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Geophys.
1.
Int.
(1994)
117,
235-256
Electromagnetic coupling and the toroidal magnetic field at the core-
mantle boundary
J.J.
Love*
and
Jeremy
Bloxham
Department
of
Earth and Planetary Sciences, Harvard University, Cambridge, Massachwetts
02138,
USA
Accepted 1993 September 22. Received 1993 June 9;
in
original
form
1992 September
11
SUMMARY
Owing
to
exchanges
of
angular momentum between the Earth’s fluid outer core and
the overlying solid mantle, the Earth’s rotational rate fluctuates on periods
of
a few
years to a few decades. However, the mechanism which allows the exchange
of
angular momentum is not understood. Here we examine the possibility that
core-mantle coupling is predominantly electromagnetic and thus responsible for the
decadal length
of
day variations. The electromagnetic couple
on
the mantle can be
divided into poloidal and toroidal parts, and, by requiring continuity
of
the
horizontal component of the electric field at the core-mantle boundary, the toroidal
couple can be divided into separate advective and leakage parts. The poloidal
couple results entirely from the interaction of the poloidal field with currents
induced by its time variation; the advective couple results from the dragging
of
poloidal field lines through a conducting mantle; and the leakage couple results from
the diffusion of toroidal magnetic field from the core’s interior into the mantle. The
poloidal and advective couples can be estimated by using models
of
the downward
continued poloidal field and models
of
the core velocity. We find that neither the
poloidal couple nor the advective couple exhibit sufficient variability to account for
the decadal length
of
day variations.
If
this is true, and if core-mantle coupling is
indeed predominantly electromagnetic, then most
of
the variability in the length
of
day must result from the leakage couple, which, unfortunately, cannot be calculated
directly from surface observations.
We assume that the horizontal component
of
the magnetic field is continuous
across the core-mantle boundary, that the frozen-flux approximation adequately
describes the time dependence
of
the horizontal component
of
the magnetic field at
the core surface, and that most
of
this time dependence results from steady core
motion. Then by treating the determination
of
the toroidal field at the core-mantle
boundary as an inverse problem, we
find
that only very strong and spatially complex
toroidal field models are consistent with both advection in the core and the decadal
length
of
day variations. We argue that strong toroidal fields are necessary to
account for the length-of-day variations since there is significant cancellation when
the magnetic stress is integrated over the core-mantle boundary
(CMB),
the
necessary time-dependent torque resulting from the slight and temporary non-
cancellation
of
magnetic stress established by a slowly varying and spatially complex
toroidal field. But, since our toroidal field models are too strong according to
dynamo theory, produce electric fields at the Earth’s surface which are stronger than
measured values, and produce ohmic heating which either exceeds or contributes an
unacceptably large fraction
of
the Earth’s surface heat flow, we deem the
toroidal-field models consistent with our analysis
of
electromagnetic coupling to be
physically unreasonable. Thus, we argue that core-mantle coupling is not pre-
dominantly electromagnetic. However, this conclusion may not hold if, for example,
*
Present address: Department
of
Earth Sciences, The University
of
Leeds, Leeds, LS2 9JT,
UK.
235
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236
J.J.
Love
and
J.
Bloxham
core
motion is highly time dependent,
or
if
a
strong
diffusive boundary layer is
present beneath the core-mantle boundary, the
presence
of
which
may
allow
for
a
significant discontinuity in the horizontal component
of
the magnetic field and the
breakdown
of
the frozen-flux approximation.
Key
words:
core-mantle boundary, core-mantle coupling, earth rotation,
geomag-
netism, length
of
day, toroidal magnetic field.
1
INTRODUCTION
That the compass needle points roughly towards geographic
north is not accident: rotation plays an important role in the
dynamo process responsible for maintaining the geomag-
netic field. It is also possible that an opposite effect occurs:
the magnetic field may influence the Earth’s rotation. The
core is the primary source of the Earth’s magnetic field, and
it
is often suggested that exchanges of angular momentum
between the core and mantle could be caused by
electromagnetic coupling; currents flowing in the electrically
conducting lowermost mantle interact with the magnetic
field to produce Lorentz torques which might cause
variations in the mantle’s rotationa! rate. Because angular
momentum is conserved, an increase in the mantle’s
rotational rate is accompanied by a decrease
in
the core’s
net rotational rate and vice versa. And since, for our
purposes, the mantle and crust function as a single rigid
unit, an observer standing on the Earth’s surface perceives
the fluctuations of the rotational rate of the mantle as
variations
in
the length
of
day (LOD).
Core-mantle coupling is just one of a host of different
geophysical processes which, over a wide range of
time-scales, affect the Earth’s rotational rate. Others include
the exchange
of
angular momentum between the solid Earth
and either the atmosphere
or
ocean, the tidal exchange of
angular momentum
in
the Earth-Moon-Sun system,
or
changes
in
the moment
of
inertia of the mantle due to
postglacial rebound (Munk
&
MacDonald 1960; Lambeck
1980; Rochester 1984; Wahr 1988; Hide
&
Dickey 1991). In
a kinematic analysis of LOD variations, one uses
measurements to account for the angular momentum budget
of
the Earth. In a dynamic analysis, one considers the
physical mechanisms responsible for the rotational varia-
tions. Whereas the kinematic problem of LOD variations is
now well resolved, analysis
of
the dynamic processes
responsible for LOD variations is far from complete. In this
analysis we consider the dynamics
of
electromagnetic
core-mantle coupling.
Since universal time (UT)
is
defined by the position
of
the
Earth relative to the stars, small differences (AT) between
UT and the theoretical uniform time-scale, ephemeris time
(ET), can be noticed through astronomical observation and
modern geodetic measurement.
For
example, the time of a
lunar occultation of a star can be accurately predicted,
provided the rotational rate of
the
Earth is constant. Small
changes in
the
rotational rate cause discrepancies between
the predicted time and the actual time
of
an occultation. A
deviation
of
1
m
s
in the Earth’s rotation rate, integrated
over a decade, amounts to about a
1s
deviation in UT.
Since the Moon’s motion against the stars is relatively rapid,
-0.5”
s-’,
by setting a pendulum clock, say, according
to
the stars and
soon
thereafter noting the time and position
when the Moon occults a particular star, discrepancies in
UT of about one second were just
on
the edge of
detectability by
a
single telescopic measurement as far back
as the middle of the last century (Stephenson
&
Morrison
1983). These small LOD fluctuations, when integrated over
several decades, can add up to an appreciable amount of
time. From Fig.
l(a)
we notice that the discrepancy between
ET and UT may be several tens
of
seconds. Thus, by
accumulating many historical astronomical measurements,
very small fluctuations in the Earth’s rotational rate can be
detected over periods of decades.
In this analysis we examine decadal variations in LOD
which have occurred over the period 1861-1988. For the
period 1861-1956, we use yearly averaged values
of
AT
deduced from over 25,000 lunar occultation times (Morrison
1979), and
for
1957-1988 we use
AT
values based
on
the
international atomic time-scale determined by the Bureau
International de I’Heure. UT may be equivalently expressed
as the sidereal displacement angle
q,
which represents the
position
of
the
Earth relative to the stars, and by
differentiating
I#
once we obtain the variation
in
the LOD
A
=
(1)
which we plot in Fig.
l(c).
By differentiating
I+!J
twice we
obtain the torque on the mantle
r
=
I,
3:q,
(2)
where
I,
is the moment of inertia of the mantle; we plot the
torque on the mantle in Fig. l(d).
Although it has been known since the time of Newcomb
(1882) that the Earth’s rotational rate is variable, the
observations
of
Glauert (1915), Brown (1926) and de Sitter
(1927) led astronomers to theorize that LOD fluctuations
consisted of occasional abrupt changes in the Earth’s
rotational rate. What followed was a fruitless search for
catastrophic events
of
sufficient magnitude to affect the
LOD variations. Not until Brouwer’s (1952) analysis of
many lunar occultation timings were the more gradual
decadal variations in the LOD definitively revealed. Since
then, it has been widely speculated that the decadal LOD
variations result from the exchange
of
angular momentum
between the liquid core and solid mantle. But despite
suggestive correlations between the changes
in
the LOD and
variations
in
the Earth’s magnetic field (Vestine 1953), only
very recently has it become clear that the decadal variations
are indeed caused by core-mantle coupling. By using the
secular variation
of
the geomegnetic field to estimate the
fluid motions
in
the core, Jault, Gire
&
Le
Moue1
(1988)
and Jackson, Bloxham
&
Gubbins (1993) have shown that
changes in the LOD, and the corresponding changes in the
angular momentum of the mantle, are roughly correlated
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Core-mantle
boundary
237
variations, torques of up to 10'xNm are necessary, and
several core-mantle coupling mechanisms have been
proposed which might produce such large torques: viscous,
topographic, gravitational and electromagnetic coupling.
Whilst viscous coupling can readily be shown to be too small
by several orders of magnitude (Rochester 1984), at least for
conventional estimates
of
the viscosity
of
the core fluid
(Poirier 1988), the relative importance
of
the other
core-mantle coupling mechanisms has yet to be determined.
Gravitational coupling arises from the attraction of
aspherical density structure in the core to that in the mantle
(Jault
&
Le Mouel
1989),
but since the aspherical density
structure of the core is not known, the gravitational couple
is difficult to assess. Topographic coupling (Hide 1969)
arises as the pressure gradients which drive core convection
act on topography,
or
bumps, on the CMB, and
is
analogous
to the topographic coupling that exists between the
atmosphere and the solid Earth, and is partially responsible
for
the seasonal fluctuations in the LOD. Using models of
the CMB topography deduced from seismological inversion
and models
of
the fluid motion at the top of the core
deduced from variations
of
the magnetic field, recent work
indicates that topographic coupling produces torques
of
more than sufficient magnitude to account for the decadal
LOD variations (Jault
&
Le Mouel 1990; Hide
et
a/.
1993).
As
a result, many in the geophysical community currently
favour the topographic-coupling hypothesis. Conversely,
many dynamo theorists (see, for example, Braginsky 1988
or
Roberts
1988)
favour electromagnetic coupling. Thus we are
motivated, in this analysis, to examine the hypothesis that
electromagnetic core-mantle coupling is the predominant
mechanism responsible
for
the decadal LOD variations.
in
30
20
10
0
-10
-20
-30
-
-
I-
111
111I1.~111
11
I
i.ir
1880 1900 1920 1940 1960 1980
2
1
0
-1
-2
w
0
3
-7
::
a
J1
1880
1900
1920
1940
1960
1980
4
Yr-rTT
-r-v;
i
-1
.
