CHAPTER 9TIDESANDTIDALCURRENTSORIGINSOFTIDES900.Introductionother natural forces. Similarly, tidal currents are super-imposed upon non-tidal currents such as normal riverTides aretheperiodicmotion of thewaters of the seaflows,floods,freshets,etcdue to changes in the attractiveforces of themoon and sunupon the rotating earth. Tides can either help or hinder a902.CausesOfTidesmariner. A high tide may provide enough depth to clear abar,whilea lowtidemayprevent entering or leaving a har-The principal tidal forces are generated by the moonbor. Tidal current may help progress or hinder it, may setand sun.The moon is the main tide-generating body.Due totheshiptoward dangersor awayfromthem.Byunderstand-its greater distance,the sun's effect is only46percent oftheing tides, and by making intelligent use of predictionsmoon's.Observed tides will differ considerably from thepublished intide and tidal currenttables andofdescriptionstides predicted by equilibrium theory since size, depth, andin sailing directions,the navigator can plan an expeditiousconfigurationofthebasin or waterway,friction,landmass-and safepassagees, inertia of water masses, Coriolis acceleration, and otherfactors are neglected in this theory.Nevertheless, equilibri-901.TideAndCurrentum theory is sufficient to describe the magnitude anddistribution of the main tide-generating forces across theThe rise and fall of tide is accompanied byhorizon-surface of the earth.tal movement of the water called tidal current.It isNewton'suniversal lawofgravitationgoverns boththenecessary to distinguish clearly between tide and tidalorbits of celestial bodies and the tide-generating forcescurrent, for the relation between them is complex andwhich occur on them.The force of gravitational attractionvariable.For the sake of claritymariners have adoptedbetween any two masses, m and m2, isgiven by:thefollowingdefinitions:Tide is the verticalriseandfallof thewater,and tidal current isthehorizontal flow.TheGm,m2F=tide rises and falls, the tidal current floods and ebbs.Thedonavigatorisconcernedwiththeamount andtimeof thetide,as it affects access to shallow ports.The navigatoris concerned with the time, speed, and direction of thewhered is the distancebetween thetwomasses, and Gistidal current, as it will affect his ship's position, speed,a constant which depends upon the units employed. Thisand courselawassumesthatm,andm2arepointmasses.NewtonwasTides are superimposed on nontidal rising and fall-ableto showthat homogeneousspherescouldbetreateding water levels, caused by weather, seismic events, oras point masses when determining their orbitsTOMOONEARTH-MOONBARYCENTERCENTER'OFMASSOFEARTHFigure902a.Earth-moonbarycenter143
143 CHAPTER 9 TIDES AND TIDAL CURRENTS ORIGINS OF TIDES 900. Introduction Tides are the periodic motion of the waters of the sea due to changes in the attractive forces of the moon and sun upon the rotating earth. Tides can either help or hinder a mariner. A high tide may provide enough depth to clear a bar, while a low tide may prevent entering or leaving a harbor. Tidal current may help progress or hinder it, may set the ship toward dangers or away from them. By understanding tides, and by making intelligent use of predictions published in tide and tidal current tables and of descriptions in sailing directions, the navigator can plan an expeditious and safe passage. 901. Tide And Current The rise and fall of tide is accompanied by horizontal movement of the water called tidal current. It is necessary to distinguish clearly between tide and tidal current, for the relation between them is complex and variable. For the sake of clarity mariners have adopted the following definitions: Tide is the vertical rise and fall of the water, and tidal current is the horizontal flow. The tide rises and falls, the tidal current floods and ebbs. The navigator is concerned with the amount and time of the tide, as it affects access to shallow ports. The navigator is concerned with the time, speed, and direction of the tidal current, as it will affect his ship’s position, speed, and course. Tides are superimposed on nontidal rising and falling water levels, caused by weather, seismic events, or other natural forces. Similarly, tidal currents are superimposed upon non-tidal currents such as normal river flows, floods, freshets, etc. 902. Causes Of Tides The principal tidal forces are generated by the moon and sun. The moon is the main tide-generating body. Due to its greater distance, the sun’s effect is only 46 percent of the moon’s. Observed tides will differ considerably from the tides predicted by equilibrium theory since size, depth, and configuration of the basin or waterway, friction, land masses, inertia of water masses, Coriolis acceleration, and other factors are neglected in this theory. Nevertheless, equilibrium theory is sufficient to describe the magnitude and distribution of the main tide-generating forces across the surface of the earth. Newton’s universal law of gravitation governs both the orbits of celestial bodies and the tide-generating forces which occur on them. The force of gravitational attraction between any two masses, m1 and m2, is given by: where d is the distance between the two masses, and G is a constant which depends upon the units employed. This law assumes that m1 and m2 are point masses. Newton was able to show that homogeneous spheres could be treated as point masses when determining their orbits. F Gm1m2 d 2 = - Figure 902a. Earth-moon barycenter
