rin t or re p c redit to the U S. Arm y Coastal E n g ineerin g R esearc h Center.

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3 c ommerc ial p rodu c ts. R e p rin t or re p u blic atio n of an y o f this mate rial shall g ive a pp ro p riate c redit to the U S. Arm y Coastal E n g ineerin g R esearc h Center. Limited f ree distribution within th e U nited States of sin g le c o p ies of this p ublic ation has been made b y this Center. available from Additio n al Co p ies are N a ti on al T e c hn ic al I n f o rma tion S e rvic e A T TN : O p e ra tion s D i vision 5285 Por t R o y al R oad S p rin gf i e l d Vir, gin ia Contents of this re p ort ar e n ot to be used f or advertisin g, publi c ati on, or p romotional p ur p oses. Citation of tr ade n ames does not c o n stitute an of f ic ial en dorsement or a pp roval of the use of suc h T he fi n di n g s in this re p ort ar e n ot to be c onstrued as an of ficial D e p art ment of the Arm y position u nless so desi g nated b y other au thorized doc uments.

4 1 6. D IS T R I B U TI O N S T A T E ME N T ( a t a m R e p ort ) Approved for pub li c re lease ; distribu ti on u n limi ted I7. D I S T R I B U T IO N S T A T E ME N T 1 9. K E Y WO R D S ( C o n t i n u e on r e m -o ol d. I f n o c o a oc ry c ad i d e n ti ty b y b l o c k n u mb e r ) Great Lakes I nlet - harbor resona n c e Lake se i c h i n g Nonti dal inlet Inlet hydraulic s 20, AS T R A C T (C a n ḡh a n m u c i t f " m I n " Id e n t i ty b y b l o c k u m b e r ) Revers i n g c u rrents i n i nlets on the Great Lakes are gen erated pri mari ly b long wave se i c h i n g modes of the lakes rather than by the a stronomi c a l tide. Fie ld measurements were c on du c ted i n at n in e harbors on the Great Lakes to : (a) Investigate the natu re of lon g wave exc i tation and the gener ating me c hanism for si gni fi c ant inlet ve locities, (b ) estab li sh te c hniques f or predi c ti n g in let- bay system response, an d ( c ) deve lop base d a t a for future p l an n i n g an d design s tudies. Data c o l lec ted inc lude c ontinuous harbor water ( c o n tinued ) DD som o u o r t n o v c s IS O B S O L E T E U NCLASSI FIED S E C U R IT Y C L AS S I F IC AT IO N O F T HRS P AGE ( h c u D a ta E n to

5 leve l meas u rements at a l l sites, in let ve loc ity meas u rements a t the primary site (Pentwater, Mi c higan ), an d c hann e l hydrographi c su rveys at the sites where more re c ent d at a were needed. Avai lab le his tori c water leve l and ve loc i ty data for some of the harbor sites were a l so u sed. Amp li fied harbor os i l s c lation an d of the highes t let gen eration in ve l oc ities aused by the He lmho lt z res a r e c onan e mode whi h c c h a s a period of to 5 rs for the hou in let - bay sys tems died A re stu. c ent ly deve loped simple meri al, nu c mode l is shown to be effe tive in predi g inlet c c tin - bay response over the ge ran of i tation periods ntered A finite exc enc ou. - di fferenc e form of the c uity ontin i s to ate ly predi t let ve equ a tion shown adequ c in loc i ties i f high - ality bay qu wa ter leve l re c ords are ava i l ab le. Se le c ted data from the stu dy si tes are presen ted to demonstrate the hydrau li c response of the in let - bay systems and the app li c abi lity of the predi c tion s c hemes. Examples to demonstrate u s e of the c onc epts an d te c hn iqu es deve loped in the stu dy are app lied to the design of a n e w in let c hann e l an d to the modi fi c ation of existing c hanne l.

6 PRE FACE Thi s report i s pub l i shed to provi de coas ta l engineers wi th an ana lys i s o f the hydrau l i c respons e of in let - bay sys tems on the Great Lakes. The work was c arri ed out under the coas tal res earch program of the U. S. Army Coas tal Engineering Res earch Center (CE RC). The report was prepared by Wi l l i am. N. Se e l i g and Dr. Rob ert M. Sor ens en, Coas tal S tructures Bran ch, Res earch Divi s i on, CERC, under the general supervi s i on o f R. P. Savage, Chi ef, Res e ar ch Divi s i on. The authors acknow ledge the e ffor ts o f the U. S. Army Engine er Di s tri ct, De troi t and the Nat i onal Oceani c and At mospheri c Admi ni s trati on, Lake Survey Center, who c o l l e c t ed mos t of the fie l d dat a, and the report revi ew and comments by C. Mas on and B. He r c h e n r ode r. Comments on thi s pub li cat i on a r e invi te d. Approve d for pub l i cati on in accordan ce wi th Pub li c Law 1 66, 79 t h Congres s, approved 3 1 Ju ly , as supp lemented b y Pub li c Law 1 72, 88th Congres s, app r oved 7 November S Co l one l, Corps of Engineers Commander and Dir ect or

7 CONTENTS CONVERS I ON FACTORS, U. S. CUSTOMARY TO METRI C (S I ). SYMBOLS AND DE FINI TI ONS. I INTRODUCTI ON I I LAK E AND INLET HYDRAULI CS. 1. Great Lakes and Inl et - Bay System Hydraul i cs 2. Predi c t i on of In let Ve lo c i ti es. I I I THE F I ELD DATA COLLE CTION PROGRAM. 1. Fi e ld Measurements. 2. Equipment 3. Data Reducti on and Ana lys i s Te c hniqu es. IV RESULTS. 1. S e i c h i n g of th e Great Lakes 2. Predi cted In l et - Bay Respons e to Monochr omati c Long Wa ve Forcing 3. Obs erved Lake Level Fluctuat i ons, Bay Respons e, and In l et Ve lo ci ti es V INLET DES I GN 1. New In let Channel 2. In let Channe l Modi fi cat ion. VI SUMMARY AND CONC LUS I ONS. LI TE RATURE CI TE D TAB LES 1 Summary of fi e ld measurements 2 In let and bay geome tri c measurements. 3 Modes o f os ci l lati on o f the Great Lakes 4 Influence o f Manning ' s n on in let - bay response at Pentwater 5 Predi cted peri ods of maximum wave amp li fi cati on and maximum in l et ve lo ci ti es 6 Predi cted Duluth Superi or maximum in l et water ve l o c i ti es for a forcing wave o f 1 hour ( a o 3 cent imeters ). 7 Numeri ca l Mode l Predi ct ion o f Pentwater respons e to Lake Mi chi gan modes of os ci l l ati on.

