Topographic Core‐Mantle Coupling and Polar Motion On Decadal Time‐Scales
ABSTRACT Associated with nonsteady magnetohydrodynamic (MHD) flow in the liquid metallic core of the Earth, with typical relative speeds of a fraction of a millimetre per second, are fluctuations in dynamic pressure of about 103 N m−2. Acting on the nonspherical coremantle boundary (CMB), these pressure fluctuations give rise to a fluctuating net topographic torque Lt(t) (i=1, 2, 3)—where t denotes time—on the overlying solid mantle. Geophysicists now accept the proposal by one of us (RH) that Li(t) makes a significant and possibly dominant contribution to the total torque Li*(t) on the mantle produced directly or indirectly by core motions. Other contributions are the ‘gravitational’ torque associated with fluctuating density gradients in the core, the ‘electromagnetic’ torque associated with Lorentz forces in the weakly electrically conducting lower mantle, and the ‘viscous’ torque associated with shearing motions in the boundary layer just below the CMB. the axial component L3*(t) of Li* (t) contributes to the observed fluctuations in the length of the day [LOD, an inverse measure of the angular speed of rotation of the solid Earth (mantle, crust and cryosphere)], and the equatorial components (Li* (t)) L* (t) contribute to the observed polar motion, as determined from measurements of changes in the Earth's rotation axis relative to its figure axis.In earlier phases of a continuing programme of research based on a method for determining Li(t) from geophysical data (proposed independently about ten years ago by Hide and Le Mouël), it was shown that longitudedependent irregular CMB topography no higher than about 0.5 km could give rise to values of L3(t) sufficient to account for the observed magnitude of LOD fluctuations on decadal timescales. Here, we report an investigation of the equatorial components (L1(t), L2(t)) = L(t) of Li(t) taking into account just one topographic feature of the CMB—albeit possibly the most pronounced—namely the axisymmetric equatorial bulge, with an equatorial radius exceeding the polar radius by 9.5 ± 0.1 km (the mean radius of the core being 3485 2 km, 0.547 times that of the whole Earth). A measure of the local horizontal gradient of the fluctuating pressure field near the CMB can be obtained from the local Eulerian flow velocity in the ‘free stream’ below the CMB by supposing that nearly everywhere in the outer reaches of the core—the ‘polosphere’ (Hide 1995)—geostrophic balance obtains between the pressure gradient and Coriolis forces. the polospheric velocity fields used were those determined by Jackson (1989) from geomagnetic secular variations (GSV) data on the basis of the geostrophic approximation combined with the assumption that, on the timescales of the GSV, the core behaves like a perfect electrical conductor and the mantle as a perfect insulator.In general agreement with independent calculations by Hulot, Le Huy & Le Mouël (1996) and GreffLefftz & Legros (1995), we found that in magnitude L (t) for epochs from 1840 to 1990 typically exceeds L3(t) by a factor of about 10, roughly equal to the ratio of the height of the equatorial bulge to that strongly implied for irregular topography by determinations of L3(t) (see Hide et al. 1993). But L (t) still apparently falls short in magnitude by a factor of up to about 5 in its ability t o account for the amplitude of the observed timeseries of polar motion on decadal timescales (DPM), and it is poorly correlated with that timeseries. So we conclude that unless uncertainties in the determination of the DPM timeseries from observationswhich we also discusshave been seriously underestimated, the action of normal pressure forces associated with core motions on the equatorial bulge of the coremantle boundary makes a significant but not dominant contribution to the excitation of decadal polar motion. Other geophysical processes such as the movement of groundwater and changes in sealevel must also be involved.
 [Show abstract] [Hide abstract]
ABSTRACT: The groundwork for a new field in the geophysical sciences—space geodesy—was laid in the 1960's with the development of satellite and lunar laser ranging systems, along with the development of very long baseline interferometry systems, for the purpose of studying crustal plate motion and deformation, the Earth's gravitational field, and Earth orientation changes. The availability of accurate, routine determinations of the Earth Orientation Parameters (EOPs) afforded by the launch of the LAser GEOdynamics Satellite (LAGEOS) on May 4, 1976, and the subsequent numerous studies of the LAGEOS observations, has led to a greater understanding of the causes of the observed changes in the Earth's orientation. LAGEOS observations of the EOPs now span 26 years, making it the longest available spacegeodetic series of Earth Orientation Parameters. Such long duration homogenous series of accurate Earth Orientation Parameters are needed for studying longperiod changes in the Earth's orientation, such as those caused by climate change. In addition, such long duration series are needed when combining Earth orientation measurements taken by different spacegeodetic techniques. They provide the backbone to which shorter duration EOP series are attached, thereby ensuring homogeneity of the final combined series.01/2002;  [Show abstract] [Hide abstract]
ABSTRACT: We present the first investigation that explores the effects of an isolated topographic ridge on thermal convection in a planetary corelike geometry and using corelike fluid properties (i.e. using a liquid metallike low Prandtl number fluid). The model's mean azimuthal flow resonates with the ridge and results in the excitation of a stationary topographic Rossby wave. This wave generates recirculating regions that remain fixed to the mantle reference frame. Associated with these regions is a strong longitudinally dependent heat flow along the inner core boundary; this effect may control the location of melting and solidification on the inner core boundary. Theoretical considerations and the results of our simulations suggest that the wavenumber of the resonant wave, LR, scales as Ro1/2, where Ro is the Rossby number. This scaling indicates that smallscale flow structures [wavenumber ?] in the core can be excited by a topographic feature on the coremantle boundary. The effects of strong magnetic diffusion in the core must then be invoked to generate a stationary magnetic signature that is comparable to the scale of observed geomagnetic structures [?].Geophysical Journal International 05/2012; 189(2):799814. · 2.85 Impact Factor  SourceAvailable from: Wei Chen[Show abstract] [Hide abstract]