144TIDESANDTIDALCURRENTSBARYCENTEROFEARTH-MOON-SUNSYSTEM2MOONOCENTEROFMASSOFSUNEARTH-MOONBARYCENTERELLIPTICALORBITCENTEROFMASSOFEARTHFigure902b.Orbit of earth-moon barycenter (notto scale)However,whencomputingdifferentialgravitationalforces,law of gravitation also predicts thatthe earth-moon bary-the actual dimensions of the masses must be taken intocenter will describe an orbit which is approximatelyaccount.elliptical about the barycenter of the sun-earth-moon sys-Using thelaw ofgravitation, it is found thatthe orbitstem. This barycentric point lies inside the sun.of two point masses are conic sections about the bary-center ofthetwomasses.Ifeither one orboth ofthemasses903.TheEarth-Moon-SunSystemarehomogeneousspheresinsteadofpointmassestheor-bits are the same as the orbits which would result if all ofThefundamental tide-generatingforceon theearth hasthemass of thespherewereconcentratedatapointatthetwo interactive but distinct components.The tide-generat-center of the sphere. In the case of the earth-moon system,ingforces aredifferential forcesbetween thegravitationalboththe earth and the moon describe elliptical orbits aboutattraction of the bodies (earth-sun and earth-moon)and thetheirbarycenter if bothbodies are assumed tobe homoge-centrifugal forces on the earth produced by the earth's orbitneous spheres and thegravitational forces of the sun andaround the sun and the moon's orbit around the earth.New-other planetsare neglected.The earth-moonbarycenter iston's Lawof Gravitationand his Second LawofMotioncanlocated74/100ofthedistancefromthecenterof theearthbecombined todevelopformulations forthedifferentialto its surface,alongthe line connecting theearth's andforce at any point on the earth,as the direction and magni-moon'scenters.tude aredependent on whereyou are on the earth's surface.Thus the center of mass of the earth describes a veryAs a result of these differential forces, the tide generatingsmall ellipse about the earth-moon barycenter, while theforces Fdm (moon)and Fds (sun)are inversely proportionalcenter of mass of the moondescribes a much larger ellipseto the cubeofthe distance between the bodies, where:about the samebarycenter.Ifthegravitational forcesoftheotherbodiesofthesolarsystemareneglected,Newton'sTOMOON-c+EARTHFigure903a.Differential forces alongagreat circle connectingthesublunar point and antipode
144 TIDES AND TIDAL CURRENTS However, when computing differential gravitational forces, the actual dimensions of the masses must be taken into account. Using the law of gravitation, it is found that the orbits of two point masses are conic sections about the barycenter of the two masses. If either one or both of the masses are homogeneous spheres instead of point masses, the orbits are the same as the orbits which would result if all of the mass of the sphere were concentrated at a point at the center of the sphere. In the case of the earth-moon system, both the earth and the moon describe elliptical orbits about their barycenter if both bodies are assumed to be homogeneous spheres and the gravitational forces of the sun and other planets are neglected. The earth-moon barycenter is located 74/100 of the distance from the center of the earth to its surface, along the line connecting the earth’s and moon’s centers. Thus the center of mass of the earth describes a very small ellipse about the earth-moon barycenter, while the center of mass of the moon describes a much larger ellipse about the same barycenter. If the gravitational forces of the other bodies of the solar system are neglected, Newton’s law of gravitation also predicts that the earth-moon barycenter will describe an orbit which is approximately elliptical about the barycenter of the sun-earth-moon system. This barycentric point lies inside the sun. 903. The Earth-Moon-Sun System The fundamental tide-generating force on the earth has two interactive but distinct components. The tide-generating forces are differential forces between the gravitational attraction of the bodies (earth-sun and earth-moon) and the centrifugal forces on the earth produced by the earth’s orbit around the sun and the moon’s orbit around the earth. Newton’s Law of Gravitation and his Second Law of Motion can be combined to develop formulations for the differential force at any point on the earth, as the direction and magnitude are dependent on where you are on the earth’s surface. As a result of these differential forces, the tide generating forces Fdm (moon) and Fds (sun) are inversely proportional to the cube of the distance between the bodies, where: Figure 902b. Orbit of earth-moon barycenter (not to scale). Figure 903a. Differential forces along a great circle connecting the sublunar point and antipode
145TIDESANDTIDALCURRENTSthe point directly below the moon, known as the sublunarpoint, and the point on the earth exactly opposite,known asthe antipode. Similar calculations aredonefor the sunGMmReGM,ReIf we assume that the entire surface of the earth is cov-FdsFdm ==ered with a uniform layer of water, the differential forcesd.3dm3may be resolved into vectors perpendicular and parallel tothe surface of the earth todetermine their effectThe perpendicular components change the mass onwhich they are acting, but do not contribute to the tidal ef-fect. The horizontal components, parallel to the earth'ssurface, have the effectofmoving the water in a horizontaldirection toward the sublunar and antipodal points until anequilibrium position is found. The horizontal componentsof the differential forces are the principal tide-generatingforces.Theseare also called tractiveforces.Tractiveforcesare zero at the sublunar and antipodal points and alongthe-TOMOON.great circle halfway between these two points.Tractive-forcesaremaximum alongthe small circleslocated45°1from the sublunar point and the antipode. Figure 903bshows the tractiveforces across the surface oftheearth.