E ~ "C ~
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!l =-+ l g' i ~ l .. c, . fj
~ ~ ~ C ai C -g :!
:e ·e
Cl C ~ E a "' ... ! e
j
Forcing Functions i ,-----, ~ .-----------I Wet/Dry 1---~
P,.tlpltaUon Climate
r---'--------l Cycles
,----;---, Watershed
Erosion
&Transport
t -" ,;; l
1• 8oundary_Condltlons .......................................... ,
! Control Cells Farfteld Grid ' ! i Accretion / N111tlng Regional
Erosion Wave Bathymetry
& Topography [
,'. llt1oralTran1port ..... :J .................................... i ............................................. . ' . l Sediment , ____ ___,~----: .. ----1 Refraction/ ;
Budgat .....___,'--....., Dlfftactlon
i Sediment · Covar •-----~ r ....................................................................................... 1 .............. · ••••••••••••••••• -.... .
•
...................................... , ••• ,1 ••••••••••••••••• 1.,, •••••••...................................
GIS Accumulation
of Profiles
Seal.evel
Change
1mag11ry or
coa·stal Morphology
Figure 32: Architecture of the Coastal Evolution Model (CEM); from Jenkins and Wasyl, (2005).
In the LCM, the coastline of the far-field computational cell is divided into a series of
coupled control cells, as illustrated schematically in Figure 33b. Each control cell is a small
coastal unit of uniform geometry where a balance is obtained between shoreline change and the
inputs and outputs of mass and momentum. The model sequentially integrates over the control
cells in a down-drift direction so that the shoreline response of each cell is dependent on the
exchanges of mass and momentum between cells, giving continuity of coastal form in the down-
drift direction. While the overall computational domain of the far-field cell remains constant
throughout time, the beach profile within the individual control cells can change shape or shift on
or offshore, as shown in Figure 33c, in response to changes in wave height or longshore transport
rates, or due to the introduction of new sediment from dredge disposal as shown in Figure 33a.
These changes are computed from time-stepped solutions to the sediment continuity
equation (otherwise known as the sediment budget) applied to the boundary conditions of the
coupled control cell mesh round Agua Hedionda Lagoon, as in Figure 34. The sediment
continuity equation is written (Jenkins, et al, 2007):
dv' o ( 8v) av -=-&--U1 -+J(t)-R(t) dt ay ay ay (5)
Where v' is the beach volume per unit length of shoreline (m3/m), 6 is the mass diffusivity, U,
is the longshore current, J(t) is the flux of new sediment into the control cell from dredge
disposal and R(t) is the flux of sediment lost from the control cell due to the tidal influx of
sediment into Agua Hedionda Lagoon. The first term in ( 1) is the surf diffusion while the second
is divergence of drift. For any given control cell that does not enclose the lagoon inlet, equation
(5) may be discretized in terms of the rate of change of beach volume, V, in time, t, given by:
dV -= qin -qout +J(f)-R(f) dt
and V = I [qin -qout + J(t)-R(t)] dt
(6)
Over any given period of time, T = nAf , comprised of n number of time steps of interval !!.t , the
volume change of the beach can be computed by discretizing equations (6) according to:
n
~v = L [qin -qout +J(n)-R(t)]nM (7)
I
Referring to the control cell schematic in Figure 33c, sediment is supplied to the control cell by
dredged beach fill, J(t), or by the influx littoral drift from up-coast sources, qin =q LI, (where
q L is the longshore transport rate on the updrift side of the control cell). Sediment is lost from the
control cell due to the action of wave erosion and expelled from the control cell by exiting
a) Accretion / Erosion Wave
J
accretion
(sand delta)
t 1
b) Coupled Control Cells
c) Profile Changes
/ closure
depth
qin-0-qout
I
T
h
1
accretion --•---
erosion ~
Figure 33: Computational approach for modeling shoreline change after Jenkins, et. al., (2007).
N
0
San Luis Rey River SCALE
2000· 6000' 10000· ---o· 4000· sooo·
ELEVATION REFERENCE TO MLLW
Oceanside
Buena Vista Lagoon
Carlsbad
Agua Hedlonda Lagoon
South Beach
Groin
Batlqultos Lagoon
Figure 34: Coupled control cell mesh of the farfield computational cell used to model the
impacts of dredge disposal and removal of the South Beach Groin.
littoral drift, q0u1 =qL2 , or by becoming ingested by the lagoon's tidal inlet, R(t). Here fluxes
into the control cell (J(t) and qL,) are positive and fluxes out of the control cell (qL2 and q,ide)
are negative. The beach sand volume change, dV/dt, is related to the change in shoreline
position, dX/dt, according to:
dV = dX .z./
dt dt
(8)
where
Here, h is the height of the shoreline flux surface equal to the sum of the closure
depth below mean sea level, he, and the height of the berm crest, Z1, above mean sea level (from
Hunt's Formula.); and l is the length of the shoreline flux surface (see Figure 33c). Hence,
beaches and the local shoreline position remain stable if a mass balance is maintained such that
the flux terms on the right-hand side of equation (2) sum to zero; otherwise the shoreline within
each control cell will move during any time step increment as:
When dredge disposal produces a large episodic increase in J, an accretionary bulge in
the shoreline (like a river delta) is initially formed (cf. t1 in Figure 33a). Over time the
accretionary bulge will widen and reduce in amplitude under the influence of surf diffusion and
advect down-coast with the longshore current, forming an accretion-erosion wave (cf. t2 & t3 in
Figure 33a). The local sediment volume varies in response to the net change of the volume
fluxes, between any given controi celi and its neighbors, referred to as divergence of drift=
q;n -q0"1 , see Figure 33c. The mass balance of the control cell responds to a non-zero
divergence of drift with a compensating shift, Ax , in the position of the equilibrium profile
whose shape is calculated from equations (1)-(4) after Jenkins and Inman, (2006). This is
equivalent to a net change in the beach entropy of the equilibrium state. The divergence of drift
is given by the continuity equation of volume flux, requiring that dqldt on the left hand side of
equation (5), is the net resultant of advective and diffusive fluxes of sediment plus the influx of
new sediment, J, from dredge disposal, per the right hand side of equation (1 ). In response to the
rate of change of volume flux through the control cell, the equilibrium profile will shift in time
according to equation (9). If the divergence of drift is positive because more sand fluxes into the
control cell due to longshore transport than leaves the cell, ( q;n -q0"1 = qL1 -q L2 > 0 ), then the
equilibrium beach profile in that cell will shift seaward. Conversely, if the divergence of drift is
negative because less sand fluxes into the control cell than is expelled from the cell by longshore
transport, ( q;n -q011, = qL1 -q L2 < 0 ), then the equilibrium beach profile in that cell will shift
landward, as diagramed schematically in Figure 33c. If a negative divergence of drift causes the
equilibrium profile to shift sufficiently landward that it intersects the basement surface of the
critical mass envelope, then the cycloidal shape of the profile is disrupted, and the equilibrium
state of the profile is lost.
The formulation for the longshore transport rate q L is taken from the work of Komar and
Inman (1970) according to:
(10)
where q L is the local potential longshore transport rate; Cn is the phase velocity
of the waves; S yx =Esinab cos ab is the radiation stress component; ab is the breaker angle
relative to the shoreline normal; E=l/8pgH; is the wave energy density; p is the density of
water; g is the acceleration of gravity; Hh is the breaking wave height; and, K is the transport
efficiency equal to:
K=2.2F:, (11)
(12)
Here c,b is the reflection coefficient which is calculated from the mean bottom slope, fJ (which
is known either from the measured profiles or from the elliptic cycloids); and, a-is the radian
frequency = 21T:!T, where T is the wave period. These equations relate longshore transport rate to
the longshore flux of energy at the break point which is proportional to the square of the
breaking wave height and breaker angle. By this formulation, the CEM computer code
calculates a local longshore transport rate for at the up-drift and down-drift sides of each side of
each control cells of the mesh in Figure 34.
Longshore transport around the inlet jetties to Agua Hedionda Lagoon causes the
sediment budget of the North Beach disposal site to be coupled to the sediment budget of the
Middle Beach disposal site, since the prevailing southward littoral drift (cf. Figure 22) results in:
(12)
where the longshore transport entering Middle Beach at its northern end is the net of longshore
transport exiting North Beach and the tidal influx into the lagoon, R(t). To calculate the tidal
influx rates we run a tidal hydraulics model known as TIDE_FEM, (Jenkins and Inman, 1998).
TIDE_FEM was built from some well-studied and proven computational methods and numerical
architecture that have been successful in predicting shallow water tidal propagation in
Massachusetts Bay [Connor & Wang, 1974] and estuaries in Rhode Island, [Wang, 1975 ], and
have been reviewed in basic text books [Weiyan, 1992] and symposia on the subject, e.g.,
Gallagher (1981). A discussion of the physics ofTIDE_FEM is given in Jenkins and Wasyl
(2003 & 2005). In its most recent version, the TIDE _FEM modeling system has been integrated
into the Navy's Coastal Water Clarity Model and the Littoral Remote Sensing Simulator (LRSS)
(see Hammond, et al., 1995). The TIDE FEM code has been validated in mid-to-inner shelf
waters (see Hammond, et al., 1995; Schoonmaker, et al., 1994). A detailed description of the
architecture and codes of the TIDE_FEM/ is given in Appendix-B.
Calibrations for determining the appropriate Manning factors and eddy viscosities were
performed by running the TIDE_FEM model on the Figure 35 bathymetry file and comparing
TIDE_FEM simulations of inlet channel velocities against measurements by Elwany et al. (2005)
during a complete spring-neap cycle of 13 -30 June 2005, as shown in Figure 36. Plant flow rates
during this lagoon monitoring period were input to TIDE _FEM according to daily recordings by
Cabrillo Power 1 LLC. Iterative selection of Manning factor n0 = 0.03011 and an eddy viscosity
of & = 6.929 ft2/sec gave calculations of inlet channel velocities that reproduced the measured
values to within 2% over the 18 day spring-neap monitoring cycle. Figures 3 7-42 map the tidal
velocities in Agua Hedionda Lagoon from hydrodynamic simulations of the spring, neap and
mean tides during the spring-neap cycle calibration period using the TIDFEM model. The flood
and ebb current maximums and minimums in the inlet channel are found to lead the high and low
ocean water levels by as much as 13.7 hours during the spring tides on 21 June 2005. Maximum
flood tide currents on this day were 5.16 ft/sec, while maximum ebb tide currents were -2.87
ft/sec; the flood tide dominance due to the scavenging effect of the power plant intake rate on the
available lagoon water volume which was operating at 501 mgd. Throughout the 18 day
monitoring period, average flood tide currents in the inlet channel were 1.91 ft/sec while average
ebb tide currents were -0.91 ft/sec while the power plant averaged an intake flow rate of 430.97
mgd. The amplitudes and degree of non-linearity in the inlet current time series simulated by the
model closely duplicate that observed in the measured currents. The maximum error in
simulating the ebb tide currents was found to be & L = +0.1 ft/sec. The maximum flood tide error
in the modeled currents relative to observations was found to be & H = -0.05 ft/sec. Using the
calibrated Manning factor n0 = 0.03011 and an eddy viscosity of & = 6.929 ft2/sec from these
2005 calibration simulations, the ocean water level forcing for the disposal period 1 January
2015 to 17 April 2015 was input into the TIDE _FEM model in order to calculate the tidal influx
term, R(t), for the following beach evolution simulations in Sections 5.1 and 5.2 below.
5.1 Calibration of the Coastal Evolution Model for Middle and South Beach:
The Coastal Evolution Modei (CEM) for Middle and South Beach was calibrated using
the measured pre-and post-dredging beach profiles for Middle and South Beach during the
2014-2015 dredging event, in conjunction with daily beach fill placement volumes as reported in
the monitoring report to the Regional Water Quality Control Board, San Diego Region, (NRG,
2015). Wave forcing for the CEM was based on shoaling wave data from Figure 8, while beach
fill grain size was based on Figure 13. Daily beach fill volumes were assumed to be laid out over
the 22 December 2014 profiles from Figures 1-6 in a standard beach fill template with a flat
backshore platform and a 1: 10 (rise over run) seaward facing beach slope extending down to 0 ft.
MLLW. Throughout the Middle and South Beach disposal period, that began on 1 January 2015
and ended on 15 April 2005, daily beach fill increments ranged from 0 yds3/day to 5,480
yds3/day and were successively added to the Middle and South Beach control cells in Figure 34,
while the wave forcing continued to rearrange those fill volume increments according to flux
balance relations in equations (5)-(9). Free parameters in the CEM, including the mass
diffusivity in equation (5) and the longshore transport efficiency in equation ( 10) were adjusted
through successive iterative simulations until the change in beach sand volume between 1
January 2015 and 17 April 2015 (when the post dredging beach surveys were done) matched the
volumetric changes of the measured profiles in Figures 1-6. These volumetric changes were
computed by the SolidWorks 3-d CAD model in Figure 43.
---+7.7 contour ---+4.0 contour ----3.0 contour -..e.o contour ----13.0contour ----18.0contour ----23.0 contour
---+7.1 contour ---+2.0 contour ----4.0 contour ----9.0 contout ----14.0 C:0111Dur ----19.0 contour ----24.0 contour
---~.4 contour ---0.0 contour ----5.0 contour ----10.0 contour ----15.0 contour ----20.0 contour ----25.0 contour
---+5.8 co moor ----1.0 contour ----6.0 contour ----11.0contour ----16.0 contour ----21.0 contour
5.6 contour ----2.0 contour ----7.0 contour ----12.0 contour ----17.0 cootour ----22.0 contour
Figure 35: Pre-dredging bathymetric survey of Agua Hedionda Lagoon prior to the 2006/2007 dredging of Agua Hedionda Lagoon
C > C, z -=
C 0 ;:: • > • iii
6
4
2
0
-2
I
Ocean Wate1r Level I
----Simulated lrdet Velocity
I + + + Measured Inlet Velocity
1
average ebb current = -0. 91 ft/sec
maximum ebb current= -2.87 ft/sec
average flood current = 1. 91 ft/sec
maximum flood current= 5.16 ft/sec
6
4
2
0
-2
-4 ____ ........, ...... _____ _..,.......,. __ ........,,_ _____ ._.,.......,. _________________ 4
13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
June,2005
u .,
i:
i .2 • > -..
.5
Figure 36: TIDE_FEM model calibration using spring--neap cycle velocity measurements from Elwany, e al., 2005. Ocean water
levels at station SO indicated in blue; hydrodynamic simulation of inlet channel velocity shown in green and compared against ADCP
inlet velocity measurements shown as black crosses.
Figure 37: Hydrodynamic simulation of tidal flow into Agua Hedionda Lagoon during spring flood tides.
Figure 38: Hydrodynamic simulation of tidal flow out of Agua Hedionda Lagoon during spring ebb tides.
Figure 39: Hydrodynamic simulation of tidal flow into Agua Hedionda Lagoon during mean flood tides.
Figure 40: Hydrodynamic simulation of tidal flow into Agua Hedionda Lagoon during mean ebb tides
Figure 41: Hydrodynamic simulation of tidal flow into Agua Hedionda Lagoon during neap flood tides.
Figure 42 : Hydrodynamic simulation of tidal flow out of Agua Hedionda Lagoon during neap ebb tides.
Erosion exposes pre-disposal
beach surface per 22 Dec. 2014 survey
Sand remaining 17 April 201 S on Middle and South Beach
= 178,584 cubic yards
Figure 43: SolidWorks 3-d CAD model of a composite surface overlay of the modeled post-
disposal Middle and South Beach surface (brown) and the pre-disposal beach surface (silver).
The SolidWorks mass properties tool calculates a volume of 178,584 yds3, representing the
volume of sand retained after placing 229,693 yds3 over a 80 day period of beach nourishment.
In Figure 43, the CEM modeled beach surface on 17 April 2015 (represented in brown) shows
numerous patches where erosion has occurred and the pre-disposal beach surface (represented in
silver) is exposed. The mass properties tool in SolidWorks calculates that the volume in the
region between the two surfaces (representing the volume of beach fill retained) is 178,584 yds3,
which matches the retention calculated directly from the pre-and post dredging beach profile
measurements in Figures 1-6.
The time stepped CEM wave-driven flux calculations which led to this calibration result
are plotted in Figure 44. Initially, as sand was being placed on Middle Beach during the first 40
days, waves were generally small, short period and mostly approaching from the west/southwest
with very little north swell (cf Figure 8b). Consequently, the calibration simulation begins with
weak northward flowing longshore transport, Figure 44a, causing some sands near Batiquitos
Lagoon to be transported northward and arrive at South Beach. During this early period, more
sand enters the South Beach control cells, than leaves Middle Beach at the south inlet jetty, and
the difference between the cumulative net transport at the south inlet jetty vs South Beach
(Figure 44 b) initially creates a positive divergence drift (Figure 44c ). Consequently, Middle and
South Beach gains sand volume during the first 40 days from wave driven transport, in addition
to the gains from Middle Beach dredge disposal. But then, circa 19 February 2015, several
west/northwest storm swells arrived with waves reaching 1.9 m in height prior to breaking ( cf.
Figure 8b ). Coinciding with these larger swells, the longshore transport reverses direction toward
the south and increases in rate, (Figure 44a) resulting in cumulative net transport toward the
south at the south groin that exceeds cumulative net transport into Middle Beach at the south
inlet jetty (Figure 44 b). The divergence of drift turns negative and remains that way throughout
the remainder of the disposal activities, resulting in a loss of 51,109 cubic yards of sand from
Middle and South Beach by 17 April 2015 due to wave-driven transport (Figure 44 c ). Negative
divergence of drift is prevalent along Middle and South Beach because of the way the prevailing
west/northwest swell and wind waves are refracted around the Carlsbad submarine canyon,
creating higher shoaled wave heights at the southern end of South Beach than found further north
nPHr thP inlPt to Ag1rn TlPnion,h T Hgoon, UThPrP. rPfrHr.tion pffp_rt« ofthP. f:HrkhHn ~11hmHrine
Canyon are weaker, (cf. Figure 14).
When the net sand loss to divergence of drift is superimposed on the incremental
sequence of beach fill being placed on Middle and South Beach, we get a look at how the sand
retention on Middle and South Beach varies throughout the disposal period (Figure 45). It
appears from the black line in Figure 44 that sand volume with the standard 1: 10 (rise over run)
beach fill template increases throughout most of the Middle and South Beach disposal period,
before falling off around 2 April 2015, about 10 days after South Beach disposal was completed
on 23 March 2015. Up until 2 April 2015, the placement of beach fill on Middle and South
Beach had exceeded or at least kept pace with the sand loss rate to negative divergence of drift
during the building north swell period. By the time the post dredging beach surveys were
performed on 17 April 2015 (108 days after beach fill placement began), about 78% of the total
volume of sand placed on Middle and South Beach remained, and the average rate of loss of
beach fill was < dV I dt >= 473 yds3/day. Based on this average loss rate, the retention time,T0 ,
for fill placed on Middle and South Beach during the 2014/2015 dredging event was
T -Vo o -<dV/dt>
229 693 = 485 days
473
(13)
_g?
C'O 0::
0.02
~-8. ~ 0.01 ~ en cu-.... (") ~ en ~ -g_ 0.00 o-.c en 0)
C
_3 -0.01
-= '5 20.00
'5-(") a.> en o -c 0.00 C >. ~o .... (/) g? -c -20.00
i5 ~
a.> en i g -40.00
C'O ~ '3-
§ -60.00
(..)
