Physics of the Earth and Planetary Interiors 118 Ž2000. 291–316
www.elsevier.comrlocaterpepi
Length of day decade variations, torsional oscillations and inner
core superrotation: evidence from recovered core surface zonal
flows
A. Pais ) , G. Hulot
Laboratoire de Geomagnetisme,
CNRS UMR 7577, Institut de Physique du Globe, 4 place Jussieu, 75252 Paris cedex 05, France
´
´
Received 22 September 1999; accepted 16 November 1999
Abstract
We consider the core surface flow derived from geomagnetic models wBloxham, J., Jackson, A., 1992. Time-dependent
mapping of the magnetic field at the core–mantle boundary. J. Geophys. Res. 97, 19537–19563.x under the frozen flux and
tangentially geostrophic assumptions, and focus on the significance of its toroidal zonal component. This component
represents a small fraction of the whole flow, but is believed to contain important information concerning core dynamics and
the way the liquid core interacts with both the mantle and the inner core. We consider the 150-year period 1840–1990 for
which the best data is available. Our results show that with the current methodology and data, recovered core surface zonal
flows can successfully be used to address three issues in core dynamics, provided that one properly takes data and
methodology uncertainties into account. The first issue deals with the possibility of testing the theory of Jault et al. wJault, D.,
Gire, C., Le Mouel,
¨ J.-L., 1988. Westward drift, core motions and exchanges of angular momentum between core and
mantle. Nature 333, 353–356.x which predicts a relationship between core surface zonal flows and length of day ŽLOD.
variations on decade time scales. We recover the known fact that this theory leads to a successful prediction of LOD
variations after 1920 and not early on, and show that this failure can entirely be attributed to the Žpartly correlated.
uncertainties affecting the t 10 and t 30 flow components required to carry on the prediction. The second issue deals with the
possibility of detecting torsional oscillations of the kind that Braginsky wBraginsky, S.I., 1970. Torsional magnetohydrodynamics vibrations in the Earth’s core and variations in day length. Geomagn. Aeron. 10, 3–12 ŽEng. transl. 1–8..x predicted
should occur in the core on decade time scales. We show that the large scale component of the relevant equatorial symmetric
zonal flow Ž t 10 , t 30 , t50 and very marginally t 70 . displays significant time variations that can be attributed to such oscillations.
But uncertainties affecting these coefficients are quite large and should therefore be taken into account. The third and final
issue deals with the possibility of identifying a surface signature of the inner core superrotation that some seismologists
claim to have detected and that is predicted by most dynamo numerical simulations. We show that the average zonal flow
recovered over the 1840–1990 period displays a strong westward rotation Žat 0.3–18ryr. within the inner core tangent
cylinder, which can possibly be interpreted as the surfacic counterpart of an opposite eastward inner core superrotation.
q 2000 Elsevier Science B.V. All rights reserved.
Keywords: Length of day decade variations; Torsional oscillations; Inner core superrotation; Core surface zonal flows
)
Corresponding author.
0031-9201r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 0 3 1 - 9 2 0 1 Ž 9 9 . 0 0 1 6 1 - 2
292
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
1. Introduction
The possibility of inverting the observed secular
variation ŽSV. and building maps of the fluid motion
just beneath the core–mantle boundary ŽCMB. has
received much attention in the past 30 years Žsee
e.g., Bloxham and Jackson, 1991, for an account of
the earlier work on the subject.. This inversion is
possible in principle if one assumes that on the
secular time scale, the Earth’s main magnetic field
changes only as a result of its advection by the flow
at the core surface ŽRoberts and Scott, 1965.. But
observing the SV of the field does not provide
enough information for this flow to be completely
recovered ŽBackus, 1968.. Some additional assumption has to be used regarding the nature of the flow.
It is now usually believed that the flow can be
assumed to be tangentially geostrophic ŽHills, 1979;
Le Mouel,
¨ 1984..
Systematic studies on how much information
about these flows can then be recovered from the
actual SV data have been carried out by Hulot et al.
Ž1992. and more recently by Celaya and Wahr Ž1996.
and Jackson Ž1997.. These papers addressed the
issue from a rather general standpoint. Celaya and
Wahr Ž1996. provided some evidence that the flow
could be assumed to be large scale at the core
surface Ži.e., that the spatial spectrum of the flow
energy could be assumed to fall off as ny2 or faster.
and Hulot et al. Ž1992. showed that if this was the
case, and if the main field spectrum could be extrapolated from its known behavior Žknown up to degree
13., then the errors linked to our limited knowledge
of the main field and of the SV would not map too
strongly on the very first degrees of the flow. In fact,
Hulot et al. Ž1992. showed that in practice and with
the data available at the present day, the components
of the flow with degree less than 5 could reasonably
be recovered whereas those with degree greater than
8 were absolutely unconstrained. In the present paper, we will not reconsider these earlier and general
results but will try to assess in more detail how much
can actually be said from the data concerning the
toroidal zonal part of the flow. This part of the flow
deserves a little more scrutiny for several reasons.
It at least partly Žand possibly mainly. reflects
rigid cylindrical flows that operate within the body
of the core as a result of the occasional breaking of
the so-called Taylor condition. This basically happens when dynamical axial torques start acting on
such cylinders, as a result of some likely core–mantle coupling Žsee e.g., Jault and Le Mouel,
¨ 1989;
Jault et al., 1996. or when intrinsic torsional oscillations of the kind proposed by Braginsky Ž1970. take
place. Such torques must then be balanced either by
a viscous torque Žas is often assumed in numerical
simulation of the dynamo, e.g., Glatzmaier and
Roberts, 1995. or by a dynamical reaction of the
cylinder Žas is assumed for torsional oscillations and
is anyway more likely to happen within the core,
e.g., Jault, 1995; Jault and Le Mouel,
¨ 1999; Jault et
al., 1988; Kuang and Bloxham, 1997.. In fact toroidal
zonal flows computed from the observed SV have
already been used to both make predictions of the
likely effect of exchange of angular momentum between the core and the mantle on the length of day
ŽLOD. Že.g., Jault et al., 1988; Jackson, 1997; Jackson et al., 1993; Le Mouel
¨ et al., 1997., and estimate
the magnitude of the field responsible for possible
torsional oscillations within the bulk of the core
Že.g., Zatman and Bloxham, 1997, 1998, 1999..
One last but important motivation for the present
study comes from the claim by Song and Richards
Ž1996. and Su et al. Ž1996. that the inner core could
be in a state of superrotation within the fluid core.
This claim is based on the observation of some
variations over the past 30 years within the traveltime of seismic rays passing through the anisotropic
inner core. Such variations can indeed be interpreted
in terms of the inner core rotating with a speed of
1–38ryr eastward relative to the mantle, which is
one order of magnitude larger than the value usually
retained for the average westward drift at the core
surface Žas inferred from the computed block t 10
toroidal zonal flow.. Although still questionable
ŽSouriau, 1998a, and Souriau et al., 1997, argue that
the quality of the data is not yet good enough to
support the claim, and Laske and Masters, 1999,
argue that Earth’s free oscillations studies rather
imply no rotation to within 0.28ryr., and of uncertain magnitude Žrelying on still another method,
Creager, 1997, found a value of about 0.2–0.38ryr.,
this superrotation could be of the same kind as the
one that has been seen in recent 3D numerical
dynamos ŽGlatzmaier and Roberts, 1995; Kuang and
Bloxham, 1997.. As these dynamos also show a
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
surfacic counterpart to their inner core superrotation,
a closer look at the zonal flows at the core surface is
clearly of some interest. Hence, our present effort to
better assess the characteristics of these flows as
inferred from magnetic observations.
293
™
straint.. The Tn0 are the elementary zonal toroidal
vector fields:
™
™
Tn0 s ycrˆ n =H Yn0 s
™
(
n Ž n q 1.
2
Pn1 Ž u . fˆ
Ž 3.
and the Wnm Ž1 F n - `; 0 - m F n. are defined by
the linear combinations:
2. Computing the flow at the CMB
™
™
™
™
™
™
™
™ ™
S m s c= Y m
ms
ms
Wnm c s Snm c y a nm Tny1
y bnm Tnq1
2.1. Stating the inÕerse problem
We make use of the method developed by Le
Mouel
¨ et al. Ž1985., Gire and Le Mouel
¨ Ž1990. and
discussed by Hulot et al. Ž1992., for determining a
large-scale tangentially geostrophic flow beneath the
CMB accounting for the observed SV Žsimilar methods have been developed by others, see e.g., Jackson
et al., 1993..
This method relies on the frozen-flux approximation ŽRoberts and Scott, 1965. which assumes that on
decade timescales the diffusive term in the radial
component of the induction equation can be neglected compared to the advective term. In that case
one may write at the CMB Ž u r s 0.:
EBr
Et
™
s y=H P Ž ™
uBr . ,
Ž 1.
where ™
u is the fluid velocity with respect to the
mantle, B™
r is the
™ radial component of the magnetic
field and =H s = y rˆŽErEr . is the horizontal gradient operator.
As flow solutions of Eq. Ž1. are highly nonunique
Žthere is a whole space of flows ™
u that do not advect
the field and cannot be tracked down by the SV;
Backus, 1968., tangential geostrophy is further imposed on the flow as an admissible constraint to
reduce the ambiguity ŽLe Mouel,
¨ 1984.:
™
=
HP
u cos u . s 0,
Ž™
™
™
™
and T m s ycr n = Y m
mc
mc
Wnm s s Snm s q a nm Tny1
q bnm Tnq1
Ž 2.
where u is the colatitude.
This constraint is implemented by expanding the
CMB flow in terms of a basis™ of tangentially
™
geostrophic elementary functions Wnm and Tn0 ŽGire
and Le Mouel,
¨ 1990, but see, e.g., Le Mouel
¨ et al.,
1985, and Jackson et al., 1993, for other possible
implementations of the tangentially geostrophic con-
Ž 4.
where n
are
ˆ
H n
n
H n
the elementary poloidal and toroidal vector fields,
forming a set of orthogonal vectors. Ynm c s
Pnm Žcos u .cos m f and Ynm s s Pnm Žcos u .sin m f are the
Schmidt semi-normalized surface harmonics of degree n, order m, Pnm Žcos u . are the associated
Schmidt semi-normalized Legendre functions and c
is the core radius. The numerical coefficients a nm and
bnm are to be found in Gire and Le Mouel
¨ Ž1990..
Our aim is then to recover the geostrophic motion:
`
™ ` n
™u s c Ý
wn0 Tn0 q Ý Ý
½
ns1
™
™
wnm c Wnm c q wnm s Wnm s
ns1 ms1
5
Ž 5.
`
™ ` n
™u s c Ý
t n0 Tn0 q Ý Ý
½
ns1
™
™
™
snm cSnm c q snm sSnm s
ns1 ms1
™
qt nm c Tnm c q t nm s Tnm s
5
,
Ž 6.
the coefficients being in radryr. From Eq. Ž4., the
two sets of flow coefficients w and Žs <t. are readily
related to each other through a matrix relationship of
the form:
Ž s <t . s Q w.
Ž 7.
Because we may also assume that the mantle is an
insulator ŽShankland et al., 1993; Mandea Alexandrescu et al., 1999., both EBrrEt and Br at the CMB
can be obtained by downward continuing the SV and
main field models computed at the Earth surface
from direct observations Žsee e.g., Bloxham et al.,
1989.. In the present paper we rely on the time-dependent model ufm1 of Bloxham and Jackson Ž1992.