8 CONTENTS TAB LES - Continued Summary o f Pentwater hydrau li c c haracteri s ti c s for s e l e c te d in le t depths FI GURES In l et - b ay syst em. Amp l i fi cat i on and phas e l ag for in l et - bay syst ems Thre e predi c ted longi t u dinal modes o f os ci l lation of Lake Mi c hi gan. Pentwater respons e to s inusoi dal wave in Lake Mi chigan. Respons e to long wave exci tati on a t Pentwater Numeri ca l mode l water l eve l predi ct ions at Pentwater. In l et s tudy s i tes Data co l l e c tion s i tes on Lake Mi chi gan. Data co l l e ct i on s i tes on Lake Superi or. 1 0 Data co l l e c ti on s i t e on Lak e Eri e, Presque I s l e, P ennsy lvani a 1 1 Dat a c o l l e c ti on s i tes on Lake Ontari o 1 2 Samp l e spe c tra o f P entwater bay water l eve ls. Measured and predi ct ed inl et ve l o c i ty cumul ative frequen c y di s tributions at P entwater, Mi c hi gan 1 4 Pentwater respons e to long wave forcing 1 5 Toronto respons e to long wave for c ing 1 6 Predi ct ed respons e of in let - bay sys tems on Lake Mi chi gan to monochromati c forci n g 1 7 Predi cted respons e of in l et - bay sys t ems on Lake S u peri or to monoc hromati c for c ing 1 8 Predi cted respons e of in let - bay sys tems on Lakes Eri e and Ontario to monochromati c for cing

9 CONTENTS F I GURE - Continued 1 9 Samp le Pentwater and Lake Mi chi gan water l eve l s and spe c tra. 20 Samp l e Pentwater and Lake Mi chi gan water l eve ls and spectra. 2 1 S amp l e Pentwater and Lake Mi chi gan water leve ls and spe c tra. 22 Samp l e water leve l fluctuati ons 23 Water l eve l fluc tuati ons in Li tt l e Lake Harbor. 24 Samp le bay l eve ls and in l et water ve loci ti es at Duluth - Superi or. 25 Pentwater in let cumu lative frequency ve loci ty di s tributions. 26 In let ve loci ty cumul ative frequency di s t ributi ons 27 Crys ta l Lake, Mi c hi gan. 28 Predi cted respons e characteri s ti cs of an in l e t for Crys ta l Lake 29 Predi cte d in let ve loci ti es at Mi chi gan in lets 30 Respons e to l ong wave exci tati on at Pentwater 3 1 Respons e to long wave exci tat ion at Pentwater 3 2 Predi cted Pentwater in l et ve loci ties for Lake Mi chi gan water leve ls re corded on 1 8 Augus t

10 C ON VE R SI ON F ACTOR S, U. S. C U STOMAR Y T O ME T R IC ( SI ) U NITS OF ME AS U R E ME NT U.S. c u stomar y u nits of me asu reme nt u sed in this r e p or t c a n be c o n ve rted t o me tric ( S I) u nits as f ollows: in c hes Mu lti p l y T o obtai n milli me te rs c e ntime te rs squ are in c hes squar e c entime ters c ubic inc he s c u bic c e n timeters c entime ters meters squ ar e f eet c ubic f eet me te rs squ are me ters kilomete rs per hour ac res he c tares n ewton meters " kilo g rams p e r s q uar e c en timeter ou n c es g rams p oun ds ton, lon g ton, short metric ton s r adian s Celsius de g rees o r K e lvin s l 1 T o obtain Celsiu s ( C) t e m p e r atu r e r e adin g s f r om Fahre n he it ( F ) r e adin g s u u se f o rmu l a : C ( F T o obtain K e lvm ( K ) r e adin g s, u se f o rmu l a : K f (F 32)

11 SYMBO LS AND DE F INI TI ONS in l et c ros s s e c tional area at the bay end bay surface area in let cros s - s e cti ona l area gri d cr os s - s ect i ona l area in let cros s - s e ctiona l area at the s ea end in l et wi dth grid ce l l wi dth wave speed gri d c e l l depth water depth acce l erat ion due to gravi ty water surfac e e l evation wat er surface e l evat ion in the b a y wat er surface e l evat ion in the bay at the previous t ime st ep wat er sur fa c e e levat ion in the s ea numb er o f channe l s in the gr i d numb er o f s e c ti ons in the gri d gr id ce l l sub s cript indi cating the channe l gri d ce l l sub s cript indi cating the cros s s ecti on mode of os ci l lat ion in l et l ength added in l et l ength in the a c c e l e r a t i on t e r ms to ac c ount for l ong wave radiation from the harb or bas in l ength gri d ce l l l ength

12 SYMB O LS AND DE FINITI ONS - Cont inue d numb er o f in lets conne c t ing a bay to a s ea Manning ' s n Manning ' s n o f a grid c e l l di s charge di s c harge for the n t h in let tot al di s c harge for al l in l ets hydrau li c radius o f a c hanne l in l et - bay system He lmho l t z peri od fri c tion l es s in l et - bay sys tem He lmho l t z peri od peri od o f free mode of os c i l lati on where the subs cript indi c ate s the mode of os ci l lati on mean ins t ant aneou s in let wat er ve l o c i ty at a c ros s s e c tion mean in l et water ve loc ity at a c ros s s ecti on over a s amp ling interval the gri d c e l l wei ght ing fun c t i on whi ch i s the fracti on o f tota l in let di s charge pas s ing through the ce l l at a time s tep di s tance a long the longi tudina l axis of the inl et di s tan ce al ong a cros s s e ction time s tep component of the bott om - s tres s tensor in the dire c t i on of flow

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14 HYDRAULI CS OF GREAT LAK ES INLETS Wi l l i am N. S e e l i g and R obe r t M: S or e n s e n I. INTRODUCTI ON Nu merous bays and harbors are conne cted to the Great Lake s by j ett ie d in le t c hanne l s. Thes e in le t channe ls are important b e caus e they a l l ow (a) acces s to commercia l shipping and re creationa l b oating, (b) migrati on of fi sh, and (c) flushing of po l lutant s from the bays and harbors. Great Lake s in le t - bay systems are general ly smal l er than thos e on the At lanti c, Paci fi c, and gul f c oas ts of the Uni ted S tates and respond p r imari ly to the long wave se i c h i n g mode s o f the Great Lakes rather than to the as tronomi cal ti des. Thes e s ei c hes have sma l ler amp li tudes and shorter peri ods than the tides on the ocean coasts. The maj or e ffort o f thi s s tudy involved the co l le cti on and ana lys is of hydraul i c data at s evera l in l et - bay sys tems throughout the Great Lakes during and Measurements at Pentwater Mi chi gan the primary.,, s tudy locati on in clude d s imu l t aneous re ording o f in l et urrent, c c ve l oc i ti es and water leve ls in the bay and in Lake Mi chi gan. At the other lo cations on ly bay water l eve l s were measured However hydrographi c,., surveys were ob tained for a l l the inl et s inves ti gated, and his tori c hydraul i c data from s e lected s our c es were ana lyzed. Thi s s tudy defines the hydraul i c me chani sms important to Great Lakes in le t - b ay sys tems, deve lops ana lyti c al techni que s for the predi cti on of in let currents and bay water leve l os ci l lations, and pres ents des i gn data and sys tem respons e curve s for s e le cted in l e ts. Fi e ld data were analyzed us ing (a) a formula for e stimating the sei che peri ods of the Great Lakes whi ch are important in producing reversing current s at an in l et ; ( b ) a mode l that us e s bay water leve l time hi s tori es to predi ct in let ve loci ti es ; and (c) a s imp li fi ed numer i cal in le t hydrauli c mode l that, when ca l ibrate d for fri cti on e ffe cts, can be us ed to predi ct in le t ve lo ci ti es and bay water l eve l os ci l l ati ons generated by lake os ci l lati ons. Thes e ana lys i s tec hniques are us ed wi th the fi e l d dat a to deve lop respons e curves and c umulative in let cur rent ve lo ci ty dis tribution curves for the in l ets s tudi ed. I I. LAK E AND INLET HYDRAULI CS 1 Great Lakes and In le t. - B a y S y s tem H drauli cs y. An in l et i s a re l ative ly narrow channe l whi ch connects a " sea " (or one o f the Grea t Lakes in this study) to a l ake or harb or (a " bay i n thi s study). The bay is large compared to the in l et the radius