0.0
0.0
---Inlet Jetty, q.., = Qu
South Beach, Cab 1-01, q0.,, = qL2
20.0 40.0 60.0 80.0
---Inlet Jetty. 'Equn41
--South Beach, Cab 1--01, Eqt2 n.r
20.0 40.0 60.0
21 February 2015
Begin South Beach
Disposal " 73,637 yds3
: At'
100.0
~ ---------------
1 January 2015
Begin Middle Beach
Disposal= 156,056 yds3 ~
'E(qL 1 -qL2) nt,,t
t
Middle and
South Beach
sand loss ___ = 51,109 ydsJ
0.0 20.0 40.0 60.0 80.0 100.0
Julian Day, 2015
Figure 44: Wave-driven fluxes during the Middle & South Beach CEM calibration: a) longshore
transport rate, b) cumulative net transport; and, c) cumulative divergence of drift.
230.00
220.00
210.00
;;;-200.00 ~ 190.00 >-180.00
0 170.00
1/l 160.00 -g 150.00 m 140.00 5 130.00
£ 120.00
~ 110.00 > 100.00
<l 90.00 a> 80.00
E 70.00
:J 60.00
50.00
-c 40.00 ffi 30.00
00 20.00
g
--Middle & South Beach Disposal, J,,/n)
--Cumulative Divergence of Drift, I:(q,_, -q,2) nt.t
--Beach Volume Retained, V(n)
21 February 2015
Begin South Beach
Disposal = 78,011 yds3
1 January 2015
Begin Middle Beach
Disposal = 151,682 yds3
/
22 March 2015. Middle & South
Beach Fill Placed = 229,693 yds3
I
17 April 2015, Retention
= 178,584 yds3
17 April 2015
Middle & South Beach
Sand Loss = 51,109 yds3
.5 10.00 Jl'=~-,:c-~"':"".=-=-::-::--:-:-:::-:--=--:~..,,, Cl> 0.00 g> -10.00
~ -20.00
(.) -30.00
-40.00
-50.00
----------------------1
-60 .00 ---=l-rrn"TTTmrrrrrHTTTTTrrrTir-rrT"TTTTTT"rrHr-rnTTTTTT"rrn.....-.-rrrTTT"rrH"TTTTTT"rrTir-rn"TTT-rrrTTT"....-r,"TTTTTT,.,.,..........,.,..,.,.T>
0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 110.0
Julian Day, 2015
Figure 45: CEM simulation of temporal variation in beach sand retention on Middle and South
Beach (black line) as a result of the net between the incremental sequence of beach fiii piacement
(green line) and the wave-driven divergence of drift (red line).
The retention time from equation (13) is less than a typical bi-annual dredging cycle for Agua
Hedionda Lagoon, suggesting that Middle and South Beach will be significantly eroded by the
time the next increment of dredged beach fill is placed there. This result was a consequence of
several factors: 1) placement of a fill on a highly eroded Middle and South Beach surface, that
had probably been eroded to the basement of the critical mass envelope, (since it had been 3
years since previous nourishment was provided by lagoon dredging disposal); and 2) the fill
placed in 2015 exceeded the critical mass carrying capacity and was laid down in a non-
equilibrium profile shape. In the following section, we evaluate potential improvements in beach
fill retention time using the composite cycloid-dune beach fill template and the appropriate
amounts of beach fill volume.
5.2 Performance Simulations of the Proposed Cycloid Beach Fill Template:
In this section, the calibrated Coastal Evolution Model (CEM) is run in long-term
simulations of the fate of beach fill placed on Middle and South Beach throughout the entire
period of record (1998-2015) that Agua Hedionda dredged sands have been disposed as beach
fill on Middle and South Beach. Wave forcing for these simulations was based on shoaling wave
data from Figure 15 for the period 1998-2016, while beach fill grain size was based on Figure
15. Daily beach fill volumes were derived from the dredge monitoring reports to the Regional
Water Quality Control Board (NRG, 2015) and from an NRG dredging data base contained in an
Excel spreadsheet< Dredge History.xis>. The spreadsheet contains volumetric time histories for
8 separate Middle and South Beach disposal events that are summarized in Appendix-A. The
CEM was first run for these 8 events assuming the beach fill volumes were distributed using the
standard 1: 10 (rise over run) template laid out over an eroded basement surface that was
specified by the 22 December 2014 profiles from Figures 1-6. These runs were used to establish
a Middle and South Beach baseline performance standard for average sand loss rate,< dV I dt >,
and retention time, T0 • This baseline was then compared against a sensitivity analysis of the
proposed composite cycloid-dune beach fill template to determine possible advantages of this
new template design and establish the optimal beach fill volume for Middle and South Beach,
given the limitations of: 1) its small wave-cut platform, and 2) intrinsically high longshore
transport rates with negative divergence of drift in the presence of the prevailing west/northwest
swell and wind waves.
5.2.1: Historic Sand Retention Baseline: The baseline CEM simulation results are
plotted in Figure 46, showing average rate of loss of beach fill< dV I dt > (red) and the retention
time, T0 , (blue) as a function of the total beach fill volume, V0 • Sand loss rates are scaled against
the left hand axis in Figure 46, while retention time is plotted relative to the right hand axis. The
solution points for the 8 historic Middle and South Beach dredge disposal events that occurred
between 1998 and 2015 are plotted with star symbols that were subsequently connected by cubic
spline fitting functions plotted as the red and blue colored lines. The cubic spline best fit curves
in Figure 46 show a minimum in the sand loss rate for Middle and South Beach fill volumes near
the critical mass Vc,;i = 200,890 yds3, which is the theoretical maximum carrying capacity of
Middle and South Beach for supporting a beach profile in equilibrium. The theoretical minimum
sand loss rate using the standard 1: 10 (rise over run) beach fill template is < dV I dt >= 431
yds3/day. When beach fill volumes are less than critical mass, V0 < Vcrit, there is insufficient sand
to establish an equilibrium profile. A beach is in its most stable state with an equilibrium profile,
Historic Middle & South Beach Fill Retention
----Historic Loss Rate, yds'/day
>, <ti 1.00 -0
----Historic Retention Time, days * * • Disposal Events, 1988-2015 520 ... Q) 510
a.
M 0.90 500
(/)
-0 490
>, -480 0 0.80
(/) 470
-0 C: 460 <ti 0.70 (/) :::::,
0 £ 0.60
450 ~ <ti 440 -0
a~n , e, ---,... . I\ .... 420 Q)
-0 5; 0,50
,::,
V
"O 0,40
410 .5 I-
400 c
0
390 ~
C: <ti (J)
0 0,30 minimum loss rate= 431 yds3/day
380 Q)
370 &
360
Q) +-' <ti 350
a:: 0.20 340 (/)
(/) 0 330
...J 0.10
Q) 320
OJ <ti 310 ... 4l 0.00 300 > <t 160.0 180.0 200.0 220.0 240.0 260.0 280.0 300.0
Total Disposal Volume, V0 ( thousands of yds3)
Figure 46: CEM simulation of average rate of loss of beach fill, < dV I dt > (red, left hand axis),
and the retention time, T0 , (blue, right hand axis), as a function of the total beach fill volume, V0 ,
for 8 historic Middle and South Beach dredge disposal events , 1998-2015. CEM solution points
are plotted with star symbols that were connected by cubic spline fitting functions plotted as the
colored lines. Simulations based on standard 1: 10 (rise over run) beach fill template assumption.
but Middle and South Beach has a prevailing negative divergence of drift so that when
equilibrium cannot be achieved due to insufficient sand volume, sand loss rates increase. For
example, the measured sand loss rates during the 2010/2011 dredging event (where V0 = 163,996
yds3) were 519 yds3/day, or 20% higher than the theoretical minimum loss rate of 431 yds3/day
(where theoretical minimum loss rate occurs when V0 ~ Vc,.u ). Consequently the retention time
in 2011 was only T0 = 316 days; but would have increased to T0 = 466 days if the beach fill
volume placed on Middle and South Beach were increased from V0 = 163,996 yds3 to Va = Ve,.;, =
200,890 yds3•
Figure 46 also shows that retention time continues to increase as the disposal volume
placed on Middle and South Beach exceeds the critical mass volume, but the sand loss rate also
increase, because the excess san volume cannot be retained on the limited wave-cut platform
which is only 600 ft. to 650. ft. in width. For beach fill volumes above critical mass the
improvement in retention time is minor, because the excess sand cannot be supported in
equilibrium on the limited wave-cut platform, which is only 550 ft. to 600 ft. wide. The excess
sand is quickly lost to the negative divergence of drift, ( cf. Figures 44 b&c ), after which an
equilibrium profile can be established, once V0 ~ Vc,.u . The initially high loss rate of the excess
sand contributes to a higher average sand loss rates than what would have otherwise occurred if
the beach fill volume were limited to the critical mass volume. Historically, what we find that
when beach fill sand volume is increased by 41 % over critical mass using the standard 1: 10
beach fill template (as occurred during the 2000/2001 dredge event when Va = 281,195 yds3 on
Middle and South Beach), the retention time retention time is only increased by 7 % from T0 =
466 days to T0 = 497 days. This is not a good return on the investment in beach fill for Middle
and South Beach because the sand loss rate increases by 31 % to < dV I dt >= 566 yds3/day, or
an increase in sand loss of 135 yds3/day over what would have otherwise occurred if the beach
fill volume were limited to critical mass.
5.2.2: Sand Retention with the Cycloid-Dune Template. The cycloid dune templates
proposed in Figures 25-30 are tested in long-term CEM simulations using the same model
calibration parameters and the same wave forcing and grain size inputs used in the historic
baseline simulations. We also assume that the proposed cycloid-dune template is laid out over an
eroded basement surface specified by the 22 December 2014 profiles from Figures 1-6, the same
assumption as was made for the baseline simulations. Because these cycloid-dune simulations
are hypothetical, we do not limit ourselves to a mere handful of options for beach fill volumes, as
was the constraint with the historic baseline simulations in Section 5 .2.1. Instead, a sensitivity
analysis is performed on beach sand loss rates and retention times using beach fill volumes that
range from V0 = 151,000 yds3 to V0 = 281,195 yds3• The dune portion of the proposed templates
in Figures 25-30 holds Vv = 49,680 yds3 which were designed for a fill volume of V0 = 200,593
yds3, approximately the same as the critical mass, Vcru = 200,890 yds3. For the modeling
scenarios that involved less than critical mass, the dune portion of the template was
proportionally reduced in volume until reaching the absolute minim fill volume scenario of V0 =
150,913 yds3 , for which no dune component remains in the template. For the modeling scenarios
where Va > Vcrit , the dune portion of the template was proportionally increased in width until
reaching the scenario of V0 = 281,195 yds3, which is the historic maximum fill volume placed on
Middle and South Beach.
Because we do not have historical guidance on the temporal beach fill placement for
these hypothetical scenarios, we have made several additional assumptions. First, we assume that
the beach fill is laid down at a constant daily rate of 4,500 yds3 per day on Middle and South
Beach, which was the average daily placement rate in the dredge monitoring reports to the
Regional Water Quality Control Board (NRG, 2015) and in the NRG Excel spreadsheet(<
Dredge History.xis>). Secondly, we assume a future condition with the South Groin removed (cf.
Figure 34 for location). Thirdly, we pose a construction scenario whereby the back-beach dune
portion of the composite cycloid-dune template is built first, starting at the south inlet jetty and
adding sections to the dredge pipeline until the build-out of the dune reaches abeam of the EPS
generating building. Building the dune first creates a "safe" reservoir of sand before the template
can be fully constructed, and sand from this reservoir is only released to the lower eroded
basement surface during periods of the highest tides and waves. After the buildout of the dune to
the EPS generating building, the cycloid portion of the template is laid out beginning from the
toe of the dune and spreading the material to down slope to MLL W, and working back towards
the south inlet jetty, removing sections pipelines as the cycloids are completed.
CEM beach evolution simulations were performed on 18 different disposal scenarios
involving cycloid-dune beach fill placement on North Beach ranging from V0 = 151,000 yds3 to
V0 = 281,195 yds3• Each cycloid-dune scenario was repeated 8 times using start dates and wave
forcing corresponding to the 8 historic Middle and South Beach dredge disposal events listed in
Appendix A. Selecting these specific start dates in the wave record in Figure 15 eliminates the
randomness effects of the historic occurrence of extreme waves, and allows for direct
comparisons with the results of the historic baseline in Section 5.2.1. Sand loss rates and
retention times for the 8 separate outcomes of each scenario were ensemble averaged to produce
the points plotted as crosses in Figure 47, and these solution points were then fitted to cubic
splines (plotted in red for sand loss rates relative to the left hand axis; and plotted in blue for
retention time relative to the right hand axis). Sand loss rates were calculated from lofting the
beach surfaces in SolidWorks 3-d CAD at a given time, t = t;, during the CEM beach evolution
simulations. The mass properties tool in SolidWorks was used to calculate the volume change
between the beach surface at t = t; and the pre-disposal basement surface (represented by the 22
December 2014 surveys). An example of this procedure is illustrated by Figures 48 and 49 for
the disposal scenario with V0 = Vcru = 280,345 yds3 of cycloid-dune beach fill, at t = +90 days
post-dredge disposal. This scenario includes 79,455 yds3 on North Beach and 200,593 yds3 on
Middle and South Beach with the South groin removed. In Figure 49, we zoom in on a cutplot
at 90 days into the CEM beach evolution simulation, showing how the dune has been eroded and
re-distributed by extreme wave runup at high tides, leaving a total residual volume on North
Beach, South Beach and Middle Beach of V, = 214,994 yds3 , while the residual volume on
Middle and South Beach is V,. = 173,053 yds3 of sand remaining between the basement surface
(silver) and the new beach surface (brown). The new beach surface at t;= 90 days conforms
closely to an elliptic cycloid profile. Given this residual sand volume the sand loss rate that has
occurred on Middle and South Beach between the start of the CEM simulation at t = t O and t = f;
= 90 days is given by:
>, i 0.50
.... a> a.
C")
1/)
"C >,
0 0.40
1/) "C C: (1)
(/) ::::,
0 .s:: ::.o.3o
i "C Si:
"C
V
..; 0.20
C: (1)
Cl) -0
* a::0.10
(/)
(/)
0 ...J
a> Cl ~
10.00
160.0
Middle & South Beach Fill Retention
Cycloid-Dune Template
----Loss Rate, yds'/day
----Retention Time, days
~ minimum loss rate = 306 yds3/day
180.0 200.0 220.0 240.0 260.0
Total Disposal Volume, V0 ( thousands of yds3)
280.0 300.0
700
690
660
670
660
650
640
630
620
610 Ill
600 >, ca 590 "C
580 1-0
570
560 ~-
550 ·-540 ....
530 5
520 ~
510 Q) -500 Q)
490 0::
480
470
460
450
440
430
420
410
400
Figure 47: Sensitivity analysis of average rate of loss of beach fill < dV I dt > (red, left hand
axis) and the retention time, T0 , (blue, right hand axis) as a function of the total beach fill
volume, V0 , using the proposed cycloid-dune templates (cf. Figures 25-30 ). Each solution point
(crosses) is an ensemble average of 8 modeled outcomes coinciding with the timing of historic
Middle and South Beach dredge disposal events, relative to the wave forcing in Figure 15.
~
a, t § ~ U1 100
'I" g~
j~
Iii• Ill ____________ ..,...,o
2000 111111
Northing, ft
Figure 48: Three-dimensional CAD model of two overlaid surfaces on North, Middle and South
Beach. The brown layer is the beach surfaces after waves re-work 280,345 yds3 of cycloid-dune
beach fill, at t = +90 days post-dredge disposal; this includes 79,455 yds3 on North Beach and
200,593 yds3 on Middle and South Beach. The silver layer is the bottom of the critical mass
envelope or basement surface (from the 22 December 2014 piofilc measurements). CAD model
shown with 10 to 1 vertical exaggeration.
0
sand remaining @ t = + 90 days
Middle+ South Beach= 173,053 cubic yards
North+ Middle+ South Beach= 214,994cubicyards
Figure 49: Cross-section of 3-d CAD model of two overlaid surfaces on North, Middle and South Beach after waves re-work cycloid-
dune beach fill, (brown) at t = +90 days post-dredge disposal; and 2) the bottom of the critical mass envelope (as delineated in silver
from the 22 December 2014 profile measurements). CAD model shown with 10 to 1 vertical exaggeration.
200,593-173,053 = 306 yds3/day
90
This gives an estimate of retention time (at t;= 90 days) of
(14)
T, Vcrit 200,593 X 90 _ 656 da S
0 (dV I dt)i=t; (200,593-173,053) y
(15)
Four such SolidWorks loftings of the CEM beach surfaces were done at four different times,
t; =t; ,t2 ,t3 ,t4 , during each of the 18 scenario simulations, and the sand loss rates and retention
times calculated at each of those times were ensemble averaged to give the points and cubic
spline curves plotted in Figure 47.
While an elliptic cycloid is an equilibrium beach surface, it does not produce a state of
zero sand loss in the presence of a negative divergence of drift, which is the persistent littoral
drift state along Middle and South Beach. Recall from Section 5.0 that when the divergence of
drift is negative (because less sand fluxes into Middle Beach at the south inlet jetty than is
expelled from South Beach abeam the EPS generating building, cf. Figure 44c ), then the
equilibrium cycloidal beach profile will shift landward, eventually intersecting the basement
surface of the critical mass envelope, (ie, the 22 December 2014 profile). Once this happens,
then the cycloidal shape of the profile is disrupted, and the equilibrium state of the profile is lost.
The concept behind the cycloid-dune template is that, as the cycloid begins approach an
intersection with the basement surface of the critical mass envelope, (under the erosional effects
of continued negative divergence of drift), it also intersects the base of the dune and receives
i:iciclitioni:t I c::P.climP.nt c.ovP.r i:tc:: thP. cl11nP. P.roriP.'< i:tncl "PrPi:trl'< 011t ciown'<lopP i:tc.roc::.'< thP c::till inti:td
cycloidal surface. So, the dune acts as a restoring mechanism that re-supplies the cycloid with
sand lost to negative divergence of drift.
Comparing the results in Figure 4 7 against the historic baseline in Figure 46, the general
trends are similar, but the sand loss rates are greatly diminished and retention times significantly
increased by using the proposed cycloid-dune template (in Figures 25-30). Again, the most
efficient use of Agua Hedionda dredged sands occurs when the cycloid-dune template is filled to
no more than critical mass ( V0 ~ Vcrit = 200,890 yds3), ~hich reduces average sand loss rates on
Middle and South Beach to an absolute minimum of< dV I dt >= 306 yds3/day, while extending
retention time to T0 = 656 days, sufficient to retain a sandy beach over a typical bi-annual dredge
cycle. This is a 35% improvement in sand retention time over historical dredge disposal practices
at Middle and South Beach, which could result in a reduction of sand influx rates into Agua
Hedionda Lagoon by a similar factor during the months following Middle and South Beach
disposal.
If the cycloid-dune template is filled to more than critical mass ( V0 > Vc,it) by adding
more sand to the back-beach dune, then Middle and South Beach retention time will increase
beyond 656 days and reach as much as T0 = 693 days if the reserve sand volume in the dune were
more than doubled to VD= 129,985 yds3 (achieving the historic maximum placement volume of
V0 = 281,195 yds3). But, again, this is not a good return on doubling the investment in reserve
beach fill for Middle and South Beach because retention time is only increased by an extra 5
weeks while the sand loss rate increases by 33 % to < dV I dt >= 406 yds3/day, (an additional
100 yds3 /day of sand loss). This inefficiency occurs because the enlarged dune encroaches
further seaward into the middle bar-berm portion of the profile that is subject to more frequent
wave attack, and the groin field formed by the inlet and discharge jetties at Middle and South
Beach is already filled to carrying capacity at the critical mass of Ve,;,= 200,890 yds3•
On the other hand, under-filling the cycloid-dune template, (by building a reduced dune),
leads to accelerated sand loss rates and reduced retention times. If the Middle and South Beach
cycloid dune templates were filled with the historic minimum beach fill of V0 = 163,996 yds3,
(by under-building the back-beach dune with only 12,786 yds3) then sand loss rates would
increase to 375 yds3/day and retention times would be reduced by to T0 = 437 days, a 33%
reduction in sand retention time relative to the ideal build using beach fill equal to critical mass.
