Simulation Model for Storm Cycles and Beach Erosion on Lake Michigan

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WILLIAM T. FOX Williams College, Williamstown, Massachusetts 01267 RICHARD A. DAVIS, JR. Western Michigan University, Kalamazoo, Michigan 49001 Simulation Model for Storm Cycles and Beach Erosion on Lake Michigan

ABSTRACT generated from the southwest forming waves and northward-flowing longshore currents. As A mathematical simulation model is used to the low-pressure front passes through the area, study the relations among storm cycles, beach the winds and waves build up and shift over to erosion, and nearshore bar migration. The the northwest, and strong southward-moving model is based on Fourier analysis of weather longshore currents are generated. During and wave data collected on Lake Michigan storms, bars and rip channels are formed near during the summers of 1969 and 1970. the shoreline (Davis and Fox, 1972). In the In the simulation of coastal processes, quiet poststorm interval, gentle waves lap barometric pressure is used as the independent across the bars which migrate slowly toward the variable with longshore current velocity com- shore. A cyclical pattern develops in which puted as the first derivative and breaker height winds accompanying low-pressure systems as a filtered version of the second derivative of generate waves and longshore currents that barometric pressure. The simulated curves are flip-flop back and forth from north to south. Nearshore bars and rip channels that are used to compute wave and longshore current formed during storm conditions respond to the energy for each storm cycle and poststorm changing patterns of winds and waves. recovery. Daily profiles across the nearshore area A computer simulation model has been provide data for topographic maps and maps of developed to study the relations between wind, erosion and deposition. For simulation, the waves, and erosion for a nontidal coastal nearshore area is broken down into five com- environment. Data for the model were col- ponents including beach, foreshore, plunge lected along the southeastern shore of Lake zone, trough, and bar. A gently sloping linear Michigan during the summers of 1969 and plus quadratic surface is used to represent the 1970. These data provide a framework for a barless topography, with bars and troughs computer program that is used to mathemat- generated by normal curves. Bar distance is ically model coastal processes and the response computed as a function of wave energy and of morphologic features in the beach and bottom slope. Position of the bar and trough nearshore environment. A mathematical sim- along the shore is determined by wave and ulation model is made feasible by employing longshore current energy. Simulated maps are a high-speed computer to perform the neces- produced for each storm cycle and poststorm sary calculations, logic operations, and graph- ical displays. The value of the simulation lies recovery. in the ability to rather quickly explore the INTRODUCTION complex interactions among many different Along a coastline, there is a dynamic inter- variables while holding some of the variables play between air, sea, and land in the form of constant. The ultimate aim of the coastal wind, waves, and beaches. Waves, longshore simulation model is to predict the rates and currents, and rip currents are constantly magnitude of coastal processes and their effects shifting sand, forming new bars and spits, and on the beach and nearshore topography. destroying beaches. The bar and trough topography developed During the summer months, low-pressure on the eastern shore of Lake Michigan is systems move through the Great Lakes region similar for the ridge and runnel system de- at 4- to 6-day intervals. As low-pressure scribed along many shorelines. King and systems approach out of the west, winds are Williams (1949) first described the ridge and

Geological Society of America Bulletin, v. 84, p. 1769-1790, 15 figs., May 1973 1769

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runnel system and restricted it to coasts with and current parameters to study longshore a considerable tidal range using the term transport of sand on tieaches. These equations "barred beaches" for nontidal areas. According have extensive possibilities for developing a to Davis and others (1972), the morphulogy deterministic computer simulation model in- and genesis are quite similar in tidal and non- volving nearshore cirrulation cells, but as yet tidal areas, and the terms "ridge" and "runnel" such a model has not been attempted. can be applied to both environments. Com- The model proposed in this paper is a com- parable ridge and runnel development has bined deterministic and probabilistic model been described from the North Sea coast of empirically fitted to the data. The evolution Germany (Reineck, 1964), Plum Island along of the model has passed through several stages. the New England coast (Hayes, 1969), and After the initial data collection, Fourier along Padre Island in the Gulf of Mexico analysis was used to synthesize and smooth (Hayes, 1967). Therefore, a model that can be the coastal-process da :a (Fox and Davis, 1970, used to describe the formation and migration of 1971a). The response of the beach and near- nearshore sand bars on Lake Michigan can shore bars to coastal processes was recorded by also be applied to other coastal areas around closely spaced daily profiles across the study the world. area. These surveys were incorporated into a sequence of topographic maps and eventually SIMULATION MODELS OF into a four-dimensioral diagram called an area- SEDIMENTARY ENVIRONMENTS time prism (Davis and Fox, 1971). Using Over the past few years, several mathemat- barometric pressure as a key variable, a con- ical simulation models of marine sedimentation ceptual process-response model has been con- have been attempted. Harbaugh (1966) structed which shows the effects of waves and devised a probabilistic simulation model to longshore currents oti the morphology of the study the behavior of sediments as they are beach and inner nearshore environment (Davis transported and deposited within a marine and Fox, 1972). From the conceptual model, a basin. Briggs and Pollock (1967) developed mathematical model was formulated for com- a deterministic simulation model of evaporite puter simulation (Fox and Davis, 1971b). In sedimentation in the Michigan basin to explain the simulation model, wave height and long- the observed distribution of Upper Silurian shore current velocily are derived from bar- salt deposits. Bonham-Carter and Sutherland ometric pressure. Wave and longshore cur- (1967) used a combined probabilistic and rent energy are computed for each storm cycle deterministic model for the simulation of and poststorm recovery period. Wave and deltaic sedimentation. Hydrodynamic equa- longshore current energy are used in mathemat- tions are used to model the velocity field and ical functions to control beach and nearshore sediment diffusion patterns at the mouth of a bar morphology through time. The develop- river. McCallagh and King (1970) developed a ment of the computer simulation model along probabilistic model to simulate the formation with listings of the computer programs and of a recurved spit on the south coast of Eng- samp'.e data is explained in Fox and Davis land. McCammon (1971) used a probabilistic (1971b). model for reconstructing the environmental It should be emphasized that the model is pattern of the Mississippi Delta region. restricted in application since it is based on Along with the above deterministic and empirical relations observed in the field and probabilistic simulation models, several workers not on hydrodynamic equations derived to have developed equations for sediment trans- explain the processes. The beauty of the model port in the nearshore area. Shepard and In- lies in its ability to mimic changes in the man (1951) defined a nearshore circulation cell beach and nearshore morphology based on consisting of two rip currents, associated long- simple weather patterns. shore currents, and drift through the breaker zone. Bowen and Inman (1969) expanded the FIELD PROCEDURES study of rip currents using laboratory and field During the summers of 1969 and 1970, time observations. Longuet-Higgins (1970) devel- series data were collected on the eastern shore oped the mathematical theory for longshore of Lake Michigan near Stevensville and Hol- currents generated by obliquely incident waves. land, Michigan (Fig. 1). Field observations on Komar and Inman (1970) made simultaneous meteorological conditions, lake and ground- measurements of sand transport rate and wave water level, wave parameters, and longshore

