Butikov Oceanic_Tides

A dynamical picture of the oceanic tides Eugene I. Butikova) Department of Physics, St. Petersburg State University, St. Petersburg 198904, Russia ~Received 5 November 2001; accepted 11 June 2002 ! Adetailed treatment of tide-generating forces is given, followed by a simplified dynamic theory oftidal waves. To clarify the underlying physics, we use a simple model of the ocean that consists ofa water shell of uniform depth completely covering the globe. The treatment is appropriate forcollege and university undergraduate students studying introductory geophysics or astronomy,general physics, or intermediate mechanics. A computer simulation is developed to aid inunderstanding the properties of sun- or moon-induced tide-generating forces and of the stationarytidal waves created by these forces in the open ocean. © 2002 American Association of Physics Teachers. @DOI: 10.1119/1.1498858 # I. INTRODUCTION All textbooks in introductory astronomy and many in physics and intermediate mechanics mention the existence ofoceanic tides as an interesting manifestation of universalgravitation. Pedagogical papers devoted to the tides ~see, for example, Refs. 1–9 !testify to the fact that many teachers are interested in this topic, but are not satisfied with the clarityand correctness of the commonly accepted explanations ofthe physics of tidal phenomena. A review of textbooks andrelated literature shows that the most important aspects of theorigin and properties of tides are often treated inaccurately oreven erroneously. Much of the confusion over generatingtides is related to the roles of the orbital motion of the moonand earth about their common center of mass and of theearth’s axial rotation. In discussing the physics behind thisphenomenon, authors usually explain ~more or less success- fully!why two tidal swells appear on the opposite sides of the globe. However, it is difficult to find a plausible expla-nation of the physical mechanism responsible for the phaseshift between the zenith of the moon and the moment of hightide, which at some places approaches 90°. Misunderstand-ings also occur in discussions about the role of tidal frictionin the retardation of axial rotations and in the evolution oforbital motions of the gravitationally coupled celestial bod-ies. To clarify the basic physics underlying the tidal phenom- ena, we suggest a rather simple but rigorous treatment of thetide-generating forces, followed by a theory of the circulat-ing tidal wave produced by these forces. This treatment usesa simplified model of the ocean consisting of a water shell ofuniform depth entirely covering the globe.Acomputer simu-lation is developed to support the analytical treatment. 10The simulation gives a dynamical picture of the forces and thetidal wave driven by these forces in the open ocean. Thispaper and the simulation are intended only to clarify thephysical background of this natural phenomenon and do notassume to describe the complete picture. The purely theoret-ical quantitative description of tides for a given location onthe earth, derived solely from first principles, is hardly pos-sible because of the extremely complex structure of theoceans, the actual system that responds with tides and tidalcurrents to the well-known tide-generating forces. The paper is organized as follows. First we discuss quali- tatively the physical nature of the sun- and moon-inducedtide-generating forces in a nonrotating geocentric frame ofreference, deriving the mathematical expressions for theseforces at an arbitrary point on the earth. Next the static ~equi- librium !distortion of the ocean surface under these forces is determined. Then we show that the same expressions for thetidal forces are applicable on the rotating earth, and we dis-cuss how these forces depend on time. We show that a uni-form rotation of the system of tidal forces coupled with theapparent motion of the sun ~moon !can be represented as a superposition of two oscillating quadrupole systems of forceswhose axes make an angle of 45° with respect to one an-other. Each of these systems of forces generates a steady-state forced oscillation of the ocean ~a standing wave !. Next we treat the tidal wave circulating around the globe as asuperposition of these standing waves. Finally the real-worldcomplications of this simplified picture are discussed briefly,as well as the role of tidal friction in the evolution of theaxial rotations and orbital revolutions of celestial bodies.II. THE TIDE-GENERATING FORCES: ANELEMENTARYAPPROACH The tides are manifested by alternating vertical displace- ments of the surface of the sea coupled with horizontalmovements of the water that are called the tidal currents .I t is well known that the tides are caused by the varying gravi-tational forces that the moon and sun exert on both the earthand its oceans. More exactly, the origin of tidal phenomena isrelated to the inhomogeneity ~nonuniformity !of the lunar and solar gravitational fields across the globe. The gravitational force the moon exerts on any body on the surface of the earth is much smaller than the gravitationalforce of the sun. However, because the moon is much closerto the earth than the sun, the inhomogeneity of the lunargravitationa lfieldacrosstheearthisconsiderabl ygreater than that of the solar field. As a result, moon-induced tidesare more than twice as great as sun-induced tides. Neverthe-less, to arrive more easily at an understanding of the physicalorigin of tide-generating forces, we begin our analysis withsun-induced tides. These are somewhat simpler to explainbecause the center of mass of the sun–earth system verynearly coincides with the center of the sun. We next divide the problem into two parts: First we dis- cuss the origin and properties of tide-generating forces, afterwhich we investigate qualitatively the much more compli-cated case of the dynamical effect that these time-varyingforces have on the ocean. We note that much of the confu- 1 1 Am. J. Phys. 70~9!, September 2002 http://ojps.aip.org/ajp/ © 2002 American Association of Physics Teachers sionintheliteratureisrelatedtothefirst~rathersimple !part of this problem, which can be completely and unambigu-ously solved using Newtonian mechanics. The earth as a whole moves with an acceleration relative to an inertial reference frame. This acceleration is producedby the gravitational attraction of the earth to the sun ~and also to the moon and to all other celestial bodies !. Although the earth travels in an almost circular orbit, its centripetalacceleration a 0in this orbital motion is generated by the gravitational pull of the sun and hence is just the accelera- tion of free fall , which is independent of the orbital velocity. The earth would move with the same acceleration were itfreely falling in the gravitational field of the sun. What isimportant in this problem is the acceleration, not the orbitalvelocity, of the earth. To better understand the tides, we first use a nonrotating geocentric reference frame.Although the origin of this framemoves approximately in a circle around the sun ~more ex- actly, around the center of mass of the sun–earth system !, the frame itself does not rotate because the directions of its axesare fixed relative to the distant stars. That is, the motion ofthis frame—revolution without rotation—is a translational~though nearly circular !motion. It reminds us of ‘‘the circu- lar motion of the frying pan’’in the hands of a cook ~see Ref. 1!. With respect to inertial space, all points of this reference frame move with an acceleration a 0whose magnitude and direction are the same for all the points.Any body of mass m whose motion is referred to this noninertial geocentric frame~for example, an earth satellite, or a drop of water in the ocean !is subject to the pseudoforce of inertia, F in52ma0, which is independent of the position of the body relative tothe earth. If the body were placed at the center of the earth,this pseudoforce would exactly balance the gravitational at-traction of the body to the sun. In other words, if we considerthe earth as a giant spaceship orbiting the sun, a body placedat the center of this ship would seem to be weightless withrespect to the gravitation of the sun, just as astronauts on anorbital station seem to be weightless in the gravitational fieldof the earth. The force of inertia, F in52ma0, experienced by a body in the freely falling geocentric frame of reference ~or in the frame that revolves without axial rotation about the sun–earth center of mass !, has the same magnitude and direction everywhere on the earth. On the other hand, the gravitationalpull of the