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4
Modeling Coastal Systems
Modeling coastal processes can be a subject worthy of extensive
discussion, and it is certainly a topic of rapidly growing interest—for
both researchers and engineers. This chapter, however, addresses
only the relation between modeling and coastal measurement and is
therefore limited in scope.
Mathematical models are useful, if not indispensable, tools in the
analysis, synthesis, and interpretation of field or laboratory measure-
ments in the coastal environment. Models are often a useful too! for
identifying significant gaps in data as well as for completing missing
data in a data series. A mode! may be based on purely statistical
considerations, on purely physical principles, or both, and generally
involves a finite number of parameters whose values must either be
stipulated or inferred from the measurements. Models used in the
analysis of data are usually, but not exclusively, diagnostic models
(see the following section), while models used in estimating future
changes are usually some form of predictive model. Engineering re-
quirements dictate what one wishes to predict, whether it be changes
in beach bathymetry or in hydrodynamic forces on a structure under
given design storm conditions. Measurements are clearly indispens-
able information for verification of models, but at the same time
models can be extremely useful in identifying the appropriate type
of data that ought to be obtained.
Modeling and measurements should work hand in hand. A model
66
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67
allows one to assess the sensitivity of some result to changes in cer-
tain input ciata. For example, a study of the sensitivity of coastal
flooding models was made by Jennings (NRC, 1983~. As expected,
the parameterization of the storm was the most critical factor. How-
ever, a number of other inputs were listed. Among these, doubling
the bottom friction factor caused errors of greater than +3 feet in
the scale flood elevation, while increases of 1 foot in all land elevators
resulted in peak flood elevation changes of only 0.3 feet.
A mode! may also indicate that measurements ought to be made
at a certain minimal spatial or temporal scale. An example of the
latter exists in the Coastal Upwelling Experiment (CUE), sponsored
by NSF during the mid-1970s, which involved field measurements of
currents and water density along the continental slope region of Ore-
gon and Northern California. The initial data were taken at rather
coarse spatial resolution, with no unusual results. Numerical mode}
experiments by Wang and Mooers (1976) predicted that a narrow
subsurface jet flowing counter to the surface currents could exist.
Subsequent field measurements at much closer spacing disclosed that
a subsurface narrow countercurrent jet indeed existed. The lesson
to be learned is that data gatherers and modelers must coordinate
their efforts to achieve the most meaningful results. If some natural
scales in space and/or time exist related to the physical phenomena
under study, then sampling must take such scales into account to
avoid biasing or misinterpreting the measurements.
When one makes measurements, either in the laboratory or in
the field, the strategy is nearly always based on some preconceived
notion of a process (i.e., a model), whether it be purely conceptual
or highly quantitative. Physical models as discussed in the following
section should be understood to be quantitative mathematical rela-
tionships among several variables and involving one or more param-
eters that characterize some physical, chemical, or biological process.
In contrast, laboratory physical models are scaled-down versions of a
prototype system in which one employs some scaling considerations.
Both field measurements and laboratory measurements are essen-
tial means of verifying, based on physical principles, a mathematical
mode} or of providing the essential parameters that are not known a
· —
prlorl.
In this section a general discussion is given of the relevance of
models to the requirements for making or improving coastal mea-
surements. This is done in the context of recognized limitations that
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68
exist in both models and measurements. Mathematical models re-
quire input data to drive them, and their predictions relate, with
varying degrees of sensitivity, to such input data. Information about
mode] sensitivity can be useful in determining the accuracy or resm
lution needs for basic measurements. By making models an integral
part of the measurement system, the strengths of measurements and
of models can offset the limitations of each. Mathematical models
are limited in accuracy by:
. the completeness and accuracy of rendition of the physical
processes that are included in the model,
the accuracy and resolution of the required data (e.g., initial
conditions, forcing data, morphological data, physical parameters),
and
. the spatial and temporal resolution of the model.
Field and laboratory measurements are limited by:
the accuracy of sensing devices and recording systems,
the accuracy of location of sensors,
the influence of the presence of some instruments on the
process they are intended to measure,
the spatial and temporal resolution of the measurements, and
the completeness of the measurements in terms of the physical
process being studied (i.e., have all the pertinent variables been
observed to characterize and interpret the process adequately?.
The second limitation listed ureter "mathematical models" is
very dependent on the measurements. The third limitation is pri-
marily related to the first. A model can usually be made to a finer
grid scale than the measurements and can sometimes indicate where
measurement locations are most elective. Where the model grid
size is a concern is in the ability of the mode! to deal properly with
physical processes that are important at scales too small for this
grid size. Such subgrid scale processes must be represented in some
ad hoc manner in the model. The classical example is the effect of
those turbulent processes that are too small in scale for the adopted
average grid scale of the model. One must always adopt some form of
closure hypothesis for the subgrid-scaTe turbulent stresses and mix-
ing processes. The adequacy of the ad hoc closure hypothesis must
rely on comparison of mode! results with measurements from spe-
cially designed experiments. This is where laboratory-scale physical
modeling methodology is of paramount importance.