+eEquilibriumwill be reached when a bulge of water has++++t,formed at the sublunar and antipodal points such that the-tractive forces due to the moon's differential gravitational-forcesonthemassofwatercoveringthesurfaceoftheearthare just balanced by the earth's gravitational attraction (Fig-ure 903c),Now consider the effect of therotation of the earth.Ifthe declination of the moon is O°,the bulges will lie on theequator.As the earth rotates,an observer at the equator willFigure903b.Tractiveforcesacross thesurfaceoftheearthnote that the moon transits approximatelyevery24hourswhereMm is themass ofthe moon and M,is themass oftheand 50minutes.Since there are twobulges of wateronthesun, Re is the radius of the earth and d is the distance to theequator, one at the sublunar point and the other at the anti-pode, the observer will also see two high tides during thismoonorsun.This explains whythetide-generatingforceofthe sun isonly46/100of thetide-generatingforceoftheinterval with one high tide occurring when the moon ismoon.Even though the sun is much moremassive,it isalsooverhead and another high tide12hours 25minutes laterwhen the observer is at the antipode.He will also experi-muchfartheraway.UsingNewton's second lawofmotion,wecan calculateence a low tide between each high tide.The theoreticalthe differential forces generated by the moon and the sun af-range of these equilibrium tides at the equator willbe lessfecting anypoint on the earth.Theeasiest calculation is forthan Imeter
TIDES AND TIDAL CURRENTS 145 where Mm is the mass of the moon and Ms is the mass of the sun, Re is the radius of the earth and d is the distance to the moon or sun. This explains why the tide-generating force of the sun is only 46/100 of the tide-generating force of the moon. Even though the sun is much more massive, it is also much farther away. Using Newton’s second law of motion, we can calculate the differential forces generated by the moon and the sun affecting any point on the earth. The easiest calculation is for the point directly below the moon, known as the sublunar point, and the point on the earth exactly opposite, known as the antipode. Similar calculations are done for the sun. If we assume that the entire surface of the earth is covered with a uniform layer of water, the differential forces may be resolved into vectors perpendicular and parallel to the surface of the earth to determine their effect. The perpendicular components change the mass on which they are acting, but do not contribute to the tidal effect. The horizontal components, parallel to the earth’s surface, have the effect of moving the water in a horizontal direction toward the sublunar and antipodal points until an equilibrium position is found. The horizontal components of the differential forces are the principal tide-generating forces. These are also called tractive forces. Tractive forces are zero at the sublunar and antipodal points and along the great circle halfway between these two points. Tractive forces are maximum along the small circles located 45° from the sublunar point and the antipode. Figure 903b shows the tractive forces across the surface of the earth. Equilibrium will be reached when a bulge of water has formed at the sublunar and antipodal points such that the tractive forces due to the moon’s differential gravitational forces on the mass of water covering the surface of the earth are just balanced by the earth’s gravitational attraction (Figure 903c). Now consider the effect of the rotation of the earth. If the declination of the moon is 0°, the bulges will lie on the equator. As the earth rotates, an observer at the equator will note that the moon transits approximately every 24 hours and 50 minutes. Since there are two bulges of water on the equator, one at the sublunar point and the other at the antipode, the observer will also see two high tides during this interval with one high tide occurring when the moon is overhead and another high tide 12 hours 25 minutes later when the observer is at the antipode. He will also experience a low tide between each high tide. The theoretical range of these equilibrium tides at the equator will be less than 1 meter. Figure 903b. Tractive forces across the surface of the earth. Fdm GMmR e dm3 = F - ds GMs Re ds 3 ; = -
146TIDESANDTIDALCURRENTSNLowWaterEQUATORTowardthemoonHighWaterHighWaterSOLIDEARTH-LowWatersFigure903c.Theoretical equilibrium configurationdue tomoon's differential gravitational forces.Onebulgeofthewaterenvelopeislocated atthesublunarpoint, theotherbulgeattheantipodeTHIPOLEN24Towardthe Moon2ASDASOLID EARTHYORTHLPOLXscBUTWTPOLEFigure903d.Effectsof thedeclinationof themoonThe heights of the two high tides should be equal at thelowwaterseachday.equator.At points northor south of the equator,an observerC.Observers at points X, Y, and Z experience onewouldstill experiencetwohighandtwolowtides,butthehigh tide when moon is on their meridian, then an-heightsofthehightideswouldnotbeasgreatastheyareattheotherhightide12hours25minutes later when atequator.Theeffects ofthedeclinationofthemoon areshownX, Y, and Z.The second high tide is the same atin Figure 903d, for three cases, A, B, and C.X'as at X.High tides at Y'and Z are lower thanhigh tides at Y and Z.A. When the moon is on the plane of the equator, theThe preceding discussion pertaining to the effects offorces are equal inmagnitudeat thetwo points on thesame parallel of latitude and 180°apart in longitudethemoon is equally valid when discussing theeffects oftheB.Whenthemoonhasnorthorsouthdeclination,thesun, taking into account that the magnitude of the solar efforces are unequal at such points and tend to causefect is smaller.Hence, the tides will also vary according toan inequality inthetwohighwaters and thetwothe sun's declination and its varying distance from the