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current velocity were measured at 2-hr inter- tion period. In time-series analysis of coastal vals for 30 days. Meteorological measurements processes, the wavelength is given in hours and include barometric pressure, air temperature, the amplitude in the observed units for each wind speed and direction, and sky condition. parameter. Lake and ground-water levels were recorded The observed data can be represented by in pipes that were placed in the beach with a an aggregate of simple wave forms that are tube connecting the lake-level pipe to an in- expressed by the amplitude of the cosine and take located beyond the inner bar. Wave sine terms, a„ and bn, respectively. Although height, period, breaker angle, type, and dis- the function of the form Z =f(t) is not known, tance from shore were measured on the shore- data points A, Zi, where ti is time and Z,- is the ward side of the nearshore bar. For part of observed parameter, are available at equal the study, a step resistance wave staff was intervals. Thus, the coefficients an and bn may located on the outer bar about 350 ft from be determined by numerical integration the shore. Longshore current velocity was methods employing equations (1) and (2) and measured parallel to the shore between the used in equation (3) to approximate the ob- nearshore bar and the plunge zone. Usually, served curve according to methods described the maximum longshore current velocity was by Tolstov (1962) and Harbaugh and Merriam found to be just on the shoreward side of the (1968): breaking waves. The techniques, instruments, and computer program and lists of the ob- 2 Ivnu a = — 2-1 A cos served data are described in Fox and Davis n K, K (1970, 1971a). n = 0, 1, 2, ..K/ 2, (1) Two surveying programs were carried out to monitor changes in the nearshore topography 2 limn sin ——- through time. First, nine profiles spaced at A. t=l A. 100-ft intervals along the shore were made each day across the beach and inner nearshore n= \,2,...,K/2, (2) area. Using the stake and horizon method, and the profiling was extended to a depth of over 4 a Imti ft except during extremely high wave condi- Zi•7 = —° +I •sr cos tions. Second, a 20-ft aluminum pipe was used L n=i K for surveys across the nearshore area. A boat , . 2irnti „ was used to stretch a rope 1,400 ft normal to + bn sm— (3) the shore. Distances were marked along the rope, and depths were read from the pole by a \ swimmer and recorded in the nearby boat. The I1 second surveying technique was carried out 1 1970 after each storm to measure any significant 0 10 20 HOLLAND topographic changes across the offshore bars. The profiles were processed on the computer miles to construct daily topographic maps and maps of erosion and deposition. Copies of the daily LAKE maps and computer programs used in process- ing the topographic data are included in MICHIGAN

Davis and Fox (1971). / MICHIGAN

1969 / FOURIER ANALYSIS OF COASTAL

PROCESSES ^^^r STEVENSVILLE Coastal processes that are cyclic in nature can be best described using Fourier analysis. In Fourier analysis, a curve can be broken down into a series of sine and cosine curves. The 20 40 observed data can be expressed in terms of the INDIANA M: fundamental harmonics which are theoretically i kilometers independent. Each harmonic has a wavelength Figure 1. Index map showing location of study that is a discrete fraction of the total observa- areas in southeastern Lake Michigan.

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where n is the degree of the term, ao is the By using Fourier analysis to study the relations mean, K is the number of sampling points, and among these curves, il is possible to derive a N is maximum degree of the series N = K/2. theoretical model that can be used for predict- In analyzing coastal processes, it is con- ing longshore current velocity and direction venient to plot each harmonic as a single sine and breaker height directly from barometric curve with a given phase and amplitude. When pressure. the sine and cosine curves are added alge- The 15-term Fourit:r curve for barometric braically, a sine curve results with a new phase pressure during July 1 970, which accounts for and amplitude. The phase pn can be deter- 95.4 percent of the total sum of squares, is mined by using an arc tangent subroutine plotted across the top of Figure 2. During the according to equation (4) (Louden, 1967): study period, the observed barometric pressure ranged from a minimum of 29.60 in. to a

pn = arc tangent — . (4) maximum of 30.21 in. of mercury. Four major an low-pressure systems passed through the area The phase expressed in radians or degrees is on July 4, 9, 15, and 19, 1970. The lows were used to determine the starting point for a sine accompanied by high wind and waves that curve for each individual harmonic. The caused beach erosion and deposition and 2 discrete power spectrum '5. • 2Q . . 25 1964):

2 2 2

The amplitude an for each harmonic is derived from the square root of the power spectrum according to equation (6):

2 (T„ = Va„ + bn* , n = 0,1,2, ...,K/2. (6) The height of the curve Zi can be computed at each sampling point using the phase and amplitude according to equation (7):

a 2 n + Z7 i = -° +_ L Vh ~) L „=1 n A. The amplitudes of the Fourier coefficients are especially meaningful because they are in the same units as the original data. Thus, if wave height is measured in feet or longshore current velocity in feet per second, the Fourier coefficients will be expressed in the same respective units. Three related families of time-series curves can be used to represent the coastal processes along the eastern shore of Lake Michigan (Fig. 2). First, the cyclical pattern of low- and high-pressure systems is shown in the curve for barometric pressure. Second, plots of the longshore component of wind and longshore current reflect the shift in wind direction as the low-pressure system passes over the area; these plots closely resemble the first derivative of the barometric pressure curve. Thirc., the curves for wind velocity and breaker neight are very similar to the inverse of, and resemble pressure, wind velocity, breaker height, longshore the second derivative of, barometric pressure. wind, and longshore current velocity during July 1970.