sun, F sun, experienced by the body diminishes with its distance from the sun and is directed to the sun, andhence both the magnitude and direction of F sundepend on the position of the body on the earth. Because the earth is anextended body, the pseudoforce F inand the force Fsunare generally unequal and not exactly opposite, except at thecenter of the earth. The combined actions of the gravitationalpull of the sun and the pseudoforce of inertia is the tidal force. In other words, the tidal force at a given position near the earth equals the vector difference of the gravitational pull thesun exerts on an object at this position and the gravitationalpull the sun would exert on this object were it at the center ofthe earth. We may avoid using a noninertial reference frameif we are not inclined to introduce the concept of the pseudo-force of inertia to students. In doing so, we can use a some-what different language in the subsequent derivation of thetidal force: Instead of discussing the vector addition of thepull of the sun and the corresponding pseudoforce of inertiaarising from the noninertial character of the reference frame,we can use instead an inertial frame, in which the tidal forcecan be found by the vector subtraction of the gravitationalforce of the sun on the body at its given location with theforce of the sun on the body were it located at the center ofthe earth. Indeed, when viewing the situation on the earthfrom the inertial frame of reference, we can apply the Gal-ilean law according to which, in the same gravitational field~here the field of the sun !, all free bodies experience equal accelerations. Hence the earth as a whole and all free bodieson the earth, being subjected to almost the same solar gravi-tational field, are very nearly accelerated toward the sun.Consequently we do not particularly notice the influence ofsolar gravitation on what happens on earth. The small differ-ences between the acceleration of the earth as a whole and ofthe earthly bodies depend on the distances of the bodies fromthe center of the earth because these differences are causedby the nonuniformity of the solar gravitational field over theextent of the earth. 11 These differential effects of gravity give rise, in particular, to solar gravitational perturbations of an earth satellite’s geo-centric orbit. The tide-generating forces slightly distort theearth’s gravitational pull that governs the satellite’s motionso that after a revolution, the satellite does not return to thesame point of the geocentric reference frame. On the surfaceof the earth, these same forces give rise to the tides. Weemphasize that tidal forces are caused not by the sun’s gravi-tational field itself, but rather by the nonuniformity of thisfield. Figure 1 illustrates the origin and properties of the tide- generating forces produced by the sun. The free-fall accel-eration of the earth Ein the gravitational field of the sun Sis a 05GMsun/R2, whereMsunis the mass of the sun, and Ris the sun–earth distance. The gravitational pull of the sun Fsun experienced by the body ~for example, a satellite !at pointA almost equals the force of inertia Finin magnitude because the distances to the sun from the body and from the center ofthe earth are very nearly equal. However, at point Athe direction of the gravitational force F sunis not exactly oppo- site to the force of inertia Fin. Thus their nonzero resultant, the tidal force FAat pointA, is directed toward the earth. Its magnitude equals ma0b5ma0(r/R)5(GmMsun/R2)(r/R), where b5r/Ris the angle between the body and the center of the earth as seen from the sun. The tidal force FBat the opposite point BequalsFAin magnitude and is also directed vertically downward to the earth. On the surface of the earth,the tidal force is directed vertically downward at all places~forming a circle !where the sun is in the horizon at that moment. The distance from the sun to the body at point Z~for which the sun is at the zenith !is smaller than to the center of the earth. Here the gravitational pull of the sun points exactly Fig. 1. Sun-induced tide-generating forces at different points A,B,Z, andN. 2 2 Am. J. Phys., Vol. 70, No. 9, September 2002 Eugene I. Butikov opposite to and is somewhat greater than the force of inertia.Hence, the tidal force F Zat this point is directed vertically upward, from the earth toward the sun. Its magnitude, FZ5GmMsun ~R2r!22ma05ma0FR2 ~R2r!221G ’ma02r R5GmMsun R22r R, ~1! is approximately twice the magnitude of the tidal forces atpointsAandB. Similarly, at the opposite point N~for which the sun is at its nadir !the force of inertia is greater than the gravitationalpullofthesun,andsothetidalforce F Natpoint Nis also directed vertically upward from the earth ~and from the sun !. In magnitude, FNapproximately equals FZ. The expressions for the tidal forces, FA5(GmMsun/R2) 3(r/R) andFZgiven by Eq. ~1!, are valid also for the tidal forces produced on the earth by the moon if we replace Msun by the mass of the moon and Rby the moon–earth distance. There is no intrinsic difference between the sun-induced andmoon-induced tide-generating forces. In both cases, the onlyimportant factor is the acceleration of the earth under thegravitational pull of the celestial body that causes the tideson the earth, not the orbital velocities of both gravitationallycoupled bodies ~the earth and the sun, or the earth and the moon !. The tidal force experienced by any object is proportional to its distance rfrom the center of the earth and inversely proportional to the cube of the distance Rto the celestial body that causes the force, and is proportional to the mass ofthe source body.As noted, lunar tide-generating forces on theearth are more than twice those of the sun ~their ratio is approximately 2.2 !because the moon is much closer to the earth.III. TIDAL FORCES AT AN ARBITRARY POINTNEAR THE EARTH The standard derivation of tidal forces uses the tide- generating potential ~see, for example, Refs. 12 and 13 !for which the mathematics is somewhat simpler. However, toemphasize the physics underlying the origin of tide-generating forces, we consider the vector addition of the rel-evant forces, just as in the elementary treatment of Sec. II.Toobtain a general mathematical expression for the tide-generating force at an arbitrary point Dover the earth ~Fig. 2!, we introduce the radius vector rof this point measured from the center of the earth, and also the vector r s5R1r measured from the center of the sun, S, whereRis the vector of the center of the earth from the center of the sun. The tidal force Ftidexperienced by a body of mass mat pointDin the noninertial, nonrotating geocentric frame isthe vector sum of its gravitational attraction to the sun, Fsun 52GmMsunrs/rs3, and the force of inertia, Fin52ma0 5GmMsunR/R3: Ftid5Fsun1Fin52GmMsunSrs rs32R R3D. ~2! We express rsin Eq. ~2!as the vector sum R1rand calcu- late the square of rs. We take into account that r!Rand write rs25~R1r!25R212~R"r!1r2’R2S112~R"r! R2D.~3! To find an approximate expression for 1/ rs3in Eq. ~2!,w e raise the right-hand part of Eq. ~3!to the power ( 23/2). If we substitute the resulting value of 1/ rs3into Eq. ~2!forFtid, we obtain: Ftid’2GmMsun R3F~R1r!S123~R"r! R2D2RG ’2GmMsun R3Fr23R~R"r! R2G. ~4! We note that the main contributions of FsunandFintoFtid, whose magnitudes are inversely proportional to R2, cancel in Eq.