Geodesy and Geodynamics 2012 , 3 ( 2) : 1 - 7

http://www. jgg09. com

Doi:10.3724/SP.J.1246.2012.00001

Effects of the Earth ' s triaxiality on the polar motion excitations

Chen Wei 1 '2 and Shen Wenhin 1

•3

1 Departmem of Geophysics, School of Ge~y and Geomatics, Wuhan University, Wuhan 430079, China 2 Shanghai Astronomical Obseroatory, ChiMse Academy of Science , Shanghai 200030 , China

3 Ly Lohoratory of Geospace Environment and Geodesy, Mini.ttry of Education, Wuhan 430079 , China

Abstract : This study aims to evaluate the significance of the Earth ' s triaxiality to the polar motion theory.

First of all, we compare the polar motion theories for both the triaxial and rotationally-symmetric Earth models,

which is established on the basis of the EGM2008 global gravity model and the MHB2000 Earth model. Then,

we use the atmospheric and oceanic data (the NCEP/NCAR reanalyses and the ECCO assimulation products)

to quantify the triaxiality effect on polar motion excitations. Numerical results imply that triaxiality only cause

a small correction (about 0. 1 - 0. 2 mas) to the geophysical excitations for the rotationally-symmetric case.

The triaxiality correction is much smaller than the errors in the atmospheric and oceanic data, and thus can he

neglected for recent studies on polar motion excitations.

Key words : Earth rotation theory ; dynamic figure parameter; atmospheric-oceanic excitation

1 Introduction

In the traditional Earth rotation theory , the Earth ' s

equatorial principal inertia moments , A and B, are as­

sumed to he equivalent to simplify the Liouville equa­

tion, and only the long-period responses in the rota­

tional motion are considered ( thus the Liouville equa­

tion is equipped with transfer constants[1'2J ) • But in

fact , the Earth is triaxial (A ;t* B) and its responses to

the perturbations vary with frequency , due to the reso­

nances near the normal modes , the mantle anelasticity

and the dynamic ocean tides[l-Sl.

Recently, Chen and Shen[8J re-estimated the Earth's

principal inertia moments (A , B and C) based on the

global gravity model EGM2008 [9 J , and further formula­

ted an extended Liouville equation incorporating the

effects of the triaxialities and deformations of both

Received :2012.03-11; Accepted :2012.03-15

Corresponding author: Chen Wei, E-mail: daniel135@ 126. com

This study is supported by the National Natural Science Foundation of

China ( 41174011).

the mantle and core , as well as the effects of the

frequency-dependent responses of the Earth. Howev­

er, including these effects ( especially the triaxiality

effect ) significantly increase the complexity of the

Liouville equation ( see Section 3 for details ) , thus

we wonder whether it is necessary to consider these

effects. The motivation of the present study is to quan­

titatively evaluate the effects of the Earth ' s triaxiality

on polar motion excitations with atmospheric and oce­

anic data.

2 Effects of the Earth' s triaxiality

2.1 Models for the stratified Earth

Chen and Shen r 8 J determined the figure parameters

(namely the principal inertia moments, the polar and

the equatorial dynamic flattenings: Tab. 1 ) for an

Earth model with a triaxial mantle and a triaxial fluid

core , on the basis of the MHB2000 Earth model [7 •101

( Tab. 2 ) and the EGM2008 gravity model [9J.

Wherein, the MHB2000 is a revision of " hydrostatic

2 Geodesy and Geodynamics Vol. 3

equilibrium + PREM ( Preliminary Reference Earth

Model[II])" with some non-equilibrium corrections

taken into account, and was adopted to establish the

IAU2000 precession-nutation model[?,IOJ; the

EGM2008 is currently the most accurate global gravity

model and has been recommended by the IERS con­

ventions ( 2010) [I2J, These bases can guaranty the

reliability and accuracy of the Earth figure parameters

presented in table 1 .

To quantify the triaxiality effects and taking into ac­

count the fact that the triaxialities of the stratified Earth

are not important in many cases, we also provide a cor­

responding version for the rotationally-symmetric

( hereafter symmetric for short) Earth ( see table 2 ) .

In table 2 , we provide the mean values ( for example,

here A is equivalent to (A + B) /2 in table 1 ) of the e­

quatorial principal inertia moments nf the whole Earth

and the fluid core.