ŽBJ92 hereafter., and for any given epoch between
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
294
1840 and 1990 write Br and EBrrEt at the CMB in
the form:
LB
Br Ž c, u , f . s
n
nq 2
a
Ý Ý
ns1 ms0
ž /
c
Ž n q 1.
= bnm c Ynm c q bnm s Ynm s
EBr
Et
LSV
Ž c,u , f . s
n
Ý Ý
ns1 ms0
a
Ž 8.
F Ž w . s M Ž w . q lj Ž w . ,
nq 2
ž /
c
Ž n q 1.
= b˙ nm c Ynm c q b˙ nm s Ynm s .
Ž 11.
where
T
Ž 9.
The coefficients bnm cŽ s. are those of BJ92 at the Earth
surface Ž r s a. for the epoch considered, and the
b˙nm cŽ s. are the time-derivatives of those coefficients
for the same epoch. The degrees of truncation L B
and LSV are mainly defined by observational constraints and will be discussed in Section 2.2.2.
Computing the flow from Eq. Ž1. then amounts to
solve the matrix equation
ḃ s A b w
that we introduce a covariance matrix C SV that
reflects the uncertainties we believe affect the SV
model, and a covariance matrix C w that determines
the form of the regularization condition we a priori
impose on the flow. The estimate wˆ that we seek is
then the one which minimizes the objective function:
Ž 10 .
for the column vector w Ždefining the flow ™
u through
the geostrophic basis, recall Eq. Ž5.., b˙ being the
column vector made of the b˙nm coefficients defining
the SV ŽEq. Ž9.. and A b being the interaction matrix
defining
™ how™much SV each elementary geostrophic
flow Tn0 or Wnm will create by interacting with the
main field defined by Eq. Ž8.. ŽMore details about
the form of the matrices A b and Q are to be found in
Gire and Le Mouel
¨ Ž1990... Because only finite
expansions of both Br and EBrrEt are accessible, not
all coefficients of w Ži.e., of the flow ™
u. can be
recovered, and expansion Ž5. has to be truncated at
some degree. This can be done provided one further
assumes the flow is mainly large scale, which is
implemented by requiring the energy of the flow to
converge following some a priori constraint.
2.2. SolÕing the inÕerse problem
2.2.1. Introducing a priori beliefs on the flow and
the magnetic data for a first set of flow inÕersions
(PH-inÕersions)
In practice, we compute an estimate of the flow
by using a Bayesian inference approach Žsee e.g.,
Gubbins and Bloxham, 1985; Backus, 1988, for an
outline of the Bayesian formalism.. This requires
˙
M Ž w . s Ž A b w y b˙ . Cy1
SV Ž A b w y b .
j Ž w . s wT Cy1
w w
Ž 12 .
and l is a damping parameter to be discussed later
on. This estimate is known to satisfy
˙
wˆ s Ayg
b b,
Ž 13 .
where
T y1
y1
Ayg
b s Ž A b C SV A b q lC w .
y1
ATbCy1
SV
Ž 14 .
is sometimes known as the generalized inverse Žsee
e.g., Menke, 1984..
In the present paper, as in Gire and Le Mouel
¨
Ž1990., the data covariance matrix C SV we will
prefer to rely on, for reasons discussed in Section
2.2.3, is chosen so that M Ž w . in Eq. Ž12. reflects the
wish to minimize the residual
SV energy at the Earth
™ ™
surface Ž1r4p a 2 .HHrs aŽ Bp y Bo . 2 d S Žwhere p stands
for ‘‘predicted’’ and o for ‘‘observed’’.. Since each
b˙ nm cŽ s. coefficient contributes through Ž n q 1.
Ž b˙nm cŽ s. . 2 to the SV energy, this implies
y1
Ž Cy1
SV . PH s WSV Ž n q 1 . I
Ž 15 .
where I is the identity matrix and WSV is a scaling
parameter Žin ŽnTryr. 2 ..
Our a priori covariance matrix C w for the flow is
otherwise chosen so that j Ž w . in Eq. Ž12. reflects
the wish that the flow spatial energy spectrum decreases like nyp as a function of the degree n ŽLe
Mouel
¨ et al., 1985, but see e.g., Jackson et al., 1993,
for other possible choices of regularizing conditions
™ ..
Because the elementary geostrophic motions Wnm do
not form a set of orthogonal vectors in the Euclidean
space of all square integrable tangent vector fields on
the CMB, the constraint on the energy does not
translate into a simple diagonal covariance matrix
C w . In our code, the constraint is therefore directly
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
applied on the poloidal and toroidal components of
the flow. C w is then defined through the relationship:
T
y1
<
<
w T Cy1
w w s Ž s t . C Žs < t . Ž s t .
Ž 16 .
where Žtaking Eq. Ž7. into account.:
T y1
Cy1
w s Q C Žs < t .Q,
Ž 17 .
Cy1
Žs < t. being now a diagonal matrix. As each
™ elemen™
tary poloidal and toroidal degree n flow Ž Snm and Tnm ,
whatever m. has an elementary kinetic energy
Ec Ž n . s
1
4p c 2
1
s
4p c
2
™m
2
™m
2
HHCMB ž S /
dS
n
HHCMB ž T /
n
dSs
n Ž n q 1.
2nq1
,
and since each elementary flow contributes independently to the total kinetic energy, the nyp behavior
requested for the total energy of the degree n component of the flow then implies that
y1 Ž pq1.
n
Ž n q 1 . I,
Ž Cy1
Žs < t . . PH s E
Ž 18 .
where E is a second independent scaling parameter
Žin Žradryr. 2 ..
Eqs. Ž15. – Ž18. define the covariance matrices
reflecting our a priori beliefs on the flow and the
magnetic data for a first set of flow inversions. From
now on, these inversions will be referred to as the
PH-inversions.
2.2.2. Choosing the truncation leÕels for the PH-inÕersions
As is well known Že.g., Langel, 1987., the Gauss
coefficients of the main field for 1980 are very well
constrained up to degree 13 but not at all for larger
degrees, because of crustal field contamination. Fortunately, this shortcoming is not too serious. Using a
statistical description for both the main field and the
flow, assuming that the behavior of the field spectrum is the same for degrees smaller and larger than
13 and that the flow complies with the nyp energetic
behavior, Hulot et al. Ž1992. have indeed shown that
the SV produced by the advection of the small scale
main field Žwith degree larger than 13. by the flow,
lies below the RMS error believed by them Žand by
us. to be associated with the SV models. This crucial
point, which will be discussed in Section 2.2.3,
295
shows that L B s 13 is an adequate choice for defining the truncation in Eq. Ž8., provided that the flow
can indeed be assumed to be large-scale. The main
field coefficients bnm cŽ s. with degree less than 13 can
otherwise be assumed to be known exactly, despite
some likely but minor crustal contamination Žsee
e.g., Jackson, 1996.. wIn principle, this contamination
could be taken into account in core motions computation, but this would require considerable additional
computational cost Žsee Jackson, 1995.x.
Also known is the fact that the SV can hardly be
trusted for degrees larger than 8 because of the poor
geographical distribution of observatories. However,
since we decided to rely on the time-dependent field
model of BJ92 which provides us with such a possibility, LSV s L B s 13 has been chosen. This in fact
makes it possible to assess the impact on the final
flows of precisely those SV coefficients with n ) 8
Žby making runs with or without truncating the BJ92
SV model at n s 8, the remaining coefficients being
set to zero..
Finally, we had to decide at what degree L u to
truncate the flow ™
u Ži.e., expansion Eq. Ž5... This
choice is a priori linked to the choice of the exponent
p in the nyp energy requirement. In the present
paper, we decided to follow the suggestion of Celaya
and Wahr Ž1996. that the data seems to require a
value of p s 2 or larger. In this case, as discussed in
some detail by Hulot et al. Ž1992., who considered
the two possibilities p s 2 and p s 3, the exact
values chosen for p and L u are not critical provided
that L u is larger than 8, and that the truncation is
properly implemented. In the present paper, all PH
computations have therefore been carried out assuming p s 3 and L u s 13.
2.2.3. Choosing optimized parameters for the PH-inÕersions
The parameters L B , LSV , L u and p having been
defined, it eventually remains to specify WSV and E,
and to properly adjust the damping parameter l, in
order to actually be able to produce a flow from Eq.
Ž13..
At this point it is important to realize that the
solution wˆ of Eq. Ž13. only depends on the global
choice of the single parameter L s lŽWSV rE . Žas
can readily be checked from Eqs. Ž11. – Ž18... This
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
296
shows that choosing absolute values for the dimensional quantities WSV and E simply has no influence
on the space of possible solutions ŽEq. Ž13.., which
can always be explored by conveniently varying the
damping parameter l. By contrast, WSV and E play
a major role in defining the criteria underlying the
physical concept of a ‘‘satisfying’’ solution.
In the Bayesian approach we rely on Žrecall, e.g.,
Backus, 1988., WSV and E indeed reflect the a priori
belief we have concerning the maximum absolute
magnitude Žin the sense of a ‘‘soft’’ bound. for,
respectively, the global weighted misfit between the
observed SV and the SV predicted by the model
flow w,
ˆ and the global weighted energy of the flow.
A ‘‘satisfying’’ solution is then a solution that leads
to a misfit and an energy that are consistent with our
a priori beliefs. This can be partly formalized by the
criteria that the RMS of the weighted Žadimensional.
misfit to each of the NSV s LSV Ž LSV q 2. coefficients of b˙ satisfies
m Ž wˆ . s
(
M Ž wˆ .
NSV
Q O Ž 1.
Ž 19 .
and that the RMS of the weighted velocity of each of
the NŽs < t. s 2 L u Ž L u q 1. q L u toroidal and poloidal
flow coefficients satisfies
e Ž wˆ . s
)
j Ž wˆ .
NŽs < t .
Q O Ž 1. ,
Ž 20 .
Since all solutions ŽEq. Ž13.. minimizing Eq. Ž11.
can be plotted on a mŽ wˆ . versus e Ž wˆ . diagram,
where they all lie on a single ‘‘trade off curve’’ Žsee
e.g., Parker, 1994., the ‘‘satisfying’’ solutions will
be those lying on that section of the curve near
enough to the origin so that both Eqs. Ž19. and Ž20.
are satisfied. This then defines the subset of relevant
values of l which may be used. The smaller WSV
and E, the smaller the subset, which may eventually
become empty, thus indicating a lack of satisfying
solutions.
In practice, we based our choice of WSV on the
data represented in Fig. 1. Our aim is to adjust WSV
1r2 Ž
in such a way that s Ž n. s WSV
n q 1.y1 r2 Žrecall
Ž
..
Eq. 15 best reflects what we assume is the standard deviation for the errors within the degree n SV
coefficients. Various estimates for such standard de-
Fig. 1. Standard deviation s Ž n., as a function of degree n,
ascribed to the errors on the spherical harmonic coefficients of the
SV models at the Earth surface. Solid lines are for curves
sJ1840Ž n. ) sJ1890Ž n. ) sJ1940Ž n. ) sJ1980Ž n., computed from the
time varying matrix for b˙ Ž t i . of BJ92, assuming rotationally
invariant errors along the lines of Jackson Ž1997, see his Fig. 2..
Triangles are for the curve sdiff Ž n. computed by taking the RMS
values of the differences between the 2 nq1 coefficients of the
BJ1980 and USGS80 models Žsee text., up to degree 8. Also
shown, our preferred curve s PH Ž n. s1.3Ž nq1.y1 r2 ŽnTryr.
