A Modified Liouville Equation for the Triaxial Earth with Frequency-dependent Responses

1. A Modified Liouville Equation for the Triaxial Earth with Frequency-dependent Responses Wei Chen & WenBin Shen Department of Geophysics, School of Geodesy & Geomatics, Wuhan University Oct. 25, 2010

2. Contents • Introduction • Effects of Earth’s Triaxiality • Model for the triaxial stratified Earth • Comparisons with Traditional Theory • Effects of Frequency-dependent Responses • Theory • Effects of Ocean Tides • Effects of Ocean Pole Tides • Effects of Mantle Anelasticty • Frequency-dependent Transfer Functions • Comparisons with Traditional Theory • Conclusions and Discussions • References

3. Introduction • Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References • The traditional Earth rotation theory • assumes that equatorial principal inertia moments A and B are equivalent (rotationally symmetric) • only models the long-period responses (constant transfer functions) • In fact • A and B are not equivalent (triaxial) • Earth’s responses to the perturbations with various frequencies are frequency-dependent (FD) • It is agreed that the traditional theory is not adequate for the current observation accuracy at the meeting of the IAU Commission 19 “Rotation of the Earth” during the IAU General Assembly 2009 in Rio de Janeiro

4. Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References • Motivations of this study • Overcoming the shortcomings of the rotational-symmetry approximation and the long-period assumption in the traditional Earth rotation theory

5. Effects of Earth’s Triaxiality • Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References • Model for the triaxial stratified Earth (Chen & Shen, 2010). Based on the MHB2000 (Mathews et al., 2002) and EGM2008 (Pavlis et al., 2008)

6. Comparisons with Traditional Theory • Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References • The traditional Liouville equation (Eubanks, 1993) • The modified Liouville equation (Chen & Shen, 2010)

7. Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References • Rotational normal modes • For the triaxial Earth (Chen & Shen, 2010) • For the rotationally symmetric Earth (Mathews et al., 2002)

8. Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References • Proofs of the new expressions of the normal modes

9. Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References • Differences between the excitations of the triaxial and biaxial Earth model (AAM+OAM) x-axis: ~0.1 mas y-axis: ~0.2 mas

10. Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References • Power spectrum density (PSD) comparisons among the excitations of the triaxial and biaxial Earth model, and the observed excitation No obvious differences!

11. Effects of Frequency-dependent Responses • Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References • The Earth is not a perfectly elastic body • The mantle is anelastic • The oceans are dissipative • Mantle anelasticity and ocean tide dissipations will lead to frequency-dependent responses (FDRs) and response phase lags (RPLs) • Rotational responses are described by the transfer functions • FDRTransfer functions varies with frequency • RPLTransfer functions own imaginary parts

12. Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References • We start with the rotational symmetric case • Due to the mantle anelasticity and ocean tide dissipations, the compliance k varies with frequency (Mathews et al., 2002)

13. Effects of Ocean Tides • Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References • For diurnal tides • The IERS Conventions 2010 has provided the corrections to the Love number k in the diurnal bands • k and k are linked by (Mathews et al., 1995) • For long period tides • Long-period tidal excitations are provided by Gross (2009), and Dickman & Gross (2010) • The compliance k for long period can be written as (Mathews et al., 2002)

14. Effects of Ocean Pole Tides • Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References • Polar motion will give rise to deformations of the ocean --- ocean pole tides • Amplitude of pole tide is proportional to that of polar motion • Polar motions can be divided into • Chandler wobble (~433 days, ~160 mas) • Annual wobble (~433 days, ~100 mas) • Wobbles at other frequencies (~10 mas or less) • Only the pole tides at the Chandler and annual frequencies need to be considered

15. Effects of Mantle Anelasticty • Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References • Mantle anelasticity is usually expressed by the power law for the FD mantle quality factor Q • The mantle anelasticty is poorly understood in most frequency bands except for the Chandler frequency • Based on the compliance increment at the Chandler frequency and the power law, we get

16. Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References • The compliance k at the Chandler frequency • The Chandler frequency can be expressed as (Mathews et al., 2002)  • Based on the theoretical and observational estimates of • Then we get