T-T-TT~
1880
1900 1920
1940 1960 1980
1880
1900
1920 1940
1960 1980
Figure
1.
Since universal time
(UT)
is defined by the position
of
the
Earth relative to the stars, small differences
(AT)
between
UT
and
the theoretical uniform time-scale, ephemeris time
(ET),
may be
noticed through astronomical observation.
AT
may be equivalently
expressed as an angle, the sidereal displacement angle,
9.
The
standard error is denoted by
6.
By fitting a smooth curve
to
9
and
differentiating once with respect to time, we obtain the variation in
Earth's rotational rate
A,
the excess LOD. The points in (c) denote
the predicted change in the LOD as determined by an analysis
of
the core's angular momentum by Jackson
et
nl.
(1992).
Differentiating once again we obtain the torque
r
on the mantle
responsible
for
the LOD variations.
A
secular trend
of
1.8
m
s
per
century (Stephenson
&
Morrison 1983) has been removed, and
for
both the
LOD
and the
torque
on the mantle we have plotted an
envelope
to
indicate the standard error.
with changes in the angular momentum of the core. In Fig.
l(c) we show the LOD as predicted by the analysis of
Jackson
et
a/.
,
which can be compared with the actual LOD
variations. The correlation is best for the most recent times,
say after 1960. Considering that the LOD variations and the
predicted LOD variations
of
Jackson
et
a/.
are based on two
entirely independent data sets, and given the number
of
assumptions implicit in the analysis of Jackson
et
a/.,
the
correlation in Fig. l(c) is remarkable, despite the differences
prior to
1930.
Hence, the kinematic aspect
of
the decadal
LOD variations is approaching resolution.
However,
of
all LOD variations, the dynamics of the
decadal LOD variations have been the most problematic.
From Fig. l(d) we note that to produce the decadal LOD
2
ELECTROMAGNETIC COUPLING: A
HEURISTIC EXAMINATION
Electromagnetic coupling (Bullard
et
a/.
1950;
Munk
&
Revelle 1952) arises as electric currents in the mantle
M
interact with the magnetic
field
B
via the Lorentz force
rH
=
JMrX
(J
X
B)
dV,
(3)
where
r
is the radius vector in a spherical coordinate system
(r,
0,
$).
The current density
J
is related to the magnetic
field through AmpCre's law
VX
B
=
pJ,
(4)
where
p
is
the magnetic permeability. Combining (3) with
(4) and applying the divergence theorem, the electromag-
netic couple on the mantle may be converted to an integral
of the magnetic stress over the CMB
C
k--+
B,B,sinOdS
CMB
(Rochester 1962), where
c
is the core radius and where we
have specialized to consider only those torques that are
parallel to the Earth's rotational axis and therefore affect
the LOD. This equation shows that the electromagnetic
torque on the mantle depends on the boundary conditions
established at the CMB. Since the magnetic field at the
CMB has a characteristic length-scale much shorter than the
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238
J.J.
Love
and
J.
Bloxham
core circumference, there is considerable cancellation in
(5)
when integrating the magnetic stress over the CMB.
Moreover, we note from Fig. l(d) that the torque
on
the
mantle changes sign
on
the decadal time-scale, brief
compared to the time-scale usually thought to characterize
core processes. Thus, if indeed electromagnetic coupling is
responsible for the decadal LOD variations, then the time
dependence
of
the torque
on
the mantle results from the
slight and temporary non-cancellation of the magnetic stress
at the CMB produced by the relatively more slowly varying
magnetic field.
2.1
Kinematics
of
electromagnetic coupling
Like the LOD variations, the magnetic field changes over a
wide range of time-scales as a result
of
a wide variety
of
geophysical processes (Parkinson 1983; Merrill
&
McElhinny 1983; Courtillot
&
Le Mouel 1988), but
for
this
analysis we shall concern ourselves with the magnetic
variations which originate interior to the Earth's surface and
which occur over periods of years to centuries, the so-called
secular variation of the Earth's magnetic field. The secular
variation
of
the poloidal magnetic field is frequently
described as a westward motion of
-lo-'('
rad
s-'
of
the non-axial dipolar features in the geomagnetic field
(Bullard
et
al.
1950). But such
a
description is an over
simplification; detailed analysis of the surface field, and its
extrapolation down to the core's surface, reveals that the
poloidal magnetic field is complex, consisting
of
static,
rapidly drifting and oscillating features (Bloxham
&
Gubbins 1985).
The time dependence of the magnetic field
B
in the core is
governed by the induction equation
1
WC
d,~
=
v
x
(U
x
B)
+-v'B,
where
u
denotes fluid velocity and
uc
is the conductivity
of
the core. The relative importance
of
advection and diffusion
in eq.
(6)
is measured by the magnetic Reynolds number
R,.
If the flow and magnetic field have length-scales
characterized by the core radius, then
R,,
=
PO,.
Uc
=
500,
(7)
where
U
is
a typical value
of
the fluid velocity of the core
(numerical values listed in Table 1). Since the magnetic
Reynolds number
is
large, advection dominates in the bulk
of the core, and it
is
common to make the frozen-flux
approximation
so
that
d,B
=
V
X
(U
X
B).
(8)
Table
1.
Typical numerical values
used
in
this
analysis.
Value
Magnetic permeability
Moment
of
inertia
of
the mantle
Moment
of
inertia
of
the core
Radius
of
the Earth
Radius
of
the core
Typical
fluid
velocity
in
the core
Typical strength
of
the radial field
at
the CMB
Electrical conductivity
of
the mantle
4n
x
lo-'
H/m
7.2
x
103' kgm'
8.5
x
1PE
kgm2
6.4
x
10'
m
3.5
x
lb
m
-
3
x
lo-'
m/s
*
3
x
lo-'
T
Rotational rate
of
the Earth
7.3
10-5
Electrical conductivity
of
the core
-
3
105
slm
51-mSJm
This
is
AlfvCn's theorem in differential form, which states
that for
a
perfect conductor, the magnetic field lines move
with fluid particles. Considering only the radial component
of
the magnetic field, eq.
(8)
may be rewritten as
d,B,
+
V,
.
(B,u)
=
0
(9)
(Roberts
&
Scott 1965), where
V,
=V-i(i.V)
is
the
horizontal gradient operator. This equation suggests that,
given the secular variation
of
the poloidal field and its
downward continuation to the CMB, it
is
possible to invert
for the velocity field at the top
of
the core (Roberts
&
Scott
1965). Such
a
program has been successfully performed
by a number of investigators (Voorhies
1986;
Whaler
&
Clarke 1988; Bloxham 1989; Lloyd
&
Gubbins 1990; Gire
&
Le Mouel 1990). Their work indicates that most
of
the
secular variation can be explained by core motions which
are consistent with the frozen-flux approximation. In fact,
using the frozen-flux approximation, most
of
the secular
variation over the past 300 years can be explained by steady
fluid motion at the surface of the core (Bloxham 1992).
On the other hand, the theory that the decadal
LOD
variations result from the exchange of angular momentum
between the core and mantle requires that core flow possess
some temporal variability. By conservation of angular
momentum
I,
6Q,
+I,.
6&,
=
0
where
I,,
I,
are the moments of inertia of the mantle and
core, each of which are assumed to be spherically
symmetric, and
SQ,,
6Q,
are the variations in the
rotational rates of the mantle and core. If we define
651,
=
651,
-
651,
as the change in the relative motion of
the core and mantle, then
Now a change in the LOD
of
1
m
s
is equivalent to a relative
change in the mantle's rotational rate of
6&,/8,
=
lo-',
and assuming, for the moment, that the motion of the core
is roughly described by westward motion, a
1
m
s
change in
the LOD would be accompanied by an
8
per cent change in
the westward drift. From a kinematic standpoint, the
decadal LOD variations are easily accommodated by
relatively minor changes in core flow.
Thus, in this heuristic examination we might represent the
fluid velocity in the
core
by the decomposition
u(t)
=
(u)
+
6u(r),
(12)
where
(u)
is a slowly varying, and for our purposes
quasi-steady, component of the flow which characterizes
large-scale convection in the core, that which Bloxham
(1992) has found to describe the greater part
of
the secular
variation; and where
6u(t)
is a smaller and more rapidly
varying component of the core flow consistent with
angular-momentum conservation, but which makes only a
minor contribution to the secular variation. If
(U)
and
6U
are typical values of
u
and
6u
respectively, then noting from
Fig.
l(c)
that the largest deviation in the
LOD
is about
4
m
s
and using the conservation of momentum argument above,
we may roughly bound the fluctuations in core motion
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Core-mantle boundary
239
necessary to account for the
LOD
variations
su
1
u3
-5-
or
bus
(U).
(14)
The
time-scale which characterizes
(U)
is roughly a
convective overturn time
tc
=
c/U
2-
400
yr,
(15)
which is long compared to the 128 yr timespan
of
our
analysis. On the other hand, the time-scale characterizing
fluctuations 6U in core motion are described by the decadal
time-scale
tLOD
=
30
years
(16)
and might be related to torsional oscillations in the core as
envisaged by Braginsky (1970), although he favours a
60
yr
periodicity. In any case,
~LOD
<<
rc.
(17)
This difference in time-scale, as we shall see, places a strong
constraint on the toroidal field at the CMB necessary to
account for the decadal
LOD
variations.
2.2
Dynamics
of
electromagnetic
coupling
To
appreciate the difficulty in estimating the electromagnetic
couple, and because it is useful in spherical coordinates, we
divide the magnetic field into its toroidal and poloidal
ingredients
B
=
B,+
BP
=
V
X
(Yr)
+
V
X
V
X
(a),
(18)
where
3
and
9?
are the toroidal and poloidal scalar
functions. AmpCre’s law shows that poloidal magnetic fields
are sustained by toroidal electric currents and vice versa. In
so
far as the exterior
of
the Earth is an insulator, the
observed potential field is a special case
of
a poloidal field
and is sustained by electric currents in the Earth’s interior.
On
the other hand, a toroidal field, being entirely
non-potential, is trapped within a conductor, and conse-
quently, toroidal magnetic fields vanish above the Earth’s
surface.
By using a perturbation expansion which considers
conductivity in the mantle, the observed potential magnetic
field can be downward continued to the CMB
so
that the
radial component
B,
and part
of
the azimuthal component
B,
in eq.
(5)
may be estimated (Benton
&
Whaler 1983).
But the toroidal field, which is tangent to a spherical
surface, also contributes to
B,
;
thus the electromagnetic
couple divides
where the couple
rp
depends only on the poloidal field,
whilst
rT
depends on the radial part
of
the poloidal field and
the axirnuthal component
of
the toroidal field. Since the
toroidal field is unknown, the total electromagnetic couple
cannot be easily evaluated.
