US. Ar, Co@ack. Cre, Cea ern IT eS Hydraulics of Great Lakes Inlets by William N. Seelig and Robert M. Sorensen TECHNICAL PAPER NO. 77-8 JULY 1977 Approved for public release; distribution unlimited. U.S. ARMY, CORPS OF ENGINEERS COASTAL ENGINEERING RESEARCH CENTER Kingman Building Fort Belvoir, Va. 22060 Reprint or republication of any of this material shall give appropriate credit to the U.S. Army Coastal Engineering Research Center. Limited free distribution within the United States of single copies of this publication has been made by this Center. Additional copies are available from: National Technical Information Service ATTN: Operations Division 5285 Port Royal Road Springfield, Virginia 22151 Contents of this report are not to be used for advertising, publication, or promotional purposes. Citation of trade names does not constitute an official endorsement or approval of the use of such commercial products. The findings in this report are not to be construed as an official Department of the Army position unless so designated by other authorized documents. MBL/WHOI MUM (QUAI 0 0301 0089976 1 il I UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) REPORT DOCUMENTATION PAGE BE EOREICONEH EN CIEOR 1. REPORT NUMBER 2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER TP 77-8 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED HYDRAULICS OF GREAT LAKES INLETS Technical Paper 6. PERFORMING ORG. REPORT NUMBER 7. AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(a) William N. Seelig Robert M. Sorensen 9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK Department of the Army UN Coastal Engineering Research Center (CERRE-CS) A31220 Kingman Building, Fort Belvoir, Virginia 22060 AREA & WORK IT NUMBERS 11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE Department of the Army July 1977 Coastal Engineering Research Center 13. NUMBER OF PAGES Kingman Building, Fort Belvoir, Virginia 22060 69 14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) 15. SECURITY CLASS. (of thie report) UNCLASSIFIED 1Se. DECLASSIFICATION/ DOWNGRADING SCHEDULE Approved for public release; distribution unlimited 16. DISTRIBUTION STATEMENT (of thie Report) . DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report) - SUPPLEMENTARY NOTES - KEY WORDS (Continue on reverse side if necessary and identify by block number) Great Lakes Lake seiching Inlet-harbor resonance Nontidal inlet Inlet hydraulics 20. ABSTRACT (Continue em reverse side if rreceacary and identify by block number) } : Reversing currents in inlets on the Great Lakes are generated primarily by long wave seiching modes of the lakes rather than by the astronomical tide. Field measurements were conducted in 1974-75 at nine harbors on the Great Lakes to: (a) Investigate the nature of long wave excitation and the gener- ating mechanism for significant inlet velocities, (b) establish techniques for predicting inlet-bay system response, and (c) develop base data for future planning and design studies. Data collected include continuous harbor water (continued) DD , fie 1473 ~—sC EDITION OF ? NOV 65 IS OBSOLETE UNCLASSIFIED SECURITY CLASSIFICATION GF THIS PAGE (Wren Data Entered) UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) level measurements at all sites, inlet velocity measurements at the primary site (Pentwater, Michigan), and channel hydrographic surveys at the sites where more recent data were needed. Available historic water level and velocity data for some of the harbor sites were also used. Amplified harbor oscillations and generation of the highest inlet veloc- ities are caused by the Helmholtz resonance mode which has a period of 0.6 to 5 hours for the inlet-bay systems studied. A recently developed, simple numerical model is shown to be effective in predicting inlet-bay response over the range of excitation periods encountered. A finite-difference form of the continuity equation is shown to adequately predict inlet velocities if high-quality bay water level records are available. Selected data from the study sites are presented to demonstrate the hydraulic response of the inlet-bay systems and the applicability of the prediction schemes. Examples to demonstrate use of the concepts and techniques developed in the study are applied to the design of a new inlet channel and to the modification of an existing channel. 2 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) PREFACE This report is published to provide coastal engineers with an analysis of the hydraulic response of inlet-bay systems on the Great Lakes. The work was carried out under the coastal research program of the U.S. Army Coastal Engineering Research Center (CERC). The report was prepared by William N. Seelig and Dr. Robert M. Sorensen, Coastal Structures Branch, Research Division, CERC, under the general supervision of R.P. Savage, Chief, Research Division. The authors acknowledge the efforts of the U.S. Army Engineer District, Detroit and the National Oceanic and Atmospheric Administration, Lake Survey Center, who callected most of the field data, and the report review and comments by C. Mason and B. Herchenroder. Comments on this publication are invited. Approved for publication in accordance with Public Law 166, 79th Congress, approved 31 July 1945, as supplemented by Public Law 172, 88th Congress, approved 7 November 1963. OHN H. COUSINS Colonel, Corps of Engineers Commander and Director IV VI CONTENTS CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI). SYMBOLS AND DEFINITIONS. INTRODUCTION . LAKE AND INLET HYDRAULICS. : 1. Great Lakes and Inlet-Bay System ‘Tavdnaniies ‘ 2. Prediction of Inlet Velocities. SAIL THE FIELD DATA COLLECTION PROGRAM. 1. Field Measurements. 2. Equipment . . 3. Data Reduction and Analysis Techniques. RESULTS. j ; 1. Seiching of the Grose takes : 2. Predicted Inlet-Bay Response to Monochromatic even Wave Forcing . 0 3. Observed Lake Level Pieecuacionen Bay ReswOnSe, and Inlet Velocities . ences oO 0 INLET DESIGN . i 1. New Inlet Ghannell - ‘ 2. Inlet Channel Nace Bloaeson, SUMMARY AND CONCLUSIONS. LITERATURE CITED . TABLES Summary of field measurements . Inlet and bay geometric measurements. Modes of oscillation of the Great Lakes . Influence of Manning's n on inlet-bay response at Pentwater . Predicted periods of maximum wave amplification and maximum inlet velocities . Predicted Duluth-Superior maximum inlet water velocities for a forcing wave of 1 hour (a, = 3 centimeters). Numerical Model Prediction of Pentwater response to Lake Michigan modes of oscillation. Page 47 48 17 18 CONTENTS TABLES --Continued Summary of Pentwater hydraulic characteristics for selected inlet depths FIGURES Inlet-bay system. Amplification and phase lag for inlet-bay systems Three predicted longitudinal modes of oscillation of Lake Michigan. Pentwater response to sinusoidal wave in Lake Michigan. Response to long wave excitation at Pentwater . Numerical model water level predictions at Pentwater. Inlet study sites Data collection sites on Lake Michigan. Data collection sites on Lake Superior. Data collection site on Lake Erie, Presque Isle, Pennsylvania . Data collection sites on Lake Ontario . Sample spectra of Pentwater bay water levels. Measured and predicted inlet velocity cumulative frequency distributions at Pentwater, Michigan . Pentwater response to long wave forcing . Toronto response to long wave forcing . Predicted response of inlet-bay systems on Lake Michigan to monochromatic forcing . Predicted response of inlet-bay systems on Lake Superior to monochromatic forcing . Predicted response of inlet-bay systems on Lakes Erie and Ontario to monochromatic forcing . Page 63 14 15 IL 21 22 24 25 28 29 30 31 36 37 40 41 44 45 46 19 20 ail 22 23 24 OS 26 27 28 29 30 31 32 CONTENTS FIGURE--Continued Sample Pentwater and Lake Michigan water levels and SPE CET ae ata leew oh ieciyie: MPa kal keccta nce nC sntc a) cam Onn Masa sonra Sample Pentwater and Lake Michigan water levels and spectra. Sample Pentwater and Lake Michigan water levels and spectra. Sample water level fluctuations Water level fluctuations in Little Lake Harbor. Sample bay levels and inlet water velocities at Duluth-Superior. Pentwater