15 o f the bay i s typi c a l ly larger than the l ength o f the in l et) and the surface area o f the b ay i s much sma l ler than the surface area of the s ea. a. Caus es o f Revers in g I nl et Current s. Mortimer ( 1 965) and Freeman, Hamb l in, and Murty ( ) show that s igni fi cant revers ing in l et current ve lo ci t ie s are caus ed by water l eve l f luctuat ions in the Great Lake s whi ch generat e a hydraul i c respons e in the in let and bay. The mo st important Great Lake s water l eve l fluctuati ons are due to the re sonant s e i c h i n g or os ci l lati on of the parti c ular l ake at it s fu ndamenta l and harmoni c periods. S ei c he s are ini tiated by s torm pres sure and wind force s on the s ea whi ch redi s tribute water in the lake t o caus e a h igher e levati on than norma l in s ome areas and lower leve l s in other area s. When gravi ty tri es to res tore the water l eve l, s ei ches are generat ed. Thes e s ei ches usua l ly continue for a numb er of cyc le s whi ch may extend over a few days after the s torm has pa s s ed. When a s ei c he i s generated in one o f the lake s, the water l eve l fluc tuat ions outs i de an in l et caus e a head di ff erence ac r os s the in l e t, whi ch, in turn, g enerat es a current in the in let. Water di s charge through the in let results in water added t o or removed from the bay s o the b ay leve l ri ses and fa l ls in a pumping fashion for mos t se i c h i n g peri ods of the lake. Astronomi ca l ti des and other n on se i c h i n g l ong wave s caus e water l eve l fluctuati ons o f the Great Lake s ; however, thes e fluctuat i ons genera l ly have in suffi ci ent amp li tudes or are at peri ods that usua l ly do not s i g n i f i c an t l y in fluence in let hydrau li cs. St o rm sur ge, parti cular ly on sha l low Lake Eri e, ma y occas i onal ly generate s trong in l et cur rents. b Mathemat i cal Des cri t ion of In l et. p - B a y H ydr au li cs The respons e. of an in let - b ay sys tem ma y b e des cribed in te r ms of the one - dimens i ona l equati on o f water moti on in the in le t and the cont inui ty equati on r e lat ing the rate of b ay l eve l change to the in l et di s char ge. The one - dimens iona l equati on o f moti on a l ong the in let channe l axi s can b e wri tten f E d i where x di s tance a long the channe l $ 3 " water I I surface l eve l A in l et cros s - s ectiona l area 1 2

16 V water ve lo ci ty in the inlet y di s tance a long the cros s s ect ion ( r zx ) z component of the s tres s tens or in the dire cti on of flow g ac c e l e rat ion due to gravi ty t time The in le t has a wi dth, B, length, L, and cros s - sec tional area, A c the b ay has a surface area, The water l eve ls in the s ea and A ba b a y are h S and h b, respe c tive ly ( F 1 %. ati on Equ ( 1 ) equates the hori zonta l driving force due to the water s rfa e s lope wi th three t u c e rms on the ri ght whi ch are the channe l fri c t i on a l res i stance the, c onve ctive ac c e l eration caus ed by ve l oci ty vari ation a long the channe l axi s, and the tempora l acc e leration (or inerti a) resulting from ve loci ty variati on at a p oint wi th time. In near ly pri smati c c hanne l s, such as many in lets on the Great Lakes, the c on v e c tive ac ce l erati on i s often ne g li gib le. The continui ty equati on, whi c h re lat es rate of bay water l eve l change, ah b / B t, to in l et di s charge, Q, i s B h b Q VA C Aba y at A s imultaneous s o luti on of equati ons ( 1 ) and ( 2 ) for a s inu s oi dal s ea l eve l fluctuati on revea ls the important re spons e characteri s ti cs o f an in le t - bay sys tem (F i g. In thi s fi gure, the phas e lag between the s ea and bay water l eve l fluctuati ons and the amp l i fi cati on o f the fc r c ing wave in the bay by the in l e t - b ay s ystem ar e p lotte d as funct ions of dimens i on les s peri od. Dimens i on les s peri od i s de fined as the fri ct i on les s in let - bay sys tem He lmho l t z peri od, TE ', divi ded by the for c ing wave peri od, T. The He lmho lt z peri od i s tha t peri od o f the forcing wave whi ch through res onan c e wi l l caus e the larges t water l eve l fluctu a ti on in the bay. The b ay water leve l remains approximate ly hori zonta l throughout thi s fluc tuati on. The in let - harbor sys tem response i s ana logous t o the re spons e of a s li ght ly damped spring - mas s sys tem or i ts ac ous ti c counterpart, the He lmho l t z res onator. The moti on o f the mas s o f water in the in l e t c hanne l corre sponds to the motion o f the mas s of the spring - mas s sys tem, and the a c ti on o f gravi ty on the ri s ing and fa l ling harbor water surface corresponds to the res training force of the spring. At value s o f TH ' l T approaching zero (l ong wave periods ) the water leve l fluc tuati ons in the bay are the s ame as thos e in the s e a wi th no I 3

17 P lo n Vle w unle f. c ross se c h onol are a A su rfac e are a P rofi le View F i g ure 1. In l et - bay sys tem. I 4

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19 phas e l ag (point A, Fi g. At values of TH ' /T approaching 3 (short forcing p e r i od s) t h, e in l et - bay sys tem stron g ly dampens inci dent waves and water leve l fluc tuat ions in the sea have l i tt le inf luence on in let hydraul i c s (point B, Fi g. At va lues of TH ' / T in the ran ge o f the amount of fri cti ona l res i s tan c e in the c hanne l has a maj or influence on the respons e charac teri s ti cs. In lets wi th hi gh fri cti on, e. g., ti da l in l ets on ocean coasts, typi c al ly have amp li fi cati ons of l es s than one. Thi s amp l i fi cati on factor de creases as the forcing wave peri od b ecomes shorte r (point C, Fi g. At most tidal inl ets, the primary ti da l period i s large compared to the in let - b ay He lmho lt z peri od. Low - fri ct ion inl ets, such as thos e on the Great Lakes, have amp li fi c ati ons greater than one and phas e lags of approximat e ly 9 0 for forcing waves wi th TH ' / Ta d (point D, Fi g. 2) which common ly o ccur. 2. Pre di ct ion of In let Ve l oci ti es. Predi c ti on o f in l et ve l ociti es r equi res (a) the time hi s tory of s ea or bay water l eve l s, (b) the geometr i es of the in let and bay, and (c) a fri cti on - ca l ibrated mode l t o re late water leve l fluctuati ons to inlet bay respons e. a. Great Lakes Water Leve l F luctuati ons. In genera l, no methods are and peri ods o f water leve l fluctuat ions at any point in one of the Gre at Lakes. There fore, water leve l s general ly mus t b e measured. However, inexpens ive s chemes are avai lab le for accurate ly predi cting s ome peri ods and re lative amp li tudes of s ei ches of the Great Lakes (Fe e, ; Rao and S chwab, K now l edge o f the exi s ting wave peri ods wi l l ai d in the des i gn of water l eve l meas uring sys tems, analys i s procedur es, and pre l iminar y in le t des i gn. The b as i c method for es timat ing one dimens i ona l fundamenta l and harmoni c s ei che peri ods, T k, i s to determine the time r equi re d for a wave to trave l twi ce the l ength o f the b as in whe r e k i s the mode o f os ci l lation, Lb the length o f the bas in in the dire cti on of the s ei che, an d c the spee d of the wave. Samp le predi cted longi tudina l s ei ches for Lake Mi chi gan, us ing the Fee (1 968) computer program, are shown in Fi gure 3 for modes k - l, 2, and 3. b. In l et and B a y Geomet r i es. Hydrographi c surveys, including Corps of Engineers dre dging r e cords, can b e us ed to determine in l et ge ometry ; i. e., l ength, wi dth, and depth fi e ld. Maps and aeri a l photos can b e us ed to de termine bay surface area. I 6