Nonetheless, an approximate upper bound may be
obtained by a heuristic argument. By ignoring possibly
significant cancellation when performing the integration
of
the magnetic stress over the CMB in eq.
(9,
we have
BT, Br
r,
c
4x2
-
P
The toroidal field
BT,
in the mantle is governed by the
diffusion equation, the solution of which requires two
boundary conditions. The first is the requirement that the
toroidal field vanishes at the top of the conducting part
of
the mantle. The second boundary condition commonly used
is continuity of the horizontal component of the electric field
across the CMB
[?xE]’=O
at
r=c.
Using Ohm’s law
J
=
a(E
+u
X
B)
(22)
and AmpCre’s law
(4),
we have
1
1
-?
X
(V
X
B,)
=
-B,u
+--i
X
(V
X
B,)
WJM
WC
at
r
=
c,
where
U,
is the conductivity
of
the mantle, and where
B,,
B,
are the toroidal magnetic fields in the mantle and core
respectively. The first term
on
the right
of
eq. (23)
represents the production
of
toroidal field through the
shearing of pre-existing poloidal magnetic field lines by
advective fluid motion
in
the core, and the second term
represents the leakage of toroidal field from the core’s
interior. Thus, the toroidal field in the mantle may be
divided into advective and leakage fields
B,
=
B,
+
B,,
(24)
which gives rise to separate advective and leakage couples
(Rochester
1960;
Roberts 1972)
rT
=
rA
+
rL.
(25)
Using the advective part
of
(23)
we can estimate the
strength
of
the advective field at the CMB
B,
=
pUYB,
(26)
rA
5
4nc’UYBf,
(27)
and an upper bound for the advective couple
where the conductance
is
where
a
is
the Earth’s radius and
6,
is a radial length-scale
typical of the thickness
of
the conducting part
of
the
lowermost mantle. Using typical values, we show
in
Table 2
estimates
of
upper bounds on the advective couple for a
variety
of
mantle conductivity profiles.
Using the leakage part
of
(23) we can estimate the
strength
of
the leakage field at the CMB
(29)
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240
J.J.
Love
and
J.
Bloxham
Table
2.
A
summary
of
the three different mantle-conductivity models used in this analysis, and the results
of
the heuristic analysis in Section
2.
Mantle Conductivity Models
#
Reference
~~(7)
=
vCMB
(:)a
(S/m)
7"
(yr)
<
0.9
1
Courtillot
et
a[.
gCMB
=
150
(1984)
0=9
-
<
2.1
2
UCMB
=
1000
a
=
15.27
<
1.6
3
Stix
&
Roberts
u~.,,,~
=
3000
(1984)
(L
=
30
and an upper bound for the leakage couple
<
0.7
0.065
<
1.5
<
1.1
(30)
where
6,
is a radial diffusive length-scale
for
the core, and
B,
is
a
typical strength
of
the toroidal magnetic field in the
core.
To
estimate an upper bound on the leakage couple, we
need an estimate
of
the radial gradient
of
the toroidal field
in
the core
-B,/6,.
This requires a consideration
of
the
dynamic processes
in
the core.
The strength
of
the toroidal field
B,
in the core depends
on the nature
of
the geodynamo. If the dynamo operates in
a strong-field regime, which now seems likely (Roberts
1987), then the Coriolis and Lorentz forces are comparable
1
2pCQ
X
u
=-
(V
X
B)
X
B,
P
where
pc
is the density
of
the core and where
Q
is the
diurnal rotational vector
of
the Earth. With such a
magnetostrophic balance we can estimate the strength
of
the
toroidal field in the interior of the core (Hide
&
Roberts
1979)
(31)
B,
=
q2-
=
40
mT,
(32)
which is about a hundred times greater than the estimated
strength
of
the poloidal field
in
the core.
Estimating the length-scale
6,
is, however, more difficult.
Although some poloidal motion is necessary for dynamo
action (Elsasser 1947), it has been suggested that core flow
may be predominantly toroidal for the case
of
stable
stratification (Whaler 1980), or for strong stratification along
with a strong Lorentz force (Bloxham 1990).
If
core motion
is large-scale and predominantly toroidal, then spatial
variations in the toroidal magnetic field, maintained by
core-wide fluid shearing of the poloidal magnetic field,
might be characterized by a radial length-scale approxim-
ately equal to the core radius
6,
=
C.
(33)
If,
on
the other hand, the core has significant poloidal
motion, fluid upwelling may concentrate the toroidal field
near the CMB; see Fig.
2.
The thickness
of
the resulting
magnetic diffusive boundary layer (Weiss 1966) is
0.24
0.36
57
5
25
5
38
5
0.9
5
3.5
5
5.2
5
0.06 56
5
0.2
5
21
5
0.3
5
31
determined
by
balancing diffusion with advection, and
instead
of
(7)
we have
so
that
-
175
km.
(35)
Because
of
the balance between diffusion and advection,
CMB
Figure
2.
Formation
of
a magnetic diffusive boundary layer and
subsequent flux expulsion into the mantle. Fluid upwelling
impinging on a magnetic field (a) concentrates the magnetic field
lines beneath the
CMB
where they may then diffuse into the mantle
(b).
The thickness
of
the resulting boundary layer
is
6,.
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Core-mantle
boundary
241
Figure
3.
Simple dynamo theories pre-suppose a core where motion
is
dominated by simple differential rotation (a).
A
poloidal dipolar
field (b) would be twisted about the rotational axis,
(c)
and (d), by
the so-called w-effect. Finally, diffusion allows the magnetic field
poloidal fluid motion will not only lead to the break down
of
the frozen-flux approximation (Allen
&
Bullard 1966;
Bloxham 1986), but by shortening the radial length-scale
characteristic
of
the toroidal field in the core, poloidal fluid
motion has an important influence on the size of the leakage
couple; see eq. (30). Using typical values we have
estimated, in Table
2,
the strength
of
the toroidal field at the
CMB and the leakage couple for a variety
of
mantle-
conductivity models, where we have ignored cancellation in
the integral of the magnetic stress and have assumed the
shorter length-scale (35)
so
as to obtain upper bounds
on
the
leakage couple.
These heuristic calculations indicate that electromagnetic
coupling could potentially produce torques
of
the magnitude
known to cause the decadal LOD variations. However, as
we have mentioned, the success
or
failure of electromag-
netic coupling hinges
on
two important issues: cancellation
in the integral of the magnetic stress at the CMB and time
dependence. These issues must be addressed explicitly
before electromagnetic coupling can justifiably be con-
sidered a viable mechanism responsible for the decadal
LOD variations.
3
PREVIOUS ANALYSES
OF
ELECTROMAGNETIC COUPLING
Inspired by the westward drift
of
the geomagnetic field, and
the zonal motion that it suggests, early workers in
geomagnetic theory, most notably Elsasser and Bullard,
argued that fluid motion in the core might be dominated by
differential rotation. For example, the angular velocity
of
the core might have the form
where
s
is the distance from the Earth’s rotational axis.
Elsasser (1947) proposed that differential rotation might
produce a strong toroidal quadrupolar magnetic field as
dipolar magnetic-field lines are wrapped around the
rotational axis via the so-called
o
effect; see Fig. 3.
In the context of axisymmetric convection, Bullard
et
af.
(1950) first invoked electromagnetic core-mantle coupling
in
an attempt to account for the westward drift. It was
argued that conservation
of
angular momentum would cause
the part
of
the core furthest from the rotational axis to
rotate slower than the part closest to the axis. Relative to
the mantle, the inner parts
of
the core would rotate
eastwards, while the outer parts rotate westwards. The
westward motion
of
the outer parts of the core would then
produce the westward drift
of
the geomagnetic field.
However, in the absence
of
any other coupling mechanism,
Bullard
et
al.
argued that viscous coupling would cause the
mantle to rotate along with the outer portions
of
the core,
thereby preventing the relative motion necessary to account
for the westward drift
of
the magnetic field. If viscous
coupling is particularly weak, then tidal deceleration
of
the
lines to close (e), resulting in a quadrupolar toroidal field in the
Earth’s core. See Parker
(1979)
for a detailed discussion of this
process.
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242
J.
J.
Love
and
J.
Bloxham
mantle by the Moon would produce an eastward drift.
Hence, in order to maintain the geomagnetic westward drift,
it was reasoned that another coupling mechanism must be
present. Bullard
et
af.
proposed that electromagnetic
coupling could allow for a geomagnetic westward drift. The
eastward differential rotation in the deep interior
of
the core
could produce a quadrupolar toroidal field
of
one polarity
while the westward motion of the outer part of the core
drags the dipole field through the mantle thereby producing
a toroidal field
of
the opposite polarity.
As
the toroidal field
produced
in
the interior leaks out to the CMB, it cancels the
toroidal field produced by shearing at the CMB
so
that there
is no net torque on the mantle:
r,+
rA
=
0.
(37)
Thus, electromagnetic core-mantle coupling might allow the
mantle and the bulk of the core to rotate in unison, even
though different parts of the core rotate differentially.
Today, the picture of core convection painted by Elsasser
and Bullard may seem overly simplistic. Studies of an
internally heated rotating sphere (Busse
&
Carrigan 1976;
Hart, Glatzmaier
&
Toomre 1986; Zhang 1991) indicate that
the pattern of convection can be complex. Moreover,
Bullard
et
af.
did not consider the dynamic effects
of
the
magnetic field; the Lorentz force probably plays an
important role in determining the pattern
of
convection. But
perhaps most importantly, the secular variation is much
more complicated than a simple westward drift and is
inconsistent with purely azimuthal fluid motion (Whaler
1982; Bloxham 1990). If we consider specific core-flow
models deduced from the secular variation, we find that an
axisymmetric quadrupolar toroidal field would quickly be
distorted into a field with structure over many harmonic
degrees (see Fig.
4).
Consequently, a simple quadrupolar
toroidal field does not exist near the CMB. Thus, whilst
useful for gaining an understanding
of
the dynamo process
(see, for example, Parker 1979), the simple dynamos
envisaged by Elsasser and Bullard are, at best, idealizations.
But their influence has been considerable, and as a result,
simple differential rotation and the concomitant quadrupolar
toroidal magnetic field are frequently invoked in analysis
of
electromagnetic coupling.
