JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 104, NO. B8, PAGES 17,735-17,745, AUGUST 10, 1999
An estimate of average lower mantle conductivity
by wavelet analysis of geomagnetic jerks
Mioara Mandea Alexandrescu,• DominiqueGibert,= Jean-LouisLe Mou/•l,
GauthierHulot,• and Ginette Saracco
=
Abstract.
It has recently been proposed that geomagneticjerks observed at
the Earth's surfacecould be viewed as singularitiesin the time behavior of the
geomagnetic
field with a regularityof about 1.5 when waveletanalyzed. Sucha
signalshouldhavesufferedsomedistortionwhen diffusingfrom the core-mantle
boundary(CMB) throughthe conductingmantle. Assumingthat the uppermantle
is an insulator and given the electromagnetictime constant of the mantle, we
compute the distortion that a pure singularity introduced at the CMB suffersas
it traversesthe mantle. We compute this distortion through its effects on the
so-calledridge functionsextracted from the wavelet transform of the signal. This
distortion is very similar to the small but significant one that we observein real
data. We therefore speculate that jerks must have been pure singularities at the
base of the mantle and infer an averageestimate for the mantle electromagnetic
time constantfrom the way the signal is distorted by fitting the synthetic ridge
functionsto the experimental ones. Assuming,for example, a thicknessof 2000 km
for a uniform lower conductingmantle, we find an electrical conductivity smaller
than 10 $ m-1. This valueis in reasonable
agreement
with valuesderivedfrom
high-pressureexperimentsfor a silicate mantle.
1.
Introduction
Courtillot and Le Mougl, 1984; Kerridge and Barraclough,1985; Gavoret et al., 1986; Gubbinsand Tomlinson, 1986; Whaler, 1987; Golovkovet al., 1989; McLeod,
Variations of the geomagneticfield are the sum of
external variations (whosesourcesare located in the
1992;Stewartand Whaler, 1992]. Theseformer analionosphereor farther out in the magnetosphere)
and yses have shown the global character and the internal
the so-called secular variation of the main field of interorigin of theseevents. In order to recovermore accunal origin. The secular variation gives information on rate informationabout theseevents(time of occurrence,
the motionsin the fluidcore[e.g.,Hulot et al., 1992]and duration,distribution,andothercharacteristics),
we reprovidestests for the different models of field genera- cently applied a wavelet analysisto geomagnetictime
tion which have been put forward. Also, and this is the
seriesfrom •0 100 observatories[Alexandrescu
et al.,
subject of the present paper, the shortest-periodcom1995,1996].We foundthat seven,andonlyseven,such
ponents of the secular variation reaching the Earth's
eventstook placeduring the 1900-1990period, two of
surfacewhich are not completelyscreenedby the manwhich(1969 and 1978) at least couldbe describedas
tle make it possibleto estimatesomeaveragevalue of
globalin character.The eventsreveala singularbehavior with a regularity(seeequation(27)) closeto 1.5. A
morerecentgeomagnetic
jerk occuringin 1991hasbeen
discussed
by Macmillan [1996]and De Michelis et al.
the secularvariation, whichhave beencalled "geomag[1998]and alsohasa worldwide
character[Le Huy et
its electrical conductivity.
Examination of geomagneticdata from worldwideobservatorieshas revealedsuddenchangesof the trend of
netic jerks" or "secular variation impulses" and have
been discussedby a number of authors [Courtillot et
al., 1978; Malin and Hodder, 1982; Malin et al., 1983;
l Institut de Physique du Globe de Paris, France.
2G•osciences
Rennes-CNRS/INSU, Universit•de Rennes
1, France.
Copyright 1999 by the American GeophysicalUnion.
Paper number 1999JB900135.
0148-0227/99/1999JB900135509.00
al., 1998].
The identification of these geomagneticjerks originatingwithin the core,observedat the top of the mantle, and havinga shorttime constantproduceda flurry
of interest in the topic of lower mantle conductivity.