20 F ir st lon g it u dinal mode I pe r iod, 9 hr ) Se c ond IOII Q II u d ll lgi mode I pe riod, 5.3 hr onh node Por tag e Mu skegon Mu ske q o n I hl f d lon g i tu d i nal mode (pe r i od. 3-5 hr ) antinode Re lat i ve Se ic he He i g ht of Wat e r ( n ormalize d b y le ve l at lhe sou t he rn l i p of Lake Mic hi g an ) Fi gure 3. Thre e predi ct ed l ongi tudinal mod es of os c i l l at i on o f Lake Mi c hi gan (modi fi ed from Mort imer, I 7

21 c. Methods of Anal y s i s of In l et B a y H y drauli cs. In let current ve loci ti es and bay water surface os ci l l ati ons may be measured to provi de ne ces s ary informati on on the hydrauli c characteri s ti cs of an in let - b ay sys tem. Te chniques us ed for the fi e ld measur ements in thi s s tudy are di s cus s ed in S ecti on I I I, 2. Severa l analyti cal methods are avai lab le for p r edi cting in let hydrauli cs, depending on the type of informati on avai l ab l e and requi red resul ts. Thr ee methods are di s cus s e d b e l ow. ( 1 ) E s tima t ion o f S ei che Peri ods Imp ortant to In let H y draul i cs. To es timate whi ch o f the Gre at Lakes s ei che peri ods, T k, may be i m portant, the fri cti on l e ss in l et - b ay He lmho l t z peri od, T fi ', may b e determined from T 3 ' 2n where L ' i s an adde d channe l length determi ned fr om L ' account s for the water mas s es in moti on beyond the ends of the in let (Mi l es, Equati ons (4) and ( 5 ) may b e i terative ly s o lved to obt ain a value o f T y '. Thi s approach p r oved to yi e l d re as onab ly accu rate es timate s for the in lets cons i dere d in thi s s tudy. As a firs t approximat ion s eiche wave periods whi ch are approximate ly, equa l to the fr ic tion les s He lmho l t z period b etwe e n and 2 time s TE ' ; see Fi g 2) wi l l prob. ab ly caus e the hi ghe st inl et revers ing currents The s ei che node - ant inode patt er n in the Great Lakes wi l l a l so i n f l u ence the importanc e of the vari ous s ei che modes on an in let - bay system. Se i ches wi th ant inodes adj a cent to the in let wi l l produce the largest wat er leve l fluctuations. S ince ends o f the lakes are antinodes for a l l modes o f o s ci l lat ion a long that axi s, bays at the end of a l ak e wi l l norma l ly b e sub j e ct to hi gher wat er l eve l fluctuations than those at other lo cat ions ; e. g., midway a long the longi tudina l axi s o f Lake Mi chigan, near Pentwater, the firs t longi tudina l mode of os ci l lation has a node (Fi g. There fore, on ly sma l l os ci l l at ions can b e generated in the lake at thi s point by thi s mode. The s e cond longi tudina l mode of osc i l lation has an antinode adj acent to P entwat er, s o large water l eve l fluc t u a t i on s in Lake Mi chi gan cou ld be generated outs i de Pentwater by thi s mode. (2) Es tima t ion o f In l et Ve loci tie s from B a y Wat er Leve l Records. A me thod of predi cting in let ve lo ci ti es, i f high - qual i ty b a y wat er l eve l -I 8

22 r e cords are avai lab le, i s to us e the c ont inuity equation. Wri tt en in finite - di fference form, equat ion ( 2 ) b e c omes where V i s the average in let current ve loci ty at a c ros s s e c t i on o f area A c over a water leve l s amp l ing interva l, and At, h b ', an d h b are mean bay water l eve ls at the beginning and end of the samp ling interva l. Measurements o f water leve l at any point in the bay wi l l be repre s entative o f the mean bay l eve l for the He lmho lt z mode of os c i l l ati on o f in let - bay sys tems. Thi s method for predi cting inl et ve loci ti es i s we l l sui t ed for Great Lakes in l ets becaus e in l et and bay geometri es are s imp le and l eve l r e corders are easy t o ins tal l in the protected b ays. The samp l ing interva l should be one - twenti eth of T3 ' or shorter and the st i l l ing we l l care ful ly des i gned f O r b es t re sults (s ee Sec. I I I ). (3) A Numeri cal Mode l. A re lative ly s imp le but extreme ly us e ful equa ti ons o f moti on and continui ty. In thi s mode l the in let channe l i s divi ded by a flow net into a gri d o f sub channe l s and cros s s e cti ons. The subs cripts i and j de s c rib e the l ocat i on of the ce l l for su b channe l s (I C number of channe ls) and gri d s e cti ons (I S number of s e cti ons ) The equati on of moti on for an in let (. e q 1 ) rewrit ten. in fini te di fference form, and inte grat ed along the axi s yi e lds (See li g, Harri s, He r c h e n r od e r, in preparati on, ) g (h s h b ) i = 1 2 A i j I S - 1 Z M ' I I J H D i j where A s and A b are the in let cros s - s e ctional areas at the s ea and bay ends of the inlet, and w i j i s a weighting fun c ti on for di s tributing flow throughout the in let. The di s charge thr ough a g ri d ce l l i s equal to t h e wei ghting functi on of t h e c e l l, W i j, time s the tota l di s charge of the in let, Q. The Manning ' s fri cti on factor, ni j, i s de termined I9