Ždiamonds..
viations have been plotted in Fig. 1. The curves
sJ t Ž n. for t i s 1840, 1890, 1940 and 1980 Žcomi
puted from the BJ92 model in the same way as
Jackson Ž1997, see his Fig. 2. represent the standard
deviations that Jackson Ž1997. and Jackson et al.
Ž1993. a priori assume for the errors within the SV
coefficients of the BJ92 model at the epochs t i . The
evolution of these curves can be understood in terms
of the improvement in the SV accuracy that BJ92
derive as a result of the larger data set they use for
modeling the recent years. Whether this improvement is as good as they assume can, however, be
slightly challenged. First, by considering the curve
sdiff Ž n. computed by taking the RMS value of the
differences between the degree n Gauss coefficients
of the 1980 SV models of BJ92 Žmodel BJ1980. and
of Peddie and Fabiano Ž1982, model USGS80.. This
curve shows that the disagreement between two SV
models for the same 1980 epoch can be larger than
predicted by sJ19 80Ž n.. Second, by noting that the true
covariance matrix derived by BJ92 through quite a
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
Fig. 2. Trade-off curves between the normalized velocity e Ž ŵ .
and the normalized misfit mŽ wˆ ., for four different epochs 1840
Žlower curve., 1890, 1940 and 1980 Župper curve., for the PH-inversion ŽC SV s 1.69Ž n q 1.y1 I ŽnTryr. 2 and C Žs < t. s 8 =
10y6 ny4 Ž nq1.y1 I Žradryr. 2 .. Each curve is the result of varying the damping parameter l between 2=10y1 0 and 2=10 7.
Open diamonds show our preferred inversion Ž l s 4.7=10y2 . for
each epoch.
complex procedure is very non-diagonal Žsee, e.g.,
Jackson et al., 1993.. We therefore decided to mainly
rely on a constant covariance matrix of the form Ž15.
1r2
s 1.3 nTryr. The corresponding curve
with WSV
s PH Ž n., also displayed in Fig. 1, shows that this
amount to puts more confidence in the 1980 SV
model than suggested by sdiff Ž n., but less than suggested by sJ19 80Ž n.. It represents a reasonable temporal average of the time varying errors used by Jackson Ž1997.. It finally reflects an admittedly arbitrary,
also challengeable, but easily readable assumption:
that the errors in the SV models mainly arise from
the errors made in directly measuring the variation of
the field at observatories on the Earth surface, these
errors having remained essentially the same throughout the time period considered.
Let us now choose E to reflect our a priori beliefs
concerning the order of magnitude of the surface
flow energy. If we rely on the classically quoted
westward drift rate of the magnetic field Ž t 10WD s
0.28yry1 . and take the corresponding energy at the
CMB as the reference for the expected toroidal
degree 1 energy, then E s 8 = 10y6 Žradryr. 2 . But
we must acknowledge that this order of magnitude
for the westward drift is itself the result of some
simplified inversion, consisting in explaining the SV
297
observations with a single t 10 flow. Therefore, request Ž20. must not be understood as some data-independent a priori information on the flow. Rather, it
corresponds to imposing that the rough inversion
leading to the estimation of t 10WD is a reasonable zero
order flow, and that the true flow should not involve
substantially more leading terms. Request Ž20. is
therefore much less stringent than Eq. Ž19. and can
only be used as a way of measuring how easy it is to
find a model satisfying these a priori beliefs.
WSV and E having been defined in the way just
described, Fig. 2 shows the trade-off curves, i.e.,
mŽ wˆ . as a function of e Ž wˆ ., for PH-inversions at
epochs 1840, 1890, 1940 and 1980, when wˆ is given
by Eq. Ž13. and l is varying from 2 = 10y1 0 to
2 = 10 7. wFor those computations, the SV coefficients up to degree 8 are being used, degrees 9 to 13
being set to zero. But we have checked that very
similar flows and results are found when using all
SV coefficients up to 13.x These trade-off curves
show that criteria Ž19. can readily be satisfied provided that criteria Ž20. is slightly relaxed. They,
however, also show that not so many values of the
damping parameter l may be chosen without leading
to some contradiction with our a priori assumptions.
Trying to be more confident in the SV data than we
actually are Ži.e., trying to reduce mŽ wˆ .. calls for a
solution with much more energy than a priori expected Ži.e., such that e Ž wˆ . becomes larger.. Fig. 3
shows the detailed misfit to each SV Gauss coefficient and the energy spectrum for three 1980 flows,
the corresponding values of l, mŽ wˆ . and e Ž wˆ . being
displayed in Table 1. It reveals that the way the flow
succeeds in better fitting the data for a smaller l is
not so much in homogeneously increasing its energy,
but rather in involving more and more energetic
small scales. This shows that trying to put more
confidence in the data leads to a solution that cannot
be reconciled with our a priori assumption by just
revising the Žrather arbitrary. choice of E. Such a
solution would more fundamentally be in conflict
with the a priori assumption that the energy spectrum
decreases like nyp . Resolving this inconsistency
would therefore require that we either change the a
priori assumption or that we increase the number of
parameters describing the flow Ži.e., increase L u ,
pushing the nyp energetic constraint towards higher
degrees of the flow.. But this would then lead to
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
298
small subset of the damping parameter l that leads
to flows that are consistent with the level of accuracy
we assume for the SV, that comply with the nyp
energetic a priori requirement for the flow, and that
do not lead to too large a RŽ n. noise Žas shown by
Hulot et al., 1992, who relied on the same assumptions and considered both possibilities p s 2 and
p s 3..
In the rest of the paper, all PH-inversions have
therefore been carried out by choosing lPH s 4.7 =
10y2 . This leads to flows that comply with all the
above requirements throughout the 1840–1990 time
period as illustrated by Figs. 2 and 3 Žall flows
displaying similar misfits and spectra..
Fig. 3. PH-inversion for epoch 1980, using three different values
of l Žsee Table 1.. Ža. Normalized energy associated with each of
the Ž L u Ž L u q1.r2.q L u degree n, order m toroidal coefficient of
the flow, as given by Ž n4 Ž nq1.r8=10y6 .wŽ t nm ,c . 2 qŽ t mm ,s . 2 x; Žb.
normalized misfit between each Ž LSV Ž LSV q1.r2.q LSV degree
n, order m observed Žo. and predicted Žp. SV coefficient, as given
by
(Ž Ž
2
2
˙
˙
x.
nq1 . r1.69 . w ˙
q ˙
In both cases, values are in order of increasing order m within
increasing degree n, and joined by lines for clarity. Long-dashed
lines correspond to model 1, solid lines to model 2 and dotted
lines to model 3, as defined in Table 1.
Ž
bnm , cŽ p . y bnm , cŽ o .
. Ž
bnm , sŽ p . y bnm , sŽ o .
.
another difficulty linked to our lack of knowledge of
the main field degrees above L B s 13. Highly energetic small scales within such a flow would necessarily interact with unknown main field degrees above
13 to produce some unknown contribution within the
observed low degree SV. Such contribution, termed
‘‘rest of the SV’’ by Hulot et al. Ž1992., would
behave as an additional source of noise RŽ n., at a
level which is directly related to the energy content
of the small scale component of the flow, and which
can exceed the observational noise within the SV,
thus rendering the whole computation meaningless.
The key issue as far as the present study is
concerned is that it is fortunately possible to find a
2.2.4. Defining an alternate set of inÕersions (J-inÕersions)
In order to assess the extent to which the results
derived in the next sections might be sensitive to the
choices made in defining our preferred PH-inversions, a second different inversion scheme has also
been considered. These J-inversions are carried out
by choosing the same covariance matrices C SV and
C Žs < t. as those on which Jackson Ž1997. and Jackson
et al. Ž1993. rely. This amounts to use
y1
Ž Cy1
Žs < t . . J s E
n3 Ž n q 1.
2nq1
3
I
Žwhich minimizes the spatial roughness of the model,
.
Bloxham, 1988. and the time varying matrix ŽCy1
SV J
derived from the BJ model in the way described by
Jackson et al. Ž1993.. This time varying diagonal
covariance matrix corresponds to the assumption that
the SV models are much better known for the recent
epoch than for earlier epochs Žas illustrated by Fig.
1.. The J-inversions no longer require that we define
Table 1
RMS normalised misfit mŽ wˆ . and RMS normalised velocity e Ž wˆ .
for three different PH-inversions for the 1980 epoch Žsee also Fig.
3.
Model
l
mŽ wˆ . ŽnTryr.
e Ž wˆ . Žradryr.
1
2
3
4.7=10y1
4.7=10y2
4.7=10y3
1.7
1.0
0.5
1.2
2.9
5.7
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
.
WSV Žwhich is now directly included in the ŽCy1
SV J
matrix., but still require some choice for E. The
same value E s 8 = 10y6 Žradryr. 2 as in the PH-inversions has been kept, amounting to essentially the
same a priori assumption on the degree 1 energy of
the flow. Fig. 4 shows the corresponding trade-off
curves at epochs 1840, 1890, 1940 and 1980, the SV
coefficients up to degree 13 being now used in order
to produce results as close as possible to those of
Jackson Ž1997. and Jackson et al. Ž1993.. ŽNote that
we again checked that setting all the degree 9 to 13
SV coefficients to zero does not lead to very significant differences in the computed flows.. Fig. 4 reveals that the 1890 trade-off curve for the J-inversion
is very similar to those obtained for the PH-inversions. This shows that the detailed shapes of the Cy1
SV
and Cy1
Žs < t. matrices do not have a strong influence on
the results, as long as their overall behaviors remain
the same. This is indeed the case here since sJ18 90Ž n.
and s PH Ž n. compare reasonably well Žrecall Fig. 1.
.
Ž y1 .
and ŽCy1
Žs < t. and C Žs < t. PH lead to the same asymptotic
behavior for large n. In fact, searching for a damping
parameter l leading to the same misfit as the one we
obtain for our preferred 1890 PH-inversion Ž l J s 8
= 10y2 ., produces a very similar energy distribution
ŽFig. 5.. By contrast, Fig. 4 shows that the 1980
J-inversion trade-off curve strongly differs from both
the 1890 J-inversion and the 1980 PH-inversion
trade-off curves. This directly reflects the additional
Fig. 4. Same as Fig. 2, but for the J-inversion: 1840 Žlower curve.,
1890, 1940 and 1980 Župper curve.. C SV Ž t i . s sJ t Ž n.I and C Žs < t.
i
s8=10y6 ny3 Ž nq1.y3 Ž2 nq1.IŽradryr. 2 .. The open diamonds
are for l s8=10y2 , which gives flows very similar to the uÕm-i
intermediate flow model of Jackson Ž1997..
299
Fig. 5. For two J-inversions, Ža. normalized energy associated
with each degree n, order m toroidal coefficient of the flow, and
Žb. normalized misfit between each observed and predicted degree
n, order m SV coefficient, computed as in Fig. 3. Two inversions
are considered, one for 1890 Ždotted lines., that closely resembles
inversion 2 of Fig. 3, the other one for 1980 Žsolid line.; l s8=
10y2 in both cases.