146 TIDES AND TIDAL CURRENTS The heights of the two high tides should be equal at the equator. At points north or south of the equator, an observer would still experience two high and two low tides, but the heights of the high tides would not be as great as they are at the equator. The effects of the declination of the moon are shown in Figure 903d, for three cases, A, B, and C. A. When the moon is on the plane of the equator, the forces are equal in magnitude at the two points on the same parallel of latitude and 180° apart in longitude. B. When the moon has north or south declination, the forces are unequal at such points and tend to cause an inequality in the two high waters and the two low waters each day. C. Observers at points X, Y, and Z experience one high tide when moon is on their meridian, then another high tide 12 hours 25 minutes later when at X’, Y’, and Z’. The second high tide is the same at X’ as at X. High tides at Y’ and Z’ are lower than high tides at Y and Z. The preceding discussion pertaining to the effects of the moon is equally valid when discussing the effects of the sun, taking into account that the magnitude of the solar effect is smaller. Hence, the tides will also vary according to the sun’s declination and its varying distance from the Figure 903c. Theoretical equilibrium configuration due to moon’s differential gravitational forces. One bulge of the water envelope is located at the sublunar point, the other bulge at the antipode. Figure 903d. Effects of the declination of the moon
147TIDESANDTIDALCURRENTSearth.Asecond envelopeof waterrepresenting the equilib-tideswould be smaller,and thelowtides correspondinglyriumtidesduetothesunwouldresembletheenvelopenot as low.shown in Figure 903c except that the heights of the highFEATURESOFTIDES904.General Featurespendent upon its dimensions.None ofthe oceans isasingleAtmostplaces thetidal changeoccurstwicedaily.The31215182191215182tide rises until it reaches a maximum height, called highHOURSAHOURStide or high water,and then falls to a minimum level called-85lowtideor lowwater6Therateof riseand fall is not uniform.From lowwa-ter, the tide begins to rise slowly at first, but at an increasingrate until it is about halfway to high water.The rate of rise雅then decreases until high water is reached, and the rise ceas-BOSTONes.Thefalling tidebehaves in a similarmanner.Theperiod5at high or low water during which there is no apparentFigure905a.Semidiurnaltypeoftidechange of level is called stand.The difference in height be-tween consecutivehigh and low waters is the range1521183691213182136369121518215中品总有员中有司HOURSHOURS-3PELHAINEW YORKFigure904.The riseand fall of thetide at NewYorkshowngraphicallyFigure 905b.Diurnal tideFigure 904 is a graphical representation of the rise andoscillating body,rather each one is made up of several sep-fall of the tide at New York during a 24-hour period. Thearate oscillating basins.As such basins are acted upon bycurvehas thegeneral form of a variable sine curvethe tide-producing forces, some respond more readilytodaily or diurnal forces,others to semidiurnal forces,and905.TypesOfTideothers almostequallyto both.Hence,tides areclassified asone of three types, semidiurnal, diurnal, or mixed, accordAbody ofwater has a natural period of oscillation,deingtothecharacteristics ofthetidal pattern.In thesemidiurnaltide,there are twohighand two lowwaterseachtidal day,withrelativelysmalldifferencesintherespectivehighs and lows.Tides on theAtlantic coast of theUnited States are of the semidiurnal type,which is illustrat-edinFigure9o5abythetidecurveforBostonHarborIn the diurnal tide, only a single high and single lowwater occur each tidal day.Tides of the diurnal type occuralong the northern shore of the Gulfof Mexico, in the JavaSea, the Gulfof Tonkin, and in a few other localities.Thetide curve for Pei-Hai, China, illustrated in Figure 905b, isanexampleofthediurnaltypeInthemixedtide,thediurnalandsemidiurnaloscilla
TIDES AND TIDAL CURRENTS 147 earth. A second envelope of water representing the equilibrium tides due to the sun would resemble the envelope shown in Figure 903c except that the heights of the high tides would be smaller, and the low tides correspondingly not as low. FEATURES OF TIDES 904. General Features At most places the tidal change occurs twice daily. The tide rises until it reaches a maximum height, called high tide or high water, and then falls to a minimum level called low tide or low water. The rate of rise and fall is not uniform. From low water, the tide begins to rise slowly at first, but at an increasing rate until it is about halfway to high water. The rate of rise then decreases until high water is reached, and the rise ceases. The falling tide behaves in a similar manner. The period at high or low water during which there is no apparent change of level is called stand. The difference in height between consecutive high and low waters is the range. Figure 904 is a graphical representation of the rise and fall of the tide at New York during a 24-hour period. The curve has the general form of a variable sine curve. 905. Types Of Tide A body of water has a natural period of oscillation, dependent upon its dimensions. None of the oceans is a single oscillating body; rather each one is made up of several separate oscillating basins. As such basins are acted upon by the tide-producing forces, some respond more readily to daily or diurnal forces, others to semidiurnal forces, and others almost equally to both. Hence, tides are classified as one of three types, semidiurnal, diurnal, or mixed, according to the characteristics of the tidal pattern. In the semidiurnal tide, there are two high and two low waters each tidal day, with relatively small differences in the respective highs and lows. Tides on the Atlantic coast of the United States are of the semidiurnal type, which is illustrated in Figure 905a by the tide curve for Boston Harbor. In the diurnal tide, only a single high and single low water occur each tidal day. Tides of the diurnal type occur along the northern shore of the Gulf of Mexico, in the Java Sea, the Gulf of Tonkin, and in a few other localities. The tide curve for Pei-Hai, China, illustrated in Figure 905b, is an example of the diurnal type. In the mixed tide, the diurnal and semidiurnal oscillaFigure 904. The rise and fall of the tide at New York, shown graphically. Figure 905a. Semidiurnal type of tide. Figure 905b. Diurnal tide