ABSTRACT: The Earth's rotation is perturbed by mass redistributions and relative motions within the Earth system, as well as by the torques from both the internal Earth and celestial bodies. The present study aims to establish a theory to incorporate all these factors perturbing the rotation state of the triaxial Earth, just like the traditional rotation theory of the axialsymmetric Earth. First of all, we reestimate the Earth's inertia tensor on the basis of two new gravity models, EIGENGL05C and EGM2008. Then we formulate the dynamic equations and obtain their normal modes for an Earth model with a triaxial anelastic mantle, a triaxial fluid core, and dissipative oceans. The periods of the Chandler wobble and the free core nutation are successfully recovered, being ∼433 and ∼430 mean solar days, respectively. Further, the Liouville equations and their general solutions for that triaxial nonrigid Earth are deduced. The Liouville equations are characterized by the complex frequencydependent transfer functions, which incorporate the effects of triaxialities and deformations of both the mantle and the core, as well as the effects of the mantle anelasticity, the equilibrium, and dissipative ocean tides. Complex transfer functions just reflect the fact that decays and phase lags exist in the Earth's response to the periodic forcing. Our theory reduces to the traditional rotation theory of the axialsymmetric Earth when assuming rotational symmetry of the inertia tensor. Finally, the present theory is applied to the case of atmosphericoceanic excitation. The effective atmosphericoceanic angular momentum function (AMF) χeff = χeff1 + iχeff2 for the present theory is compared with the AMF χeffsym = χeff1sym + iχeff2sym for the traditional theory and the observed AMF χobs = χ1obs + iχ2obs; we find that the difference between χeff and χeffsym is of a few milliseconds of arc (mas) and can sometimes exceed 10 mas. In addition, spectrum analyses indicate that χeff is in good agreement with χeffsym and, further, show an increase of coherency with χobs especially in the lowfrequency band. The obvious advantage of χeff in the lowfrequency band with respect to χeffsym is the critical support of the present theory. However, still better performance of our theory can be expected if the models of the mantle anelasticity and oceanic dynamics were improved. Thus we conclude that the traditional Earth rotation theory should be revised and upgraded to include the effects of the Earth's triaxiality, the mantle anelasticity, and oceanic dynamics. The theory presented in this study might be more appropriate to describe the rotation of the triaxial Earth (or other triaxial celestial bodies such as Mars), though further studies are needed to incorporate the effects of the solid inner core and other possible influences.Journal of Geophysical Research Atmospheres 01/2010; 115. · 3.44 Impact Factor
Page 1
Geophys. J. Int. (1996) 125,599607
Topographic coremantle coupling and polar motion on decadal
timescales
R. Hide,’q2 D. H. Boggs,’ J. 0. Dickey,’ D. Dong,’ R. S. Gross’ and A. Jackson3
Space Geodetic Science and Applications Group, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91 1098099, USA
Department of Physics (Atmospheric, Oceanic and Planetary Physics), University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU
School ofEarth Sciences, University oJLeeds, Leeds LS9 NJT
Accepted 1996 January 10. Received 1996 January 9; in original form 1995 June 21
SUMMARY
Associated with nonsteady magnetohydrodynamic (MHD) flow in the liquid metallic
core of the Earth, with typical relative speeds of a fraction of a millimetre per second,
are fluctuations in dynamic pressure of about lo3 N m2. Acting on the nonspherical
coremantle boundary (CMB), these pressure fluctuations give rise to a fluctuating net
topographic torque Li(t) (i = 1,2,3twhere t denotes timeon the overlying solid
mantle. Geophysicists now accept the proposal by one of us (RH) that L,(t) makes a
significant and possibly dominant contribution to the total torque LT( t ) on the mantle
produced directly or indirectly by core motions. Other contributions are the ‘gravi
tational’ torque associated with fluctuating density gradients in the core, the ‘electromag
netic’ torque associated with Lorentz forces in the weakly electrically conducting lower
mantle, and the ‘viscous’ torque associated with shearing motions in the boundary
layer just below the CMB. The axial component L;(t) of LT(t) contributes to the
observed fluctuations in the length of the day CLOD, an inverse measure of the angular
speed of rotation of the solid Earth (mantle, crust and cryosphere)], and the equatorial
components (Lf(t), L;(t)) = L*(t) contribute to the observed polar motion, as deter
mined from measurements of changes in the Earth’s rotation axis relative to its
figure axis.
In earlier phases of a continuing programme of research based on a method for
determining Li(
t) from geophysical data (proposed independently about ten years ago
by Hide and Le Mouel), it was shown that longitudedependent irregular CMB
topography no higher than about 0.5 km could give rise to values of L3(t) sufficient to
account for the observed magnitude of LOD fluctuations on decadal timescales. Here,
we report an investigation of the equatorial components (L,(t), L2(t)) = L ( t ) of Li(t)
taking into account just one topographic feature of the CMBalbeit
pronouncednamely the axisymmetric equatorial bulge, with an equatorial radius
exceeding the polar radius by 9.5 kO.1 km (the mean radius of the core being
3485 2 km, 0.547 times that of the whole Earth). A measure of the local horizontal
gradient of the fluctuating pressure field near the CMB can be obtained from the local
Eulerian flow velocity in the ‘free stream’ below the CMB by supposing that nearly
everywhere in the outer reaches of the corethe ‘polosphere’ (Hide 1995tgeostrophic
balance obtains between the pressure gradient and Coriolis forces. The polospheric
velocity fields used were those determined by Jackson ( 1989) from geomagnetic secular
variations (GSV) data on the basis of the geostrophic approximation combined with
the assumption that, on the timescales of the GSV, the core behaves like a perfect
electrical conductor and the mantle as a perfect insulator.