Wind velocity, duration, and fetch are the curve for longshore current velocity and the driving forces that control wave period and curve for the longshore component of the breaker height; therefore, a phase lag would be wind (Fig. 2). The longshore wind in miles expected in the Fourier curves with wind per hour is approximately 10 times the long- velocity increasing first, followed by wave shore current in feet per second. Therefore, period and breaker height. The average wind along the eastern shore of Lake Michigan, it velocity over a 2-hr interval was used in com- would be possible to use the longshore com- piling the Fourier curves. The maximum wind ponent of the wind to get a rough approxima- velocity of 32.2 mi per hr was recorded during tion of longshore current velocity. the passage of the first low-pressure system at 8:00 a.m. on July 4. CONCEPTUAL MODEL OF There is a close relation between breaker NEARSHORE SEDIMENTATION height and wind velocity. The four major Before proceeding to a mathematical simula- peaks in the breaker height curve correspond tion model, it is useful to develop a conceptual closely to the peaks in the wind velocity curve model in an attempt to understand the rela- (Fig. 2). On Lake Michigan high waves are tions between storm cycles and sand-bar generated by local winds, and the peaks in the migration (Davis and Fox, 1972). The model breaker height curve are aligned with the peaks can be broadly thought of in terms of energy in the wind velocity curve. The maximum transfers that take place at the air-water and breaker height of 3.5 ft was observed during water-sediment interfaces. When a front the passage of the third low on July 20. The accompanying a low-pressure system moves curve for breaker height accounts for 79.1 through an area, gradient and geostrophic percent of the total sum of squares. winds blow in a counterclockwise direction The reversal of wind direction with a pas- around the center of the low (Cole, 1970). As a sage of a low-pressure system plays an impor- low-pressure system moves over the surface of tant role in controlling coastal processes. a body of water, waves are generated by the During the summer months, low-pressure winds, resulting in an energy transfer across the centers generally pass to the north of the study air-water interface. When the waves approach area. As a low-pressure system moves into the the shore and break at an angle to the beach, area, winds circulate in a counterclockwise some of the wave energy is dissipated as heat direction around the low. The southwest winds in turbulent motion, and some is transformed generate waves that break on the north-south into longshore current energy. When the shoreline at an angle forming northward-flow- conditions are right, the longshore currents ing longshore currents. In like manner, after are shunted offshore and rip currents are the passage of the low-pressure system, winds produced which form nearshore circulation shift over to the northwest generating waves cells (Shepard and Inman, 1951). A second out of that direction. With the shift in wind energy transfer takes place when the waves direction and breaker angle, the longshore and currents sweep the bottom and lift the current is reversed and flows to the south. bottom sediment into suspension or move it Waves and wind approaching from the north- by saltation and traction. Sand is transported west and the southward-flowing longshore down the beach and offshore by currents to current are considered positive, while wind and form shoals and sand bars in the nearshore waves from the southwest with northward- area. The process portion of the model is flowing longshore current are negative (Fig. 2). concerned with the first energy transfer from air to sea in the form of wind to waves, and the Plots of the longshore component of the response portion is concerned with the second wind and longshore current velocity are given energy transfer from waves and currents to the in Figure 2. In the study area located at Hol- bottom sediment resulting in erosion and land, Michigan, the shoreline is oriented in a deposition of sand bars. north-south direction with land on the east side (Fig. 1). Wind direction was recorded as During a storm cycle, large amounts of an azimuth measured in a clockwise direction sediment are lifted into suspension by waves from true north. In order to compute the or moved by bed load and transported by longshore component of the wind, a cosine longshore and rip currents. Large waves break transformation of wind direction was used. near the shore at an angle with the beach, and There is a striking resemblance between the strong longshore currents are developed

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parallel to the shore (Sonu, 1972). Nearshore processed by the computer to produce topo- bars tend to confine the longshore current to graphic and erosion and deposition maps. The the trough between the bar and the shore. computer-printed map for July 17, 1970, has Where the current is deflected oils tore by a three-times oflViore exaggeration to show protuberances or large cusps in the beach face, detail in the nearshore area (Fig. 3). A series a portion of the longshore current may spill of contour maps lor July 13 through July 27 over the bar forming a rip current. When a showing progressive changes in the nearshore persistent rip current is deflected over a sand area is given in Figure 4. bar, it may cut a saddle or channel through The sequence of nearshore maps (Fig. 4) the bar. During a storm, a series of pro- starts out on July !.3 with a typical bar and rip tuberances develop along the beach face channel development following a poststorm separated by crescent-shaped embi.yments recovery. Two bars enclosed by the 2-ft con- spaced at more or less regular interval along tour line lie roughly parallel to the shore with the shore. Bars develop adjacent to the saddles marking rip channels between the bars. embayments and rip channels in front oi the The shoreline has been eroded at the head of protuberances. Therefore, the position of bars the old rip channel, and a protuberance has along the shore is often closely related to the built out behind the bar. sinuous pattern of large cusps seen along the The next map in the sequence (Fig. 4, 7—15— shoreline. 70) shows the beach and nearshore topography After the front moves out of the area and shortly after a storm passed through the area. the storm conditions subside, the wind slowly The storm is shown in Figure 2 by the drop shifts back to the southwest producing small in barometric pressure accompanied by high waves from that direction. The relatively calm winds, strong longshore currents, and high period between storms has been labeled the waves on July 15. As the storm approached, the poststorm recovery. During the poststorm southwest waves generated strong northward- recovery, small waves move across the bar, flowing longshore currents which were de- and sand is transported shoreward by saltation flected by the shoreline topography forming and traction. Sand lifted from the lakeward rip currents at locations 300 and 700. During side of the bar cascades down the prograding the storm, the bar at the north end of the map slip face as the bar migrates slowly toward the was eroded, and a prominent rip channel was shore (Miller and Zeigler, 1964; Sonu, 1969; developed in its place. Hayes, 1969). The low waves tend to break The maps for July 17 and July 19 display the over the shallow sand bars but pass unimpeded shoreward migration of the bar during the next through the inactive rip channels. The sand poststorm recovery (Fig. 4, 7-17-70 and bars act like detached breakwaters that inter- 7-19-70). Between July 15 and July 17, the cept the waves and create protected areas of bar migrates about 25 ft toward the shore and relatively calm water between the bars and the builds up so that its crest is less than 1 ft below shore (Johnson and Eagleson, 1966). Refrac- the mean lake level. During the migration, the tion and defraction of waves in the channels shoreline builds behind the bar and is eroded between the bars set up weak currents from the channels to the quiet areas behind the bars. Therefore, sand is eroded at the head of the old rip channels and deposited in the protected areas behind the bars. As the bars continue to migrate toward the beach, the shoreline builds out behind the bars and may eventually extend to the bars. During a poststorm recovery, new protuberances build out behind the bars, and the old protuberances that existed between the bars are eroded back. The beach, bar, and rip channel development in the conceptual model is based on a Figure 3. Computer-printed map for July 17,1970. series of daily maps from Holland, Michigan Contours are in feet, and there is a 3:1 offshore exag- (Davis and Fox, 1971). Nine profiles spaced geration with a 100-ft spacing between profiles. North at 100-ft intervals along the beach were is to the left.