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69
The real strength of the measurements, aside from inherent oh
servational errors and limited ability to handle small-scale features,
lies in the simple fact that the measurements contain all of the
physics. The strength of the model, when properly tuned, is that it
can be used as a diagnostic too! to help interpret the discrete mea-
surements. This is particularly true of dynamic systems with many
variables. For example, not only does the model provide a rational
interpolator, it also provides the means to compute quantities usually
not measured directly, like stresses, transport rates, energy fluxes,
and energy density.
PHYSICAL MODELS
Physical models, including hydraulic models, are generally used
when flow conditions of a prototype system are not amenable to
mathematical analysis. The inability to mathematically mode! a
prototype system may be due to (1) the nonlinear character of the
equations of motion, (2) an inability to characterize turbulence or
dissipation, (3) complex geometries or interconnected flow passages,
or (4) hydraulic elements that are not susceptible to physical de-
scription. Physical models are applied for prediction of prototype
behavior or for studying details of a system that are not easily oh
served in nature.
Two major problems encountered in physical modeling are (1)
maintaining equivalence between mode! and prototype and (2) mak-
ing proper interpretation of mode! data. These two problems are
inherently linked because both geometric and dynamic similarity
must be achieved before mode! data can be considered quantita-
tively valicl. Obtaining geometric similarity (all dimensions scaled
proportionally) does not guarantee dynamic similarity. Therefore,
the assumption that a physical mode] that looks like a small-scale
version of the prototype does in fact respond in a dynamically cor-
rect sense is unsound. In many cases, scaling one of the dimensions
(say the vertical dimension) differently from the others will actually
improve the model's performance.
An acIditional problem in physical modeling is the limited rem
resentation of the forcing function. In many cases, the directional
spread of waves is omitted or no wind is included.
Once similarity is established in a model, the model must then be
calibrated. Calibration is the procedure of adjusting model parame-
ters until the mode! can satisfactorily reproduce measured prototype
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70
behavior. Model calibration is, therefore, directly dependent on the
type and quality of measured prototype field data. furthermore,
mode! verification requires a suitable quantity and range of protm
type data so that the mode! can be checked against ciata not used
for calibration.
Initial construction of a physical model to achieve geometric sim-
ilarity is highly dependent on the density and resolution of bathymet-
ric data available. Usually physical models are constructed with hor-
izontal dimensions constrained by the space available for the model.
In an undistorted model, this constraint usually results in small
model depths, especially at the coast and in harbors. Unless the
bathymetric data from the prototype had a high vertical resolu-
tion, calibration of the mode! may be impossible. If a distorted
model is constructed where vertical exaggeration is imposed with re-
spect to prototype dimensions, the necessary resolution of prototype
bathymetry may be relaxed somewhat. However, the lack of rapid
high-resolution bathymetric measurement systems usually results in
physical models being poorly calibrated- or, if calibrated, they are
seldom verified owing to the low number of detailed surveys.
Calibration and verification of similarity between mode} and pro-
totype requires~detailed field measurements of the spatial and tempm
ral distribution and variability of velocity fields. Present technology
such as acoustic-Doppler current meters may be able to acquire the
necessary prototype data at a point, but large numbers of these
sensors are required to verify mode! similarity.
In most models pressure and average velocity are in similitude
(Froude similitude). The problem is in achieving similitude of forced
turbulence and boundary layers, which affect quantification of sedi-
ment transport; at present, providing similitude of suspension is not
possible. Lacking the ability to use similitude models, the sediment
transport must be empirically estimated though calibration of the
model by comparison with observed fuB-scale results in nature that
are obtained through improved field measurement systems.
MATHEMATICAL MODELS
/
[ong-Period Waves and Currents
With the possible exception of tsunamis, all models of waves
having periods in excess of five minutes neglect effects of vertical
acceleration of the fluid, although allowing for slow rise or fall of
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71
water level. The Coriolis force associated with the earth's rotation
is also important for periods greater than a few hours. Tides and
storm surges differ primarily in the nature of the forcing; both are
adequately modeled using the averaged-over-depth equations of mo-
tion with appropriate incorporation of bottom stress. Some models
of tides or storm surges allow for vertical variation of velocity in a
parametric manner; others are full three-dimensional models.
Models of storm surges and of slowly evolving circulation on the
continental shelf or in estuaries demand adequate information on
the spatial distribution of wind stress and on barometric pressure
versus time in order to drive them. The modeling of hurricane
induced storm surges generally employs an auxiliary storm model for
the inclusion of the wind stress and pressure fields associated with
such storms (NRC, 1983~. The main difference between the storm
surge and wind-driven circulation models is that the latter generally
allows for effects of water density stratification. For shelf waters
or estuaries with both horizontal and vertical variations of density,
proper rendition of currents demands a full three-dimensional mode}
in which density, as well as the current structure, are dependent
variables.
Aside from the foregoing differences in physics, the various mod-
els of low-frequency hydrodynamics differ mainly in domain (area
of effective coverage), resolution, optimization, and numerical imple-
mentation. With the exception of global tide models and ocean-scale
circulation models (excluded from consideration here), the mode!
domain is generally of limited area with open lateral boundaries, at
which appropriate forcing and/or radiation of energy can be allowed.