inlet cumulative frequency velocity distributions. a6 .0 0 Inlet velocity cumulative frequency distributions Crystal Lake, Michigan. Predicted response characteristics of an inle® for Crystal Lake . Predicted inlet velocities at Michigan inlets Response to long wave excitation at Pentwater . Response to long wave excitation at Pentwater . Predicted Pentwater inlet velocities for Lake Michigan water levels recorded on 18 August 1967. .. . Page 49 50 51 53 54 55 57 58 59 60 62 64 65 66 CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) UNITS OF MEASUREMENT U.S. customary units of measurement used in this report can be converted to metric (SI) units as follows: Multiply by To obtain inches 25.4 millimeters 2.54 centimeters square inches 6.452 square centimeters cubic inches 16.39 cubic centimeters feet 30.48 centimeters 0.3048 meters square feet 0.0929 square meters cubic feet 0.0283 cubic meters yards 0.9144 meters square yards 0.836 square meters cubic yards 0.7646 cubic meters miles 1.6093 kilometers square miles 259.0 hectares knots 1.8532 kilometers per hour acres | 0.4047 hectares foot-pounds 1.3558 newton meters millibars 1.0197 x 10°? kilograms per square centimeter ounces 28.35 grams pounds 453.6 grams 0.4536 kilograms ton, long 1.0160 metric tons ton, short 0.9072 metric tons degrees (angle) 0.1745 radians Fahrenheit degrees 3/9 Celsius degrees or Kelvins! CE ———— ———————————————_—_—_—S—OorcqCceeeeeeeeeeeeeEeeEeEeEeEeEeEeEeEeee To obtain Celsius (C) temperature readings from Fahrenheit (F) readings, use formula: C = (5/9) (F — 32). To obtain Kelvin (K) readings, use formula: K = (5/9) (F — 32) + 273.15. L' bp. SYMBOLS AND DEFINITIONS inlet cross-sectional area at the bay end bay surface area inlet cross-sectional area grid cross-sectional area inlet cross-sectional area at the sea end inlet width grid cell width wave speed grid cell depth water depth acceleration due to gravity water surface water surface water surface water surface elevation elevation elevation elevation in the in the in the number of channels in the grid number of sections in the grid grid cell subscript indicating grid cell subscript indicating mode of oscillation inlet length bay bay at the previous time step sea the channel the cross section added inlet length in the acceleration terms to account for long wave radiation from the harbor basin length grid cell length <| tj Coe) 2 SYMBOLS AND DEFINITIONS--Continued number of inlets connecting a bay to a sea Manning's n Manning's n of a grid cell discharge discharge for the mth inlet total discharge for all inlets hydraulic radius of a channel inlet-bay system Helmholtz period frictionless inlet-bay system Helmholtz period period of free mode of oscillation where the subscript indicates the mode of oscillation time mean instantaneous inlet water velocity at a cross section mean inlet water velocity at a cross section over a sampling interval the grid cell weighting function which is the fraction of total inlet discharge passing through the cell at a time step distance along the longitudinal axis of the inlet distance along a cross section time step component of the bottom-stress tensor in the direction of flow ron | mana wy fi ia belie | HYDRAULICS OF GREAT LAKES INLETS by Willtam N. Seeltg and Robert M. Sorensen I. INTRODUCTION Numerous bays and harbors are connected to the Great Lakes by jettied inlet channels. These inlet channels are important because they allow (a) access to commercial shipping and recreational boating, (b) migration of fish, and (c) flushing of pollutants from the bays and harbors. Great Lakes inlet-bay systems are generally smaller than those on the Atlantic, Pacific, and gulf coasts of the United States and respond primarily to the long wave seiching modes of the Great Lakes rather than to the astronomical tides. These seiches have smaller amplitudes and shorter periods than the tides on the ocean coasts. The major effort of this study involved the collection and analysis of hydraulic data at several inlet-bay systems throughout the Great Lakes during 1974 and 1975. Measurements at Pentwater, Michigan, the primary study location, included simultaneous recording of inlet current veloc- ities and water levels in the bay and in Lake Michigan. At the other locations, only bay water levels were measured. However, hydrographic surveys were obtained for all the inlets investigated, and historic hydraulic data from selected sources were analyzed. This study defines the hydraulic mechanisms important to Great Lakes inlet-bay systems, develops analytical techniques for the prediction of inlet currents and bay water level oscillations, and presents design data and system response curves for selected inlets. Field data were analyzed using (a) a formula for estimating the seiche periods of the Great Lakes which are important in producing reversing currents at an inlet; (b) a model that uses bay water level time histories to predict inlet velocities; and (c) a simplified numer- ical inlet hydraulic model that, when calibrated for friction effects, can be used to predict inlet velocities and bay water level oscillations generated by lake oscillations. These analysis techniques are used with the field data to develop response curves and cumulative inlet current velocity distribution curves for the inlets studied. II. LAKE AND INLET HYDRAULICS 1. Great Lakes and Inlet-Bay System Hydraulics. An inlet is a relatively narrow channel which connects a "'sea"™ (or one of the Great Lakes in this study) to a lake or harbor (a “bay" in this study). The bay is large compared to the inlet (i.e., the radius of the bay is typically larger than the length of the inlet) and the surface area of the bay is much smaller than the surface area of the sea. a. Causes of Reversing Inlet Currents. Mortimer (1965) and Freeman, Hamblin, and Murty (1974) show that significant reversing inlet current velocities are caused by water level fluctuations in the Great Lakes which generate a hydraulic response in the inlet and bay. The most important Great Lakes water level fluctuations are due to the resonant seiching or oscillation of the particular lake at its fundamental and harmonic periods. Seiches are initiated by storm pressure and wind forces on the sea which redistribute water in the lake to cause a higher elevation than normal in some areas and lower levels in other areas. When gravity tries to restore the water level, seiches are generated. These seiches usually continue for a number of cycles which may extend over a few days after the storm has passed. When a seiche is generated in one of the lakes, the water level fluctuations outside an inlet cause a head difference across the inlet, which, in turn, generates a current in the inlet. Water discharge through the inlet results in water added to or removed from the bay so the bay level rises and falls in a pumping fashion for most seiching periods of the lake. Astronomical tides and other nonseiching long waves cause water level fluctuations of the Great Lakes; however, these fluctuations generally have insufficient amplitudes or are at periods that usually do not sig- nificantly influence inlet hydraulics. Storm surge, particularly on shallow Lake Erie, may occasionally generate strong inlet currents. b. Mathematical Description of Inlet-Bay Hydraulics. The response of an inlet-bay system may be described in terms of the one-dimensional equation of water motion in the inlet and the continuity equation relat- ing the rate of bay level change to the inlet discharge. The one-dimensional equation of motion along the inlet channel axis can be written: Yo any ple ah, av a 1 J (‘2x )a 8 + 85x * 3p (1) Y1 where distance along the channel =r il} water surface level > iT} inlet cross-sectional area l2 V = water velocity in the inlet y = distance along the cross section = component of the stress tensor in the direction of flow g = acceleration due to gravity t = time The inlet has a width, B, depth, d, length, L, and cross-sectional area, Ag; the bay has a surface area, A . The water levels in the sea and bay are hg and hp, respectively (Fig. 1). Equation (1) equates the horizontal driving force due