Lecture 1: Introduction to ocean tides

1 Lecture1: Introductionto oceantidesMyrlHendershott1 IntroductionThephenomenonof oceanictideshasbeenobserved andstudiedby humanity localizedtidalpredictionandin thegeneralunderstandingof tidalpropagationin oceanbasinsledtothebeliefthatthiswasa wellunderstood phenomenonandnolongerof interestforscienti ,recent decadeshave seena renewalof interestforthissubjectby thescienti ccommunity. Thegoalis now tounderstandthedissipationof tidalenergyin donein theseventiessuggestedthatratherthanbeing mostlydissipatedoncontinentalshelves andshallow seas,tidalenergycouldexcitefartravelingi nternalwaves in ,topographicfeaturesor withotherwaves,thesecouldtransferenergyt o rstlectureis introductoryandits aimis to reviewthetidalgeneratingmechanismsandto arrive at a TideGeneratingForcesTidaloscillationsare theresponseof theoceanandtheEarthto theirmovement relative to theEarth,thisgravitationalpullchangesin time,andbecauseof the nitesizeof theEarth,it alsovariesin spaceover ,thetemporalresponseof theoceanis verylinear.

2 Allowingtidalrecordsto be interpretedas thesuperpositionofperiodiccomponents withfrequenciesassociatedwiththemovement s of in uencedby thepresenceof continents andbottomtopography, andis a showsa two month tidalrecordfromPortAdelaide, cantlyfordi erent coastallocations,thisonein particularcanbeconsideredtypicalin thatit clearlyshowscharacteristicsof tidaloscillationsthatcanbedirectlyrelate dto rstfeatureto standoutis thesemi-diurnalcomponent, two hightidescanbe seento occuroneach day. A closerlookrevealsa modulationof theamplitudeof thesemi-diurnaloscillation,roughlyover a onemonth period. Intervalsof highamplitudeareknownas springtideswhilethoseof lower amplitudesareknownas the gure,thesprings-neapscycleis associatedwiththephasesof a sameday, thereis oftena di erencein theamplitudeof thetwo knownas the1 Figure1: Two monthsof canseein thisrecordsomefeaturesthatcanbe directlyaccountedforby thedetailsof theastronomicaltidalforc-ing,such as thesprings-neapscycle,thedailyinequality andtheabsenceof dailyinequalitywhentheMoonis \dailyinequality"and,as indicatedin the gure,disappearswhentheMoonis over willtryto clarifybelow willalsotryto explaintheoriginof theforcingtermsresponsibleforcausingthet idalrecordfeaturesdescribedabove.

3 Finallywe willattemptto explainthederivationof TidalForcingEventhoughsmallcomparedto theplanetsandespeciallyto thesun,theMoonis by farthecelestialbodyclosestto ,theMoon exertsthebiggestin uenceover theEarth,contributingthemostto theformationof willbeginby consideringitse massof theEarthandtheMoon orbitaroundthecommoncentreof massof theEarth-Moon aresuch thatcentrifugalforcecounterbalancesgravi tationalattractionat theindividualcentresof a rigidbody, soeverymaterialpoint in it executesanidenticalorbit,andis thereforesubjectto thesamecentrifugalforce,as illustratedin willvarybecausethedistancebetweenthesepo ints to theMoonmay varyby upto centrifugalforceonthehemisphereclosestto theMoonandcentrifugalforcewillprevail on thehemispherefurthestto netforcesin oppositedirections,causingtheoceanto guration,two.

4 Thecentreof theEarth,shownas a lleddot,rotatesaboutthecentreof massoftheEarth-Moon system,indicatedby a `x' theorbitalmovementof thepoints shownby a `o'markonEarth'ssurfaceandthecenterof commonsourceof confusionregardingtheargument above is to supposethatthecentrifugalforcerelevant to theproblemis dueto thespinningof rstsight becausethisforceis constant foreverylatitudecircle,allowingforanimba lancewithlunarattraction,which is longitudedependent atany giveninstant. However,thecentrifugalforcedueto theEarth'sspinhaspermanentlydeformedtheE arth'ssurfaceinto a spheroid(asopposedto thesphericalshape thatwouldensuefromself-gravitationonly). Thatis to say, thiscentrifugalforceis compensatedby3theEarth'sowngravitational mnemonicphraseto keepin mindis thattidesarecausedby theactionofothercelestialbodiesover thattidesarepartlytheresultof thevariationof centrifugalforceover thesurfaceof confusionarisesbecausethedistancebetween thecentreof theEarth-Moonsystemis smallerthanoneEarthradiusandthereforethi spoint lies\within" thiswerea xedmaterialpoint, like thecentreof theEarth,aroundwhich theplanetrevolved,therewouldindeedbe a variationof ,thecentreof massoftheEarth-Moonsystemis justa point in space,andas theEarthrevolves aroundit,asindicatedin gure2 noneof itsmaterialpoints are constant eld.

5 Thecentrifugalforcedueto therevolutionof theEartharoundthesystem'scentreof massis essentialto thesemi-diurnality of canillustratethisby consideringthesituationin which thecentreof theEarthis thiscasetheonlyforceactinguponit is ,it pullseverypoint onEarthtowardstheMoon,causingthewaterto bulgeonthehemispherecloserto theEarthwouldthereforemake everypoint onit experiencea diurnaltidecycle,insteadof a illustratedin gure3h MmFigure3: Tidaldeformationthatwouldensueif fromlunarattractionif theEarth'scentreof masswere thattheelevationwouldbe such thatnetterrestrialgravitationalforcewoul dexactlycompensateforthelunarforceat 'sgravitationalforce:GM(a+h)2 GMa2' GMa22ha:NetEarth'sgravitationalforce:GMa 2(ar)2mM;whereMis themassof theEarth,ris thedistancebetweenEarth'sandMoon'scentre sofmass,ais theEarth'sradius,h is theseasurfacedeformationat thesublunarpoint andGis theuniversalgravitationalconstant.