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in front of the rip channel. Between July 17 rip channel is E.:companied by deposition and 19, the bar migrates about 10 ft farther behind the bar. E'uring the next storm cycle, toward the shore, and the shoreline ontirues the longshore current is shunted offshore by to be built out behind the bar and erodec in the protuberance forming a strong rip current. front of the rip channels. During th: 4-day A nearshore circulation cell is established as poststorm recovery, the bar migrated about the rip current erodes a channel through the 35 ft toward the shore, while the protuDerance previously existing bar. New bars are formed built out about 20 ft behind the bar. between pairs of rip currents near the center The most dramatic changes in bar and shore- of the circulation cell. During a storm, the line configuration took place on July 20 (Fig. 4, lateral boundaries of circulation cells are 7-20-70). At that time, a front passed through defined by protuberances along the shore. the area accompanied by strong winds from The idealized conceptual model consists of the northwest which generated high waves and two storm cycles eac.i followed by a poststorm a powerful southward-flowing longshore cur- recovery period. As with the ideal cyclothem rent (Fig. 2). The longshore current was (Weller, 1960), the conceptual model with two deflected offshore by the protuberance at B storm cycles can be interrupted at different and formed a rip current at location 500. The stages in its development. If an extended post- bar that was located at the middle of the map storm recovery takes place following a storm on July 19 was eroded by the rip current. A cycle, a tombolo will form between the shore rip channel marked by the 3-ft conto.r at and the bar, and the bar will eventually mi- location 500 provided further evidence of the grate and weld onto the beach. Examples of erosive power of the longshore and rip currents. the bar-welding process are given by Davis New bars developed between the rip currents and Fox (1971) and Hayes (1969). If one storm at locations 200 and 700. Sediment that was BAROMETRIC PRESSURE excavated from the trough and bars by the longshore and rip currents was deposited to form the new bar. The maps for July 23 through July 27 (Fig. 4, 7-23-70 and 7-27-70) show the beginning stages of another poststorm recovery. An idealized conceptual model showing the relation between barometric pressure, longshore current, and breaker height for two storm cycles and two poststorm recovery periods is given in Figure 5. In the model, two low-pressure cells pass through the area forming two storm cycles separated by a poststorm recovery. In the ideal summer storm cycle on the eastern shore of Lake Michigan, the wind velocity reaches a peak a short time after the minimum in barometric pressure. As the front moves through the area, the wind direction shifts from southwest to northwest, and the longshore current reverses direction from north to south. As barometric pressure falls, breaker height increases and reaches a peak a few hours after the low has passed. In the conceptual model, the bar form oscil- lates back and forth during storms changing positions with the rip channels (Fig. 6). The poststorm recovery period with the shoreward migration of sandbars and the deposition and erosion of protuberances along the shore plays pressure, wind velocity, longshore current, and breaker an essential role in this model. During the height during two storm cycles (from Davis and Fox, poststorm recovery, erosion at the head of the 1972).

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WAVE ENERGY H,0H ^wmmmmmv - —» LOW 41 \ \ \ \mm\ BEACH [g§| SAND BAR Figure 6. Idealized sequence of maps showing cycles shown in Figure 4 (from Davis and Fox, 1972). beach erosion and bar migration during the two storm cycle is followed closely by a second storm influenced the rip current in the first storm cycle, the intervening poststorm recovery will cycle is still present, the rip current generated be bypassed. With the omission of the post- by the second storm cycle will follow the same storm recovery, there is not sufficient time for path. If the bar has started to migrate toward the protuberance behind the rip channel to the shore, the leading edge of the bar will be be eroded and for a new protuberance to form eroded by the longshore current, but the bar behind the bar. Since the protuberance that will not change its position significantly along

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the shore. An example of closely spaced storm closeness of fit of the model to the empirical cycles is seen on the maps for July 4 and July data. Using this method, it is possible to adjust 8, 1970, in Davis and Fox (1971). The second the parameters in t le model so that predicted storm followed so closely on the heels of the values fit as closely as possible to the observed first that it acted as an extension of me first data. storm cycle. The conceptual model explains the migration Simulation of Longshore Current and of nearshore sand bars as environmental condi- Breaker Height tions and coastal processes change through Several equation:; have been derived to pre- time. A definite cyclical pattern is present with dict longshore current velocity from breaker essentially three steps to each cycle (Figs. 5 angle, breaker height, and wave period. and 6). A pair of storm cycles and poststorm Equations used to predict longshore current recovery periods are necessary for the shore- velocity are based on the conservation of line to return to its original configuration. It is momentum (Putnam and others, 1949; Eagle- interesting to note that the bar form itself is son, 1965; Longuet-Higgins, 1970), conserva- not migrating along the shore but is being tion of mass (Bruun, 1963; Inman and Bagnold, bisected and reformed by rip currents during 1963), or empirics.! correlation of the data storms. The net result is a longshore transport (Inman and Quinn, 1951; Harrison and of sediment in the direction of the dominant Krumbein, 1964; Harrison, 1969). With these longshore current with the bar forms osc.llating equations, it is necessary to have field measure- back and forth. The relations between coastal ments of breaker height, wave period, breaker processes and bar migration presented in the angle, and bottom slope before a prediction of conceptual model are used as a framework for longshore current can be made. Since field the computer simulation model. measurements are costly and sometimes difficult to obtain, a new method of longshore COMPUTER SIMULATION MODEL current prediction is proposed using barometric In the computer simulation model, the wave pressure data thai: can be obtained from and longshore current energy during storm weather maps. cycles and poststorm recoveries is used to From an inspection of the curves for baro- modify the nearshore topography t'irough metric pressure and longshore current velocity time. Several intermediate steps are necessary reproduced in Figure 2, the curve for longshore in order to accomplish the final simulation. current velocity closely resembles the first First, a method is developed for simulation of derivative of the curve for barometric pressure the time-series curves for coastal processes. according to equation (8): Second, a set of mathematical functions is formulated for analyzing and simulating the maps of beach and nearshore topography. Third, the wave and longshore current energies where V is the longshore current velocity, p within storm cycles and poststorm recovery is the barometric pressure, and t is time. periods are calculated from the time-series This basic mathematical relation can be curves for coastal processes. Fourth, empirical explained in terms of the physical processes equations are derived to relate wave and long- operating within the area. As the low-pressure shore current energy to the parameters that system approaches, the barometric pressure control the simulation of topographic maps. curve has a negative slope. The steepness of the Fifth, all the previous steps are brought to- slope is a function of the intensity of the low- gether into a single model for predicting shore- pressure system and its rate of movement. A line changes through time from barometric longshore current flowing to the north is pressure data. generated by the southwest winds and waves Before the model can be used for predicting as a low-pressure sys tem approaches. Following shoreline changes, it is necessary to check the the passage of the low-pressure system, the simulation model against the observed data barometric pressure curve rises, and the wind at each stage in its development. For each step and waves shift over to the northwest generat- up to the final prediction model, it is possible ing longshore current to the south. Therefore, to compute the total sum of squares accounted a negative longshore current accompanies for by the model to get an indication of the falling barometric pressure, and a positive

longshore current occurs with a rising baro- of a Fourier series can be expressed as equation metric pressure. Since the derivative is a (10): function of the slope of the curve, the curve for longshore current resembles the derivative Zi = F\U) of the curve for barometric pressure. 2.«(|±ii) + 2^y (10) = V2s n,Jn sin (I - Fourier analysis is based on the summation n=1 \ of a series of sine and cosine curves; therefore, it is a fairly simple matter to take the derivative In equation (10), the lim/K term is added to of barometric pressure to predict longshore shift the phase of the sine term by 90° to form current velocity. The Fourier series given in a cosine with the same phase. equation (3) can be differentiated term by term The first two curves in Figure 7 show baro- to yield equation (9): metric pressure and longshore current velocity for July 1970, and the third curve shows the i / lirnti . 2irnti\ nb = 2-, I n cos —— — nan sin —— J . (9) derivative of the barometric pressure curve which is used for simulating longshore current