to the water surface slope with three terms on the right which are the channel fric- tional resistance, the convective acceleration caused by velocity vari- ation along the channel axis, and the temporal acceleration (or inertia) resulting from velocity variation at a point with time. In nearly prismatic channels, such as many inlets on the Great Lakes, the convec- tive acceleration is often negligible. The continuity equation, which relates rate of bay water level change, dhp/ot, to inlet discharge, Q, is: dhp Qe WAS S Apay Gram (2) A simultaneous solution of equations (1) and (2) for a sinusoidal sea level fluctuation reveals the important response characteristics of an inlet-bay system (Fig. 2). In this figure, the phase lag between the sea and bay water level fluctuations and the amplification of the forc- ing wave in the bay by the inlet-bay system are plotted as functions of dimensionless period. Dimensionless period is defined as the friction- less inlet-bay system Helmholtz period, Ty', divided by the forcing wave period, T. The Helmholtz period is that period of the forcing wave which through resonance will cause the largest water level fluctuation in the bay. The bay water level remains approximately horizontal throughout this fluctuation. The inlet-harbor system response is analogous to the response of a slightly damped spring-mass system or its acoustic counterpart, the Helmholtz resonator. The motion of the mass of water in the inlet channel corresponds to the motion of the mass of the spring-mass system, and the action of gravity on the rising and falling harbor water surface corresponds to the restraining force of the spring. At values of Ty'/T approaching zero (long wave periods) the water level fluctuations in the bay are the same as those in the sea with no 13 Plan View inlet cross- sectional area=Ac BAY sad a oA surface area = Apay Profile View Figure 1. Inlet-bay system. 180° Increasing Friction Phase Lag O Z Increasing Friction Amplification Figure 2. Amplification and phase lag for inlet-bay systems. phase lag (point A, Fig. 2). At values of Wy approaching 3 (short forcing periods), the inlet-bay system strongly dampens incident waves and water level fluctuations in the sea have little influence on inlet hydraulics (point B, Fig. 2). At values of Ty'/T in the range of 0.25<(Ty'/T)<2, the amount of frictional resistance in the channel has a major influence on the response characteristics. Inlets with high friction, e.g., tidal inlets on ocean coasts, typically have amplifications of less than one. This amplification factor decreases as the forcing wave period becomes shorter (point C, Fig. 2). At most tidal inlets, the primary tidal period is large compared to the inlet-bay Helmholtz period. Low-friction inlets, such as those on the Great Lakes, have amplifications greater than one and phase lags of approximately 90° for forcing waves with T,'/Tx1 (point D, Fig. 2) which commonly occur. 2. Prediction of Inlet Velocities. Prediction of inlet velocities requires (a) the time history of sea or bay water levels, (b) the geometries of the inlet and bay, and (c) a friction-calibrated model to relate water level fluctuations to inlet- bay response. a. Great Lakes Water Level Fluctuations. In general, no methods are presently available for inexpensively predicting all important amplitudes and periods of water level fluctuations at any point in one of the Great Lakes. Therefore, water levels generally must be measured. However, inexpensive schemes are available for accurately predicting some periods and relative amplitudes of seiches of the Great Lakes (Fee, 1968; Rao and Schwab, 1974). Knowledge of the existing wave periods will aid in the design of water level measuring systems, analysis procedures, and preliminary inlet design. The basic method for estimating one-dimensional fundamental and harmonic seiche periods, Tz, is to determine the time required for a wave to travel twice the length of the basin: Ue © ae (3) where k is the mode of oscillation, 1, the length of the basin in the direction of the seiche, and c the speed of the wave. Sample predicted longitudinal seiches for Lake Michigan, using the Fee (1968) computer program, are shown in Figure 3 for modes k=1, 2, and 3. b. Inlet and Bay Geometries. Hydrographic surveys, including Corps of Engineers dredging records, can be used to determine inlet geometry; i.e., length, width, and depth field. Maps and aerial photos can be used to determine bay surface area. First longitudinal mode ere: ( period ,Qhr ) & Second longitudinal mode | eee ( period,5.3 hr ) antinode Portage Portage u Ludington. Pentwater Ludington Pentwater Muskegon ——- oi da Holland Holland aa ae Ak Third longitudinal mode ( period ,3.5 hr ) antinode Relative Seiche Height of Water (normalized by level at the southern tip of Lake Michigan ) Figure 3. Three predicted longitudinal modes of oscillation of Lake: Michigan (modified from Mortimer, 1965). c. Methods of Analysis of Inlet-Bay Hydraulics. Inlet current velocities and bay water surface oscillations may be measured to provide necessary information on the hydraulic characteristics of an inlet-bay system. Techniques used for the field measurements in this study are discussed in Section III, 2. Several analytical methods are available for predicting inlet hydraulics, depending on the type of information available and required results. Three methods are discussed below. (1) Estimation of Seiche Periods Important to Inlet Hydraulics. To estimate which of the Great Lakes seiche periods, Ty, may be im- portant, the frictionless inlet-bay Helmholtz period, Ty', may be determined from: [CL+L") A Way! = 27 Sac, Seen (4) Aa g where L' is an added channel length determined from: -B 1B L! = — In |———], (5) w % | gd | L' accounts for the water masses in motion beyond the ends of the inlet (Miles, 1948). Equations (4) and (5) may be iteratively solved to obtain a value of Ty'. This approach proved to yield reasonably accu- rate estimates for the inlets considered in this study. As a first approximation, seiche wave periods which are approximately equal to the frictionless Helmholtz period (e.g., between 0.5 and 2 times Tyg'; see Fig. 2) will probably cause the highest inlet reversing currents. The seiche node-antinode pattern in the Great Lakes will also influ- ence the importance of the various seiche modes on an inlet-bay system. Seiches with antinodes adjacent to the inlet will produce the largest water level fluctuations. Since ends of the lakes are antinodes for all modes of oscillation along that axis, bays at the end of a lake will normally be subject to higher water level fluctuations than those at other locations; e.g., midway along the longitudinal axis of Lake Michigan, near Pentwater, the first longitudinal mode of oscillation has a node (Fig. 3). Therefore, only small oscillations can be generated in the lake at this point by this mode. The second longitudinal mode of oscil- lation has an antinode adjacent to Pentwater, so large water level fluc- tuations in Lake Michigan could be generated outside Pentwater by this mode. (2) Estimation of Inlet Velocities from Bay Water Level Records. A method of predicting inlet velocities, if high-quality bay water level 18 records are available, is to use the continuity equation. Written in finite-difference form, equation (2) becomes: hy > hy 7 2 Samy iE a) (6) where V is the average inlet current velocity at a cross section of area Ag over a water level sampling interval, and At, hp', and hp are mean bay water levels at the beginning and end of the sampling interval. Measurements of water level at any point in the bay will be representative of the mean bay level for the Helmholtz mode of oscillation of inlet-bay systems. This method for predicting inlet velocities is well suited for Great Lakes inlets because inlet and bay geometries are simple and level re- corders are easy to install in the protected bays. The sampling interval should be one-twentieth of Ty' or shorter and the stilling well care- fully designed for best results (see Sec. III). (3) A Numerical Model. A relatively simple but extremely useful method of modeling