The prevailing negative divergence of drift across Middle and South Beach causes the initial
cycloid profile in the lower portion of the template to shift landward, and once intersection with
the basement surface of the critical mass envelope occurs, there are insufficient sand reserves in
the reduced dune to resupply the cycloid in the presence of continued negative divergence of
drift. Once the reserve sand supply in the dune is exhausted, the cycloidal shape of the profile is
disrupted, and the equilibrium state of the profile is lost. Even so, if the cycloid-dune template on
Middle and South Beach were filled to a volume equivalent to the 2104/2015 disposal event (V0
= 229,693 yds3) by using a dune containing only 78,483 yds3, then sand retention times are still
significantly better than what was achieved using the standard 1: 10 (rise over run) template.
With this over-built dune in combination with the cycloid, retention times following the
2014/2015 dredge cycle could have been T0 = 676 days with sand loss rates reduced to 340
yds3/day, an improvement of 39% over what was achieved using standard Middle and South
Beach disposal practices.
6) Summary and Conclusions:
A detailed set of beach profile surveys at Middle and South Beach in Carlsbad CA were
provided by Cabrillo Power I LLC, delineating beach surfaces before and after the 2104/2015
dredging of Agua Hedionda Lagoon, (AHL), which placed 229,693 yds3 between the south inlet
jetty to Agua Hedionda Lagoon and the Encina Power Station (EPS) generating building. The
surveys were accurately performed by Noble Engineers using differential GPS and known
historic benchmarks. Three-dimensional CAD models were lofted from the measured points
along the three (3) Middle Beach survey range lines (Cab 1-04 -Cab 1-06) and three (3) South
Beach survey range lines (Cab 1-01 -Cab 1-03) to delineate the beach surfaces immediately
before beach dredge disposal (based on the 22 December 2014 profile measurements) and
immediately after dredge operations were completed (based on the 17 April 201 7 profile
measurements). When these two surfaces were lofted together in a common reference frame, it
was determined that 178,584 cubic yards of beach fill have been retained after placing 229,693
cubic yards on Middle and South Beach between 1 January 2015 and the post-dredge survey on
17 April 2015. This calculates to an average sand loss rate of 473 yds3/day and projects a sand
retention time of 485 days. This is significantly longer than the retention time at the North Beach
disposal site where retention time projected to only 33 days.
To understand the reasons for the contrasting sand retention characteristics of North
Beach vs. Middle and South Beach, a baseline beach evolution study was conducted using the
Coastal Evolution Model (CEM) to hindcast the fate of beach fill placed on the three receiver
beaches. The CEM was developed at the Scripps Institution of Oceanography with a $1,000,000
grant from the Kavli Foundation, (see bttp://repositories.cdlib,org/sio/techreport/58/ ), and is
based on latest thermodynamic beach equilibrium equations published in the Journal of
Geophysical Research. Inputs to the CEM baseline study were based on measured shoaling wave
data, grain size data for the dredged sands, and daily beach fill volumes were derived from the
dredge monitoring reports to the Regional Water Quality Control Board (Cabrillo, 2015) and
from Cabrillo dredging data bases.
Between 1998 and 2015 there have been eight (8) different events when Agua Hedionda
dredged sands have been disposed concurrently on all three receiver beaches (North, Middle and
South Beach). Historic dredged sand volumes placed on North Beach ranged from 62,030 yds3 to
161,525 yds3, while beach fill placed on Middle and South Beach ranged from 163,996 yds3 to
281,195 yds3 • The CEM beach evolution simulations of these events determined that the
minimum sand loss rate occurs when beach fill volumes on the three receiver beaches are equal
to the critical mass, which was found to be Vc,u = 79,471 yds3 for North Beach, (see companion
North Beach report, Jenkins, 2017), and Vc,it = 200,890 yds3 for Middle and South Beach. The
critical mass is the theoretical maximum carrying capacity of a beach fill site for supporting a
beach profile in equilibrium. The carrying capacity of a beach is limited by the width of the
wave-cut platform in the bedrock on which beach sands have accumulated over geologic time
scales. The wave-cut platform at North Beach is only 550 ft. to 600 ft. in width and 600 ft. to 650
ft. at Middle and South Beach. Many of the beaches throughout north San Diego County are
perched on narrow wave-cut platforms. The platforms are narrow because they were carved by
wave action into erosion resistant bedrock formations during the present high-stand in sea level,
and these narrow wave-cut platforms physically cannot hold large quantities of beach sand; and
often become fully denuded during periods of high-energy winter waves.
Another contributing factor to the limited carrying capacity of the three receiver beaches
is that they are exposed to a prevailing negative divergence of drift caused by the way the
bathymetry surrounding the Carlsbad Submarine Canyon produces variable wave shoaling along
the length of these beaches. The presence of the Carlsbad Submarine Canyon creates a bright
spot in the shoaling wave pattern that diminishes in intensity with increasing distances toward
the north. For example, wave heights are locally higher at the inlet jetties than further to the
North around Maple Avenue. The prevailing littoral drift transports beach sand southward
throughout the entire Oceanside Littoral Cell; but the alongshore imbalance in shoaling wave
height causes higher southerly longshore transport rates of sand at the southern end of each of the
receiver beaches than at the northern ends. Consequently more sand exits each receiver beach at
its southern end due to longshore transport, than enters at the northern ends from sand sources
further updrift. This inequality in sand transport rates between the north and south ends of the
receiver beaches is referred to as divergence of drift, and when the sand transport rates are higher
at the down-drift end of the receiver beaches, it becomes a constant loss system referred to as
negative divergence of drift. So, when beach fill volumes exceed the critical mass of the receiver
beaches, the excess sand cannot be supported in equilibrium on its narrow wave-cut platform and
is quickly lost to the negative divergence of drift. However, this effect is somewhat muted at
Middle and South Beach because the AHL inlet jetties and the EPS discharge jetties produce a
groin field which impedes the longshore transport at both the updrift and downdrift ends of the
Middle/South Beach complex, and provides extra storage capacity for sand on the wave cut
platform. Consequently retention times for beach fill on Middle and South Beach is longer than
at North Beach.
Historically, the CEM baseline study finds that when a standard 1: 10 (rise over run)
beach fill template on North Beach is filled to critical mass, the theoretical minimum sand loss
rate to negative divergence of drift is 1,495 yds3/day, and the sand retention time is 53 days (see
the companion North Beach report, Jenkins, 2107). By contrast, the standard 1: 10 beach fill
template at Middle and South Beach historically achieved minimum sand loss rates of 431
yds3/day, and the sand retention times of 466 days. But, when beach fill sand volumes at Middle
and South Beach were increased by 41 % over critical mass ( as occurred during the 2000/2001
dredge event when 281,195 yds3 were placed on Middle and South Beach), the retention time is
only increased by 7 % from T0 = 466 days to T0 = 497 days. In contrast, over-filling the North
Beach receiver site produces an even worse return on beach fill investment. During the
2002/2003 dredge event, 161,525 yds3 were placed on North Beach, (103% increase over critical
mass), but the retention time increased by only 26 % from T0 = 53 days to T0 = 67 days, while
the sand loss rate increased by 61 % to 2,411 yds3/day. This is an increase in sand loss rates at
North Beach of 916 yds3/day. Unfortunately, such increases in sand loss rates at North Beach
correlate with proportional increases of sand influx rates into Agua Hedionda Lagoon.
The 2010/2011 survey data show that AHL sand loss rates also increase when the fill
volumes are less than the critical mass. Sand influx rates in 2010/2011 were 519 yds3/day when
only 163,996 cubic yards were placed on Middle and South Beach (36,894 yds3 below critical
mass requirements). Bear in mind that the critical mass is the minimum volume of sand required
to establish an equilibrium beach profile on a wave-cut platform; and a beach is in its most stable
state with an equilibrium profile. But with a prevailing negative divergence of drift along Middle
and South Beach, equilibrium cannot be achieved due when there is insufficient sand volume,
and consequently sand loss rates increase with a destabilized, non-equilibrium profile.
Following CEM beach evolution analysis of the Middle and South Beach historic
baseline, attention was given to finding a more effective beach fill template that could increase
sand retention using beach fill from Agua Hedionda Lagoon dredging. Beach fill has typically
been placed on Carlsbad beaches using a standard beach fill template with a flat backshore
platform and a 1: 10 (rise over run) seaward facing beach slope extending down to Oft. MLL W.
This convention dates back to the Regional Beach Sand Project, (AMEC, 2002). However,
stable beach profiles in Nature have a much more gradual, curving profile with slopes that range
between 1 :50 to 3:100. Formulations of equilibrium beach profiles are found in the U.S. Army
Corps of Engineers Shore Protection Manual and later the Coastal Engineering Manual; and the
latest most advanced formulation is known as the elliptic cycloid. The elliptic cycloid
formulation can account for continuous variations in the equilibrium beach profile due to
variability in wave height, period and direction when occurring in combination with variations in
beach sediment grain size and beach sand volume. Therefore, a new beach fill template has been
proposed here for Middle and South Beach referred to as the cycloid-dune template (see Figures
25-30). The shape of the template is based on the extremal elliptic cycloid which is the
equilibrium profile for the highest wave in the period of record. But the extremal elliptic cycloid
extends below the MLL W tide line and earth moving equipment which spread out the beach fill
cannot work below MLL W. So, the template truncates the extremal elliptic cycloid at MLL W
and places the residual volume of critical mass (totaling 49,680 yds3) in a back-beach dune that
stretches 3,680 ft. from the south inlet jetty to Agua Hedionda Lagoon to the north end of the
EPS generating building.
While an elliptic cycloid is an equilibrium beach surface, it does not produce a state of
zero sand loss in the presence of a negative divergence of drift, which is the persistent littoral
drift state along Middle and South Beach. When the divergence of drift is negative, the
equilibrium cycloidal beach profile will progressively shift landward as it loses sand to negative
divergence of drift, eventually intersecting the basement surface of the critical mass envelope.
Once this happens, then the cycloidal shape of the profile is disrupted, and the equilibrium state
of the profile is lost. The concept behind the cycloid-dune template is that, as the cycloid begins
to approach an intersection with the basement surface of the critical mass envelope, (under the
erosional effects of continued negative divergence of drift), it also intersects the base of the dune
and receives additional sediment cover as the dune erodes and spreads out downslope across the
still intact cycloidal surface. Thus, the dune acts as a restoring mechanism that re-supplies the
cycloid with sand lost to negative divergence of drift.
The construction method envisioned for the cycloid-dune template begins with building
the back-beach dune portion first, starting at the south inlet jetty and adding sections to the
dredge pipeline until the build-out of the dune extends beyond the South Groin abeam of the
north end of the EPS generating building. Building the dune first creates a "safe" reservoir of
sand before the template can be fully constructed, and sand from this reservoir is only released to
the lower eroded basement surface during periods of the highest tides and waves. After the
buildout of the dune to the southern end of South Beach, the cycloid portion of the template is
laid out beginning from the toe of the dune and spreading the material down slope to MLL W,
and working back across Middle Beach to the south inlet jetty, removing pipeline sections as the
cycloids are completed
CEM beach evolution simulations of the Middle and South Beach cycloid-dune templates
were run for future conditions with the South Groin removed, and show significant
improvements in sand loss rate and retention time relative to the historic baseline. Again, the
most efficient use of Agua Hedionda dredged sands occurs when the cycloid-dune template is
filled to no more than critical mass (200,890 yds3 for Middle and South Beach), which reduces
average sand loss rates on Middle and South Beach to an absolute minimum of 306 yds3/day,
while extending retention time to 656 days. This is a 35% improvement in sand retention time
over historical dredge disposal practices at Middle and South Beach. If the cycloid-dune template
is filled to more than critical mass by adding more sand to the back-beach dune, then Middle and
South Beach retention times will increase beyond 656 days. If the reserve sand volume in the
dune on Middle and South Beach were increased by a factor of 2.6 to 129,985 yds3 (producing a
cycloid dune equivalent to the historic maximum placement volume of V0 = 281,195 yds3) then
retention time could be extended to a maximum of 693 days. But, again, this is not a good return
on doubling the investment in reserve beach fill placed in the back-beach dune because retention
time is only increased by 5 weeks while the sand loss rate on Middle and South Beach would
increase by 33% to 406 yds3/day, (an additional 100 yds3/day of sand loss). This inefficiency
occurs because the enlarged dune encroaches further seaward into the middle bar-berm portion
of the profile that is subject to more frequent wave attack, and the groin field formed by the inlet
and discharge jetties at Middle and South Beach is already filled to carrying capacity at the
critical mass of Ve,;,= 200,890 yds3•
On the other hand, under-filling the cycloid-dune template, (by building a reduced dune),
leads to accelerated sand loss rates and reduced retention times. If the Middle and South Beach
cycloid dune templates were filled with the historic minimum beach fill of V0 = 163,996 yds3,
(by under-building the back-beach dune with only 12,786 yds3) then sand loss rates would
increase to 375 yds3/day and retention times would be reduced by to T0 = 437 days, a 33%
reduction in sand retention time relative to the ideal build using beach fill equal to critical mass.
The prevailing negative divergence of drift across Middle and South Beach causes the initial
cycloid profile in the lower portion of the template to shift landward, and once intersection with
the basement surface of the critical mass envelope occurs, there are insufficient sand reserves in
the reduced dune to resupply the cycloid in the presence of continued negative divergence of
drift. Once the reserve sand supply in the dune is exhausted, the cycloidal shape of the profile is
disrupted, and the equilibrium state of the profile is lost. Even so, if the cycloid-dune template on
Middle and South Beach were filled to a volume equivalent to the 2104/2015 disposal event (V0
= 229,693 yds3) by using a dune containing only 78,483 yds3, then sand retention times are still
significantly better than what was achieved using the standard 1: 10 (rise over run) template.
With this over-built dune in combination with the cycloid, retention times following the
2014/2015 dredge cycle could have been T0 = 67 6 days with sand loss rates reduced to 340
yds3/day, an improvement of 39% over what was achieved using standard Middle and South
Beach disposal practices.
7) References:
AMEC, 2002, "Regional Beach Sand Project Post-construction Monitoring Report for Intertidal,
Shallow Subtidal and Kelp Forest Resources", submitted to SANDAG,
http://www.sandag.org.
Cabrillo, 2015, '"'Order 96-32: First Quarter 2105, Second Quarter 2015 and Final Monitoring
Report for Agua Hedionda Lagoon Dredging", submitted to California Regional Water
Quality Control Board, 30 pp.
CDIP, 1984-1988, "Coastal data information program, monthly reports," U.S. Army Corps of
Engineers, California Department of Boating and Waterways, Monthly Summary Reports
#97-#150.
CDIP, 1976-1995, "Coastal Data Information Program, Monthly Reports," U.S. Army Corps of
Engineers, California Department of Boating and Waterways, SIO Reference Series, 76-
20 through 95-20.
CDIP, 1993-1994, "Monthly Summary Report," SIO Reference Series (93-27) through (94-19).
CDIP, 2016, "Coastal Data Information Program" http://cdip.ucsd.edu/
Elwany, M. H. S., A. L. Lindquist, R. E. Flick, W. C. O'Reilly, J. Reitzel and W. A.
Boyd, 1999, "Study of Sediment Transport Conditions in the Vicinity of Agua Hedionda
Lagoon," submitted to California Coastal Commission, San Diego Gas & Electric, City
of Carlsbad.
Elwany, M. H. S., R. E. Flick, M. White, and K. Goodell, 2005, "Agua Hedionda Lagoon
Hydrodynamic Studies," prepared for Tenera Environmental, 39 pp.+ appens.
Ellis, J.D., 1954, "Dredging Final Report, Agua Hedionda Slough Encina Power Station," San
Diego Gas and Electric Co., 44pp.
Inman, D. L. and B. Brush, 1970, "The coastal challenge" Science, vol38, no. 5 pp36-45.
Inman, D. L. & S. A. Jenkins, 1985, "Erosion and accretion waves from Oceanside
Harbor," p. 591-593, in Oceans '85: Ocean Engineering and the Environment,
IEEE and Marine Technology Society, v. 1,674 pp.
Inman, D. L. and Masters, P. M., 1991, "Coastal sediment transport concepts and
mechanisms," Chapter 5 (43 pp.) in State of the Coast Report, San Diego
Region, Coast of California Storm and Tidal waves Study, V. S. Army Corps
of Engineers, Los Angeles District Chapters 1-10, Appen. A-I, 2 v.
Inman, D. L., M. H. S. Elwany and S. A. Jenkins, 1993, "Shorerise and bar-berm profiles on
ocean beaches," Jour. Geophys. Res., v. 98, n. Cl 0, p. 18,181-199.
Inman, D. L., S. A. Jenkins, and M. H. S. Elwany, 1996, "Wave climate cycles and
coastal engineering practice," Coastal Eng., 1996, Proc. 25th Int.
Conf,(Orlando), Amer. Soc. Civil Eng., Vol. 1, Ch. 25, p. 314-327.
Inman, D. L. & S. A. Jenkins, 1997, "Changing wave climate and littoral drift along the
California coast," p. 538-549 in 0. T. Magoon et al., eds., California and the
World Ocean '97, ASCE, Reston, VA, 1756 pp
Inman, D. L. & S. A. Jenkins, 1999, "Climate change and the episodicity of
sediment flux of small California rivers," Jour. Geology, v. 107, p. 251-270.
Inman, D. L. & S. A. Jenkins, 2004, "Scour and burial of objects in shallow
water," p. 1020-1026 in M. Schwartz, ed., Encyclopedia of Coastal Science,
Kluwer Academic Publishers, Dordrecht, Netherlands.
Jenkins, S. A. and D. W. Skelly, 1988, "An Evaluation of the Coastal Data Base Pertaining to
Seawater Diversion at Encina Power Plant Carlsbad, CA," submitted to San Diego Gas
and Eiectric, Co., 56 pp.
Jenkins, S. A., D. W. Skelly, and J. Wasyl, 1989, "Dispersion and Momentum Flux Study of the
Cooling Water Outfall at Agua Hedionda," submitted to San Diego Gas and Electric, Co.,
36 pp.+ appens.
Jenkins, S. A. and J. Wasyl, 1993, "Numerical Modeling of Tidal Hydraulics and Inlet Closures
at Agua Hedionda Lagoon," submitted to San Diego Gas and Electric, Co., 91 pp.
Jenkins, S. A. and J. Wasyl, 1994, "Numerical Modeling of Tidal Hydraulics and Inlet Closures
at Agua Hedionda Lagoon Part II: Risk Analysis," submitted to San Diego Gas and
Electric, Co., 46 pp.+ appens.
Jenkins, S. A. and J. Wasyl, 1995, "Optimization of Choke Point Channels at Agua
Hedionda Lagoon using Stratford Turbulent Pressure Recovery," submitted to San
Diego Gas and Electric, Co., 59 pp.
Jenkins, S. A. and J. Wasyl, 1997, "Analysis of inlet closure risks at Agua
Hedionda Lagoon, CA and potential remedial measures, Part II," submitted to San
Diego Gas and Electric, Co., 152 pp. + appens.
Jenkins, S. A. and J. Wasyl, 1998a, Analysis of Coastal Processes Effects Due to the San
Dieguito Lagoon Restoration Project: Final Report, submitted to Southern California
Edison Co., 333 pp.
Jenkins, S. A. and J. Wasyl, 1998b, Coastal Processes Analysis of Maintenance Dredging
Requirements for Agua Hedionda Lagoon, submitted to San Diego Gas and Electric Co.,
176 pp.+ appens.
