A dynamical picture of the oceanic tides

American Journal of Physics

Volumn 70, Issue 10, 2002, Pages 1001-1011

  Butikov, Eugene I ^a  

^a ST PETERSBURG STATE UNIVERSITY   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 23044532486     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1498858     Document Type: Article

References (23)

1. E. Tsantes, "Note on the tides," Am. J. Phys. 42 (4), 330-333 (1974).
2. P. Seligmann and M. Steinberg, "Simple hydrodynamic treatment of ocean tides," Am. J. Phys. 43 (12), 1106-1108 (1975).
3. E. Horsfield, "Cause of the earth tides," Am. J. Phys. 44 (8), 793-794 (1976).
4. A. B. Arons, "Basic physics of the semidiurnal lunar tide," Am. J. Phys. 47 (11), 934-937 (1979).
5. G. M. Kapoulitsas, "On the generation of tides," Eur. J. Phys. 6 (3), 20-207 (1985).
6. L. M. Celnikier, "Tidal braking of the Earth's rotation-a study in cubism," Eur. J. Phys. 11 (1), 60-62 (1990).
7. J. M. Woosley, "Satellites and tides," Phys. Educ. 29 (3), 177-179 (1994).
8. H. Härtel, "The tides-a neglected topic," Phys. Educ. 35 (1), 40-45 (2000).
9. J. Viiri, "Student's understanding of tides," Phys. Educ. 35 (2), 105-110 (2000).
10. E. Butikov, The Ocean Tides, a set of Java applets at http://www.ifmo.ru/butikov/Projects/Tides0.html.
11. A very clear physical explanation of tidal forces in which the concept of pseudoforces 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?" A similar approach to the effects of solar gravity on the orbiting earth is related to Einstein's principle of equivalence: The situation in a noninertial reference frame that is falling freely in a gravitational field is equivalent to what would happen in an inertial frame in the absence of this gravitational field. A freely falling closed cabin of an elevator is often used in gedanken experiments that appeal to the principle of equivalence. Because the earth in its orbital motion is falling freely in the gravitational field of the sun, our planet is similar to such a falling cabin in the sense that the sun doesn't interfere much due its gravity in our earthly dealings. However, the freely falling cabin is exactly equivalent to an inertial reference frame only when the gravitational field is uniform. In a nonuniform gravitational field, such as the field of the sun, the equivalence holds approximately only for a small enough cabin. Finite dimensions 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 the equivalence principle.
12. D. E. Cartwright, "Oceanic tides," Rep. Prog. Phys. 40 (6), 665-708 (1977).
13. F. D. Stacey, Physics of the Earth (Wiley, New York, 1969), 2nd ed.
14. As we have emphasized, only the translational acceleration of the earth is essential for the explanation of tides: The tidal forces would have been the same 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 acceleration of 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 can also 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. The centrifugal acceleration that results from this whirling varies with distance from the center of mass across the planet. This effect, together with the differential force of the satellite's gravity across the planet, is responsible for the rising of two tidal bulges." However, only the second effect is responsible for producing the tides, while the variation of centripetal acceleration 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.
15. P. Goldreich, "Tides and the earth-moon system," Sci. Am. 226 (4), 42-52 (1972).
16. The equatorial bulge of the earth produced by the time-independent centrifugal force of inertia is responsible for the gradual cyclic change in the direction 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 far more obvious daily rotation and annual revolution. Precession is caused by the gravitational influence of the sun and the moon acting on the earth's equatorial 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.
17. Encyclopedia Britannica CD, see http://www.eb.com.
18. A. G. Alenitsyn, E. I. Butikov, and A. S. Kondratyev, Concise Handbook of Mathematics and Physics (CRC Press, Boca Raton, FL, 1997).
19. The redistribution of energy during tidal evolution is also a subject of some confusion in the literature. In Ref. 15 the motion of the moon in its orbit is erroneously indicated as one of the sources of the mechanical energy dissipated by tidal friction. However, as the moon's orbit expands, the mechanical energy of the orbital motion increases. This mechanical energy, as well as the energy dissipated by tidal friction, also has its source in the energy of axial rotation of the earth.
20. M. Zeilik, Astronomy: The Evolving Universe (Harper and Row, New York, 1982), 3rd ed.
21. E. Butikov, Planets and Satellites (computer simulation programs), User's Manual (American Institute of Physics, Physics Academic Software, New York, 1998), pp. 39-40.
22. The variation of the orbital motion during the tidal evolution is also interesting in the sense that it disproves a common belief that internal forces cannot influence the motion of the center of mass of a system. This statement is true only for a system moving in a homogeneous external field. The aforementioned expanding of the earth's orbit during the tidal evolution is actually caused (though indirectly) by internal forces, namely by gravitational forces between the ocean water and the hard body of the earth (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 external gravitational force exerted on the system. These variations of the external gravitational force modify the orbital motion of the system. During tidal evolution, the internal forces of mutual gravitation together with friction cause the redistribution of masses (of the earth and tidal bulges) that move in the external nonhomogeneous central field.
23. George H. Darwin, The Tides and Kindred Phenomena in the Solar System (Freeman, San Francisco, 1962).