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Triaxial Earth's rota.on: Chandler wobble, free core nuta.on and diurnal polar mo.on
Rong Sun1, WenBin Shen1,2 1Key Laboratory of Geospace Environment and Geodesy, School of Geodesy and Geoma?cs, Wuhan University, Wuhan, China; 2 State Key Laboratory of Informa?on Engineering in Surveying, Mapping and Remote Sensing, Wuhan University, Wuhan 430079, China Correspondence: WenBin Shen (wbshen@sgg.whu.edu.cn)

1 Introduc.on

For a biaxial elas?c Earth, the free mo?ons CW and FCN are circular mo?ons. When non-
rota?onal-
symmetric effects are taken into considera?on, these free mo?on would become ellip?cal ones rather than circular ones. An ellpi?cal mo?on can be decomposed into two circular mo?ons with opposite frequencies. So in the frequency domain, for every free mo?on, CW or FCN, there are two resonant frequencies. If the excita?on func?on is very near to these two resonant frequencies, resonance will happen. In this study, we used a matrix based method to study the influence of triaxility on the frequencies of CW and FCN and on the diurnal polar mo?on.

a
0.2 3.706 3.705 3.704 3.703 3.702 3.701 -1.0024 -1.0023 -1.0022

x 10

-3

b

|

s

|m /

0

|m /

s

s

0.1

s
1.0022

|

1.0023

1.0024

/
x 10 3
-5

/

c
0.03 0.02 0.01 0 -1.0023 -1.0022 1.0022

d
| |m /
o o

2 Influence of triaxil.y on frequnecies of CW and FCN

| |m /
o o

2 1 0

To es?mate the frequencies of CW and FCN, we treat the Euler-
Liouville's equa?on as a set of linear differen?al equa?ons with constant coefficients. Then the frequencies can be derived by using the theory of solu?on of a set of linear differen?al equa?ons with constant coefficients. We find that the influence of triaixlity on CW is very small, which is in accordance with the previous stuides. When it comes to the influence of triaixlity on FCN, we find that our results suggest that considering triaxilty of the core would make the FCN period shorter, which is in contradic?on with the conclusion given by van Hoolst and Dehant (2002). However, the theore?cal system used by van Hoolst and Dehant(2002) is different from the theore?cal system used in this study, which is the same as that used by Mathews et al., (2002) and that used by Chen and Shen (2010). The lateral inhomogeneity of the Earth is modelled as triaxial buldge in the study of van Hoolst and Dehant(2002), which means that the expression of FCN contains contribu?on of lateral inhomogeneity. Aaer we consider the contribu?on of lateral inhomogeneity, the period of FCN is also increased as shown in table 1. Table 1 Influence of triaxilaity on the period of FCN
Biaxial Core van Hoolst and Dehant (2002) This study (triaxility alone) This study 430.941420 (traixility+lateral inhomogeneity) 0.004864
b

-1 -1.0024

1.0023

1.0024

/

/

Figure 1 Resonances near FCN (a,c) and nega?ve FCN(b,d). The resonances in (b,c,d) won't exist if the Earth mode is biaxial.

Table 2 Difference of polar mo?on amplitude between biaxial case and triaxial case. The closer the excia?on is to the nega?ve FCN, the larger the difference between the biaxial case and triaxial case.

Tide Q1 O1

Frequencycpsd 0.891 0.927 0.995 1.000 1.002288790

Triaxial-biaxial(µas) -0.002 -0.018 -0.089 -0.954

Triaxial Core 0.018699 -0.007421

P1 K1 Negavtive FCN

429.613213 430.941420

4 Conclusion

3 Resonance at nega.ve FCN

We solve the polar mo?on equaiton for a two-
layer triaxial Earth model. The wobble can be decomposed into two parts: the same frequency O S S m response that is excited by excia?on process of same frequency and O m opposite frequency response that is excited by excita?on process of opposite frequency. The resonance can be shown in Figure 1. To find out how large the resonaces shown in figure 1 would be, we use the ocean ?de model C of Chao(1996) as the excita?on func?on. We focus on the resonance strength at nega?ve FCN the resonance at nega?ve FCN(1d) is only slightly samller than resonance at FCN (1a). The results are shown in Table 2.
This study is supported by Na?onal 973 Project China (grant No. 2013CB733305), NSFC (grant Nos. 41174011, 41210006, 41128003, 41021061, 40974015).

Triaxility would alter the frequencies of CW and FCN by a small amount. Triaxility also makes the free mo?on of CW and FCN become ellip?cal mo?ons rather than circular mo?ons as suggested by biaxial Earth model. Since an ellip?cal mo?on can be decomposed into two circular mo?on, there is an extra resonance frequency at complex Fourier's spectrum. At nega?ve FCN frequency, we find that this resonance would bring a difference of the order of 1as.
References
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20163, doi:10.1029/96JB01649. Chen, W., and W. Shen (2010), New es?mates of the iner?a tensor and rota?on of the triaxial nonrigid Earth, Journal of Geophysical Research, 115(B12), B12419, doi: 10.1029/2009JB007094. Mathews, P. M., T. A. Herring, and B. A. Buffeh (2002), Modeling of nuta?on and precession: New nuta?on series for nonrigid Earth and insights into the Earth's interior, Journal of Geophysical Research, 107(B4), 2068, doi:10.1029/2001JB000390. van Hoolst, T., and V. Dehant (2002), Influence of triaxiality and second-
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