6 Equatingthetwo forcesandisolatinghwe get:h=a32r2mM= 10:7m:(1)4 Thisis we now allow theEarth'scentreof massto accelerate,thecentrifugalforceduetothism otionwillcompensatefortheMoon'sgravity at thatpoint. We cananticipatethattidaldeformationwillbe smaller,sinceit willbe a responseto a smallerresultant ,predominanceof lunarattractiononthehemispherefacingtheM oonandof thecentrifugalforceontheoneopposingdefor msthesurfaceinto thiscasewe (r a)2 Gmr2'GMr22ar;(2)wherethe rsttermontheright is lunargravity, thesecondis thecentrifugalforceandthetermontheright is thenetgravitationalforceof get:h=a4r3mM= 35:8cm(3)TheMoonrotatesaroundtheEarthin thesamedirectionas theEarthspins, takes slightlylongerthana day fortheMoontobe directlyover thesamepoint ontheEarth'ssurface,as illustratedin gure4 thisis calleda lunarday.

7 Likewise,theperiod betweentwo hightidesis halflunarday. Apartfromthesemi-diurnaltides,we canexpectthepresenceof theMoonto anorderzeroe ectcalledthepermanent : theMoonwithrespectto a xedpointontheEarth'ssurfaceafteronerevol utionis now,we have consideredtheorbitof is however elliptic,withtheEarth-Mooncentreof massbeingoneof 'sgravitationalforceover theEarthwillbe modulatedover theperiod of oneanomalisticmonth ( gure5),aswillthetidalcomponents it summary, tidalforcingby theMoonalonecanbe representedby thefollowinghar-monics:LunarSemi-Diurnal Tide(M2) 2= (N2)2=LD 1= (Mm)1= (anomalisticmonth)The2=LD+ 1=perigeetermwas leftoutbecauseit hasa theamplitudeofM2is representedby theinteractionofM2andN2, which is constructiveonceeach.

8 AstheMoonrotatesaroundtheEarth, ,aftereach orbitalperiod (anomalisticmonth),theMoonis in di erent positionin itsorbitwithrespectto theabove picture,we represent ontopaninitialpositionof theEarthandMoon,below it Moonin thiscaserepresents thepositionit wouldhave to be in to exhibitthesamephaseas in theinitialcon mentionedabove arevalidforany othercelestialbodywhich might bereasonablyconsideredto forma two bodysystemwiththeEarth,forwhich bodyis thesun,whosetidale ectover theEarth(whenthetwobodysystemis consideredin isolation)canbe reducedto theharmoniccomponents (S2)2= (N2)2=SD 1=anom: ()1= (anomalisticyear)WhenMoon andSunarealignedwiththeEarth,theirsemi-d iurnalcomponents interfereconstructively, givingriseto tidesof largeramplitude,knownas quadrature,theinterferenceis exactlydestructive, givingriseto smalleramplitudetidalvariations, arrangement of theEarthSunandMoonis perceived ontheEarthas thephasesof theMoon,andthereforethesprings-neapscycl ehasa period of onelunar(synodic) lunarmonthis thedurationrequiredfortheMoonto returnto a xedpositionin itsorbitin relationto theSunas illustratedin now we have assumedthattheorbitsof theEarth(aroundtheSun)

9 AndMoon arecoplanarto thespinningof theEarthat all instants of reality theseplanesintersectat ectthishasover thetideis illustratedin gure6. AstheEarthrotates,6it perceives thetidalsurfaceas being\tilted"in relationto termsof harmonics,thisis representedby a dailycomponent, which gives riseto thedailyinequality. Whenthetidegeneratingbodiesintersectthee quatorialplane,thedailyinequality thePortAdelaiderecord( gure1),we canseethatdailyinequality disappearswhentheMoonis : TheMoon'sorbitis in general\tilted"withrespectto theEquatorgivingriseto thedeclinationalcomponents dependsontheangleof thebodiesorbitto , 'sorbitprecessesover ,itsangleto theEclipticvaryingbetween 5 080and5 080.

10 In relationto theEarth'sequatorialplanethevariationis between23 270 5 080and23 270+ 5 ,theusefulnessof decomposingthetidegeneratingforceinto har-monicsis dueto thelinearity of theoceansresponseto it in fact,we have takenthisforgrantedin theprecedingsectionwhenwe explainedthesprings-neapscyclepurelyas theresultof theinterferenceof two determinetheimportant frequenciesin theirharmonicexpansions, usedonlyto determinetheamplitudesof localresponseto SpatialStructureof the TidesAstheEarth'sspinningunderthetidegen eratingpotentialis feltas thepropagationofthetidalwave. However,thispropagationis obstructedby thepresenceof continents andbottomtopography. Realco-tidallinesthereforelook nothinglike theconstant planewave entersa basin,it feelsthee ectof theEarth' thecasewithnorotationdegeneratesinto a nodalpoint, oftheoceanicbasinsandof bottomtopography disruptthepropagationanda precisemapof tidalpropagationcouldonlybe obtainedaftertheadvent of satellitealtimetry.