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
Conclusions and Discussions
References

3. Introduction The traditional Earth rotation theory In fact
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. Motivations of this study
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. Rotational normal modes
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. Proofs of the new expressions of the normal modes
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. No obvious differences!
Power spectrum density (PSD) comparisons among the excitations of the triaxial and biaxial Earth model, and the observed excitation
Introduction
Effects of Earth’s Triaxiality
Effects of Frequency-dependent Responses
Conclusions and Discussions
References
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. We start with the rotational symmetric case
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 For diurnal tides For long period 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. The compliance k at the Chandler frequency
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. improvements: ~5 mas Introduction Effects of Earth’s Triaxiality
Effects of Frequency-dependent Responses
Conclusions and Discussions
References
improvements: ~5 mas

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

21. Improvements in other frequency bands
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 x PSD(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. The IB model is adopted in the atmospheric-oceanic excitations
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: /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. Effects of Earth’s Triaxiality
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: /2001JB
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 attention NOTE: 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: /2009JB007094, in press. Chen, W., and W. B. Shen (2010), Frequency-dependent responses in the polar motion excitations, Geophys. Res. Lett., submitted.