Evidence for Self—Similarities in the Harmonic Development of Earth Tides

  • Hans-J. Kümpel


Recent models of the earth’s tidal potential include more than 103 individual harmonic constituents (Tamura 1987, Xi Qinwen 1987, 1989), enough to screen these data for the existence of self—similarities. It is found that the number of constituents surpassing a certain amplitude level shows a fractal distribution over at least four orders of magnitude. Its fractal dimension is close to 0.5. Accordingly, a refined tidal model which is complete should include about 10 times as much harmonic constituents as a coarser model, when the smallest amplitude of the refined model is 100 times smaller than that of the coarse one. Whether this feature masks an inherent structure in the spatial distribution or in the dynamics of celestial bodies governing the tidal forces is yet unknown.


Fractal Dimension Celestial Body Fractal Distribution Earth Tide Tidal Force 
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  1. Bronstein IN, Semendjajew KA (1970) Taschenbuch der Mathematik. Harri Deutsch, Zürich.Google Scholar
  2. Büllesfeld F-J (1985) Ein Beitrag zur harmonischen Darstellung des gezeitenerzeugenden Potentials. Dt Geodät Kommission, Bayer Akad der Wiss, Reihe C, vol 314.Google Scholar
  3. Cartwright DE, Edden AC (1973) Corrected tables of tidal harmonics. Geophys J Roy Astron Soc 33: 253–264.CrossRefGoogle Scholar
  4. Cartwright DE, Tayler RJ (1971) New computations of the tide—generating potential. Geophys J Roy Astron Soc 23: 45–74.Google Scholar
  5. Dahlen FA (1993) Effect of the earth’s ellipticity on the lunar tidal potential. Geophys J Int 113: 250–251.CrossRefGoogle Scholar
  6. Doodson AT (1921) The harmonic development of the tide—generating potential. Proc Royal Soc London Ser A100: 305–329.CrossRefGoogle Scholar
  7. Melchior P (1978) Tides of the planet earth. Pergamon Press, Oxford.Google Scholar
  8. Merriam JB (1992) An ephemeris for gravity tide predictions at the nanogal level. Geophys J Int 108: 415–422.CrossRefGoogle Scholar
  9. Merriam JB (1993) A comparison of recent tide catalogues and the consequences of catalogue error for tidal analysis. In: Melchior P (ed) Bull d’Inform Marrees Terrestres. Brussels, pp 8515–8535 (vol 115).Google Scholar
  10. Schroeder M (1991) Fractals, chaos, power laws. Freeman, New York.Google Scholar
  11. Tamura Y (1987) A harmonic development of the tide—generating potential. In: Melchior P (ed) Bull d’Inform Marrees Terrestres 99, Brussels, pp 6813–6855.Google Scholar
  12. Turcotte DL (1989) Fractals in geology and geophysics. PAGEOPH 131: 171–196.CrossRefGoogle Scholar
  13. Wenzel H-G (1992) Program ETGTAB, Version 920107. Geodätisches Institut, Universität Karlsruhe.Google Scholar
  14. Wilhelm H, Zürn W (1984) Tidal forcing field. In: Landolt—Börnstein, Zahlenwerte und Funktionen aus Naturwissenschaft und Technik, Neue Serie, Group V (vol 2). Springer, Berlin.Google Scholar
  15. Xi Qinwen (1987) A new complete development of the tidegenerating potential for theepoch J2000.0. In: Melchior P (ed) Bull d’Inform Marrees Terrestres 99. Brussels, pp 6786–6812.Google Scholar
  16. Xi Qinwen (1989) Datafile of tidal model Xi_1989 (unpublished).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Hans-J. Kümpel
    • 1
  1. 1.Geologisches InstitutRheinische Friedrich-Wilhelms-Universität BonnBonnGermany

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