Jenkins, S. A. and D. L. Inman, 1999, A Sand transport mechanics for equilibrium in tidal inlets,
Shore and Beach, vol. 67, no. 1, pp. 53-58.
Jenkins, S. A. and J. Wasyl, 2001, Agua Hedionda Lagoon North Jetty Resoration Project: Sand
Influx Study, submitted to Cabrillo Power LLC., 178 pp. + appens.
Jenkins, S. A. and J. Wasyl, 2003, Sand Influx at Agua Hedionda Lagoon in the Aftermath of the
San Diego Regional Beach Sand Project, submitted to Cabrillo Power LLC., 95 pp. +
appens
Jenkins, S. A. and J. Wasyl, 2005, Hydrodynamic Modeling of Dispersion and Dilution of
Concentrated Sea Water Produced by the Ocean Desalination Project at the Encina Power
Plant, Carlsbad, CA. Part II: Saline Anomalies due to Theoretical Extreme Case
Hydraulic Scenarios, submitted to Poseidon Resources, 97 pp.
Jenkins, S. A. and J. Wasyl, 2005, "Oceanographic considerations for
desalination plants in Southern California coastal waters," Scripps Institution
of Oceanography Tech. Rpt. No. 54, 109 pp + appendices.
http://repositories.cdlib.org/sio/techreport/54/
Jenkins, S. A. and J. Wasyl, 2005, "Coastal evolution model," Scripps Institution of
Oceanography Tech. Rpt. No. 58, 179 pp+ appendices.
http://repositories.cdlib.org/sio/techrepmt/58/
Jenkins, S. A. and D. L. Inman, 2006, "Thermodynamic solutions for equilibrium
beach profiles", Jour. Geophys. Res., v.3, C02003, doi:10.1029, 21pp.
Jenkins, S. A., Inman, D.L., Michael D. Richardson, M.D., Thomas F. Wever, T.F. and J.
Wasyl, 2007, "Scour and burial mechanics of objects in the nearshore", IEEE
Jour. De.Eng, vol.32, no. 1, pp 78-90.
Jenkins, S. A. and J. Wasyl, 2011, "Hydrodynamic Approach to Wetland Restoration by
Optimization of Bridge Waterways", Department of Transportation, District 11
Environmental Division, MS-242, Technical Report# 1 lAl 766, 258 pp.
Jenkins, S. A. and S. Taylor, 2015, "Storm and Tidal Conditions Determination
for Coastal Drainage Design," submitted to Office of Highway Drainage Design
Division of Design, MS 28, California Department of Transportation
Jenkins, S. A., 2017, "Beach Equilibrium Analysis of North Beach Disposal Options for Dredged
Sands from Agua Hedionda Lagoon, Carlsbad, CA," submitted to Cabrillo Power I LLC,
66 pp.
Merkel, 2008, " Agua Hedionda Outer Lagoon Flood Shoal Maintenance Dredging Sediment
Characterization Report, Tech Rpt # ACOE-2001100328-SKB, submitted to Cabrillo
Power, LLC., 45 pp.
NOAA, 1998, A Verified/Historical Water Level Data@
http://www.opsd.nos.noaa.gov/data res.html
NWS, 2009, "National Weather Service Daily Climate Reports,"
http://www.wrh.noaa.gov/sgx/obs/rtp/carlsbad.html
U.S. Army Corps of Engineers, 1985, "Littoral zone sediments, San Diego Region,
October 1983 -June 1984", Coast of California Storm and Tidal Wave Study,
CCSTWS 85-11.
U.S. Army, Corps of Engineers (USACE), 1991, "State of the Coast Report, San Diego
Region," Los Angeles District, CA: Coast of California Storm and Tidal Waves
Study, Final Report 1.
U. S. Army, Corps of Engineers (USACE), 2006, "Coastal Engineering Manual,"
Engineering Manual 1110-2-1100, U.S. Army, Corps of Engineers, Washington, DC, (in
6 volumes).
U.S. Department of Commerce National Ocean Service, 1986, "Tide tables 1986, high and low
water predictions for west coast of North and South America", 234 pp.
USGS, 1997, "USGS Digital Data Series DDS-37 at INTERNET URL,"
http://wwwrvares.er.usgs.gov/wgn96cd/wgn/wq/region18/hydrologic unit code.
Van der Meer, J.W., 2002. Wave Run-up and Overtopping at Dikes. Technical Report,
Technical Advisory Committee for Water Retaining Structures (TAW), Delft, the Netherlands
Appendix-A: Maintenance Dredging History for Agua Hedionda Outer Lagoon
Dredging Disposal
Year Date Volume cubic Influx Volume Comments
Basin *Location
Start Finish yard Days Yds3/Day cubic yard
1955 Aug-55 Sep-55 90,000 Outer 90,000 s Maintenance
1957 Sep-57 Dec-57 183,000 822 223 Outer 183,000 s Maintenance
1959-60 Oct-59 Mar-60 370,000 821 451 Outer 370,000 s Maintenance
1961 Jan-61 Apr-61 227,000 396 573 Outer 227,000 s Maintenance
1962-63 Sep-62 Mar-63 307,000 699 439 Outer 307,000 s Maintenance
1964-65 Sep-64 Feb-65 222,000 703 316 Outer 222,000 s Maintenance
1966-67 Nov-66 Apr-67 159,108 789 202 Outer 159,108 s Maintenance
1968-69 Jan-68 Mar-69 96,740 700 138 Outer 96,740 s Maintenance
1972 Jan-72 Feb-72 259,000 1,067 243 Outer 259,000 s Maintenance
1974 Oct-74 Dec-74 341,110 1,034 330 Outer 341,110 M Maintenance
1976 Oct-76 Dec-76 360,981 731 494 Outer 360,981 M Maintenance
1979 Feb-79 Apr-79 397,555 851 467 Outer 397,555 M Maintenance
1981 Feb-81 Apr-81 292,380 731 400 Outer 292,380 M Maintenance
1983 Feb-83 Mar-83 278,506 699 398 Outer 278,506 M Maintenance
1985 Oct-85 Dec-85 403,793 1,006 401 Outer 403,793 M Maintenance
1988 Feb-BB Apr-88 333,930 852 392 Outer 333,930 N,M,S Maintenance
1990-91 Dec-90 Apr-91 458,973 1,095 419 Outer 458,973 M,S Maintenance
1992 Feb-92 Apr-92 125,976 366 344 Outer 125,976 N Maintenance
1993 Feb-93 Apr-93 115,395 365 316 Outer 115,395 M Maintenance
Outer 74,825 N ,
1993-Dec-93 Apr-94 158,996 365 436 Outer 37,761 M Maintenance 1994
Outer 46,410 s
Outer 106,416 N
1995-96 Nov-95 Apr-96 443,130 731 606 Outer 294,312 M Maintenance
Outer 42,402 s
1997 Sep-97 Nov-97 197,342 579 341 Outer 197,342 M Maintenance
Dec-97 Feb-98 59,072 92 642 Middle 59,072 M Modification
1998 120,710 M
Feb-98 Jul-98 214,509 150 1,430 Inner Modification
93,799 s
1999 Feb-99 May-99 155,000 304 510 Outer 155,000 N Maintenance
141,346 N
2000-01 Nov-00 Apr-01 422,541 701 603 Outer 195,930 M Maintenance
85,265 s
2002-03 Dec-02 Apr-03 354,266 730 485
2004-05 Jan-05 Mar-05 348,151 704 495
2006-07 Jan-07 Apr-07 333,373 763 437
2006-09 Dec-08 Apr-09 299,328 733 408
2010-11 Der.-10 Apr-11 226,026 736 307
2014-15 Dec-10 Apr-11 294,661 736 400
TOTAL 8,528,842
MAINTENANCE TOTAL 8,255,261
*Location: N= North Beach; M = Middle Beach; S = South Beach
Green= pre-back-passing sand influx rates
Red = post-back-passing sand influx rates
161,525 N
Outer 131,377 M Maintenance
61,364 s
100,487 N
Outer 170,515 M Maintenance
77,149 s
149,166 N
Outer 121 ,036 M Maintenance
63,167 s
104,141 N
Outer 102,000 M Maintenance
93,185 s
62,030 N
Outer 93,696 M Maintenance
70,300 s
64,968 N
Outer 156,056 M Maintenance
73,637 s
8,526,640
APPENDIX-B: Details of the TIDE_FEM Tidal Transport Model
A finite element approach was adapted in preference to more common finite
difference shallow water tidal models, e.g., Leendertse (1970), Abbott et al (1973), etc.
Finite difference models employ rectangular grids which would be difficult to adapt to
the complex geometry of the systems of channels of the Agua Hedionda. It is believed
that large errors would accumulate from attempting to approximate the irregular
boundaries of the Agua Hedionda system with orthogonal segments. On the other hand,
finite element methods allow the computational problem to be contained within a domain
bounded by a continuous contour surface, such as the Sf contours stored within the
bathym file. The finite element mesh used to model the tidal fluxes at Agua Hedionda
Lagoon using the TIDE _FEM model is shown in Figure B-1.
TIDE_FEM employs a variant of the vertically integrated equations for shallow
water tidal propagation after Connor and Wang (1975). These are based upon the
Boussinesq approximations with Chezy friction and Manning's roughness. The finite
element discretization is based upon the commonly used Galerkin weighted residual
method to specify integral functionals that are minimized in each finite element domain
using a variational scheme, see Gallagher (1981). Time integration is based upon the
simple trapezoidal rule [Gallagher, 1981]. The computational architecture of
TIDE_FEM is adapted from Wang (1975), whereby a transformation from a global
coordinate system to a natural coordinate system based on the unit triangle is used to
reduce the weighted residuals to a set of order-one ordinary differential equations with
constant coefficients. These coefficients (influence coefficients) are posed in terms of a
shape function derived from the natural coordinates of each nodal point. The resulting
systems of equations are assembled and coded as banded matrices and subsequently
solved by Cholesky's method, see Oden and Oliveira (1973 and Boas (1966).
We adapt the California coordinates as our global coordinate system (x, y) to
which the nodes are referenced, with x (easting) and y (northing). The vertical
coordinate z is fixed at 0.0 ft NGVD and is positive upward. The local depth relative to
0.0 ft NGVD is h and the mean surface elevation about 0.0 ft NGVD is IJ. The total
depth of water at any node is
H = h + '1· The vertically averaged xy-components of velocity are (u, v). The continuity
and momentum equations may be written from Connor and Wang, (1974), as:
(Bl)
Extreme, High Water (EHW) 7.65 ft MLLW
---0 500 1000 1500 2000 2500
HORIZO TAL SCALE IN FEET
Extreme High Water (EHW) 7 .65 ft MLL W
Extreme High Water (EHW) 7.65 ft MLLW
DR. SCOTT A. JENKINS CONSULTING Agua Hedionda Finite Element Mesh SCOTT A. JENKINS PhD & JOSEPH WASYL
Figure B-1: Finite element mesh used to model tidal transport at Agua Hedionda Lagoon
Here qx, qy are mass flux components
1/
qx=pfudz
-h
1/
qy =pfvdz
-h
(B2)
(B3)
and qi is the mass flux through the ocean inlet due to water surface elevation changes in
the estuary:
(B4)
Fp is the pressure force resultant and Fxx, Fxy, Fyy are "equivalent" internal stress
resultants due to turbulent and dispersive momentum fluxes
,, pgH2
F = fpdz =--
P 2 -h
(BS)
o o Fyx = Fxy = e(-qy +-qJ oy ox
and c is the eddy viscosity. Bx and By are the bottom stress components
oh Bx =rx +pgH-ox
oh BY =ry + pgH oy
(B6)
In Equation (B6), 'tx and Ty are the bottom shear stress components that are quasi-
linearized by Chezy-based friction using Manning's roughness factor, no:
(B7)
where Cz is the Chezy coefficient calculated as:
C = l.49 H 116 (B8)
z no
Boundary conditions are imposed at the locus of possible land/water boundaries, Sr in the
bathym file and at the ocean inlet, So. Flux quantities normal to these contours are
denoted with "n" subscripts and tangential fluxes are given "s" subscripts. At any point
along a boundary contour, the normal and tangential mass fluxes are:
1/
qn = f pundz=a.xqx+anyqy
-h
1/
qs = f pusdz = -anxqx + anyqy
-h
a nx = cos(n, x)
any= cos(n,y)
(B9)
Components of momentum fluxes across a boundary are equivalent to internal force
resultants according to:
Fnx =anxCFxx -FP)+anyFyx
Fny =an/Fyy -FP)+anxFxy
On land boundary contours, the flux components are prescribed
on land
(BIO)
(Bl 1)
On the ocean boundary, the normal boundary forces (due to sea surface elevation) are
continuous with ocean values, and the mass exchange is limited by the storage capacity
of the estuary. Hence
qnm = qi at inlet (Bl2)
In the problem at hand Fnn is prescribed at the inlet by the ocean tidal elevation, 170 , and
the inlet sill depth, ho according to
-pg( )2 Fnm = -1/o + ho 2
on So (B13)
Ocean tidal forcing/unctions 170 were developed in Section 3. The ocean
boundary condition as specified by Equation (B12) places a dynamic boundary condition
on the momentum equations and a kinematic boundary condition on the continuity
equation that is constrained by the storage rating curve. Solutions are possible by
specifying only the dynamic boundary condition, but then mass exchanges are controlled
by the wetting and drying of individual grid cells with associated discretization and
interpolation errors which threaten mass conservation. The technique of over specifying
the ocean boundary condition with both a dynamic and kinematic condition is discussed
in the book by Weiyan (1992).
The governing equations and the boundary conditions are cast as a set of integral
functionals in a variational scheme, [Boas, 1966]. Within the domain of each element of
the mesh, Ai the unknown solution to the governing equations is simulated by a set of
trial functions (H, q) having adjustable coefficients. The trial functions are substituted
into the governing equations to form residuals, (RH, Rq). The residuals are modified by
weighting functions, (~H, ~q). The coefficients of the trial functions are adjusted until
the weighted residuals vanish. The solution condition on the weighted residuals then
becomes:
A,
Jf Rq~qdA = 0
A,
By the Galerkin method of weighted residuals, [Finlaysen, 1972], the weighting functions
are set equal to nodal shape functions, <N>, or:
The shape function, <N>, is a polynomial of degree which must be at least equivalent to
the order of the highest derivative in the governing equations. The shape function also
provides the mechanism to discretized the governing equations. The shape function
polynomial is specified in terms of global (California) coordinates (Figure B2) for the
first nodal point, N 1 of a generalized 3-node triangular element of area Ai,. Wang (1975)
obtained significant numerical efficiency in computing the weighted residuals when the
shape functions of each nodal point, Ni, are transformed to a system of natural
coordinates based upon the unit triangle, giving Ni ~ Li, see Figure 8b. The shape
functions also permit semi-discretization of the governing equations when the trial
functions are posed in the form:
Specifying the Shape Function <N>
for any 3-Node Triangular Element
a) Global (California) Coordinates
y
.,__ __________ X
<N> = (N1, N2, N3)
Coordinate ______ ;_..> Ni = Li
Transform
N1 =[(X2Y3 -X3Y2 )+(y2 -y3 )x+(x3-Y2 )y ] /2Ai
2Ai =(x1-x3)(Y2 -y3 )-(x2-x3)(Y1 -y3)
1.0
b) Natural Coordinates
y
L1 L3 ____________ X
1.0
x=L1 x 1 +L2 x 2 +L3 x 3
y=L1 Y1 +L 2Y2 +L3y3
L1 +L2+L3=l.O
Figure B2: Shape function polynomial and transform to natural coordinates for a generalized 3-node triangular element; (a) 3-
node element in California coordinates; (b) 3-node dement in natural coordinates.
H(x,y,t) = LH/t)N;(x,y)
I
q(x,y,t) = Lq1(t)N/x,y)
J
(B14)
Discretization using the weighting and trial functions expressed in terms of the nodal shape
functions allows the distribution of dependent variables over each element to be obtained from
the values of the independent variables at discrete nodal points. However, the shape function at
any given nodal point, say N1, is a function of the independent variables of the two other nodal
points which make up that particular 3-node triangular element. Consequently, the computations
of the weighted residuals leads to a series of influence coefficient matrices defined
aii = ~-fJ N;N1dA ,
I ff 8N . S--=-N .-1 dA
lJ A, I ax
1 ff oN. t,,=-N .-1 dA
!, A,.. I 8y (B15)
g .. k =-1 ff NN. aNk dA
" A ' 1 a· i X
h.k = _!_ff N.N. oNk dA
" A; ' 1 By
The influence coefficient matrices given by equation (B 15) are evaluated in both global and
natural coordinates. Once the influence coefficients have been calculated for each 3-node
element, the weighted residuals reduce to a set of order-one ordinary differential with constant
coefficients. The continuity equation becomes:
I( a !i d:,) = -~~[guk(Hiqxk + HkqJ+hyk(H;qyk +Hkqyj
I( a, d!;1 )--~~[g,,(q,,q, )+ h,,(q,,q.,, )]+ N, ~ N1S, + g~s,H,
L(al} ddqy}) = -LL [gijk (qxjqyk )+ hijk (qyJqxk )]+ N; L NJS 17 + g LfuH;
{ j k J I
(B16)
Equations (B 16) are essentially simple oscillator equations forced by the collection of algebraic
terms appearing on the right hand side; and are therefore easily integrated over time. The time
integration scheme used over each time step of the tidal forcing function is based upon the
trapezoidal rule, see Gallagher 1981) or Conte and deBoor (1972). This scheme was chosen
because it is known to be unconditionally stable, and in tidal propagation problems has not been
known to introduce spurious phase differences or damping. It replaces time derivatives between
two successive times, ~t = tn+l -tn, with a truncated Taylor series. For the water depth it would
take on the form:
dH -= 17(t) dt
Hn+I -Hn = ~ fon+l +17J+EM
E=__!__(M)2 d217
12 dt2
(B17)
To solve equation (Bl 7), iteration is required involving successive forward and backward
substitutions.
The influence and friction slope coefficient matrices together with the trapezoidal rule
reduce equations to a system of algebraic equations [Grotkop, 1973] which are solved by
Cholesky's method per a numerical coding scheme by Wang (1975). For more details, refer to
the TIDE_FEM code in Appendix-I of Jenkins and Wasyl (1996), and Gallagher (1981) or Oden
and Oliveira (1973).
APPENDIX-C: Equilibrium Beach Profile Algorithms:
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. Ill, C02003, doi:10.1029/2005JC002899, 2006
Thermodynamic solutions for equilibrium beach proftles
Scott A. Jenkins 1 and Douglas L. Inman2
Received 26 January 2005; revised 17 November 2005; accepted 10 December 2005; published 11 February 2006.
[1] Solutions are developed for beach profiles using equilibrium principles of
thermodynamics applied to simple representations of the nearshore fluid dynamics.
Equilibrium beaches are posed as isothermal shorezone systems of constant volume that
dissipate external work by incident waves into heat given up to the surroundings. By
the maximum entropy production formulation of the second law of thermodynamics (the
law of entropy increase), the shorezone system achieves equilibrium with profile shapes
that maximize the rate of dissipative work performed by wave-induced shear stresses.
Dissipative work is assigned to two different shear stress mechanisms prevailing in
separate regions of the shorezone system, an outer solution referred to as the shorerise and
a bar-berm inner solution. The equilibrium shorerise solution extends from closure depth
(zero profile change) to the breakpoint, and maximizes dissipation due to the rate of
working by bottom friction. In contrast, the equilibrium bar-berm solution between the
breakpoint and the berm crest maximizes dissipation due to work by internal stresses of a
turbulent surf zone. Both shorerise and bar-berm equilibria were found to have an
exact general solution belonging to the class of elliptic cycloids. The elliptic cycloid
allows all significant features of the equilibrium profile to be characterized by the
eccentricity and the size of the ellipse axes. These basic ellipse parameters are evaluated
by process-based algorithms and empirically validated parameters for which an extensive
literature already exists. The elliptic cycloid solutions displayed wave height, period
and grain size dependence and demonstrate generally good predictive skill in
point-by-point comparisons with measured profiles.