23 He r c h e n r od e r (in preparati on, during calibrati on o f the mode l (s ee See li g, Harri s, and He r c h e n r od e r, in preparati on, For M in lets conne cting the b ay to the sea, the total di s charge for a l l in le ts, Q T i s M Q T = m 1 Qm The continui ty equation i s wri tten as Bay l eve ls and in let current ve loci ti es are de termined by s o lving the s imultane ous di fferenti al equati ons (7) and ( 9 ) us ing a Ru nge - K utta - Gi l l fourth order fini te - di fference te chnique in c onj un c ti on wi th ini tia l con di t i on s and the time hi s t ory o f water leve ls in the s e a Derivati on and. samp l e app li cati ons o f thi s mode l are given in See li g Harri s and,, To ob tain respons e characteri s ti cs s imi lar to Fi gure 2 for a spe ci fi c in l e t - bay sys tem, the mode l c an b e run b y as suming s inu s oi dal s eawater leve l fluctuat i ons wi th a typi ca l amp l i tude ; e. g., 3 cent imeters foot) at peri ods cover ing the anti cipate d range of lake osci l l ati on modes. Each run o f the mode l wi l l give predi c te d b ay leve ls and in let current ve l oci ti es for the wave peri od us ed Samp le mode l results for Pentwater. in let are shown in Fi gure 4 The s ea leve l predi cte d bay l eve l and.,, in l et ve lo ci ty are shown in the l ower part o f Fi gure 4 ; the importan ce o f e ach o f the terms in the equati on of mot ion, normal i zed by divi ding by the magni tude o f the lar ges t term at e ach time s tep, i s shown in the u ppe r part o f the fi gure. F or thi s condi ti on, the b a y l eve l f luctuati on i s l arger than the s ea leve l fluctuati on due to inerti a in the sys tem and the b ay leve l lags the s ea l eve l b y 84 (Fi g. P l otting resul t s from many r uns s imi l ar to Fi gure 4, but wi th many di ffer ent forcing peri ods, wi l l give the respons e characteri s ti cs of the in l et - b ay sys tem. Thes e curve s for P entwater (Fi g. 5) show that the He lmho l t z peri od wi th fri cti on, T3, i s hours, waves wi th pe r i ods o f 1 to 3 hour s wi l l be amp l i fie d b y the sys tem, and waves with a per i od of hours wi l l gen e r a t e the hi ghes t in l et curr ent ve loci ti es (3 - centimete r wave amp li tude as sume d). The effe ct of fr i cti on on T3 i s demons trated b y the dash l ine in Fi gure 2. The fr i cti on les s He lmho l t z peri od i s a ls o coinci den ta l ly hours (fr om eqs. 4 and The ca lcul ated bay amp li fi cati on and channe l ve lo ci ty in Fi gure 5 are for a Manning ' s n The nume ri ca l mode l usua l ly had to b e r u n for three or four cyc les for the b ay respons e to bui l d to equi lib r ium. In the prototype harbor i t i s l ike ly that equi libr ium (ful l amp li fi cati on) i s never ful ly achi eve d. Thus, the ca lib rati on curve in Fi gur e 5 for ms the upper enve lope of measur ed prototype data.

24 F ric hon He ad Dif f e re nc e T em p. Ac c ele rat io n Time (hr) Figure 4. Pentwater respons e to s inus oi dal wave in Lake Mi chi gan (peri od hours ; amp li tude foot). 2 I

25 u ou oou udm v p u o ( s m) ( gr o om u mu n xow

26

27

28

29 Le g end 0 level rec order level rec orde r 9 67 A ve loc lt y m e te r Pe ntwater Pen twater Lake, Mi c h. Portage Lak e, Mich. Portage Figure 8. Dat a co l lec tion si tes on Lake M i c hi gan.

30 mi Ludington, Mi ch. Lu din gton Wh 1 t e Lake, Mi ch. Fi gu r e 8. D a ta co l l ection s i tes on Lake Mi chi gan, - Continue d. 27

31 Muske gon, Mi c h Mu ske q o n Hol l and, Mi ch. Fi gure 8. Data co l l e cti on si tes on Lake Mi chi gan - Cont inued. 2 8

32 Du luth - Superi or Wi s. e r Le ve l R e c o rd e r c ul o n L/f f/e L Litt l e Lake, Mi c h. Du lu th Shlp Channel (I nlet 2) ale r Le ve l Re c o rder oc ahon SU DGI IOI nle t E nt r y I 525 m Fi gure 9. Data co l le cti on s i tes on Lake Su per i or. 2 9

33 Lake Erie Wate r Level Rec order Loc ahon I, OOO m Fi gure 1 0. Data c o l l e cti on s i te on Lake Eri e, Pre sque I s l e, P ennsy lvania. 3 0

34

35 T abl e 1 S u mmar y of f i e l d me asu re me n t s. Lake c hi g e n Pe n t wa t e r Lake Su p e ri or la ke Eri e 1 Na t i on al Oc e anic and At mos phe ri c Admi n i strat i on, Lake Su rve y Ce nt er. 2 A S - n m u t e samol i n g in t e rval. 3 Co rn s of E n e m e e r s re c ord s. Thi s st u dv, f all A 2 - n l nu t e sampl i ng i n t e rval. 3 2

36

37 Sma l l - amp l i tude s ea leve l fluctuati ons wi th a peri od o f approximate ly the He lmho l t z peri od of the in l et - b ay sys tem may generate re l at ive ly hi gh in l et water ve l oci ti es as shown in numeri ca l mode l ca l cul ati ons ( s ee S e c. There fore, a water l eve l re cording sys tem mus t be c are ful ly des i gned for e ach locati on to measure sma l l amp li tude but p ot enti al ly, important l ong waves At the s ame time thi s re cording sys tem shoul d., e liminate any short - peri od l arge - amp l i tude noi s e wind wave s, ) that may mask the l ong waves in the record For examp le at Pentwater re.,, c ords should me asure the low - amp li tude waves wi th a peri od of 1 hour or l onger, a n d shoul d exc lude wind wave s and other noi s e wi th peri ods of 1 minute or l es s. One me thod of des i gning a s ti l l ing we l l t o meet thes e requi rements i s to us e the line ar damping we l l des i gn (Noye, Thi s s ti l l ing we l l cons i s ts o f a verti cal cy linder wi th a s ea le d bottom and open to the l ake thr ough a long, thin tub e. Fri ction in the tube and the re l ative c ros s - s e cti ona l are as of the tube and s ti l ling we l l caus e the sys tem to re spond dire ct ly to l ong waves outs i de the we l l and to dras ti cal ly dampen the short - per i od noi s e. Des i gn of s ti l l ing we l l s i s di s cus s ed by Se e li g Fi sher Porter s eri es di gi tal float - type water leve l r e corders wi th a verti c a l res o lut i on of 3 mi l l ime ters foot) and s amp ling interva l s of 2 or 5 minutes wer e us e d t o me asure water l eve ls in the s ti l ling we l l s Data were co l l e cted on pun ched tape Water leve l s we re.. measured for s everal months at e ach l ocati on (s ee Tab le In le t ve lo ci ti es at P entwater were measur ed during and us ing a Bendix current meter suspended by a cab l e approximate ly mi dway a l ong the channe l, me ters ( 1 5 feet) fromthe north wa l l at mi d depth. Ve loci ty data were re corded on a s trip chart and later di gi ti zed for analys i s at the s ame time interva l as the water leve l data. 3. Data Re ducti on and Ana l y s i s Te chni q ues. Ini ti a l ly, the He lmho l t z peri od of e ach in let - b ay sys tem and the free s e i c h i n g mode p e r i od s. we r e c a l culate d for the lak es and bays surveyed in thi s s tudy ( s ee procedures in S e c. I I ). Thes e ca l culati ons, in c on j u n c t i on wi th a survey of pub l i she d data on Great Lake s res onance charact er i st i c s, gave an indi cati on o f the peri od and magni tude of imp ortant long waves that could b e expe cted at e ach l ocati on. The in formati on was us ed in the des i gn o f water leve l me as urement equipment (di s cus s ed in previ ous s e ction). When the fi e l d data co l l e ct i on pr ogram was comp leted, the di gi t al punched - t ape water l eve l r e cords wer e me chani ca l ly conve r ted to p u n c h c a r d s for computer ana lys i s. The fi rs t procedure for s tudying thes e dat a i n cluded p lotting the re cords for vi sua l inspe cti on. Then, a fas t F ouri er 3 4