inlet-bay hydraulics is to simulataneously solve the equations of motion and continuity. In this model the inlet channel is divided by a flow net into a grid of subchannels and cross sections. The subscripts z and j describe the location of the cell for sub- channels (IC = number of channels) and grid sections (IS = number of sections). The equation of motion for an inlet (eq. 1) rewritten in finite-difference form, and integrated along the axis yields (Seelig, Harris, Herchenroder, in preparation, 1977): d 1 ffl 1 $+ (bee) + 20 1s-1f )) L, 4/IC ener! = IC Hail | OY AgE jal IS-1 > ee ee 2.208 D, (7) i=1 7 Aig) j=1 j=l where A, and Ap are the inlet cross-sectional areas at the sea and bay ends of the inlet, and W;; is a weighting function for distributing flow throughout the inlet. The discharge through a grid cell is equal to the weighting function of the cell, Wea times the total discharge of the inlet, Q. The Manning's friction factor, njj, is determined 19 during calibration of the model (see Seelig, Harris, and Herchenroder, in preparation, 1977). For M inlets connecting the bay to the sea, the total discharge for all inlets, Qins is: M 5 Qe nh: (8) m=1 The continuity equation is written as: oh Q ERR eet 4 (9) Bay levels and inlet current velocities are determined by solving the simultaneous differential equations (7) and (9) using a Runge-Kutta-Gill fourth order finite-difference technique in conjunction with initial con- ditions and the time history of water levels in the sea. Derivation and sample applications of this model are given in Seelig, Harris, and Herchenroder (in preparation, 1977). To obtain response characteristics similar to Figure 2 for a specific inlet-bay system, the model can be run by assuming sinusoidal seawater level fluctuations with a typical amplitude; e.g., 3 centimeters (0.1 foot) at periods covering the anticipated range of lake oscillation modes. Each run of the model will give predicted bay levels and inlet current velocities for the wave period used. Sample model results for Pentwater inlet are shown in Figure 4. The sea level, predicted bay level, and inlet velocity are shown in the lower part of Figure 4; the importance of each of the terms in the equation of motion, normalized by dividing by the magnitude of the largest term at each time step, is shown in the upper part of the figure. For this condition, the bay level fluctuation is larger than the sea level fluctuation due to inertia in the system and the bay level lags the sea level by 84° (Fig. 4). Plotting results from many runs similar to Figure 4, but with many different forcing periods, will give the response characteristics of the inlet-bay system. These curves for Pentwater (Fig. 5) show that the Helmholtz period with friction, Ty, is 1.8 hours, waves with periods of 1 to 3 hours will be amplified by the system, and waves with a period of 1.4 hours will gen-: erate the highest inlet current velocities (3-centimeter wave amplitude assumed). The effect of friction on Ty is demonstrated by the dash- line in Figure 2. The frictionless Helmholtz period is also coinciden- tally 1.4 hours (from eqs. 4 and 5). The calculated bay amplification and channel velocity in Figure 5 are for a Manning's n = 0.036. The numerical model usually had to be run for three or four cycles for the bay response to build to equilibrium. In the prototype harbor it is likely that equilibrium (full amplification) is never fully achieved. Thus, the calibration curve in Figure 5 forms the upper envelope of measured prototype data. 20 Friction Head Difference Temp. Acceleration Pentwater Bay ake Level Michigan Level Velocity (ft/s) Water Level (ft) Time (hr) Figure 4. Pentwater response to sinusoidal wave in Lake Michigan (period = 1.5 hours; amplitude = 0.1 foot). 2 | *(200F T°0 = epnzt[due orem) LoyemjUeg 3@ UOTIEITOXE AEM BUOCT 02 asuodsoy (4Y) Poised 6) | G € v G UONDIIJdWYy pajDINI/09 joes M1010 WNWIXDI pajDjNI109 KeZ "¢ o1n3 Ty UOIJDIJGwWy PUD (S/J}) AjoojaA wWNWIXxOW 22 If water level fluctuations just outside the inlet are known, the numerical model can be used to predict the resulting channel velocities and bay level fluctuations. Lake Michigan levels just outside of Pent- water were measured by Duane and Saylor (1967) during August 1967 (Fig. 6). Although the record appears confused, a spectral analysis shows that the record is composed of a large number of clearly defined seiche modes of Lake Michigan. Using this record to force the model of Pentwater, a bay level time history is predicted which adequately agrees with measured bay levels (Fig. 6). The inlet-bay system responds primarily to waves with periods of 1 to 3 hours (near the Pentwater Helmholtz period) ; shorter period waves are damped. This gives the bay level record a smoother appearance than the Lake Michigan record. The maximum predicted inlet current velocity during this episode is 60 centimeters (2 feet) per second. Other models that neglect temporary acceleration, e.g., Keulegan (1967), should not be used for inlets on the Great Lakes where temporal acceleration, head difference, and friction are important during the response cycle (see Fig. 4). III. THE FIELD DATA COLLECTION PROGRAM Measurements were made at a number of inlet-bay systems throughout the Great Lakes during 1974 and 1975 (Fig. 7). These study sites were chosen because the inlets are typical on the Great Lakes, are of special economic importance, or have maintenance problems. 1. Field Measurements. Pentwater, Michigan, located midway along the longitudinal axis of Lake Michigan, was selected as the primary study location because the inlet-bay system at this location has a simple, fairly common geometry. Good historical field data are also available at Pentwater from Duane and Saylor (1967) who simultaneously measured water levels in Pentwater bay and Lake Michigan as well as inlet velocities during July and August 1967. The U.S. Army Engineer District, Detroit, provided inlet geometry data of the inlet from hydrographic surveys taken twice a year. Field measurements during this study (Fig. 8) included water level measurements at two locations (east and west ends of Pentwater bay) in 1974, and at one other location in 1975. Current velocities in the inlet were measured concurrently with water levels during both 1974 and 1975. Hydrographic surveys were also taken at Pentwater. Other field data collection sites were: Portage Lake, Ludington, White Lake, Muskegon, and Holland on Lake Michigan (Fig. 8); Little Lake and Duluth-Superior on Lake Superior (Fig. 9); Presque Isle on Lake Erie (Fig. 10); and North Pond and Little Sodus on Lake Ontario (Fig. 11). Field measurement types, locations, dates, and data sources for all the study sites are summarized in Table 1; approximate dimensions of these inlet-bay systems are summarized in Table 2. 23 *I9}eM}USg 1B SUOTIOTpoerd [OAST 10}ZBM TOpoOW [eITIOUNN (44) awiy \ paoipaig—~” © *g omn3Ty (43) |@Ae7] 400M 24 *soqts Apnjs eTuy, “ZZ eansTy ajST enbsalg © wy 191 lw OO! pud||OH B40] OU 0}u010] J8jDM}U8d gOe uoybuipn’| 26Dj40q N By] asst] 2 Jolsadns DJDG |DNN0JSIH ~4sNjNg S}UBWOINSDA|’ GJ6] O SjuawainsDaw vZl6| © puabe7 JOIIBONS a 25 Legend O e@ level recorder 1974-75 oa © level recorder 1967 SiR & velocity meter 1974-75 Lake Michigan Pentwater Pentwater Lake, Mich. ~~ a 610 m | a 2000 ft Portage Lake, Mich. Weg fe. ae) Lake Michigan Portage Figure 8. Data collection sites on Lake Michigan. 26 Outer 0.5 mi Jetties Eone m Jettiés /:-: 2 2 Lake Michigan Ludington, Mich. Ludington White Lake, Mich. Lake White Lake Michigan Figure 8. Data collection sites on Lake Michigan--Continued. rad Muskegon, Mich. Muskegon Lake 1000 m Michigan | mi | | Holland, Mich. ee SON Lake Michigan Figure 8. Data collection sites on Lake Michigan--Continued. 28 Lake Superior “-... ,Water Level Recorder rg Location i |; Little Lake, Mich. Little Lake ee 153 m | a= 500 ft Duluth Ship Channel (Inlet 2) | Water Level Recorder \\ Location W Duluth-Superior, Wis. Superior Entry «— (Inlet |) Figure 9. Data collection sites on Lake Superior. ag Lake Erie Water Level Recorder Location Erie Harbor Figure 10. Data collection site on Lake Erie, Presque Isle, Pennsylvania. 