Citation: Jenkins, S. A., and D. L. Inman (2006), Thermodynamic solutions for equilibrium beach profiles, J. Geophys. Res., lll,
C02003, doi:l 0.1029/2005JC002899.
1. Introduction
[2] Comprehensive reviews of the previous geomorphi-
cally and statistically based attempts to define and param-
eterize equilibrium beach profiles are contained in Dean
[1991], Inman et al. [1993], and Short [1999]. Here we
follow the prior conceptual treatment of beach profiles,
where an equilibrium profile is defined as the shape attained
by a beach in response to steady wave forcing over long
periods of time. Field measurements and hydraulic models
show that beach profiles change shape with changing
intensity of wave action, but given sufficient time
under steady forcing the beaches attain a constant form
[e.g., Shepard and La Fond, 1940; Bagnold, 1947;
USACE, 1947; Inman and Filloux, 1960; Nordstrom and
Inman, 1975; Inman et al., 1993]. These findings led to
the temlS "summer/winter" beach cycles and the concept
of an equilibrium beach profile [Inman, 1960; Inman
and Bagnold, 1963].
1Marine Physical Laboratory, Scripps Institution of Oceanography,
Unlv.rslly ofCallfoinlt, SM Dicto, La Jolla, C~Hfo,nla, USA. 11n!egrmivc Oce<\nog:rnphy Division, Scripps lnilitullon of Oceanogra-
phy, University of California, San Diego, La Jolla, CaUfomia, USA.
Copyright 2006 by lhe American Geophysical Union.
014 8-0227 /06/2005JC002899$09 .00
[ 3] The first general formulation for a continuous train
of waves that break 011 a beach appears to be that of
Inman and Bagnold [1963]. Their equilibrium formulation
was based on a balance between the net upslope gradient
in wave energy versus the downslope gradient in potential
energy due to gravity. The energy gradient for their model
was subsequently refined [Bailard and Inman, 1981] by
including velocity skewness moments for the upslope
wave energy gradients. In order to calculate equilibrium
profiles, these models require an a priori set of velocity
time series with their spectral moments.
[4] A mass flux balance formulation of the equilibrium
condition was introduced by Bowen [1980] and later
applied by Stomis et al. [2002] to barrier beach migration
in the Caspian Sea. Bowen's formulation requires a
balance between the upslope and downslope transport rates
using relations of bedload and suspended load transport
after Bagnold [1963]. Bowen's transcendental solutions had
several asymptotic f01ms of interest. One was based on the
assumption of vanishing suspended load in the presence of
wave asymmetry from the Stokes second harmonic and
had the fonn, h = .4.'t:us. h1 these limiting fonns, the profile
factor A increases with increasing sediment fall velocity
(or grain size). Like the Bailard and Inman [1981]
fonnulation, these solutions depend on numerous point-
dependent variables that are seldom known.
C02003 I of 21
C02003 JENKINS AND INMAN: THERMODYNAMIC SOLUTIONS FOR BEACH PROFILES C02003
[s] Dean [1977] developed an equilibrium profile fotmu-
lation h = Ax215 based on the assumption that the dissipation
of wave e11e-18y per unit area of beach is cons_tallt, and the
form h = Ar whe11 the dissipation of wave e11ergy per unit
volume of water is constant. Later, Dean [1991], following
the work of Bruun [1954], elaborated on the h = .Ax213
solution, indicating that A is an increasing function of grain
size. This derivation was based on the assumption that the
wave heigh!' decreases linear!.y with depth. This assumption
limits the domain of the .?3 solution to the inner 1>m-tion of
the beach profile, shoreward of the wave break point
(although it is commonly applied to the entire profile in
engineering practice).
[6] Inman et al. [1993] developed an equilibrium profile
consisting of two conjoined curves, both of the fonn h =
Ax"' that were best-fit to 60 separate beach profiles. They
considered distinct summer and winter equilibrium states
because these correlate with seasonal wave climate and give
recurrent curves for beach profiles taken in the field In the
region between the breakpoint and the berm crest ("bar-
benn"), it was found that only one exhibited a curvature
exponent of m -213, and that was for the transitional
profile between the winter and summer equilibrium profiles.
The best fit results produced variations in curvature between
slllllmer and winter equilibria of 0.29 S: m S: 0.55 in the
bar-benn, and 0.21 S: m S: 0.5 in the shorerise. Inman et
al. [1993] show that the conjoined shorerise and bar-benn
curves gave significantly better matches to the data, with
rms en-ors in depth generally under 35 cm nud volume
errors ranging from 13 to 134 m3/m. Comparable en·ors for
the single fitted curves using the Deau [199 I] formulation
were 90 cm and 225 to 321 m3/m [l11ma11 el al., 1993).
2. Thermodynamic Formulation of Beach
Equilibrium
[7] Equilibrium states in natural systems are governed by
tlle second law ofthennodynamics [Moore, 1962; Halliday
and Resnick, 1967; Anderson, 1996]. Thermodynamics has
been used to solve equilibrium problems in other geophys-
ical processes such as volcanism, mineral fonnation, and
melts [Newton et al., 1981; Salje, 1988; Anderson, 1995],
thermohaline and mesoscale ocean circulation [Sverdrup et
al., 1942; Eckart, 1962; Imawaki et al., 1989; Val/is et al.,
1989; Salmon, 1998], atmospheric and marine layer dynam-
ics [Bane, 1995; Bohren and Albrecht, 1998], global climate
models [ Ozawa et al., 2003] and heterogeneous fluid and
sediment systems [ Grinfeld, 1991 ; Casas-Vazquez and Jou,
1991]. The longshore sand transport solutions oflnman and
Bagnold [1963] and Komar and Inman [1970] are also
thennodynamic applications, where steady longshore trans-
port is an equilibrium response to the rate of work
perfonned by longshore directed radiation stress. Further,
thennodynamics offers the advantage of providing solu-
tions from measurable fundamental properties (macroscopic
variables). These properties typically represent averages
over time of the gross characteristics of a system. In the
shorezone they include incident wave height Hoo, wave
period T, beach sand sizes D, and the location of the profile
relative to some benchmark, X1•
[s] We begin with a simple representation of the universe
consisting of two domains, the shorezone system bounded in
green (Figure la), and everything outside the shorezone
system referred to as the surroundings. In the system the
sand elevation h is measured positive downward from mean
sea level and cross-shore position x is positive in tlle
offshore direction (Figure 1 b ). The system boundaries are
stationary, enclosing a constant volume V that contains a
fixed volume of sand v. sufficient for equilibrium to occur
in the presence of a maximum incident wave height if 00•
Equilibrium profile states are fully contained within the
system boundaries. The seaward poundary is a vertical
plane at the critical closure depth he corresponding to the
maximum incident wave [e.g., Kraus and Harikai, 1983].
The landward boundary is a veLtical plaite at U1e bean crest
(cross), a distance ,1'1 from a bench mark. The cross-shore
length of the system from the benn crest to closure depth is .:t. The distance from the point of wave breaking to closure
depth is Xc2 such that Xe = Xc2 + X2, where .X2 is the
distance from the benn crest to the origin of the shorerise
profile near the wave breakpoint. We consider equilibrium
over time scales that are long compared with a tidal cycle
and profiles that remain in the wave dominated regime
where the relative tidal range (tidal range/H) < 3 [Short,
1999]. Under these conditions, the curvilinear coordinate
that defines the profile referenced to mean sea level (MSL)
vertical datum is,
d~ = /(dx)2 + (dh)2 = V1 + _;,2 dh = Vl+hddx
x = dx. h' = dh (1)
dh' dx
where d~ is calculated separately for inner (bar-benn, d~1)
and outer (shorerise, d(i) portions of the conjoined profile.
[9] Fluxes of energy into and out of the shorezone system
are shown by arrows crossing the system boundaries in
Figure la. Work W per unit length of shoreline .:ly is
performed on the system by the incident waves that provide
energy to the system at a rate given by
where E = pgH2/8 is energy per unit longshore surface
area, p and g are water density and acceleration of gravity,
Cg is wave group velocity, and His local rms wave height.
The waves shoal and break inside the shorezone system,
dissipating wave energy into an increment of heat dQ.
This evolution of heat produces an incremental entropy
change
dS=dQ r.
where Ta is absolute temperature. Heat is removed from
the shorezone system to tlle surroundings primarily by
advection and turbulent diffusion in tlle nearshore circula-
tion system [Inman et al., 1971] and secondarily through
heat of vaporization in sea spray.
[10] The second law of thermodynamics (often referred
to as 'the entropy law') is a necessary condition for
equilibrium and requires that a natural process that starts
in one equilibrilllll state and ends in another will cause the
2 of 21
C02003 JENKINS AND INMAN: THERMODYNAMIC SOLUTIONS FOR BEACH PROFILES C02003
a. Entropy lncntaee of surroundings, dS = dQ IT
SuR"OUndlnga dQ ( heat of vaporization l
llllfft)
dW ( ~,k~) =g
dQ
( latent heat l
-systlffland
volume, Y.
I
b. fa: 1:
Shorllrise 'I' Bffenn Xci
x,, -~
"'
x~
-5 "z
0 1 MSl E
..; 5 " l,o Ito
Figure 1. Definition schematic for (a) coordLnates of a thermodynamic system in the shorezone, taken
as n rnp1·ese11tative cross section ofa unifonn three-dimensional control cell, and (b) physical Cartesian
coordinates [after Inman et al., 1993] that apply to the thennodynamic system. Note that symbols with a
circwnflex (e.g., X.,) extend to system boundaries while !hose without occur within boundm·ies.
entropy of the system plus its surroundings (universe) to
increase,
{)S) _ {)S) + {)S) > O (2)
8t UNIVERSE -{)/ SYSTEM {)t SURllOUNDING
[11] The dcci ive i ue with 1· pect to the thcnnodynamic
state of the shorezone system is the fate of the heat evolved
within it and whether the entropy increase associated with
tbat .heat evolution is l'Ctained by the system or expo1ted 10
its surroundings. We adopt a heat transport/entropy produc-
tion f01mulation for the shorezone system that is an ana-
logue of that used to describe dissipation in global climate
state models [Paltridge, 1975, 1978; Ozawa et al., 2001,
2003]. The derivation for entropy production in a fluid
system is found in de Groot and Mazur [ 1984] and Landau
and Lifshitz [1980] and can be written,
~~)UNIVERSE=/* [{)(~;Ta)+ v' · (pcvT,u) + pv' · u] dV
+Jq·n, dA (3) r.
where cv is the specific heat at constant volume, u is the
fluid velocity, p is the fluid pressure, q is the diabatic heat
flux taken as positive when oco1.1rring outward ncross
system boundaries and n, is the unit normal vector on the
system boundary. The volume integral in (3) is the rate of
change of entropy of the fluid system and represents the first
L 11 on the right side of (2). The surface integral. in (3) is
taken over the system boundary (green line in Figure la)
and represents the discharge rate of entropy into the
surrow1dings due to heat flu: across the system bowid&y.
When a fluid system is in a steady state, Chandrasekhar
[1961] has shown that the first law of thermodynamics
reduces the volume integral in (3) to:
{)S) I I I 4' -=--v'·qdV+ -dV {)t SYSTEM T, T, (4)
where qi is the dissipation function representing the rate
of viscous dissipation of kinetic energy per unit volume.
The first term on the right side of (4) represents the
entropy change due to transport of latent heat by advection
and diffusion. If latent heat is transported out of the system
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C02003 JENKINS AND INMAN: THERMODYNAMIC SOLUTIONS FOR BEACH PROFILES C02003
(q positive), the system losses entropy. Because entropy is
a state function of the system, it must remain constant if
the system is to achieve equilibrium [Landau and Lifshitz,
1980]. Therefore, any equilibrium fluid system that exports
latent heat to its surroW1dings must compensate for the
associated entropy loss through the production of new
entropy at an equivalent rate.
[12] When heat fluxes across the system boW1dary, the
entropy of the surroW1dings changes at a rate given by the
surface integral in (3). This surface integral can be
expressed in terms of a volume integral through the appli-
cation of Gauss's theorem, giving the entropy change of the
surroW1dings in terms of heat fluxes within the system
volume
8S) ;q·n, JI -= -dA= -'v·qdV 8t SUAAOUNDING Ta T,
+ J q·'v(i)dv (5)
The first term on the right side of (5) is the entropy
change occurring in the surroW1dings due to the latent
heat that was imported from the system, while the second
is due to heat conduction along temperature gradients
formed within the system between regions of hot and
cold Entropy changes in the surrouridings due to latent
heat transpo1t are equal and opposite in sign to those
occurring in the system (4), and taken together, produce
no net change in the total entropy of the universe.
hlstead, the entropy of the universe can only be changed
by temperature gradients and viscous dissipation occurring
within a fluid system, as foWld after inserting (4) and (5)
in (2),
88) = f .! dV + J q · 'v(_!_)dv > 0 (6) 8I UNIVERSE T. T.
[13] When applying (6) we assume the shorezone system
is isothe1mal and hence the second term on the right is
vanishingly small. We support this assumption by noting
that the body of empirical data from the field has never
shown warmer water Wlder breaking waves than foWld
elsewhere in the shorezone, nor have episodes of high
waves been correlated with episodes of elevated surfzone
temperatures. If the shorezone is isothermal, then no entro-
py production is possible from the heat conduction mech-
anism and the second law by (6) reduces to,
8S) =f_! dV>O 81 UNIVERSE T,
(7)
[14] Under these circumstances the first law of thermo-
dynamics (4) requires the rate of entropy production by
viscous dissipation inside the system to be in balance with
the rate at which entropy is discharged to the surroW1dings
by latent heat transport,
!_! dV = f .!...'v · qdV T. T. (8)
This balance maintains constant entropy inside the system,
8S) _0 8t SYSTEM -
(9)
The particular value at which the shorezone entropy remains
constant is determined by the number of grains of sand
contained within that system (Appendix A).
[1s] If the entropy flux balance between system and
surroW1dings in (8) were not upheld, then equilibrium states
would not be possible in the shorezone because the system
could not maintain constant state function as in (9). Without
latent heat transport from the shorezone, it will W1dergo a
progressive build up of entropy as the train of incident
waves continually work on the system, adding more and
more heat while those waves are dissipated. In this circum-
stance, the shorezone becomes the thermodynamic equiva-
lent of Joule's experiment [Zemansfy and Van Ness, 1966]
for which the first law of thermodynamics requires dQ =
dW. Assuming a constant bottom slope as a lowest order
approximation, this equation can be solved for the increase
in local water temperature l!i.Ta that would occur in time /!;.t
as a consequence of continuous working by a steady train of
incident waves:
Here Hb is breaker height, 'I is a factor relating the depth
of wave breaking hb to breaker height Hb = '/hb, tan~0 is
the mean beach slope, and c" = 3,941.3 J/kg °K is the
specific heat of sea water at 29l°K (18°C) and 35%0
salinity [Cox and Smith, 1959]. For a breaking wave of
height Hb = 1 m with 'I -4/5 and a nomirial beach slope
of tanj30 = 0.025, the temperature of the nearshore waters
would increase continuously by about 3°K (or 3°C) every
24 hours. Such warming is not observed in nature, and
surf temperatures are not known to increase with wave
height, indicating that shorezones do not violate (8). The
fact that shorezones appear to be isothe1mal (when solar
flux is negligible) suggests that thermodynamic equili-
brium is a common and persistent state for these systems
in nature.
[16] Systems like the shorezone that achieve equilibrium
through the dissipation of external work into the heat of a
reservoir in the surroW1dings, belong to the general ther-
modynamic system known as external mechanical irrevers-
ibility, [Zemansfy and Van Ness, 1966]. While (7) and (9)
are conditions for equilibrium and (8) defines the type of
equilibrium, these are not sufficient conditions to define a
unique equilibrium state for the shorezone system. To obtain
unique solutions, we adopt the criteria of maximum entropy
production (MEP) that has been successfully applied to
certain steady state equilibrium climate states by Dewar
[2003], and Ozawa et al. [2003]. The MEP criteria is a
particular form of the second law (3), that requires the
entropy of the Wliverse not only increases when the system
proceeds from one equilibrium state to the next, but that the
entropy increase is a stationary maximum. The validity of
the MEP criteria is based on observational and numerical
evidence showing that in general, non-linear systems having
many degrees of freedom for dynamic equilibrium tend to
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those states, among all possible states under the second law,
that maximize entropy production.
[17] When the MEP criteria is applied to (7) subject to
(9), we neglect reflection and talce the state variable asso-
ciated with the external work done by incident waves as the
independent v01iable 011d then seek a stationary maximum
for the viscous dissipation, which becomes the dependent
state variable. Since the shorezone system volume 011d
sediment volume have been fixed, the fluid/sediment inter-
face (bottom profile) is the only remaining thermodynamic
coordinate that is unrestrained 011d available to maximize
dissipation. From Batchelor [1970], the average rate of
dissipation of mechanical energy per unit volume in a 2-
dimensional, incompressible fluid system is <I>= 2µ(w · w) =
('i' · w), where ( ) denotes time averaging; Gi is the fluid
vorticity generated by the action of viscosity µ 011d ;-is the
time varying wave induced shear stress, including bottom
shear stresses, internal shear stresses 011d granular friction at
the fluid sediment interface [Bagnold, 1956; Inman and
Bagnold, 1963]. We assume that no vorticity or dissipation
(due to bottom ventilation) occurs within that portion of the
system occupied by the sediment mass. Let dV' = aydA'
represent a volume increment of the remaining portion of
the system that contains the fluid vorticity, where dA' is 011
increment of area bounded by the closed contour C arow1d
the fluid portion of the system (red contour, Figure la) and
ay is a unit length of shore.line. Applying these assumptions
and definitions to Stokes theorem, fw · n,dA' =fit· dC, the
average dissipation rate of the system becomes:
[1s] When (10) is used to maximize entropy production
in (7), only the segment of contour integration talcen along
the bottom profile produces a ch011ge in the state variables
of heat and work, as all remaining segments are comprised
of fixed system boundaries. fu the shorezone system, the
bottom profile defines the pathway along which heat 011d
work are evolved 011d both state variables are path depen-
dent in an irreversible process. Hence, the MEP formulation
of the second law in (7) reduces to maximization of a simple
line integral:
8~ = ~Yj(-r. u)d(;. > o 8t) UNIVERSE Ta
(maximum) (11)
[19] Our equilibrium problem now becomes that of find-
ing the profile curve C = C(h, x, x') after (1) that makes the
integral in (11) a stationary maximum. This c011 be accom-
plished with calculus of variations using a ch011ge of
variables in the integrand of (11) in terms of a generalized
functional F(h, x, x') written
F(h,x,x') = (f · u): (12)
With the functional in (12) the entropy integral in (11) is
maximized by solving the Euler-Lagt311ge equation [Boas,
1966],
~8F _ 8F =O
dh8x' 8x (13)
General solutions to (I 0) are given in section 3 for the
shorerise 011d bar-berm profiles, while particular solutions
are found in section 4.
3. General Solutions
[20] We will pose separate formulations for the viscous
dissipation in the shorerise (shoaling zone) 011d bar-berm
(surf zone) portions of the shorezone system (Figure lb).
When applied to (13), these separate formulations will yield
general solutions for the shorerise profile C2 011d bar-be1m
profile C1 that happen to belong to the same class of
equation. The solutions for the shorerise and bar-berm are
conjoined at the wave break point.