Using the phase pn from equation (4) and the velocity. The change from negative to positive amplitude crn from equation (6), the derivative longshore current velocity occurs at the same position in both simulated and observed curves. The small saddle seen in the simulated curve on July 9 is also present in the curve for the BAROMETRIC PRESSURE longshore component of the wind in Figure 2. The amplitude of the southward-flowing longshore current on July 15 and 17 is somewhat higher in the simulated curve than in the observed curve, but the general pattern of the two is very close. The derivative of barometric pressure accounts for 61.2 percent of the total sum of squares of longshore current velocity. Although better results have been obtained using regression equations, the derivative equation accounts for a greater percentage of the sum of squares than equations based on conservation of momentum or mass (Cartnick, 1971). The loss in accuracy is more than compensated for by the simplicity of the model. The curve for breaker height in Figure 3 resembles the second derivative of barometric pressure as given in equation (11):

H = ti (H) dt2 where H is breaker height, p is barometric pressure, and t is time. Equation (12) is used for taking the second derivative of a Fourier series where pn is the phase and an is the amplitude of a sine curve:

N ( 2wn(pn + h) 47T«\ Zi" X) "2 0"n sin . + (12) n=l \ K The second derivative can also be obtained Figure 7. Cumulative curves for the first 15 Fourier by taking a phase shift of 180° times the square harmonics for barometric pressure, longshore current, and breaker height. Simulated curves for longshore of the harmonic number. A better fit is ob- current and breaker height based on the first derivative tained by performing a filtering operation and and filtered version of the second derivative of baro- substituting the harmonic number in place of metric pressure. the square of the harmonic number in equa-

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tion (12). Therefore, the peaks and troughs in I X, 1 X, the filtered curve would be in the sam; position as the second derivative of barometric pressure, but the amplitude of the curve is different as given in equation (13):

f . /2Tn(pn + tj) Wv Figure 8. Generalised profile across the beach, fore- Zi = 2_, n

to each component in the simulate^, carve. slop:; Yp = drop in the plunge zone; Y, = drop in the This phase shift represents the l:g time plunge shelf; Xt = horizontal distance from the trough necessary for wave build-up following a wind to the bar; Yb = vertical drop from the bar to the shift. The 15-term Fourier curve based on the trough. observed breaker height is plotted as tl-s fourth curve in Figure 7, and the simulated curve for the cusps and protuberances that existed along breaker height based on equation (13) is the shore. The ba r distance, defined as the plotted across the bottom of Figure 7. In distance from the plunge zone to the bar crest, comparing the simulated curve for breaker varies considerably through time (Fig. 9b). height with the observed curve, tie four Following the storm on July 15, a bar developed prominent peaks occur at the same position 78 f; from the shore and migrated to within with the same relative height in both curves. 32 ft before being- interrupted by another Although the fit of the two curves is not storm. After the storm on July 20, the bar exact, the overall shape of the two curves is was located approximately 60 ft from the shore- close, and the predicted wave heights art: of the line. same magnitude. Bar height is marked by the vertical distance Since barometric pressure is easily obtained between the trough and the bar as plotted in from local weather bureau records or satellite Figure 9c. Following the storm on July 9, bars data, it would be possible to calculate breaker were present at the: 300- and 700-ft profiles. height and longshore current velocity in a During the storm or. July 15, the bar moved to beach erosion study. It is possible to generate location B profiles. From the July 20 storm to future hypothetical barometric pressure curves the end of the observation period, the bars and to predict longshore current velocity and were centered on the 100- and 600-ft profiles. breaker height by using past weather datr. and a These bars reached a maximum height of 1.95 random number generator. ft. Since the final result is a map, it is necessary Simulation of Nearshore Maps to model changes across a single profile as In order to adapt the map data given in well as along the shore. For modeling purposes, Davis and Fox (1971) to a computer simulation the individual profile: is broken down into five model, it is necessary to analyze the changes component parts: beach, foreshore, plunge that take place in each of the map parameters. zone, trough, and bar (Fig. 8). In simulating a The parameters used in the analysis are plotted map of the nearshore area, the plunge zone in Figure 8 (see Fig. 8 caption). provides a critical boundary. The width of the For each parameter a table was print;d by beach and foreshore is controlled by the dis- the computer giving the observed valus for tance of the plunge zone from the base line. each day along each profile. The profiles are In the model, berm height is read in as a con- arranged across the table with a time sequence stant, and foreshore slope is computed as a from the bottom of the table to the top. iach function of bar distance and bar height. table was then contoured so that it was poj-sible The plunge zone is divided into two parts: to visualize the changes that were taking ;nlace the plunge step and the plunge shelf. In making in each parameter through time. The t.iblcs an analogy between the plunge zone and con- and contour diagrams for each parameter are tinental margin, the foreshore would corres- included in Fox and Davis (1971b). pond to the continental shelf, the plunge step Contour diagrams showing the changts in would be equivalent io the continental slope, plunge distance, bar distance, and bar height the plunge shelf would correspond to the across the map and through time are given in continental rise, and the bottom of the trough Figure 9. The chart for plunge distance snows would be the abyssal plain.

C 700 600 500 B 300 200 100 A distance (/)&), distance from the bar crest to the shore (Xb), and the height to the bar (Yb) are used as independent variables. The foreshore slope is computed as follows:

F = 0.16 - 0.004 Dp 4- 0.0005 Xb - 0.03 Yb . (14) The drop in the plunge zone is computed according to the following equation:

Yp = 0.52 - 0.002 Dp - 0.0004 Xb

+ 0.044 Yb . (15) The drop in the plunge shelf is computed as follows:

Y. = 0.6 - 0.007 Dp - 0.0034 Xb PLUNGE DISTANCE + 0.113 Yb. (16) A simulated profile across the nearshore area can be constructed by using a normal curve k to represent the bar and an inverted normal curve for the trough (Fig. 10). The linear plus quadratic curve in Figure 10A is used to represent the barless topography extending lakeward from the plunge shelf in Figure 8. The inverted normal curve that is used to simulate the trough is given in Figure 10B. The nearshore bar is plotted as a normal curve (Fig. IOC). By combining the normal curve generated for the bar with the curve generated for the trough, an asymmetrical bar and trough curve is formed as shown in Figure 10D. In this example, the trough depth is less than the bar height, giving a steep face on the landward side of the bar. When the combined curve for bar and trough is superimposed on the linear plus quadratic curve, the simulated nearshore profile is formed (Fig. 10E). By varying the parameters for the bar and trough, it is possible to generate a wide variety of nearshore profiles. Once a single profile had been reconstructed, a sine curve was selected for representing the longshore variation. For each of the profile parameters, the mean, phase, amplitude, and wavelength of the sine curve are given as input data. To use the plunge distance as an example, a

BAR HEIGHT mean plunge distance of 38.8 ft is used based Figure 9. Contoured diagrams of plunge distance, on the mean observed values plotted in Figure bar distance, and bar height. Distance along the shore 9a. The phase is estimated by measuring the is plotted along the horizontal axis, and time is the distance in feet from the left-hand edge of the vertical axis for each diagram. diagram to the first maximum. The amplitude Regression equations have been derived of the sine curve is then estimated from the empirically from data for computing foreshore fluctuation of the plunge zone in onshore and slope (F), drop in the plunge zone (Yp), and offshore directions. The wavelength is deter- drop in the plunge shelf (Ys). In these equa- mined by measuring the distance between tions, the deviation around the mean plunge maxima along the shore. By varying the mean.