~4!. This cancellation corresponds to the aforementioned state of weightlessness that we experience on the spaceshipEarth with respect to the sun’s gravity. For points AandBin Fig. 1,ris perpendicular to R, and hence the scalar product ~R"r!is zero. Therefore at these points the tidal force is directed opposite to r~that is, vertically downward !, and its magnitude equals GmM sun(r/R3). For points ZandN, the tidal force is directed along r~that is, vertically upward !, and its magnitude 2 GmMsun(r/R3) is two times greater than at pointsAandB. We see that at these four points, the general result given by Eq. ~4!agrees with the simpler calculations of Sec. II.IV. HORIZONTALAND VERTICAL COMPONENTSOF THE TIDAL FORCE The sun-induced tide-generating forces exerted on the earth have a quadrupole character: They stretch the earthalong the sun–earth line, and squeeze the earth in the direc-tions perpendicular to that line. Because of the axial symme-try with respect to the sun–earth line, the vertical and hori-zontal components of the tidal force depend only on theangle ushown in Fig. 2 ~and on the distance rfrom the center of the earth !. The angle udetermines the position of the mass point mon or near the surface of the earth mea- sured from this line. Figure 3 shows how the tidal forces are directed at differ- ent points near the earth. Because of axial symmetry aboutthe sun–earth line, Fig. 3 applies to any plane passingthrough the sun–earth line. The horizontal ~tangential to the surface !components of the tidal forces are much more influential on the ocean tidesand on the orbits of earth satellites than are the vertical ~ra- dial!components, which only modify slightly the earth’s gravitational force. For the horizontal component of the tidal Fig. 2. For calculation of the tide-generating force at arbitrary point D. 3 3 Am. J. Phys., Vol. 70, No. 9, September 2002 Eugene I. Butikov force at an arbitrary point D, whose geocentric position is determined by the two coordinates randu~in the plane shown in Fig. 2 !, Eq. ~4!yields: ~Ftid!hor523GmMsun R3rcosusinu 523Fsunr Rcosusinu523 2Fsunr Rsin2u,~5! whereFsun5GmMsun/R2is the gravitational pull of the sun on the body. The horizontal component of the tidal force iszero at points AandBand at all other points of the plane orthogonal to the line sun–earth ~for which u590°!, as well as at points NandZ~for which u50° and u5180°!. The horizontal component of the tidal force has its maximumvalue (3/2)( r/R)F sun5(3/2)(r/R)GmMsun/R2at all points on the earth for which u545° and u5135°. This maximal horizontal component of the solar tide-generating forcecauses a deviation of the plumb line from the direction of theearth’s own gravity only by 0.008 9. If we take the scalar product of the right-hand side of Eq. ~4!forFtidwith the unit vector r/r, we obtain the depen- dence of the vertical component ( Ftid)vertof the tidal force on the angle ubetweenRandr: ~Ftid!vert5GmMsun R3r~3 cos2u21! 53 2GmMsun R2r RScos2u11 3D. ~6! The last term on the right-hand side of Eq. ~6!is indepen- dent of uand is thus independent of time on the spinning earth. It can therefore be dropped as far as the tides areconcerned. This term in the vertical component of the tidalforce is the same everywhere on the earth ~for a given value ofr!and adds only a tiny constant value to the vertical force of the earth’s gravity ~about ten million times smaller than mg!. Thus, the vertical and horizontal components of the tidal force exerted on a body of mass mlocated at a position determined by angle uand radius rare given by: Fvert5~3/2!~r/R!Fsuncos2u, ~7!Fhor52~3/2!~r/R!Fsunsin2u, whereFsunis the total gravitational pull of the sun experi- enced by the body anywhere on the earth.This representationof the tide-generating force is especially convenient becauseEq.~7!defines a tidal force vector whose magnitude (3/2) 3(r/R)F sun5(3/2)GmMsunr/R3is independent of the angle u: The tidal forces at all points that lie at a given distance rfrom the earth’s center are equal in magnitude and differ onlyin direction. Equations ~5!–~7!also are valid for the tidal forces pro- duced by the moon, provided we replace the mass of the sun M sunby the mass of the moon Mmoonand the sun–earth distanceRby the moon–earth distance. In this case the angle uin Eq. ~7!determines the position of the body relative to the moon–earth line. The tide-generating force of the moon, Ftidal 5(3/2)GmMmoonr0/R3, experienced by a body of mass m on the surface of the earth ~r0is the earth’s radius !is very small compared to its weight—the earth’s force of gravityF grav5mg5GmMearth/r02. If we let the ratio Mmoon/Mearth 51/81 and the mean earth–moon distance R560r0~actually this distance varies between 57 r0and 63.7r0because of the elliptical shape of the moon’s orbit !, we obtain Ftidal/Fgrav5~3/2!~Mmoon/Mearth!~r0/R!3’8.631028. ~8! Although the maximal lunar tidal force on the surface of the earth is only about 1027of the earth’s gravitational force, its effect on the ocean water can be considerable because ofits horizontal component, which is orthogonal to the earth’sgravitational field and varies with time periodically becauseof the earth’s axial rotation. The horizontal component shiftsthe ocean water around the globe.V. THE STATIC DISTORTION OF THE WATERSURFACE To estimate the static ~equilibrium !distortion of the ocean’s surface due to the tidal forces, we can use the hypo-thetical situation of a nonrotating planet on which the tide-generating forces are nearly time-independent. From thesymmetry of tidal forces, Eq. ~7!, we can assume that the distorted surface has an ellipsoidal shape given by r ~u!5r01acos2u, ~9! where 2a!r0is the difference in the static maximal and minimal levels at points ZandA~see Fig. 3 !. Hence we can write for the small inclination aof the water surface with respect to the horizon: a51 rdr~u! du’22a r0sin2u. ~10! We see that the water surface is horizontal ( a50) at u50 andu590°~pointsZandA!. The angle ais maximum and equals 2a/r0atu545° and at u5135°, where the tidal force is directed horizontally. In equilibrium the distortedwater surface is orthogonal to the plumb line.The plumb lineshows the direction of the vector sum of the earth’s gravityand the tidal force.Asmall departure of the plumb line fromthe direction of the earth’s gravity is caused by the horizontalcomponent of the tidal force. Therefore, the angle aequals the ratio of the horizontal tidal force Fhorto the force of the earth’s gravity Fgrav5mg. If we equate a52a/r0atu 545° toFhor/Fgravand take into account that for sun- induced tides, Fhor/mg5(3/2)(Msun/Mearth)(r03/R3), we find for the maximal static level difference 2 aat points Z andA: 2a5~3/2!r0~Msun/Mearth!~r03/R3!. ~11! Fig. 3. Directions of the tidal forces at different equatorial points near theearth. 4 4 Am. J. Phys., Vol. 70, No. 9, September 2002 Eugene I. Butikov Equation ~11!yields 2a50.24 m. A similar expression is valid for the static distortion of the ocean surface due to thelunar tidal force, and yields 2 a50.54 m for the moon- induced static distortion. In Sec. VII the equation for thisstatic distortion is also derived from the tide-generating po-tential.VI. TIDAL FORCES ON THE ROTATING EARTH In the above we have used a revolving but nonrotating geocentric reference frame. The origin of this frame movesin a circle around the sun–earth ~moon–earth !center of mass, but the frame itself does not rotate because the direc-tions of its axes are fixed relative to the distant stars. That is,the frame moves translationally in a circle. This referenceframe is convenient for the analysis of a motion of an artifi-cial satellite. If we ignore the perturbations caused by tidalforces, the earth satellite traces out a closed elliptical orbitrelative to this reference frame. To introduce tidal forces on the rotating earth, we must use a true geocentric frame of reference that takes part in thedaily rotation of the earth. This frame is noninertial, andhence we should be concerned with the acceleration of itsdifferent points.We can consider the motion of the earth ~and of the geocentric reference frame !as consisting of two com- ponents. The first is the component considered above,namely translational motion ~revolution without rotation ! about the sun–earth ~moon–earth !center of mass. The sec- ond component is a uniform daily rotation ~spin!of the earth about an axis passing through the center of the earth. Both these motions of the earth are important in the prob- lem of tides, but the roles they play are quite different. Theacceleration a 0related to the translational motion is respon- sible for the origin of the uniform pseudoforce of inertiaF in52ma0, whose action on a body on the earth, combined with the nonuniform gravitational pull of the sun ~moon !,i s described by the tidal force Ftidconsidered previously. We note again that only the acceleration a0of this translational motion is important, not the orbital velocity of the earth.14To avoid confusion often encountered in the literature ~see, for example, Ref. 15 !, we must be careful with definitions. In discussing tides, we should be concerned only with thosegravitational and inertial forces that depend on the apparentposition of the celestial body that produces the tide. Theaxial rotation of the earth is related to the centripetal accel-eration and gives rise to centrifugal forces that increase inproportion to the distance from the earth’s axis. The centrifu-gal force of the earth’s daily rotation generally is muchgreater in magnitude than tidal forces. Because of the cen-trifugal forces, the equilibrium shape of the earth differsslightly from an ideal sphere—it is approximately an ellip-soid of rotation whose equatorial diameter is a bit greaterthan the polar diameter ~see, for example, Ref. 13 !. The cen- trifugal effect of the earth’s daily rotation causes an equato-rial bulge, which is the principal departure of the earth fromits spherical shape. 