38 trans form and c os ine be l l fu ncti on were us ed to ob tain a spe c tra l ana lys is of each record (Harri s, A re c ord l ength of point s was used in thes e ana lys es to obtain detai l ed spe c tra l l ine res o lution. Spectra l ana lys i s indi cated the peri od and amp li tude of l ong waves o f intere st to 5 hours) in t h e re cord. Thi s an alys i s i s n e c es sary becaus e s evera l long waves are genera l ly s imu l taneous ly pres ent and the superpos i t ion o f the waves give s the impres s i on o f a c onfus ed re c ord. Examp les o f spe c tra for P entwat er bay water leve ls re c orded during 5 to 1 4 May are shown in Fi gure 1 2. I f wat er leve l re cords were of good qua li ty, leve l s measured in the bay were then us ed t o predi ct in l et water ve loci ti e s us ing the fini te di fference continui ty equation The res o lut ion of the continuity equat i on should b e checked to j udge the us efulnes s of the predi cted ve loci tie s ; e. g at P entwater the l eve l.,, re corder has a verti cal reso luti on o f 3 mi l l imeters the samp l ing int erva l, is 5 minutes ( 300 s econds ) and the rat io, A b a y / A s o the ve lo city c predi cti on reso luti on, V bas ed on equat i on i s r V " foot or centimeters per s econd ( 1 0 ) Thus, the ve loci ty wi l l b e expres s ed as mu lt ip l es of centimeters foot ) per s e cond whi ch may be adequate for many purpos es. For examp l e, i f A b a y / A c was 1 0 5, then wi th the given verti ca l re so luti on, ve loci ti es could on ly b e expres s ed as multip l es of 1 00 centime ters (3 feet) per s e cond, whi ch i s inadequate for mos t purpos es. At Pentwater, the me asured ve l oci ti es in the in let were di gi ti zed at a samp ling r ate of 5 minutes s o that a dire ct compar i son coul d b e made o f measured and predi cte d ( e q 6) ve l oci ti es Cumu lative frequency.. di s tributi ons of measur ed and predi cted ( e q 6) in let ve lo ci ti e s a r e. shown in F i gure 1 3 The. 2 months o f record in show that ve loci ti es predi cted by continui ty are s li ght ly hi gher than measur ed ve l oci ti es, but adequate for many des i gn purpos es. I V. RESULTS 1. S e i c h i n g o f the Great L akes. Free modes of os ci l lati on o f the Great Lakes have been i denti fi ed by spe ctra l ana lys i s of water leve l re cords in thi s s tudy and others (Mortimer, ; Mortimer and Fee, ; Hamb lin, ; Rao and Schwab, and have been pre di cted us ing numer i ca l tec hniques (Rockwe l l, ; Mortime r, ; Bi rchfi e ld and Murty, ; Rao and S chwab, Tab l e 3 summari zes the known mode s o f os ci l l ati on of Lakes Mi chi gan, Superi or, E r i e, an d On t ari o. 3 5

39 Pe rnod (hr) Fi gure 1 2. Samp le spe ctra o f Pentwater bay water leve ls (May ) 3 6

40 Measu red Pred ic ted (aa. 6) l nle l c hanne l ve loc rr y (f t/s) 1 3. Measur ed and predi ct ed in l et ve loci ty cumulative fr equency di s tribut i ons at Pentwate r, Mi c hi gan (Octob er - Novemb er 3 7

41 Tab l e 3. Modes of os c i l lati on of the Gre at Lake s l. 1 Ob s erved, computed by Fee ( 1 968) program and Rockwe l l (1 966) d a t a, an d compi led from many sources. 2 L l ongi tudina l, T transvers e ; other ob s erved peri ods, mode unknown. TWO - dimens i ona l modes ar e not cons i dered. 3 Va lue unknown. 3 8

42

43 3 2 Wave Period (hr) Fi gure 1 4. P entwater respons e to long wave forcing (n 4 0

44 Freeman Hamblin and Time (hr ) Fi gu r e 1 5. Tor onto r e s p o n s e t o wave forcing.

45 Va lue s o f n were es timated for other Great Lakes inlets bas ed on, experi ence at Pentwater and Toronto. Th e n, t h e fri ct ion les s He lmho lt z peri od was e s tima ted for ea c h in l et - bay sys tem us ing e qpa t i o n s (4 ) and and the numeri c al mode l meri a l mode l and fri ctionl es s. Nu c He lmho l t z peri od resul t s are s u mmari z ed in Tab l e 5. The amp l i tude respons e curves and pre di cted maximum ve l oci ti es are shown for s e l e cted in lets on Lake Mi chi gan ( Fi g. Lake Superi or (Fi g. and Lake s Eri e and Ontari o ( Fi g. A 3 - cent imeter mono chromati c forcing wave amp l i tude wa s us ed in thes e mode ls. For waves o f di fferent amp li tudes, t h e maximum in l et ve lo ci ty i s approximat e ly proporti ona l to amp li tude. Water leve l changes throughout the forcing cyc le caus e non l ine ar e ffe cts ab /ao i s s li ght ly di fferent at hi gh and l ow water), s o that the mean o f ebb and fl ood condi ti ons i s us e d in the respons e curves in thi s s tudy. Thi s analys i s shows that al l o f the j etti ed in let sys tems s tudi e d have s i gni fi cant inerti al e ffe cts becaus e long waves at or near the He lmho l t z peri od o f each sys tem have higher amp li tudes in the bay than in the Great Lakes. The in l et - bay sys tems mode led have a wi de vari ati on in respons e characteri s ti cs from on e sys tem to another b ecaus e of the comp li cated interacti ons b etween the four terms in the equati on of moti on o f the in l et and the respons e o f the b ay to the in let. Pentwat er, for examp l e, has a moderate amount o f wave amp l i fi cat ion and produce s in let ve lo ci ti es greater than 3 0 centimeters p e r s e c ond (1 foot per s econd) for forcing waves o f 3 - centimet e r amp l i tude and per i ods ranging from to hours (Fi g. Whi te Lake has les s wave amp l i fi cati on, but the interact ion b etween the in l et and b ay produc es hi gher ve loci ti es over a wi der range o f forcing peri ods (greater than 3 0 cent imeters p e r s econd for per i ods of 1 to h ou r s) ( F i g. Since Li tt l e Lake and Pre sque I s le have the capaci ty to gener ate r eve r s ing cur r ents in on ly a narrow window of forcing p eri ods, i t i s un l ike ly that s i gni fi cant revers ing cur rents wi l l be frequent ly generat ed (F i gs. 1 7 and Duluth - Superior has the hi ghest capaci ty for generat ing revers ing currents for a given wave amp l i tude wi th maximum ve loci ties oc c urring at a theoret i ca l for c ing peri od of hours. The mean ve loci t ie s in Duluth (in l et are approximate ly t imes l arger than in Superi or (in let 1 ) (F ig A unique featur e o f the Duluth - Superior system i s that the. mode l predicts a net ' f l OW i n t O the harbor through the Duluth in let and a net out fl ow through the Superior in let when the forcing period i s near 1 hour (Tab l e Thi s asymmetry in fl ow throughout the forcing cycl e wi l l generate a sma l l count erc lockwi s e net fl ow throughout the inl et - bay system at Duluth - Superi or. North Pond in, had two short nat, ur al in lets conne cting a re l ative ly large b ay to Lake Ontario North Pond does not amp l i fy long. waves becaus e the mas s of water in the in lets i s sma l l compar ed to bay s i ze, and fri cti on in the in lets i s hi gh due to the sha l low - water depths (Fi g. Since fri ct ion i s hi gh, North Pond b ehaves like a tradi ti onal ti da l in let wi th a b al ance b etween he ad and fr i ct i on in the inl ets. 4 2

