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On “pathological oscillations” of rotating fluids in the theory of nutation

Summary

"Pathological oscillations" of rotating fluids occur when the period of oscillations with respect to a mobile reference frame exceeds half of a rotational period. The oscillations of such kind play an important role in the theory of nutation of a real Earth model with compressible, heterogeneous outer liquid core and solid inner core. In this paper we try to present a sufficiently rigorous description of such oscillations.

The full system of two-dimensional solutions is found numerically. Results are presented in form of infinite matrices which connect the coefficients of spherical harmonic expansions of the boundary conditions, pressure, and horizontal components of tidal velocities. The substitution of these known solutions in the three-dimensional equations of motion results in an infinite set of ordinary differential equations which describe the conditions of equilibrium in the radial direction. Below we analize"pathological" oscillations of different kind which are described by this set of equations.

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References

  1. Aldridge K D (1972) Axisymmetric inertial oscillations of a fluid in a rotating spherical shell. Mathematica 19: 163–168, Phys. Earth Plan. Int. 49: 176–176

  2. Bryan B A (1888) The waves on a rotating liquid spheroid of finite ellipticity. Acta Mathematica A.CLXXX 7: 187–219

  3. Crossley D I and M G Rochester (1980) Simple core undertones. Geophys. J. Roy. astr. Soc. 60: 129–161

  4. Crossley D I (1993) The gravity effect of core modes for a rotating Earth. J. Geomag. Geoelectr. 45: 1371–1381

  5. De Vries D and Wahr J M (1991) The effects of the solid inner core and nonhydrostatic structure on the Earth's forced nutations and Earth tides. J. Geophys. Res. 96:8275–93

  6. Dziewonski A M and Anderson D L (1981) Preliminary reference Earth model. Phys. Earth. Plan. Interiors 25 (4): 297–356

  7. Friedlander S (1985) Internal oscillations in the Earth's fluid core. Geophys. J. R. Astr. Soc. 80: 345–61

  8. Greenspan H P (1964) On the transient motion of a contained rotating fluid. J. Fluid Mech. 20 (4): 673–696

  9. Jeffreys H (1978) On some difficulties in the theory of nutatin. Geophys. Journ. Roy. astr. Soc. 54: 711–712

  10. Lamb H (1932) Hydrodynamics, 6th ed. Cambridge Univ. Press, 738

  11. Leibenson L S (1947) The theory of elasticity. ogiz-gostehizdat, Moscow (in Russian).

  12. Mathews P M, Buffett B A, Herring T A, Shapiro II (1991a) Forced nutations of the Earth: Influence of inner core dynamics 1. Theory. J. Geophys. Res. 96:8219–42

  13. Mathews P M, Buffett B A, Herring T A, Shapiro II (1991b) Forced nutations of the Earth: Influence of inner core dynamics 2. Numerical results and comparisons. J. Geophys. Res. 96:8243–57

  14. Mathews P M, TA Shapiro II (1992) Nutations of the Earth. Ann. Rev. Earth and Plan. Sc. 20

  15. Melchior P (1986) The physics of the Earth's core. Pergamon Press, Oxford, N.Y., p 256

  16. Molodensky S M (1989) Asymptotic behaviour of solutions to Laplace's tidal equations at low frequencies. Geophys. J. Int., 97: 459–469

  17. Molodensky S M and Sasao T (1995a) A new approach to the theory of nutation. 1. basic equations. Physics of the Solid Earth 12, in print

  18. Molodensky S M and Sasao T (1995b) A new approach to the theory of nutation. 2. Numerical results. Physics of the Solid Earth 12, in print

  19. Rickard J A (1973) Free oscillations of a rotating fluid contained between two spheroidal surfaces. Geophys. Fluid Dyn. 5: 369–383

  20. Rieutord M (1991) Linear theory of rotating fluids using spherical harmonics. Geophys. Astr. Fluid Dyn. 59: 185–208

  21. Sasao T, Okubo S, Saito M (1980) A simple theory of the dynamical effects of a stratified liquid core upon nutational motion of the Earth. In: Fedorov E P, Smith M L, Bender P L (ed). Proc. IAU Symp. 78, Reidel, Hingham, Mass., pp. 165–83

  22. Smylie D E and X Jiang (1993) Core oscillations and their detection in superconducting gravimeter records. J. Geomag. Geoelectr. 45: 1371–1381

  23. Stewartson K and Roberts P H (1963) On the motion of a liquid in a spherical cavity of a precessing rigid body. I. J. Fluid Mech. 17: 1–20

  24. Stewartson K and Rickard J R A (1969) Pathological oscillations of rotating fluid. Journ. Fluid Mechanics 35: 759–773

  25. Valette B (1998a) Spectre des vibrations propres d'un corps élastique, autogravitant, en rotation uniforme et contenant une partie fluide. C.R. Acad. Sci. Paris, t. 309, Serie I: 419–422

  26. Valette B (1989b) Etude d'une classe de problèmes spectraux. C.R. Acad. Sci. Paris, t. 309, Serie I: 785–788

  27. Wahr J M (1981) Body tides on an elliptical rotating, elastic and oceanless Earth. Geophys. J. Roy. Astr. Soc. 64: 677–703

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Correspondence to E. Groten.

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Molodensky, S.M., Groten, E. On “pathological oscillations” of rotating fluids in the theory of nutation. Journal of Geodesy 70, 603–621 (1996). https://doi.org/10.1007/BF00868223

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Keywords

  • Reference Frame
  • Radial Direction
  • Horizontal Component
  • Earth Model
  • Inner Core