16 But we are not concerned here with this constant distortion of the earth because this distortion is independent of the ap-parent position of the celestial body that produces the tides.Therefore, the centripetal acceleration of the axial rotationadds nothing to tidal forces. However, the daily rotation ofthe earth makes tidal forces time-dependent because the pat-tern of tidal forces on the earth is coupled to the apparentpositions of the sun and moon. A dynamical response of theoceanic waters on the spinning earth to these time-dependentforces is the essence of the phenomenon of tides. Thus, in the problem of tides, expressions for the tide- generating forces F horandFvertin Eq. ~7!are applicable also to the true geocentric frame of reference, which takes part inthe daily axial rotation of the earth. The system of tidalforces shown in Fig. 3, being coupled to the apparent posi-tion of the sun ~moon !, rotates rigidly together with the earth–sun ~earth–moon !line. For simplicity, we shall con- sider the case in which the source celestial body ~the sun or moon !occurs in the equatorial plane of the earth. Although the system of tidal forces rotates as a whole with the angularvelocity Vof the earth’s axial rotation, that is, with a period of 2 p/V, the true period of variation of the tidal forces on the earth equals half this value ( T5p/V) because of the quad- rupole symmetry of the system of forces ~thesemidiurnal tide!. For the sun-induced tidal forces the period equals 12 h. For the moon-induced tidal forces the period is 12 h 25min—the difference between the periods is due to the orbitalmotion of the moon. If we fix a point on the equator of theearth, the local tidal force vector executes a uniform rotationin the vertical plane, making two complete revolutions dur-ing a day. The simulation clearly shows how the daily rota-tion of the whole system of tidal forces produces this doublyfast uniform rotation of the tidal force at a given equatorialpoint, as seen by an observer on the spinning earth. 10Be- cause of this periodic dependence on time, the tidal forces, inspite of their small magnitude compared even to the centrifu-gal force of inertia, produce the oceanic tides. To find analytical expressions for the time dependence of the tidal forces at a given point in the equatorial plane of thespinning earth, we substitute u5Vtin Eq. ~7!. This substi- tution yields the following expressions for the point of theequator at which the sun culminates ~passes through its ze- nith!att50: F vert~t!5Arcos2Vt, ~12!Fhor~t!52Arsin2Vt, whereA5(3/2)Fsun/R5(3/2)GmMsun/R3. At any other equatorial point of the earth, the tidal force vector also ro-tates in the vertical plane with angular velocity 2 V. That is, all the vectors at different points rotate synchronously butwith different phases.VII. THE POTENTIAL FUNCTION FOR TIDALFORCES An approach often used in deriving an expression for the tidal force is to begin with the potential energy of a bodyunder the influence of tide-generating forces. This approachis simpler than that presented above. However, we have cho-sen the above approach because it does not obscure the un-derlying physics and consequently may be considered advan-tageous to physics instructors. Nevertheless, forcompleteness, we introduce here the potential function,U tides(r,u), and show how it can be used in calculating the equilibrium shape of the surface of the ocean and the staticdistortion of the water under tidal forces. The components of the force that lie in the equatorial plane are given in Eq. ~7!and are the negative gradients of the potential function U tides(r,u): 5 5 Am. J. Phys., Vol. 70, No. 9, September 2002 Eugene I. Butikov Fvert5Arcos2u52]Utides~r,u!/]r, ~13!Fhor52Arsin2u52~1/r!]Utides~r,u!/]u. Therefore, the potential function for the tidal forces can bewritten as: U tides~r,u!52~1/2!Ar2cos2u 52~3/4!~GmMsun/R3!r2cos2u. ~14! The restoring forces that limit the tidal distortion of the water’s surface are due to the earth’s gravity. If the earthwere not rotating relative to the earth–sun line, the staticdistortion of the water surface covering the globe would bethe surface of equal total potential: U ~r,u!5U0~r!1Utides~r,u!5const, ~15! whereU0(r)5mgris the spherically symmetric potential function of the earth’s gravity which yields the radial com-ponent of the earth’s gravitational force 2dU 0(r)/dr 52mg. Thus, U~r,u!5mgr2~1/2!Ar2cos2u. ~16! In particular, at points ZandA~see Fig. 3 !of the water surface, the values of the total potential function, Eq. ~16!, are equal: U(rZ,p)5U(rA,p/2), from whence we obtain mgrZ2~1/2!ArZ25mgrA1~1/2!ArA2, ~17!mg~rZ2rA!5~1/2!A~rA21rZ2!. We can use this condition to determine the static equilibriumdistortion under the tidal forces of the otherwise sphericalocean surface. Let the radii of the distorted water surface atpointsZandAber Z5r01aandrA5r02a, respectively, wherer0is the radius of the undistorted surface. Then 2 ais the static level difference at points ZandAin which the level is maximum and minimum, respectively. Thus, fromEq.~17!we have 2 mga 5(1/2)A(r Z21rA2)’Ar02, and for 2 a we obtain: 2a5Ar02/~2mg!5~3/2!r0~Fsun/mg!~r0/R!. ~18! We note that Fsun/mg5(Msun/Mearth)(r02/R2), so that the static distortion of the ocean surface under the sun-inducedtidal forces can also be expressed as: 2a5 ~3/2!r0~Msun/Mearth!~r03/R3!. ~19! This expression is the same as Eq. ~11!derived by requiring that in equilibrium the surface of the ocean be orthogonal tothe vector sum of the earth’s gravitational force and the tidalforce.VIII. THE NATURAL WAVE AND THE DRIVINGTIDAL FORCES Most authors oversimplify the problem of tides and con- sider ~after Newton and Bernoulli !only the so-called static ~orequilibrium !theory of tides, which treats the ocean sur- face as a liquid ellipsoid stretched alongthe earth–moon ~earth–sun !line, as if this surface were always in equilib- rium under the earth’s force of gravity and tidal forces pro-duced by the moon ~sun!. In this approach, the tidal bulges are aligned with the earth–moon ~or earth–sun !axis. There- fore on the spinning earth the moments of high water at agiven location should coincide with the upper and lower cul-minations of the moon ~sun!, that is, when the moon ~sun! passes through its zenith and nadir. Observations do notagree with this prediction. Instead, almost the opposite isusually observed: the moments of low tide occur approxi-mately at the culminations of the moon. Acomplete theory of the tides should take into account the dynamical response of the ocean to the time-dependent gen-erating forces. The dynamical theory of tides ~first suggested by Laplace and developed by Airy !treats the tides as a steady-state forced motion ~under varying tidal forces !of a dynamical system ~the ocean !. 17Such a theory predicts a resonant growth of the steady-state amplitude in cases whenthe driving period approaches the period of natural oscilla-tions. To avoid the complications related to the three- dimensional character of the problem and to explain thephysical aspect of the dynamical theory using the simplestpossible model, we imagine, followingAiry, water in a widecanal of uniform depth engirdling the entire earth along theequator. Imagine the water surface in this canal being dis-torted statically under the tide-generating forces so that twobulges form on opposites sides of the earth, changing theshape of the surface from circular to elliptical. If the forcesmaintaining this shape suddenly vanish, the earth’s gravitywould make the distorted surface restore its equilibrium, cir-cular shape. The water would start to flow and the bulgesdisappear so that after a time, namely a quarter period, thewater surface would become circular. But because the watercontinues to move, after another quarter period the bulgesreappear in new positions showing an elliptical distortion ofthe surface along the line perpendicular to the line of theoriginal distortion. Then the motion repeats itself in reverse.This motion of water in the circular canal is a gravitationalstanding surface wave whose wavelength equals half-circumference of the globe. Such a mode of oscillation ischaracterized by a certain natural period. The superposition of two such standing waves whose phases differ by p/2 and whose elliptical axes are separated by 45° produces a circulating ~traveling !wave of constant elliptical shape and a wavelength equal to half of the earth’scircumference. The two opposite bulges in the water surfacetravel with this wave around the globe preserving theirheight and shape. 