30 Newer Inlet 1~> Water level Je 2:0: Older Inlet 2% recorder location North Pond, N.Y. Lake Ontario North Pond 9,000 ft 1,924 m Lake Ontario :.. Little Sodus, N.Y. ZN 9 900 ft vival ee ZZ — ee 610 m Figure 11. Data collection sites on Lake Ontario. 3| Table 1. Summary of field measurements. Water Inlet Water level Data source level Channel geometry Location and dates quality velocities (date measured) Lake Michigan Portage Portage Lake? This study, fall 1974 Good Sept. 19733 (Oct.-Nov. 1974 Ludington |Pere Marq. Lake? This study, 1975 Good Saas June 19743 Pentwater Lake Michigan and Duane and Saylor (1967) Good Analog June 19673 Pentwater Lake record East Pentwater This study, 1975 Good Analog Oct. 1974" Lake? record Oct.-Nov. 1974 West Pentwater This study, fall 1974 Fair Analog Sept. 1975" Lake? This study, 1975 record (Oct.-Nov. 1974, July-Oct. 1975 White Lake |White Lake This study, fall 1974 Good --- May, Aug. 19743 Oct.-Nov. 1974 Muskegon Muskegon Lake* This study, fall 1974 Apr., Aug. 19743 Oct.-Dec. 1974 Holland Lake Macataw? National Oceanic and Mar. 19743 Nov. 1974 Atmospheric Administration (analog) Tecords Lake Superior Little Lake|Little Lake® June 19753 May-Aug. 1975 Duluth- Duluth Harbor? National Oceanic and Analog re- |] Oct. 19743 Superior j|June 1973 Atmospheric Administratio cord gages (analog) records had not Tecently been cali- brated? Lake Erie Harbor levels? This study, 1975 May-Oct. 1975 ace) MN CEN | National Oceanic and Poor Atmospheric Administratio records ue re \National Oceanic and Atmospheric Administration, Lake Survey Center. 2a S-minute sampling interval. 3Corvs of Engineers records. “This study, fall 1974. 5A 2-minute sampling interval. Presque Isle Bay levels? May-Nov. 1972 Little Bay levels? May-Oct. 1975 North Pond 5) 2 Sept. 19753 1975 several surveys made by University of Buffalo under contract Chart No. used to determine bay area! 768 767 763 92 332 22 0°26 c° Ce 8° tT 0°09T CV OW EL 0°728T 9°16 T°8T T°8t 0°6T (g0T x 235) Aeq fo eore 90eF Ins SL6T *8ny SL6L AeW SL61I *3dag pL6T °2°0 GL6I ounr pL6l pL61 ‘Sny-*ady pL6L ‘“8ny-Aey vL6T °390 L961 oun pL6T oun EL6T ‘das ‘Lew (35) eole WNUTUTU 18 YIPIM JOTUT (235) eole [T BUOTII9S-SSOID ZOTUT WNUTUTpW *sjUuoWsInseow ITIZOWO0eS Aeq pue YoTUI ‘Z aTqeIL (Z 2eTUT) (1 29 TUT) puod yI10N snpos e[331TT SITS] enbserg (Z 2°TuT) (I 20TUT) Zotzedns-y3ntng OYT 913977 pueT Toy uoseysny oye] 93 TUM 197eM}U9g 197eM4U9g u0}SUTpPNT 938 410g woeasks Aeq-}9T UT 33 Pre Equipment. Small-amplitude sea level fluctuations with a period of approximately the Helmholtz period of the inlet-bay system may generate relatively high inlet water velocities as shown in numerical model calculations (see Sec. II). Therefore, a water level recording system must be carefully designed for each location to measure small amplitude, but potentially important long waves. At the same time, this recording system should eliminate any short-period, large-amplitude noise (e.g., wind waves) that may mask the long waves in the record. For example, at Pentwater, records should measure the low-amplitude waves with a period of 1 hour or longer, and should exclude wind waves and other noise with periods of 1 minute or less. One method of designing a stilling well to meet these requirements is to use the linear damping well design (Noye, 1974). This stilling well consists of a vertical cylinder with a sealed bottom and open to the lake through a long, thin tube. Friction in the tube and the relative cross-sectional areas of the tube and stilling well cause the system to respond directly to long waves outside the well and to drastically dampen the short-period noise. Design of stilling wells is discussed by Seelig (97/77) Fisher-Porter series 1500 digital float-type water level recorders with a vertical resolution of 3 millimeters (0.01 foot) and sampling intervals of 2 or 5 minutes were used to measure water levels in the stilling wells. Data were collected on punched tape. Water levels were measured for several months at each location (see Table 1). Inlet velocities at Pentwater were measured during 1974 and 1975 using a Bendix current meter suspended by a cable approximately midway along the channel, 4.5 meters (15 feet) from the north wall at mid- depth. Velocity data were recorded on a strip chart and later digitized for analysis at the same time interval as the water level data. 3. Data Reduction and Analysis Techniques. Initially, the Helmholtz period of each inlet-bay system and the free seiching mode periods.were calculated for the lakes and bays surveyed in this study (see procedures in Sec. II). These calculations, in conjunc- tion with a survey of published data on Great Lakes resonance character- istics, gave an indication of the period and magnitude of important long waves that could be expected at each location. The information was used in the design of water level measurement equipment (discussed in previous section). When the field data collection program was completed, the digital punched-tape water level records were mechanically converted to punchcards for computer analysis. The first procedure for studying these data in- cluded plotting the records for visual inspection. Then, a fast Fourier 34 transform and cosine bell function were used to obtain a spectral analysis of each record (Harris, 1974). A record length of 512 points was used in these analyses to obtain detailed spectral line resolution. Spectral analysis indicated the period and amplitude of long waves of interest (Q.5 to 5 hours) in the record. This analysis is necessary because several long waves are generally simultaneously present and the Superposition of the waves gives the impression of a confused record. Examples of spectra for Pentwater bay water levels recorded during 5 to 14 May 1975 are shown in Figure 12. If water level records were of good quality, levels measured in the bay were then used to predict inlet water velocities using the finite- difference continuity equation (6). The resolution of the continuity equation should be checked to judge the usefulness of the predicted velocities; e.g., at Pentwater, the level recorder has a vertical resolution of 3 millimeters , the sampling interval is 5 minutes (300 seconds), and the ratio Aba /Aa = 10"; so the velocity prediction resolution, V,, based on equation (6), is: = 10" (2:01) = 0.33 foot or 10.1 centimeters per second. (10) U5 300 r Thus, the velocity will be expressed as multiples of 10.1 centimeters (0.33 foot) per second which may be adequate for many purposes. For example, if Abay/Ke was 10°, then with the given vertical resolution, velocities could only be expressed as multiples of 100 centimeters (3 feet) per second, which is inadequate for most purposes. At Pentwater, the measured velocities in the inlet were digitized at a sampling rate of 5 minutes so that a direct comparison could be made of measured and predicted (eq. 6) velocities. Cumulative frequency distributions of measured and predicted (eq. 6) inlet velocities are shown in Figure 13. The 2 months of record in 1974 show that velocities predicted by continuity are slightly higher than measured velocities, but adequate for many design purposes. IV. RESULTS 1. Seiching of the Great Lakes. Free modes of oscillation of the Great Lakes have been identified by spectral analysis of water level records in this study and others © (Mortimer, 1965; Mortimer and Fee, 1974; Hamblin, 1975; Rao and Schwab, 1974), and have been predicted using numerical techniques (Rockwell, 1966; Mortimer, 1965; Birchfield and Murty, 1974; Rao and Schwab, 1974). Table 3 summarizes the known modes of oscillation of Lakes Michigan, Superior, Erie, and Ontario. 39 Normalized Variance and 6 May 7 and 8 May 9 and 10 May 1.25 1.44 Wave Period [2 and 13 May wl 44 Period (hr) Figure 12. Sample spectra of Pentwater bay water levels (May 1975). 36 50 @ Measured O Predicted (eq. 6) ine) oO (e) Pct of time velocity (equaled or exceeded) (S)) 2 (SG) 0.1 0.5 1.0 1.5 2.0 Inlet channel velocity (ft/s) Figure 13. Measured and predicted inlet velocity cumulative frequency distributions at Pentwater, Michigan (October-November 1974). 