One of the troublesome aspects of mode! domains with open bound-
aries is in making sure the boundary conditions employed at the open
boundaries are compatible with physical processes being simulated
within the interior of the mode! domain.
The state of the art in modeling of tides and storm surges for
limited domains is generally adequate. The needs are prunarily in
the adequacy of data for verification and ~ fine tuning of such
models. Some deficiencies of course do exist; these were alluded to in
Chapter 3 In connection with storm surges. One ~ the need for proper
modeling of wave setup with attendant need for data to verify this
component of water-lever anomaly. Another is the need to properly
address the coupling of short- and long-wave dynamos (wave/current
interaction), which is important in nearshore circulation and in the
quantification of bottom stress.
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72
Modeling of circulation at estuarine or shelf scale ~ still a devel-
oping field of research. Here the requirement for adequate measure-
ments of currents in three dimensions, for verification and tuning of
such modem, ~ vastly more demanding on available resources. The
problem is compounded by joint forcing due to winds, tide/current
interaction, and effects of density stratification an controlled by river
input and seaward boundary conditions.
Of all the phenomena that fall in the low-frequency category,
tsunarn~s are unique in the sense that the forcing ~ unknown for
most events. While the epicenter of a given event is known, the
spatial configuration and amplitude of the ground motion at the
seabed is generally not known. The modeling techniques are fairly
well advanced, including allowance for weak dispersion effects that
are important in long-range propagation from the source to distant
coastlines. In principle, one could use measured response at tide
stations to aback out" what the unknown source characteristics are;
this is similar to what geophysicists do in estimating the structure
of the earth's interior from seismic data. Such inverse methods for
recovery of cause from effect demand receiver data that have minima]
contamination by local effects. Unfortunately, nearly all tide station
data (with the exception of benthic pressure gauges) are heavily
contaminated (distorted) by local resonant effects that render inverse
methods virtually meaningless because the available models cannot
be that site specific.
Table ~1 summarizes the types and objectives of models perti-
nent to low-frequency motions, the existing techniques, needs, and
example references.
Waves and Wave-~duced Flows
Estimation of nearshore waves is the design information most
often needed for addressing coastal engineering problems. Waves
are a primary consideration for measuring design forces on ocean
structures; waves are also the most important agent for littoral sed-
iment transport and are responsible for driving the currents in the
nearshore. The primary requirement for accurate prediction of lit-
toral transport and nearshore currents is high-resolution direction e]
wave information. Adequate input is obtained by directly measuring
radiation stress in shallow areas using a slope array.
The drawback of this approach is that its measurement loca-
tion is site-specific, making it costly to obtain information over a
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73
TABLE 4-1 Modeling of Low-Frequency Motions
Model Type Objectives Technique Needs References -
Storm force Wind and pressure Empirical,
Parametric
Drag coefficient Empirical,
Parametric 8
Storm surge Water level
Current profile
Inundation
Tide
Water level and
volume nux
Current profile
Global model
Tsunami Water level
Generation
Propagation
Local response
Source energy
Ware/current Wave-induced
interaction
currents
Bottom stress
Wave setup
Estuarine
circulation
Shelf
circulation
density
Currents
Currents and
2. 3
Numeric 2-D 2, 4
Parametric, 3, 7
Numeric 2-D 1, 6, 7
Numeric 2-D 2, 4
Numeric 3-D 3, 7
Numeric 2-D 2
Analytic 8
Numeric 1, 6, 7
2-D
2-D
Graham and Nunn, 1959;
Schwerdt et al., 1979.
Forristall, 1980.
Garratt, 1977.
Huang et al., 1986; Reid and
Whitaker, 1976; Wu, 1985.
Butler, 1980; Jelesnianski,
1972.
Butler, 1980; Forristall,
1974, 1980; Forristall et
al., 1977.
Numeric 3-D 7 Heaps, 1974; Leendertse and
Liu, 1975; Sheng, 1983.
Butler, 1980; Jelesnianski
and Chen, 1981.
Butler, 1980; Leendertse,
1984.
Heaps, 1974; Leendertse and
Liu, 1975; Sheng, 1983.
Hendershott, 1977;
Schwiderski, 1980.
Many
(2-D), Hwang and Di~roky,
1972.
(3-D ), Mader, 1984.
Hwang and Di~roky, 1972;
Kim et al., 1987.
Houston, 1978; Vastano and
Reid, 1970.
Van Darn, 1984.
2
2, 7
Inferential 1, 6, 7
Numeric
Analytic 3, 6, 7
Empirical 1, 6, 7
Numeric 2-D 2, 6, 7
Numeric S-D 2, 6, 7
1,2,6,7 Vermulakonda et al., 1985;
Wu et al., 1985.
Grant and Madsen, 1979.
CERC, 1984.
Butler, 1980.
Blumberg, 1975; Leendertse
and Liu, 1975; Sheng, 1982.
Numeric 2-D 2, 6, 7 Brink and Chapman, 1985;
Brink et al., 1987.
Numeric 3-D 2, 6, 7 Leendertse and Liu, 1975;
Liu and Leendertse, 1987.