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phase, amplitude, and wavelength for each cf Research (ONR) technical report (Fox and the nearshore parameters, it is possible to Davis, 1971b). duplicate a wide variety of map patterns. The map simulations in Figures 11 and 12 The map for July 20 is used to demonstrate were accomplished without regard to storm the stages used in simulating bar and trough cycles or poststorm recovery periods. The next topography (Fig. 11 A). The parameters used step in the simulation model is the wedding of in simulating each successive map are given in environmental parameter simulation and map Fox and Davis (1971b). simulation into a single nearshore simulation The comparison between the simulated maps model. and the observed map for July 20 in Figure 11 shows broad areas of agreement, but not a Energy Distribution during Storm Cycles perfect fit. The bars and troughs are in their The analysis of energv distribution during correct position, and the protuberance occurs storm cycles and poststorm recovery periods in the right place along the shore. The final provides a link between the environmental simulated map in Figure 1 IF accounts for 96.35 parameters and map simulation. The curves for percent of the total sum of squares for the barometric pressure, breaker height, and long- observed map. Since the model is restricted shore current velocity for the observed and to normal curves in the offshore direction and simulated data in Figure 7 can be broken down sine functions along the shore, it is difficult to into storm cycles and poststorm recovery perfectly simulate minor fluctuations in the periods. By integrating the area under the observed map. breaker height curve, it is possible to compute A series of four pairs of observed and sim- total wave energy during each storm cycle. It is ulated maps is given in Figure 12. The weather also possible to compute: the longshore current conditions that prevailed during the formation energy for the north and south components of of the bars and troughs in the maps are given the current. A listing of program STORM in Figure 2. The parameters used to generate which is used for computing wave and long- the maps are listed in the Office of Naval shore current energies is given in Fox and Davis (1971b). Wave and longshore current energy is computed for each storm cycle and poststorm recovery. The total wave energy within a A. BARLE S S PROFILE - LINEAR PLUS QUADRATIC single wave, Ew, where

B. TROUGH - INVERTED NORMAL CURVE = (17)

-Xb 1 The total wave energy within a storm cycle or poststorm recover)' can be computed by integrating the area under the wave energy C. BAR - NORMAL CURVE curve for the time interval, T, represented by the storm cycle according to equation (18): ? Z 0.46H . (19) E. SIMULATED PROFILE Figure 10. Sequence of profiles showing the stages Longshore current energy for a northward- in bar and trough simulation. flowing current, En, or for a southward-flowing

Figure II. Sequence of maps showing stages in simulation of nearshore topography for July 20, 1970.

current, Es, can be computed according to STORM to compute wave energy and long- equation (20), where M is the mass of the shore current energy within each storm cycle water and V is the velocity of the longshore and poststorm recovery period (Fox and Davis, current: 1971b). The Fourier curves for barometric pressure, breaker height, and longshore current velocity are broken down into storm cycles and post- For computing longshore current energy, storm recovery periods in Figure 13. The total it is assumed that the cross-sectional area of wave energy within the first storm cycle was the longshore current is a triangle extending 17.35 million ft-lb per lin ft of beach according from the shoreline for a distance of 100 ft to to equation (19). The energies for the north- a water depth of 2 ft, and that the average ward-flowing and southward-flowing longshore longshore current velocity is 0.6 of the max- currents are obtained by integrating the areas imum velocity. The longshore current energy on either side of the zero line in the longshore to the north, E„, within a single storm cycle current curve according to equation (21). The can be computed by equation (21) in foot- remainder of the energy computations for the pounds per linear foot of beach: succeeding storm cycles and poststorm recovery T periods during the month are given in Table 1. E„=X;0.72 F2. (21) The maximum wave energy of 51.94 million i—i ft-lb per lin ft of beach was recorded during Equations (19) and (21) are used in program the fourth storm cycle.

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A OBSERVED MAP - 9 JULY 1970 3 SIMULATED MAP - 9 JULY 1970 —-J C 700 600 800 B 300 200 100 * —J C 700 600 900 i 300 200 100 A • IIII< iii

q | OBSERVED MAP - 15 JULY 1970 p | SIMULATED MAP - 13 JULY 1970

1 — ' 3 —-

c 1 OBSERVED MAP - 19 JULY 1970 p | SIMULATED MAP - 19 JULY 1970 ^ 1 i i i i i i ill

Q| OBSERVED MAP - 23 JULY 1970 H SIMULATED M.1P - 23 JULY 1970

ISO- - ^^ ___- 4 """ -

Figure 12. Pairs of observed and simulated maps for July 9, 15, 15, and 23, 1970.

To test the program for computing wave height. For the first storm cycle, using the energies, it is possible to compare the energies observed data, the total wave energy was 38.8 derived from the observed longshore current million ft-lb per lin ft of beach; using the and wave curves with those from the simulated simulated curves, the total wave energy was curves for longshore current energy and wave 42.5 million ft-lb per lin ft of beach. The long-

JULY 1970 5 10 15 20 2 5 previous storm cycle or poststorm recovery, Pp-i, and the longshore current energies according to equation (23):

Pp = Pp-1 + 10 (En - E.) . (23) The phase for the nearshore bar, P&, is offset half a wavelength from the phase for the plunge zone according to equation (24):

Pb = PP+^. (24)

The position of the trough in respect to the bar is controlled by rip currents generated during storms. If the waves are approaching out of the southwest, the rip channel forms on the north side of the bar, and the trough is offset to the north. This is demonstrated by the offset rip channel for July 20 in Figure 12. The phase of the trough, P(, is computed as a function of the phase for the bar, the wave- S-o|p-ol S-l I S-2 | S-3 |p-3l S-4 I P-4 mSTORM CYCLE | |POSTSTORM RECOVERY length, and the longshore current energies according to equation (25): Figure 13. Barometric pressure, longshore current, and breaker height during storm cycles and poststorm recovery periods, July 1970. shore current energies for each of the storm The bar distance, X&, from the plunge zone cycles, using the observed and simulated data, to the crest of the bar is computed as a func- are also quite close. tion of wave energy and bottom slope, S, according to equation (26): Simulation of Beach Parameters