10 An essential point in explaining the steady-state phase shift between the moments of high tide and the culminationof the moon ~sun!is the relation between the natural period T 0of this circulating wave and the period Tof the tide- generating driving forces. It is possible to estimate T0as the time taken by the circulating surface wave to travel alonghalf the globe. In the limiting case of very long waves on thesurface of shallow water ( l@h) the speed of wave is deter- mined by the earth’s gravity gand depth h, and is indepen- dent of l. From hydrodynamics we know that this speed equals (gh) 1/2~see, for example, Ref. 18, p. 405 !. We as- sume that the mean value hof the ocean depth is 3.5 km. During a period T0, the wave travels half the circumference of the globe pr0, and hence T05pr0/(gh)1/2’30 h. Thus, the approximately 12-h driving external period Tis less than the natural period T0of the free oscillation. An essential point in explaining the steady-state phase shift between the moments of high tide and the culminationof the moon ~sun!is the relation between the natural period T 0of this circulating wave and the period Tof the tide- 6 6 Am. J. Phys., Vol. 70, No. 9, September 2002 Eugene I. Butikov generating driving forces. It is possible to estimate T0as the time taken by the circulating surface wave to travel alonghalf the globe. In the limiting case of very long waves on thesurface of shallow water ( l@h) the speed of wave is deter- mined by the earth’s gravity g and depth h, and is indepen- dent of l. From hydrodynamics we know that this speed equals (gh) 1/2~see, for example, Ref. 18, p. 405 !. We as- sume that the mean value hof the ocean depth is 3.5 km. During a period T0, the wave travels half the circumference of the globe pr0, and hence T05pr0/(gh)1/2’30 h. Thus, the approximately 12-h driving external period Tis less than the natural period T0of the free oscillation. We emphasize that it is the shape of the surface ~the wave ! that circulates around the globe, not the water itself . Relative to the earth, points on the surface of the ocean execute os-cillatory motions in closed paths that are considerablystretched horizontally. On the average, the water is stationaryin the geocentric frame. To obtain the dynamical picture of tides on the rotating earth, we should use the reference frame that rotates with theearth. Relative to this frame, the quadrupole system of tide-generating forces, being coupled to the position of the sun~moon !, rotates as a whole while the sun ~moon !travels along its apparent daily path around the earth. This rotationof the forces occurs at an angular velocity V, the angular velocity of the earth’s daily rotation ~or the difference be- tween Vand the angular velocity of the moon in its orbit for moon-induced tides !. Such a uniform rigid rotation of the system of mutually fixed vectors can be represented as asuperposition of two oscillating quadrupole systems of forces~with a frequency v52V!that do not rotate and whose axes make an angle of 45° to one another. At each point one ofthese forces oscillates along the radial ~vertical !direction, while the other force—along the tangential ~horizontal !di- rection. The oscillations of these orthogonal components oc-cur a quarter period out of phase. At any given point in theequatorial plane, the vector sum of these mutually orthogonaloscillating forces produces a force of constant magnitudewhose direction rotates uniformly following the apparentmotion of the sun ~moon !, but with angular velocity v 52V.10For different points on the earth, the phases of these rotating vectors differ.IX. THE TIDES AS FORCED OSCILLATIONS OFTHE OCEAN What is really of interest is the steady-state forced oscil- lation of the ocean surface due to the time-dependent tidalforces. Each of the two oscillating systems of forces de-scribed above excites a mode of forced oscillation of thewater in the equatorial canal, specifically the mode of thesame symmetry as is characteristic of the corresponding sys-tem of driving forces. These modes have elliptical shapes,much like the natural oscillations considered above, namely,the elliptical standing waves whose axes make an angle of45° with one another. Nevertheless, we can consider thesemodes to be orthogonal in the sense that their spatial formsare described by eigenfunctions forming an orthogonal basisin the function space. The two forced oscillations in thislinear system, each excited by one system of oscillating driv-ing tidal forces, are independent of one another, and the re-sulting forced motion is a superposition of these forced os-cillations.Any steady-state forced oscillation occurs exactly with the period of the driving force. The amplitude and phase lag ofthe oscillation depend on the amplitude of the driving force,on the damping factor, and, more importantly, on the relationbetween the driving and natural periods. The two systems ofoscillating driving tidal forces are characterized by equal am-plitudes and frequencies. Also the natural frequencies anddamping factors of both excited modes are equal. Hence bothexcited modes also have equal amplitudes and equal phasedelays behind the corresponding driving forces. The super-position of these modes produces a forced circulating ~trav- eling!elliptical wave that has the same phase relation with the rotating driving forces as is characteristic of forced os-cillations in general. If weignore friction ~dissipation of mechanical energy in the excited wave motion !, the forced motion occurs exactly in phase with the driving force, provided the driving period is longer than the natural period. Otherwise the forced mo-tion occurs in the opposite phase with respect to the driving force. For the simplified model of tides in the equatorialcanal of uniform depth ~and also for an earth covered every- where by an ocean of uniform depth !, the natural period of free oscillation is longer than the 12-h driving period. Thusthe dynamical theory predicts in this case a stationary circu-lating elliptically shaped wave whose axis ~the line of tidal bulges !isperpendicular to the earth–sun ~earth–moon !line. On the other hand, the natural period of an elastic wave in the crust of the earth is shorter than the 12-h period of thetidal forces. Hence, in the frictionless model, bulges in theearth’s crust are oriented alongthe earth–sun ~earth–moon ! line. Observations show that the solid body of the earth ac-tually experiences twice-daily tides with maximum ampli-tude of about 30 cm whose bulges lag approximately 3°behind the earth–moon line. 17 X. MATHEMATICAL DESCRIPTION OF THEFORCED OSCILLATIONS Each of the partial forced oscillations can be described by a differential equation of a linear oscillator. Let q 1(t) be the normal coordinate describing the first forced oscillationwhose elliptical shape is characterized by a major axis ori-ented along the earth–sun line ~and in the perpendicular di- rection after a half period !, and let q 2(t) be the normal co- ordinate describing the second oscillation with the axisinclined 45° to the earth–sun line.Adisturbance of the watersurface caused by the first oscillation can be described byDr 1(u,t)5q1(t)cos(2 u), which gives the small vertical dis- placement of the surface at an arbitrary point ( r0,u) of the equator. Similarly, the second oscillation causes a distortionof the surface described by Dr 2(u,t)5q2(t)sin(2 u). The forced oscillations experienced by the normal coordinatesq 1(t) andq2(t) are periodic ~steady-state !partial solutions of the two differential equations: q ¨112gq ˙11v02q15v02acosvt, ~20!q ¨212gq ˙21v02q25v02asinvt. Here v0is the natural frequency of the corresponding mode (v052p/T052(gh)1/2/r0),gis the damping constant, v 52Vis the driving frequency, and ais the magnitude of the equilibrium distortion of the ocean surface under the staticsystem of tidal forces ~that is, the distortion for the planet 7 7 Am. J. Phys., Vol. 70, No. 9, September 2002 Eugene I. Butikov whose axial rotation is synchronized with its orbital revolu-tion!. The theoretical value of a is given by Eq. ~11!or~18!. Although the values of vandaare fairly well known, the situation is quite different regarding the values of v0andg. In the limiting case of extremely slow rotation of the earth, the steady-state solution of Eq. ~20!isq1(t)5acosvt, q2(t)5asinvt. This solution describes the quasistatic ellip- tical distortion whose axis follows adiabatically the slowlyrotating earth–sun ~earth–moon !line. The major axis of the ellipse at any moment is oriented along this line. The dis-placement of the water level from its mean position in theequatorial plane in this limiting case is given by Dr ~u,t!5Dr1~u,t!1Dr2~u,t! 5q1~t!cos2u1q2~t!sin2u 