Transactions on the Built Environment vol 70, © 2003 WIT Press, www.witpress.com, ISSN 1743-3509
A coastal morphodynamic
model for cross-shore sediment transport
F. Li, C. Dyt & C . Griffiths
Predictive Geoscience, CSIRO Petroleum, Australia
Abstract
A numerical model to represent the mechanisms of cross-shore sediment transport
induced by wave and storm events has been established. The model consists of three
sub-models (wave transformation, morphodynamic movement, and sedimentation)
capable of representing the transport and redistribution of four grain sizes over a
coastal area. It is designed to simulate the morphological evolution caused by crossshore transport. This model can thus be superimposed on existing long-shore
sediment transport models. An application to a well-documented South Australian
coastline suggests that the model is able to reproduce erosion and accretion trends
observed in nature. The short computer run-time permits efficient model calibration
and verification.
1 Introduction
In coastal sediment transport models, representing the day-by-day cross-shore
transport by individual wave events is difficult because, very often, long shore drift
overshadows their contribution to general shoreline evolution, and the strength of
the cross-shore transport varies widely. There is also a lack of high quality field data
to verify the model. This'is due to the rapid removal of large quantities of sediment
making the collection of field data a difficult task. Therefore, in most wave-induced
transport models an equilibrium beach profile is assumed, i.e. cross-shore transport
is viewed as high frequency swings from an equilibrium beach profile and assumed
to be temporary and marginal to long-term beach developments. In such profiles the
onshore transport of large grains is supposed to be balanced by the offshore
transport of small grains.
However, in nature, the profiles of beaches are rarely in equilibrium, instead,
they are constantly modified by the ever changing environment. A single storm may
change the beach considerably. Once a bar was created during an episode of storms,
it may be located at such great depth that very little or almost no sand transport
activity takes place until another storm at such scale occurs. Many of these bars are
Transactions on the Built Environment vol 70, © 2003 WIT Press, www.witpress.com, ISSN 1743-3509
336 Coastal Engineering V1
well preserved in stratigraphic records of ancient coastal environment. As an
important part of coastal systems, cross-shore transport plays an indispensable role
in the construction of coastal morphology.
In the last three decades, the morphodynamic change of coastal areas caused by
transport normal to the shoreline has been intensively investigated in field and
laboratory studies worldwide, leading to great improvements in understanding the
mechanisms of cross-shore sediment transport by a number of high quality, longterm field investigations (Noda [l], Lee [2], Aagaard [3]).
Along with field and laboratory studies, several cross-shore morphological
models have been proposed to predict beach evolution (Kriebel and Dean [4],
Bailard[S]). Some of them deal with fairly long time periods (10 to 100 years). Most
of those models are 2DV models, or beach profile models, assuming water depth
contours being straight and parallel to each other, or low spatial variability along the
shoreline.
In this paper, a two-dimensional depth-averaged cross-shore sediment transport
model is presented. The intense sediment transport associated with wave breaking at
surf zone is modelled. The essence of this model is to obtain approximate averaged
predictions at appropriate time and spatial scale rather than striving to achieve
precise predictions at all scales.
Considering the fact that longshore transport has been much better understood
and well developed, this model is focused on representing the morphological
evolution induced by cross-shore transport. Therefore, in situations when both longshore and cross-shore transport are important it can be easily superimposed on
existing long-shore sediment transport models.
2 Wave transformation model
A computational efficient parabolic version of the mild slope equation is used
(Radder,1979) in the wave transformation model. Conditions for this equation are:
a) the reflection wave field is negligibly small, so that only forward travelling waves
are considered; b) the main effects are in the direction of propagation. One grid
coordinate x has to be approximately parallel to the predominant wave direction.
This parallel condition creates problems if there exist a wide range of wave incident
angles. Therefore a special technique to remove this restriction has been introduced
and will be addressed later.
Besides that, other innovations include: a) Allow an ever-fluctuating sea water
level to represent the semi-diurnal tidal effect on waves; b) Random storm return
time as an extra option for pre-defined wave input. Given an averaged storm
frequency the present model allows a random storm return intervals and a predefined deviation in the incident direction.
2.1 Matching algorithm
The model input includes the general dip direction of the major shoreline relative to
North, edip, deep water wave heights, H,, angle of incidence, 8,, and wave period,
To. For the user's convenience the y coordinate is always oriented towards North,
and all the input angles are relative to North.
Transactions on the Built Environment vol 70, © 2003 WIT Press, www.witpress.com, ISSN 1743-3509
Coastal Engineering V1
337
To remove the restriction on X coordinate on the parabolic version of the mild
slope equation, four types of discrete modes are developed to suit the individual
coastline orientation in a fixed Cartesian coordinates. The choice of the
computational mode is decided by the value of the general dip direction of the major
Mode I
2 2 5 ~ 3 ,<~3 15"
Mode I11
135°<0,,<
Mode I1
225"
Mode IV
OO<Od,,< 45"
315"<8d,, < 360"
45~3,,< 135.
