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 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 1770 FCX AND DAVIS 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 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 SIMULATION MODEL FOR STORM CYCLES, LAKE MICHIGAN 1771 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.