37 Table 3. Modes of oscillation of the Great Lakes|. Mode? Michigan Superior Erie Ontario 1L 8 4.4 4.9 2L 8 Soil 25) 3L .6 5.8 2.4 4L ? 4.1 1.6 Bh .0 3.6 ILS <37/ 6L 2 3.0 Zo) 7L of 08) 0.9 8L 38) 22 0.85 OL DP 2.0 AGS 10L .14 1.8 ---- TA 2T : ? : 685 ? .66 to 0.68 ? : 58 ? .52 to 0.53 4 .49 1Qbserved, computed by Fee (1968) program and Rockwell (1966) data, and compiled from many sources. 2 = longitudinal, T = transverse; ? = other observed periods, mode unknown. Two-dimensional modes are not considered. 3Value unknown. 38 2. Predicted Inlet-Bay Response to Monochromatic Long Wave Forcing. The response of each of the inlet-bay systems (Table 2) to uniform long wave forcing was evaluated using the numerical model. In this analysis, a sinusoidal wave was used to force the model for several cycles (generally four) until the inlet-bay system response became periodic. The results will give an upper limit of wave amplification and inlet velocities because the prototype will generally not reach a periodic condition. Each inlet was modeled by using a grid system with one to three channels and two to seven cross sections. The complexity of the inlet determines the number of grids used to model the friction; e.g., Pent- water with a constant width and only slight changes in depth along the length of the inlet, was modeled using one channel and four cross sections. Little Lake, with a more irregular inlet, was modeled using three channels and five cross sections. Pentwater, Michigan, and Toronto, Canada (Freeman, Hamblin, and Murty, 1974), are the two harbor systems on the Great Lakes with water levels recorded simultaneously inside and outside the harbor to provide the necessary information for calibration of the numerical model. These models were calibrated by varying the value of the Manning's friction factor, n, so that the model long wave amplification is a best-fit upper envelope of prototype measurements. Values of n of 0.036 and 0.062 were found for Pentwater and Toronto, respectively (Figs. 14 and 15). The numerical model as used in these analyses did not explicitly account for energy losses due to radiation of long waves into the sea or entrance and exit losses. These losses are incorporated into the model in the form of bottom friction through model calibration. Including these losses in the friction term means that Manning's n calibrated for Great Lakes inlets is higher than that used in open channel flow compu- tations. In the numerical model, the magnitude of Manning's n determines the amount of energy dissipated. Larger values of n will cause higher amounts of energy loss which results in less wave amplification in the bay and lower inlet velocities. The influence of n_ on inlet-bay response to long wave forcing at Pentwater is shown in Table 4. Table 4. Influence of Manning's n on inlet-bay response at Pentwater. Vereen Che) Se tor Ely S Wodl sae) Se) 2.0 Numerical 1.5 5 4 3 2 | 0 Wave Period (hr) Figure 14. Pentwater response to long wave forcing (n = 0.036). 40 2.0 0.5 Figure 15. Predicted (n= 0.062), Observed Freeman, Hamblin, and Murty, 1974) 3 Time (hr) Toronto response to long wave forcing. 4 | Values of n were estimated for other Great Lakes inlets, based on experience at Pentwater and Toronto. Then, the frictionless Helmholtz period was estimated for each inlet-bay system using equations (4) and (5), and the numerical model. Numerical model and frictionless Helmholtz period results are summarized in Table 5. The amplitude response curves and predicted maximum velocities are shown for selected inlets on Lake Michigan (Fig. 16), Lake Superior (Fig. 17), and Lakes Erie and Ontario (Fig. 18). A 3-centimeter mono- chromatic forcing-wave amplitude was used in these models. For waves of different amplitudes, the maximum inlet velocity is approximately proportional to amplitude. Water level changes throughout the forcing cycle cause nonlinear effects (i.e., ap/a, is slightly different at high and low water), so that the mean of ebb and flood conditions is used in the response curves in this study. This analysis shows that all of the jettied inlet systems studied have significant inertial effects because long waves at or near the Helmholtz period of each system have higher amplitudes in the bay than in the Great Lakes. The inlet-bay systems modeled have a wide variation in response characteristics from one system to another because of the complicated interactions between the four terms in the equation of motion of the inlet and the response of the bay to the inlet. Pentwater, for example, has a moderate amount of wave amplification and produces inlet velocities greater than 30 centimeters per second (1 foot per second) for forcing waves of 3-centimeter amplitude and periods ranging from 0.9 to 2.5 hours (Fig. 16). White Lake has less wave amplification, but the interaction between the inlet and bay produces higher velocities over a wider range of forcing periods (greater than 30 centimeters per second for periods of 1 to 5.6 hours) (Fig. 16). Since Little Lake and Presque Isle have the capacity to generate reversing currents in only a narrow window of forcing periods, it is unlikely that significant reversing currents will be frequently generated (Figs. 17 and 18). Duluth-Superior has the highest capacity for generating reversing currents for a given wave amplitude with maximum velocities occurring at a theoretical forcing period of 1.1 hours. The mean velocities in Duluth (inlet 2), are approximately 1.5 times larger than in Superior (inlet 1) (Fig. 17). A unique feature of the Duluth-Superior system is that the model predicts a net flow into the harbor through the Duluth inlet and a net outflow through the Superior inlet when the forcing period is near 1 hour (Table 6). This asymmetry in flow throughout the forcing cycle will generate a small counterclockwise net flow throughout the inlet-bay system at Duluth-Superior. North Pond, in 1975, had two short natural inlets connecting a relatively large bay to Lake Ontario. North Pond does not amplify long waves because the mass of water in the inlets is small compared to bay size, and friction in the inlets is high due to the shallow-water depths (Fig. 18). Since friction is high, North Pond behaves like a traditional tidal inlet with a balance between head and friction in the inlets. 42 "INd90 JOU SeOp UOTIeOTFITdwe Jey, OS JUeUTWOP ST UOTIITIY, pereiqrred, * (SZ90q0N1T] USD s) 2005 T'0 = %e ‘opnzttdwe yzIM oAeM ITIeWOIYIOUOM eB BUTS, UeSTYITW eAe] [epow (s) pue (7) *sbo u s,suTUURBW Teo TLounN SSOTUOTIITIA (zy) potzed z1ToyWT eH (Tepow [eotzoumu) (ay) ATOOTOA UNUTXeU FO poTI9g *, SOTPTOIOTOA T ZOTUT uUNUTXeW pue uoTeOTFITdwe oAeM UNUTxeW FO spotzed poeydtperg 03U0LO], puog YiZON Snpos 213311 eS] enbseirg Zotzedns -yantng oye] 9T33TT PueT OH uoseysny] sxe] 93 TUM 1978M}U9 q u07ZS3UTPNT 938110g UOT}BIOT "g OTqeL 43 — — Portage —--— Ludington —-— Pentwater ----— White Lake Muskegon —-— Holland Op/d, 6 5 4 3 2 | (0) Wave Peroid (hr) — — Portage —--— Ludington —-— Pentwater ——-—- White Lake Muskegon —-— Holland Vox (ft/s) 6 5 4 3 2 | (0) Wave Period (hr) Figure 16. Predicted response of inlet-bay systems on Lake Michigan to monochromatic forcing (a, = 0.1 foot). 44 Duluth - Superior —-— Little Lake Superior —---- Duluth —-— Little Lake Vmox, (ft/s) 6 5 4 3 2 | (0) Wave Period (hr) Figure 17. Predicted response of inlet-bay systems on Lake Superior to monochromatic forcing (a, = 0.1 foot). 45 ———-— Presque Isle Little Sodus —-— North Pond 3 Wave Period (hr) ———— Presque Isle Little Sodus —-— North Pond, North Inlet —--— North Pond, South Inlet Vmax, (ft/s) Wove Period (hr) Figure 18. Predicted response of inlet-bay systems on Lakes Erie and Ontario to monochromatic forcing (a, = 0.1 foot). 46 Table 6. Predicted Duluth-Superior maximum inlet water velocities for a forcing wave of 1 hour (a, = 3 centimeters). 