Figure 1: Computation Modes to suit different shore orientations
shoreline,edi, , which guarantees a shoreward matching algorithm. Then, the wave
parameters are always computed from deepwater wave boundary conditions towards
the unknown territory of the shore. Therefore, no inshore boundary conditions are
required because of the forward march solution method. Figure 1 illustrates the
relationship between the shore orientation and the computational mode.
3 Morphodynarnic model for cross-shore transport
3.1 Mechanism of cross-shore sediment transport
During a storm the cross-shore sediment transport is responsible for the formation of
breaker bars. In the surf zone, strong energy dissipation and bed friction usually
generate a circulation with an offshore-directed undertow near the bed and an
onshore mean flow near the surface. The former brings suspended sediment
offshore. Outside the surf zone, the wave energy dissipation and turbulence are
mainly confined to the near bed boundary layer, where sediment transport is much
weaker and with a tendency to be in the onshore direction. In the following
paragraphs the surf zone is defined as the region between the incipient wave break
point and the limit of the back-rush, where mainly broken waves exist.
As the offshore bar grows, the waves break farther offshore, causing the location
of the maximum transport rate and the bar to be moved offshore. This process
continues until the storm is over or a stable beach profile is achieved which
dissipates storm-wave energy without significantly changing in its own shape.
After incipient breaking the broken waves reform and reach a stable wave
height. Dissipation of energy by the reformed waves decreases, implying a
corresponding decrease in the transport rate. In the present study a linear decrease of
transport rate from the breaking point to the run-up limit on shore is assumed.
Transactions on the Built Environment vol 70, © 2003 WIT Press, www.witpress.com, ISSN 1743-3509
338 Coastal Engineering V1
After a storm or a series of storms, the fair-weather waves tend to move the
offshore sediment towards the shore. The tide level changes help to shift the surf
zone between offshore and onshore. This slow but long-lasting process facilitates
beach recovery and the build-up of the so-called berm profiles.
The present model is confined between shoreline, the upper limit of the territory
affected by waves, and offshore wave breaking line.
The coastal strip between these two boundaries is split up into blocks by the lines
perpendicular to each segment of the shoreline. As shown in Figure 2, the present
model takes into account only the cross-shore component of the wave-induced
transport, which takes place at each block. Therefore, the direction of the onshoreoffshore transport in an individual block is defined by its shoreline dip direction.
The model assumes that the net
offshore-directed transport rate
peaked at the offshore edge. Inside
the surf zone a linear rate
distribution is assumed with null
transport at shoreline boundary and
maximum rate at offshore breaking
line. An onshore-directed transport
rate is assumed to be constant in the
surf zone, which results in the
sediment, picked up at the offshore
boundary, being dumped at the
Incipient break line
Shoreline
onshore boundary. The criterion for
Figure 2: Definition of cross-shore
the local net transport direction has
transDort and surf zone
been derived.
3.2 Direction of transport: onshore (berm) or offshore (bar)
Given a set of wave parameters, whether or not the beach is going to be eroded can
be decided by the direction of sediment transport. Field observation (Larson [6])
shows that beach morphology is closely related to the direction of net cross-shore
transport. Typically, if a bar forms, the main direction of net cross-shore transport is
offshore and if a berm is present, the sediment is moving onshore.
The empirical criteria for the occurrence of barlberm profiles is [6]
where W is the grain settling velocity, H. and L. are wave height and length at deep
water, the empirical coefficient M=0.00070
3.3 Magnitude of net cross-shore transport rate
Various arguments have been made for different relationships between wave
conditions and cross-shore transport rate [4], [ 5 ] .
Transactions on the Built Environment vol 70, © 2003 WIT Press, www.witpress.com, ISSN 1743-3509
Coastal Engineering V1
339
Detailed beach analyses show that the net cross-shore transport rate q is closely
related to energy dissipation. Based on those analyses the net cross-shore transport
rate at any point within the surf zone can be expressed as (Kriebel, 1985)
q(X)=K(D-D,)
(2)
where D is the actual energy dissipation per unit volume, D* is the equilibrium value
of the D. K is a transport rate coefficient.
where X is the offshore direction coordinate. h is local average water depth, E is
wave energy, c, is group velocity, the speed of propagation of wave energy, Odif is
the angle between the shore dip direction and the wave incidence.