In general agreement with independent calculations by Hulot, Le Huy & Le Mouel
(1996) and GreffLefftz & Legros (1995), we found that in magnitude L ( t ) for epochs
from 1840 to 1990 typically exceeds \L3(t)j by a factor of about 10, roughly equal to
the ratio of the height of the equatorial bulge to that strongly implied for irregular
topography by determinations of L3(t) (see Hide et al. 1993). But L (t) still apparently
possibly the most
599
0 1996 US Government
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600
R. Hide et al.
falls short in magnitude by a factor of up to about 5 in its ability to account for the
amplitude of the observed timeseries of polar motion on decadal timescales (DPM),
and it is poorly correlated with that timeseries. So we conclude that unless uncertainties
in the determination of the DPM timeseries from observationswhich
discusshave been seriously underestimated, the action of normal pressure forces
associated with core motions on the equatorial bulge of the coremantle boundary
makes a significant but not dominant contribution to the excitation of decadal polar
motion. Other geophysical processes such as the movement of groundwater and
changes in sealevel must also be involved.
we also
Key words: coremantle boundary, dynamics, Earth's core, Earth's interior, Earth's
rotation, geomagnetism, planetary oblateness, topography.
INTRODUCTION
Electric currents generated in the Earth's liquid metallic
core are responsible for the main geomagnetic field and its
secular changes (see e.g. Merrill & McElhinny 1983; Jacobs
19871991). The currents are produced by dynamo action
involving irregular MHD flow in the core. Concomitant
dynamical stresses acting on the overlying mantle have been
invoked by geophysicists as the main source of excitation of
the socalled 'decadal' fluctuations in the speed of rotation of
the 'solid Earth' (mantle? crust and cryosphere). It is possible
that core motions make a detectable contribution to polar
motion (i.e. the movement of the rotation axis of the solid
Earth relative to the figure axis) on these timescales. Studies
of rotational manifestations of core motions have important
implications, for they bear directly on the improvement of
models of the structure, composition and dynamics of the
Earth's deep interior, which have to reconcile a wide variety
of geophysical observations (for references see Loper & Lay
1995).
Consider a set of bodyfixed axes xi (i = 1,2,3) aligned with
the principal axes of the solid Earth and rotating about the
centre of mass of the whole Earth with angular velocity
6i = Ciji( t ) = ((21, &, 63)
= Q(tft,, h2,
where t denotes time and 0 the mean speed of rotation of the
solid Earth in recent times (0.7292115 x
Munk & MacDonald 1960; Lambeck 1980; Rochester 1984;
Moritz & Mueller 1987; Wahr 1988). Over timescales that are
short compared with those characteristic of geological pro
cesses, the rotation of the Earth departs only slightly from
steady rotation about the polar axis of figure, so that ml, m2, m3
are all very much less than unity and lhil<<Q, where
hi = dmi/dt. Periodic variations in wi on timescales less than
a few years are forced by periodic lunar and solar tidal torques
and related changes in the moment of inertia of the solid
Earth. Irregular subdecadal variations are evidently produced
largely by atmospheric and oceanic torques due to tangential
stresses in boundary layers and normal (pressure) stresses
acting on surface topography. These fluctuating stresses are
associated with seasonal, intraseasonal and interannual fluctu
ations in the total angular momentum of the atmosphere (for
references see Hide 1986; Hide & Dickey 1991; Eubanks 1993;
Rosen 1993). When these rapid variations have been removed
from the observational data, the smoothed timeseries,
(1)
1 + A3),
rad sC1) (see e.g.
that remains (see Figs 1 and 2) represents the contributions of
decadal variations to Q,(t). Errors and uncertainties in these
determinations of m, ( t ) and mz(
It is convenient at this stage to introduce the vector L: (t),
defined as the hypothetical torque acting on the solid Earth
that would account for observed Earthrotation fluctuations
on decadal timescales in the absence of other processes, such
as changes in the inertia tensor of the solid Earth associated
with the stresses responsible for L+(t) and also with the
movement of groundwater, melting of ice, etc. Using standard
methods (see e.g. Munk & Macdonald 1960; Lambeck 1980)
it is readily shown that, to sufficient accuracy here (cf: Hide
1989),
t ) are discussed in Appendix A.
(3)
where C(")  A(m1 = 2.63 x
principal moments of inertia of the solid Earth (about polar
and equatorial axes through the Earth's centre of mass
respectively). C(") is 0.895 times the polar moment of inertia
of the whole Earth, including the metallic core. Timeseries of
L: (t) and L: (t) are given in Fig. 2.