.
confidence that ŽCy1
SV J imposes on the SV data for
this epoch Žagain recall Fig. 1.. However, it also
shows that this covariance matrix makes it very
difficult to find a ‘‘satisfying’’ 1980 J-solution in the
way defined in Section 2.2.3. To best illustrate this
point, just consider the 1980 flow obtained with
l s l J . This flow already fails to satisfy the request
Ž19.. It also has too energetic a small scale content
ŽFig. 5.. Changing l in order to reduce mŽ wˆ . to
mŽ wˆ . s 1 would not improve the situation, but would
rather again lead to the inconsistency we described
in the previous section. Nevertheless, and in order to
compare our own results with those of Jackson Ž1997.
and Jackson et al. Ž1993., we did produce a set of
flows throughout the 1840–1990 period, based on
the J-inversion with l s l J Žthis value of l in fact
leads to flows extremely similar to the uÕm-i intermediate flow model of Jackson, 1997..
300
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
3. Recovering information concerning the zonal
component of the flow
The purpose of the present paper being to assess
how much information one may recover about the
toroidal zonal component of the flow at the core
surface, this is the part of the flow on which we will
now focus our study.
Fig. 6 displays the results of our preferred PH-in0
version Žwith l s lPH . for the 13 t 10 to t 13
zonal
coefficients of the flow. In what follows this flow
will be termed ‘‘Flow0 ’’, its zonal component,
‘‘TZ 0 ’’ and its complementary non-zonal component, ‘‘NZ 0 ’’. For comparison, Fig. 6 also displays
the same components obtained for the J-inversion
Žwith l s l J .. These two flows compare quite well
with each other. wNote, however, that for degrees
n G 9, the J-inversion produces higher estimates than
the PH-inversion, especially for the period from
1940 onward. This simply reflects the different convergence of the spectra of the two flows we discussed in the previous section.x For this reason, the
tests reported in the present section have been carried out on our preferred PH-inversion. An obvious
feature of the zonal flow revealed by Fig. 6 is that it
is strongly time-dependent. This is not a new result.
It has been implied by Jault et al. Ž1988., explicitly
noticed by Hulot et al. Ž1993. and definitively confirmed by Jackson Ž1997.. But it shows that the
question of the robustness of the recovered zonal
flow cannot be addressed by relying on a single
one-epoch case study. For this reason, all tests have
been carried out over the whole 1840–1990 period,
allowing for some possible time variability within
the robustness of the results we wish to assess.
3.1. Mapping, attenuation and noise
The flow wˆ provided by Eq. Ž13. is a Bayesian
estimate of the first L u s 13 degrees of the ‘‘true’’
flow. This flow wtrue is such that
ḃ s A b wtrue q e SV
Ž 21 .
where the error e SV is assumed to follow a Gaussian
law characterized by the covariance matrix C SV Žsee
e.g., Backus, 1988.. wRecall that e SV is assumed to
describe both the observational errors one believes
affect b˙ Ži.e., the observed SV., and the contribution
Fig. 6. Estimates of the zonal toroidal coefficients of the flow for
the period 1840–1990, together with the corresponding 95% CIs
Župper and lower curves., relying on our preferred PH-inversion
Ž l s 4.7=10y2 , solid lines. and on the J-inversion Ž l s8=10y2 ,
dotted lines..
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
of degrees larger than L u s L B s 13 of the true flow
and main field Ži.e., the ‘‘rest of the SV’’ discussed
in Section 2.2.3.. All other possible sources of errors
Ži.e., failure of the tangentially geostrophic and frozen
flux assumptions. are implicitly ignored.x Setting Eq.
Ž21. in Eq. Ž13. shows that the estimate wˆ is related
to the true flow through
yg
wˆ s Ayg
b A b wtrue q A b e SV
s Rwtrue q e w ,
Ž 22 .
where the matrix R s Ayg
b A b is known as the ‘‘Resolution matrix’’ Že.g., Backus, 1988..
Eq. Ž22. makes it clear that wˆ cannot exactly
reflect the true flow for three reasons. One is that R
is not purely diagonal, so that each coefficient ŵi
depends not only on the value of the corresponding
true coefficient wtrue i , but also on the values of the
coefficients wtrue j/ i . This is what we will refer to as
the ‘‘mapping’’ of the components j / i on the
component i. A second reason is that the diagonal
terms of R are not exactly unit coefficients. This is
because the regularization condition j Ž w . Žrecall Eq.
Ž12.. constrains the solution wˆ to produce small
coefficients and also because part of the original
signal wtrue i might be mapped onto other coefficients
wtrue j/ i . This effect is what we will broadly refer to
as ‘‘attenuation’’. Finally, wˆ cannot exactly reflect
wtrue also because of e w , which reflects the way the
noise in the SV blurs the estimate Rwtrue to produce
the actual estimate w.
ˆ This we will refer to as the
effect of ‘‘noise’’. The flow wˆ we recover from Eq.
Ž13. is thus not the true flow wtrue , but rather a
biased Žbecause of mapping and attenuation. and
noisy estimate of this flow. Understanding the effect
of this bias and noise on the zonal component of the
flow is what we now intend to do.
3.2. Effect of mapping and attenuation on the zonal
flow
Studying the effect of mapping and attenuation
cannot be done by shear inspection of the R i j coefficients, since it also requires some a priori knowledge
of the wtrue j coefficients defining the true flow. The
exact values of these coefficients are, of course,
unknown. But an insight of this effect can nonetheless be gained by making use of wˆ j in place of wtrue j
Ži.e., of Flow0 in place of the true flow..
301
3.2.1. Attenuation
In a first test, we started from TZ 0 , computed the
SV it produces when interacting with the BJ92 main
field model, and truncated it at degree 8 to give
SVT Z 0 . SVT Z 0 was then inverted for FlowA . The
zonal component of this flow, TZ A shown in Fig. 7,
was then expected to mainly reflect TZ 0 to within
the effect of attenuation. Strictly speaking this may
not exactly be the case, because mapping between
the various degrees of the toroidal zonal flow can
also potentially occur. To make sure that this is not
so and to also derive a quantitative measure of the
effect of attenuation, the following ‘‘correlation’’
analysis has been carried out. For each t n0 coefficient, a parameter k AŽ n. was sought in order to
minimize the ‘‘disagreement’’ DAŽ n.:
H
2
k A Ž n . t n0 Ž TZ A . y t n0 Ž TZ 0 . d t
DA Ž n . s
H
2
.
t n0 Ž TZ 0 . d t
Fig. 8 shows the k AŽ n. obtained in this way, together
with the corresponding DAŽ n.. This figure clearly
confirms what could already be seen in Fig. 7,
namely that t n0 ŽTZ A . and t n0 ŽTZ 0 . compare very
well, and that little attenuation and interzonal mapping seems to occur, so long as n F 6. In fact strong
attenuation only starts occurring for degrees above 7.
But for degrees up to 11, as shown by DAŽ n.,
attenuation explains most of the signal, suggesting
that interzonal mapping does not have too strong an
impact on these coefficients. By contrast, DAŽ12. and
0 Ž
0 Ž
DAŽ13. clearly show that t 12
TZ A . and t 13
TZ A .
cannot be understood in terms of attenuation. They
are clearly dominated by mapping Žinterzonal mapping, in that case..
3.2.2. Mapping
In a second test, the flow NZ 0 has been used
together with the BJ92 main field model to produce
some SV through Eq. Ž10. which, once truncated at
degree 8, gave SVNZ 0 . SVNZ 0 was then inverted for
FlowM . The zonal component of this flow, TZ M , is
of course entirely due to some mapping of the
non-zonal component of Flow0 through the inversion
scheme. Plotting the evolution of TZ M together with
the estimate TZ 0 of the zonal flow, as is also being
done in Fig. 7, thus provides an estimate of how
302
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
Fig. 8. Fitting of Flow0 by a degree-dependent scaled version of
flow TZ A ; Ža. values of the factor kAŽ n. by which t n0 ŽTZ A . must
be multiplied to best fit t n0 ŽTZ 0 .; Žb. normalised misfits DAŽ n..
much of TZ 0 could be due to mapping of the
non-zonal component of the true flow. Except possibly for degree 1 which does display a non-negligible
mapping effect, it appears that up to degree 6 this
effect seems to only play a marginal role in biasing
TZ 0 . By contrast degrees above 12 are clearly
strongly affected by the mapping of non-zonal terms.
The case of degrees 7 to 11 deserves a more
careful discussion. Fig. 8 shows that for these degrees, TZ A can reasonably well recover TZ 0 if we
allow for some quite strong attenuation k AŽ n. , 3.
But Fig. 7 also shows that, for these degrees, TZ A
and TZ M are comparable in magnitude and that not
only TZ A but also TZ M are often comparable in
shape to TZ 0 . This visual impression can be quantified in the same way as we quantified attenuation,
i.e., by searching a parameter k M Ž n. best minimizing
for each coefficient the disagreement D M Ž n.:
H
2
k M Ž n . t n0 Ž TZ M . y t n0 Ž TZ 0 . d t
DM Ž n. s
H
Fig. 7. Results of the tests on the zonal component of the flow, as
explained in Section 3.2. Shown are the t n0 components of flow
TZ 0 Žsolid lines., flow TZ A Ždot–dashed lines. and flow TZ M
Ždotted lines..
2
.
t n0 Ž TZ 0 . d t
Fig. 9 displays the results of this analysis which was
carried out for all degrees n. It first confirms the
lack of connection between the mapping TZ M and
the original flow TZ 0 for degrees up to 6 Žfor these
degrees the normalized disagreement D M Ž n. always
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
Fig. 9. Same as Fig. 8, for the fitting of Flow0 by a degree-dependent scaled version of flow TZ M ; Ža. scale factors k M Ž n.; Žb.
normalised misfits D M Ž n..
remains large.. It next shows that degrees 12 and 13
of TZ 0 can be interpreted in terms of the mapping
TZ M to within, however, a factor k M Ž n. of the order
of 2 to 3. It finally confirms that, for degrees 7 to 11,
TZ M is comparable in shape to TZ 0 , and shows that
the two flows can be matched reasonably after application of the amplifying factors k M Ž n. to TZ M .
Of course, had we started from the full Flow0
Ži.e., NZ 0 q TZ 0 ., produced a SV, truncated it at
degree 8 and proceeded to invert for a new flow,
Flow1 , then the toroidal zonal component TZ 1 of this
flow would have been TZ 1 s TZ A q TZ M . The previous tests thus show that contrary to degrees 1 to 6
for which our tests clearly evidenced little mapping
and attenuation effect, contrary also to degrees 12
and 13 which have been shown to be a pure product
of mapping, degrees 7 to 11 of the toroidal zonal
flow TZ 1 recovered from a PH-inversion of the SV
produced by Flow0 , appear to be a combination of
two correlated Žand comparable in magnitude. flows:
TZ A , which is an attenuated version of TZ 0 , and
TZ M which results from some mapping of the nonzonal components of Flow0 and which also appears
to be an attenuated version of TZ 0 .
3.2.3. Double mapping
It thus appears that degrees 7 to 11 of the toroidal
zonal flow recovered through a PH-inversion cannot
303
blindly be taken as representative of the original
flow. This conclusion applies to TZ 1 , when starting
from Flow0 . But it does not necessarily apply in the
same way to TZ 0 when starting from the true flow
because Flow0 may not be fully representative of
this true flow. It may be for instance, that the part of
NZ 0 which is responsible for the degrees 7 to 11 of
TZ M , is itself the result of some mapping of the
toroidal zonal component of the true flow on the
non-zonal component of Flow0 . This issue can easily
be addressed in the following final test. Starting from
TZ 0 as in the first test, considering NZ A , the nonzonal component of FlowA , computing the SV produced by NZ A and reinverting for FlowMM , makes it
possible to compare TZ MM , the zonal component of
FlowMM , to TZ A ŽMM stands for mapping from
zonal to non-zonal and back to zonal. in the same
way as we compared TZ M to TZ 0 . Computing parameters D MM Ž n. and k MM Ž n. in the same way as
D M Ž n. and k M Ž n. leads to Fig. 10.