In the current model, at each simulation block the magnitude of transport rate is
represented by averaging eqn (2) over the surf zone.
where q,
is the average cross-shore transport rate, B is the width of surf zone,
D*,,, is the averaged D*,the subscript b denotes the location of the offshore edge of
surf zone.
Studies have revealed that during storm weather the maximum, q,, , appears to
be located in the vicinity of the wave plunge point where maximum energy
dissipation occurs. Materials supplying the offshore bar are mainly taken from the
region of the inner surf zone. Therefore, a linear triangle shaped distribution of the
transport rate is specified in the model when the direction of sediment transport is
towards offshore (see Figure 3). On the other hand, when the wave conditions
produce onshore transport a rectangular shaped distribution is assumed. In the
current numerical model the transport rate coefficient functions largely as a
calibration parameter to give the proper time scale of profile change.
The equilibrium value of the energy dissipation, D*,is difficult to specify, but its
role in the model is acting as a safety net preventing the unrealistic correction of the
beach surface. To avoid the difficulty the present model uses a slope factor in the
surface updating scheme represent the equilibrium effect implicitly.
3.4 Morphological changes updating scheme
Finally the bed mass continuity equation is solved over time. Simulation of the
beach profile change is achieved by repetitive computations of the wave height,
sediment transport rate and bottom surface levels.
Transactions on the Built Environment vol 70, © 2003 WIT Press, www.witpress.com, ISSN 1743-3509
340 Coastal Engineering V1
where X denotes the cross-shore coordinate, z is the local beach surface level, q is
the sediment transport rate in the cross-shore direction, S is the rate of a source
adding sediment into the system.
accretion
erosion
(a) Stormy weather
erosion
(b) Fair weather
Figure 3: Cross-shore transport rate distmbution and its morphological effects
for a time step at (a) stormy weather, (b) fair weather conditions
The major morphological changes in the coastal region are delivered by breaking
waves. By applying the wave refraction and diffraction model, the wave height
across the region can be calculated. Then, apply eqn (4) to obtain the local net
sediment transport rate.
When the erosive wave conditions prevail the sudden drop in energy dissipation
rate at the offshore edge of the surf zone causes a cessation of sediment transport.
As a result, the sediment eroded from the surf zone will be deposited here. During
periods of fair-weather the sediment is picked up offshore and moved onshore. The
assumption of the uniform transport rate
will send all the sediment picked up
from offshore edge to the onshore limit
of the beach.
3.5 Sedimentation and sediment-size
grading
The present model is able to account for
the changing size distribution at the bed
surface during a re-suspension process.
It can take account of up to four different
grain sizes which make up the sediment
at each cell in a simulation grid. And
each cell is layered by the sequence of
deposition.
Figure 4: Location of the Brighton
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Coastal Engineering V1
341
4 Model application to Brighton Beach, Adelaide
Brighton beach, Adelaide, South Australia forms a part of the East coast of Gulf St
Vincent, Figure 4, which is generally sheltered from the direct influence of deep
ocean waves. However, west to southwest waves are able to get through from the
entrance of Gulf St Vincent. The beach lies within a piece of low to moderate
energy coastline and the micro to meso-tidal range of 1.2 to 3.3 m. The tidal flow in
the &ion is relatively
g 30
weak and the sediment
El relative time of wind v s direction
.transport is dominated by
25
waves and storms.
50 20
The beach system is
Z 15
undergoing changes in
10
response
to natural
processes and human
5
impacts. It was reported
0
that
storm
erosion
-Max height
regularly
damaged
Average height
foreshore roads, car
parks and community
('J)
facilities. In order to
/
\
I
\
understand the beach
behaviour, from 1992 to
1997 a serial of regular
beach profile surveys
were conducted along
0
45
90 135 180 225 270 315 360
Direction of wind
the
metropolitan
beaches. The shoreline at
Figure 5: (a) Distribution of wind directions for the
Brighton is generally
waves
with their significant wave heights exceeding 1.0
north south oriented, the
south boundary of the metre, (b) the corresponding maximum and averaged
significant wave height
beach is a rocky cliff that
disrupts sediment supply
from south.
The wind direction data were sourced from Adelaide airport, situated 12 km to
the north-northeast. The wave height data were measured every six minutes and
sourced from Port Stanvac Jetty, situated 11 km to the south-southwest
The analysis of the waves for which significant wave heights exceed 1.0 metre
shows that those relatively big waves are predominately from Southwest (225") to
West (270"). And the maximum wave height also falls between 225" - 270°, shown
in Figure 5. The geometry of the local shoreline also confirms that the energetic
waves, which
the most significant role in beach morphodynamic evolutions, are
from those directions.