In this paper we investigate, as part of a continuing pro
gramme of research on Earthrotation fluctuations, the extent
to which the observed DPM can be accounted for by the
fluctuating topographic torque L (t) = [L,(t), L2(t)] associ
ated with normal pressure forecasting on the equatorial bulge
of the CMB. Our findings (see below) are in general agreement
with those of independent and parallel studies by Hulot, Le
Huy & Le Moue1 (1996) and GreffLefftz & Legros (1995)
extending earlier work by Hinderer et a/. (1987), in which
effects of irregular latitudedependent topography as well as
electromagnetic coupling and induced changes in the inertia
tensor of the imperfectly rigid mantle are also investigated.
kg mz, C'"' and A'"' being the
THE TOPOGRAPHIC TORQUE EXERTED
BY THE CORE ON THE MANTLE
The fluctuating torque Ll(t) exerted by the core on the
overlying mantle is produced by:
(a) tangential stresses at the CMB associated with shearing
motions in the thin (probably less than 1 m) viscous boundary
layer just below the CMB;
(b) normal dynamic pressure forces acting on the equatorial
C 3 I996 US Government, GJI 125, 599607
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Coremantle coupling on decadal timescales
60 1
'vu
200 
100 
0 
2 100:
E 200 
300 
400 
500 
600 !
A,
m, 
400
300
200
a
E 100
200
300
uu 
0
m __
2
4
I I
I
I I
I
,
I I
c
IT
~~
7
I I
1
L
I
I
I840 1860
1880 1900
1920 1940 1960 1980
I I
I
I
I
I
I I
1840
1860 1880 1900 1920 1940 1960 1980
6 '
4 
1
I I I
I
I
A, = 8.64 x 107 ms
I
1
r

A rn
0 3
4 
6 .
I
I
I
I
I I
1 1
L
1840 1860
1880 1900
1920 1940 1960
1980
Yea
Figure 1. Timeseries (solid lines) of the observed fluctuations in the Earth's rotation vector &=RCA,(t), ffi2(f), I +n^13(t)] [see eq. ( 1 ) and
Appendix A]. The top panel shows the xcomponent of polar motion, fil(t), defined to be positive towards the Greenwich meridian, the middle
panel shows the ycomponent of polar motion, m2(t), defined to be positive towards 90"E longitude, and the bottom panel shows changes in the
length of day &t), which is related to A3(t) by A(t) = hoA3(t), where A, is the nominal length of day of 86 400 s. The dashed lines show the
decadal variations obtained by applying a lowpass filter (cutoff period = 10 years) to the observed series (see also Fig. 2).
bulge and other (possibly) smaller and more irregular
departures from sphericity of the shape of the CMB;
(c) Lorentz forces due to the flow of electric currents in the
weakly conducting lower mantle generated by the electro
motive forces associated with the geodynamo processes in the
core; and
(d) gravitational forces associated with horizontal density
variations in the core and mantle and especially with CMB
topography (Jault & Le Mouel 1993; Eubanks 1993).
Dynamical arguments show that the pressure coupling associ
ated with quite modest CMB topography could predominate
over other effects (Hide 1969, 1986, 1989). While the contri
bution made by viscous stresses is negligible if, as is likely, the
effective coefficient of kinematic viscosity of the core is less
than about lo4 mz s', electromagnetic coupling might be
quantatively significant if the (unknown) electrical conductivity
of the lower mantle were sufficiently high. Gravitational coup
ling may turn out to be significant as well (see Jault & Le
Mouel 1993; Hulot et al. 1996).
The axial component L3(t) of the topographic torque and
its manifestation in decadal LOD variations have been con
sidered in previous studies (e.g. Jault & Le Mouel 1990; Hide
et d. 1993). Here we discuss the equatorial components L,(t)
and L2(t) and their contribution to the decadal polar motion
rn,(t) and rnz(t), a problem which, as noted above, has also
been studied independently by Hinderer et a/. (1987), Hulot
et a/. (1996) and GreffLefftz & Legros (1995), see also Hide
(1989), with findings in general agreement with those of the
present study. If ps is the dynamic pressure associated with
core motions u =us =(us, us, w,) in the free stream just below
the viscous boundary layer at the CMB (u being the Eulerian
flow velocity relative to a reference frame fixed to the solid
Earth), and the CMB is the locus of points where the distance
from the Earth's centre of mass is r = c + h(0, $) (c being the
mean radius of the CMB and ( ( I , & )
longitude of a general point P), then
L,(t) = 2 L'l,' (r x p,V,hli sin 0 dB d+
the colatitude and
(4)
0
1996 US Government, GI1 125, 599607
Page 4
602
R. Hide et al.
36
h 24
2 2 12
6 12
z 24
x
W
o
I
,
I I
I
1 1
1840 1860 1880 1900 1920 1940 1960 1980
36
n 24
2
0 12
K
W
6 12
z 24
O
4
36
0.16
n 0.12
2
3 0.01
E 0.04
0.08
x 0.00
W
* 0.08
0.12
0.16
I
I
I
t
I
I
I I
1
L2’(t) 
L2(t) 
t
1840 1860 1880 1900 1920 1940 1960 1980
60
40 
2o 8
0 %
20 E
40
cd
60
60
40 h
20 E?
20 2
40
0 %
60
1840 1860 1880 1900 1920 1940 1960 1980
Yea
Figure 2. Timeseries of the components (L: (t), L: ( t), L: (t)) of the equivalent torques implied by decadal components of (A,( t), mz( t), 1 + m,( t))
for a perfectly rigid mantle [see eqs (2) and (3) and Hide (1989)l. Also given (dashed lines) are the estimated timeseries of the contribution made
to (&,(t), Az(t)) by the equatorial torque (Ll(t), L,(t)) (see eq. 23) due to the action on the equatorial bulge of the CMB o f normal pressure forces
associated with core motions in the case of a perfectly rigid mangle (see Fig. 3). A lowpass filter with a cutoff period of 10 years has been applied
to all time series.