73 (2.4) 64 (2.1) 43 (1.4) 49 (1.6) A unique feature of the North Pond inlets is that the maximum velocity in the inlets is predicted to be approximately the same over a wide range of forcing periods because of the approximately linear relation between wave amplitude propagation in the bay and wave period (Fig. 18). The velocities in the northern inlet at North Pond, the most recently formed inlet, are predicted to be 1.4 times larger than those in the older inlet. The numerical model for Pentwater was run using monochromatic forc- ing waves with the various modal periods of oscillation of Lake Michigan (Table 3). The predicted amplification and maximum velocity for a forc- ing wave of a, = 0.1 foot are listed in Table 7. From this analysis, the sixth through ninth longitudinal modes of oscillation of Lake Michigan are predicted to cause the largest wave amplification and generate the highest relative velocities. However, analysis of the node-antinode pattern of Lake Michigan shows that only even modes of oscillation will have antinodes, and cause significant water level fluctuations adjacent to Pentwater (Fig. 3). Since odd modes of oscillation have a node near Pentwater (Fig. 3), even the presence of one or more of the odd modes of oscillation in Lake Michigan will cause only small water level fluctu- ations near Pentwater. This means that the sixth and eighth longitu- dinal modes of oscillation of Lake Michigan will probably have the largest influence on the hydraulics of Pentwater. 3. Observed Lake Level Fluctuations, Bay Response, and Inlet Velocities. The first obvious characteristic of Great Lakes water level fluc- tuations is that they are not uniform (as assumed in the previous section). Therefore, the monochromatic analysis can be used to obtain an upper esti- mate of bay wave amplification and iniet velocities; however, a complete analysis is necessary for an accurate estimate of response to a particular Great Lakes water level time history. Sample water level records from Lake Michigan and Pentwater bay, along with spectral analysis of 42-hour records to show typical bay response, are plotted in Figures 19, 20 and ZA Figure 19 shows a storm event on Lake Michigan when several modes of oscillation were excited by meteorological effects. The second longitu- dinal mode of Lake Michigan (5.3 hours) and a 0.65-hour wave are partic- ularly dominant. All of the 5.3-hour wave propagates into Pentwater 47 Table 7. Numerical model prediction of Pentwater response to Lake Michigan modes of oscillation. OF 1.0 Bo 1.0 0.2 3. 1.2 0.5 So 1.3 0.7 2. 1.4 0.9 Bo 1.6 iodl ite 1.7 1.6 1. 1.6 1.9 alte 1.5 2.0 1. 1.3 1.9 0. 0.8 1.5 0. 0.6 1.1 Notes--L = longitudinal modes of oscillation of Lake Michigan. transverse modes of oscillation of Lake Michigan. 4 i} ? = observed oscillation,:mode unknown. X = wave has‘a node near Pentwater. Modes 6L to 9L have modes of oscillation with large ap,/a, and Vmax for Pentwater. 48 Period (hr) 0.3 0.4 Pentwater Lake Lake Michigan 0.5 0.7 1.0 1.4 5.3 Forcing Period (hr) A0){0) 0) QIUDIIDA 1.78 2.13 2.48 2.84 3.19 3.54 Frequency (c/hr) 0.38 0.73 Lake Michigan ee Pentwater Lake — See — — — = == —<——-—-—-y == == == == ‘ = LT Sa o as E : 3 a N oe “6 fe) ro) ( (43) |8A87 48}0M Sample Pentwater and Lake Michigan water levels and spectra. Figure 19. 49 Period (hr) 0.3 0.4 Pentwater Lake 1.0 0.7 0:5 ----- Lake Michigan 1.4 20.0 3.0 QUDIJOA [Ou eal Se 2:48 8 25845059 so 4 Frequency (c/hr) 1.43 |.08 OL} O78) Lake Michigan Pentwater Lake _-—— _ Time, 4hr ——_ 6) -0. | (0), (43) 18487 49)0M Sample Pentwater and Lake Michigan water levels and spectra. Figure 20. 50 Period (hr) Z2O)(0) BO) ks ol) 0.7 0.5 0.4 OS) 1.44 Wave Period (hr) S Seltrs Lake Michigan io) = Pentwater Lake > AMAL ss \ y , AA 1 ult rx 0.38 O73 1.08 1.43 1.78 2.13 2.48 2.84 3.19 3.54 Frequency (c/hr) 0.2 Lake Michigan Pentwater Lake Water Level (ft) Time, 4 hr S| Figure 21. Sample Pentwater and Lake Michigan water levels and spectra. 3 | with an amplification factor of 1.0, as predicted by the numerical model, because the wave period is much longer than the Pentwater Helmholtz period of 1.8 hours. However, the 0.65-hour wave, which reaches heights of 15 centimeters (0.5 foot), has a negligible effect on the harbor because it is much shorter than the Helmholtz period of 1.8 hours. Waves of 1.44 and 1.25 hours are slightly amplified by the harbor (shown by the spectral analysis in Fig. 19), but these waves are difficult to distinguish in the record because of the mixing of individual wave components. The storm event in Figure 20 shows that a different set of modes of oscillation of Lake Michigan is present. The 1.44-hour wave is the highest, is amplified the most, and probably generates the highest per- centage of significant reversing inlet current velocities. A 1.8-hour period wave is also present and is amplified. Waves shorter than 1 hour are damped by the harbor. An unusual water level fluctuation at Pentwater where only the 1.44- hour wave is dominant in Lake Michigan, is shown in Figure 21. As predicted previously, the 1.8- and 1.44-period waves which are the sixth and. eighth longitudinal modes of oscillation of Lake Michigan, cause the highest current velocities. Figure 22 shows the wide variation of water level fluctuations occurring in three different harbors along the eastern shore of Lake Michigan at the same time (Pentwater and Ludington are only 2.3 kilometers (11 miles) apart). The reasons for the differences are that the forcing waves outside each location are different as a result of the node-anti- node pattern of seiching in Lake Michigan (see Sec. II) and because each harbor responds differently to the forcing that is present (see Sec. IV); e.g., the 1.44- and 1.28-hour waves in Pentwater and Ludington are not noticeable in Muskegon harbor which has a Helmholtz period of 5 hours. The forcing of harbors on the other Great Lakes will be completely different because the system of seiching varies from lake to lake; e.g., on Lake Superior, wave periods of 0.59, 0.68, 0.95, 1.14, and 1.7 hours occur in Little Lake Harbor (Fig. 23). Shorter period waves may also occur in Lake Superior, but are not observed in the harbor because the harbor dampens waves shorter than approximately 0.4 hour. The 1.7-, 1.14-, and 0.95-hour waves on Lake Superior (the 7th, 10th, and 11th longitudinal modes of oscillation) were observed to cause high re- versing currents and associated navigation problems at Duluth-Superior; e.g., on 10 June 1973, a 1.7-hour wave with a height of approximately 30 centimeters (1 foot) in the harbor, in conjunction with small 0.95- hour period waves, affected Duluth-Superior. Velocities as high as 200 centimeters per second (6.5 feet per second) were generated in Duluth inlet and 140 centimeters per second (4.5 feet per second) in Superior (Fig. 24). High velocities are generated in these inlets because of the large forcing waves in Lake Superior at this location which have periods 52 *SUOT}ZENJONTF TOAST Leqem ofdmes °77 omMBTy == = = 76 | WOH AON == == l 2 ———— = 18) HONE Ge ay G= 4, uobaysnw ro) wnjog Aspsjyiquy Jy QQ’) = 4] JayDMyuad D8) 1.14 Variance 3.0 Variance 3.0 0.59 28 Aug 1975 = Observed Water Levels = 0.2 em (0); au = -0. | = -0.2 1.0 06 04 Wave Period (hr) a) Typical Fluctuations Computed Spectra fa} Typ 0.68 19 July 1975 Observed Water Levels Water Level (ft) oo _ 99S hm —- O —Pp 10 06 O04 Wave Period (hr) Computed Spectra (b) Extreme Event Figure 23. Water level fluctuations in Little Lake Harbor. 54 Bay Level al Duluth Inlet Velocity (ft/s) Superior Inlet Velocity (ft/s) Figure 24. Sample bay levels and inlet water velocities at Duluth-Superior (inlet velocities are approximate; gages had not recently been calibrated). 59 near the harbor Helmholtz period. Forcing waves are large because the harbor is located on the converging end of the lake, which will always be an antinode of longitudinal oscillations. Maximum water velocities observed in other inlets are much lower than in Duluth-Superior; e.g., at Pentwater, all measurements and pre- dictions show that, velocities are less than 60 centimeters per second (2 feet per second) for 99.5 percent of the time (Fig. 25). Predicted inlet velocities for other locations show that Portage, Ludington, and Pentwater have similar velocity distributions; Presque Isle, Muskegon, and Little Lake have still lower velocities (Fig. 26). V. INLET DESIGN Great Lakes inlet design problems generally fall into one of two classes: (a) a pond or lake to be connected to one of the Great Lakes by a new channel, and (b) an existing inlet channel to be modified. The concepts and techniques developed in this study can be used to aid the design of an inlet in either class. An example application for each class is given below. 