The on-going beach replenish program dumped about 180,000 m3 of sand onto
the area each year. Each beach rehabilitation process usually took 2-3 months to
complete. A series of high quality beach profile surveys have been conducted from
1992-1997 annually at the end of January to early February. The sediment samples
=
L
-
Transactions on the Built Environment vol 70, © 2003 WIT Press, www.witpress.com, ISSN 1743-3509
342 Coastal Engineering V1
were collected from Brighton Beach at low, mid and high tidemarks. It consists
largely medium to fine sand and poorly sorted.
The simulation area covers 2.0x5.0 km from onshore to offshore -lOm water
depth. The grid size is set as 50 m. Four types of different sized grains, with the
diameter of 0.75 mm, 0.375 mm, 0.1875 mm and 0.1 mm, represent the beach
sediment. The density of the sediment particles was set to 2650 kg/m3. A uniform
sand layer, 9.1 m thick, was draped over the initial bathymetry (1993) except at the
southern rocky cliff. The base level (water mark) of the simulation area was
adjusted accordingly. In the Brighton beach area, both the cross-shore and the
longshore transport of sediment by waves are important. The present cross-shore
model is incorporated into the existing long-term process-based stratigraphic
forward model Sedsim[7][8]. In Sedsim, the wave-induced longshore sediment
transport is calculated by the sub-routine WAVE.
In brief, WAVE employs the wave-power equation (Martinez [9]) to estimate the
rate of longshore transport. It treats longshore transport rate as a function of power
available at the moment that waves break.
Figure 6:
Brighton beach morphological change between
1993 and 1997, (a) Observed, (b) Simulated
(positive dz for accretion, negative dz for erosion.)
Previous investigations have confirmed that there is fairly steady longshore drift
transport of sediment towards the north, which was considered to be responsible for
the slow loss of sand from the beach. Figure 6 plots the topographic change between
the 1993 and 1997 surveys and the numerical model result. However, the longshore
Transactions on the Built Environment vol 70, © 2003 WIT Press, www.witpress.com, ISSN 1743-3509
Coastal Engineering V1 343
transport alone cannot explain the swing of erosion and accretion exhibited in the
annual beach topographic surveys.
The cross-shore morphological model, along with the long shore transport model,
is calibrated by the survey data obtained in 1992 to 1997. The results, shown in
Figure 6, indicate a fairly good agreement between the model and the survey. The
cross-shore transport rate parameter found during calibration test to be 1.2 X 10.~
m4/N. Based on those calibrated coefficients the models can provide coastal
engineers with a useful tool to predict the topographic change and to evaluate the
possible damage given a storm scenario. Figure 7 shows a predicted seabed response
to a hypothetical storm episode, similar to the one on 12-14 September 1992.
The predicted distribution of the grain sizes indicates a fining trend offshore
shown in Figure 7. The figure also gives the data of storm wave heights and
directions. As a typical storm event its angle of incidence changed over 130" in
about two days time. The storm wave height peaked at the time when the waves
approach the Brighton beach from the west-to-southwest sector; consequently, the
northern beach will suffer from heavy erosion. And most of the sand reworked will
be deposit offshore. During a storm the surf zone width is changing with wave
height, local beach slope, and water level. At Brighton beach the most destructive
storm is the one that strikes at high tide, and the most vulnerable part of the beach is
the northern part as biggest waves are most likely approaching from southwest
sector.
Coarse --* Fine
Angle ot Incidence
1902699
1'332701
19927m
1992705
1).
Figure 7: Predicted Brighton Beach erosion and distribution of sediment
grain sizes after a real life storm, which occurred on 12-14
September 1992.
Transactions on the Built Environment vol 70, © 2003 WIT Press, www.witpress.com, ISSN 1743-3509
344 Coastal Engineering V1
5 Conclusions
A numerical model to represent the process of cross-shore sediment transport
induced by wave and storm events has been established. The model lies between the
small-scale process-based model and the large-scale long-term conceptual model.
Given the topography of a coast the model is able to find a shoreline/shorelines
and to define the directions of offshore and onshore transport automatically, which
is independent of the orientation of the coordinates. This has made it easier to apply
the model to any complicated geometry of coast, to adapt to existing coastal
morphological models, and to model coastlines under changing water levels.
The model is robust and the short computer run-time permits efficient model
calibration and verification.
The model is a first step towards representing the cross-shore sedimentation
process. Further verification and calibration are needed. Most of the fine details of
the erosion/accretion pattern have been simplified because of their high sensitivity to
the irregularity of hydrodynamic input. In practice, the more useful information
from the model often lies at a more synoptic level, in the overall transport pattern
and its variation with the forcing conditions.
Acknowledgements
We would like to thank the Office for Coasts and Marine, Coast Protection Board,
the Environmental Protection Agency of South Australia for supporting the
simulation and providing data.
Reference
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