if Ihl<< c and IV,hl<< 1. Here V, = c’(ba/aO, 6 cosec &?/ad,),
where 6 and I$ are unit vectors in the direction of increasing 8
and d, respectively, and r is the vector distance from the Earth’s
centre of mass. Because r x V,(hp,) is a toroidal vector, with
no radial components, its integral over the whole CMB is
equal to zero (Hide 1989). This leads to a more useful
expression for Li( t ) in the present context (see below), namely
Li( t ) = c2 l’ [
L, = c2 r[ h (sin 8 sin d
ae
(r x ~VJJ,)~ sin 8 d8 dd, ,
(54
with components
,~
aP + cos 8 cos
L, = cz jo2‘[
h(sin 8 cos 4 aps cos 8 sin
ae
L, = c2
h( sin 0%) d8 dd,.
Given h(8,d,) and determinations of p,(8, d,, t) on the CMB,
Li(t) could be calculated directly using either eq. (4) or eq. (5).
However, as in the case of the Earth’s core and in other
situations where ps is not known from direct measurements
but other information is available, such as us(& 4, t), it is
still possible to estimate Li(t) by using the equations of
fluid dynamics to relate the horizontal pressure gradient to
‘observable’ quantities. Owing to the Earth’s rotation, Coriolis
forces exert a strong influence on core flow (Elsasser 1939;
Frenkel 1945; Inglis 1955). They should be in close ‘geostrophic’
balance with horizontal pressure gradients nearly everywhere
within those regions of the core where Lorentz forces, which
produce the strongest ‘ageostrophic’ effects (Hide 1956), are
comparatively weak, and in ‘magnetostrophic’ balance in any
regions where Lorentz forces are comparable in magnitude
with Coriolis forces. Now the magnetic field in the core can
be decomposed into ‘toroidal’ and ‘poloidal’ parts [Elsasser
1947; c $ eq.(12) below], where the former has no radial
component and may exceed the latter in magnitude by as
much as a factor of about 10 and give rise to magnetostrophic
flow throughout most of the liquid corethe ‘torosphere’ (see
Hide 1995tbut not in the outer reaches of the corethe
‘polosphere’where by definition the toroidal magnetic field
0 1996 US Government, GJI 125, 599607
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Coremantle coupling on decadai timescales
603
is no stronger than the poloidal field (see Le Mouel 1984; Hide
1995). With a fractional error of no more than about lo’,
polospheric flow can be assumed geostrophic nearly every
where, satisfying the equation
2p,Rc0~ ~[w,,u,] = ~~[ap,/a0,cosece ap,/a4],
where p, is the average density over the spherical surface where
u = us. Here (us, w,), the (0, $) components of us, are typically
much greater in magnitude than us, the rcomponent of us. It
follows from eqs ( 5 ) and (6) that on the timescales of interest
here, over which u may change significantly but h does not,
[w),
(6)
WI= 2 m c 3 jazz [ #)Cfi(6 4, t), f’(0, 4, t)l
x sinBcosHdOd4,
(74
where
[ f l , f2] = [u, cos 0 cos q ! ~  w, sin 4, v, cos 0 sin 4 + w, cos 41
(7b)
(Hide 1989, 1995; Jault & Le Mouel 1989; Voorhies 1991).
The basic theoretical relationships needed here are given by
eqs (37). The integral on the righthand side of eq.(7a)
involves the CMB topography h(0,4). The &independent mean
equatorial bulge h,(0) (say) of the CMB corresponds to a
9.5 & 0.1 km difference between the equatorial and polar radii
of the CMB (see Denis & lbrahim 1981; Gwinn, Herring &
Shapiro 1986; Herring et al. 1991; cf: Moritz & Mueller 1987).
If this is substantially larger than the typical vertical amplitude
of irregular topographic features of the CMB [and there is
evidence that this might be so from the study of decadal
variations in the length of day, involving the evaluation of
L,(t); see Jault & Le Mouel (1990); Hide et al. (1993)], then
to a good first approximation we can replace h(O,4) in eq. (7)
by ho(B) [see eqs (17) and (18) below]. The other quantity
needed in the evaluation of Li(t) is either the field of dynamical
pressure p,(e, 9, t) [see eqs (4) and ( 5 ) and Appendix B] or the
field of horizontal flow (us(& 4, t), w,(0, 4, t)), which is related
to ps through eq. (6). Geomagnetic secular variation data have
been used by various workers to infer ( v , , ~ , ) by a method
that invokes geostrophic balance in combination with the
equations of electrodynamics appropriate to the case when the
mantle can be treated as a perfect electrical insulator of uniform
magnetic permeability and the core as a perfect conductor, as
we shall now discuss.