Considering only the case of coupling with the dipole
poloidal and the quadrupolar toroidal fields, and a relatively
weak mantle conductivity, Bullard
et
af.
concluded that
electromagnetic coupling is inadequate to account for the
decadal LOD variations. Munk
&
Revelle (1952) revived
the electromagnetic coupling hypothesis, finding that it
could be responsible for the decadal LOD variations,
provided the lowermost mantle had a conductivity greater
than 100
S
m-I. Variations in the LOD might result from
slight non-cancellation between the leakage and advective
couples. Since Munk
&
Revelle’s reappraisal, numerous
investigations
of
electromagnetic coupling have been
performed. Most workers have conducted forward calcula-
tions
of
the advective couple, and then assumed an
approximate balance between the advective and leakage
couples, eq.
(37),
to deduce the strength
of
the toroidal field
in the core’s interior.
Rochester (1960), in an extension
of
the work by Bullard
et
af.,
calculated the advective torque
rA
with higher
harmonics
of
the poloidal field with a core surface that was
assumed to rotate as a rigid spherical shell at a rate
Velocity
at
CQR
surface
Figure
4.
A
demonstration that simple quadrupolar toroidal fields cannot by sustained near the
CMB.
A toroidal model
(a)
of
fluid motion at
the core surface as determined by inverting the poloidal magnetic secular variation (Jackson
&
Bloxham
1991)
acts on an initially quadmpolar
toroidal magnetic field (b). After
128
yr, the time-span
of
this analysis, the toroidal magnetic is distorted into a complex field (c), which is
no
longer simply quadrupolar.
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Core-mantle boundary
243
Toroidal
field
before
advection
To~t~iidaP
field
after
advecthn
Fire
4.
(Continued
.)
representative
of
the westward drift. Roden
(1963)
found
that the advective couple was enhanced by concentrating the
conductivity in a thin highly conducting layer adjacent the
CMB,
and that an advective couple larger than the
10l8
Nm
necessary could be affected. Similar results were obtained by
Braginsky
&
Fishman
(1976),
again assuming that core
motion consists
of
rigid rotation. In a particularly elaborate
treatment, Stix
&
Roberts
(1984)
deduced time-dependent
azimuthal motions at the core surface from the secular
variation. But although they admitted that such motions are
not entirely justified by the complexity of the secular
variation, they proceeded
to
calculate a time-dependent
advective torque. The advective torque
of
Stix
&
Roberts
exhibits fluctuations
of
the magnitude known to cause the
LOD
variations, but the correlation
is
poor. In fact, the
advective couple is always negative, the result
of
a strong
component
of
core flow causing the westward drift
of
the
geomagnetic field. Since the decadal torque
on
the mantle
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244
J. J.
Love
and
J.
Bloxham
fluctuates about zero, they suggested, as Bullard
et
al.
proposed, that the offset may be due to the leakage couple.
In order to calculate the leakage couple
rL,
it has usually
been assumed, consistent with the theories
of
Elsasser and
Bullard, that the toroidal field in the core is essentially
quadrupolar axisymmetric. By assuming that the leakage
couple and the advective couple approximately balance, eq.
(37), Rochester estimated the maximum strength
of
a
quadrupolar toroidal field in the core at
-30mT,
a result
consistent with
our
magnetostrophic estimate (32). Stix
&
Roberts used a similar approach to estimate the radial
gradient
of
a quadrupolar toroidal field, finding that several
hundred kilometres beneath the CMB, the toroidal field
strength approximates that determined by Rochester.
However, as was certainly well known to previous
investigators, electromagnetic coupling occurs between
poloidal and toroidal fields
of
all harmonics; each
toroidal-field harmonic of degree
I
couples with the poloidal
harmonic degrees
1
+
1
and
1
-
1.
A qualitative understand-
ing of the magnitude of the coupling as a function
of
harmonic degree may be gained from Fig.
5,
where we plot
the maximum possible toroidal torque as a function
of
harmonic degree. For a given strength of the poloidal and
toroidal fields, the strongest possible couple occurs between
low
degree fields, particularly between the dipole poloidal
field and the quadrupolar toroidal field. But significant
coupling can occur between other toroidal-poloidal
components, and for high harmonic degrees the magnitude
of
the electromagnetic couple levels
off
at a constant for a
given strength
of
the toroidal field. The importance
of
coupling over many harmonic degrees has previously been
emphasized by Braginsky
&
Fishman.
However, all
of
these analyses have used both velocity
fields at the core surface which, as we have discussed, are
not
satisfactorily compatible with the poloidal secular
N
b
E
E
\
z
m
-
0
Poloidal
Degree
L-1
200
E.
Poloidal
Degree
L+l
1
i
100
1
'
1
I
I
'
11
'
I
I
'
I
'
1
2
3
4
5
6
7
8
9
1011121314
TOROIDAL
DEGREE
(e)
Fire
5.
The toroidal torque depends on the coupling
of
the
toroidal field harmonics
I
with the poloidal field harmonics
I
f
1.
For
each harmonic degree, the exact couple depends on the
strength
of
the magnetic fields and cancellation between the
different harmonic orders
m.
Here we plot the maximum torque
for
given strengths
of
the toroidal and poloidal fields.
variation, and models
of
the toroidal field which, as we have
demonstrated, are incompatible with velocity fields deduced
from the secular variation. Although it has been shown that
electromagnetic coupling may produce torques
of
sufficient
magnitude to account for the LOD variations, a result
compatible with our simple heuristic analysis in the previous
section, it is important to realize that in none
of
these
analyses do the time-dependent variations
of
the calculated
torques correlate with the fluctuating torque responsible for
the LOD variations. indeed, it
is
not clear that such a
correlation should even exist since
it
has never been
demonstrated that core-mantle coupling is predominantly
electromagnetic.
4
TIME-DEPENDENT ELECTROMAGNETIC
COUPLING: THE FORWARD PROBLEM
To evaluate the electromagnetic couple we consider an
electrically conducting mantle where the behaviour of the
magnetic field is governed by the diffusion equation
Magnetotelluric analysis
of
externally induced magnetic
variations are used to constrain the mantle's conductivity
down to depths of about a thousand kilometres (Lahiri
&
Price 1939; Parker 1970; Achache, Le Mouel
&
Courtillot
1981; Parkinson
&
Hutton 1989). These studies indicate that
conductivity increases with depth, but electromagnetic
coupling is most dependent
on
deep-mantle conductivity,
knowledge
of
which comes from two sources: analysis
of
rapid variations,
or
jerks, in the magnetic field, and
high-pressure laboratory measurements. Geomagnetic jerks
are abrupt changes
of
the field such that the second time
derivative is nearly discontinuous.
All
magnetic variations
originating in the core, including jerks, are filtered by the
conducting mantle before they are observed at the Earth's
surface (Backus 1983); the mantle filter both smooths and
delays magnetic signals. By analysing geomagnetic jerks,
Courtillot, Le Mouel
&
Ducruix (1984) found that the time
delay
t,,
is less than about three years and the smoothing
time
tS
is less than about one year. They concluded that at
least 97 per cent
of
the volume
of
the mantle has a
conductivity
of
less than a few hundred Sm-'. Moreover,
high-pressure laboratory measurements
of
the constituents
thought to comprise the mantle confirm the conclusion that
the bulk
of
the mantle is only weakly conducting
(Peyronneau
&
Poirier 1989; Wood
&
Nell 1991). However,
it is possible that the iron
of
the core and the silicates
of
the
mantle chemically react at the CMB (Knittle
&
Jeanloz
1986), producing a thin highly conducting layer at the base
of
the mantle (Li
&
Jeanloz 1987). This chemically reactive
layer may be associated with the -100 km thick seismically
anomalous zone
in
the mantle adjacent the CMB, the
so-called D" layer (see, for example, Young
&
Lay 1987).
But this is speculative, and despite recent progress, the
conductivity
of
the lowermost mantle remains ill
constrained.
The analyses
of
Jault
et
al.
(1988) and Jackson
et
al.
(1993), who used downward continued potential magnetic
fields to estimate the decadal angular momentum variations
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Core-mantle boundary
245
we need to consider the perturbing effects of mantle
conductivity.
For each mantle-conductivity profile considered, the time
delay and smoothing time are much less than the
10-30
yr
time-scale of the decadal
LOD
variations, thus for
our
purposes, the poloidal secular variation is only slightly
affected
by
the intervening conductivity. Accordingly, we
make the low-frequency perturbation expansion (Roberts
1972),
where we assume that the poloidal field
B,
in the
mantle is only slightly different from a potential field
B,,
Bp
=
B,
+
B,
+
. . . .
(41)
After substituting this expansion into the diffusion equation
(38)
we obtain, to first order,
of the core, appear to rule out long mantle-delay times
zD.
In particular, the time delays
of
13
years (Backus
1983)
and
18
years (Paulus
&
Stix
1989)
are difficult to reconcile with
the results of Jackson
el aZ.
(1993).
The predicted
LOD
variations
of
Jackson
et
al.,
shown
in
Fig. l(c), were
obtained with the assumption that the time delay is short
compared to the decadal time-scale. The correlation in Fig.
l(c) substantiates this assumption, and the apparent offset
prior to
1930
is probably not a phase shift due to mantle
filtering since that would imply a negative time delay;
although Le Moue1
et
al.
(1981)
argued for such a delay. In
this study we compare power-law conductivity models of the
form
where
ucMs
is the conductivity at the base of the mantle.
The details of each conductivity model are summarized in
Table
2
and Fig.
6.
As
discussed
in
Section
2,
we can divide the
electromagnetic couple
(5)
into poloidal, advective and
leakage couples
ra
=
rp+
rA
+
r,
(40)
C
P
CMB
=
-
-
f
(Bpe
+
B,,
+
B,,,)B,
sin
0
dS.
The observed potential field
is
usually extrapolated down to
the CMB assuming a perfectly insulating mantle (Gubbins
&
Bloxham
1985;
Shure, Parker
&
Langel
1985).
But to
perform a forward calculation of the electromagnetic couple
10000
E
1000
A
-\
m
t
\\
i
1
I
-4
.o
1
2000
1000
0
DEPTH
km
Figure
6.
A
comparison
of
the different mantle-conductivity models
used
in
this analysis
(see
Table
2).
We
have used constructed
mantle model
2
to
simulate a highly conducting region near the
CMB,
but which
is
also
in
general agreement with the upper mantle
conductivity model
of
Parker
(1970),
denoted by
P,
as determined
by magnetotelluric inversion and with the high-pressure measure-
ments
of
Peyronneau
&
Pokier
(1989).
denoted
by
PP. Although
there are significant differences between the various mantle-
conductivity profiles,
our
results are only slightly affected by each
particular model.
which, for a particular mantle conductivity model, may be
solved for
B,,
subject to continuity
of
the field at the Earth's
surface (Benton
&
Whaler
1983).