Wave and longshore current energy is used (26) to generate map parameters for the beach and nearshore area. The wave energy and north The remaining parameters including the and south components of longshore energy distance, depth, and width of the trough and for each storm cycle and poststorm recovery the height and width of the bar are computed are punched on cards by program STORM. as functions of bar distance. The amplitude for These values are used as input data for pro- each of the parameters, which is used in gen- gram PARAM which computes the map erating a sine curve for the longshore variation, parameters. A listing and explanation of is also a function of bar distance. With an in- program PARAM is given in Fox and Davis crease in wave energy or a decrease in bottom (1971b). slope, the bars occur farther offshore and have The mean, phase, and wavelength for the a greater longshore periodicity. plunge distance are included as part of the During a poststorm recovery, the phase of initial input for program PARAM. These the plunge zone shifts half a wavelength as values are used to simulate the initial shoreline erosion takes place in front of the rip channel configuration at the beginning of the modeling and deposition occurs behind the nearshore interval. The wavelength in feet for the long- bar. The rate of migration of the bar toward shore harmonic, L, is computed according to the beach is a function of wave energy during equation (22): the poststorm recovery and bottom slope. The beach parameters which were computed L = 100 4E~w • (22) in program PARAM are punched onto cards Sine curves used in modeling the longshore that are listed in Fox and Davis (1971b). variation in the plunge, bar, and trough are Figure 13 shows a poststorm recovery that illustrated in Figure 14. During each storm follows the initial incomplete storm cycle; cycle, the phase of the plunge zone, Pv, is however, storm cycles 1, 2, and 3 follow one computed as a function of the phase for the another and are not separated by poststorm

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TABLE 1. WAVE ENERGY AND LONGSHORE CURRENT ENERGY IN MILLIONS OF FOOT-POUNDS PER LINEAR FOOT OF BEACH rip current and rip delta as shown by the 4-ft contour. Wave Longshore current Longshore current The map for July 12 following the second energy energy - north energy - soutii storm cycle is given (Fig. 15D). The wave Storm cycle 0 17 .35 4.14 0.00 energy for storm cycle 2 is less than the wave Poststorm 0 3,.5 1 0.96 0.00 energy lor storm cycle 1; therefore, the bar is Storm cycle 1 38..8 1 0.22 10.19 somewhat closer to shore. From the energy Storm cycle 2 32..6 9 3.44 7.83 distribution listed in Table 1, the longshore Storm cycle 3 44. 18 7.95 0.93 current energy from the south is stronger than Poststorm 3 3. 92 2.34 0.00 the current from the north; therefore, the Storm cycle 4 51. 94 0.66 19.01 trough between the bar and the shore is Poststorm 4 8. 03 2.57 0.24 slightly offset to the south. The bars also Totals 200. 43 22.28 38.20 migrated somewhat to the south by erosion of the north end of the bar and deposition at the recoveries. Storm cycle 3 is separated from south end. In storm cycle 2, a portion of a storm cycle 4 by a poststorm recovery, and second bar is present at the north end of the storm cycle 4 is followed by a poststorm map. The simulated map for storm cycle 2 cor- recovery. responds to the observed map for July 13 in Figure 4. Beach and Nearshore Simulation Model Storm cycle 3 follows closely on storm cycle A series of maps given in Figure 15 shows 2 without the interruption of a poststorm the beach and nearshore bottom configuration recovery (Fig. 13). For storm cycle 3, the following each storm cycle and poststorm north longshore current energy is stronger recovery period. The computer program than the south; therefore, the trough is offset to BSIM5, which is used to generate the maps, the north of the bar (Fig. 15E). The sim- and instructions for operating the program ulated map for storm cycle 3 (July 16) re- are included in Fox and Davis (1971b). sembles the observed map for July 15 in Figure The simulated map for June 30 in Figure 4. The bar has migrated somewhat to the north 15A shows a slightly sinuous shoreline with bars with the development of a rip channel, as at locations 200 and 700. Since the first storm shown by the trough on t oe north side of the cycle was incomplete, the wave energy was bar. lower than the total wave energy for the storm A quiet interval follows storm cycle 3 cycle; therefore, the simulated bar was formed giving rise to a poststorm recovery (Fig. 13). closer to shore than the observed bar (Davis The map for the poststorm recovery shows the and Fox, 1971b). Since it is necessary to have shoreward migration of the bar with a build- a complete storm cycle to get the accurate out of the protuberance behind the bar and offshore position of the bar, the initial map erosion at the head of the rip channels (Fig. in the sequence does not provide too good a 15F). The map for the poststorm recovery fit. corresponds to the map for July 19 in Figure 4. The map for July 2 portrays the poststorm The map for storm cycle 4 shows the sim- recovery following the initial storm cycle (Fig. ulated shoreline configuration on July 21 15B). During the poststorm recovery, the bars (Fig. 15G). In this map, the southward-flowing migrated toward the shore, and the protuber- longshore current was dominant, so the trough ances built out behind the bars. The trough formed on the south side of the nearshore bar. behind the bar was partially filled in as the bar The 3-ft-deep trough is centered at location migrated toward the shore, and a rip channel 500; the bars are at locations A and 700. The was present between the two bars. protuberance that built out during poststorm The map following storm cycle 1 simulates recovery 3 deflected the longshore current the shoreline configuration on July 6 (Fig. 15C). during storm cycle 4 generating a rip current During the storm, a new bar was formed near through the middle of the map. A channel was the middle of the map with a trough offset to cut by the rip current, and bars formed in the the south. The longshore current in storm middle of the nearshore circulation cells. cycle 1 was predominantly southward flowing Again, there is a close correspondence between and excavated a trough near the south end of the map for July 20 and the simulated map for the nearshore bar. The longshore current was storm cycle 4. shunted offshore at location 100 forming a A poststorm recovery followed storm cycle

4 (Fig. 15H). The bars had migrated toward During a storm cycle, rip currents formed by the shore and the sinuosity of the shoreline was the deflection of the longshore current cut considerably straightened out. The simulated channels through the nearshore bars, and new poststorm recovery map resembles the map for bars are formed between the rip channels. July 27 in Figure 4. Since the poststorm During a poststorm recovery, the bars migrate recovery was not complete, the protuberances toward the beach, and erosion takes place at had not had an opportunity to build out the head of the rip channels. Sand transported behind the bar in anticipation of the next storm down the beach builds out protuberances be- cycle. hind the bar. Following two storm cycles, the The maps that were generated by the beach bar returned to its original position with rip simulation program closely resemble the maps channels on either side and a protuberance that were surveyed in the field. At its present along the shore. stage of development, the simulation model Fourier analysis of barometric pressure is prints out one map for each storm cycle and used to simulate wave height and longshore poststorm recovery. In reality, the beach and current velocity. Longshore current velocity nearshore configuration is constantly changing can be simulated by the first derivative and throughout the storm cycles and poststorm wave height by a filtered version of the second recovery periods. With modification, the beach derivative of barometric pressure. The curves simulation program would be able to generate for breaker height and longshore current maps at any stage during the storm cycle or velocity are used to compute wave and long- poststorm recovery. shore current energies within each storm cycle. In simulating maps of the nearshore area, SUMMARY AND CONCLUSIONS wave and longshore current energy is used to A mathematical model based on empirical control map parameters. Bar distance is com- data from eastern Lake Michigan has been puted from wave energy and bottom slope. developed to simulate the effects of wind, The longshore position of the bar and trough waves, and longshore currents on beach and and the sinuosity of the shoreline are calculated nearshore topography. A conceptual model is from the wave and longshore current energy. A used to show the relations between storm cycles map is produced by the model for each storm and nearshore topography. During a typical cycle and poststorm recovery period. storm cycle, barometric pressure drops and The model should be useful for estimating wind builds up waves from the southwest changes in beach and nearshore topography on generating a northward-flowing longshore a short-term basis where they are related to current. As a storm passes, the barometric storm cycles. By averaging past weather data, pressure rises, and the wind and waves shift it would be possible to use the model for long- over to the northwest generating currents to term prediction of beach erosion and stability. the south. As a high-pressure system moves The model must now be tested using dif- into the area, the wind and waves decrease and ferent shoreline conditions and with storms shift back to the southwest. approaching from different directions. Such tests are underway along the New England coast, the gulf coast of Texas, and the western shore of Lake Michigan where storms move in an offshore direction. The influence of tides that constantly change the position of the plunge zone must also be brought into the model. A tidal model is being developed with plans to test it on the New England coast and the coast of Oregon. The model proposed in this paper apparently works quite well but has a few potential drawbacks. The simple relations between barometric pressure, longshore current velocity, and breaker height add to the attractive- Figure 14. Generalized map for storm cycle simulation of beach and nearshore bar showing bar distance, ness of the model. However, the model makes no provision for large swells that are unrelated Xt ; plunge phase, Pp ; bar phase, Pi, ; trough phase, Pt ; and wavelength, L. to local wind conditions. Also, tides and storm