VELOCITY AND PRESSURE FIELDS IN
THE CORE
Denote the value of the main geomagnetic field at a general
point (I, 0,#) at time t by B(r, @,d, t), and the geomagnetic
secular variation (GSV) by B = aB/at. Determinations of B
made at and near the Earth’s surface at various epochs can be
used to infer us, the Eulerian flow velocity just below the CMB
(for references see Bloxham & Jackson 1991; Hulot, Le Mouel
& Wahr 1992). In turn, estimates of the associated horizontal
pressure gradient c’(ap,/a0, cosec 0 ap,/a&) there can be
deduced by using eq. (6). The first of the three reasonable key
assumptions that underlie the method used is that the electrical
conductivity of the mantle and magnetic permeability gradients
there are negligible, so that B can be written as the gradient
of a potential V satisfying Laplace’s equation V2 V = 0. This
facilitates the downward extrapolation of the observed field at
and near the Earth’s surface in order to obtain B and B at the
CMB (see e.g. Jacobs 19871991). The second assumption
concerns the timescales of the GSV and the electrical conduc
tivity of the core. Dynamo theory requires high but not perfect
electrical conductivity, for it is impossible to change the
magnetic flux linkage of a perfect conductor (Bondi & Gold
1950), in accordance with Alfven’s ‘frozen flux’ theorem. When
dealing with fluctuations in B on timescales very much less
than that of the ohmic decay of magnetic fields in the core
(which is several thousand years for globalscale features),
however, AlfvCn’s ‘frozen flux’ theorem, in our notation
aB/& = V x (U x B),
(8)
should provide a good leading approximation, implying that
the lines of magnetic force emerging from the core are advected
by the horizontal flow (us, w,) just below the CMB (Roberts
& Scott 1965; Backus 1968). Accordingly if B = (Br, B,, B+),
then B, at the CMB satisfies
(9)
The last equation alone does not permit the unique deter
mination of us from knowledge of B and B at the CMB, so a
third physically plausible assumption is needed. Effective
uniqueness can be secured by making use of the assumption
expressed by eq. (6) above, namely that, to a first approxi
mation, polospheric flow is in geostrophic balance (Hills 1979;
Le Mouel 1984; Le Mouel, Gire & Madden 1985; Backus &
Le Mouel 1986; Gire & Le Mouel 1990; see also Bloxham &
Jackson 1991; Hulot et al. 1992). This gives the additional
equation
cosO[g] +v,=o,
,=c
c
sin 0
which is obtained by eliminating p, from eq. (6) and using the
mass continuity equation
v  u = 0
for flow of an effectively incompressible fluid.
Various groups of geomagnetic workers have produced
maps of (us, w,) based on this (and other) assumptions, and
have investigated the errors and uncertainties encountered in
practice (Bloxham & Jackson 1991; Hulot et al. 1992). Fields
of B, and B, at r = c deduced by Jackson (1989) for epochs
going back to the year 1840 AD provided the basic B and B
input data for the present study. These were used to produce
hypothetical ‘geostrophic’ flow fields us, constructed using
spherical harmonic expansions (see eqs 1620) up to degree
and order 14, which account for more than 90 per cent of the
observed GSV (Jackson 1989).
(11)
TORQUE ON THE EQUATORIAL BULGE
For an effectively incompressible fluid, the Eulerian flow field
u is solenoidal, by virtue of eq. (ll), and can therefore be
decomposed into a toroidal part (with no radial component)
and a poloidal part as follows:
+ v x v x (P*i),
where i is a unit vector in the direction of increasing r (ct eq. 4).
u=u,+u,=v x (T*i)
(12)
0
1996 US Government, GJI 125, 599607
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604
R. Hide et nl.
If (u, v, w) are the (r, 0,b) components of u, then
where S* = aP*/&. If u = us =(us, us, w,) in the free stream
just below the CMB, where, for the purpose of this part of the
calculation (but see Hide 1995, eq. 4..5), u, can be set equal to
zero and r equal to c, the mean radius of the CMB (cj: eq. 4),
and T* = T and S* = S on r = c, then we have
1 as
csln 0 84 c 80
(7s
w =+
c ?O
c sin O 34 '
1
dT
1 c?T
1
By the geostrophic relationship eq. (6),
us=
+  ,
aPs

an
dT
cos 0 dS
= (2pRc cos 0)w, = (2pn)
_
84
a ' s   ( 2 p ~ c cos 0 sin @)us
c?T
,
from which it follows that T and S satisfy
aT
a4
as
aU
a2s COSQ a2s
+ 7
i?o2
sin@
sin 0
= (cos2 0  sin2 0) + cos 0 sin 0

(154
The scalar quantities T(8,4, t ) and S(H, 4, t) and the topo
graphy h(0,d) can be expressed as spherical harmonic
expansions:
4 = 1 1
n=O m=O
(q;,,m cos m4 + 4SI.m sin m4)Pn,rn(cos 0) 9
where
mn
( 16)
Here Pn,,,
degree n and order m with the Schmidt seminormalization
(cos 0) are the associated Legendre polynomials of
P,,,(cos fI)P,,,,,,(cos 0) sin t) dU = 2( 2  fim,0)Sn,n,/(2n + I),
where 6,,,, = 1 when n = n', and fin.,, = 0 when n # n'. (See
Appendix B for a discussion of the spherical harmonic expan
sion of the pressure field ps in terms of the coefficients of the
expansions of T and S.)
In the case when the equatorial bulge is the sole topographic
feature of the CMB to be considered, we have h(0, 4) = h,(@),
where
ho(0) = &.,P~.o(COS 0) = h;,o[f(3
COS2  1) . 1
(21)
This gives 3hC,,,/2 for the difference between the equatorial
and polar radii (9.5 f 0.1 km, see above), so that
h;.,=
(6.3+0.l)x IO'm.