The iterative technique
using the expansion
(41)
is
justified
so
long as
(43)
which is a condition satisfied throughout this analysis. Using
the spherical harmonic model
of
Bloxham
&
Jackson
(1992),
and the solution
of
(42)
to downward continue a
non-potential poloidal field to the CMB, we can evaluate
the poloidal couple
rp
(Stix
&
Roberts
1984).
For each
mantle-conductivity model, we have calculated the poloidal
torque
rp,
as
shown
in
Fig.
7.'
We find that the poloidal
torque is always negative from
1861
to
1988
and that it does
not exhibit the same degree
of
temporal variability as the
mantle torque
r;
both
of
these observations have previously
been made by Stix
&
Roberts
(1984)
and Paulus
&
Stix
(1989).
Thus, aside from a slowly varying offset, the poloidal
torque does not contribute significantly to the decadal
LOD
variations.
Since the toroidal field is entirely non-potential, for a
low-frequency perturbation expansion
of
the form
(41),
the
toroidal field in the mantle satisfies, to first order,
(44)
which can be solved by requiring that the toroidal field
vanish at the top of the conducting part of the mantle and by
using continuity of the horizontal component
of
the electric
field at the CMB. For the advective field, we solve
(44)
by
1880 1900 1920 1940 1960 1980
Figure
7.
The poloidal torque for each mantle-conductivity model
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246
using the advective part
of
the boundary condition (23),
which, in addition to a model
of
mantle conductivity,
requires a model
of
the velocity field at the core surface.
However, the inversion for core motion is non-unique
(Roberts
&
Scott 1965; Backus 1968), that is,
if
one solution
can be found, then an infinity
of
solutions exist. For
example,
if
u
satisfies eq.
(9),
then there exists another
velocity
u’
which also satisfies
(9)
where
J.
J.
Love
and
J.
Bloxham
B,u’
=
B,u
+
V
X
(Wr)
(45)
and
Y
is a scalar function. This so-called toroidal ambiguity
cannot be resolved from surface observations alone. To help
resolve the non-uniqueness, additional information must be
applied to the inversion. For instance, assuming that the
flow is either geostrophic (Backus
&
Le Mouel 1986)
or
toroidal (Lloyd
&
Gubbins 1990), part
of
the ambiguity is
removed, and
if
the flow is assumed to be steady (Voorhies
&
Backus 1985), then the non-uniqueness is removed
entirely. Unfortunately, considerable differences exist
between velocity models depending
on
the
a
priori
assumptions (Bloxham
&
Jackson 1991), and thus, the
toroidal ambiguity cannot be well resolved.
These differences highlight an important point: through
the application of the boundary condition (23) the advective
couple, which is derived
By,
is dependent on the
regularization used in determining the velocity field, a point
well emphasized by Jault
&
Le Mouel (1991a). Therefore, in
order to judge the sensitivity
of
our analysis to the
particularities
of
the core-velocity models, we compare three
steady models, those determined by Jackson
&
Bloxham
(1991): an unconstrained mixed toroidal-poloidal model, a
toroidal model, and a geostrophic model, each calculated
for the interval 1960-1980.
All
three
of
these velocity
models are predominantly toroidal, as we have argued is
roughly consistent with the frozen-flux approximation.
Noting from (23) that the advective production
of
toroidal
magnetic field from the pre-existing poloidal magnetic field
is explicitly dependent
on
the fluid motion at the core
surface, a time-varying advective couple
r,(t)
may be
roughly characterized by a decomposition similar to (121,
namely
‘A(‘)== (rA)
+
6rA(f)j
(46)
where
(r,)
is a roughly steady advective torque which
results from the advective dragging of poloidal magnetic
field lines through the conducting lowermost mantle by the
steady part of the core motion
(u);
whilst
6rA(t)
is a
smaller, but more rapidly varying, component
of
the
electromagnetic couple, resulting primarily from the
fluctuating advective motion
6u.
Thus, like eq. (14)
(47)
Since both (23) and
(40)
also depend on
the
poloidal
magnetic field, some time dependence in the advective
torque
r,(t)
may result from the time-varying poloidal
magnetic field, even for a steady velocity field
(u)
,
but since
most of the secular variation of the poloidal magnetic field
occurs over time-scales
tC
much longer than the decadal
time-scale
tL.OD,
this effect is probably insignificant.
With the steady velocity fields of Jackson
&
Bloxham
(1991) and the downward-continued poloidal field model of
Bloxham
&
Jackson (1992), we can calculate a steady
advective field
(BA)
by solving
(44)
and then calculate the
steady advective torque
(r,)
using eq. (40). In Table 3 we
show
(r,)
calculated for all three velocity models
of
Jackson
&
Bloxham and for all three conductivity models
of
Table 2. Like the poloidal couple, the steady advective
couple is negative. This result is similar to that obtained by
Stix
&
Roberts (1984) and Jault
&
Le Mouel (1991a), and,
as we have mentioned,
is
caused by core motion which
exhibits a strong westward-flow component, giving rise to
the rough westward drift of the observed potential field, and
hence an advective torque which is negative.
But calculating the time-varying advective torque
6r,
is
more difficult, not in the least because only the largest
patterns
of
time-dependent flow can currently be resolved,
despite the regularizing constraints placed on the flow
Table
3.
Average core velocity and estimated fluctuations in core velocity from
1960-1980
as required by the exchange
of
angular momentum between the core and
mantle
for
the three core velocity models
of
Jackson
&
Bloxham
(1991).
And the
average advective torque and its estimated fluctuations
for
each core velocity and
mantle-conductivity model.
Unconstrained
Toroidal
Geostrophic
(U)
f
6U
10.1
f
1.3
(WYr)
(U)
f
6U
13.9
f
1.3
(WYr)
(U)
f
6U
(WYr)
10.2
f
1.3
Model
1
-0.42
f
0.05
4.61
f
0.06
4.40
f
0.05
Model
2 -1.80
f
0.23 -2.63
f
0.27
-1.68
f
0.21
-4.19
f
0.39
-2.65
f
0.34
Model
3 -2.84
f
0.36
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Core-mantle boundary
247
field. Diffusion could also allow a discontinuity in the
horizontal field across the relatively thin diffusive boundary
layer, thus making the horizontal field effectively discon-
tinuous across the CMB.
A
complete consideration
of
diffusion in the core
necessitates solution
of
eq. (6), which requires knowledge of
whole-core convection. However,
a
realistic model of core
convection cannot be obtained without a much better
understanding of the geodynamo. In other words, it is
currently impossible to model magnetic diffusion in the core
realistically. Whilst we keep in mind the objections
of
those
who have suggested that the frozen-flux approximation does
not accurately describe the time dependence
of
the
horizontal field and that horizontal field may be effectively
discontinuous across a diffusive boundary layer beneath the
CMB,
in
order to make progress we proceed, for now,
assuming that the frozen-flux approximation is reasonably
accurate. Our motivation for assuming that the frozen-flux
approximation holds for the toroidal field is simply one of
expediency. We shall, at the end of our analysis, return to
examine the effects of these assumptions with the benefit of
hindsight.
Before then, however, it is worthwhile
to
examine
qualitatively the processes by which
a
diffusive boundary
layer could be maintained adjacent the CMB. A diffusive
boundary layer adjacent the CMB must exist to allow the
magnetic field in the core, which has a non-potential
ingredient, to adjust
so
as to match onto the essentially
potential field in the weakly conducting mantle. The
non-potential field in the core is maintained by electric
currents, which by virtue of Ampere's law (4), can be
equivalently expressed
as
a
curl in the magnetic
field.
The
diffusion term in the induction eq.
(6)
allows for the decay
of currents and concomitant relaxing of the curl in the
magnetic field, while advective shearing of the magnetic
field will, in general, produce currents and recurl the
magnetic field. The issue, then, is whether or not advective
motion
in
the core is such as to produce significant shearing
of
the magnetic field and thereby maintain currents in
a
thin
boundary layer despite the destructive effect of diffusion.
Consider first the simple case where initially there is no
discontinuity
of
the horizontal field, and a vertical magnetic
field
B,
threads through the core into an insulating mantle.
Subsequently, horizontal fluid, primarily toroidal, shearing
motion in accordance with the frozen-flux advection eq.
(8)
will act
on
the vertical field and induce
a
horizontal field just
beneath the CMB. If we assume that the fluid motion has
a
shear with
a
characteristic vertical length-scale
L,
then after
some time
t
the vertical gradient in the induced horizontal
field will be
models needed
to
resolve the toroidal ambiguity (Bloxham
&
Jackson 1991). We can, however, place bounds on the
variability of
6rA
by using the conservation of angular
momentum argument
of
Section 2.1. During the interval
1960-1980 used by Jackson
&
Bloxham to construct their
velocity models, we note from Fig. l(c) that the LOD
fluctuated by -1.5ms, which, from eq. (ll), requires a
change
in
the bulk motion of the core
6UJ(
CJ)
-9-12
per
cent; see Table
3.
From
eq. (27) such variations in core
motion require roughly proportional variations in the
advective torque,
so
that
From Fig. l(d) we note that during the interval 1960-1980
the torque fluctuated by
-0.8
X
10''
Nm, but by com-
parison, the fluctuations in the advective torque
(r,)
from
Table
3
are small for all of our mantle-conductivity models.
Thus, it seems that very high mantle conductivity adjacent
the CMB is required if the advective torque
rA
is to exhibit
the required variability in time to account for the decadal
LOD
variations, much higher than might have been
supposed from the heuristic estimates of Section 2.2 which
did not account for temporal variability. And, if we consider
a mantle conductivity model consistent with analysis of the
geomagnetic jerk, namely model
I
of
Courtillot
et
al.
(1984), then the temporal variability in the advective torque
is altogether inadequate. Jault
&
Le Moue1 (1991a) reached
a
similar conclusion using a different argument.
For the leakage toroidal field, we might solve (44) by
using the leakage part
of
the boundary condition (23).
Unfortunately,
as
we have stressed in Section
2.2,
this
couple requires knowledge of the radial gradient of the
toroidal field in the core,
a
quantity which cannot be
deduced directly from surface observations. Without making
assumptions about the toroidal field in this boundary layer it
is impossible to complete a forward calculation of the
electromagnetic core-mantle couple. It
is
important to note
that even though the leakage part of (23)
is
not explicitly
dependent on the velocity field, the toroidal field in the
core, embedded in a boundary layer beneath the CMB, will
change with time
in
a manner determined by the induction
equation. If the frozen-flux approximation holds for the
horizontal component of the magnetic field, then steady
advection may dominate the time dependency of the
toroidal magnetic field near the CMB.