For degrees up to 6, k MM Ž n. shows much larger
values than the corresponding k AŽ n.. This result
shows that no significant such double mapping can
possibly apply to these degrees, and is fully consistent with the fact that these degrees are properly
recovered through the PH-inversion as illustrated in
Fig. 10. Testing the possibility that the correlation between TZ M
and TZ 0 for degrees 7 to 11 could be the result of a double
mapping of a purely zonal flow; Ža. scale factors k MM Ž n.; Žb.
normalised misfits D MM Ž n.. Scale factors k M Ž n. and normalised
misfits D M Ž n. are also shown in grey diamonds for comparison.
304
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
Fig. 8. For degrees above 7, the situation is very
different. Fig. 10 reveals that for these degrees, and
especially for degrees 7 to 11, D MM Ž n. is very
small. Thus, double mapping of a pure ŽTZ 0 . zonal
flow can indeed produce a toroidal zonal flow
ŽTZ MM . very closely correlated to the directly recovered ŽTZ A . zonal flow, to within the factor k MM Ž n..
The fact that such a double mapping effect could be
responsible for the remarkable correlation we observe between the degrees 7 to 11 of TZ M and TZ 0
then becomes obvious. The pattern we observe for
these degrees on Figs. 9 and 10 are indeed very
similar. The k M Ž n. are of exactly the same order of
magnitude Ž; 3. as the k MM Ž n., and the D M Ž n. are
almost as close to zero Žespecially for n s 7 and 8.
as the D MM Ž n.. wIn fact, the slightly larger values of
D M Ž n. compared to D MM Ž n., for degrees 9 to 11,
would simply testify for the fact that not exactly all
of TZ M can be attributed to double mapping.x As far
as the true flow is concerned, these results thus show
that for degrees 7 to 11, both TZ 0 and TZ M may
have been created by a true toroidal zonal flow. If
that is the case, TZ 0 may then be interpreted as
reflecting this true zonal flow to within the factors
k AŽ n..
Of course, and this is an important provision to be
made before drawing any further conclusions, we
must acknowledge that the previous interpretation of
the observed behavior of TZ 0 and TZ M is not necessarily the only one that can be found. It may for
instance be that both TZ 0 and TZ M are in fact
produced by an analogous double mapping effect
starting from a pure non-zonal true flow. This possibility cannot be discarded. The previous double mapping interpretation thus amounts to seek the degrees
7 to 11 of the zonal flow in terms of the largest
possible ‘‘true’’ zonal flow accounting for the observed SV data.
3.3. Effect of the noise
The previous discussion dealt with the possibility
of recovering some information about the true flow,
assuming that the observed SV is exactly the one
produced by that flow. However, we know that this
is not the case and that the observed SV contains
some noise. This noise is formalized by e SV in Eq.
Ž21. and is assumed to follow a Gaussian law characterized by the covariance matrix C SV we discussed at
length in Sections 2.2.1 and 2.2.3. This noise in the
SV translates into some noise e w in the estimate w,
ˆ
as described by Eq. Ž22.. Since e SV is assumed
Gaussian, e w s Ayg
b e SV is also Gaussian and the
effect of this noise can readily be described by the
covariance matrix
T
yg
C w s Ayg
b C SV Ž A b . .
This matrix is not purely diagonal, because Ayg
is
b
itself non-diagonal. This is a property one should
keep in mind when considering the combination of
zonal flows Žas we will for instance do when computing LOD predictions in Section 4.. Nevertheless,
and in the absence of specific properties of the flow
to test, the possible effect of noise on each component wˆ i of the estimate wˆ can be materialized by
plotting the 95% confidence intervals ŽCI. defined
by "2 s Ž wˆ i . where
s 2 Ž wˆ i . s w C w x i i .
This is what has been done in Fig. 6 Žfor PH-inversion flows but also for J-inversion flows, relying on
the relevant covariance matrices ŽC SV . PH and ŽC SV . J ,
respectively; note that the CIs resulting from the
J-inversion do show, as expected, a much more
pronounced time variation and lead to significantly
narrower intervals than the PH-inversion for the
most recent period..
For degrees 1 to 6 that have been shown to be
reasonably recovered by the inversion scheme, the
error s Ž wˆ i . is quite large, which results in that even
for these large scale zonal flows it quite often happens that the recovered signal does not differ very
significantly from zero. This is for instance clearly
the case for degree 6 and degrees 1 and 3 prior to
1900. Fortunately, errors otherwise do not affect too
severely degrees 2, 4 and 5.
Turning to larger degrees, we first note that for
degrees 12 and 13 which have already been shown
not to be significant, the noise level makes it even
clearer that no information can be extracted from
these terms. Unfortunately, the noise level within the
degrees 9, 10 and 11 is also very large. In fact, only
degrees 7 and 8 seem to contain some signal above
the noise level. This we believe is not fortuitous, this
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
signal being possibly linked to interesting features of
the flow as will be discussed in Section 4.
4. Discussion
Bearing the previous conclusions in mind, we will
now first reconsider a number of published results
dealing with the prediction of the LOD variations on
decade time scales and the possible existence of
torsional oscillations, and next proceed in showing
that the time-averaged recovered zonal flow strongly
suggests an increase in angular velocities at high
latitudes. This, we will argue, may well be the
dynamic signature at the top of the core of the
presence of a superrotating inner core.
4.1. Consequences for LOD predictions
The observed time variations within the toroidal
zonal components t 10 and t 30 have been used by
several authors to predict the effect of the axial
angular momentum exchange between the liquid core
and the mantle on the LOD decade variations. We
will not discuss the theory of Jault et al. Ž1988.
justifying the fact that all that is needed to carry on
such a computation is a good knowledge of both t 10
and t 30 at the core surface Žassumed to reflect bulk
geostrophic cylindrical motions within the core, see
Jault et al., 1988; Jackson et al., 1993; Le Mouel
¨ et
al., 1997.. We will only recall that the prediction is
being made through the following formula Že.g.,
Jackson et al., 1993.:
dT s
T02
2p
ž
Ic
Ic q Im
/ž
dt 10 q
12
7
dt 30
/
Ž 23 .
where Ic and Im are the moment of inertia coefficients of the core and of the mantle, T0 s 86400 s is
the reference value for the length of day and dT s T
y T0 is the predicted LOD variation which may be
compared to the LOD variation deduced from astronomical observations Ž dTobs ., after correcting for the
effect of the tidal friction and of the post-glacial
rebound Žthis correction consists in subtracting a
linear trend of q1.8 msrcentury, centered at 1840,
see e.g., Stephenson and Morrison, 1990.. The nota-
305
tions dt 10 and dt 30 mean that the values used for the
prediction are to be taken relatively to some constant
reference flow. In practice this means that we may
use t 10 and t 30 within Eq. Ž23. provided we remove a
constant arbitrary value dTarb . Using our t 10 and t 30
estimates in units of radryr, we thus have to compute
dT s 3.98 t 10 q
ž
12
7
t 30 y dTarb
/
Ž 24 .
to obtain the predicted LOD variations in seconds,
dTarb being a free parameter. Fig. 11a shows the
observed LOD variations together with such a prediction based on the PH time varying flow. Fig. 11b
shows the same result based on the J-flows. Also
shown on both figures are the 63% and the 95% CIs
of the estimate Ž24., i.e., "s Ž dT . and "2 s Ž dT .,
based on the formula
s 2 Ž dT . s Ž 3.98 .
24
q
7
2
s 2 Ž t 10 . q
C w Ž t 10 t 30 . ,
12
ž /
7
2
s 2 Ž t 30 .
Ž 25 .
which takes the covariance C w Ž t 10 t 30 . between the
two parameters t 10 and t 30 into account.
As can be seen, and as had been noted by all
previous authors, the agreement between the two
curves is quite impressive starting from 1920 and
onward, but quite poor before 1920. Le Mouel
¨ et al.
Ž1997. suggested that this could possibly be because
part of the signal in t 10 and t 30 is not as immediately
related to the bulk geostrophic reactions as invoked
by Jault et al. Ž1988.. Fig. 11 is however suggesting
an alternative explanation. The observed disagreement could simply be due to the limited accuracy of
the geomagnetically inferred prediction ŽEq. Ž24...
Based on the assumption that the errors in the SV
models can be time-independently described by the
covariance matrix ŽEq. Ž15.., our PH-inversion leads
to a prediction that departs from the observed LOD
to an amount that remains well within the 63% CI
after 1920, and that is just on the limit of this CI
before 1920 ŽFig. 11a.. This can be interpreted in
terms of the realized noise within the SV to be more
306
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
Fig. 11. Annual means of the excess length of day DLOD as
published in the International Earth Rotation Service ŽIERS.
Annual Reports, corrected for a linear trend of 1.8 msrcy to
account for tidal friction and post-glacial rebound effects Žgrey
squares.. Solid line is for the predicted DLOD, bold dashed lines
are for the 63% CIs and thin dashed lines are for the 95% CIs;
results for Ža. the PH-inversion; Žb. the J-inversion.
model, see his Fig. 11., reveals that this could indeed
be the case. The departure of this prediction with
respect to the observed LOD is now as often reaching the limits of the CIs prior to 1920 Ža period
during which the CIs are large. and after that date Ža
period with narrower CIs because of the higher
confidence put in the SV models for that inversion..
It is however interesting to note that the prediction
obtained when putting more confidence in the SV
models ŽFig. 11b. leads to a poorer agreement to the
observed LOD after 1920 than the prediction assuming our preferred stationary confidence ŽFig. 11a..
This apparently paradoxical result, we believe, is
related to the fact discussed in Section 2.2.4 that the
J-inversion is too optimistic in terms of the quality of
the SV, and thus slightly inconsistent. A detailed
discussion of this matter would require a third set of
inversions to be carried on taking into account the
improvement of the SV models with time, but with
somewhat less optimistic assumptions than in the
J-inversions. For the purpose of the present paper,
we however felt that Fig. 11a and b were enough to
make clear that core flow inversions are fully consistent with LOD data for the whole 1840 to 1990 time
span, and are particularly good at predicting the
LOD variation after 1920, based on the theory of
Jault et al. Ž1988.. Finally, these results also provide
a reassuring explanation for the reason why some
authors, such as Holme Ž1998. and Holme and
Whaler Ž1998., have been able to construct core
flows simultaneously constrained by SV data and
LOD data.
4.2. Consequences for torsional oscillations studies
unluckily constructive in producing a bias in the
LOD close to its maximum possible value before
1920 than after that date. In fact, what seems more
surprising in Fig. 11a is less the departure of the
prediction from the observed LOD before 1920, than
the very good agreement observed after that date!
This suggests that the PH-inversion may have been a
little too pessimistic with respect to the quality of the
SV for the recent years. Turning to Fig. 11b which
shows the prediction based on the J-inversion Žwhich
we recall is very similar to the prediction made by
Jackson Ž1997. based on his uum-i intermediate flow
One of the striking features noted by Jackson et
al. Ž1993. in their studies of LOD predictions Žsee
also Jault et al., 1996. is that the contributions of t 10
and t 30 in Eq. Ž24. tend to cancel each other. This
shows that only a fraction of the angular momentum
of the individual bulk geostrophic cylindrical motions within the core is actually exchanged with the
mantle, and that most of the exchanges apparently
occur inside the core, among different cylinders.