The general expressions for (Ll, L2) obtained when eqs (7)
are combined with eqs (1216) are quite complicated and
will not be written down here (but see Appendix B). When
h(0,4) = h,(U) given by eq. (21), the expressions simplify to the
following:
(L,, L,) = 27CpRC2h;.,
(22)
The polar motion due to the action of geostrophic pressure
forces acting on the equatorial bulge of the CMB in the case
when the mantle is perfectly rigid could be derived by com
bining this %equation with the equation obtained by setting
(Lx,
L2)=(L:, L;) in eq. (3). Alternatively, we can compare
the timeseries'of (Ll(t), L,(t)) as given by eq. (23) with those
of (L: (t), L;(t)) implied by the observed polar motion
through eq. (3). The results are shown in Figs 2 and 3, where
in evaluating ml(t) and rn,(t) we have taken
p = 0.99 x lo4 kg m3,
c = 3.48 x 10' m,
(see e.g. Stacey 1992; Eubanks 1993), so that
27cPR~~h;,~
and
= 3.48 x l O I 7 kg s'
R = 7.29 x
rad s',
kg m2
C""" A('") = 2.63 x
[Q2((c("')  A("))]' = 8.12 x
kg' m' s2
are the numerical values of these factors in eqs (23) and (3)
respectively.
DISCUSSION AND CONCLUDING
REMARKS
The axial component L3(t) of the net torque Li(t) on the
mantle due to the action on topographic features h(0,4) of
the CMB of normal (pressure) stresses associated with core
motions could, as shown in previous work, make a significant
and possibly dominant contribution to observed decadal LOD
fluctuations, even with longitudinal variations in h that are no
bigger than 1 km in vertical amplitude and possibly even
slightly less (Hide 1969; Jault & Le Moue1 1990; Hide et al.
1993). The equatorial bulge of about 10 km is likely to be the
main topographic feature involved in producing the equatorial
component L (t) of Li( t), and it is not surprising therefore, as
the calculations presented in this paper show (see Fig. 2 and
Hide et al. 1993), that IL(t)l typically exceeds IL3(t)l by a
large factor. However, it is clear from Fig. 2 that L (t) is about
five times smaller in magnitude than the equatorial torque
L+( t ) inferred from the observed polar motion. Further
analysis reveals that the series are uncorrelated as well.
In our study, the equatorial bulge is taken (for simplicity)
to be the sole topographic CMB feature, and the resulting
expression for the equatorial torque involves only the second
and fourthdegree spherical harmonic coefficients of the veloc
ity field (see eq. 23). The seconddegree harmonic clearly
Q 1996 US Government, GJI 125, 599607
Page 7
Coremantle coupling on decadal timescales
605
4.8 
3.6 
2.4 
n
2 0 1.2 
3
x 0.0 
2
2.4 
3.6 
W
1 I I
,
I
I
I
,
 8
 6
 4
41 ”..’
L,: total...
S,,  S
4.8 
3.6 
2.4 
,.
1.2 
x 0.0 
2
I.,
.........................
...................
n
2
W 2
1.2
2.4 
3.6 
4.8 
 8
1840 I860 1880 190 1920 1940 1960 1980
.,
I
I I
I
I I
I
I
I I
I
I
I
I
1
I
 8
 6
 4
I ” 2
0 %
Ly total
S,,  S 41 ‘.”’
1.2
4.8 
I
4
1
I I
I I I
dominates (Fig. 3), so it is unlikely that uncertainties in the
velocity fields used could account for the discrepancy. Our
general findings concerning the inadequacy of topographic
coupling in the excitation of decadal polar motions agree
qualitatively with those of the abovementioned work by Hulot
et al. (1996) and GreffLefftz & Legros (1995), where effects
due to irregular latitudedependent topography, gravitational
torques associated with the nonspherical shape of the CMB,
and changes in the inertia tensor of the solid Earth produced
by fluctuating horizontal pressure variations in the upper
reaches of the core are also taken into account. The fluctuating
electromagnetic torque on the mantle associated with core
motions (see above) could, of course, be stronger than the
topographic torque, but only under extreme assumptions con
cerning the (unknown) distribution and magnitude of the electri
cal conductivity of the lower mantle and of the toroidal part of
the geomagnetic field just below the coremantle boundary.
It is an old suggestion that the movement of air and water
at and near the Earth’s surface on relevant timescales is
involved in the excitation of DPM (see Wilson 1993), but
quantitative studies are hard to make. The spectral character
istics of water movement are similar to that of DPMred,
and increasing sharply at decadal periodswhereas
mass excitation spectrum is white, i.e. flat (Kuehne & Wilson
the air
1 2 ;
4
 6
; 8
1991). DPM observations imply an excitation showing linear
polarization along the direction that would result from a
uniform rise or fall of sealevel, implying that forcing is due,
at least in part, to the redistribution of water mass (Wilson
1993). It will be necessary in future work, as better data
become available, to reinvestigate all possible excitation mech
anisms, for it seems that the decadal polar motion is most
likely caused by a variety of geophysical processes, with the
topographic torque induced by the equatorial bulge producing
a significant but not a dominant contribution.
ACKNOWLEDGMENT
This paper presents the results of one phase of research carried
out at the Jet Propulsion Laboratory, California Institute of
Technology, sponsored by the National Aeronautics and Space
Administration. We also acknowledge helpful comments by
Dr. Gauthier Hulot and an anonymous referee.
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Page 9
Coremantle coupling on decadal timescales
607
systematic errors on the decadal variability evident in this
series may affect the shape of the variations, but not the
amplitude.
geodesy and geodynamics, pp. 385392, eds Babock, A.F. & Wilkins,
G.A., Reidel, Dordrecht.