LEGEND:
Need:
1 Major development needed
2 Improve information detail
3 Improve physics
4 Improve efficiency
5 Improve tuning
6
7
8
~ — v
Special data needed
Verification needed
None
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74
large section of shoreline. As pointed out in Chapter 3, an alter-
native approach is to measure or hindcast waves in deep water and
to refract the wave information to shallow-water locations. This
approach requires highly accurate deep-water wave directional infor-
mation (which we are not presently capable of measuring or h~nd-
casting) and refraction of the waves to shallow waters using models
that have not been adequately verified. Another approach is to use
inverse techniques of refracting a number of shallow-water wave mea-
surements (not necessarily with high directional resolution) back out
to deep water to determine the deep-water wave directional spec-
trum. The distributed shallow-water wave sensors act as a wave
antenna. The completeness of directional information in deep water
increases with the number of shallow-water wave locations. Again,
the inverse technique requires an accurate refraction model. There-
fore, to predict nearshore waves and wave-driven currents, models
are required for wave forecasting and wave refraction and diffraction
and for nearshore dynamical models, as described in the following
. .
c .lscusslon.
There has been considerable renewed interest in wave forecasting
in the last decade. This interest primarily stems from the requirement
for improved forecasting for offshore of! facilities. Wind/wave predic-
tion was recently reviewed by Sobey (1986) and an intercomparison of
various models was accomplished by the Sea Wave Modeling Project
(SWAMP) (Sea Wave Modeling Project Group, 1985~.
Wave-forecasting models can be divided into three categories:
(1) the older empirical approach (based on dimensional analysis),
(2) the modern numerical discrete spectra, and (3) parametric am
preaches (both based on the radiative transfer equation). The numer-
ical models can be further categorized in terms of how the nonlinear
wave/wave interactions are treated. The first-generation numerical
models, evolved in the early 1960s, decouple the wind/sea genera-
tion and propagation of the directional wave spectrum. They do not
redistribute energy nonlinearly within the spectrum. These models
are still in use today for global wave prediction (e.g., the U.S. Navy)
because of computational efficiency.
Extensive field measurement of wave growth under carefully se-
lected uniform fetch-limited wave conditions became available in the
late 1960s and early 1970s (Mitsuyasu, 1969; Hasselman et al., 1973~.
The analysis of these data showed the importance of the nonlinear
wave/wave interaction to feed energy into the low-frequency end of
the spectrum, rather than direct wind forcing. The second generation
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75
models evolved to include the nonlinear coupling. The newer genera-
tion models either parametrically redistribute the wind/sea spectral
energy and use a discrete spectral representation for swell compo-
nents or use a discrete representation for both the swell and w~nd/sea.
The models were tested for various wind conditions by SWAMP
(1985~. It was concluded that the first- and second-generation mod-
els gave significantly different answers In the development of the
w~nd/sea variances for fetch and duration cases and in the shape
of the spectrum. Deficiencies exited in establishing the nonlinear
energy transfer parameters that largely control the shape and the
growth of the w~ndtsea spectrum. The deficiencies were particularly
evident for extreme conditions of rapidly changing winds. Under
these conditions, the present parameterization contained insufficient
information to describe the wide variety of spectral distributions that
arose (SWAMP, 1985~.
A group of international scientists formed the Wave Mode} De-
velopment and Implementation Group (WAMDI) in response to the
SWAMP recommendations. A third-generation mode! resulted that
integrates the basic transport equation describing the evolution of
the tw - dimensional ocean wave spectrum without additional ad
hoc assumptions regarding the spectral shape. The source functions
describing the-wind input, nonlinear transfer, and white-capping
dissipation are prescribed explicitly. The only tuning was two pa-
rameters in the dissipation functions set to reproduce the observed
fetch-lirn~ted growth of the fully devolved Pierson-Moskowitz spec-
trum (a spectrum widely employed in dee~water wave analysis).
Improved agreement over the earlier modem was obtained compar-
ing the mode! output to a variety of North Atlantic and North Sea
storms and Gulf of Mexico hurricanes (WAMDI, 1988~.
Shallow-water wave modeling has not received the same attention
as deep-water forecasts but appears to give comparable results. The
greatest deficiencies appear to be in the transition between deep
and shallow water, where generation and refractive effects on the
propagation and wave damping are important. A spectrum (referred
to as the TMA Spectrum) has been developed by Bouws et al. (1985)
for this purpose and requires further testing.
Wave Refraction and Diffraction Modeling
As a train of swell waves is propagated into shallow water, the
wave speed changes with varying water depth. Along the crest of a
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76
wave, the part of the wave in deeper water moves faster than the
part in shallower water. As a result, the wave bends to align with the
contours, and wave refraction occurs. Refraction and shoaling due to
depth are important for determining the distribution of wave energy
along a coast. Observations indicate that refraction has significant
influence on the erosion and deposition of beach sediments. Wave
trains can also be refracted by ocean currents.
The refraction problem has been modeled using severe am
preaches. Based on the theoretical formulation, the four commonly
used methods are (1) Snell's law, (2) ray theory, (3) mild-sIope elliptic
equation, and (4) parabolic equation approximation.
