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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 interestfor 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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80 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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81 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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82 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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83 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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84 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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85 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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86 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.