46

47 Wa ve Pe roid (hr ) Wove Pe r iod (h r ) Fi gure 1 6. Predi cted r e spons e o f in le t - b a y sys tems on Lake Mi chi gan to mono chromati c forcing ( a o foot). 4 4

48 Wave Pe r i od (hr ) Wave Pe r i od (hr ) Fi gur e 1 7. Predi cted respons e of in let - bay sys tems on Lake Super i or to monochromati c forcing ( a o 4 5 foot).

49 Wav e Pe r i od (hr ) Wav e Pe r i od (hr ) Fi gure 1 8. P r edi cte d r espons e o f in let - bay systems on Lakes Eri e and Onta r i o to monochromati c forcing ( a o foot). 4 6

50

51 Tab le 7. Numeri cal model predi cti on of Pentwater re spons e to Lake Mi chi gan modes o f os ci l lati on. Mode Notes - L l ongi tudina l modes of os ci l l ati on o f Lake Mi chi gan. T transvers e mode s of o sci l l ation of Lake Mi chi an g. obs erved os c i l l a t i on, «mod e unknown. x wave has ' a node near Pentwat er. Modes 6L to 9 L have modes o f os c i l lat ion wi th large and V f or Pentwate r. 4 8

52 I.4 I.O Pe r i od (hr ) I. 4 3 I. 78 I3 2. F r e q u e n c y (c / hr ) Pe nt wat e r La ke Mi c h i g a n Time, 4 hr Fi g ure Sam p l e Pen twater and Lake Mi chi g an water leve ls and spe ctra. 4 9

53 Pe r iod (hr ) L4 l I. 78 F r e qu e n c y (c / hr ) Lake Mic h i g a n Pe ntwate r l_ 0 ke T ime, 4 hr Fi gure 20. Samp l e Pentwater and Lake Mi chi gan water leve l s and spe ct ra. 5 0

54 Pe r io d (hr ) L4 I.O 0 38 l.0 8 l.4 3 I.7S 2 l3 3. l9 F r e q u e n c y (c / h r ) Lake Mic hi g a n Pe ntwate r Lake Time, 4 h r Fi gure 21. S amp le Pentwater an d Mi chi gan water leve l s an d spe ctra. 5 I

55 with an ampl i fi cation fac tor o f as predi c t ed by the numeri c al mode l, b e c au s e the wave peri od i s muc h l onger than the Pentwater He lmho l t z peri od of hours However the r wave whi - hou.,, c h reac hes hei ght s of 1 5 c ent imet ers foot ) has a negl i gib l e e ffect on the harbor b ecaus e it, i s mu c h shorter than the He lmho l t z period o f hours Waves of. and hours are s l i ght ly ampl i fi ed by the harbor (shown by the spectra l ana lys i s in Fi g. b u t thes e waves are di fficul t to di s tingui sh in th e re c ord b e c s e of the mixing o f individual wave component s au The s torm event in Fi gure 20 shows that a di fferent s et o f modes o f os ci l lati on of Lake Mi chi gan i s pres ent The - hou. r wave i s the hi ghes t i s amp li fi ed the mos t and prob,, ab ly generates the hi ghes t per cent age of s i gni fi cant revers ing in l et cur rent ve lo ci ties. A -hour peri od wave i s a l s o pre s ent and i s amp l i fi ed Waves sho. r ter than 1 hour are damped by the harb or. An unusua l water leve l fluctuati on at Pentwater where on ly the hour wave i s dominant in Lake Mi chi gan, i s shown in Fi gure 21. As predi cted previ ous ly, the and - peri od wave s whi ch are the s ixth and ei ghth longi tudinal modes of os ci l l ati on of Lake Mi chi gan, caus e the hi ghes t current ve l oci ti es. Fi gure 22 shows the wi de variat ion of water leve l f luctuat i ons occurring i n three di ffe r en t harbor s a long the eas te r n shore of Lake Mi chi gan at the same time ( Pentwater and Ludington a r e on ly ki l omete r s (1 1 mi les ) apart) The r eas ons f ar the di ffer ences ar e that the forcing. waves outs ide each l ocati on are di fferent as a result of the node - anti node pattern of s e i c h i n g in Lake Mi chi gan (see S e c. I I ) and b e c ause each harbor responds di ffe r ent ly to the for cing that i s pres ent (s ee Sec. IV) ; e. g., the and - hour waves in Pentwater and Ludington a r e not noti ceab le in Muskegon harb or whi ch has a He lmho lt z pe r i od o f 5 hours. The forcing of harbor s on the other Great Lakes wi l l b e comp lete ly di ffer ent b e caus e the sys tem of se i c h i n g vari es fr om lake to lake ; e. g., on Lake Super i o r, wave peri ods of and hours o ccur in Li tt le Lake Harb o r (Fi g. Shorte r pe r i od waves may al so o ccur in Lake S u per i or, but are not obs er ve d in the har b or b e cause the harb or dampens wave s shor ter than app r oximate ly hour. The and - hour waves on Lake Supe r i or (the 7t h, l ot h, an d 1 1 t h longi tudina l modes of os ci l l at i on) wer e ob s er ved to cause hi gh r e vers ing currents and as s oci ated navi gati on p r ob lems at Duluth - Su per i or ; e. g., on 1 0 June , a - hour wave wi th a height of ap p r oximate ly 30 centime te r s ( 1 foot ) in the har b or, in conj unct ion wi th smal l hour per i od waves, affe cted Duluth - Su per i or. Ve lo ci ties as hi gh as 200 centime ters pe r s e cond feet p e r s e cond) were gener ate d in Duluth in le t and centimeters p e r s e cond fe et per s e cond) in Super i or (Fi g. Hi gh ve lo ci ti es are gener ated in thes e in lets becaus e of the l arge f b r c i n g waves in Lake Super i or at thi s lo cati on whi ch have per i ods 5 2

56

57 28 Au g I 975 O bse rve d Wate r Le ve ls (0 ) T y p i c al F lu c tu ations I9 Ju l y l975 O bse rve d Wate r Le ve ls Wave P e riod (hr ) Com p u te d S p e c tra (b) E xtre me E ve nt Fi gure 23. Wate r leve l fluctuati ons in Li tt le Lake Harbor. 5 4