Snell's law modeling employs classical optical refraction prin-
ciples and generally is applied to simple geometries (straight and
parallel bottom contours) Ray theory follows a certain element of
the incident wave field (a section of wave crest for regular waves) an
it approaches the shore along a "ray, which is its trajectory. This
method determines the wave height and direction of wave propa-
gation along the ray in an efficient manner. The drawback in ray
models is their inability to incorporate the diffraction process. The
ray models may cross and cause locally large (infinite) wave heights
which are not realistically attainable. The mild-slope equation is a
2-D approximation to the equation of motion that can be solved for
combined refraction and diffraction. The forward parabolic approxi-
mation method is a fast wave mode} limited to forward-propagating
waves and does not account for reflection. For steep slopes, an
iterative scheme must be used to solve the parabolic equation. Re-
cently, practical alternatives for linear wave propagation problems
have been developed and applied to solve the mild-sIope equation
(Ebersole, 1985; Kirby and Dairymple, 1983; Liu and Tsay, 1983; Ito
and Tanimoto, 19723.
If combined refraction/diffraction is the dominant physical pros
cess of wave propagation and wave transformation, the foregoing
mathematical models may give adequate results for different engi-
neering purposes. Due to the respective applicability of the models,
a series of mode! verification tests Is needed. A practical problem is
specifying the grid spacing in the models. The determination of am
propriate grid scale for bathymetry in refraction models is unsolved
and can be critical to the results.
For simple bathymetries, the wave refraction models have been
verified using detailed laboratory data (WhaTin, 1972; Tsay and Liu,
1982~. On the other hand, there has not been an adequate mode!
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77
verification of wave propagation using field data to date. Given
the importance of wave refraction analysis to coastal engineering
projects, this is a major deficiency.
Breaking Waves
Dynamical wave forcing occurs when there ~ a change in mo-
mentum, which is primarily due to wave breaking. Wave breaking
is identified by foam and overturning of the wave face. Nearshore
breaking is a transient process associated with shoaling of the waves
that results in a steepening and eventual instability of the wave.
Breaking waves are a principal mechanism for the redistribution of
mass and momentum over both the vertical and the horizontal. The
kinematics and dynamics of breaking waves are highly complex and
are only crudely modeled. The exact criteria for the onset of break-
ing are not understood. Kinematic instability occurs when the water
particle velocities near the crest exceed the speed of the crest and
the wave ~ observed to break. The influence of infragravity wave
velocities and wave reflection on wave breaking is unknown. Wave
breaking is one of the least understood phenomena (Thornton and
Guza, 1986) ant] is critical to the various dynamical models and
wave-force problem. While substantial progress has been made in
understanding the mechanisms that cause breaking in deep water
(Tanalta, 1985; Longuet-Higgins, 1984, 1985), the theoretical basis
for analysis of shallow-water breaking is not well understood. Wave-
breaking processes need to be studied both in the field and laboratory,
and theoretically. ~ particular, understanding of the processes has
been slowed by the lack of quantitative measurements, particularly
in the field.
Wave-~duced Nearshore Currents
Under the action of waves, nearshore currents, including long-
shore currents, mass transport, undertow, and rip currents, are
formed. These currents are responsible for the direction and magni-
tude of the net coastal sediment transport. As waves refract and shoal
and eventually break, there is a change in wave-induced momentum
that must be balanced by a slope in the mean water level or an
increase in the bottom shear stress and concomitant nearshore cur-
rents. The wave-induced momentum is commonly referred to as radi-
ation stress (Longuet-Higgins and Stewart, 1962~; recent advances in
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78
nearshore dynamics are based on this concept. Analytic solutions for
longshore current (Bower, 1969; Thornton, 1970; Longuet-Higgins,
1970) have established that the driving force for the longshore cur-
rent is a component of the radiation stress tensor, Slays On the other
hand, rip currents can be produced by longshore variation In wave
setup (owing to variable wave height and resulting normal radia-
tion stress terms). These earlier nearshore current models were for
monochromatic incident waves over simple topographies.
The nearshore environment is complex. Waves are random and
the beach topography is three-dimensional; a numerical mode! ~ nec-
essary for predicting two-dimensional or three-~rnensional nearshore
dynarn~cs. Using a finite difference scheme, Nocia (1974) and oth-
ers developed steady linear nearshore current modem. ~cluding the
nonlinear inertial terms and the turbulent Denting terms, successful
modeling results were obtained (Ebersole and Dairymple, 1979; Wu
and Liu, 1985~. The latter mode! was verified with field data (Wu et
al., 1985) for waves with a narrow spectrum on a beach with nearly
parallel bottom contours. A random wave description using a prom
abilistic wave height distribution was used in a longshore current
mode} by Thornton and Guza (1986), which agreed well with field
measurements. All these modeling and exper~rnental results are for
the two-dimensional case, and the secondary currents, due to the ver-
tical nonuniformity of the wave-induced velocities as observed ~ the
surf zone and near breaking, are not considered. Due to imbalances in
the momentum fluxes in the vertical and nonuniform flow, a strong
seaward flow (undertow) is generated above the bottom boundary
layer. This undertow appears to exist between the shoreline and the
breaker line.