17. FD Transfer Functions • Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References Nutation Polar motion Definition of polar motion (IERS Conventions 2010)

18. Comparisons with Traditional Theory • Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References Well agreements with the traditional theory and observations

19. Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References improvements: ~5 mas

20. Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References Obvious improvements in the low frequency bands

21. Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References Improvements in other frequency bands

22. Statistical Information • Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References Note:RMS(x) means root mean square of xPSD(x) means power spectrum density of x

23. Conclusions and Discussions • Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References • The present theory agrees better with the observations than the traditional one does • The effects of FDR is more significant than that of triaxiality of the Earth • FDR: ~5 mas / obvious improvement in the frequency domain • Triaxiality: ~0.1 mas (x axis), ~0.2 mas (y axis) / -- • FDR is more adoptable than the traditional constant response and should be included • Triaxiality should be taken into account when the needed accuracy is very high

24. Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References • The IB model is adopted in the atmospheric-oceanic excitations • The oceans react as “Inverted Barometer” in front of the pressure variations • However, the oceans’ reaction to the pressure variations should also be frequency dependent Dynamic Barometer (DB) • Further improvements in the numerical results are expected when the DB model is adopted • DB model is needed for a full description of the frequency dependent responses in the polar motion excitations

25. References • Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References • Chen, W., and W. B. Shen (2010), New estimates of the inertia tensor and rotation of the triaxial nonrigid Earth, J. Geophys. Res., doi:10.1029/2009JB007094, in press. • Chen, W., and W. B. Shen (2010), Frequency-dependent Responses in the Polar Motion Excitations, Geophys. Res. Lett., submitted. • Dickman, S. R., and R. S. Gross (2010), Rotational evaluation of a long-period spherical harmonic ocean tide model, J. Geod., 84, 457–464. • Eubanks, T. M. (1993), Variations in the orientation of the Earth, in Contributions of Space Geodesy to Geodynamics: Earth Dynamics, Geodyn. Ser., vol.24, edited by D. E. Smith and D. L. Turcotte, pp. 1–54, AGU, Washington, D. C.. • Gross, R. S. (2009), An improved empirical model for the effect of long-period ocean tides on polar motion, J. Geod., 83, 635–644. • Hartmann, T., and H. Wenzel (1995), The HW95 tidal potential catalogue, Geophys. Res. Lett., 22: 3553–3556. • Kudryavtsev, S. (2004), Improved harmonic development of the Earth tide-generating potential, J. Geod., 77, 829–838.

26. Introduction • Effects of Earth’s Triaxiality • Effects of Frequency-dependent Responses • Conclusions and Discussions • References • Mathews, P. M., and P. Bretagnon (2003), Polar motions equivalent to high frequency nutations for a nonrigid Earth with anelastic mantle, Astron. Astrophys., 400, 1113–1128. • Mathews, P. M., T. A. Herring, and B. A. Buffett (2002), Modeling of nutation and precession: new nutation series for nonrigid Earth and insights into the Earth’s interior, J. Geophys. Res., 107(B4), 2068, doi: 10.1029/2001JB000390. • Pavlis, N. K., S. A. Holmes, S. C. Kenyon, and J. K. Factor (2008), An Earth gravitational model to degree 2160: EGM2008, Presented at the 2008 General Assembly of the European Geosciences Union, Vienna, Austria, April 13–18. • Vicente, R. O., and C. R. Wilson (1997), On the variability of the Chandler frequency, J. Geophys. Res., 102, B9, 20439–20446. • Wahr, J., and Z. Bergen (1986), The effects of the mantle anelasticity on nutations, Earth tides, and tidal variations in the rotation rate, Geophys. J. R. Astron. Soc., 87, 633–668.

27. Thanks for your attentionNOTE: Please cite the following papers when the results of this report is used.Chen, W., and W. B. Shen (2010), New estimates of the inertia tensor and rotation of the triaxial nonrigid Earth, J. Geophys. Res., doi:10.1029/2009JB007094, in press.Chen, W., and W. B. Shen (2010), Frequency-dependent responses in the polar motion excitations, Geophys. Res. Lett., submitted.