Ducruixet al. [1980],Achacheet al. [1980, 1981],
Backus[1983], Courtillotet al. [1984],and McLeod
[1994]tookthis observation
as a stimulusto reconsider
mantle conductivity estimates by the theory of electro-
magneticdiffusionthroughthe mantle[Runcorn,1955].
Ducruixet al. [1980]and Achacheet al. [1980,1981]
17,735
17,736
MANDEA ALEXANDRESCU ET AL.' MANTLE ELECTRICAL CONDUCTIVITY
used the secular variation impulse to constrain deep
mantle conductivity "from the bottom" and the 11-yr
responseto the external field to constrain conductivity
"from the top." They proposed that the conductivity is
of sphericalharmonicsY• (0, •),
p(r,t) - • Z s• (r,t)Y•"•
(0,g),
n
(4)
m
everywhere
smallerthan a fewhundredS m-1 Backus where%mcoefficients
lnust satisfy'partial differential
[1983]developedand applieda mantlefilter theory,con- equationsderivedfrom (2) and (3). In the conducting
cluding that an arbitrarily large value of the conductivshell(r• < r _<to) we have
ity is still allowed in the deepest layers of the mantle.
Courtillot et al. [1984],usingthe formalismof Backus
[1983] in relation to the 1969 geomagneticjerk, concluded that, the electrical conductivity is probably lower
than a few hundred S m- • in > 97% of the mantle vol- and in the insulatingshell (to < r_< R•),
ume. Finally, McLeod[1994]deriveda globalgeomagnetic responsefunction consistent with a conductivity
1a
"'("+
o
ofabout,
3 Sm-1 at,thecore-mantle
boundary
(CMB).
Assuminga field of internal origin whosesourcesare
locatedinsidethe spherer- '%, (6) gives
2. Diffusion Through the Conductive
Mantle
n+l
In this section our work is complementary to that.
of previousinvestigators[Smylie, 1965' Backus, 1983'
Bentonand Whaler,1983]who alsoexaminedthe case
of the diffusionof a rapid impulsivechangeproducedat
the CMB. In the following, a similar approach is used
with the view of computing the responseof the mantle
to a jerk signal (seesection5) with a nonintegerregularity. The two-layer mantle model is spherically symmetric and consistsof an internal shell (r• < r _<to) of
conductivity •r and an insulating outer shell (to < r<_
RE) where r• and RE are the radius of the outer core
•,• (/•, t)
ro
C
t).
(7)
•"* (to , t) is more
The link between s•'" ('r•.,t) and %
complicatedto derive due to the presenceof the time
variable in the diffusionequation (5). It, can be shown
[S'mglie,1965]that,the relationshipsoughtis a convolution integral:
m
s•" (r•, •)k•
•Fc
•0
oc
Sn (l'cr,t) -- --
(t - •)d•,
(8)
where the causalconvolutionkernelsk• (t) have the foland of the Earth, respectively. For the sake of sim- lowing form:
plicity, we consider the conductivity of the lower shell
uniform; in fact, what. we are interested in is the electro-
magnetictime constant• tt•r('ro- 're)• of the mantle,
i
Tn,i
where/• is the magnetic permeability of free space.
n•,i and the time constantsv,•,iare such
The problem we address is to obtain the magnetic The coefficients
that
field at the Earth's surface from the time-varying magM• (r•,7-,•-]•)
- 0
(10)
netic field given at. the CMB. The poloida] component
(which is the one of interest,in the presentproblem) of and
-1
the magnetic field B(r, t) in the mantle may be stud(11)
ied independently of the toroidal component and can be
'
ro
dq
derivedfrom a scalar potential p (r, t) [e.g., Le Mou•'l,
1976;Backuset al., 1996]'
1' thefunction/1/I,•
(r, q) is,withina mulwhereq - r-.