This situation is of the kind one would expect if
torsional oscillations occur within the core. But if
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
that is the case, there is a priori no reason why
torsional oscillations would only show up in the
large scale t 10 and t 30 zonal flows. Jault et al. Ž1996.
noted this point and in fact suggested, based on some
core flows derived by Le Huy Ž1995., that smaller
scale zonal flows of large magnitude could also be
generated by torsional oscillations.
This led Zatman and Bloxham Ž1997; 1998; 1999.
to go one step further, to try and invert the zonal
307
flows for the shape of the torsional waves, and to
derive an estimate of the bulk magnetic field responsible for them. To carry on these studies, Zatman and
Bloxham relied on the toroidal zonal equatorial symmetric component Ži.e., t n0 with n odd. of a tangentially geostrophic core flow model similar to those
computed here. Although they did assess the confidence with which they could explain this specific
flow model in terms of torsional oscillations Žby
Fig. 12. A comparison of the torsional oscillation model of Zatman and Bloxham Ž1999, Table 1. with the flow recovered from the
PH-inversion. Having removed the mean zonal flow for the 1840–1990 period, we represent the 95% CIs for each zonal coefficient Žthin
solid lines.. Dashed and bold solid curves correspond to the zonal flow components associated with the torsional oscillations model when
only using wave A or a combination of the two waves A and B.
308
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
testing the null assumption that any ‘‘random’’ core
flow parameters could as easily be explained in
terms of torsional oscillations, Zatman and Bloxham,
1998., they did not question the quality of this
model. But based on the present study, it is clear that
uncertainties affecting the zonal coefficients should
be taken into account. Doing so and duplicating the
full analysis of Zatman and Bloxham would be
beyond the scope of the present paper. But we may
easily assess their claim that the equatorial symmetric zonal flows can accurately be described in terms
of one or two damped harmonic waves and a steady
flow.
This is what is done in Fig. 12. To make this
figure, we first produced an average flow t n0 Žcoefficients are provided in Table 2. from our PH-inversion. We then removed this average flow from Flowo
and plotted the 95% CI previously derived Žsame CI
as in Fig. 6.. Finally, we plotted the coefficients
describing the torsional oscillations which Zatman
and Bloxham Ž1999, see their Table 1. claim properly account for the observed time varying toroidal
zonal equatorial symmetric flow between 1900 and
1990. According to their results, two predictions can
be plotted; one is based on a main wave A, the other
on the superposition of wave A and a second wave B
which they felt was also required to properly fit the
data. Fig. 12 shows that the combination of waves A
and B leads to a flow that is remarkably consistent
with our flows and CIs, thus confirming the fact that
Table 2
Average values t n0 of the computed t n0 over the 1840 to 1990 time
period ŽPH-inversion, l s 4.7=10y2 .. Units are in 1=10y4
radryr
Degree, n
t n0
1
2
3
4
5
6
7
8
9
10
11
12
13
y5.208
6.307
y0.807
y3.715
y1.266
y0.046
y2.125
0.589
y0.415
0.053
y0.110
0.004
0.013
SV data supports the assumption of torsional oscillations occurring within the core. More precisely, Fig.
12 shows that degrees t 10 , t 30 and t50 display significant time variations Žthe variations within t 70 being
0
0
poorly significant, those within t 90 , t 11
and t 13
being
totally unresolved.. Although wave B is well improving the fit, it is quite clear that slightly increasing the
period of wave A would have also produced a good
fit. With this respect, our results do confirm the
warning made by Zatman and Bloxham Ž1998. and
based on their ‘‘null’’ assumption test, that a second
wave does not seem to be absolutely required by the
data. But these results also show that even the characteristics of wave A are not perfectly resolved. Fig.
12 finally suggests that core flows extending as far
back as 1840 Žrather than 1900. could also be used
to provide additional constraints on torsional oscillations.
4.3. Inner core signature?
4.3.1. Features resolÕed within the aÕerage flow
Torsional oscillations thus seem to be able to
account for much of the time varying toroidal zonal
equatorial symmetric flow. But this is not the only
type of zonal flow that we observe. Fig. 6 and Table
2 further suggest the presence of a non-negligible
average zonal flow t n0 over the 150-year period
considered. Such a flow could provide some insight
about the way the dynamo process acts on intermediate timescales Ži.e., on timescales less than the typical diffusion timescale of the order of several 1000
years and larger than the decade timescales typical of
torsional oscillations riding on top of the full dynamo process..
Plotting the average flow of Table 2 immediately
reveals two interesting features Žsee Fig. 13a and b..
One is a strong high latitude equatorial symmetric
zonal flow emerging in both hemispheres, mainly
confined within the shadow of the inner core. The
other is a low latitude rather antisymmetric large
scale flow. Fig. 13c, which represents the angular
velocity v N Ž u .:
N
vN Ž u . s
Ý
ns1
(
n Ž n q 1 . Pn1 Ž u .
2
sin u
t n0
Ž 26 .
Žsee Eqs. Ž3. and Ž6.., as a function of the colatitude
u for various degrees N of truncation, makes the
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
309
Fig. 13. Mean zonal flow for the 1840–1990 period ŽPH-inversion.. Maps of the velocity zonal field as seen from Ža. the North pole and Žb.
the South pole. Žc. Angular velocities as a function of colatitude for different degrees of truncation of Eq. Ž26. Žsee text.: N s 5 Žopen
circles.; N s 6 Žopen triangles.; N s 7 Žgrey squares.; N s 8 Žgrey triangles. and N s 13 Žgrey circles.. Žd. Spectrum of the flow
Žequatorial symmetric components, open diamonds; equatorial anti-symmetric components, filled diamonds.. Also shown in Ža. and Žb. the
trace of the inner core tangent cylinder.
310
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
case even clearer and shows that the low latitude
antisymmetric flow is mainly due to the degrees
n - 6, while the high latitude symmetric flow is
mainly due to n s 7. This result is further confirmed
in Fig. 13d, which shows that the flow is the superposition of two contributions: a strong antisymmetric
Ž n even. large scale component which dominates the
spectrum for n - 5 and a strong symmetric Ž n odd.
component which dominates the spectrum for n G 5
and displays a peak at n s 7. Because Fig. 6 clearly
shows that t 20 , t 40 and t 70 are the terms displaying the
most significant systematic off-set with respect to a
zero average value, we believe this dual average
flow to be a real feature of the true Earth, at least for
the period considered Ž1840 to 1990..
In the present paper, we will make no special
attempt to interpret the very well resolved antisymmetric large scale component of the average flow,
although this flow likely contains some interesting
information concerning long time scale core dynamics. Rather, we will focus on the symmetric medium
scale average flow which is especially worth discussing for at least two reasons. First because many
recent studies have suggested that such a flow is
very likely to occur at the core surface as a result of
the presence of a superrotating inner core. Second,
because this flow is mainly revealed by a t 70 component which reflects the ‘‘true’’ flow only to the
provision made in Section 3.2. We will come back to
this point in the last part of the present section.
4.3.2. Suggestions from numerical studies
A number of theoretical and numerical studies on
magnetohydrodynamics and convection-driven geodynamos strongly suggest that some direct link could
possibly exist between the behavior of the inner core
and that of the average zonal flow at the core
surface.
Proudman Ž1956. and Stewartson Ž1957. studies
for a rapidly rotating spherical shell filled with some
fluid have for instance shown that when a small
superrotation D V of the inner boundary is imposed
in the limit of vanishing viscosity Žwhich is relevant
to the Earth’s core., the fluid inside the imaginary
cylindrical surface coaxial with the shell’s rotation
axis and touching the inner boundary at its equator
Žthe tangent cylinder. no longer corotates with the
outer boundary Žcontrary to the fluid outside the
tangent cylinder., but superrotates in a way which,
for the present purposes, could be approximated as a
block rotation at speed D Vr2. In such a simple
experiment with no magnetic field, observing the
flow at the outer boundary of the shell would make it
possible to detect the presence, size and speed of a
superrotating inner boundary.
Taking the magnetic field into account Žand assuming that the fluid and inner shell are conductors.,
Hollerbach Ž1997. showed that if an axial magnetic
field crossing the inner boundary inside the tangent
cylinder is imposed, then the previous inner boundary shearing zone disappears and the fluid inside the
tangent cylinder starts superrotating at the same speed
D V as the inner boundary. Dormy et al. Ž1998.
showed that if the imposed field is further assumed
to be dipolar with internal sources, so that field lines
also cross the tangent cylinder, then the flow gets
even simpler and becomes a block superrotation of
the whole fluid Žat speed D V . with respect to the
outer boundary. This is simply because the flow
generally avoids creating regions of shearing of the
field lines.
Although these experiments are not immediately
relevant to the Earth’s core, they provide us with
some hints of what may occur inside the real core:
either the dynamo manages to create a field which is
mainly parallel to the tangent cylinder in its vicinity,
in which case the inner core would be allowed to
superrotate with some strong zonal flows occurring
inside the tangent cylinder up to the CMB, or else
the dynamo produces a field frankly crossing the
tangent cylinder and the inner core is refrained from
superrotating with respect to the rest of the core Žin
which case no special features revealing the inner
core would be seen in the flow.. This admittedly
simplistic way of classifying possible geodynamos is
in reasonable agreement with recent computations of
self-consistent numerical dynamos, quite independently of the parameters and approximations being
used in these studies Že.g., Glatzmaier and Roberts,
1995, 1996; Kuang and Bloxham, 1997; Sarson et
al., 1998.. It thus suggests that if any superrotation is
to be suspected for the inner core Žas has been
suggested by Song and Richards, 1996 and Su et al.,
1996., a significant surfacic toroidal zonal flow
within the tangent cylinder at the CMB ought to be
found.
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
All self-consistent simulations up to date have
however shown that the zonal flow within the tangent cylinder would not be a simple bloc co-rotation
with the inner core. This is because the driving
mechanism for the superrotation of the inner core is
of complex nature and likely to lie within the tangent
cylinder itself. In fact, if the inner core is to superrotate, by simple counter-reaction a zonal flow of
opposite direction could more naturally be expected
to develop elsewhere inside the tangent cylinder,
especially near the CMB. This has for instance been
observed in dynamo simulations displaying inner
core superrotation, where an ‘‘X’’ structure within
the tangent cylinder develops as a result of some sort
of ‘‘thermal’’ wind which provides the energy setting on the superrotation ŽGlatzmaier and Roberts,
1996; Aurnou et al., 1998; Sarson et al., 1998..
However, as recently pointed out by Hollerbach
Ž1998., some slight Žmagnetic. coupling could still
exist between the material inside the tangent cylinder
Žincluding the inner core. and the material outside
this cylinder. Such coupling would drive a global
geostrophic flow inside the tangent cylinder, resulting in an additional zonal velocity term with respect
to the mantle. In any case, for situations relevant to
the Earth, it appears Žboth in fully consistent dynamo
calculations and in the studies of Aurnou et al.
Ž1998. and Hollerbach Ž1998.. that if an Žeastward.
rotation of the inner core is to exist, a strong westward zonal flow is to be expected within the tangent
cylinder, the amplitude of this flow being at least
comparable to the Žopposite. rotation rate of the
inner core, if not larger by a factor of up to 5.
4.3.3. Reconciling the seismically inferred inner core
superrotation and the flow obserÕed at the CMB
Whether any such inner core superrotation can
currently be detected by seismology is still a hotly
debated issue Že.g., Richards et al., 1998; Souriau,
1998b.. Clearly, claims for an eastward superrotation
of about 38ryr ŽSu et al., 1996. have not resisted
further testing ŽSouriau et al., 1997., and the original
claim of 1.18ryr made by Song and Richards Ž1996.
should now rather be seen as a very upper limit.
