Izvestiya, Atmospheric and Oceanic Physics

, Volume 54, Issue 8, pp 911–931 | Cite as

Tidal Love Numbers of Degrees 2 and 3

  • E. A. SpiridonovEmail author


In the coming years, the high-precision processing of large arrays of modern observational data from the Global Satellite Navigation System (GNSS) will require knowledge of the theoretical values of the tidal numbers h and l of degrees 2 and 3 with a relative error no worse than 10–4. This will make it possible to predict vertical and horizontal tidal displacements of the earth’s surface at the modern level. The paper presents the values of Love numbers h, k, and l of degrees 2 and 3 calculated for an inelastic ellipsoidal self-gravitating rotating earth without an ocean. Twelve different models are considered that differ from each other by the inclusion or exclusion of individual factors that affect the result. In particular, two different versions of the earth’s structure model are used, corrections for the relative and Coriolis accelerations are considered, and the values of the Love numbers are determined considering their latitude dependence. Lame elastic parameters were recalculated for the periods of semidiurnal and diurnal tidal waves using the logarithmic creep function to account for dissipation. The normalization of the obtained values of the Love numbers corresponds to International Earth Rotation and Reference Systems Service (IERS) Conventions regarding the calculation of tidal displacements of the earth’s surface. The values are compared with the results of widely known works of other authors. Currently, the results are used to develop a new version of the ATLANTIDA3.1 software for predicting earth tides.


Love numbers earth tidal displacements GNSS observations earth tide prediction ATLA-NTIDA3.1 earth-tide prediction software 



This work was supported by the State Assignment of the Institute for Physics of the Earth of the Russian Academy of Sciences.