5
ELECTROMAGNETIC COUPLING
AS
AN
INVERSE PROBLEM
5.1
Diffusive boundary layers and continuity
of
the
toroidal magnetic field
In this analysis we assume that the frozen-flux approxima-
tion describes the greater part of the time dependence
of
the
toroidal field in the core near the CMB. We emphasize that
the validity of this assumption has not been established.
Indeed, several investigators (Backus
1968;
Braginsky
1984;
Jault
&
Le Mouel 1991b; Braginsky
&
Le
Mouel 1993) have
suggested that diffusion in a boundary layer adjacent to
the CMB
may
lead to
a
breakdown of the frozen-flux
approximation for the horizontal component of the magnetic
6B,
-
UB,t
6r
L~
'
(49)
Thus, over the thickness of the fluid shear layer the
discontinuity in the vertical field is
UB,t
L
6B,
=
-
Jault
&
Le Mouel (1991b) argued that
(SO)
represents the
discontinuity in the horizontal field across
a
diffusive
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248
J.J.
Love
and
J.
Bloxham
boundary layer
of
thickness
Whilst admittedly our argument may be overly simplistic,
we note that
(SO)
is the result
of
fluid shear over a layer
of
thickness
L,
not over the diffusive length-scale
6,.
Thus,
it appears that Jault
&
Le Mouel's argument needs
modification, a better approximation to the discontinuity
of
the horizontal magnetic field across the diffusive boundary
layer is instead
where we have assumed that a typical length-scale
characterizing shear in the core is the core radius c. Relative
to the strength
of
the poloidal field, (52) is an
inconsequential difference
of
about
1
per cent. We note that
if we consider shorter time-scales, like those used by Jault
&
L,e
Mouel, this conclusion is only reinforced. On the other
hand, if fluid shearing occurs over a relatively short radial
length-scale, then the discontinuity in the horizontal field
may be larger than we have estimated.
Alternatively,
as
we noted in Section 2.2, a diffusive
boundary layer, determined by a balance between diffusion
and advection and characterized by the length-scale (35),
may result from poloidal fluid upwelling acting on the
horizontal, primarily toroidal, magnetic field of the core.
The development
of
the diffusive boundary layer also leads
to the breakdown
of
the frozen-flux approximation for the
radial component
of
the magnetic field. The importance of
the radial length-scale in determining the validity
of
the
frozen-flux approximation, that is, whether the length-scale
is represented best by (33) or by (39, has been recognized
previously by Roberts
&
Scott (1965) and Backus
&
Le
Mouel
(1986). Thus, some measure
of
the importance
of
the
diffusive boundary layer, and a concomitant discontinuity in
the horizontal component
of
the magnetic field near the
CMB, might be made by testing the frozen-flux
approximation.
Analysis
of
the poloidal secular-variation data itself
indicates that the frozen-flux approximation might be
tentatively rejected over certain regions
of
the CMB,
particularly beneath southern Africa where a reverse-flux
patch has developed (Bloxham
&
Gubbins 1986).
Nonetheless, most
of
the secular variation is consistent with
the frozen-flux approximation (Constable, Parker
&
Stark
1993). And, as we have mentioned in Section 2.1, it is
possible to construct models
of
core flow, assuming the
frozen-flux approximation, which account for most of
secular variation. Indeed, most
of
these core-flow models
are predominantly toroidal (for a review see Bloxham
&
Jackson 1991), which, we have suggested in Section 2.2,
may be the flow most consistent with the frozen-flux
approximation. Furthermore, the models
of
core flow and
shear constructed by Jackson
&
Bloxham (1991) are
consistent with whole-core convection. Such observations
might lead one to tentatively conclude that the radial
length-scale characteristic
of
the toroidal field
in
the core is
approximated by the core radius (33), rather than the
shorter diffusive boundary layer length-scale (35). If this
were true, then it might also be thought that the horizontal
component
of
the magnetic field is effectively continuous
from the base
of
the mantle to the top
of
the free stream
just beneath the Ekman layer
[iXB]T=O
at
r=c,
(53)
as has been suggested by Roberts
&
Scott (1965), and that
the time dependence
of
the magnetic field is well
approximated by the frozen-flux approximation (8). For the
case
of
an insulating mantle, Barraclough, Gubbins
&
Kemdge (1989) found that, after allowing for expected
errors, the secular variation
of
the poloidal magnetic field is
consistent with (53).
As
Barraclough
et
al.
have noted, such
consistency arguments do not,
of
course, prove that (53)
actually holds, but merely that observations do not
contradict the hypothesis that the horizontal field is
effectively continuous across the fluid layer beneath the
CMB
.
In a theoretical analysis, Braginsky
&
Le Mouel (1993),
assuming that the flow near the CMB is essentially poloidal,
concluded that the horizontal field is discontinuous across
the CMB. We too have suggested that poloidal motion can
lead to the development
of
a diffusive boundary layer and
perhaps to a discontinuity of the horizontal field near the
CMB, but, as we have mentioned, predominantly poloidal
motion is not required to explain the secular variation.
We note that the toroidal field generated over the 128
year time-span
of
our analysis just beneath the CMB by fluid
shear, eq. (52), is feeble in comparison to our heuristic
estimates
of
the toroidal field from Section 2.2; see Table 2.
Since the mantle is probably a weak electrical conductor,
the above discussion
of
the continuity
of
the horizontal field,
which assumed an insulating mantle, is hardly affected,
although,
of
course, a non-vanishing, but probably very
weak, toroidal field may exist at the CMB for the case
of
a
conducting mantle. Some generation
of
this toroidal field
may result from the advective motion
of
the core relative to
the mantle, but we have argued that this
is
primarily a
steady field and not
likely
to produce a significant
time-dependent electromagnetic torque. Thus, assuming the
frozen-flux approximation, over the time-span of our
analysis the time dependence
of
the toroidal field is
dominated by the advection of pre-existing toroidal field.
the production
of
toroidal field by shearing
of
poloidal field
lines making an insignificant contribution.
In our opinion, more work is required before we can
understand the nature
of
the diffusive boundary layer and
make reliable assertions as to the possible discontinuity
of
the horizontal field and the validity
of
the frozen-flux
approximation as a description
of
the time dependence of
the horizontal magnetic field. But as we have emphasized, in
order to make progress, we assume for the time being that
the horizontal magnetic field is effectively continuous near
the CMB and that the frozen-flux approximation is a
reasonable means
of
modelling the time dependence
of
the
toroidal field.
5.2
Formulation
of
the inversion
The success
of
the electromagnetic-coupling hypothesis
hinges on the ability
of
the toroidal couple
r,
to affect the
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Core-mantle boundary
249
necessary
LOD
variations. But since we cannot directly
calculate the leakage couple we adopt a different approach.
We investigate the hypothesis that core-mantle coupling is
predominantly electromagnetic by treating the determina-
tion of the toroidal field at the CMB as an inverse problem.
If core-mantle coupling is indeed predominantly
electromagnetic
r
=
rB,
(54)
r
-
rp
=
rT.
(55)
then the toroidal couple is approximately
Since
r
is deduced from the decadal
LOD
variations and
rp
is calculable (Stix
&
Roberts
1984),
the left-hand side is
known, and henceforth we need
only
consider the integral
equation
B,B,
sin
0
dS,
where
BT@
is unknown.
A
combination
of
diffusion and
advection
in
a boundary layer beneath the CMB produces a
time-varying toroidal field. We have assumed that advection
is dominant and that the time dependence of the toroidal
field is described by the frozen-flux approximation. Thus,
instead of making assumptions about the toroidal field in the
core
so
that we may use the boundary condition
(23)
to
solve the diffusion equation and obtain the toroidal field at
the CMB, we invert
(56)
for the toroidal field necessary to
account for the decadal
LOD
variations, and with the
application
of
continuity
of
the horizontal component of the
magnetic field
(53),
we require that the time dependence
of
the toroidal field be consistent with advection in the core
(8).
Since steady models of core motion account for most
of
the secular variation, and since the steady part of the motion
is likely to be larger than the fluctuating part, we assume
that most
of
the temporal dependence
of
the toroidal field
adjacent the CMB is determined by a steady velocity field.
Indeed, we recall from Section
4
that the fluctuations in the
advective torque arising from fluctuations in core motion are
likely small compared to the fluctuations in the observed
torque on the mantle.
We express the poloidal and toroidal magnetic fields in
terms
of
their scalar functions
(18),
and then expand the
scalar functions by spherical harmonics
(57)
where
Y;"
is a Schmidt-normalized spherical harmonic, by
which we mean
4n
inYr1Yz2
dw
=
-
6:
21,
+
1
(59)
where the integration
is
over all solid angles
dw.
These
expansions allow the toroidal torque
(56)
to be expressed as
a linear matrix equation of the form
where
(61)
(Rochester
1960;
Roberts
1972).
The function
e(t)
describes
observational errors in the
LOD,
theoretical errors, due to
truncation of the expansions
(57)
and
(58),
and errors in the
downward-continued poloidal field
B,.
The matrix eq.
(60)
can be more compactly written in boldface form
rT(t)
=
qT(t)t(t)
+
e(t)t
(62)
where the superscript 'T' denotes transpose.
At the CMB the radial velocity vanishes, and
if
the core
fluid is assumed to be incompressible, the toroidal-poloidal
decomposition reduces to
u=uT
+
up,
(63)
uT=VX("P"r)
(64)
up
=
V&W).
(65)
where
Expanding the velocity scalar functions by spherical
harmonics and assuming time independence of the velocity
field, we have
As we discussed in the previous section, we assume that
over the time-span
of
our analysis the advective shearing
of
poloidal field lines does not substantially contribute to the
time dependence
of
the toroidal field near the CMB. Thus,
in our analysis, the time dependence of the toroidal field is
dominated by
a
quasi-steady fluid motion which advects a
pre-existing toroidal field. Specializing the induction eq.
(8)
to consider
only
the advection
of
toroidal magnetic fields at
the CMB,
(8)
can be expressed as a matrix equation
(Bullard
&
Gellman
1954)
where ATlzmZ is a matrix which depends only on the velocity;
where the real Adams-Gaunt and Elsasser integrals are
respectively
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250
J. J.
Love
and
J.
Bloxham
The matrix eq. (68) can be more compactly written in
boldface form
a,t(t)
=
At(t). (72)
After finite differencing
in
time we have, with some
rearrangement
(73)
where
ti,
t,+l,.
. .
denote particular instances
in
time such
that
ti
<
ti+,
for all
i
and the time-step
6t
is constant, that is,
6t
=
ti+l
-
ti
for all
i.