5a~cos2Vtcos2u1sin2Vtsin2u! 5acos2 ~Vt2u!. ~21! To find the distortion of the water surface for an arbitrary value of v, we can use the relevant well-known steady-state solution to Eq. ~20!for the normal coordinates q1(t) and q2(t): q1~t!5q0cos~vt2d!,q2~t!5q0sin~vt2d!,~22! where their common amplitude q0and phase lag dare given by q05v02a A~v022v2!214g2v2, ~23! tand52gv v022v2. ~See, for example, Ref. 18, pp. 372–373. !Therefore the re- sultingdistortionofthewatersurfaceunderthetidalforcesisgiven by Dr ~u,t!5Dr1~u,t!1Dr2~u,t! 5q1~t!cos2u1q2~t!sin2u 5q0@cos~2Vt2d!cos2u1sin~2Vt2d!sin2u# 5q0cos2 ~Vt2d/22u!. ~24! We see from Eq. ~24!that at any time t the maximum ~high water !of the tidal wave circulating around the earth is located at the position defined by the angle umax5Vt2d/2. That is, the position of the maximum lags behind the sun~moon !by the angle d/2. If g!v, it follows from Eq. ~23! that this retarding angle is almost zero if v,v0. In other words, the marine tide would be nearly the equilibrium tidewith the high-water time coinciding with culminations of thesun~moon !if the natural period of the circulating wave were less than the 12-h driving period ~that is, if T 0,T!. How- ever, for our model of the ocean, we estimate the naturalperiod to be close to 30 h. Therefore the situation corre-sponds to v.v0, when the steady-state forced oscillations occur nearly in the opposite phase relative to the drivingforce. In this case the tide should be inverted with respect tothe equilibrium one. The retarding angle d/2 approaches p/2 according to Eq. ~23!, which means that for a given equato- rial point, the high water occurs when the sun ~moon !is almost at the horizon ~rather than at zenith or nadir !.At any given place on the equator, it follows from Eq. ~24! that the water level ~above the average value !varies with t according to z(t)5q0cos(2 Vt2d), where t50 corresponds to the culmination of the sun ~moon !at the place in question. We can expect that for the model of a water canal of uniformdepth, the value of q 0given by Eq. ~23!is more or less reliable because hydrodynamics allows us to estimate thenatural frequency v052p/T052(gh)1/2/r0by using the known speed n5(gh)1/2of very long gravitational waves. However, considerable uncertainty is related to the dampingfactor g. If we assume that the damping is small ( g!v0), we can conclude that the orientation of the tidal bulges de-viates only slightly from the line perpendicular to the sun–earth ~moon–earth !line, but the particular value of this de- viation remains indefinite. In the above discussion, we considered only the steady- state oscillation of the ocean surface ~the stationary wave !, assuming that the transient is already over. For this steadymotion to establish itself, some friction ~even if very small ! is necessary. In the problem under consideration, we are con-cerned with the water motion caused solely by the eternaltidal forces, and therefore we have had centuries and evenmillennia to wait for the fading away of the transient. There-fore our use of the steady-state solution is appropriate fortides. We also emphasize that in the dynamical theory oftides, the driving tide-generating forces are perfectly wellknown, so that most uncertainties originate primarily from avery poor correspondence between the simple model of thedynamical system and the real oceans of the earth.XI. REAL-WORLD COMPLICATIONS The pattern of tide-generating forces is coupled to the po- sition of the moon ~and the sun !with respect to the earth. For any place on the earth’s surface, the relative position of themoon has an average periodicity of 24 h 50 min. The lunartide-generating force experienced at any location has thesame periodicity. When the moon is in the plane of the equa-tor, the force runs through two identical cycles within thistime interval because of the quadrupole symmetry of theglobal pattern of tidal forces. Consequently, the tidal periodis 12 h 25 min in this case ~the period of the semidiurnal lunar tide !. However, the lunar orbit doesn’t lie in the plane of the equator, and the moon is alternately to the north and tothe south of the equator. The daily rotation of the earth aboutan axis inclined to the lunar orbital plane introduces anasymmetry in the tides. This asymmetry is apparent as aninequality of the two successive cycles within 24 h 50 min. Similarly, the sun causes a semidiurnal solar tide with a 12-h period, and a diurnal solar tide with a 24-h period. In acomplete description of the local variations of the tidalforces, still other partial tides play a role because of furtherinequalities in the orbital motions of the moon and the earth.In particular, the elliptical shape of the moon’s orbit pro-duces a 40% difference between the lunar tidal forces at theperigee and apogee of the orbit. Also the inclination of themoon’s orbit varies periodically in the interval 18.3°–28.6°,causing a partial tide with a period of 18.6 yr. The interfer-ence of the sun-induced tidal forces with the moon-inducedtidal forces ~the lunar forces are about 2.2 times as strong ! causes the regular variation of the tidal range between spring tide, when the range has its maximum ~occurring at a new moon and at a full moon, when the sun and moon are in thesame or in the opposite directions !, andneap tide , when the 8 8 Am. J. Phys., Vol. 70, No. 9, September 2002 Eugene I. Butikov range has its minimum ~which occurs at intermediate phases of the moon !. The amplitude of a spring tide may be 2.7 times the amplitude of a neap tide. Because the earth is not surrounded by an uninterrupted water envelope of equal depth, but rather has a very irregulargeographic alternation of land and seas with complex floorgeometry, the actual response of the oceans and seas to thetidal forces is extremely complex. In enclosures formed bygulfs and bays, the local tide is generated by an interactionwith the tides of the adjacent open ocean. Such a tide oftentakes the form of a running tidal wave that circulates withinthe confines of the enclosure. In some nearly enclosed seas,such as the Mediterranean, Black, and Baltic seas, a steady-state oscillation in the form of a standing wave, or tidal se-iche, may be generated by the tidal forces. In these seas, thetidal range of sea level is only on the order of centimeters. Inthe open ocean, it generally is on the order of decimeters. In bays and adjacent seas, however, the tidal range may be much greater because the shape of a bay or adjacent sea mayfavor the enhancement of the tide inside. In particular, theremay be a resonance response of the basin concerned with thetide. Tides are most easily observed along seacoasts, wherethe amplitudes are exaggerated. When tidal currents run intothe shallow waters of the continental shelf, their rate of ad-vance is reduced, the energy accumulates in a smaller vol-ume, and the rise and fall are amplified. The details of tidalmotions in coastal waters, particularly in channels, gulfs, andestuaries, depend on the details of coastal geometry andwater-depth variation over a complex sea floor. Tidal ampli-tudes and phase lags, the contrast between spring and neaptides, and the variation of times of high and low tide allchange widely from place to place. For the aforementioned reasons, a purely theoretical cal- culation of the times and heights of tides at a particular lo-cation is practically impossible. Nevertheless, for a givenplace on a coast, the tides can be quite successfully predictedon the basis of accumulated long-term observations of thetides at the place concerned. The analysis of the observationsrelies on the fact that any tidal pattern in time is a superpo-sition of variations associated with periodicities in the mo-tions of the moon and the sun relative to the earth. Theperiods involved are the same everywhere on the earth, butthe relative amplitudes and phases of their contributions arehighly variable from one place to another. Observations overa sufficient time make it possible to calculate which contri-butions are significant at a particular location and, thus, toforecast tidal times and heights. It is common that 40 har-monic components may be significant for practical calcula-tions at one location. 17 XII. THE EVOLUTION OF ORBITAL MOTIONSAND SPINS OF CELESTIAL BODIES INDUCED BYTIDAL FORCES When the forced motion occurs exactly in the same or opposite phase with respect to the driving force, no energyexchange occurs on average between the external source andthe oscillatory system. To explain the secular variation ~the retardation !of the earth’s axial rotation under the tidal forces, we have to take friction into account. One may wonder why the dissipation of mechanical en- ergy in the tides has a scale that seems very modest. Thepoint is that only the wavecirculates around the globe, not the water itself. The phase lag dof the steady-state forcedoscillation behind the periodic driving force is determined byEq.