It' s worthy pointing out that due to the accuracies of

the EGM2008 and other relevant parameters , uncer­

tainties occur in the last two digits in the values of the

principal inertia moments for the whole Earth , while

uncertainties for the other values are difficult to esti­

mate since these values are inferred from the MHB2000

Earth model, which did not provide model uncertainty

( see references [ 7 , 8 , 10] for details) .

3 Theoretical comparisons

Considering the perturbations of the Celestial Interme­

diate Pole ( CIP) from mass displacements and relative

angular momentums (denoted by c = c13 + i13 and h = h 1 + ih2 respectively ) , the Liouville equation for the

triaxial and symmetric Earth models can be written in

the forms listed in table 3. Wherein, p denotes the

motion of the CIP , u c the Chandler frequency, x dff the

Table 1 Parameters for the triaxially stratilied Earth

Parameter

Principal inertia moments of the whole Earth ( x 1037 kgm2 )

Principal inertia moments of the mantle ( x 1037 kgm? )

Principal inertia moments of the core ( x 1036 kgm? )

Principal inertia moments of the fluid outer core ( x 1036 kgm2)

Principal inertia moments of the solid inner core ( x 1034 kgm2)

Polar ellipticity of the whole Earth

Equatorial ellipticity of the whole Earth

Polar ellipticity of the fluid core

Equatorial ellipticity of the fluid core

Polar ellipticity of the solid core

Equatorial ellipticity of the solid core

A

B

c

A.

B.

c.

A,

B,

C,

• •

•• ••

Value

8.0100829

8. 0102594

8.0364807

7. 0985591

7.0987194

7.1225467

9.1152379

9.1153996

9.1393399

9.0567171

9.0568779

9.0806773

5.8520816

5.8521767

5. 8662552

3. 2845161 X 10-3

2. 2033010 X 10 -s

2. 6455744 X 10-3

I. 7746896 X 10 -s

2. 4219766 X 10-3

I. 6246969 X 10 ->

No.2 Chen Wei ,et al. Effects of the Earth's triaxiality on the polar motion excitations 3

Table 2 Parameters for tbe rotationally-symmetric Earth: a comparison

Parameter Value Value of MHB2000'

A 8. 0101711 8. 0115 Principal inertia moments of the whole Earth ( x 1037 kgm2

)

c 8.0364807

A. 7.0986392 7.0999 Principal inertia moments of the mantle ( x 1037 kgm2

)

c. 7.1225467

A, 9. 1153188 9.1168 Principal inertia moments of the core ( x 1036 kgm2

) C, 9. 1393399

A, 9.0567975 9.0583 Principal inertia moments of the fluid outer core ( x 1036 kgm?) c, 9.0806773

A, 5.8521291 5.8531 Principal inertia moments of the solid inner core ( x 1034 kgm2

) C, 5.8662552

Polar ellipticity of the whole Earth e 3. 2845161 X 10 "3 3. 2845479 X 10

"3

Polar ellipticity of the fluid core •r 2. 6455744 X 10-3 2. 6456 X 10-3

Polar ellipticity of the solid core e, 2. 4219766 X 10-3 2.422x10-3

# All the listed values of MHB2000 Earth model are from reference [ 10] except for those of e and er, which are fitted from the VLBI

nutation data by Mathews et al [7] •

Table 3 Rotation theories for tbe triaxial and symmetric Earth models

Dyoamic equation

Transfer function

Effective excitation

i . -=P+xeii=xem +~em <Tc

For triaxial Earth (All parameters take the values listed in table 1 )

""'"' I ( ')' C-A C-A

{

1' =K i+k C.-A.C-(i+K)A

..VNL •• C-B C-B 12 =K(i+k) C.-B. C-(i+K)B

K= [ [(I +K)B- (I +r)B1][ C- (I +K)B]l [(I +K)A- (I +r)A1][ C- (I +K)A]

{

1'!' c" 1': h, X.m=' C-A+ 'll(C-A)

""' c., ... h., x..,=,, C-B+ 12 1l(C-B)

Chandler frequency Ucw =11

[C (I +Kcw)AJ[C (I +Kcw)B] [(I + Kcw )A- (I +rcwAr ][I + KcwB- (I + 1'cw )B,]