Downloaded from http://pubs.geoscienceworld.org/gsa/gsabulletin/article-pdf/84/5/1769/3443674/i0016-7606-84-5-1769.pdf by guest on 26 September 2021 Figure 15. Simulated maps giving the nearshore topography following storm cycles and poststorm recoveries given in Figure 14.

surges that affect the position of the plunge Massachusetts Inst. Technology, Hydro- zone are ignored in the model. In the early dynamics Lab. Tech. Rept. 82, 31 p. stages of any model, it is necessary to make Fox, W. T., and Davis, R. A., Jr., 1970, Fourier simplifying assumptions in order to get the analysis of weather and wave data from Lake model to work. As work on the model pro- Michigan: Tech. Rept. no. 1, ONR contract 388-092, 47 p. gresses and it becomes more sophisticated, it 1971a, Fourier analysis of weather and wave should be possible to include some of these data from Holland, Michigan, July, 1970: other factors in the model. The present model Tech. Rept. no. 3, ONR contract 388-092, is analogous to a model A Ford, and hopefully 79 p. it will form the basis for future generations of 1971b, Computer simulation model of coastal coastal simulation models that will evolve processes in eastern Lake Michigan: Tech. through time. Rept. no. 5, ONR contract 388-092, 114 p. Harbaugh, J. W., 1966, Mathematical simulation ACKNOWLEDGMENTS of marine sedimentation with IBM 7090/7094 computers: Kansas Geol. Survey Computer This project was supported by the Geog- Contr. 1, 52 p. raphy Branch of ONR, contract NR 388-092. Harbaugh, J. W., and Merriam, D. F., 1968, Com- Some of the field equipment was supplied by puter applications in stratigraphic analysis: a grant from the Sloan Foundation to Williams New York, John Wiley & Sons, 282 p. College. Student field assistants included Harrison, W., 1969, Empirical equations for fore- Ronald DeWitt, Randall T. Kerhin, Robert shore changes over a tidal cycle: Marine D. LoPiccolo, and Kent M. Murray of Geology, v. 7, p. 529-551. Western Michigan University, and Edward Harrison, W., and Krumbein, W. C., 1964, Inter- C. Cartnick, John D. Cunningham, Henry actions of the beach-ocean-atmosphere system Flint, and Paul S. Willis of Williams College. at Virginia Beach, Virginia: Washington, D.C., Coastal Eng. Research Center, Tech. Memo, no. 7, 52 p. REFERENCES CITED Hayes, M. 0., 1967, Hurricanes as geological Bonham-Carter, G. F., and Sutherland, A. J., agents: Case studies of hurricanes Carla, 1961, 1967, Diffusion and settling of sediment at and Cindy, 1963: Texas Univ. Bur. Econ. river mouths: A computer simulation model: Geology Rept. Inv. no. 61, 56 p. Gulf Coast Assoc. Geol. Socs. Trans., v. 17, 1969, Coastal environments, northeastern p. 326-338. Massachusetts and New Hampshire: Amherst, Boxven, A. J., and Inman, D. L., 1969, Rip cur- Massachusetts Univ., Coastal Research Group, rents, 2, Laboratory and field observations: Contr. no. 1, 462 p. Jour. Geophys. Research, v. 74, p. 5479-5490. Inman, D. L., and Bagnold, R. A., 1963, Littoral Briggs, L. I., and Pollack, H. N., 1967, Digital processes, in Hill, M. N., ed., The sea, v. 3: model of evaporite sedimentation: Science, The earth beneath the sea: New York, Inter- v. 155, no. 3761, p. 453-456. science, p. 529-553. Bruun, P. M., 1963, Longshore currents and long- Inman, D. L., and Quinn, W. H., 1951, Currents shore troughs: Jour. Geophys. Research, v. 68, in the surf zone: 2d Conf. Coastal Eng. Proc., p. 1065-1078. Berkeley, Univ. California Council on Wave Cartnick, E. C., 1971, An analysis of longshore Research, p. 24-36. current on Lake Michigan [honors thesis]: Johnson, J. W., and Eagleson, P. S., 1966, Coastal Williamstown, MA, Williams College, 56 p. processes, in Ippen, A. T., ed., Estuary and Cole, F. W., 1970, Introduction to meteorology: coastline hydrodynamics: New York, Mc- New York, John Wiley & Sons, 388 p. Graw-Hill, p. 404-492. Davis, R. A., Jr., and Fox, W. T„ 1971, Beach and King, C.A.M., and Williams, W. W., 1949, The nearshore dynamics in eastern Lake Michigan: formation and movement of sand bars by Tech. Rept. no. 4, ONR contract 388-092, wave action: Geog. Jour., v. 113, p. 70-85. 145 p. Komar, P. D., and Inman, D. L., 1970, Longshore 1972, Coastal processes and nearshore sand sand transport on beaches: Jour. Geophys. bars: Jour. Sed. Petrology, v. 42, p. 394-403. Research, v. 75, no. 30, p. 5914-5927. Davis, R. A., Jr., Fox, W. T., Hayes, M. O., and Longuet-Higgins, M. S., 1970, Longshore currents Boothroyd, J. C., 1972, Comparison of ridge generated by obliquely incident sea waves: and runnel systems in tidal and non-tidal Jour. Geophys. Research, v. 75, no. 33, p. environments: Jour. Sed. Petrology, v. 42, 6778-6801. p. 404-412. Louden, R. K„ 1967, Programming the IBM 1130 Eagleson, P. S., 1965, Theoretical study of long- and 1800: Englewood Cliffs, NJ, Prentice- shore currents on a plane beach: Cambridge, Hall, 433 p.

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