Eq. (73) is a matrix equation
of
the
form
t(ti+,)
=
Dt(fi). (74)
We note that since we are using a steady velocity field
u
at
the CMB, the matrix
D
is independent of time. By recursive
nesting
of
eq. (74) we obtain
t(ti)
=
DD
. . .
Dt(to), (75)
t(ti)
=
M(ti)t(to), (76)
which we represent more compactly as
where the matrix M is calculable. And thus, for an initial
toroidal field t(to) the subsequent evolution
of
the toroidal
field for all subsequent times
t,
is determined.
We seek models of the toroidal field
t;"(t,,)
that
satisfactorily account for the LOD variations via (62) and
are simultaneously consistent with advection in the core as
described by (76). Combining these two equations we obtain
rT(fj)
=
g"(t;)M(t;)t(t<,)
e(t;).
(77)
To test the plausibility
of
electromagnetic coupling, we
might, for example, seek toroidal field models of minimum
strength which also satisfactorily account for the LOD
variations through eq. (77). We measure the agreement with
the data by the misfit
M,
proportional to
x2,
as the root
mean square of the weighted discrepancies, and thus from
(77)
where the standard error of the torque on the mantle is
Er(t).
The time-averaged strength
of
the toroidal field at the
CMB is
9,
where
1
Ti
=
-
C
tT(ti)Nt(ti)
and where
N
is a diagonal matrix
(79)
With eq. (76) we have
(81)
1
@
=
TC
[M(r,)t(t,,)l"[M(t,)t(r,,)l.
I
Thus, the time integral of the strength
of
the toroidal field at
the CMB can be expressed as a function
of
the initial
toroidal field t(to). A suitable model
of
the toroidal field
may be obtained by simultaneously minimizing a weighted
average
of
both misfit and field strength,
so
we obtain
solutions
t;"(t,)
to (62) by minimizing the functional
M2
+
€42.
(82)
By adjusting the trade-off parameter
E
we obtain a
continuum
of
toroidal-field models ranging from those that
are strong and fit the data well, to those that are weak and
fit the data poorly. Deciding which models are acceptable
depends on the amount of misfit we are willing to tolerate,
and on
our
a
priori
expectation
of
what is a physically
reasonable toroidal-field strength.
6
INVERSION RESULTS
To test the plausibility of electromagnetic coupling we seek
time-dependent models
of
the toroidal magnetic field at the
CMB which satisfy the LOD data, are consistent with the
velocity models
of
Jackson
&
Bloxham (1991), and which
satisfy three additional criteria. First, the strength
of
the
toroidal field cannot exceed an approximate upper bound
provided by dynamo theory. Second, the electric currents
which maintain the toroidal field, and which leak to the
Earth's surface, cannot exceed measured values. Third,
ohmic dissipation in
the
mantle cannot exceed a small
fraction of the observed surface heat flow.
The trade-off between the strength
of
the toroidal field
and misfit is shown in Fig. S(a), for a qualitative idea
of
misfit see Fig. 9. The toroidal field necessary to account for
the LOD variations differs only slightly with different
conductivity models, but toroidal field strength
is
highly
dependent
on
misfit. For acceptable misfits, the toroidal
field at the CMB is necessarily very strong; a misfit
of
-1
requires a toroidal field strength
of
-20mT. Are such
toroidal field strengths reasonable? In our heuristic analysis
in
Section 2.2, where we assumed a strong-field dynamo, we
estimated upper bounds on the strength
of
the toroidal field
at the CMB; see Table
2.
In comparison, the toroidal field
strength required for a misfit
of
-1
is
excessive. For a given
toroidal field strength at the CMB we may use (29) to
estimate the field strength in the interior
of
the core.
For
B,
=
20 mT and using the relatively short length-scale (35),
then for mantle conductivity model
3,
the toroidal field in
the interior
of
the core is some 70 times greater than our
magnetostrophic estimate (32), namely
B,
=
2900 mT. This
is greater than the upper bound of -600 mT for the average
field strength set by Backus (1975) using thermodynamic
arguments, and exceeds conventional estimates
-10
mT
based on numerical models of strong-field dynamos (see, for
example, Levy
&
Pearce 1991). For mantle-conductivity
models with less conductance than model 3, or for a weak
dynamo, the conclusion that excessive field strengths are
necessary for a misfit of
-1
is
only
reinforced. If we are
willing to tolerate a misfit
of
-2, then the toroidal field
strength at the CMB must be
-2.5
mT. Such field strengths
are just
on
the edge
of
acceptability, but only
for
mantle
conductances exceeding that
of
model
3.
We thus conclude
that toroidal field models consistent with our analysis
of
electromagnetic coupling are prohibitively strong.
Using a technique first introduced by Runcorn (1964), in
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Core-mantle boundary
251
I
-
-
-
-
-
-
-
-
(b)
-
-
8..
-
-
i~
-
-
\.
E
“\3
-
Measured
Electric
Field]
Strengtp
-
1000
100
10
1
.1
1000
100
10
1
.1
I
-1
1880
1900
1920
1940 1960
1980
Figure
9.
Misfit as defined by eq.
(51)
is equal
to -1
if
on average
the difference between the torque on the mantle and the torque
predicted by
our
toroidal field model is approximately equal to the
standard error of the mantle torque; we have a misfit
-2
if,
on
average, the difference is twice the standard error, etc. Thus, a
misfit
of
-1
is very good, whilst a misfit
of
-3
if
quite poor.
which the potential drop across transoceanic telecom-
munication cables is measured, a surface electric field
strength
of
0.18
mV
km-’
has been determined by
surface electric fields which exceed this measured value.
To
extrapolate the electric field produced by the toroidal field
to the Earth’s surface we combine Ampere’s law
(4)
with
Ohm’s Law (22)
to
obtain
1
P*M
%=
-V
X
B,.
However, the upward continuation of the electric field from
the
CMB
to the surface is
a
sensitive function of mantle
conductivity (Roberts
&
Lowes 1961). The extrapolation is
Figure
8.
The trade-off curves
of
(a) toroidal field strength at the
CMB,
(b)
electric-field strength at the Earth’s surface and (c) ohmic
dissipation in the mantle, each as required by electromagnetic
core-mantle coupling. Strong toroidal fields, which produce strong
electric fields at the Earth’s surface and excessive amounts
of
ohmic
heating in the mantle, are necessary
to
accurately account
for
the
decadal
LOD
variations in
our
analysis. The numbers on each curve
indicate the particular mantle-conductivity model.
0
1
2
3
4
Misfit
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252
J.J.
Love
and
J.
Bloxham
unstable;
if
the conductivity
of
the mantle is not a
monotonically decreasing function
of
radius, then continua-
tion
of
the electric field up from the CMB might yield
surface electric field strengths that do not exceed surface
measurements (Backus 1982). Indeed,
if
the mantle contains
an electrically insulating layer, then the upward-continued
electric field would vanish altogether. Of course the actual
conductivity profile
of
the mantle has not been well
resolved, see Fig.
6,
and whilst we keep
in
mind the
difficulties noted by Backus, for this analysis we consider
only
relatively simple power-law conductivity profiles.
Neither model
1
nor model
3
approximate upper mantle
conductivity, therefore we have constructed conductivity
model
2;
see Table
2
and Fig.
6.
This model, with high
conductivity near the CMB, approximates upper mantle
magnetotelluric models
(for
example, that
of
Parker 1970)
and is
in
agreement with high-pressure laboratory
measurements (Peyronneau
&
Poirier 1989). In Fig.
8(b)
we
show the trade-off between electric-field strength and misfit.
For all
of
the models, the surface electric-field strength
greatly exceeds measured values.
Finally, we consider our third criterion, the heat budget
constraint. Of the observed surface heat flow,
4
x
W
(Gubbins, Masters
&
Jacobs 1979; Sclater, Jaupert
&
Galson 1980), about three-quarters
of
which comes from
radioactivity in the mantle (Stacey 1977); the remainder
comes from a wide variety
of
sources, including remnant
heat
of
accretion, tidal dissipation, latent heat release, and
ohmic dissipation in the core (Gubbins
et
al.
1979). It is
virtually certain, however, that ohmic dissipation in the
mantle contributes only slightly to the Earth's surface heat
flow. We may use the toroidal field strengths obtained
in
Section
2.2
to estimate the ohmic dissipation in the mantle
The heuristic estimates
of
ohmic dissipation
in
the mantle
are shown in Table 2, where we see that the ohmic
dissipation in the mantle should contribute much less than
1
per cent of the observed surface heat flow. These results
should be compared with the inversion calculations in Fig.
8(c). Ohmic dissipation is highly dependent
on
mantle
conductivi'ty, but
for
all three conductivity models the ohmic
dissipation
in
the mantle is prohibitively large for misfits
of
-1.
In fact, for all mantle conductivity models, the ohmic
dissipation exceeds the surface heat flow.
For
more modest
misfits
of
-2, only the most highly conducting mantle
models allow acceptable amounts
of
ohmic heat compared
to the surface heat flow. Moreover, the ohmic dissipation
produced by the toroidal magnetic determined by our
inversion greatly exceeds our heuristic upper bounds for all
mantle conductivity models. Thus, toroidal fields consistent
with our analysis of electromagnetic coupling produce
excessive amounts
of
ohmic dissipation.
To
examine the sensitivity
of
this inversion to differences
in
the models of the core-velocity field, we have performed
the inversion for time-dependent toroidal fields using all
three
of
the velocity models of Jackson
&
Bloxham (1991),
the trade-off curve of strength versus misfit
for
all three
velocity models being shown in Fig.
10.
Since
our
results are
affected
so
slightly by differences
in
the three velocity
l0O0
:
100
10
1
.1
0
1
2
3
4
Misfit
Figure
10.
The trade-off curves
of
toroidal field strength at the
CMB
for the three cases: purely toroidal fluid motions at the core
surface, mixed poloidal-toroidal motions, and geostrophic. Notice
that there is very little difference between the three cases, hence we
conclude that
our
results are not strongly dependent
on
the details
of
fluid motion at the core surface.
models
of
Jackson
&
Bloxham, we conclude that most
of
the
temporal variation of the toroidal field is governed by the
large-scale quasi-steady pattern
of
core flow, the pattern
which is common to all three velocity models. Although
future investigators may wish to re-examine our inversion
technique with time-dependent core-flow models, the results
shown
in
Fig. 10, together with the fact that most
of
the
secular variation
of
the poloidal field can be explained by
steady-core motion, lead us to believe that relatively small
differences in core-flow models will not substantially change
our conclusions.
Why do we find that such strong toroidal fields are
necessary to account for the decadal variations in the LOD?