  1. 1.
    Dehant, V., Tidal parameters for an inelastic Earth, Phys. Earth Planet. Int., 1987a, vol. 49, pp. 97–116.CrossRefGoogle Scholar
  2. 2.
    Dehant, V., Integration of the gravitational motion equations for an elliptical uniformly rotating Earth with an elastic mantle, Phys. Earth Planet. Int., 1987b, vol. 49, pp. 242–258.CrossRefGoogle Scholar
  3. 3.
    Dehant, V., Defraigne, P., and Wahr, J.M., Tides for a convective earth, J. Geophys. Res., 1999, vol. 104, no. B1, pp. 1035–1058.CrossRefGoogle Scholar
  4. 4.
    Mathews, P.M., Buffet, B.A., and Shapiro, I.I., Love numbers for a rotating spheroidal Earth: New definitions and numerical values, Geophys. Res. Lett., 1995a, vol. 22, pp. 579–582.CrossRefGoogle Scholar
  5. 5.
    Mathews, P.M., Buffett, B.A., and Shapiro, I.I., Love numbers for diurnal tides: Relation to wobble admittances and resonance expansion, J. Geophys. Res., 1995b, vol. 100, pp. 9935–9948.CrossRefGoogle Scholar
  6. 6.
    Mathews, P.M., Love numbers and gravimetric factor for diurnal tides, J. Geod. Soc. Jpn., 2001, vol. 47, no. 1, pp. 231–236.Google Scholar
  7. 7.
    McCarthy, D.D., IERS conventions (1996), Paris: Int. Earth Rotation Serv., 1996.Google Scholar
  8. 8.
    Molodenskii, M.S., Elastic tides, free nutation and some issues of the Earth structure, Tr. Geofiz. Inst. Akad. Nauk SSSR, 1953, no. 19, pp. 3–52.Google Scholar
  9. 9.
    Molodenskii, M.S. and Kramer, M.V., Zemnye prilivy i nutatsiya Zemli (Earth Tides and Nutation of the Earth), Moscow: AN SSSR, 1961.Google Scholar
  10. 10.
    Molodenskii, M.S., Prilivy, nutatsiya i vnutrennee stroenie Zemli (Tides, Nutation, and Inner Structure of the Earth), Moscow: IFZ RAN, 1984.Google Scholar
  11. 11.
    Petit, G. and Luzum, B., IERS Conventions (2010), Frankfurt, 2010.Google Scholar
  12. 12.
    Smith, M.L., The scalar equations of infinitesimal elastic gravitational motion for a rotating, slightly elliptical Earth, Geophys. J. R. Astron. Soc., 1974, vol. 37, pp. 491–526.CrossRefGoogle Scholar
  13. 13.
    Smith, M.L., Translational inner core oscillations of a rotating, slightly elliptical Earth, J. Geophys. Res., 1976, vol. 81, pp. 3055–3065.CrossRefGoogle Scholar
  14. 14.
    Smith, M.L., Wobble and nutation of the Earth, Geophys. J. R. Astron. Soc., 1977, vol. 50, pp. 103–140.CrossRefGoogle Scholar
  15. 15.
    Spiridonov, E.A., ATLANTIDA3.1_2014software for analysis of Earth tide data, Nauka Tekhnol. Razrab., 2014, vol. 93, no. 3, pp. 3–48.Google Scholar
  16. 16.
    Spiridonov, E.A., Certificate of State Registration of Computer Program no. 2015619567, 2015.Google Scholar
  17. 17.
    Spiridonov, E.A., Love number corrections with respect to relative and Coriolis accelerations, Geofiz. Protsessy Biosfera, 2016a, vol. 15, no. 1, pp. 73–81.Google Scholar
  18. 18.
    Spiridonov, E.A., Amplitude delta-factors of the second order and their dependence on latitude, Geol. Geofiz., 2016b, no. 4, pp. 796–807.Google Scholar
  19. 19.
    Spiridonov, E.A., Results of comparison of predicted Earth tidal parameters and observational data, Seism. Instrum., 2016c, vol. 52, no. 1, pp. 60–69.CrossRefGoogle Scholar
  20. 20.
    Spiridonov, E.A., How dissipation and selection of the Earth model affects the quality of the Earth tidal prediction, Seism. Instrum., 2016d, vol. 52, no. 3, pp. 224–232.CrossRefGoogle Scholar
  21. 21.
    Spiridonov, E.A., Amplitude delta-factors and phase shifts of tidal waves for the Earth with the Ocean, Geofiz. Protsessy Biosfera, 2017, vol. 16, no. 2, pp. 5–54. doi 10.21455/GPB2017.2-1Google Scholar
  22. 22.
    Spiridonov, E.A. and Vinogradova, O.Yu., The results of integrated modeling of the oceanic gravimetric effect, Seism. Instrum., 2018, vol. 54, no. 1, pp. 43–53. doi 10.21455/si2017.1-5CrossRefGoogle Scholar
  23. 23.
    Spiridonov, E., Vinogradova, O., Boyarskiy, E., and Afanasyeva, L., ATLANTIDA3.1_2014 for Windows: A software for tidal prediction, Bull. Inf. Marées Terrestres, 2015, no. 149, pp. 12063–12082.Google Scholar
  24. 24.
    Spiridonov, E.A., Yushkin, V.D., Vinogradova, O.Yu., and Afanas’eva, L.V., Software for tidal prediction ATLANTIDA3.1_2014: New version, Nauka Tekhnol. Razrab., 2017, vol. 96, no. 4, pp. 19–36. doi 10.21455/ std2017.4-2Google Scholar
  25. 25.
    Wahr, J.M., The tidal motions of a rotating, elliptical, elastic and oceanless Earth, Ph.D. Thesis, University of Colorado, Boulder, 1979.Google Scholar
  26. 26.
    Wahr, J.M., Body tides on an elliptical, rotating, elastic and oceanless Earth, Geophys. J. R. Astron. Soc., 1981a, vol. 64, pp. 677–703.CrossRefGoogle Scholar
  27. 27.
    Wahr, J.M., A normal mode expansion for the forced response of a rotating Earth, Geophys. J. R. Astron. Soc., 1981b, vol. 64, pp. 651–675.CrossRefGoogle Scholar
  28. 28.
    Wahr, J.M. and Bergen, Z., The effects of mantle and anelasticity on nutations, Earth tides, and tidal variations in rotation rate, Geophys. J., 1986, vol. 87, pp. 633–668.CrossRefGoogle Scholar
  29. 29.
    Zharkov, V.N. and Molodenskii, M.S., Corrections of shear modulus of ice for Love numbers, Izv. Akad. Nauk SSSR, Fiz. Zemli, 1977, no. 5, pp. 17–20.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Institute for Physics of the Earth, Russian Academy of SciencesMoscowRussia

Personalised recommendations