In a recent study on the interaction of the undertow and the
boundary layer flow, Svendsen et al. (1987) proposed a two-layer
mode} in a surf zone. It was found that field measurements were
needed to supply information about the breaker heights and the em-
pirical parameters for the solution. It appears that further progress
depends on measurements to obtain a better understanding of the
vertical structure of waves and turbulence.
In~agravity Waves
As waves traverse the surf zone, the frequency of the peak shifts
to lower frequency. As the sea-swell waves dissipate due to breaking,
the low-frequency infragravity waves (see Figure 3-1) are amplified.
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79
The infragravity waves are identified as either reflected long waves
(surf beat) or edge waves propagating alongshore. The infragravity
waves are omnipresent. They have been shown to be important
for nearshore processes because of potentially large velocities in the
swash area (Guza and Thornton, 1982) and have been hypothesized
as the cause of morphological changes alongshore and of cross-shore
bars (e.g., Carter et al., 1973; Hoirnan and Bowen, 1982~.
Generation mechanisms proposed could be the result of preferen-
tial forcing offshore by nonlinear wave/wave interaction (Gallagher,
1971), surf-beat forcing (Longuet-Higgins and Stewart, 1962), gener-
ation by time-varying break-point (Symonds et al., 1982), or through
resonant tuning by the bar from a broad-spectrum offshore forcing
(Symonds and Bowen, 1984~. Existing models for infragravity wave
generation in the surf zone are not well verified and need to be
developed to incorporate better physics for infragravity wave dissi-
pation. The interaction of sea-swell waves with infragravity waves in
the nearshore needs to be investigated. Although some generation
mechanisms have been proposed and utilized in initial models, the
level of understanding is not well developed and field measurements
are required for mode! development and verification. Measurement
of longshore variation in setup, radiation stress, and related mor-
phology is necessary for further development of these modem.
Swash Zone
A highly dynamic area is the swash zone, the area where the
water edge runs up and down the beach face. Here the amplitude of
the infragravity waves is at a maximum, the waves finally dissipate,
and the bottom is highly variable due to the energetics of the fluid
motion. To mode! the swash properly, the moving edge of the water
surface running up and down a sloping bottom needs to be included.
Permeability, the internal pressure, and the flow field withm the
sandy bottom of the swash zone influence the awash dynamics and
are import ant to the sediment movement.
Sediment Transport
Mathematical modeling of sediment transport has progressed
significantly over the past decade, but basic theoretical and mod-
eling questions still remain. Predictive mathematical models for
sediment transport are in routine operational use; however, lack of
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verification over a broad range of conditions makes their predictions
questionable. Coastal engineers need improved models of nearshore
sediment transport to perform engineering tasks effectively and to
bring this aspect of their discipline up to the level attained by our
understanding of other processes (surface gravity waves, infragravity
waves).
Major field experiments have been directed at improvement of
our knowledge of sediment transport (NSTS*, C2S2**, for instance),
yet we still are not able to predict this transport with sufficient accu-
racy to meet many engineering needs. This difficulty arises from the
interaction of the driving forces twaves, currents) with the sediment
that they are moving. Once the sediment moves and achieves a new
form (profile, roughness), this affects the driving forces, changing
their characteristics and thus changing the equilibrium form of the
sediment. This feedback makes modeling difficult. Further compli-
cations arise because of the nonlinear terms in the sediment and fluid
flow equations. For instance, in shallow water, bottom friction plays
an important role in the momentum balance. The form of the bot-
tom friction term is nonlinear; that is, friction is not linearly related
to velocity. In addition to velocity, bottom friction is related to the
grain size of the bed, to the roughness of the bed, to the interaction
of steady or quasi-steady currents with waves, and to their relative
directions. Although models have been developed to account for
some of these nonlinear effects (e.g., Smith, 1977; Grant and Mad-
sen, 1979, 1982), these models assume the nonlinear interaction of a
linear wave and a current. In shallow water, waves themselves be-
come nonlinear, creating a nonlinear interaction between a nonlinear
wave and a current. No adequate models for these effects have yet
been developed. In summary, the prediction of sediment transport
in shallow water is complicated by
interaction of the sediment with the driving forces,
the mobility of the bed (it is not stationary),
the nonlinear character of waves in shallow water,
lack of understanding of turbulent momentum balances in
shallow water, and
lack of incorporation of cohesion or biological binding effects.
*National Sediment Transport Study.
**Canadian Coastal Sediment Study.
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In spite of these limitations, progress has been made ~ modeling
sediment transport In the nearshore zone. Some of these models are
describer} below.
Sed~rnent Entrainment
Sedunent resting on the bed must be entrained by the fluid before
being transported. The entrainment potential can be represented as
a critical shear stress at the bed required before the sediment leaves
the bed. Alternatively, this entrainment quantity can be nondi-
mensionaTized as a critical criterion for the threshold of initiating
transport. The threshold function depends on grain size, grain size
distribution (whether unimodal or multimodal), biological activity
(organisms can be either stabilizing or destabilizing), and cohesion
(both mineralogical and other chern~cal effects).