The answer depends
on
the cancellation
of
the magnetic
stress when integrated over the CMB. In Fig. 11 we show
the spectrum
of
the toroidal field, and in Fig. 12 we show
the spectrum
of
the toroidal torque. Notice that the toroidal
field has structure over many harmonic degrees, the toroidal
field consistent with our analysis
of
electromagnetic coupling
is not simply quadrupolar as is sometimes assumed. As we
demonstrated in Section
3,
advective motions in the core
necessarily produce a complex toroidal field at the CMB,
thus we have no
a
priori
reason to expect simple toroidal
fields. Moreover, for a misfit
of
-1, the average torque for a
particular harmonic degree is substantially greater than the
10" Nm necessary to account for the decadal LOD
variations. The observed torque can only be reproduced by
substantial cancellation when integrating the magnetic stress
over the CMB. A time-dependent electromagnetic torque,
which results from slight non-cancellation
of
the spatially
complex magnetic stress, can thus only be obtained by a
very strong toroidal field.
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Core-mantle
boundary
253
decadal LOD variations. Indeed, steady models
of
core flow
successfully account for most
of
the secular variation
in
the
poloidal magnetic field over the 128 yr time-span
of
our
analysis. The same steady core motions are probably also
responsible for most
of
the time dependence of the toroidal
field.
For
simple toroidal field models, steady motions would
produce a steady trend in the LOD data. But the lack
of
such a trend unaccounted for by tidal deceleration and
postglacial rebound implies that if core-mantle coupling is
predominantly electromagnetic, then the toroidal field must
be complex enough
so
that most
of
the magnetic stress
cancels when integrated over the CMB. Since the toroidal
field
is
relatively slowly varying, a sufficiently large and
variable electromagnetic torque, which results from
only
slight and temporary non-cancellation
of
the magnetic
stress, can only result from a toroidal field
of
considerable
strength.
In an important respect, electromagnetic coupling is
fundamentally different from topographic coupling: even
though time-dependent motions must exist in the core,
time-dependent electromagnetic coupling is probably domi-
nated by processes involving quasi-steady fluid motion at the
core surface, whilst time-dependent topographic coupling,
and for that matter, gravitational coupling, can be
accomplished only by time-dependent fluid motion.
The
quasi-steady flow which, by virtue of our assumption
of
the
frozen-flux approximation, dominates the temporal variabi-
lity
of
the toroidal field, and can only produce the relatively
more rapid fluctuations in the LOD through the slight
non-cancellation of the magnetic stress which results from a
complex and strong toroidal field. In our view, electromag-
netic coupling is an inefficient means
of
producing the
decadal LOD variations, and hence we conclude that it is
probably not the dominant mechanism responsible for these
variations.
Our results should be compared with those
of
Paulus
&
Stix (1989) and Stewart (1991). Paulus
&
Stix assumed
a
priori
that core-mantle coupling is predominantly electrom-
agnetic and used a technique which allowed them to
extrapolate the observed poloidal field through a highly
conducting mantle without making the approximate
expansion (41). Then with a time-dependent azimuthal
velocity field
like
that
of
Stix
&
Roberts (1984), they
calculated a time-dependent advective couple. After
subtracting a constant offset, due presumably to a
steady-leakage couple, they found a mantle conductivity
for
which the sum
of
the advective and poloidal couples
correlate with the torque producing the decadal LOD
variations. However, given that Paulus
&
Stix used a
velocity field which is inconsistent with the secular variation
of the poloidal field, and the arbitrary assumption that the
leakage couple
is
constant in time, their conclusion, that
electromagnetic coupling may account for the decadal LOD
variations, is surprising. Stewart found a correlation
between the torque responsible for the LOD variations and
the sum
of
the poloidal and advective couples. Here too, the
assumption was made that the leakage couple is roughly
constant over the decadal time-scale, and therefore only
contributes a constant offset to the total torque
on
the
mantle, Stewart also subtracted a linear trend from the
torque in order to obtain a correlation, but the physical
basis for this trend is unclear.
b
E
A
c
m
_.
AaAAAAAaA
MISFIT
W
m
2
D:
01
i
4
11
11iI
I
I I
I
I
I
I
I I!
1
2
3 4
5
6
7
8
9
1011121314
TOROIDAL DEGREE
(E)
Figure
11.
The spectrum
of
the
root
mean square
of
the
time-varying leakage field necessary
to
account
for
the
LOD
variations in our analysis. The numbers denote the misfit.
This explanation seems even more reasonable when one
considers the time-scales characterizing the variations
in
the
magnetic field and fluctuations
in
the LOD (see also
Voorhies 1991 and Jault
&
Le Moue1 1991a). Obviously, by
conservation
of
angular momentum, if the LOD is variable,
then fluid motion in the core must be time dependent (Jault
et
al.
1988; Jackson
et
al.
1993). However, as discussed in
Section 2.1, the changes in the motion
of
the core necessary
to accommodate the decadal LOD variations are small
compared to the overall motion of the core. This is hardly
surprising
if
we envisage a core where convection occurs
essentially independently
of
core-mantle coupling.
Moreover, as we discussed, the convective process in the
core changes much more slowly than the relatively rapid
E
z
m
3
0
3
h
L-
W
m
z
e:
t
0
k.
.
1
a
n
MISFIT
01
.
A
3
a
.
na
.
A
0
.
A
i
4
-
3
i
12
3
4
5
6
7
8
9
1011121314
TOROIDAL
DEGREE
(U)
Figure
12.
The spectrum
of
the
root
mean square
of
the
time-varying toroidal toique necessary
to
account
for
the
LOD
variations in
our
analysis The numbers denote the misfit
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254
J.J.
Love
and
J.
Bloxham
Finally, although we agree with the conclusion
of
Jault
&
Le Mouel (1991a) that core-mantle coupling is not
predominantly electromagnetic, we find their argument to
be unsatisfactory. Their conclusion is based
on
time-scales:
they found that the advective couple does not fluctuate
sufficiently to account for the decadal LOD variations, and
that the time dependence
of
the leakage field is dominated
by radial diffusion. Implicitly, they have assumed that
advection does not affect the lateral morphology of the
toroidal field within the core. Indeed, this was one
motivation behind the assumption of quadrupolar axisym-
metric toroidal fields and purely zonal core motion: the
toroidal magnetic field would not be distorted by advection.
As
we
have discussed
in
Section
3,
such simple toroidal
fields and core motions almost certainly do not exist at the
core surface. And if
the
frozen-flux approximation holds
near the CMB, the time dependence
of
the toroidal field in
the core is probably dominated by advection, not diffusion.
Thus, the time dependence
of
the toroidal couple might be
more accurately determined by the horizontal advection
of
toroidal field lines embedded
in
a boundary layer adjacent
the CMB, a process explicitly modelled in our inversion.
7
CONCLUSIONS
Our three-pronged analysis indicates that for electromag-
netic coupling
to
account for the decadal LOD variations
the necessary toroidal fields are too strong, produce
excessive electric fields, and produce an unacceptable
amount
of
ohmic heating. A skeptic might doubt some
aspect
of
any one of these points,
or
might point to
time-dependent core motions
or
diffusion in the core,
neither
of
which we have considered. Indeed, core motion
with time dependency in excess of our estimates in Section
2.1 might produce large advective torques. And as we have
discussed, some investigators (Jault
&
Le Moue1 1991b;
Braginsky
&
Le Mouel 1993) have argued that diffusion may
allow a violation of our assumption that the time
dependence
of
the toroidal field near the CMB is essentially
described by the frozen-flux approximation.
The
importance
of a diffusive boundary layer,
or
even its very existence, has
not been
fully
resolved.
How would consideration of diffusion affect
our
conclusion that core-mantle coupling is not predominantly
electromagnetic? Since the toroidal field is likely to be
relatively slowly varying, a sufficiently large and rapidly
variable electromagnetic torque, which results from only
slight and temporary non-cancellation
of
the magnetic stress
at the CMB, can
only
result from a toroidal field of
considerable strength. In short, the toroidal field does not
appear to change rapidly enough
to
produce the LOD
variations. With the benefit of hindsight, we believe that the
explicit consideration
of
diffusion would probably not
substantially change this conclusion: diffusion may produce
an even slower variation
in
the toroidal field as field lines
slip past moving fluid particles, thus making the rapid LOD
variations more difficult to affect.
Topographic (Hide 1969), and perhaps gravitational
coupling (Jault
&
Le Mouel 1989), probably contribute
significantly to the decadal LOD variations, with many
investigators favouring the topographic mechanism. Howe-
ver, for reasonable mantle conductivities, some electromag-
netic coupling must occur: we have explicitly calculated the
poloidal couple and have estimated the advective couple,
and an acceptably weak toroidal field probably also
contributes a leakage couple. But if electromagnetic
coupling does not by itself account for the decadal LOD
variations, it may play an important role in maintaining a
rough time-averaged equilibrium between the core and
mantle. For example, topographic torques may act to
accelerate the mantle relative to the core, but the Faraday
tension
of
magnetic field lines will act to restore the core
and mantle to rotational equilibrium. Moreover, the
dissipative effects
of
the magnetic field would act to damp
core oscillations (Roberts 1972).
The nature of the coupling between the core and mantle is
of considerable importance to geodynamo theory. The
Taylor-type dynamos (Taylor 1963) assume that there is
no
coupling between the core and mantle; given that the
decadal LOD variations are due to core-mantle coupling,
such dynamo models, whilst of theoretical interest, may not
be completely suitable for the geodynamo. On the other
hand, the model-Z-type dynamos
of
Braginsky (1975, 1990)
depend explicitly
on
core-mantle coupling. However, it
is
important to note that investigations of model-Z dynamos
have, thus far, only considered viscous and electromagnetic
coupling. If, as we believe, core-mantle coupling has a
strong topographic component, then future dynamo
calculations will need to consider the effects
of
an aspherical
CMB. If gravitational coupling is important, then density
heterogeneities in the mantle might play a role in the
dynamo process. Allowance for these effects will be a
difficult task, but may ultimately be necessary in order to
obtain a complete understanding
of
the process responsible
for the Earth’s magnetic field.
ACKNOWLEDGMENTS
While at Harvard, JL was supported by a NASA Graduate
Student Research Fellowship, JB has been supported by the
Packard Foundation and by a NSF Presidential Young
Investigator Award. This work was also supported by
NASA Award NAG5-1369 and by NSF Award EAR90-
18620. We thank Andrew Jackson for inciteful conversa-
tions, and for providing the LOD results in Fig. l(c)
deduced from the core’s angular momentum. And we thank
Paul Roberts for providing numerous helpful comments on
an early verison
of
this manuscript.
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