148TIDESANDTIDALCURRENTStions areboth importantfactors and thetide is characterizedlow waters are about the same.At Seattle the greater ine-by a large inequality in the high water heights, low waterqualities aretypicallyin the lowwaters,while at Honoluluit is thehigh waters that have the greater inequalitiesheights, or inboth.There are usually twohigh and two lowwaters each day,but occasionally the tide may become di-906.SolarTideurnal.Suchtides areprevalentalongthePacificcoast of theUnited States and in many other parts of theworld.Exam-The natural period of oscillation of a body of waterples ofmixed types oftide are shown in Figure 905c. At LosAngeles, it is typical that the inequalities in the high andmay accentuate either the solar or the lunar tidal oscilla-8728286728812181212P#ESHOURSHOURSHOURSF.-4F-7-10F6LUEE2nNFoF4F4LOSANGELESSEATTLEHONOLULUFigure 905c.Mixed tidetions. Though as a general rule the tides follow the moon,riod.Thepractical effect is tocreate alonger periodof standthe relative importanceofthesolareffect varies indifferentat high or lowtide.Thetidetables list theseand other pecu-areas.Thereareafewplaces,primarily in the SouthPacificliaritieswheretheyoccur.and the Indonesian areas, where the solar oscillation is themore important,andatthoseplaces thehigh and lowwaters908.VariationsInRangeoccur at about the same time each day.At Port Adelaide,Though the tide at a particular place can be classifiedAustralia the solar and lunar semidiurnal oscillations areequal andnullifyone anotherat neaps.as to type, it exhibits many variations during the month(Figure 908a).The range of the tide varies according to the907.SpecialTidalEffectsintensity ofthetide-producingforces,though theremaybea lag ofa day or two between a particular astronomic causeand thetidal effectAs a waveentersshallowwater,its speed isdecreased.The combined lunar-solar effect is obtained by addingSince the trough is shallower than the crest, it is retardedmore,resulting in a steepening of the wave front.In a fewestuaries,the advanceof thelow water trough is so muchretarded that the crest of the rising tideovertakes the low,and advances upstream as a breaking wave called a bore.Bores that are large and dangerous at times of large tidalranges maybe mere ripples at thosetimes of the monthwhentherangeis small.ExamplesoccurinthePetitcodiacRiver in theBay of Fundy,and at Haining,China, in theTsientang Kaing. The tide tables indicate where boresoccurOther special features are thedoublelowwater (as atHoek Van Holland)and the double high water (as atSouthampton, England).At such places there is often aslight fall or rise in the middle of the high or low water pe-
148 TIDES AND TIDAL CURRENTS tions are both important factors and the tide is characterized by a large inequality in the high water heights, low water heights, or in both. There are usually two high and two low waters each day, but occasionally the tide may become diurnal. Such tides are prevalent along the Pacific coast of the United States and in many other parts of the world. Examples of mixed types of tide are shown in Figure 905c. At Los Angeles, it is typical that the inequalities in the high and low waters are about the same. At Seattle the greater inequalities are typically in the low waters, while at Honolulu it is the high waters that have the greater inequalities. 906. Solar Tide The natural period of oscillation of a body of water may accentuate either the solar or the lunar tidal oscillations. Though as a general rule the tides follow the moon, the relative importance of the solar effect varies in different areas. There are a few places, primarily in the South Pacific and the Indonesian areas, where the solar oscillation is the more important, and at those places the high and low waters occur at about the same time each day. At Port Adelaide, Australia the solar and lunar semidiurnal oscillations are equal and nullify one another at neaps. 907. Special Tidal Effects As a wave enters shallow water, its speed is decreased. Since the trough is shallower than the crest, it is retarded more, resulting in a steepening of the wave front. In a few estuaries, the advance of the low water trough is so much retarded that the crest of the rising tide overtakes the low, and advances upstream as a breaking wave called a bore. Bores that are large and dangerous at times of large tidal ranges may be mere ripples at those times of the month when the range is small. Examples occur in the Petitcodiac River in the Bay of Fundy, and at Haining, China, in the Tsientang Kaing. The tide tables indicate where bores occur. Other special features are the double low water (as at Hoek Van Holland) and the double high water (as at Southampton, England). At such places there is often a slight fall or rise in the middle of the high or low water period. The practical effect is to create a longer period of stand at high or low tide. The tide tables list these and other peculiarities where they occur. 908. Variations In Range Though the tide at a particular place can be classified as to type, it exhibits many variations during the month (Figure 908a). The range of the tide varies according to the intensity of the tide-producing forces, though there may be a lag of a day or two between a particular astronomic cause and the tidal effect. The combined lunar-solar effect is obtained by adding Figure 905c. Mixed tide
149TIDESANDTIDALCURRENTSAuE9N<-鑫oHE022MNSMLWSFMHSHLWSTADELADFMHHA10MLLWMINNMEW.FHHWWw*NOLULMHHWMLLWPEI-HATeLLWOrewnoonDfetaartO.tilmeeClautcusterE,ontheFohNSmaostarthestrerthotsouhottheEouteerpeign:Oy.sunatahumnalegsinaxter,A,P,ToochartdnumFigure908a.Monthlytidalvariationsatvariousplaces
TIDES AND TIDAL CURRENTS 149 Figure 908a. Monthly tidal variations at various places