Previous work on sediment entrainment has been mostly empir-
ical (e.g., Madsen and Grant, 1976; Komar and Miller, 1975, 1973;
and Inman et al., 1976~. The latter investigators all assumed sedi-
ment of uniform grain size, although some discussion of mixtures of
grain sizes has been included in the literature (e.g., Kamphuis, 1975;
Madsen and Grant, 1975, 1976~. Little comprehensive treatment of
nonuniform grain sizes has been available for oscillatory flows. Fi-
nally, some recent work has expanded on the role of biological effects
and cohesion on sediment entrainment (Jumars and Nowell, 1984;
Nowell et al., 1981~.
Near-Bed Sediment Transport
Near-bed (or bed-Ioad) transport occurs near the bottom and
incorporates as a dynamically important element grain-to-grain con-
tact and collisions for its sustenance. Since the precise definition of
bed-Ioad transport has been debated, we refer instead to near-bed
transport where gra~n-to-gra~n interactions are important dynam-
ically. Available models for near-bed transport take many forms.
Although it is not possible to review all models of transport, some
are summarized briefly here. These models all suffer from lack of in-
clusion of adequate physics of near-bed turbulence, the interaction of
waves and currents, and field verification. Major steps forward in this
modeling require improved understanding of momentum exchanges
within the thin (order of 10 cm) wave boundary layer (the layer
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closest to the bottom, which is stationary at the bed and moving at
approximately the wave-driven flow at the top).
Near-bed transport equations come in many different forms.
Near-bed transport integrated ~ the cross-shore direction ~ exempli-
fied by the commonly used equations to predict total longshore sed-
iment transport under wave-induced currents. Here, quasi-empirical
and often dimensionally incorrect equations are derived to represent
the integrated transport across the surf zone. The most commonly
used of these is the CERC longshore transport equation (COE, 1984),
as well as those of Komar and In man (1971) and others. These in-
tegrated forms rely heavily on field observations to set one or more
coefficients. In practice, the U.S. Army Corps Districts modify these
coefficients, which are intended to be universal, to match some sum
jective input, such as position of a nodal point in longshore transport
or total transport trapped in a tidal inlet.
A second popular approach to sediment transport has been the
energetics approach (Bagnold, 1963; Inman and Bagnold, 1963;
Ballard, 1981~. These models assume that a fraction of the dissi-
pated wave energy is available to move sediment. By setting the
fraction of dissipated energy involved in sedunent transport, one can
calculate the total transport. As shown by Aubrey (1978) and others,
incorporation of a nonlinear wave or a steady current can result in
net sed~rnent transport (instead of just oscillatory, zero-net transport
under a linear wave).
A third type of approach to sediment transport has been to relate
various sediment parameters to the driving forces, using empirical re-
lationships. Popular are those relating transport rate to the threshold
parameter (Meyer-Peter and Muller, 1948; Ackers and White, 1973~.
These methods have been tested in a variety of field settings, against
themselves and against other transport formulations. Although the
authors commonly reach conclusions about the advantages of one
particular method, field data generally are inadequate to draw true
conclusions. CERC uses the Ackers and White formulation for much
of their sediment transport calculations (e.g., Vermulakonda et al.,
1985), in spite of lack of good field or laboratory evidence supporting
its use in oscillatory flow.
A fourth method used to calculate near-bed transport is the
probabilistic method derived by Brown (1950) and Einstein (1972)
for steady flows. Madsen and Grant (1976) expanded this work to
include oscillatory flow. Although just as well tested perhaps as other
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methods, few models use the probabilistic approach of Einstein and
Brown.
Suspended Sediment Transport
The debate about the relative unportance of near-bed versus
suspended-Ioad transport In the nearshore zone continues unabated.
Field data are only now beginning to quantify the importance of
suspended-Ioad transport, although lack of concurrent measurements
of near-bed transport make it Biscuit to examine relative quantities
of near-bed versus suspended load. As with all sediment-transport
models, various approaches have been applied to examine suspended-
load transport. The basic concept behind much of the modeling is
that fluid turbulence is required to keep sediment in suspension; the
primary force acting to remove sediment from suspension is grav-
ity. Whereas in near-bed transport gra~n-to-grain interactions are
important, for suspended load, fluid/grain interactions predominate.
Models diner in how they represent mathematically the turbulence
or maintenance forces.
A heuristic mode} of suspended-Ioad transport was derived by
Dean (1973) to examune suspended sediments in the surf zone. More
complicated models involving turbulence explicitly have been pros
posed by a large number of investigators (Beach and Sternberg,
1988; Grant and Madsen, 1979; Glenn, 1983; Smith, 1977; Souisby,
1988~. These models use eddy diffusion to represent the turbulence
that maintains the sediment in suspension. More complicated mod-
els incorporate different representations of turbulence. Prime among
these are the so-called higher-order closure models, where turbulence
production is calculated (e.g., Mellor and Yamada, 1974; Adams
and Weatherly, 1981; Sheng, 1982~. Computational complexity in-
creases with the higher order closure models. So far, too little data
have been collected to evaluate these models adequately, particu-
larly for shallow-water, nonlinear wave conditions. Most application
and evaluation has taken place on midcontinental shelf areas or in
estuaries.
Another common method for modeling suspended-sediment
transport is based on observations. Some representation of the driv-
ing force is related empirically to observed sediment concentrations.