1. New Inlet Channel. The procedures for analysis of a new channel that is to connect a lake to one of the Great Lakes are: (a) determine the approximate inlet dimensions (length, width, and depth) based on physical limitations such as the desired navigable depth and width and the distance between the lake and Great Lakes; (b) estimate a Manning's n for the proposed channel (see Sec. IV, 2, for typical values of n); (c) use the numerical model to obtain monochromatic response characteristics of the harbor for the range of expected lake seiching periods and a typical amplitude; (d) compare the results to those of other nearby harbors; and (ce) apply the numerical model to predict inlet velocities, discharge, and bay levels for the period of record (if Great Lakes water level fluctuation records are available in the vicinity of proposed site). For example, suppose an inlet is to be designed to connect Crystal Lake to Lake Michigan (Fig. 27). Crystal Lake, located on the eastern shore of Lake Michigan 35 kilometers (22 miles) north of Portage Lake, has a bay surface area of 4.12 x 107 square meters (4.44 x 108 square feet). The inlet at this site would be approximately 1,200 meters (4,000 feet) long. Assume that the inlet would be 61 meters (200 feet) wide and 5.5 meters (18 feet) deep. Since the inlet is similar to Pent- water (see Table 5), the Manning's n for this channel is estimated to be 0.036. The numerical mode] was run for Crystal Lake using Lake Michigan seiche periods of 9.3, 5.3, 3.5, 2.2, 1.85, and 1.4 hours with an amplitude of 3 centimeters. The predicted response characteristics of this inlet-bay system are shown in Figure 28. The model predicts that 56 50 a a. @ Summer |967 oN Q Fall 1974 » @ Summer 1975 XN 20 “ > oO = 2 23s = Co) ( x 3) cb} ES Eo a co oS =} ao OL 1.0 1.5 20) Inlet Channel Velocity (ft/s) Figure 26. Inlet velocity cumulative frequency distributions (based on equation 6 and bay water level records). 58 Crystal Lake Portage Muskegon 000 ft = eS eee m OP Or ome Gein A poy = 4-44x10 ft” Coke Mj chigan Crystal Lake, Michigan. Figure 27. 519 Vmax (G,= 0.1 ft) 0 10 9 8 7 6 5 4 3 2 | 0 Wave Period (hr) Figure 28. Predicted response characteristics of an inlet for Crystal Lake (L = 4,000 feet, A = 3,600 square feet, D = 18 feet, B = 200 feet). 60 the wave amplitude will be smaller in the bay than in Lake Michigan, pri- marily because the bay surface area is much larger than the inlet design cross-sectional area (a ratio of approximately 10°). The model also predicts that monochromatic seiches with an amplitude of 3 centimeters will generate maximum velocities of 43 centimeters per second (1.4 feet per second) for wave periods of 4 to 9 hours (Fig. 28). Since maximum velocities decrease for waves shorter than 4 hours, a wave with a 1-hour period produces insignificant inlet velocities. The first three modes of oscillation of Lake Michigan (9.3-, 5.3-, and 3.5-hour waves) will generate the highest velocities for the Crystal Lake inlet design. Portage is located near Crystal Lake; therefore, forcing amplitudes of the first three modes of oscillation of Lake Michigan will be similar at both locations (Fig. 3). The predicted vel- ocities for a given wave are different at Portage and Crystal Lake (Fig. 29) due to differences in inlet and bay geometry. However, Portage water level data can be used to estimate Crystal Lake inlet velocities. To predict inlet velocities at Crystal Lake using Portage data, the measured Portage bay level fluctuations must first be adjusted to esti- mate the nearby Lake Michigan wave amplitudes. These amplitudes are then used to predict velocities at Crystal Lake; e.g., a measured Portage bay level fluctuation has a period of 3.5 hours and amplitude of 0.15 foot (4.6 centimeters). This wave was amplified by a factor of 1.3 by Portage harbor (Fig. 16), so the Lake Michigan wave amplitude was 0.15/1.3 = 0.12 foot. A 0.1-foot wave amplitude in Lake Michigan produces a maximum velocity of 1.3 feet per second at Crystal Lake inlet (Fig. 28); there- fore, the 0.12-foot wave produces 1.3 (0.12) = 1.6 feet per second maxi- mum velocity. This procedure could be followed for other seiche modes to estimate the maximum velocities expected at Crystal Lake. If a complete analysis of inlet velocities is required, water levels should be measured in Lake Michigan adjacent to Crystal Lake for at least several months. These levels can be used as the forcing function in the numerical model to produce a predicted time history of inlet velocities, discharge, and bay levels for the period of record. 2. Inlet Channel Modification. Procedures for investigating the effect of a modification to an inlet are: (a) Determine the geometry of the present system and obtain proto- type hydraulic data (i.e., concurrent bay levels, Great Lakes levels, and inlet velocities); (b) calibrate the numerical model; (c) obtain monochromatic response characteristics of the inlet-bay system, (d) modify the model geometry to reflect the proposed inlet change and pre- dict the response characteristics of the new condition; and (e) use the water level records in the Great Lakes to force the model to produce a time history of inlet velocities, discharge, and bay levels for the proposed design. 6 | ——_, -— >) = = Bf oom Crystal Loke be “wo ~ = So i= > <9 2.0 5 4 3 2 | Wave Period (hr) Figure 29. Predicted inlet velocities at Michigan inlets (a, = 0.1 foot). 62 For example, assume that a prediction of inlet velocities and the amplitude of bay level fluctuations at Pentwater is desired if (a) the inlet was deepened to 7.3 meters (24 feet), and (b) the inlet was allowed to shoal to a depth of 1.8 meters (6 feet). The monochromatic response of the inlet-bay system for these two inlet modifications is predicted by changing the inlet geometry in the cali- brated model of Pentwater. Each model was run with sinusoidal wave periods between 0.5 and 5 hours and an amplitude of 3 centimeters to predict am- plification of the wave in the harbor and maximum velocity in the inlet. Figures 30 and 31 show the amplification and maximum velocities, respec- tively, for the 1967 inlet geometry, and for inlet depths of 1.8 and 7.3 meters. The results are summarized in Table 8. Table 8. Summary of Pentwater hydraulic characteristics for selected inlet depths. (85/85) maze T, (hr) Vi a (ft/s)! 1 ee (hr) lFor ago = 0.1 foot. These predictions show that deepening the Pentwater channel from 1.8 to 7.3 meters causes the peak amplification and inlet velocity to in- crease, and the Helmholtz period and period of maximum velocity to de- crease. Comparison of the modes of oscillation of Lake Michigan (Table 3) with the predicted velocities (Fig. 31) suggests that the 0.85-, 0.97-, 1.1-, 1.25-, and 1.44-hour waves will generate the highest reversing currents at Pentwater if the inlet was deepened to 7.3 meters. The Lake Michigan water levels recorded on 18 August 1967 (Fig. 6) were used to force the lumped parameter model for the selected depths (Fig. 32). The model predicts that for these Lake Michigan level fluctu- ations, the maximum velocity for an inlet 1.8 meters deep would be 46 centimeters per second (1.5 feet per second); a 7.3-meter-deep inlet would have a velocity of 107 centimeters per second (3.5 feet per second). VI. SUMMARY AND CONCLUSIONS 1. Meteorologically generated seiches cause most of the significant reversing currents at Great Lakes inlets. Seiche periods and node-anti- node patterns can be predicted numerically. However, water levels must 63 Amplification Figure 30. 1967 Condition (I | ft a) " ; i i=? ve * s a conht t tet Westhgitaals eae Be ah “ Me : eTOg Ra baa acct )- t yarA Ge a ree vet y BY (thovied 1 4x08 ceed iy. Pee ar AAT ee ef co | : (SHS 504 alban Spree aaa Tt a vabia ey “ aod Blige. 29) ate OM Lagreus ; ay 4 ova? 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