[21] The simplest surrogate for the shorerise is one that is
uniform in the alongshore direction in the region between
closure depth 011d the wave breakpoint (Figure I b ). The
fluid dynamics in this region are approximated by the linear
shoaling transformation of the shallow water Airy wave,
u = Um cos( at -lex)
_H(x) CF
Um -2 y h(x)
R ( )1/4 H(x) = vTci h(x)
k=-(J-
Jgh(x)
(14)
Here, k = 21t/wavelength is local wave number, CJ = 21t/
period is radi011 frequency, um(x) is velocity amplitude at
the sea floor boundary layer, 011d H00 is incident wave
height. Local wave height 011d depth, HM and h~"), are
talcen with respect to local curvilinear coordinate of the
bottom profile Ci, as shown in Figure 1 b. The Airy
approximation in (14) has been shown in Mei [1989] to be
valid over sloph1g bottoms if the following mild slope
condition is satisfied:
(15)
where tan ~ = dhldx is the local bottom slope. Exactly
how much smaller th011 unity (15) must be is not
definite, but its largest value is at the breakpoint where
kh = <J(H,J-y g)1'2• We assume there is some N » 1
such that (15) is satisfied everywhere in the shorerise by
requiring
dh ~ Edx ~ O(L)
E = ~ (Hb) 1/2"" rtlS (Hoo) 2/5
N yg 21/SN 8'Y
(16)
where L is a characteristic length scale, 1;. is a stretching
factor proportional to the Airy wave mild slope factor N.
3.1. Shorerise Profile
[ 22] futemal she01· stresses are neglected in the shorerise
due to the absence of wave breaking. A simple power law
formulation is used to prescribe the bottom shear stress,
f = T0 cos(crl -lex+ <p)
To = pqu;;, = pK,.U::,
(17)
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where 'To is the shear stress amplitude; i,p is the phase angle
between the bottom shear stress and the oscillatory potential
flow velocity from Airy theory; c1 is the quadratic drag
coefficient and n is the shear stress velocity exponent
referred to as shear stress linearity. The particular value of n
varies with the dependence or c1 011 pnrnmeters of dynamic
similitude e.g., oscillatory Reynolds nwnber (It.= U:,,lcrv),
grain Reynolds number (Reg= urr:J)lv), Keulegan-Carpenter
(inverse Strouhal) number (St= u.,/cr,i,), where 11. is bottom
rougiu1ess, [ e.g., Taylor, 1946; Ke11lega11 011d Carpenter,
1958; S/eath, 1984). !fwe generalize er.-(.R'~88;'), then
n = 2 + 2} + l + w, and the shear stress amplitude can be
written in terms of a proportionality factor KT that is
independent of Um and consists of a collection of other
factors contained in R0, Reg and S, that make (17)
dimensionally correct.
[23] From (14) and (17), the dissipation rate per unit
length of profile varies with depth h as
(--) AV /H·l 'tU '!"' II = l'"T COS<p I'm = /rJ(.+1)/4
(18)
pK.,. COS !p H'.'+I gJf,H+l )/4
1:u-"" -80!•+11/2
where n = fJ/L°' and the second integration constant is
-1i/4n. The first root is given by,
R = R, = (2J2} [4!1h" -4!121?".
+-2-(1 -4!1h" + 4!12112")] !+CY.
and the second root by,
R = Rb =(21~1,)
2
[1 -4!1h" +4n2h2".
+-2-(4!1/i" -4!12h2")] !+CY.
(22a)
(22b)
Here f.1l, f.2l are elliptic integrals of the first and second
kind, respectively.
[24] The general solution given by (22) belongs to a class
of equations known as elliptic cycloids [Boas, 1966]. We
can show that by making a transformation into polar
coordinates (r,0) with a substitution of variables:
where the work factor -ro is independent of h, x and x'. When 9 = arc cos(! -2!1h") (23)
we select from (I) an infinitesimal arc length of the
shorerise profile having the fonn, from which we get
dC.2 = vi+ x'2dh
then the integral in (I 1) for which we seek a stationary
maximum becomes:
(19)
In terms of nond:imensional variables denoted by an
underscore, x = x/L; h = h/L, the following functional F is
collected from the order-I te1111s in (19) for use in the Euler-
Lagrange equation in (13)
(20)
Because fJFlfJ!, = 0, the first integration of(13) using (20)
gives,
q=f (21)
where ex = 3 ( n + 1 )/2 and il is the first integration
constant. We can rationalize the integrand of (21) using
two separate Euler substitutions after Gradshteyn and
Ryzhik, [1980, sections 2.261 and 2.264). These provide a
general solution with two roots that has the following
dimensional form,
x=------h2"+-arccos(l-2!11i") 0c ... -1i, .. [ ~;,--1 ]
&./R n 2n (22)
~ -h2" = -1-sin2 9 n 4!12 (24)
With (23) and (24), equation (22) reduces to two types of
elliptic cycloids having the general polar coordinate form:
2ri'ki•)
x=x2 =-0-(9-sin9) m (25)
where r is the radius vector measured from the center of an
elli1,se whose sem:imajor and semiminor ax.es are a, b and e·~) is fhc elliptic integral of the first or second kind (k1.z>
depending on which of the two cycloid types we resolve
from (22a) and (22b). We limit our discussions to the
solutions for the type-a cycloids that result from the first
root in (22a) because these were shown to be in good
agreement with field data. The polar equivalent of the type-a
cycloid from (25) has a radius vector whose magnitude is
[
ti21l _l 112 av1-=-e2
r = r. = n2 6i112 9+ b2 cos2 OJ = Jsin2 9 + (! -e2) cos2 9
(26)
where e is the ~enu'icity of the ellipse given by e =
.jl -(b2 / a2). TI1e polar fo1m of the type-a cycloid in
(22.a) is based on the elliptic integral of the second kind
that has an analytic approximation, P.1 = (1i12)./{2 -e2)/2,
see Hodgman [1947]. The integration constant n in (22) is
determined from the dimensions of the ellipse axes by noting
when 0 = arc cos (1 -2nh"') = 1i, then nh"' = 1. This gives:
(type-a cycloid) (27)
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type-a cycloid
\
\
\
....
Boundary Conditions
x@P(J), Xi =X1
(8=0) Xa =Xa
x@P(3),
(8=1t)
P(3)
...
h
Figure 2. EUiptic cycloid solution (~·ed line) fo1· the equilibriwn be~h profile traced by a point on a
rolling ellipse with semimajor and semiminor axis a, b, eccentricity e = J 1 -(b2 / a2) polar COOl'dinafe
vector r, and angle ofrotation e, and e: = 1.0.
rt is api>are.itt lhat tlie root R in (22a) is equivaJei,t to ,-f/•~)/b
in (25) when
2
I -e2 = (1 + a)
Hence, the eccentricity of the elliptic cycloid is govcmed by
the shear stress lil1eority, 11
(28)
The inverse of(22) subject to (25) gives the companion polar
equation for the elliptic cycloid,
h = h2 = 1'rEX2 (l -c~so) = r(I -cos9)
21,(1ti,,) 9-smO (29)
[2s] A geometric representation of the type-a elliptic
cycloid used in the general solution for the shorerise profile
is shown in Figure 2 as traced by an ellipse having
eccentricity e = 0.75 and e: = 1. The equilibrium beach
profile is given by the trajectory of a point on the semimajor
axes of an ellipse that rolls seaward in the cross-shore
direction under the plane of h = h2 = 0. This trajectory
defines the el.liptic cycloid and tl1e segment traced by the
first half of a rotation cycle (0 < 0 < 1'r) of the rolling ellipse
is the equilibrium beach profile (solid red curve). The depth
of water at the seaward end of the profile (0 = 1t) is h = 2a in
the case of the type-a cycloid. The length of the profile Xis
equal to the semi-circumference of the ellipse,
X = 2aJJ21 e!!! 1'r a J2 -e2
e E 2 at 9 = 1'r (type-a cycloid)
(30)
The detemunation of the particular values for a or b
semi.axes and e, will be established from boundary
conditions and field measurements in section 4.
3.2. Bar-Berm Profile
[26] The bar-berm spans the region of shorezone between
the wave break poh1(. aud the becm crest. (Figure lb) where
the surf zone dissipation may be represented by several
simple analytic formulations. The simplest is a dissipation
formulation based on a depth-limited bore,
('f ·u) = !scg
I E=g pgH2
Cg =ygh
H = K. h~
(31)
where K. is the bore decay factor relating bore height to some
powel' Tl of the local wat<:r dep1h after formul:1tio1ts derived
from measurements in open channel flow detailed in Chow
[1959] and He11rle1·s011 [1966). From (I) we lake the
altemative represemation for nn infmitesinlal arc length of
the bar-bemi profile,
(32)
With (31) and (3 2), the integral in (11) that must be
made stationary to maximize entropy production
becomes,
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C02003 JENKINS AND INMAN: THERlvIODYNAMIC SOLUTIONS FOR BEACH PROFILES C02003
where
_!_EC = K h(4<,tJ/2 dh1 (lX[ g I [ dxt
K _ 411+ I l/2 2
I -16 pg K,
In terms of nondimensional variables, the order-I terms
collected from (33) give the following form for the
functional to be used in the Euler-Lagrange equation in
(13) as,
(
h'2
) I+ =-e:2 (34)
Since {)F/0!_1 = 0, the first integration of (13) using (34)
yields
q=J (35)
where CJ is the dimensionless integration constant. The
indefinite integral in (35) for the depth-limited bore is
equivalent to (21) when a= 411 -1 and integrates into an
elliptic cycloid of the form (22) having the alternative
polar form (23)-(25) with an eccentricity,
where 11 is defined in (31 ).
(21] The integration constant is CJ= fi,J/L4'fl-J = (2a)-4T1+1_
For a linear bore (11 = 1) after Bowe11 et al. [1968],
the eccenllicity _of the bnr-benn cycloid becomes e =
./ffj."" 0.707 While the integ,:ation COllStant is C1 "'(2ar3
for the type-a bar-berm cycloid.
[2s] An empirical approach to the formulation of surf
zone dissipation is obtained from the work of Thornton and
Guza (1983] that produced estimates of the intemal dissi-
pation per unit area due to wave breaking. By fitting
empirical relations to measured breaking wave distributions,
they formulated two separate dissipation functions. Elliptic
cycloid solutions can be derived for the bar-berm of these two
relations using calculus similar to that outlined in (32)-(35).
The details of these derivations are given in Appendix B. The
first of these two empirically based solutions follows from the
hypothesis that waves break in proportion to the distribution
of all waves and produces a type-a bar-benn ·cycloid widt
ecceutri.cily e = ;j97IT ,..,., 0.904 based 011 a dimensional
integratio11 constant of (20)-10. Alten1atively, Tliomton 011d
Guza [1983] developed a second dissipation function from
the assumption that waves break in proportion to the distri-
butio11 of the lUl'gost waves. This asswnption resulted in a bar-
bemi cycloid having llll eccentricity e = J'ffi "" 0.845 and
integration constalll of(2a)-6 for the type-a cycloid.
(29] These three possible general solutions have been
developed here and in Appendix B for the bar-berm
equilibrium profile from separate formulations of surf zone
dissipatioJL Each of these solutions is represented by type-a
elliptic cycloids of a specific eccentricity but arbitrary size,
(a, b). In the following section, field data will be used to
resolve the cycloids that are most commonly found in beach
surveys, while boundary and matching conditions will help
resolve the dimensions of these cycloids.
4. Particular Solutions
[3o] In this section we apply bow1dary and matching
conditions to the general solutions developed in the previ-
ous section to obtain particular solutions for the shorerise
and bar-berm profiles that conjoin at the breakpoint, X3
(Figure 1 b ). The general solutions developed in the previous
section admit to an arbitrary number of equilibriwn profiles
depending on the type of elliptic cycloid and its eccentricity.
Here we use field measurements to resolve the eccentricity
and select the best-fit cycloid that conforms to natural
beaches. A Taylor series expansion of (33) about x = 0
gives a simple analytic approximation to the general elliptic
cycloid solution that is equivalent to the equilibrium profile
formulations developed earlier by Dean [1977, 1991] and
Inman et al. [1993]. The leading order terms of this Taylor
series expansion are,
h =Ax"'+ 0(£!/(l+<>))
where terms O (e:111+") are neglected and
2 2(1 -i:2)
m = (2 + ~) = (3 -el)
(36)
(37)
For the type-a cycloid, the profile factor A in (36) becomes
A-3n(2a) •
[
_ op] 2/(2+a)
-4/J2J (38)
[31] Both the profile factor A and the curvature exponent
m of the classical paranietric representation in (37, 38) are
functions of the eccentricity of the elliptic cycloid. How-
ever, only the profile fuctor A varies with cycloid size. We
use these dependencies in combination with the extensive
data base on (A, m) derived from best-fits to beach profile
measurements [Inman et al., 1993] to establish a criteria
for the selections for a and e, that give particular solutions.
4.1. Shorerise
[n] Beginning with the shorerise profile, the origin of the
cycloid at P(l) in Figure 2 is positioned at mean sea level
where 0= 0 811d h = h2 = 0 atx =x2 = 0. Itis apparent that the
elliptic cycloid must converge on closure depth he within
one-halfrevolution of the cycloid wheel, h = h2 --+ he as 0 --+
it. Hence, the size of the shorerise ellipse ax.es are given by
a= a2 = hc/2 (39)
This means that the closure depth formulation is decisive in
achieving a particular solution to the shorerise equilibrium
profile. However, the quantification of closure depth
appears to be somewhat vague in the literature.
(33] The general notion of closure depth he is the max-
imum depth at which seasonal changes in beach profiles are
measurable by field surveys, most commonly using fathom-
eters [Inma11 and Bagnold, 1963]. Closure depth for sea-
sonal profiles repeated over a period of a year or more is
usually taken as the depth of closure of the envelope of
profile changes, e.g., where the depth change vs depth
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C02003 JENKINS AND INMAN: THERMODYNAMIC SOLUTIONS FOR BEACH PROFILES C02003
4.0
3.5
a 3.0
tl:18
I 2.5
I 2.0
I 1.6
1.0
0.5
a..In1i211D1,1'111
100 160 200 250 300
I
2 4 11 I 10 12 14 111
Clomre depth,\,,.
360 400
Figure 3. Closure depth he dependence on grain siz.e D2
and incident wave height H00 for waves of 15 sec period
('I' -0.33, K0 -2.0, D0 -100 µm).
decreased to a common background error [e.g., Kraus and
Harikai, 1983; Inman et al., 1993). When observations ere
limited to comparison of two or three surveys, the closure
depth becomes the depth ai which ihe survey iines converge
with depth, a point of some uncertainty [ e.g., Shepard and
Inman, 1951; Nordstrom and Inman, 1973; Birkemeier,
1985). Hallermeier [1978, 1981) derived a relation for
closure depth, by assuming a relationship for the energetics
of sediment suspensions based on a critical value of the
Froude number, giving
he"" 2.28H,, -6.85(H;,/gT2) (40)
where H.. is the nearshore stonn wave height that is
ex:ceeded only 12 homs eaoh year and T is the associated
wave period.
[34) Birkemeier [1985) suggested different values of the
constants in equation ( 40) and found that the simple relation
h0 -1.57 H,. provided a reasonable fit to his profile
measurements at Duck, North Carolina. Cowell et al.
[1999) reviews the Hallermeier relation for closure depth
he and limiting transport depth h, and extends the previous
data worldwide to include Australia. Their calculations
indicate that he ranges from 5 m (Point Mugu California)
to 12 m (SE Aus!rnlia), while h, ranges from 13 m (Nether-
lands) to 53 m (La Jolla, California). They conclude that
disorepenoies in date ond oo.loulation procedures ma..'<e it
"pointless to quibble over aocuracy of prediction" in h0 and
h;. In the contex:t of planning for beach nourishment, Dean
[2002) obseives that "although closure depth ..... is more of
a concept than a reality, it does provide an essential basis for
calculating equilibrated ... beach widths."
[,s) While it may be reasonable to apply (40) or its
simpler form after Birkemeier [1985] to the shorerise
boundary condition (39), comparisons with the Inman et
al. [1993) beaoh profile data set show that these relations
tend to underestimate closure depth. We propose an alter-
native closure depth relation. This relation is based on two
premises: (1) closure depth is the seaward limit of non-zero
net transport in the cross-shore direction; and (2) closure
depth is a vortex: ripple regime in which no net granular
exchange occurs from ripple to ripple. Inman [1957) gives
observations of stationary vortex ripples in the field and
Dingler and Inman [1976] establish a parametric relationship
between dimension~ of stationary vorte]( ripples and the
Shield's parameter 0 in the range 3 < 0 < 40. Using the
inverse of that parametric relation to solve for the depth gives,
h, = K.Hoo (Do)\'¥
sinhkh, \D1 (41)
where K. and 'I' Bl'e nondimensional empirical parameters,
D2 is the shorerise mediun grain size; and D0 is a reference
grain size. With Ke ~ 2.0, \jl ~~ 0.33 and D0 N 100 µm, the
empirical closure depths reported in Inman et al. [1993] are
reproduced by (41). Figure 3 gives a contour plot calculated
from (41) showing the rates at which closure depth
increases with increasing wave height and decreasing grain
size. Because of the wave number dependence, closure
depth also increases with increasing wave period. Figure 3
is based on T-15 sec, typical of storm induced waves on
exposed high-energy coastlines.
[36) Using (41), the distance to closure depth X02 can be
obtained from (30),
L ,(2) -L f:7-·2 k=~C:it~ _-_e-_
E 2£ 2 (42)
where Xc2 is measured from the origin of the shorerise
located a distance X2 from the berm and a distance X3 X2
inside the breakpoint (Figure 2). This will be determined
subsequently from the matching condition. It is apparent
from (41) and (42) that the shorerise profile dimensions
grow with increasing wave height and period, and with
decreasing grain size.
[,1] A family of shorerise equilibrium profiles calculated
from (25)-(29) for a type-a cycloid is plotted in Figure 4a
using the closure depth formulation in (41) for 15 second
period incident waves. Toe aroitrary constunt in (16) that
satisfies the mild slope condition in (15) is set at N -10.
This particular selection has been found suitable for the
narrow-shelf beaches of southern California reported in
Inman et al. [1993]. Although tan ~ --+ oo as x2 -, 0, the
shorerise bottom slope remains less than the angle of repose
(~, :::a 33°, tan ~r :::a 0.65) at the breaker depth (hb -Hbfy),
where the shorerise profile must match the bar-berm profile.
In fact, shore1.i.se bottom slopes are tOD ~ ~ 0.1 for nny
depth h > 1.0 m. Altogether, the family of cycloids shown in
Figure 4 offer considerable diversity in the potential size
and shape of the equilibrium shorerise profile.
[ ,8] It is interesting to note that when cx -1 the general
solution (22) reduces to that for a trochoid, a cycloid
9 of 21
C02003 JENKINS AND INMAN: THERMODYNAMIC SOLUTIONS FOR BEACH PROFILES
a) profllea: eccentricity and 1hear ltrell llneartty
----e=0.845;n=3
e = 0.798; n = 2
----e=0.107;n=1
----e=0.447;n=0
----e = O ; n = -0.33 ; brachllltochrone IOlullon
700
700
800 500 400 300 200
Cross-Shore Distance x2, m
b) slope: eccentricity and shear atreaa llneartty
----e=0.845;n=3
e = 0.798 ; n = 2
----e=0.707;na 1
----e=0.447;n=-0
----e = 0 ; n = -0.33 ; brachlstochrone BOlutlon
800 500 400 300 200
Cross-Shore Distance x2, m
0
2
4
8 ~
a
8 ,c;..