Indeed, studies based on different approaches and
data sets have either confirmed this upper-bound
ŽSouriau, 1998c. or argued in favor of a superrotation of an even smaller magnitude Ž0.2–0.38ryr,
311
Creager, 1997, or maybe even less, Laske and Masters, 1999..
For the purpose of the present study, we may thus
summarize the situation by assuming that the seismically inferred inner core superrotation lies somewhere in between 0 and 18ryr, the preferred rate
being of order 0.28ryr. Assuming that this superrotation Žtypically estimated over a 30-year period of
time. is representative of an average super-rotation
over centuries, it may then be related to the superrotation seen in the numerical simulations we discussed in the previous section. This in turn leads to
the conclusion that: Ž1. if inner-core superrotation
were to be a pure seismological artefact, strong zonal
flows within the shadow of the inner core would be
unlikely to be seen; Ž2. an eastward superrotation of
0.28ryr Žresp. - 18ryr. would lead to a typical
westward zonal flow of order 0.2–18ryr Žresp. 58ryr. within the shadow of the inner core at the
CMB.
Clearly, Fig. 13 suggests that this is the situation
in which we are. Computing the average rotation rate
of the mean flow t n0 over the north and south inner
core shadows for various degrees of truncation leads
to the values in Table 3. This table shows typical
values of 0.3–0.58ryr which are nicely compatible
with the ‘‘predictions’’ of item Ž2. above.
However, three final comments ought to be made
before concluding. The first one deals with the reliability of the recovered t 70 term which is essential in
defining the Žequatorial-symmetric and strong. zonal
flow in the shadow of the inner core ŽFig. 13, Table
3.. From the results presented in Section 3, it indeed
appeared that although this term could reasonably be
assumed preserved from the blurring effect of noise
ŽSection 3.3., it does suffer from at least attenuation
Table 3
Angular velocity values averaged over the core surface within the
tangent cylinder, for the 1840–1990 mean zonal flow of Table 2.
Units are in 8ryr, and a negative algebraic sign means a westward
flow relative to the mantle
Truncation, N
Northern Hemisphere
Southern Hemisphere
5
6
7
8
13
y0.237
y0.241
y0.498
y0.414
y0.491
y0.063
y0.058
y0.315
y0.400
y0.498
312
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
and possibly also biasing. We noted that the possibility that the recovered t 70 flow could entirely be due
to some bias in the PH-inversion Ži.e., to a non-unique
interpretation of the true SV., could not be dismissed. However, we also showed that if the true
flow were to display any strong degree 7 to 11 zonal
terms, the recovered corresponding terms could then
be interpreted as an attenuated version of the true
flow Žby the factor k AŽ n., recall Section 3.2..
This precisely brings us to our second comment.
Reasons for believing that a strong zonal flow could
develop within the inner core shadow have been
presented and need no further discussion. But what is
worth stressing is that given the current size of the
inner core, the low degree Ž n F 13. spatial spectrum
of such a flow would have a shape very much
resembling that of the recovered equatorial symmetric average flow Ži.e., n odd on Fig. 13d.. Fig. 14
Fig. 14. Spectral signature at the CMB of the rotation of a
cylinder coaxial with the Earth’s rotation axis and defined by the
colatitude u 0 . An arbitrary rotation of 18ryr is assumed for three
different values of u 0 . Only the first 13 degrees of the flow are
shown for comparison with Fig. 13d.
illustrates this point. This figure shows the spectrum
of a simple flow model that corresponds to a block
rotation at angular speed 18ryr of the fluid within
the tangent cylinder Ždefined by the colatitude u 0 .,
the rest of the fluid remaining still. Three values of
u 0 Ž188, 20.58 and 258. have been tested. It appears
that for u 0 close to 20.58 Žcorresponding to colatitude of the tangent cylinder., a maximum is obtained
for n s 7 just as in the real spectrum ŽFig. 13d, n
odd.. This maximum shifts smoothly as u 0 changes,
and is clearly sensitive to this parameter. When
carrying out additional tests involving zonal flows
with more complex structure within the tangent
cylinder, we have found that the shape of the spectrum could vary quite a lot for high degrees Žwhich
are required to describe the details of the flow., but
relatively little for the intermediate degrees that are
relevant to us. The ‘‘envelope’’ of the flow, that is
the requirement that it be zero outside the tangent
cylinder, is in fact the main feature responsible for
the peak of the spectrum at n s 7, when u 0 s 20.5, a
feature that we recover in our average flow. This we
consider as a serious reason to believe that for
degrees 7 to 11, our recovered average zonal flow
could really be an attenuated version of the true
zonal flow. Taking the corresponding attenuation
factor k AŽ n. into account then leads to a possible
rotation rate of up to 18ryr within the inner core
shadow. This value is again compatible with the
predictions of item Ž2. above.
Our third and final comment then consists in
noting that our estimates are reasonably consistent
with the 0.68ryr found by Olson and Aurnou Ž1999.
who relied on a local inspection of the behavior of
the field in the northern shadow of the inner core
Žbut made no attempt to discuss the impact of the
limited quality of the field models they rely on, and
of the possible non-uniqueness problems underlying
their interpretation..
This consistency is worth mentioning because of
the very different nature of their study. Olson and
Aurnou Ž1999. indeed not only relied on a local
analysis but also relaxed the tangentially geostrophic
and Žpartly. the frozen flux assumptions we relied on
for our own analysis. By contrast however, they
impose the flow to be axisymmetric. In doing so,
they note that the flux is reasonably frozen Žthe flow
they recover explains most of the observation., and
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
find a flow which is dominated by a vortex Žsimilar
to our zonal flow. but also has a significant meridional flow. This weaker meridional flow reflects an
upwelling along the rotation axis which is inconsistent with the tangentially geostrophic assumption. As
pointed out by a referee, this result thus suggests that
this assumption does not hold exactly within the
tangent cylinder. That this could be the case does
not, however, remove the validity of our approach,
which mainly requires the assumption to hold as a
first order approximation over most of the core, and
can anyway detect the kind of vortex which dominates the flow Olson and Aurnou see. That both
methods detect this vortex is thus rather encouraging.
Possibly more annoying is the fact that Olson and
Aurnou find less evidence of a similar vortex in the
Southern Hemisphere. This, however, could be because of a lack of enough resolved small scale
features for their local method to well sense such a
vortex. Our method being sensitive to average scale
features Žas just outlined in the previous comment.,
this might explain why by contrast it better detects
such a southern vortex.
In any case, this shows that any method better
fitted to detect such features are clearly worth developing. Methods based on global approaches of the
kind considered here are unlikely to produce more
accurate results than the one we found. By contrast,
local methods such as the one developed by Olson
and Aurnou are only beginning to emerge Žsee also
Chulliat and Hulot, 2000.. It will be very interesting
to see how much of the flow will eventually be
resolved in the dynamic region enclosed within the
tangent cylinder.
5. Conclusions
The main thrust of the present paper has been to
assess in some detail how much information can
actually be recovered regarding the toroidal zonal
flows computed at the CMB from SV data. This
question is of some importance, because such toroidal
flows are expected to testify for major aspects of the
dynamics of the core. Mechanical core–mantle interactions are expected to produce t 10 and t 30 zonal
313
flows carrying the core angular momentum required
to explain the LOD decade variations ŽJault et al.,
1988.. Smaller scale zonal flows are also expected to
be produced as a result of torsional oscillations
ŽBraginsky, 1970., and it has been suggested that
some superrotation of the inner core could exist
ŽSong and Richards, 1996., raising the question of
the possible signature that such a feature would
produce at the CMB.
Our results show that, in spite of relatively large
errors bars affecting our estimates, some features
within the zonal flow may be trusted and used to
provide important information.
We showed that the disagreement between the
observed LOD decade variations and the zonal
flow-based prediction of Jault et al. Ž1988. before
1920 could be attributed to our inability to properly
resolve the required combination of t 10 and t 30 zonal
terms before that epoch. After 1920 the theory of
Jault et al. Ž1988. leads to a remarkable prediction.
In fact, what makes these predictions so difficult is
that the t 10 and t 30 zonal terms taken individually are
apparently more sensitive to angular momentum exchanges that occur within the liquid core, than to
core–mantle angular momentum exchanges. Such
internal exchanges are indeed expected to occur as a
result of torsional oscillations of the kind proposed
by Braginsky Ž1970. and recently studied in some
detail by Zatman and Bloxham Ž1997; 1998; 1999..
Our study shows that the zonal flow coefficients Ž t n0 ,
n odd. required to detect and analyse such oscillations display significant time variations Žfor n s 1, 3,
5 and marginally 7, but not for larger degrees..
These time variations are compatible, within the
error bars we estimated, with the torsional oscillation
model proposed by Zatman and Bloxham Žespecially
their combined A q B wave. from 1920 onwards.
But we noted that a proper assessment of the characteristics of the oscillations would require that errors
be taken into account. Fortunately, our study further
shows that the full 1840–1990 data set could be used
to much better constrain those models Žthere are no
reasons to restrict attention to the 1900–1990 period,
as has been done in Zatman and Bloxham, 1997,
1998, 1999..
Finally, our study revealed an interesting and
significant average flow which we argue is the superposition of two components of different origins.
314
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
One of them is an equatorial-antisymmetric low latitude large scale flow Ždominated by degrees 2 and
4., the origin of which we made no attempt to
discuss. The other one is an equatorial-symmetric
flow, confined in the shadow of the inner core,
which can be interpreted as the medium scale signature of a strong zonal flow occurring at the top of the
core inside the tangent cylinder, in connection with
some possible superrotation of the inner core. We
indeed pointed out that the typical amplitude recovered for this flow Žof order 0.3–18ryr, in agreement
with the study of Olson and Aurnou, 1999, based on
a different approach. is compatible with both the
results of numerical studies of the dynamic signature
of a superrotating inner core, and the constraint
imposed by seismological evidence on the rate of
this superrotation. At a time when the seismological
evidence is severely questioned and a good case has
been made that the inner core might be gravitationally locked with respect to the mantle ŽBuffett, 1996;
Laske and Masters, 1999., it thus appears that independent geomagnetic evidence would rather point
out at a slightly superrotating inner core Žsee also
Voorhies, 1999, for a discussion on additional possible large scale geomagnetic consequences of such a
superrotating inner core.. Despite its small size, the
inner core already thought to be instrumental in
controlling the general morphology of the geomagnetic field Že.g., Gubbins and Bloxham, 1987., the
dominant non-zonal flow Že.g., Hulot et al., 1990,
Fig. 3b. and possibly core–mantle interactions Že.g.,
Buffett, 1996., does now appear to also have a
strong imprint on the CMB zonal flow.
Acknowledgements
We thank D. Jault, P. Olson and P. Cardin for
fruitful discussions and warm encouragements during the long process of completing this study, the
preliminary results of which have been presented at
the IAGA 1997 meeting in Uppsala ŽPais and Hulot,
1997.. One of us ŽAP. was partly funded by Fundaçao
ˆ
para a Ciencia
e Tecnologia ŽPortugal., grant
ˆ
PRAXIS-XXIrBDr2815r94. This is IPGP contribution no. 1650 and DBT contribution no. . . .
References
Aurnou, J., Brito, D., Olson, P., 1998. Anomalous rotation of the
inner core and the toroidal magnetic field. J. Geophys. Res.