Free core [ nutation (J' FCN = - fi frequency

AB[C,-(1+,8., Kc..][C1 -(i+.Bm,-Kc..)B1 ] +I] 1A -;-tlJf'['( Io'+~K~""'---c-) A~--"'7.( 1'"'+-r-.,'"'"'"')Ar;-ol,-;['7( :o-1 +~K-FCN~) B'Oo--__.( IC'+~r-FCN'-'-;-) B;occ, J

* From references [ 7 , 10]

For symmetric Earth ( All parameters take the values listed in table 2)

i -p+p=x .. <Tc

T"NL =(I +k' )' ~ _e_ Cm-Ame-K

L C NL h x .. =T C -A +T .O.(C -A)

<Tm~ = -n[: (e, -.Bro.) +I]•

4 Geodesy and Geodynamics Vol. 3

effective excitation function, {}, the mean sidereal rate,

K , f3 and 'Y the compliances ( see reference [ 13 ] . The

subscript CW or FCN means K ,{3 and 'Y take the values

valid for the Chandler or the free core nutation frequen­

cy) and K ' the degree 2 order 1 load Love number; /J

equals 1 or 0 , depending on whether the concerning

terms load the Earth or not (corresponding to the su­

perscripts L and NL respectively) .

It is worth pointing out that all the equations for the

symmetric Earth can he obtained from those for the tri­

axial Earth by setting A = B, A, = B, and neglecting the

core deformation terms yA, and yB,. Please refer to

Chen and Shen [Sl for some proofs.

One can see that the Earth ' s tria.xiality will lead to

asymmetric transfer functions to the Liouville equation,

that is, the transfer functions for the x- and y-axes are

not equivalent to each other in the case of a symmetric

Earth. If the tranafer functions don ' t vary with fre­

quency, the triaxiality will cause small corrections to

the excitations, and these corrections are approximately

proportional to the excitations themselves. In addition,

the triaxiality also complicates the expressions of the

normal modes: since the Earth' s equatorial ellipticity

is much smaller than its polar ellipticity, some previ­

ously ignored small terms ( such as the deformation of

the core expressed by yA, and yB,) now should be in­

cluded when considering the Earth ' s triaxiality.

With the Earth rotation theories tabulated in table 3 ,

we can choose some major excitation sources (such as

the atmospheric and oceanic excitations ) to detennine

the numerical differences between the two theories.

4 Effects of triaxialities of the strati­

fied Earth

Due to the strong mobility forced primarily by the sub

- seasonal and seasonal cycles , the atmosphere and o­

cean are the primary excitation sources for polar motion

on intra- seasonal and seasonal timescales[ 14-

171• It is

reasonable to take the atmospheric - oceanic excitation

as an example to get a rough understanding of the trivi­

ality effects on the polar motion excitation.

We have adopted the global atmospheric angular mo­

mentum ( AAM) calculated from NCEP/NCAR ( Na­

tional Centers for Environmental Prediction/National

Center for Atmospheric Research) re-analyses["·"],

and the global oceanic angular momentum ( OAM, data

file ECCO _kf080. oam) computed from the products of

the ECCO ( Estimating the Circulation and Climate of

the Ocean) ocean model[l4.l9] which is based on the

MIT -GCM (Massachusetts Institute of Technology-Gen­

eral Circulation Model) and is driven by the NCEP/

NCAR re-analyses of surface wind stress, heat and

freshwater fluxes, but not surface pressure. The ECCO

OAM data are consistent with the NCEP/NCAR AAM

ones , and have assimilated sea surface height data and

are free of tidal contaminations ( see reference [ 19 ] for

more details) .

Since the ECCO model run at the Jet Propulsion La­

boratory ( JPL) is not forced by atmospheric surface

pressure, it is common to assume the inverted barame­

ter ( IB) model to account for the effects of atmospher­

ic pressure over the oceans when using the ECCO o­

cean model [20J. The IB model is a realistic approxima­

tion of the oceans' response to surface pressure varia­

tions at long periods , and is generally thought to break­

down somewhere between a few and 10 days and is not

valid at short periods such as 1 -2 days[:ro.2l] (our re­

cent analyses of polar motion excitations estimated from

various geophysical fluid models might suggest a break­

down period of about 1 week["]). Thus we have cho­

sen the IB model to combine the atmospheric and oce­

anic excitations estimated from the NCEP/NCAR AAM

and the ECCO OAM data. The data used throughout

this study range from Jan. 1, 1993 to Aug. 31, 2009.