150TIDESANDTIDALCURRENTSthemoon'stractiveforcesvectoriallytothesun'stractiveFavThreQuarteQvar(A)E(B)Priamirg accurswhen essee hbeteee sewae firv(A)qearteraadbetversfoasdhirdqeartncHighideoocursbeferetrawsitwaocn.Figure 908b. (A) Spring tides occur at times of new and fullmoon. Range of tide is greater than average since solar andlunar tractiveforces act in samedirection.(B)Neap tidesoccur at times of first and third quarters.Range of tide islessthanaveragesincesolarand lunartractiveforcesactatright angles.(B)Lagirigoecurswhen msee s beteecnfintguaraadfelandbetverethindqaartrsndarkHightidoucersahertramsitefmoas.Figure 908c. Priming and lagging the tides.forces.The resultant tidal bulge will be predominantly lu-nar with modifying solar effects upon both the height of thetide and the direction of the tidal bulge. Special cases of in-terest occur during the times of new and full moon (Figure908b).Withthe earth,moon,and sunlyingapproximatelyon the same line, the tractiveforces ofthe sun are acting inthesamedirectionasthemoon'stractiveforces(modifiedby declination effects).The resultant tides are called springtides, whose ranges are greater than average.Between the spring tides, the moon is at first and thirdquarters.Atthosetimes,thetractiveforces of the sun areacting at approximately right angles to the moon's tractiveforces.Theresults aretides calledneaptides, whoserangesarelessthanaverageWith the moon in positions between quadrature andnew or full, the effect of the sun is to cause the tidal bulgeto either lag or precede the moon (Figure 908c). These ef-fects are called priming and lagging the tides.Thus, when the moon is at the point in its orbit nearestthe earth (atperigee),the lunar semidiurnal range is increasedand perigean tides occur.When the moon is farthestfrom
150 TIDES AND TIDAL CURRENTS the moon’s tractive forces vectorially to the sun’s tractive forces. The resultant tidal bulge will be predominantly lunar with modifying solar effects upon both the height of the tide and the direction of the tidal bulge. Special cases of interest occur during the times of new and full moon (Figure 908b). With the earth, moon, and sun lying approximately on the same line, the tractive forces of the sun are acting in the same direction as the moon’s tractive forces (modified by declination effects). The resultant tides are called spring tides, whose ranges are greater than average. Between the spring tides, the moon is at first and third quarters. At those times, the tractive forces of the sun are acting at approximately right angles to the moon’s tractive forces. The results are tides called neap tides, whose ranges are less than average. With the moon in positions between quadrature and new or full, the effect of the sun is to cause the tidal bulge to either lag or precede the moon (Figure 908c). These effects are called priming and lagging the tides. Thus, when the moon is at the point in its orbit nearest the earth (at perigee), the lunar semidiurnal range is increased and perigean tides occur. When the moon is farthest from Figure 908b. (A) Spring tides occur at times of new and full moon. Range of tide is greater than average since solar and lunar tractive forces act in same direction. (B) Neap tides occur at times of first and third quarters. Range of tide is less than average since solar and lunar tractive forces act at right angles. Figure 908c. Priming and lagging the tides
151TIDESANDTIDALCURRENTSthe earth (at apogee),the smaller apogean tidesoccur.WhenThe cycle involving the moon's distance requires an anom-the moon and sunare in line and pulling together,as at newalisticmonth of about271/2days.The sun'sdeclinationand full moon, spring tides occur (the term spring has noth-and distance cycles are respectively a half year and a yearing to dowith the season of year);when themoon and sunin length.An important lunar cycle, called the nodal peri-oppose each other, as at the quadratures, the smaller neapod,is 18.6years (usuallyexpressed in round figuresas 19tides occur. When certain of these phenomena coincide,years). For a tidal value, particularly a range, to be consid-perigean spring tides and apogean neap tides occur.eredatruemean, it mustbe eitherbasedupon observationsThese are variations in the semidiurnal portionof theextended over this period of time, or adjusted to take ac-tide.Variations in the diurnal portion occur as the moon andcount of variations known to occur during the nodal periodsun changedeclination.When themoon is at its maximumsemi-monthly declination (either north or south),tropic910.TimeOfTidetides occur in which the diurnal effect is at a maximum;When itcrosses the equator,the diurnal effect is a minimumSincethe lunar tide-producingforcehas thegreatestand equatorialtides occur.effect in producing tides at most places,the tides“followWhen therange of tide is increased, as at spring tidesthe moon."Because the earthrotates, high water lags be-thereismorewater availableonlyathightide:atlowtidehindboth upperand lowermeridianpassageof themoonthere is less,for the high waters rise higher and the low wa-The tidal day, which is also the lunar day, is the time beters fall lowerat these times.There ismore water at neaptweenconsecutivetransitsofthemoon,or24hours and50lowwaterthan at springlowwater.Withtropic tides,thereminutes on the average.Where the tide is largely semidi-isusuallymoredepthatonelowwaterduringthedaythanurnal intvpe,thelunitidal interval(theinterval betweenat the other.While it is desirable to know the meanings ofthe moon's meridiantransit and a particularphase of tide)theseterms,thebest way of determiningtheheight of theisfairly constant throughout themonth,varying some-tideat anyplaceand timeisto examinethetidepredictionswhat with the tidal cycles.There are many placesfor the place as given in the tide tables, which take all thesehowever,where solar ordiurnal oscillationsare effectiveeffects intoaccount.in upsetting this relationship.The interval generallygivenis the average elapsedtimefrom themeridiantransit (up-909.Tidal Cyclesper or lower) of the moon until the next high tide.Thismay be called mean high waterlunitidalinterval or cor-Tidal oscillations go through a number of cycles.Therected(ormean)establishment.Thecommonshortestcycle,completed inabout12hoursand25minutesestablishment is the average interval on days of full orfor a semidiurnal tide,extendsfromany phase ofthe tide tonew moon, and approximates the mean high water luniti-thenext recurrenceof the samephase.Duringa lunar daydal interval.