~23!. For the mode of oscillations in which we are inter- ested, this phase-frequency characteristic is almost a stepfunction ~zero for v,v0, that is, for T.T0, and 2poth- erwise !. Only near resonance ( v’v0) is this step slightly smoothed over. Therefore the displacement of the tidal waterbulges from the line perpendicular to the sun–earth ~moon– earth!axis is very small. However, this displacement, which destroys the symmetry of the system ~Fig. 4 !, is absolutely necessary in principle in order that the driving tidal forces be capable of maintainingthe circulating tidal wave ~that is, of preventing it from damping out !. If the earth is taken as the reference frame, we can see that by virtue of this phase shift and the correspond-ing displacement of bulges, the tidal forces exert a retardingtorque relative to the earth’s axis and thus do nonzero network on the system.This work compensates for the frictionallosses experienced by the tidal traveling surface wave andmeasures the gradual reduction of the mechanical energy ofthe system. The energy is provided by the axial rotation~spin!of the earth. Hence the spin secularly slows down and the angular momentum of the axial rotation diminishes. Looking at the whole system from the inertial reference frame, we should remember that the sun ~moon !interacts with the earth only by its central gravitational force. If thebulges were oriented exactly along or perpendicularly to thesun earth ~moon–earth !axis, this gravitational force would not exert a torque on the earth. If we consider the gravita-tional forces F 1andF2~Fig. 4 !exerted on the bulges, we conclude that the retarding torque about the earth’s axis,which slows down the axial rotation, is due to the above-mentioned displacement of the bulges which destroys thesymmetry of the system with respect to the earth–sun~earth–moon !line. However, the total torque of the central gravitational field of the sun ~moon !exerted on the earth and the bulges of its liquid shell, measured relative to the sun ~or to the moon for moon-induced tides !,i szero. Hence the total angular mo- mentum of the system is conserved, as it should be in anyclosed system. The diminishing of the earth’s spin due totidal friction means that the orbital momentum of the systemslowly increases during the tidal evolution. The earth’s orbitgradually expands. The lack of symmetry ~produced by tidal friction !does not influence the conservation of total angular momentum, although it causes a slow secular redistributionof the angular momentum between the spin and the orbitalmotion. As the orbit expands, the mechanical energy of theorbital motion also increases. This additional mechanical en-ergy, as well as the dissipated energy, is borrowed from theenergy of axial rotation. 19 This conclusion about expanding the moon’s orbit, derived from the conservation of angular momentum, is often en- Fig. 4. Gravitational interaction between the moon and the tidal bulges. 9 9 Am. J. Phys., Vol. 70, No. 9, September 2002 Eugene I. Butikov countered in the literature ~see, for example, Ref. 20 !. Al- though quite convincing, it nevertheless leaves the actualmechanism unexplained. To understand the physical reasonfor this phenomenon, it helps to take the forces into account.If we consider the properties of the gravitational forces F 18 andF28~see Fig. 4 !that are exerted on the moon by the earth’s tidal bulges and their influence on the orbital motion,we draw attention to a subtle peculiarity that deserves dis-cussion. While the orbit expands, the orbital velocity of themoon diminishes. However, from the asymmetry in the con-figuration that is responsible for the evolution, we can con-clude that the resultant gravitational force exerted on themoon by the tidal bulges is directed forward, in the direction of the orbital motion. How can this accelerating force slowdown the orbital motion? All authors who write about tidalevolution leave this question unanswered. This situation is similar to the widely known paradox of an earth satellite in a circular orbit that gradually descends inthe rarified upper atmosphere: Intuitively we expect that theweak atmospheric drag should slow down the satellite, butinstead, the satellite gains speed as its orbit gradually de-creases. Because of air resistance, the satellite is acceleratedin the direction of its motion, as if the retarding force of airresistance were pushing the satellite forward.An explanationof this so-called aerodynamical paradox of the satellite canbe found in Ref. 21. To understand the slowing down of the moon during tidal evolution, we must take into account that the moon graduallyspirals away from the earth and its orbit spreads out, so thatthe actual motion of the moon occurs along an expandingspiral. A portion of this trajectory ~with a strongly exagger- ated expansion !is shown schematically in Fig. 5. Because of this expansion, the perpendicular to the trajectory is directednot to the center of the earth but rather slightly in front of thecenter. Therefore the main gravitational pull Fexerted on the moon by the earth has a retarding tangential component F t directed back along the trajectory. This component is greaterin magnitude than the forward-directed tangential componentofF 18andF28~see Fig. 4 !that are exerted on the moon by the tidal bulges ~this component is not shown in Fig. 5 !. Hence the total tangential acceleration of the moon is directedagainst the velocity. Generally, in order to explain tidal evolution, that is, the reduction of spin and the secular variation of the orbits ofgravitationally coupled celestial bodies, it is necessary totake into account both the dynamic distortion of the sphericalshape of the body ~and of its liquid shell, if any !under the tidal forces, and the additional displacement of the bulgescaused by tidal friction. The nonuniform gravitational fieldof one body in an orbit about another distorts the shape ofthe second. The dissipation of energy stored in the resultanttidal distortions leads to a coupling that causes secularchanges in the orbit and in the spins of both bodies. Retar-dation of the axial rotation and evolution of the orbit willcontinue until the axial rotation is synchronous with themean orbital revolution. This effect is vital to an understanding of the history of the earth and moon. That the moon always keeps the same faceturned toward the earth is attributed to the past effects oftidal friction in the moon. The dissipation of tidal energy onthe earth results in a slowing of the earth’s axial rotationwhile the moon’s orbit is gradually expanding. Both the cur-rently observed increase in the length of the day of 0.0016s/century and the recession of the moon of 3 to 4 cm/yr areunderstood as consequences of the tides raised by the moonon the earth. Billions of years from now the moon will be sofar from the earth that the duration of the month will beequal to the duration of the day. The tidal evolution of thesystem ends with synchronization of the axial rotation ofboth orbiting bodies with their orbital revolution. The lengthof both the day and month in this final state of coherentrotation will be approximately 50 present days, as can becalculated on the basis of angular-momentum conservation~see, for example, Ref. 13 !. Similarly, tidal effects on the earth influence its axial rotation and its orbital revolutionaround the sun. 22 Tidal dissipation accounts for the current states of axial rotation of several planets, the spin states of most of theplanetary satellites, and the spins and orbits of close binarystars. For example, all the major and close planetary satel-lites in the solar system ~with the exception of Saturn’s sat- ellite Hyperon !are observed to be rotating synchronously with their orbital motion. The distant planet Pluto and itssatellite Charon are the pair in the solar system that has al-most certainly reached the end point where further tidal evo-lution has ceased. In this state the orbit is circular, with bothbodies rotating synchronously with the orbital motion andboth spin axes perpendicular to the orbital plane. Similarly,many close binary stars are observed to have circular orbitsand synchronized spins, providing numerous examples ofevolution under tidal forces elsewhere in the MilkyWay.Therole of tides in the cosmogony was first recognized by theastronomer George Darwin, who developed a theory of theheavenly evolution under tidal friction. 