10 !
12
14
100 0
0.1
0.08
cc.
0.08 j
}
0.04 fll
0.02
0
100 0
Figure 4. Family of type-a elliptic cycloid solutions in the shoreise: (a) profile; (b) slope. Cycloids
scaled for H00 = 4 m; N = 10; T = 15 sec; D2 = 100 µm; h0 = 14.4 m.
C02003
produced by a rolling circle. The trochoidal solution in polar
fonn after (25)-(29) has an eccentricity e = 0, and a radius
r =a= b = 1/20 = h/2. This fonn of the shorerise solution
is the equivalent of the brachistochrone solution, one of the
first known problems in calculus of variations [Baas, 1966].
Its analytic appmximation from (36)-&38) gives the popular
beach profile fom1uJation h = Ax21 after Dean [1991,
2002]. From (28) and (37) the relation between shorerise
profile exponent m2 and shear stress linearity n is
bottom 51.ress that decreased with increasing wave height.
Jirom this we conclude that the h = Ax213 formula does not
represent an equilibrium state in the shorerise portion of
the shorezone, nor does any profile for which m2 > 0.571,
for these would likewise require a shear stress inversion
(n < 0). Accordingly, we reject any shorerise cycloid
solution having an eccentricity e < 0.447.
[39] To select a preferred set from the remaining cycloid
solutions for the shorerise, we examine the relation
between (37) and the best-fit profiles to measured shore-
rises. Figure 5a gives a histogram of the shorerise profile
exponent m2 derived from best fits to 51 measured
profiles from nine beaches comprising the basic data set
in Inman et al. [1993]. Of these 51 measured profiles, 20
were reported to represent summer equilibrium profiles, 20
4 m2 =--7+3n
4 7 or, n=---3m2 3 (43)
Consequently, a profile for which m2 = 2/3 corresponds to
n = -1/3, an insensible outcome since it would require a
10 of21
C02003 JENKINS AND INMAN: THERMODYNAMIC SOLUTIONS FOR BEACH PROFILES C02003
( eccentricity, e ]
profile exponent, 111
20 a) 20 c)
~ 16 16 I,. 12
'a 8 8
1 4 4 z
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8
m2 e m 1 e
10 b) (iiia;,;protl~le~fa~ctori;,7A) 10 d)
8
8
4
2
0 0
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5
A2 A1
Flgttre S. Histograms of profile exponent m (red) and profile factor A (blue) from best fit to beach
surveys [after Inman et al., 1993], corresponding eccentricity of elliptic cycloid (green).
represented winter equilibrium, while 11 were said to be
non-equilibrium "translational" profiles. The red histo-
gram bars in Figure 5a are numbers of realizations of m2
from the best fits and the green bars are the corresponding
eccentricities per (37). Only two of the 11 non-equilibrium
profiles reported for the basic data set in Table I of Inman
et al. [1993] produced best fits with m2 > 0.571 (outside
the theoreticnl limit for shorerise equilibrium); while all 40
of the equilibrium profiles resulted in best fit m2 < 0.571.
The mean shorerise profile exponent for the entire ensem-
ble of 51 profiles (equilibrium and non-equilibrium alike)
was m2 = 0.362, which corresponds to a shear stress
linearity of n = 1.35 and a cycloid eccentricity of e =
0. 74 7. The mean m2 of the 40 equilibrium profiles is "iih =
0.365, giving n = 1.32 and e = 0.744. BRsed on these
averages (3 7) suggests thnL the bottom shear stress
amplitude over the shorerise typically varies :i..~ -r0 ~ u~3
in (17), coincident with the formulation of Kajiura [1968]
for bottom friction due to oscillatory flow over rough
beds when St= u,,./crTJ, < 30. This scale regime is consistent
with the vortex ripple regime reported by Dingler and
Inman [1976] when the roughness height TJs is taken to be
equivnlem to the ripple heigh1.
[4o] Neglecting the two anomalous translational profiles
in Figure 5a (m2 > 0.571) the range of variation in the
shorcrise exponent for the remaining 49 profiles was 0.21 ~
m2 ~ 0.5. AL the upper limit of this range, as m2 -0.5,
the bottom shear stress varies weakly as -r O ~ u;/,3 and
the shorerise cycloid has an eccentricity of e = 0.577.
Al; m2 -0.4, the bottom shear stress becomes linear (n = 1),
and the resulting cycloid solution for the shorerise has an
eccentricity of e = 0. 707. Linear bottom shear stress -r O ~ u,.,
would be expected with a laminar Stokes oscillatory bound-
81)' 1.-iyer common to smaU amplitude oscillati.011s over a
perfectly smooth, impenneable bed, for which c1 ,... R; 112•
[Stow, 1851 ; Lm11b, 1932; Batchelar1 1970· Sleatl,, 1984].
Intermediate in the upper portion of the range of variability,
where m2 ~ m2 ~ 0.4, the periodic bottom shear stress
11 of 21
C02003 JENKINS AND INMAN: THERMODYNAMIC SOLUTIONS FOR BEACH PROFILES C02003
a)
4.0
a :::L 3.5
Ja.o
,2.5 ·
u 2.0 iu
J 1.0
) 0.15
0.0 0.4 0.8 1.2 2.0
Shandie Pmile PIOIOr A.J, ,,,OA"
b) Breaker coefficient, y
0.2 0.8 1.0 1.4 1.8
4.0
[ 3.6
~8 3.0
f 2.5 = I 2.0
1.5 I 1.0
0.5
o.o 0.4 0.8 1.2
Figure 6. (a) Shorerise profile factor A2 versus incident wave height H00 and median grain size D2
based on closure depth per Figure 3 with m1 = 0.365 and N= 10. (b) Bar-berm profile factor A1 versus
incident wave heightH00 and wave breaker coefficient-y for T= 15 sec; N= 10; r = 0.76; A= 0.81; and
mi= 0.4.
can be characterized by a shear stress relation for smooth
granular beds after Taylor [1946] that uses a drag coef-
ficient fonnulation, c1 ~ R-;g115S;315, and gives a shear
stress linearity of n = 6/5, corresponding to m1 = 0.377
with a cycloid eccentricity e = 0.731. In the lower potion
of the range of variability below the mean, say m1 ~
0.25, tho shear stress becomes cubic -r O ~ ~ giving a
cycloid eccentricity of e = 0.845. At the lower limit
m1 ~ 0.21, the non-linearity of the shear stress increases
to n = 4 and the cycloid eccentricity becomes e = 0.875.
Generally, over the lower portion of the range of vari-
ability, 0.21 s m1 s m2, the shear stress takes on a
relatively high degrees of non-linearity, typical of form
drag over non-stationary rippled beds at high Reynolds
numbers R. 2': 104, [Sleath, 1982, 1984], or ventilated
oscillatory boundary layers over porous beds [Conley and
Inman, 1994 ].
[41] From the above consideration of the empirical
evidence, we conclude that the shorerise equilibrium profile
is a set of elliptic cycloids whose eccentricity is limited to
the range 0.447 s e s 0.875, depending on bed roughness
and dynamic scale regime, with semimajor and semiminor
axes specified by (39) and (41). The most common out-
come has an eccentricity e = 0.744, that can be represented
by an analytic J!Pproximation given by (36) with m = m2 =
0.365 and A= A2 calculated from (38) using a= 3.479. The
wave height and grain size dependence of this A2 solution is
shown in Figure 6a for type-a cycloids. The shorerise
12 of 21
C02003 JENKINS AND INMAN: THERMODYNAMIC SOLUTIONS FOR BEACH PROFILES
a) profiles, IOluUon and eccentr1clty -----.-w dllllll)llllon allerlhom1on & Gura (1983); •., 0.904
mmnum-wave dlalpatlon an. Thomlon a Gia (1983); • = 0.846
----lln_. bonl d_,pallon after Bowen et Ill. (1968); • • 0.707
----depth-lmltld bore dllllpatlon ., a-(1969);" = o.858 ; • = o.491
----~aalullon;,aO
-4
-2 ~
o a
4
.___,__.__~_.___,__.___,__.___,__.___,__.___,__.___,__.___,___, 6
450 400 350 300 250 200 150 100
Cross-Shore Distance, x1 , m
b) elope, aolutlon and accentrlclty
----IV8fWOI ,_ dlNlpatlon aftarThomtan & Gim (1113) ; r = 0.804
fflUffl.1-dlalpallan llllar ThortDn & Qua (1983) ; • • 0.846
----1,-boradlNlpallon lftlr 8-1 atal. (11188); • • 0.707
----deplh,6nlacl bora dallpallon alllr Chow (1959); 11 • 0.8611: • • 0.491
----~ aolullan ; r = 0
450 400 350 300 250 200 150 100
Cross-Shore Distance x1 , m
50 0
0.1
0.08
C0.
0.08 j
0.04 ! l;l.l
D.02
D
50 0
Figure 7. Family of type-a elliptic cycloid solutions in the bar-berm: (a) profile; (b) slope. Cycloids
scaled for: H00 = 4 m; T= 15 sec; N= 10; 'Y = 0.8; r = 0.76; A= 0.81.
C02003
profile factor A2 increases with increasing wave height and
decreasing grain size. A2 also increases with increasing
wave period because its dependence on the ellipse axes a
(3 8) is stronger than its dependence on 1:.. The magnitude
and dynamic range of the A2 solutions in Figure 6a are in
general agreement with the best-fit re.suits to beach surveys
from Inman et al. [1993] shown in Figure 5b. The distri-
bution in Figure 5b has a mean of 1.097 with a standard
deviation of 0.457.
Figure 2 is elevated above mean sea level at h1 = Z1
(Figure lb). Accordingly, the bar-berm cycloids from (35),
(B4) and (BS have U1e foHowlng vertically offset polar
form
4.2. Bar-Berm
[42] General solutions to the bar-berm in section 3.2 and
Appendix B follow from a fa-st integral m the Euler-
Lagrange equation (13), and consequently admit to the
addition of an arbitrary constant [Boas, 1966]. This we
must do in the bar-berm because the origin at P(l) in
'l\"£ X1 ( [ -COS Q) h=h1 =-( -) -0 , 9 +Z1 =r(l -cos0) +Z1 2l/'~ -sm
(44)
wl1ere r is given by (26). We prescribe the vc1tical oflsol· by
the maximum runup elevation from Hunt's Formula [Hunt,
1959· G111t.a a11d 711onuo11, 1985; Raubenheimer and G11za,
1996],
(45)
where r is the runup factor taken herein as r = 0. 76.
The bar-berm cycloid must converge on the breaker
13 of21
C02003 JENKINS AND INMAN: THERMODYNAMIC SOLUTIONS FOR BEACH PROFILES C02003
depth hh Hhl"Y within one-half revolution of the
cycloid wheel,
(46)
Because of the vertical offset to the bar-berm profile
origin, the ellipse axes must span the distance between Z1
and hh as 0 -+ 1'<:. Consequently, the breakpoint boundary
condition in (46) leads to the following sizing of the bar-
berm type-a cycloid ellipse axes,
(47)
where A is the shoaling factor from (B5). The field work
of Raubenheimer et al. [1996] shows that 'Y has an
observed range of variability 0.2 < 'Y < 1.6. Enforcing the
breakpoint boundary condition in (47) on (44) gives the
distance to the break point X3 from (30),
X3 = (11b -Z1)/,(2>;;,,, ~Hoo(~+ r) /z~ (4S)
e. 2AE '/ v-2
[ 43] Figure 7 gives a family of particular solutions for the
bar-berm equilibrium profiles calculated from the various
surf zone dissipation assumptions that produced the sepa-
rate cycloid solutions in (35), (B4) and (B8). These sol-
utions were applied to (44), subject to the boundary
conditions in (47). The solution envelope includes cycloids
produced by the linear bore (e = 0. 707), the maximum wave
dissipation model (e = 0.845) and the average wave
dissipation model (e = 0.904) after Thornton and Guza
[1983] in Appendix B. Each dissipation model produces
both type-a and type-b cycloids, but we focus our discus-
sion primarily on type-a cycloids. The brachistochrone
solution is included for·comp111ison, because it-is equivalent
to the solution h = Ax').13 by Dean [1991, 20021 and because
it is a particular case of the depth limited bore solution from
(35) with TJ = 0.5. Intermediate between the brachistochrone
and the linear bore solutions is another particular case of the
depth-limited bore solution based on TJ = 0.659 that
produces a cycloid with eccentricity of e = 0.491. These
solutions are calculated for the same incident wave con-
ditions (H00 = 4m, T = 15 sec) with the same arbitrary
constant (N = 10) used for the shorerise in Figure 4. The
shoaling factor assumed for these bar-berm solutions (A =
0.81) was based on uniform shoaling of the incident
wave conditions, while a mean value was chosen for
gamma ('Y = 0.8) from the data reported by Raubenheimer
et al. [ 1996].
[44] Although the equilibdum profiles from the bar-be1m
cycloids in Figure 7 are shorter than the shorerise cycloids
in Figure 4 calculated for the same conditions, many
qualitative features remain common to both, with profile
length increasing with decreasing eccentricity for the
type-a cycloids. Local slopes along the bar-berm profiles
in Figure 7b are generally comparable to slopes found
along the shorerise profiles in Figure 4b. All bar-berm
cycloids in the solution envelope trend to a flat bottom
(zero local slope) at the break point where matching with
the shorerise is enforced.
[4s] TI1e mean value ofm1 for the 51 best fits to the bar-
berm profiles reported in the data set of Inman et al. [1993]
is m1 = 0.411 (Figure 5c), of which only one (a translational
profile) gave m1 -+ 0.66, the equivalent of the brachisto-
chrone solution. The mean m1 of the 40 equilibdum profiles
was m1 = 0.400, which from (37) con·esponds to a cycloid
with eccentricity e = 0.707, coincident with the linear bore
solution (35). The range of variability for the 40 equilibrium
profiles was 0.29 :S m1 :S 0.55, corresponding to cycloids
with 0.491 :S e =S 0.813. The cycloid dedved from the
maximum wave dissipation formulation (e = 0.845) corre-
sponds roughly to the lower limit (m1 -+ 0.29) of this
statistical spread. This limit can also be obtained from the
depth-limited bore solution in (35) using TJ = 1.47 to
produce a cycloid with eccentricity of e = 0.813, exactly.
The upper limit of the 40 equilibrium profiles (m1 -+ 0.55)
can be obtained with a depth limited bore solution using TJ =
0.659.
[46] The data in Figure 5c indicates that the envelope of
paiticular solutions for the bar-berm are limited to a set of
type-a and type-b cycloids of the form in (44) derived from
the depth-limited bore (0.5 :S TJ ::; 1.47) and maximum
wave dissipation formulations from (35) and (B8), respec-
tively. Within this envelope of solutions, the most commonly
occurdng is the linear bore (e = 0. 707) that can be
approximated by (36) with m = m1 = 0.400 and A = A1
calculated from (38) using ex= 3.0. Figure 6b shows the
dependence of this A I solution on incident wave height and
the gamma ("() factor over the observational range of
vadability reported in Raubenheimer et al. [1996]. This
solution is based on the same wave pedod (T= 15 sec) and
scaling assumption (N = 10) as the shorerise solution for A2
in Figure 6a. Runup and shoaling factor asswnptions (I' =
0.76; A = 0.81) are the same as those chosen for the bar-
berm cycloids in Figure 7. The resulting magnitude a11d
dynamic range of the A I solutions in Figure 6b are consis-
tent with the best-fit results to beach surveys from Inman et
al. [1993] shown in Figure 5d. The mean of these field
results was A1 = 0.868 with a standard deviation of 0.386.
The bar-berm profile factor A I increases with increasing
wave height and decreasing 'Y. Unlike the shoredse, the
ellipse axes a and bin (47) are independent of frequency in
the bar-berm. Consequently A1 decreases with increasing
wave period due to the frequency dependence of thee factor
in (38). Therefore the mean bar-berm slopes become flatter
with longer period waves.
[47] The bar-berm A1 solution in Figure 6b shows very
similar compound variation with wave height and 'Y as the
shorerise A2 solution. A relation arises between 'Y and beach
grain size through the matching condition discussed below.
The basis for this relation is a finding in Raubenheimer et
al. [1996] who show that 'Y increases with increasing bottom
slope at the break point, tan I¾,
(49)
where K,, K1 are empirical constants taken as K0 = 0.2 and
K1 = 6.0. The bottom slope at the break point, tan flx, is
given by the slope of the shorerise profile at x2 = X3 -
2X2•
4.3. Matching the Shorerise and Bar-Berm
[4s] To complete the particular solutions, we must match
the bar-berm and shorerise profiles at the break point
14 of 21
C02003 JENKINS AND INMAN: THERMODYNAMIC SOLUTIONS FOR BEACH PROFILES C02003
(Figure 1 b ). The point of conjoinment is the seaward end of
the bar-berm profile at the break point, a distance X3 from
the origin (48). Here, the shorerise and bar-berm solutions
must both equal the breaker depth,
(50)
at x2 = X3 -X2 and x1 = X3, respectively. To obtain this
match, the origin of the shorerise profile must be located a
distance X3 -X2 landward of x1 = X3• fu terms of the polar
form of the shorerise cycloid, this distance is,
2r2J(l) X1 -Xi= ---;:--(Bi, -sin Bi,)
(51)
Bi,= arccos[l -2(~7,.) ~]
where r2 is given by (26) and evaluated at 0 = 0i,, and ~ is
related to shorerise cycloid eccentricity by (28). With (51)
the shorelise bottom slope at the breakpoint can be solved,
sinflt, +e2(cos8b -l)sin9t,cos~
tanf3x, = I -cosl\+~(siuBt,-O.,)sin!li,tosOi, (52)
Because Iii, is grain size dependent through the closure depth
relation in (41), 'Y in (49) also becomes grain size
dependent. This in tum makes the bar-berm profile sensitive
to grain size variability through (47) or by its alternative
analytic approximation (36)-(38). In terms of the analytic
approximation, the matching condition (50) is satisfied by,
X2=X1-1 1 "" [
A X"" -rn /h] 11""
Ai
giving the bottom slope at the break point as,
tan~.lz =m2A2(Xi-Xir'-1
(53)
(54)
where X3 is given by (48). Point-by-point comparisons of
type-a and type-b cycloids show that type-a provided the
best fits to measured beach profiles.
5. Discussion
(49] The cycloid solutions were derived from 2-dimen-
sional representations of the shorezone as a consequence of
the assumption that external work is done on the system by
a train of uniform, normally incident waves. Under this
assumption, viscous dissipation was maximized along a
boundary surface within the system, in the cross-shore
normal plane (Figure la). However, on natural beaches,
waves are seldom uniform and normally incident, resultu,g
in nearshore circulation cells with longshore currents. Near-
shore circulation cells are entirely compatible with equilib-
rium conditions defmed by (8) and (11) because they
provide a mechanism to ventilate latent heat into the
surroundings. There is no violation of thermodynamic
equilibrium principles through partitioning heat and work
evolution between the cross-shore and longshore shear
stress components, respectively. By this paititioning, the
cross-shore directed shear stresses perform the dissipative
work that satisfies the maximum entrnpy production for-
mulation on the second law (11 ), while the longshore
directed radiation stresses remain in balance with the long-
shore directed shear stresses driving the nearshore circula-
tion. The only consequence of such partitioning is that the
rate of working by longshore directed stresses can not alter
the thermodynamic coordinate ~ that defines the cross-shore
profile.