n,,,
tiplicative factor, the Laplace transform of %mand sat-
n•i:r•[dM,•(r•,q)
B (r,t)-
V x [V x rp (r,t)],
(1) isfies the differential equation
where r is the radius vector with spherical coordinates
(r, 0, ½). The potential p satisfiesa diffusionequation
dUM,•
n(n+1)M,•(rq)- t_,o'qM,•
(r,q) (12)
in the internal conducting shell:
subject to the boundary conditions'
vP-
Op
< <
(2)
M• (%,q)
-
1,
(13)
and the Laplace equation in the insulating upper part
of the mantle'
vp-0
(3)
dM•)
n
dr r---r•r _ l'cr
(14)
We can find an analytical expressionfor the function
In both shells the potential may be written as a series M• (q,p) [Petiau, 1955;Staylie,1965]'
MANDEA
ALEXANDRESCU
ET AL.' MANTLE
ELECTRICAL
CONDUCTIVITY
s,,'•(r•, t)- g•' (r•)j (t),
17,737
(•9)
wherethe g•,"(re) coefficients
definethe geometryof
the field emitted at the CMB and the function j (t)
describesthe time behavior (i.e., the jerk in sections3
and 5) of this field.
where
3,2- -/•rrq,
H••) andH0(2)are
Hankel
functionsThus the convolutionproduct may be removedfrom
of the first and secondkinds, respectively.
the summationover m, and (18) becomes
By combining(7) and (8), we obtainthe relationship
betweenthe sphericalharmonic coefficientsof the potential a.t the CMB
B(r,t) - E
and at the surface of the Earth:
j(•)k,•(t-•)d•.
n
•
.'•(•,t) -
--
•
ß X7x
•,• (,.•, •).
,'•
ß•.,•(t - •) d•.
(•6)
We nowderiveexpressions
for the magneticfield measured at the Earth's surfaceand created by a magnetic
jerk occurringat the CMB. From the resultsjust de-
again with the condition r > r•. Defining in (r > to),
Pn aS
rived (equations(4) and (11)) we have
u (r,t)
-
X7xr
v x Iv x rp(r,t)]
- (?'rr)n+l
--pn(r)
rcE •,,m(,'•) Um(0, •),
(•7)
(2•)
and the poloidalfield of degreen, B, (r), by
B(r,t) - EEK 7x K7xr --
n. (•) - v • Iv • •p. (•)],
s•"'(r•, •)k• (t - •)d•
(18) (20) may be rewritten as
],
subjec[ [o the condi[ion [hat [he field is computed ou[side [he conductingshell, i.e., r > r•.
We shall assume thai the time variation
(22)
n
of [he coeffi-
At the surfaceof the Earth, (23) reducesto
cien[ss• a[ the CMB is independentof bo[h degree
and order m. This hypothesis is based on [he observations thai [he regulari[y of [he signalsrecorded at
observa[ories
is [he same all over the Earth
n
wi[hin
ß
uncertainty estimates. We can wri[e
j (g) k. (t - g)dg. (24)
0.20
n=l
................n=2
0.15
n=3
n=4
•0.10
0.05
'--,'::4 ....
0.00
0
10
20
................
30
40
50
60
Time (months)
Figure1. Kernels
k. forn - 1,2,3,4, ro- rc- 2000km,andrr- 16Sm-•
17,738
MANDEA ALEXANDRESCU ET AL- MANTLE ELECTRICAL CONDUCTIVITY
a
3
-2
0
500
1000
1500
2000
Time (months)
Figure2.