(averaging24 hours and 50minutes)therearetwo highsIn the ocean, the tide may be in the nature ofa progres-andtwolows(twooftheshortercycles)forasemidiurnalsivewavewith the crestmoving forward,a stationary ortide.The moon revolves around the earth with respect to thestandingwavewhichoscillatesinaseesawfashion.oracomsun ina synodical month of about291/2days,commonlycalled the lunar month.The effectofthephasevariation isbination of the two.Consequently,caution should beused ininferring the time of tideat aplacefrom tidal data fornearbycompleted in one-half a synodical month or about 2 weeksas the moon varies from new to full orfull to new.The ef-places.In a river or estuary,the tide enters from the sea andfect of the moon's declination is also repeated in one-halfis usually sent upstream as a progressive wave so that the tideofatropicalmonthof271/3daysoraboutevery2weeks.occurs progressively later at various places upstream.TIDALDATUMS911.LowWaterDatumssoundings taken at all stages of the tide can be reduced to acommon soundingdatum.Soundings on charts showdepthsA tidal datum is a level from which tides are mea-belowa selectedlowwaterdatum(occasionallymeansealevel),and tide predictions in tidetables show heights above andsured.Therearea numberofsuchlevelsofreferencethatbelowthe same level.Thedepth ofwater available at any timeare important to the mariner. See Figure 911.is obtained byadding algebraicallytheheight ofthetide at theThemostimportant level ofreferencetothemariner isthesounding datum shown on charts. Since the tide rises and fallstime in question to the charted depthcontinually while soundings are being taken during a hydro-By international agreement, the level used as chart da-tum should be low enough so that low waters do not fallgraphic survey,the tide is recorded during the survey so that
TIDES AND TIDAL CURRENTS 151 the earth (at apogee), the smaller apogean tides occur. When the moon and sun are in line and pulling together, as at new and full moon, spring tides occur (the term spring has nothing to do with the season of year); when the moon and sun oppose each other, as at the quadratures, the smaller neap tides occur. When certain of these phenomena coincide, perigean spring tides and apogean neap tides occur. These are variations in the semidiurnal portion of the tide. Variations in the diurnal portion occur as the moon and sun change declination. When the moon is at its maximum semi-monthly declination (either north or south), tropic tides occur in which the diurnal effect is at a maximum;. When it crosses the equator, the diurnal effect is a minimum and equatorial tides occur. When the range of tide is increased, as at spring tides, there is more water available only at high tide; at low tide there is less, for the high waters rise higher and the low waters fall lower at these times. There is more water at neap low water than at spring low water. With tropic tides, there is usually more depth at one low water during the day than at the other. While it is desirable to know the meanings of these terms, the best way of determining the height of the tide at any place and time is to examine the tide predictions for the place as given in the tide tables, which take all these effects into account. 909. Tidal Cycles Tidal oscillations go through a number of cycles. The shortest cycle, completed in about 12 hours and 25 minutes for a semidiurnal tide, extends from any phase of the tide to the next recurrence of the same phase. During a lunar day (averaging 24 hours and 50 minutes) there are two highs and two lows (two of the shorter cycles) for a semidiurnal tide. The moon revolves around the earth with respect to the sun in a synodical month of about 29 1/2 days, commonly called the lunar month. The effect of the phase variation is completed in one-half a synodical month or about 2 weeks as the moon varies from new to full or full to new. The effect of the moon’s declination is also repeated in one-half of a tropical month of 27 1/3 days or about every 2 weeks. The cycle involving the moon’s distance requires an anomalistic month of about 27 1/2 days. The sun’s declination and distance cycles are respectively a half year and a year in length. An important lunar cycle, called the nodal period, is 18.6 years (usually expressed in round figures as 19 years). For a tidal value, particularly a range, to be considered a true mean, it must be either based upon observations extended over this period of time, or adjusted to take account of variations known to occur during the nodal period. 910. Time Of Tide Since the lunar tide-producing force has the greatest effect in producing tides at most places, the tides “follow the moon.” Because the earth rotates, high water lags behind both upper and lower meridian passage of the moon. The tidal day, which is also the lunar day, is the time between consecutive transits of the moon, or 24 hours and 50 minutes on the average. Where the tide is largely semidiurnal in type, the lunitidal interval (the interval between the moon’s meridian transit and a particular phase of tide) is fairly constant throughout the month, varying somewhat with the tidal cycles. There are many places, however, where solar or diurnal oscillations are effective in upsetting this relationship. The interval generally given is the average elapsed time from the meridian transit (upper or lower) of the moon until the next high tide. This may be called mean high water lunitidal interval or corrected (or mean) establishment. The common establishment is the