[so] The cycloid solutions result in vanishing bottom
slope at the seaward limit of both the shorerise and bar-
berm. It could be argued that field survey techniques are not
s11fficiently accllfate to determine whether this is a realistic
outcome, especially since the flattening at the end of the
cycloid trace is highly localized. Figures 4b and 7b show
that cycloid slopes depart from zero at small distances
inshore of the seaward limit of both the shorerise and bar-
berm. The seaward boundary of the shorezone system at
closure depth excludes any dynamics associated with the
shelf slope from entering the equilibrium formulation of the
shorerise. The slope discontinuity formed at the motch point
between the shorerise and bar-berm cycloids produce a
reasonable representation of a breakpoint bar. The validity
of the foregoing comments and resulting equations for
equilibrium beach profiles are supported by the goodness
of fit in the following detailed comparisons.
[s1] The following point-wise comparisons between par-
ticular shorerise and bar-berm solutions from thermody-
=ic theory with profiles measured by the U. S. Army
Corps of Engineers are made to determine the predictive
skill of cycloid solutions. The profiles were measured at
beach survey range PNl 180, near Oceanside, CA during a
six year period between March 1981 and September 1987
(USACE, 1985, 1991]. This range and setting are described
in Inman et al. (1993].
[s2] Wave climate was measured in 6 hour intervals by
the pressure sensor array located off Oceanside, CA [ CDJP,
19 80-198 8]. These directional wave data were back
refracted into deep water from their measurement location
and forward refracted to 10 m depth in the neighborhood of
the PNl 180 beach range to correct for local shelf and island
sheltering effects [O'Reilly and Guza, 1991, 1993]. Under
the hypothesis that equilibrium is dete1mined by the persis-
tent large waves, the refracted wave time series were filtered
for the highest 5% waves and then time-averaged over each
survey period to provided the forcing history of the H=, T,
needed to calculate temporal variability of the boundary
conditions on each profile. Grain size data from Inman
(1953], Inman and Rusnak (1956] and Inman and Masters
(1991] were used in these calculations. All empirical
parameters used to quantify boundary conditions were set
according to those used in Figures 3, 4, and 7. The
horizontal location of the bar-berm origin relative to the
survey benchmark X1 (Figure 1 b) was taken directly from
the database of each survey.
[ 53] A procedure for achieving a high resolution fit of the
cycloid solutions to field data is detailed in Appendix C. A
point-by-point comparison between tl1ese high resolution
solutions and the six measured beach profiles surveyed at
PNl 180 is shown in Figure 8 for the type-a cycloids. The
legend of each panel gives the average significant wave
15 of21
C02003 JENKINS AND INMAN: THERMODYNAMIC SOLUTIONS FOR BEACH PROFILES C02003
a) Surw, Ran .. PN 1111 lllroh 1N1
,._1911 prullle; H. = 22 m; T= 18 NC
---type-a cydold; "., 0.71M
---type.a cydold; er= 0.702
2
4
8
8
10
'---------'---'---'------'-_.__...____...._____._ _ __,.___,,2
1000 800 800 400 200
b) lurw, Ran .. PN 11IO JulJ 1112
ffllMUred pRJllle; H_ = 2A m; r .. 13,5 NC
---type.a cydold; .... 0.1101
---type.a cydold; • = 0.855
.... ,.,..-
,,/ ,, ,,,
,/
/ ,'
i'i'
' ,
/,'
0
-2
0
2
4
8
8
10
'---------'---'---'------'--'---'--'------........ --'---'12
1000 800 800 400 200
11) Burwy Ran .. PN 1tl0 Apll 1-
m-.ired praftle· H = 2.25 m· T" 15 eac
---type-e~;e=0.751
---type-a cydold; e • 0.743
0
4
8
8
10
"'--___._ _ __,.__.._____. _ _.__.....___._____.__......___.12
1000 800 800 400 200 0
~red prullle; H." 1.5 m; T= 18 NC
---type.a cydold; "= 0.71M
---type.a cydold; • • 0.788
.,, .....
,r'
/
,;'' ,
/ ,,
I -2
I , 0 /' ,, 2
4
8
8
10
._____.__.......__..________.. _ _.__...____..___.__.....____,12
1000 800 800 400 200
a) lurw, ~ PN t1IO Aprll 1N1
~ prollle; n. • 2.8 m; T• 13 NC
type,e cydold; • = 0.788
---type,e cyckJld; • = 0.700
0
10
._____._ _ __,.__.,_____. _ _.__....._____...._____._ _ __,.___,12
1000 800 800 400 200
meaeured prullle;H. • 1.7m; T• 18.5 NC
---type.a c:yclold; • ., 0.846
---type.a c:yclold; "= 0.758
0
2
4
8
8
10
..._____.. _ _.__..._____,'-----_.__.....__._____._ _ __,.___. 12
1000 800 800 400 200 0
Cross-Shore Distance x, m
Fignre 8. Comparison of beach profile survey data (gray) from Oceanside, CA, versus the elliptic
cycloid equilibrium solutions for the shorerise (blue dashed) and bar-benn (red dashed). Calculated
profiles based on CDIP wave monitoring data and median grain size Dz= 100 µm withN = 10, and 'Y = 0.8.
height and period of the highest 5% waves that occurred
between successive survey periods. The type-a cycloids
(Figure 8) gave a predictive skill factor of R = 0.83 to
0.95, where R is adapted from Gallagher et al. [1998] and
applied to the mean-square error in depth. The eccentricity
of the type-a shorerise cycloids varied over a relatively
narrow range 0.655 ~ e ~ 0.766 with a mean ofe = 0.721,
generally consistent with the shear stress fonnulations for
smooth to moderately rough beds after Kajiura [1968] and
Taylor [1946], and in overall agreement with the larger
ensemble statistics that were inferred from the analytic
approximation in Figure 5a. The type-a bar-berm cycloids
displayed a somewhat wider range of variability in eccen-
tricity, 0.704 ~ e ~ 0.901 with a mean of e = 0.785. The
range spanned all theoretical possibilities between the
depth limited bore fonnulation (35) and the average wave
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C02003 JENKINS AND INMAN: THERMODYNAMIC SOLUTIONS FOR BEACH PROFILES C02003
Sarvq Ranpa PN 1180 ud PN 1MO, Ocnndde, CA -4
meuun:d profiles (12 ea, 1950-1987) -2
---cycloid; H00= 3.0 m; T= 14 sec
0
---cyc;loid; H00= 2.0 m; T= 15 sec
---cyc;loid; H00-1.0 m; T-15 sec
8
10
1000 600 600 400 200 0
CTOM-Shore Distaneo .r, m
Figure 9. Envelope of variability of 12 measured bench J>IOfilos {1950-1987) at Oceanside CA (gray)
compared with the ensemble of type-a elliptic cycloid solulions (colored) for solcct.ed incident. wave
heights (llld periods with D2 = 100 µm, N = 10, 'Y = 0.8, (llld A= 0.81.
dissipation fonnulation (B4), while the mean was still
within the main peak of the large ensemble distribution
found in Figure 5c. Therefore, the point-by-point accmacy
of the type-a cycloid solution is not only good, but was
achieved with ellipse parameters that are compatible with
both process-based and empirically-based computations of
those parameters.
[s4] If tho ]>rocoduro in Appendix C is abbreviated ot the
first step and the cycloids are simply computed using the
mean of the empirical distributions in Figure 5 (e = 0.744 in
the shorerise and e = 0.707 in the bar-berm), then t..1ie
predictive skill of the type-a cycloid solutions is less but
still acceptable for most engineering applications, ranging
from R = 0.76 to R = 0.88.
[55] The type-b cycloids (not shown) produced a skill
factor generally less than 0.5. The type-b cycloid solutions
perfonned poorly on a point-by-point basis at PNl 180
beca11se Chey give a broad bench or trough and bar for
eccentricities e ~ 0.65.This feature is not generally found
on Jlall'OW shelf beaches such as those along the southern
California coast. However, on wide shelf beaches such as
the U. S. Gulf Coast, this feature may improve the predic-
tive skill of the type-b cycloid.
[56] When a series of type-a cycloid solutions for a
broad range of wave heights are overlaid 011 an ensemble
of many beach profile measurements, a well defined
envelope of variabi.lity becomes apparent as illusmned in
Figure 9. This figure combines 12 profiles meiism·ed over
a 37 year period from two adjacent beaches near Ocean-
side that have geomorphic equivalence. Such comparisons
with the cycloid solutions suggest that the volume of sand
associated with long term beach profile variations are
directly calculable by integration of the cycloid solution
between the limits of wave climate vaiiation for a partic-
ular site. Irt this exnmple, the wa.vo height integrated
cydoids indicate that 1,180 m3 of sand per merer of
shoreline are involved with long tenn beach variability.
At lowest order, this volume represents the minimum sand
volume that the beach must retain in the long te1m in order
for it to maintain a sus1oinable equilibrium ,vith clu111ging
seasons.
6. Conclusions
[ 57] Solutions for equilibrium beach profiles are obtained
from the maximum entropy production formulation of the
second law ofthcnnod;,Twinics, when applied to a..vi isotJ1er-
mal shorezone system of constant volume and constant
wave forcing. The entropy of the shorezone system remains
constant while that of the surroundings is increased. The
equilibrium profile was found to be the shape that max-
imizes the discharge of entropy into the surronndings by
maximizing the dissipation rate within the shorezone
system. This formulation of thennodynrunic equilibrium,
expressed in shorezone coordinates (Figures la and lb),
provides exact solutions for the shorerise and bar-be1m
portions of the c011joined beach profile. The shorerise, is
an outer zone of wave shoaling where dissipation arises
from the rate of working by bottom shear stresses. The
bar-betm inner zone nnderlies the zone of broken waves
where turbulent dissipation results in a progressive loss of
bore height with decreasing water depth, ultimately pro-
duci11g runup over the beach face and fonnation of tl1e
berm crest. It is notable that thennodynamic solutions for
these zones of widely differing mechanics provide one of
the rare instances of a geophysical problem having an
exact solution.
[58] The thermodynamic solution to the equilibrium
beach problem belongs to a class of analytic functions
known as elliptic cycloids, fonned by the trajectory of a
point 011 the perimeter of a rolling ellipse [Boas, 1966]. The
general shape of the equilibrium profile is determined by the
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C02003 JENKINS AND INMAN: THERMODYNAMIC SOLUTIONS FOR BEACH PROFILES C02003
eccentricity and starting position of the ellipse (Figure 2).
Two types of cycloids are produced, resulting in two
distinctly different classes of profile shape. Type-a cycloids
are formed when the rolling ellipse starts with the semi-
major axis (a) aligned with the vertical axis of the coordi-
nate system; while type-b cycloids start with the semiminor
axis (b) aligned with the vertical axis. While equilibrium
profile solutions were developed for both types of cycloids,
only the type-a cycloids were found to be well correlated
with beach profile measurements from southern California
that typically do not exhibit complex or large-amplitude bar
formations.
[ 59] The aspect of profile shape controlled by eccentricity
is related to the degree of non-linearity of the bottom shear
stress in the shorerise, and to the rate of bore height decay in
the bar-berm. As the shear stress in the shorerise becomes
increasingly non-linear, the eccentricity of the ellipse
increases, and the profile develops more variability in
curvature and bottom slope (Figure 4 ). The shear stress
formulation of Kajiura [1968] for oscillatory flow over
rough beds was found to give the best general average of
measured shorerise profile curvature for fine sand beaches.
In the bar-be1m, the profile eccentricity increases with
increasing rate of bore height decay with depth. The linear
bore after Bowen et al. [1968] and the maximum wave
dissipation formulation of Thornton and Guza [1983]
produced bar-berm solutions with the most commonly
observed curvature.
[60] The overall dimensions of the equilibrium beach
profile are governed by the dimensions of the semimajor
and semiminor axes (a and b) of the cycloid ellipse
(Figure 2). These dimensions are resolved from boundary
conditions that require the cycloids converge with closure
depth at the end of the shorerise profile, and converge
with the breaker depth at the end of the bar-berm profile.
The formulations used to specify these two depth limits
allow a variety of transport physics to be incorporated
in the particular solutions. We have demonstrated the
viability of a new closure depth formulation that gives
the shorerise solution wave height, period and grain size
dependence and resulted in a high degree of predictive
skill in simulating measured profiles (Figure 8). We
have shown generally that profile dimensions grow with
increasing wave height and period, but decrease with
increasing grain size. The grain size dependence of the
shorerise is imprinted on the bar-berm through the
matching condition and the entropy associated witl1 both
increases with decreasing grain size as S ~ D-3• These
solutions are easily integrated over any given range of
seasonal wave height to estimate the minimum volume
of sand that the beach must retain in order to maintain
a sustainable equilibrium for a given wave climate
(Figure 9).
[ 61] The elliptic cycloid solution allows all the signifi-
cant features of the equilibrium profile to be characterized
by the eccentricity and the size of one of the two ellipse
axes. This general formulation is especially versatile
because these two basic ellipse parameters can be related
to either process-based algorithms or empirically based
parameters for which an extensive literature already exists.
This feature gives the cycloid solution a modular charac-
ter, allowing it to be continuously upgraded by new
science, or modified by alternative formulations according
to user preference.
Appendix A: Shorezone Entropy
[ 62] In its statistical mechanics formulation, entropy is
shown to be a measure of the disorder or randomness of a
system [Fermi, 1936]. From Halliday and Resnick [1967]
and Landau and Lifshitz [1980] we write:
S =KBln(3) (AI)
where K8 = 1.38 x 10-23 J/°K is the Boltzmann constant
and S is a disorder parameter that gives the probability that
the system will exist in a particular state relative to all
possible states.
[ 63] The probability of finding a single grain of sand in
the systems is:
where J is a constant defined in (A5) that depends on the
geometty of the system boundaries. The number of sand
grains in the system is
(A3)
where N0 ~ 0.6 is the volume concentration equal to
I-porosity, and D is a representative diameter of the mobile
sand in the shorezone system of volume v. (Figure 1).
When ~ is applied to the bar-bean and shorerise portions of
Vs, D is a representative median diameter. lnserting the
probability state function (A2) into (Al) and collecting
te1ms from (A3) gives the beach entropy:
where the system boundary scale factor is
J=f~ Xe (AS)
The integral in (A5) defines the boundary dependent
constant in (A2). With equation (A4), the equilibrium state
of the beach may be written in tenns of a state integral.
Beach entropy increases with decrnasing grain size because
greater numbers of grains permit larger numbers of possible
states for the system. For the same reason, beach entropy
increases with increasing size of the shorezone system.
Appendix B: Bar-Berm Solutions From Surfzone
Dlssipation
[64] The first of two empirically derived dissipation
functions developed by Thomton and Guza [1983] is based
on the hypothesis that waves break in proportion to the
distribution of all waves, yielding a surf zone dissipation per
unit area Eb of:
(--) 8 EC K.,. Eb= T·U =-;,-g=p
</Xi '1
(Bl)
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C02003 JENKINS AND INMAN: THERMODYNAMIC SOLUTIONS FOR BEACH PROFILES C02003
where
K -3J7i 3 7 ave-16-y"pgB crH00
The dimensionless functional in (13) recovered from
order-1 terms in (B6) is
FIJl.b!'.) = ft-3 /1 + (E.t.'/ (B7)
'.111-d ~ ~ !he 0(1 breaker lru:tor. We select from (1) an
mfimtesunal arc length of the bnr-berm piofile having the With (B7) the first integration of (13) gives
form,
which gives 1:1te following form of the integral in (11) that
we make stationary to maximize entropy prudul-tiun,
From (B2) the dimensionless fhnctiona! in t.11e Euler-
Lagrange equation (10) becomes
F(ll,~,i_) = 11-5 VI+ (q_)2 (B3)
Inserting (B3) in (13) we get oFl&x = 0 and the first
integration of (13) gives -'
d ,,,o
l + dh 1 -if ,,10 --1-
(B4)
where £,11 is the dimensionless form of the integration
constant. The bar-berm profile given by (B4) is equivalent
~o (22) when ex "' IO and integra,es to an ell~cycloid as
m (23) ru1d ~29) witJi an eccemricity e = y19/1 I "'0.904.
The d1menst0nal form of the integration constant is c; =
!ti!L10 = (2a)-10 for tl1c type-a bar-benn cycloid and c( =
(2b)-10for U1c type-b cycloid.
[6s] T~e second form of the dissipation function put
forward m Thornton and Guza [1983] is based on the
hypothesis that waves break in proportion to the distribution
of the largest waves, giving,
where
Kmax = :~B3pgcrH!,{I -I s/z } (! -i:ti.2)
A. = 22/s H';J,s ( cr2 / FrY) i;s
and A. is the shoaling factor relating breaker height to
incident wav_e height A =. HoclHb, fur a shoaling Airy
wave. By a smnlar formulation to (B2), the integral in (11)
that we make stationary to maximize entropy production is
(B8)
where !f{ is a dimensionless integration constant. The
bar-berm profile given by (B8) is equivalent to (22) when
ex = 6, ru1d gives an elliplic cycloid wilh an eccentricity
e = -[ffi -0.845 according to the general Cruiesiw1
relations in (22). The dimensional fonn of the integration
constant is c'; = c'{IL6 = (2a)-6 for the type-a har-bel'.Dl
cycloid and c'( = (2b -6 for tJ1e type-b cycloid.
Appendix C: Matching Field Data to Cycloid
Solutions
[66] The cycloid solutions utilize boundary conditions
based on several empirical parameters. We fix these param-
eters iliroughout the iterative procedure described below
?Ccor'!ing to characteristic values assigned by previous field
111vest1gators. These parameters include: the Airy wave mild
slope factor N after Mei [1989] set atN= 10 in (16); closure
depth coefficients in (41) taken as 1.(,, = 2.0, D0 = 100 µm,
'I'= 0.33 after Dingler and Inman [1976); the runup factor r = 0.76 used in (45) based on Hunt's Formula [Hunt,
1959; Guza and 111ornlon, 1985; Raubenheimer and
Guza, 1996]; and the slope factors set at K0 = 0.2, K1 =
6.0 in the breaking wave criteria for 'Y in (49) after
Raubenheimer et al. [1996). These values were used to
calculate Figures 3, 4, and 7. The shoaling factor A
was calculated from the time-averaged wave climate data
(Hoo, 1) over each survey period based on shoaling Airy
wave theory as written in (B5).
[ 67] In addition, the best fit cycloid solutions were
calculated by the following series of iterative steps:
[ 68] 1. Shorerise profiles were calculated from (25), (26),
(29), (39), and (41) using an initial assumption of a cycloid
eccentJ:icity of e = 0.744, per the Kajiura [1968) shear stress
fonnulation;
[69] 2. Bar-berm profiles were calculated from (25), (26),
(44), and (47) using initial assumptions that 'Y = 0.8 and e =
0.707, per the linear bore dissipation f01mulation;
[10] 3. The initial profiles from steps 1 and 2 are matched
by the X3 -X2 shift of the shorerise profile calculated
from (51);
[11] 4. The bottom slope at the breakpoint is calculated
from (52) and used to correct the initial assumption of 'Y by
means of (49);
[_n] 5. The bar-berm profile is re-calculated per step 2
usmg the corrected value of -y;
[73] 6. Repeat steps 1-5 making iterative adjustments to
the eccentricity to minimize the mean squared error between
the cycloid solutions and the measured profile.
(B6) [ 74] Ackoowledgmeoh. This study was earned out as a collaborative
research project under contract with the Kavli Institute, Santa Barbara,
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C02003 JENKINS AND INMAN: THERMODYNAMIC SOLUTIONS FOR BEACH PROFILES C02003
California, with additional support provided by the Office of Naval
Researoh (code 321 CG and code 321 OE). The authors appreciate
discussions with John Miles, Rick Salmon, Meryl Hendershott, Robert
Guza, and Falk Feddersen on principles of thermodynamics and shallow
water wave theory. We acknowledge Joseph Wasyl for contributions in
computer programming and graphics. We also thank the reviewers and
Associate Editor for comments and guidance that helped us to clarify the
thermodynamic formulation of the problem and its practical purview.
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