Thicksolidlineis synthetic
signal
composed
of threejerkslocated
at to -
(500,
1000,
1500)
withregularities
c•= (1.4,1.7,1.5)andamplitudes
• = (+l.0,-0.5,+1.5).The
thinsolidlinesareextrapolations
ofthe signal
obtained
byextinguishing
several
jerks'linea is
whatthesignal
would
have
been
in theabsence
ofjerksl, 2, and3' lineb iswhatthesignal
would
havebeenif jerks2 and3 wereabsent,
andlinec is whatthe'signal
wouldhavebeenif
jerk 3 was absent.
wavele[
[ransform
6
_10 4
0
500
1000
1500
2000
Time (months)
Figure3. (top)Wavelet
transform
ofthe(bottom)
signal
shown
inFigure
2. Theanalyzing
wavelet
isthethirdtimederivative
ofa Gaussian
(see
Alexandrescu
etal. [1995]
fordetails).
Observe
thethreecone-like
patterns
converging
toward
thestarting
dates
ofthethree
jerks
present
in theanalyzed
signal.Thesolidlinesarelinesofextremaof thewavelet
transform
from
whichthe ridgefunctions
shownin Figure4 areextracted.
MANDEA
ALEXANDRESC[
ET AL
MANTLE
ELECTRICAL
CONDUCTIVITY
17,739
3. Ridge Functions
The data usedin this study are a set of ridgefunctions
(seesection4), andwhat we actuallyhaveto produce
for comparison
are syntheticridgefunctions.A ridge
function is a subset of the continuous wavelet trans-
form of a signalf (t) whichcontainsthe information
usefulfor characterizing
abruptvariationsin the signal
3
[Grossmann,
1986].The continuous
wavelettransform
1
of f (t) is definedby the convolution
product
-1
[;•'f
[[,
a]--/_-1-•
f(•)•[;a
(t--•)d•, (25)
,.
wherethe wavelets½'a(t) are obtainedby dilating an
analyzingwavelet 0 (t),
.
l½(t)
(26)
-- .,
with a dilation a > 0. In the present study the ana-
lyzingwaveletis the third time derivativeof a Gaussian as already used by Alexandrescu
et al. [1995,
1996]. Amongthe numerousattractivepropertiesof
the wavelettransform(see.for instance,Holschneider
-1
[1995]),weexploitthe fact that the wavelettransform
acts like a mathematical zoom lens useful for analyz-
ingthe localhomogeneity
characteristics
(i.e., the local
1 500
.-
self-similarbehavior) of a signal.
In the presentstudy we shallfocuson so-calledjerk
signalswhosegeneralexpression
is
1 515
j(tl•,•,t,,)
-1
1
2
3
4
5
6
Log2(dilation)
-
/3n(t-to)(t-to)
•,
(27)
where H (t) is the Heavisidedistribution, fi is an amplitude factor, t0 is the starting date of the jerk, and a
is its regularity.The larger the regularity,the smoother
the variation of the fi•nction.
Let, us, for instance, con-
Figure 4.
Ridge functionsr (a) extracted from the sider a signal f (t) composedof three jerks:
wavelettransformshownon Figure 3 and corresponding
to thejerks labeled(top) l, (middle)2, and (bottom)3
f (t) : j (tl•.4, t, 500)+ j (tll.7,-0.5, looo)+
in Figure 2. The solid lines are the ridge functionslo+j (tll.5, 1.5,1500).
(28)
cated on the left part of the conepattern of the wavelet
transform, and the dashed lines are the ridge functions
located on the right. The numbers are the slopesob- This signal is shown on Figure 2, and its wavelet transtained by least squaresfitting of a straight line. Observe form is displayed on Figure 3. We observe that the
that these slopesfall very near the theoretical values wavelet transform has three cone-like features point(1.4, 1.7, and 1.5 respectively).