58

59 near the harbor He lmho l t z period. Forcing wave s are large becaus e the harbor is located on the converg ing end of the l ake, which wi l l a lways be an ant inode o f longi tudina l o sci l lat ions. Maximum water ve locities obs erved in other in l e ts are much l ower than in Duluth - Superi or ; e. g., at P entwat er, a l l measurement s and p r e di c ti ons show t h a t. v e l o c i t i e s are l es s than 60 cent imet ers p e r s econd ( 2 fe et per s econd ) for percent o f the t ime (F ig. Predicted in let ve loci ti es for other l ocat ions show that Portage, Ludington, and Pentwater have s imi lar ve loc ity di stributions ; Presque I s l e, Muskegon, and Li tt l e Lake have s t i l l lower ve l ocit ies (Fi g. V. IN LET DESIGN Great Lakes in let des i gn prob l ems genera l ly fal l into one of two c las s es : (a) a pond or lake to be connect ed to one of the Great Lakes by a new channe l and (b) an exi sting in l et channe l to b e modi fi ed,. The concepts and te c hniques deve loped in thi s study can b e us ed to ai d the des i gn o f an in let in e ither cla s s. An examp le app l i cati on for each clas s is given be low. 1. New In let Channe l. The procedure s f ar ana lys i s o f a new channe l t hat i s to connect a lake to one o f the Great Lakes are : (a) de termine the approximat e in l et dimens ions (l ength, width, and depth) b ased on phys ical l imitat ions such a s the des ired navi gab le depth and wi dth and the di stance b etween the lake and Great Lakes ; ( b) es t imate a Manning ' s n for the propos ed channe l (s ee S e c IV., 2 for typi ca l va lues o f n), ; (c) us e the nume ri cal mode l to obtain monochromati c re spons e char acteri stics o f the harbor for the range o f expe cted l ake se i c h i n g peri ods and a typi cal amp litude ; (d) c ompare the resu lts to thos e o f other near by harb ors ; and (e) app ly the numeri ca l mode l to predi ct in l et ve locities, di s char ge, and bay l eve l s for the peri od o f re c ord (i f Great Lake s water l eve l fluctuat ion re cor ds are avai lab le in the vi c i n i t y of proposed s i te). For examp le, suppos e an inl et i s to be de si gned to connect Crysta l Lake to Lake Mi chi gan (Fi g. Cryst al Lake, l ocat ed on the eas tern shore o f Lake Mi chi gan 35 ki l ometers ( 22 mi l es ) north o f Portage Lake, has a bay sur face area o f x squar e me ters x squa r e feet). The in let at thi s s i te woul d b e approximat e ly me ters feet) long. As sume that the in let wou l d be 6 1 meters ( 200 fe et ) wi de and meters ( 1 8 feet ) deep. S ince the in l et i s s imi lar to P ent water (s ee Tab le the Manning ' s n for thi s channe l i s es t imat ed to be The numeri ca l mode l was run for Crys t al Lake using Lake Mi chi gan s ei che peri ods of and hours wi th an amp l i tude o f 3 cent imeters. The predi ct ed respons e characteri s ti cs of thi s in l et - bay sys tem are shown in F igure 28. The mode l predi cts that 5 6

60 I nlet c hannel ve loc it y P entwate r in l e t cumulative frequen c y ve loci ty di s tributi ons ( 1 967, , ) 5 7

61 I n le t Ch a n n e l Ve loc it y (f t/ s) Fi gure 26. In l et ve lo c i ty cumulative frequency di s tributi ons (b as ed on equati on 6 and bay water leve l re cords ) 5 8

62 Cr ystal Lake Portage Mu ske g on ( 000 Fi gure 27. Cr ys ta l Lake, Mi chi gan. 5 9

63 V max(a c : 0 I f t) Wave Pe rnod (hr ) Fi gure 28. Pre di cted re spons e c haracteri s ti cs of an in let Crys t al Lake ( L feet, A square feet, D 1 8 feet, B 200 feet). 60

64 the wave ampl it u de wi l l be smal l er in the bay than in Lake Mi c higan, pri mari ly b e c aus e the bay surfac e area i s mu c h larger than the inl et des i gn c ros s - s e c tional area (a ratio o f approxima t e ly The mode l a lso predi ct s that monochromati c s ei c hes wi th an amp l itude o f 3 centimeters wi l l generate maximum ve loci ti es of 4 3 cent imeters per s e cond feet per s e cond) for wave peri ods of 4 to 9 hours (Fi g. Since maximum ve l oci ties de creas e for waves shorter than 4 hour s, a wave wi th a 1 - hour peri od produces ins i gni fi cant in le t ve lo ci ti e s. The fi rs t three mode s o f os ci l lati on o f Lake Mi chigan and - hour waves ) wi l l generate the highest ve loci ties for the Crys tal Lake in let de s i gn. Por t age i s lo cated near Crystal Lake ; there for e, forcing amp l i tudes o f the first three modes of os ci l lat ion of Lake Mi chi gan wi l l b e s imi lar at both l ocati ons (Fi g. The predi cted ve l g c i t i e s for a given wave are di fferent at Portage and Crys t al Lake (Fi g. 29) due to di fferences in in let an d bay ge ometry. However, Portage water l eve l data can b e us ed to es t imate Crys t al Lake in let ve loci ti es. To pr edi ct inlet ve lo ci ti es at Crys t al Lake us ing Portage dat a, the measur ed Portage b ay leve l fluctuati ons mus t fi rs t b e adj us ted to es ti mat e the nearby Lake Mi chi gan wave amp litudes. Thes e amp li tudes are then us ed to predi ct ve lo ci ti es at Crys ta l Lake ; e. g a measured Port age bay., leve l f luctuat i on has a peri od of hours and amp li tude of foot centimeters ). Thi s wave was amp l i fi ed by a factor of by Portage harb or (Fi g s o the Lake Mi chi gan wave amp li tude was. foot -. A foot wave amp l i tude in Lake Mi chi gan produc es a maximu m ve loci ty of fee t p e r s e cond at Crys ta l Lake in l et (Fi g there. fore the, O. I Z - foot wave produces feet p e r s e cond maxi mu m ve l oci ty Thi s procedure cou ld b e fo l l owed for other s ei che modes. to es timat e the maximum ve loci ti es expected at Cr yst al Lake. I f a comp le te analys i s of in let ve l oci ti es i s required, water leve l s shoul d b e measured in Lake Mi chi gan adj acent t o Crys t al Lake for at l eas t S evera l months. Thes e leve l s can b e us ed as the forcing fu nct ion in the numeri ca l mode l to produce a predi cted time hi s tory of inlet ve loci ti es, di s charge, and bay l eve ls for the per i od of record. 2. In let Channe l Modi fi cati on. Pr ocedur es for inves ti gating the e ff e ct o f a modi f i cat i on to an in l et are : (a) Dete r mine the geometr y of the pres ent sys tem and obt ain proto type hydraul i c data concurr ent bay leve ls Great Lakes leve ls,, an d in l et ve loci ti es ) ; (b ) cal ibrat e the numer i ca l mode l ; ( c) obt ain monochr omati c r espons e char acteri s ti cs of the in l et - b a y sys tem, (d) modi fy the mode l geometry t o re fl ect the p r opos ed in let change and p r e di ct the re spons e cha r acteri s t i cs of the new condi ti on ; and (e) us e the water leve l re cor ds in the Great Lakes to for ce the mode l to pr oduce a time hi s tory of in let ve loci ties, di s char ge, and bay leve ls for the propos ed des ign. 6 I