These measurements are always time-integrated, because of the dif-
ficulty some researchers have in sampling rapidly enough. Examples
of these methods include Inman et al. (1980) and Kraus et al. (1988~.
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Beach Morphology
Prediction of the morphology of a beach under given conditions,
and of changes in that morphology as driving forces change, has been
a major activity of nearshore researchers because of its importance
to coastal engineering. For designing structures in the nearshore,
evaluating potential for amphibious landings, or designing beach
restoration projects, the behavior of the beach profile is critical.
This long history of interest has resulted in a variety of approaches
to profile prediction, some of which are well-tested under limited
conditions.
The initial work on beach morphology was empirical (Bascom,
1951; Aubrey, 1978, 1979; Birkemeier, 1985), relating changes in
profiles to some aspect of the driving force. Wright et al. (1985)
describes the changes in the general shape of a beach in response to
the driving force. Closely related to this empirical modeling is infer-
ential modeling, where hydrodynamic patterns are related loosely to
possible profile configurations. An example of inferential modeling is
an article by Holman and Bowen (1982) that relates theoretical inter-
ference patterns of surface gravity and infragravity waves to possible
shoreline configurations.
Energetics models of beach configuration have been developed
based on the initial work of Bagnold and coworkers (Bagnold, 1963;
Inman and Bagnold, 1963~. These later models (e.g., Aubrey, 1978;
Bowen, 1980; Ballard, 1981) relate the equilibrium slope to a po-
tential transport of sedunent related to the driving forces by an
energetics argument
Other models of beach planform change have incorporated a
variety of assumptions. Early analytical models were derived by
Pelnard-Considere (1956) and later discussed for more general sit-
uations by Larson et al. (1987~. Dean (1977) has derived a model
that has been expanded (Perlin and Dean, 1983) to enable prediction
of shoreline changes out to various depths, based on an equilibrium
profile concept. Swart (1974, 1977) has derived extensive models for
profile response under varying wave conditions based on observations
in the North Sea. CERC recently has implemented its own shoreline
response model based on a number of these previous studies (e.g.,
Perlin and Dean, 1983~. Sunamura and Horikawa (1974), Watanabe
(1982), Kraus and Harikai (1983), and Nishimura and Sunamura
(1987) propose different models of beach morphology change.
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Bed Forms
Bed forms are responsible for much of the transport of sediment
in the inner continental shelf-nearshore zone, and as such might be
incorporated in the earlier section on near-bed transport. However,
bed forms are important contributors to the momentum balance of
nearshore circulation, so they deserve an independent mention. Bed
forms are the response of a deformable sand bed to hydrodynamic
shear stresses applied at the surface of that bed. Rarely is a bed per-
fectly flat; instead it wiB have some scale of structure super~rnposed
on it. Prediction of these bed forms, and how they respond to differ-
ent wave and current forcing, is essential for predicting circulation
and sediment transport in the nearshore zone.
Bed-form prediction schemes are largely empirical in nature and
have addressed both steady and oscillatory flows. Early work includes
that of Southard (1971), Clifton (1976), Komar (1974), and Rubin
and McCulloch (1979~. Later work used a threshold-of-transport
representation to examine stable conditions for the existence of bed
forms. Included in this work is that of and Miller and Komar (1980),
Greenwood and Sherman (1984), and Dingier and Inman (1976~. Fi-
nally, dunensional analysis has been applied to bed-form prediction
to obtain stability criteria. Included in this aspect is work by Din-
gler and Inman (1976) and Yalin (1977), among others. Dynamical
models for bed forms are sorely lacking. Existing bed-form models
are useful for limited scales of bed forms (ripples, dunes, and sand
waves). Some of the largest-scale bed forms are poorly predicted
(large sand waves or sandbanks, submarine bars), leaving a large gap
in the ability of coastal engineers to make accurate calculations in
certain environments.
Bed-form prediction ability also is weak in combined steady ~d
oscillatory flows. The effects of combined waves and currents of
various relative magnitudes and directions on bed forearms have yet to
be modeled theoretically or observed adequately in the field.
MODELING FORCES ON STRUCTURES
The state of the art in modeling of wave forces on offshore com-
mercial platforms is a highly developed technology. A review of
the literature with respect to wave forces on fixed tubular mem-
bers and submerged tanks of dimensions comparable to the wave
length is given by Dean and DaIrymple (1984~. Both deterministic
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and stochastic spectral methodologies exist for estunation of wave
loading. The newer technology addresses nonlinear coupled models
of wave/structure interaction for compliant structures in deep water
and Boating tethered structures (e.g., Crandall, 1985; Jeffreys and
Patel, 1982; Basu, 1983; Nie~zwecki and Sandt, 1986~.
In striking contrast, the design of breakwaters, jetties, and groins
is based largely on highly empirical methads ~d past experience
(CERC, 1984~. The internal fluid/solid and solid/solid dynamic
stresses created by large waves striking and possibly overtopping such
structures is poorly understood. At present there is no capability
for measuring and modeling the internal dynamics of rubble-mound
structures.
Representative terms from entire chapter:
bed forms