ing (zooming) toward the starting dates to of the three
jerks composingthe analyzed signal. It can be shown
[Alcxandrescu
et al., 1.9.95;Holschneider,1.9.95]that
each cone-like pattern possessesself-similar properties
and that (24) may be usedto accountfor the geometry controlled by the homogeneity of its related jerk and
of a time-varying field measured on the Earth. In this that. usefulinformation can be recoveredby simply takway we incorporatethe spatial structure of the 1969jerk ing the absolutevalue of the wavelet transform along
into our synthetic data. This is important in practice only one of the lines of extrema of the cone-likepattern
since, depending on the geographicalposition on the (Figure 3). The set of absolutevaluesof the wavelet
Earth, the relative amplitudes of the Bn can vary and transform on a given line of extrema and ranked with
the temporal variations of the field B may differ from respect to the dilation a defines what will be referred
one place to another, since the kernels k,• display some to as a ridge functionr (c•). When plotted on a log-log
diagram, the ridge functions are straight lines whose
dependenceupon the degree n, as shown in Figure 1.
Note that Bn are also taken at the surface of the Earth
17,740
MANDEA ALEXANDRESCU
ET AL.' MANTLE ELECTRICAL
CONDUCTIVITY
Log2(dilation)
Log2(dilation)Log2(dilation)Log2(dilation)
Figure 5. Thin linesareobserved
ridgefunctions
for the 36 observatories
considered
in this
study.Solidlinesare partsof the ridgefimctionsusedin the inversion.
slopesequal the regularity c• of the corresponding
jerk
(Figure4).
wavelet transform computed for a linear combination of
the X (north) and Y (east)components
of the field,and
the regularity a was found to vary between 1.5 and 1.7.
4.
Data
The data set used in this study is composed of the
ridge functions (Figure 5) correspondingto the 1969
jerk and obtained through the wavelet analysisof the
monthlymean seriesof 36 observatories
[Alezandrescu
The uncertaintyassociatedwith estimatingthe regular-
ity in thiswetyis about10%[Alexandrescu
et al., 1996].
The analyzed signal for the lth observatoryis
Ht (t) = qt ß Bt (t),
(29)
½tal., 1995, 1996].We focuson the 1969eventbecause where Bt (t) is the magneticfield at the /th observait is the best documented and appears to be less af- tory and qz is a unit horizontal vector pointing in the
fectedby fieldsof externalorigin than the others[see direction of the jerk vector, estimated with an error of
--• 5o [seeAlcxandrescu
et al., 1996];the directionof
Alexandrescu
et al., 1995, 1996]
this
vector
is
given
in
Table
1 for each observatory.
The geographicalcoordinatesof the observatories
and
their International Association of Geomagnetism and
5. Synthetic Jerks
Aeronomy (IAGA) codes are listed in Table 1, and
their global distribution is shownin Figure 6. For each
Alexandrescu
et al. [1995, 1996]supposedthat the
observatorythe ridge function was extracted from the jerks observedat the Earth's surfacewere pure singu-
MANDEA
ALEXANDRESCU
ET AL.- MANTLE
ELECTRICAL
CONDUCTIVITY
17,741
Table 1. ObservatoriesConsideredin the Present Study
Code
a
Name
'Xb
•
c
•3
d
AQU
L'Aquila
42.383
13.317
80
BFE
Brorfelde
55.625
11.672
80
CLF
COl
Chambon
Coimbra
2.260
90
EBR
Ebro
la Forat
48.023
40.222
40.820
351.583
90
0.493
100
80
ESK
Eskdalemuir
55.317
356.800
FRD
Fredericksburg
38.205
282.627
50
FUR
GDH
Furstenfeldbrfick
Godhavn
48.165
11.277
80
69.252
GNA
Gnangara
306.467
115.950
80
355.517
284.670
104.450
90
80
60
HAD
Hartland
HUA
IRK
KNY
Huancayo
Pattony
Kanoya
LER
-31.783
50.995
-12.045
52.167
90
31.420
130.882
110
Letwick
60.133
358.817
100
LNN
LRV
Voyeykovo
Leirvogur
59.950
30.705
80
64.183
338.300
90
LVV
Lvov
49 900
23.750
90
MEA
Meanook
54 617
MMK
MOS
NGK
NUR
ODE
PAG
Loparskoye
Krasnaya Pakhra
Niemegk
Nurmijarvi
Stepanovka
Panagyuriste
68 250
246.667
33.083
110
100
55 467
37.312
110
52 072
12.675
90
60 508
24.655
90
46 783
30.883
100
SIT
Sitka
42 512
57.058
24.177
80
224.675
110
SJG
San Juan
18.113
293.85
90
SOD
Sodankylii
67.368
26.6300
90
SUA
TFS
TRD
TUC
VAL
Surlari
Dusheti
Trivandrum
Tucson
Valentia
44.680
26.'253
90
42.09'3
44.705
76.950
70
160
249.167
349.750
80
100
WIK
WIT
Wien Kobenzl
Witteveen
48.265
52.813
16.318
6.668
90
WNG
Wingst
53.743
9.073
90
8.483
32.247
51.933
80
aAccording to the IAGA convention.