103, 9721–9738.
Backus, G.E., 1968. Kinematics of geomagnetic secular variation
in a perfectly conducting core. Philos. Trans. R. Soc. London,
Ser. A 263, 239–266.
Backus, G.E., 1988. Bayesian inference in geomagnetism. Geophys. J. 92, 125–142.
Bloxham, J., 1988. The dynamical regime of fluid flow at the
core–mantle boundary. Geophys. Res. Lett. 15, 585–588.
Bloxham, J., Gubbins, D., Jackson, A., 1989. Geomagnetic secular variation. Philos. Trans. R. Soc. London, Ser. A 329,
415–502.
Bloxham, J., Jackson, A., 1991. Fluid flow near the surface of
Earth’s outer core. Rev. Geophys. 29, 97–120.
Bloxham, J., Jackson, A., 1992. Time-dependent mapping of the
magnetic field at the core–mantle boundary. J. Geophys. Res.
97, 19537–19563.
Braginsky, S.I., 1970. Torsional magnetohydrodynamics vibrations in the Earth’s core and variations in day length. Geomagn. Aeron. 10, 3–12, Eng. transl. 1–8.
Buffett, B.A., 1996. A mechanism for decade fluctuations in the
length of day. Geophys. Res. Lett. 23, 3803–3806.
Celaya, M., Wahr, J., 1996. Aliasing and noise in core–surface
flow inversions. Geophys. J. Int. 126, 447–469.
Chulliat, A., Hulot, G., 2000. Local computation of the geostrophic
pressure at the top of the core. Phys. Earth Planet. Inter. 117,
309–328.
Creager, K.C., 1997. Inner core rotation rate from small-scale
heterogeneity and time-varying travel times. Science 278,
1284–1288.
Dormy, E., Cardin, P., Jault, D., 1998. MHD flow in a slightly
differentially rotating spherical shell, with conducting inner
core, in a dipolar magnetic field. Earth Planet. Sci. Lett. 160,
15–30.
Gire, C., Le Mouel,
¨ J.-L., 1990. Tangentially geostrophic flow at
the core–mantle boundary compatible with the observed geomagnetic secular variation: the large-scale component of the
flow. Phys. Earth Planet. Inter. 59, 259–287.
Glatzmaier, G.A., Roberts, P.H., 1995. A three-dimensional convective dynamo solution with rotating and finitely conducting
inner core and mantle. Phys. Earth Planet. Inter. 91, 63–75.
Glatzmaier, G.A., Roberts, P.H., 1996. Rotation and magnetism of
Earth’s inner core. Nature 274, 1887–1891.
Gubbins, D., Bloxham, J., 1985. Geomagnetic field analysis: III.
Magnetic fields on the core–mantle boundary. Geophys. J. R.
Astron. Soc. 80, 695–713.
Gubbins, D., Bloxham, J., 1987. Morphology of the geomagnetic
field and implications for the geodynamo. Nature 325, 509–
511.
Hills, R.G., 1979. Convection in the Earth’s Mantle Due to
Viscous Shear at the Core–mantle Interface and Due to
Large-scale Buoyancy. PhD Thesis, N.M. State Univ., Las
Cruccs, 1979.
Hollerbach, R., 1997. The influence of an axial field on magneto-
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
hydrodynamic Ekman and Stewartson layers, in the presence
of a finitely conducting inner core. Acta Astron. Geophys.
Univ. Comenianae XIX, 263–275.
Hollerbach, R., 1998. What can the observed rotation of the
Earth’s inner core reveal about the sate of the outer core?
Geophys. J. Int. 135, 564–572.
Holme, R., 1998. Electromagnetic core–mantle coupling: I. Explaining decadal changes in the length of day. Geophys. J. Int.
132, 167–180.
Holme, R., Whaler, K., 1998. Issues in core-flow modelling and
core angular momentum. In: 1998 AGU Fall Meeting, abstracts volume, F231, San Francisco, 1998.
Hulot, G., Le Huy, M., Le Mouel,
¨ J.-L., 1993. Secousses Žjerks.
de la Variation Seculaire
et Mouvements Dans le Noyau
´
Terrestre. C. R. Acad. Sci. Paris 317 ŽII., 333–341.
Hulot, G., Le Mouel,
¨ J.-L., Jault, D., 1990. The flow at the
core–mantle boundary: symmetry properties. J. Geomagn.
Geoelectr. 42, 857–874.
Hulot, G., Le Mouel,
¨ J.-L., Wahr, J., 1992. Taking into account
truncation problems and geomagnetic model accuracy in assessing computed flows at the core–mantle boundary. Geophys. J. Int. 108, 224–246.
Jackson, A., 1995. An approach to estimation problems containing
uncertain parameters. Phys. Earth Planet. Inter. 90, 145–156.
Jackson, A., 1996. Bounding the long-wavelength crustal magnetic field. Phys. Earth Planet. Inter. 98, 283–302.
Jackson, A., 1997. Time-dependency of tangentially geostrophic
core surface motions. Phys. Earth Planet. Inter. 103, 293–311.
Jackson, A., Bloxham, J., Gubbins, D., 1993. Time-dependent
flow at the core surface and conservation of angular momentum in the coupled core–mantle system. In: Le Mouel, J.-L.,
Smylie, D.E., Herring, T. ŽEds.., Dynamics of Earth’s Deep
Interior and Earth Rotation, IUGG Vol. 12. AGU Geophysical
Monograph 72, pp. 97–107.
Jault, D., 1995. Model Z by computation and Taylor’s condition.
Geophys. Astrophys. Fluid Dyn. 79, 99–124.
Jault, D., Gire, C., Le Mouel,
¨ J.-L., 1988. Westward drift, core
motions and exchanges of angular momentum between core
and mantle. Nature 333, 353–356.
Jault, D., Hulot, G., Le Mouel,
¨ J.-L., 1996. Mechanical core–mantle coupling and dynamo modelling. Phys. Earth Planet. Inter.
98, 187–191.
Jault, D., Le Mouel,
¨ J.-L., 1989. The topographic torque associated with a tangentially geostrophic motion at the core surface
and inferences on the flow inside the core. Geophys. Astrophys. Fluid Dyn. 48, 273–296.
Jault, D., Le Mouel,
¨ J.-L., 1999. Comment on ‘‘On the dynamics
of topographical core–mantle coupling’’ by Weijia Kuang and
Jeremy Bloxham. Phys. Earth Planet. Inter. 114, 211–215.
Kuang, W., Bloxham, J., 1997. An Earth-like numerical dynamo
model. Nature 389, 371–374.
Langel, R.A., 1987. The main field. In: Jacobs, J.A. ŽEd.., Geomagnetism, Vol. 1. Academic Press, New York, pp. 249–512.
Laske, G., Masters, G., 1999. Limits on differential rotation of the
inner core from analysis of the Earth’s free oscillations. Nature
402, 66–69.
Le Huy, M., 1995. Le Champ Geomagnetique,
les Mouvements
´
´
315
du Fluide a` la Surface du Noyau, et les Variations Decennales
´
de la Rotation de la Terre. PhD thesis, Universite´ Paris 7,
Paris.
Le Mouel,
¨ J.-L., 1984. Outer core geostrophic flow and secular
variation of Earth’s magnetic field. Nature 311, 734–735.
Le Mouel,
¨ J.-L., Gire, C., Madden, T., 1985. Motions at core
surface in the geostrophic approximation. Phys. Earth Planet.
Inter. 39, 270–287.
Le Mouel,
¨ J.-L., Hulot, G., Poirier, J.-P., 1997. Core–mantle
interactions. In: Crossley, D. ŽEd.., Earth’s Deep Interior: The
Doornhos Memorial. Gordon and Breach.
Mandea Alexandrescu, M., Gibert, D., Le Mouel,
¨ J.L., Hulot, G.,
Saracco, G., 1999. An estimate of average lower mantle
conductivity by wavelet analysis of geomagnetic jerks. J.
Geophys. Res. 104, 17735–17745.
Menke, W., 1984. Geophysical Data Analysis: Discrete Inverse
Theory. Academic Press.
Olson, P., Aurnou, J., 1999. A polar vortex in the Earth’s core.
Nature 402, 170–173.
Pais, A., Hulot, G., 1997. Could some axial differential rotation at
the core surface be detected by geomagnetic observations? In:
IAGA abstract volume, page 11, Uppsala.
Parker, R.L., 1994. Geophysical inverse theory. Princeton Univ.
Press, Princeton, NJ.
Peddie, N.W., Fabiano, E.B., 1982. A proposed international
geomagnetic reference field for 1965–1985. J. Geomagn. Geoelectr. 34, 357–364.
Proudman, I., 1956. The almost-rigid rotation of viscous fluid
between concentric spheres. J. Fluid Mech. 1, 505–516.
Richards, P.G., Song, X., Li, A., 1998. Detecting possible rotation
of Earth’s inner core. Science 282, 1227a.
Roberts, P.H., Scott, S., 1965. On analysis of the secular variation.
J. Geomagn. Geoelectr. 17, 137–151.
Sarson, G.R., Jones, C.A., Longbottom, A.W., 1998. Convection
driven geodynamo models of varying Ekman number. Geophys. Astrophys. Fluid Dyn. 88, 225.
Shankland, T.J., Peyronneau, J., Poirier, J.P., 1993. Electrical
conductivity of the Earth’s lower mantle. Nature 366, 453–455.
Song, X.D., Richards, P.G., 1996. Seismological evidence for
differential rotation of the Earth’s inner core. Nature 382,
221–224.
Souriau, A., 1998a. Earth’s inner core: is the rotation real?
Science 281, 55–56.
Souriau, A., 1998b. Detecting possible rotation of Earth’s inner
core. Science 282, 1227.
Souriau, A., 1998c. New seismological constraints on differential
rotation of the inner core from Novaya Zemlya events recorded
at DRV, Antarctica. Geophys. J. Int. 134, F1–F5.
Souriau, A., Roudil, P., Moyot, B., 1997. Inner core differential
rotation: facts and artefacts. Geophys. Res. Lett. 24, 2103–
2106.
Stephenson, F.R., Morrison, L.V., 1990. Long-term fluctuations in
the Earth’s rotation: 700 BC to AD 1990. Philos. Trans. R.
Soc. London, Ser. A 351, 165–202.
Stewartson, K., 1957. On almost rigid rotations. J. Fluid Mech. 3,
299–303.
Su, W.-J., Dziewonski, A.M., Jeanloz, R., 1996. Planet within a
316
A. Pais, G. Hulot r Physics of the Earth and Planetary Interiors 118 (2000) 291–316
planet: rotation of the inner core of Earth. Science 274,
1883–1885.
Voorhies, C.V., 1999. Inner core rotation from geomagnetic westward drift and a stationary spherical vortex in Earth’s core.
Phys. Earth Planet. Int. 112, 111–123.
Zatman, S., Bloxham, J., 1997. Torsional oscillations and the
magnetic field within the Earth’s core. Nature 388, 760–763.
Zatman, S., Bloxham, J., 1998. A one-dimensional map of Bs
from torsional oscillations of the Earth’s core. In: Gurnis,
E.K.M., Wysession, M.E., Buffet, B.A. ŽEds.., The Core–
Mantle Boundary Region. AGUrIUGG, Washington, pp.
183–196.
Zatman, S., Bloxham, J., 1999. On the dynamical implications of
models of Bs in the Earth’s core. Geophys. J. Int. 138,
679–686.