Based on the rotation theories for the triaxial and

symmetric Earth model tabulated in table 3 , we can

obtain the differences between the atmospheric-oceanic

excitations estimated by the two theories. These differ­

ences are found to be quite small ; about 0. 1 mas

(milli-arcsecond) and about 0. 2 mas for the x- andy­

a.xes, respectively (Fig. 1) .

Now let us examine the uncertainties af the atmospher­

ic and oceanic excitations. In fact, there are different ver­

sions af the AAM and OAM time series calculated on the

basis of various models for numerical weather prediction

from different institutes. Besides the above mentioned

NCEP/NCAR AAM and ECCO OAM, the ECMWF (Eu­

ropean Centre for Medium-Range Weather Forecasts) also

releases reanalyzed and operational AAM and OAM data

called ERA40 (available from 1958 to 2001 ) , ERAinter-

No.2 Chen Wei, et al. Effects of the Earth' s triaxiality on the polar motion excitations 5

im ( available smce 1989 ) and opECMWF ( available

since 2000) [Ill. The above mentioned data can be down­

loaded from the websites of the Special Bureaus for At­

mosphere and Ocean, respectively.

The numerical methods and assumptions adopted by

different institutes are not always the same and so are the

global atmospheric and oceanic models. Thus, in figure

2, we made some time-domain comparisons among

I "

I >t

~ g ><

'<1

~ g ><

'<1

100

-100

0.25 1-Tri-Syml

0.2 0.15

0.1 0.05

0 -0.05

-0.1 -0.15 -0.2

-0.25

1994 1996 1998 2000 2002 2004 2006 2008 2010

0.25 0.2

0.15 0.1

0.05 0

-0.05 -0.1

-0.15 -0.2

-0.25

1994 1996 1998 2000 2002 2004 2006 2008 2010 Epoch(year)

Figure 1 Differences between the atmospheric-oceanic excitations for the triaxial and symmetric

Earth (denoted by Tri and Sym, respectively)

2004

100

2004.5 2005 2005.0 2006 2006.5 2007 Epoch(year)

2004 2004.5 2005 2005.0 2006 2006.5 2007

Epoch(year)

2004.5 2005 2005.0 2006 2006.5 2007 Epoch(year)

2004 2004.5 2005 2005.0 2006 2006.5 2007

Epoch(year)

Figure 2 Time-domain comparisons among the matter and motion terms of AAM and OAM derived from

different atmospheric and oceanic models

6 Geodesy and Geodynamics Vol. 3

the matter and motion terms of AAM and OAM to show

how large the differences are ( ECMWF, ERA, NCEP

and ECCO denote the operational ECMWF, ERAinter­

im, NCEP/NCAR and ECCO data, respectively). To

make the plots more clear, all the data are displayed

only within a three-year time span.

From figure 2, one can see that the differences be­

tween different versions of AAM and OAM are quite

large ( especially for the OAM) and often exceed

10 mas. In fact, these differences can be viewed as

the uncertainties of the AAM and OAM data. Thus,

the uncertainties of AAM and OAM are two orders of

magnitude larger than the triaxiality effect, and the tri­

axiality effects on the polar motion excitations can usu­

ally be ignored.

5 Discussion and Conclusion

In this study , we provided the dynamic figure parame­

ters for both the triaxial and symmetric Earth models

with the mantle and the fluid core, and discussed the

effects of the Earth' s tria:xiality on the polar motion ex-

citation , which is proved to be quite small, that is

about 0. 1 and about 0. 2 mas for the x- and y-axes, re-

spectively. The triaxiality effect is much smaller than

the uncertainties of the AAM and OAM data which of-

ten exceed 10 mas. Thus we might conclude that the

Earth ' s triaxiality is unimportant in the study of geo-

physical excitations of polar motion at present and in

the near future. Without loss of accuracy, one can a­

dopt the polar motion theory for the symmetric Earth,

which is much simpler than the one for the triaxial

Earth.

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