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