23 Another interesting manifestation of the tidal forces is the Roche limit , the minimum distance to which a large ~natural ! satellite can approach its primary body without being tornapart by tidal forces.To evaluate this critical distance R c,w e can equate the vertical tidal force, Eq. ~6!, exerted on a mass point located at u50o ru5pon the surface of a satellite of radiusrsatand mass msatby its primary of mass M, and the force of self-gravitation of the satellite ~that is, the force of gravitational attraction of this mass point mto the satellite !: 2GmM Rc3rsat5Gmmsat rsat2, whence Fig. 5. The main ~central !gravitational pull of the earth exerted on the moon. 10 10 Am. J. Phys., Vol. 70, No. 9, September 2002 Eugene I. Butikov Rc5rsatA32M msat5rplanetA32r rsat. ~25! In Eq. ~25!rplanetis the radius of the primary, ris its mean density, and rsatis the satellite’s mean density. If the satellite and its primary are of similar composition ( r’rsat), the the- oretical limit is about 21/351.26 times the radius of the larger body. The famous rings of Saturn lie inside Saturn’sRoche limit and may be the debris of a demolished moon.ACKNOWLEDGMENTS The author is grateful to Professor Hermann Ha ¨rtel of Kiel University for initiating an interesting discussion regardingthe common understanding of the physics behind the tidalphenomena. This discussion stimulated the development ofthe simulation program 10that visualizes and clarifies the dy- namical approach to the oceanic tides. I also appreciate thegenerous helpful assistance of Professor Stefan Machlup ofCase Western Reserve University and Professor Robert Bre-hme of Wake Forest University. a!Electronic mail: [email protected] 1E. Tsantes, ‘‘Note on the tides,’’Am. J. Phys. 42~4!, 330–333 ~1974!. 2P. Seligmann and M. Steinberg, ‘‘Simple hydrodynamic treatment of oceantides,’’Am. J. Phys. 43~12!, 1106–1108 ~1975!. 3E. Horsfield, ‘‘Cause of the earth tides,’’ Am. J. Phys. 44~8!, 793–794 ~1976!. 4A. B. Arons, ‘‘Basic physics of the semidiurnal lunar tide,’’Am. J. Phys.47~11!, 934–937 ~1979!. 5G. M. Kapoulitsas, ‘‘On the generation of tides,’’ Eur. J. Phys. 6~3!, 20–207 ~1985!. 6L. M. Celnikier, ‘‘Tidal braking of the Earth’s rotation—a study in cub-ism,’’ Eur. J. Phys. 11~1!,6 0 –6 2 ~1990!. 7J. M.Woosley, ‘‘Satellites and tides,’’Phys. Educ. 29~3!,177–179 ~1994!. 8H. Ha¨rtel, ‘‘The tides–a neglected topic,’’ Phys. Educ. 35~1!,4 0 – 4 5 ~2000!. 9J. Viiri, ‘‘Student’s understanding of tides,’’ Phys. Educ. 35~2!, 105–110 ~2000!. 10E. Butikov, The Ocean Tides , a set of Java applets at http://www.ifmo.ru/ butikov/Projects/Tides0.html. 11A very clear physical explanation of tidal forces in which the concept ofpseudoforces is avoided can be found in the textbook Physics, S. Machlup ~Wiley, New York, 1988 !. See the essay on pp. 125–128 on ‘‘Are we lighter at noon than at midnight?’’Asimilar approach to the effects of solargravity on the orbiting earth is related to Einstein’s principle of equiva-lence: The situation in a noninertial reference frame that is falling freely ina gravitational field is equivalent to what would happen in an inertialframe in the absence of this gravitational field. A freely falling closedcabin of an elevator is often used in gedanken experiments that appeal tothe principle of equivalence. Because the earth in its orbital motion isfalling freely in the gravitational field of the sun, our planet is similar tosuch a falling cabin in the sense that the sun doesn’t interfere much due itsgravity in our earthly dealings. However, the freely falling cabin is exactlyequivalent to an inertial reference frame only when the gravitational fieldis uniform. In a nonuniform gravitational field, such as the field of the sun,the equivalence holds approximately only for a small enough cabin. Finitedimensions of the earth ~the freely falling ‘‘cabin’’ !cause departures from the exact equivalence, which are revealed as tidal forces. In other words,we can treat tidal forces as a manifestation of the local nature of theequivalence principle. 12D. E. Cartwright, ‘‘Oceanic tides,’’ Rep. Prog. Phys. 40~6!, 665–708 ~1977!.13F. D. Stacey, Physics of the Earth ~Wiley, New York, 1969 !, 2nd ed. 14As we have emphasized, only the translational acceleration of the earth isessential for the explanation of tides: The tidal forces would have been thesame in an imaginary scenario of the earth and sun ~earth and moon !freely falling toward their common center of mass under mutual gravitation,without the orbital motion or revolution about this center. The accelerationof the earth caused by its rotation ~related both to the monthly revolution about the earth–moon center of mass or to the daily spinning motion !adds nothing to the tidal forces. A misunderstanding of the role of rotation canalso be a cause of confusion. For example, in Ref. 15 it is written that‘‘The planet and the satellite orbit around their mutual center of mass. Thecentrifugal acceleration that results from this whirling varies with distancefrom the center of mass across the planet. This effect, together with thedifferential force of the satellite’s gravity across the planet, is responsiblefor the rising of two tidal bulges.’’ However, only the second effect isresponsible for producing the tides, while the variation of centripetal ac-celeration across the planet influences only the nonuniform but constant~time independent !centrifugal force, which only insignificantly modifies the earth’s gravity ~modifies only the static equilibrium shape of the earth ! and thus cannot be responsible for the origin of moving tidal bulges. 15P. Goldreich, ‘‘Tides and the earth-moon system,’’Sci.Am. 226~4!,42–52 ~1972!. 16The equatorial bulge of the earth produced by the time-independent cen-trifugal force of inertia is responsible for the gradual cyclic change in thedirection of the earth’s axis of rotation ~with a period of almost 26,000 yr !. This precession was the third-discovered motion of the earth, after the farmore obvious daily rotation and annual revolution. Precession is caused bythe gravitational influence of the sun and the moon acting on the earth’sequatorial bulge. For tides, the time-independent oblateness of the earth~caused by the daily axial rotation of the earth !is inessential, and the earth can be taken as ideally spherical. 17Encyclopedia Britannica CD, see http://www.eb.com. 18A. G. Alenitsyn, E. I. Butikov, and A. S. Kondratyev, Concise Handbook of Mathematics and Physics ~CRC Press, Boca Raton, FL, 1997 !. 19The redistribution of energy during tidal evolution is also a subject ofsome confusion in the literature. In Ref. 15 the motion of the moon in itsorbit is erroneously indicated as one of the sources of the mechanicalenergy dissipated by tidal friction. However, as the moon’s orbit expands,the mechanical energy of the orbital motion increases. This mechanicalenergy, as well as the energy dissipated by tidal friction, also has its sourcein the energy of axial rotation of the earth. 20M. Zeilik, Astronomy: The Evolving Universe ~Harper and Row, New York, 1982 !, 3rd ed. 21E. Butikov, Planets and Satellites ~computer simulation programs !, User’s Manual ~American Institute of Physics, Physics Academic Software, New York, 1998 !, pp. 39–40. 22The variation of the orbital motion during the tidal evolution is also inter-esting in the sense that it disproves a common belief that internal forcescannot influence the motion of the center of mass of a system. This state-ment is true only for a system moving in a homogeneous external field.The aforementioned expanding of the earth’s orbit during the tidal evolu-tion is actually caused ~though indirectly !by internal forces, namely by gravitational forces between the ocean water and the hard body of theearth ~earth’s self-gravity !, and frictional forces. The internal forces change the configuration of the system ~consisting of the earth and the ocean water !that moves in the nonhomogeneous external gravitational field. Therefore these internal forces change ~indirectly !the resultant ex- ternal gravitational force exerted on the system. These variations of theexternal gravitational force modify the orbital motion of the system. Dur-ing tidal evolution, the internal forces of mutual gravitation together withfriction cause the redistribution of masses ~of the earth and tidal bulges ! that move in the external nonhomogeneous central field. 23George H. Darwin, The Tides and Kindred Phenomena in the Solar System ~Freeman, San Francisco, 1962 !. 11 11 Am. J. Phys., Vol. 70, No. 9, September 2002 Eugene I. Butikov