average interval on days of full or new moon, and approximates the mean high water lunitidal interval. In the ocean, the tide may be in the nature of a progressive wave with the crest moving forward, a stationary or standing wave which oscillates in a seesaw fashion, or a combination of the two. Consequently, caution should be used in inferring the time of tide at a place from tidal data for nearby places. In a river or estuary, the tide enters from the sea and is usually sent upstream as a progressive wave so that the tide occurs progressively later at various places upstream. TIDAL DATUMS 911. Low Water Datums A tidal datum is a level from which tides are measured. There are a number of such levels of reference that are important to the mariner. See Figure 911. The most important level of reference to the mariner is the sounding datum shown on charts. Since the tide rises and falls continually while soundings are being taken during a hydrographic survey, the tide is recorded during the survey so that soundings taken at all stages of the tide can be reduced to a common sounding datum. Soundings on charts show depths below a selected low water datum (occasionally mean sea level), and tide predictions in tide tables show heights above and below the same level. The depth of water available at any time is obtained by adding algebraically the height of the tide at the time in question to the charted depth. By international agreement, the level used as chart datum should be low enough so that low waters do not fall
152TIDESANDTIDALCURRENTSvery far below it.At most places, the level used is one de-Indian tide plane orharmonic tide plane,is a lowwaterterminedfrom amean ofanumberof lowwaters(usuallydatum that includes the spring effect of the semi-diurnalovera19year period),therefore,somelowwaters canbeportion of the tide and the tropic effect of the diurnal por-expected to fall below it.Thefollowing are some ofthe da-tion.It is about the level of lower low water of mixed tidestums ingeneral useat the timethatthe moon'smaximum declination coincideswith the time of new orfull moonMean low water (MLW) is the average height of alllowwatersatagivenplace.AbouthalfofthelowwatersMean lower lowwater springs (MLLWS)is theav-fall below it, and half above.erage level of the lower of the two low waters on the daysMean low water springs (MLWS), usually shortenedof spring tides.tolowwater springs,is the average levelofthe low watersSome still lower datums used on charts are determinedthat occur atthetimes of spring tidesfrom tide observations and some are determined arbitrarilyMean lower low water (MLLW) isthe average heightandlaterreferredtothetide.Mostofthemfallclosetooneortheotherof thefollowingtwodatumsof thelowerlowwatersofeachtidaldayTropic lower low water (TeLLW) is the averageLowest normal low water is a datum that approxi-height of the lowerlowwaters(or of thesingle daily lowmates the averageheight of monthlylowest lowwaters.waters if thetide becomesdiurnal)that occurwhen thediscarding any tides disturbed by storms.moon is nearmaximumdeclination andthediurnal effect isLowest lowwaterisanextremelylowdatum.Itconformsmost pronounced. This datum is not in common use as a tid-generallytothelowesttideobserved,orevensomewhatloweral reference.Oncea tidal datum is established, it is sometimesretained forIndian spring lowwater (ISLW), sometimes calledan indefinite period, even though it might differ slightly fromMEANHIGHWATERSPRINGS本-MEANHIGHWATERI11--MEANHIGHWATERNEAPSFhueySouesobuedHALF-TIDELEVELouuew1aMEANLOWWATERNEAPSeondean-★MEANLOWWATERMEANLOWWATERSPRINGSINDIANSPRINGLOWWATER(CHARTSOUNDINGDATUM)CHARTEDDEPTHFigure 911.Variations in the ranges and heights of tide where the chart sounding datum is Indian Spring Low Water
152 TIDES AND TIDAL CURRENTS very far below it. At most places, the level used is one determined from a mean of a number of low waters (usually over a 19 year period); therefore, some low waters can be expected to fall below it. The following are some of the datums in general use. Mean low water (MLW) is the average height of all low waters at a given place. About half of the low waters fall below it, and half above. Mean low water springs (MLWS), usually shortened to low water springs, is the average level of the low waters that occur at the times of spring tides. Mean lower low water (MLLW) is the average height of the lower low waters of each tidal day. Tropic lower low water (TcLLW) is the average height of the lower low waters (or of the single daily low waters if the tide becomes diurnal) that occur when the moon is near maximum declination and the diurnal effect is most pronounced. This datum is not in common use as a tidal reference. Indian spring low water (ISLW), sometimes called Indian tide plane or harmonic tide plane, is a low water datum that includes the spring effect of the semi-diurnal portion of the tide and the tropic effect of the diurnal portion. It is about the level of lower low water of mixed tides at the time that the moon’s maximum declination coincides with the time of new or full moon. Mean lower low water springs (MLLWS) is the average level of the lower of the two low waters on the days of spring tides. Some still lower datums used on charts are determined from tide observations and some are determined arbitrarily and later referred to the tide. Most of them fall close to one or the other of the following two datums. Lowest normal low water is a datum that approximates the average height of monthly lowest low waters, discarding any tides disturbed by storms. Lowest low water is an extremely low datum. It conforms generally to the lowest tide observed, or even somewhat lower. Once a tidal datum is established, it is sometimes retained for an indefinite period, even though it might differ slightly from Figure 911. Variations in the ranges and heights of tide where the chart sounding datum is Indian Spring Low Water