APPENDIX A: OBSERVATIONS OF POLAR
MOTION
Studies of decadal variations in polar motion (see Fig. 1 ) rely
heavily upon daily observations of the latitude of each of the
five stations of the International Latitude Service (ILS), which
are obtained by the technique of optical astrometry (see e.g.
Yumi & Yokoyama 1980). Determinations of polar motion
(m(t), rnz(t)) (see eq. 2) from monthly averages of these obser
vations span the 80year period 18991979. Analyses of this
ILS polarmotion timeseries show clear evidence of variability
on decadal time scales (e.g. Wilson & Vicente 1980; Dickman
1981). The reality of this variability, however, has been called
into question out of a concern about the contaminating effects
of possible systematic errors occurring at individual stations
(see e.g. Eubanks 1993). In particular, Zhao & Dong (1988)
noticed that, when observations from the Ukiah ILS station
are not used, the recovered DPM variations, although still
present with about the same amplitude, are not as regular as
when the polar motion is determined from all available obser
vations. From this, they conclude that the apparent regularity
of the decadal polar motion variations (i.e., the socalled
30year Markowitz wobble) is an artefact of systematic errors
in the Ukiah observations. We emphasize here that the study
of Zhao & Dong (1988), although widely cited as concluding
that the DPM variations are not real, in fact only questions
the regularity of the observed variations, not their presence.
Gross (1982. 1990), recognizing the corrupting influence of
systematic errors, developed a technique in which polar motion
is recovered from the ILS variation of latitude observations
by simultaneously solving for stationdependent systematic
errors, leaving the polarmotion parameters free of such effects.
The decadal variations evident in the polarmotion series that
he obtained by this technique are nearly identical to that
exhibited in the ILS series, leading him to conclude that these
variations are real and not an artefact of systematic errors in
the observations.
Spacegeodetic determinations of polar motion began with
the launch of the Lageos I satellite in 1976, May, and now
span nearly 20 years. The POLE93 polar motion series
analysed here is a Kalman filterbased combination of the ILS
polarmotion series (spanning 18991979), the BIH (Bureau
International de I’Heure) optical astrometric series (spanning
19621982), and spacegeodetic polar motion measurements
made by the techniques of SLR (satellite laser ranging) (span
ning 19761993), VLBI (very long baseline interferometry)
(spanning 19791993), and the GPS (global positioning system)
(spanning 19911993). Since 1982, the polar motion values in
POLE93 have therefore been based solely upon modern space
geodetic measurements. As seen in Figs 1 and 2, there is clear
evidence of decadalscale variability in the POLE93 polar
motion series since 1982. Furthermore, the post1983 variability
is consistent with similarscale variability evident in this series
during earlier epochs, thereby giving credence to this earlier
variability.
The POLE93 series is used here since it is the most complete
polar motion series currently available. The above discussion
and previously cited studies indicate that the influence of
APPENDIX B: SPHERICAL HARMONIC
EXPANSION OF PRESSURE FIELD
By virtue of the equivalence of eqs (4) and (5), which lead to
eq. (7) for L,(t) in terms of h(B, 4) and us(@, 4, t), it is possible
to determine L,(t) without going through the stage of
evaluating ps(O, 4, t) directly (see Hide 1989). For the sake of
completeness and other reasons, however, it is useful to relate,
through eq. (6), the coefficients in the spherical harmonic
expansions for us(O, 4, t ) (see eqs 14, 17 and 18) to those of
the following expression for p,(B, 4, t):
mn
ps(Q, 4 9 t ) = C
(pi,,m(t) cos m4 + p:,m(t) sin m4)
n = O m=O
x Pn,m(COS 0) 3
(B1)
where the P,,,(cos 0) are defined by Schmidt normalization
given by eq. (20). The coefficients piqrn and p;,, can be expressed
in terms of the spherical harmonic coefficients of us(@, 4, t ) as
follows [see eqs (17) and (18); I$ Gire & Le Moue1 (1990)l:
P S , ~ = 2~Q(cb;,’s:z,m + Cb:!ns:,m
p;,m = 2Pa(CbT,)s:2,m + Cn,m~n,m + Cn.mSn+Z,m),
when m > 0, and
+ cb?S:+2,m),
( + I c
(B2)
(B3)
(0) s
the coefficients s:,~, s;,~, T ; , ~ , T ; , ~ being equal to zero when
n < rn and rn > 0, in eqs (B2) and (B3).
c(J = (n  l)(n  2)[(n  m)(n + m)(n  1  m)(n  1 + m)]’”
m(2n  1)(2n  3)
n,m 
(B5)
+
(2n1) 1 ’
1 + m) (n  m)(n + m)
c;:; = m(2n + 1)
and
c(+J= (n + 2)(n + 3)[(n + 1  m)(n + 1 + m)(n + 2  m)(n + 2 + m)]’’2
n,m 
m(2n + 3)(2n + 5)
(B7)
In terms of the coefficients of pressure field, the torque
acting on the main bulge at the CMB can be expressed as
follows:
(see Hinderer et al. 1990). It is readily demonstrated, using eqs
(B2) and (B3), that eq. (B8) is consistent with the expression
for the torque in terms of the velocity field given by eq. (23).
Taking h2,,, as 2caJ3, where a, denotes the ellipticity at
CMB, our eq. (B8) is identical to the formula (22b) of Hulot
et af. (1996).
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