bLatitude of the observatory,in degrees.
CLongitudeof the observatory, in degrees,positive eastward.
aJerk direction,in degrees,positive eastward.
larities of the form given by (27) with straight ridge a straight line when plotted in a log-logdiagram. An
exampleof this situation (computedfor the Chambon
functions(Figure 5), one observesa small downward la For•t observatory coordinates, see below) is shown
curvature for many observatories. Such a curvature in Figure 7 for ro- r• = 2000 km, a = 1.5, and
that the ridgefunccould be due to diffusionof the signal through the con- •r = 1,4, 16,64S m-• We observe
functions. Actually, when lookingat experimental ridge
ductive mantle. We therefore speculate that the signal tions present a downward curvature that increaseswith
j (t) emittedat the CMB is a purejerk asgivenby (27) increasing•r. An intuitive assessmentof this feature
and that the curvatures observedin the ridge functions can be made by observingthat the transfer function is
of the surface data are caused by diffusion modeled by a localizedfilter. Then, for su•ciently large dilations of
the convolutionequation (24). The transfer functions the wavelet transform the filter is well approximated by
k, (t) are causalpositivefunctions(Figure l) and act a Dirac distribution, and the ridge function is asympas low-passfilters whosecutoff frequencyis controlled totically a straight line whoseslope is the exponent a of
by the electromagnetic
time constant• /•r(ro - r•) 2 the jerk produced at the CMB. For smaller dilations inand by the degree n of the elementary harmonic con- volvingshortertimescalesthe finite duration of the filter
sidered. The larger the electromagnetictime constant, cannot be ignored, and the signal is both smoothedand
the stronger the filtering, and the ridge functions ob- damped. Accordingly, this smoothing results in a larger
tained through a waveletanalysiswill depart more from regularity at the timescalesconsidered,and the corre-
17,742
MANDEA ALEXANDRESCU ET AL.: MANTLE ELECTRICAL CONDUCTIVITY
... • ..........................................
•'•...................... 45
45'
Figure 6. Distributionof the 36 observatories
considered
in the presentstudy.
spondingpart of the ridgefunctionhas a larger slopeat
the correspondingdilations. The ridge function of the
surfacejerk is therefore a curved line located below its
asymptotewhich is the linear ridge function of the pure
singularity jerk at the CMB.
In order to completethe expressionof the synthetic
tion for two reasons:(1) the analysisis performedfrom
the data of the observatories,includingthe diffusioneffect whichwe are trying to characterize;and (2) owing
to the small number of observatories, the coe•cients
of the sphericalharmonic expansionare not very accurate. Nevertheless,this expansionis acceptablefor the
jerk to be comparedwith the observedone, through needsof the presentinvestigation(a two-stepcomputatheir respectiveridge functions,we need the harmonic tion couldbe performed,i.e., recomputingthe B, after
field coe•cients B, We take as B,the results of the
a first determination of the singularity at the CBM,
analysisof Le Hu.y ½t al. [1998]. This study is basi- but this appearsto be unnecessary).The expansionof
cally a sphericalharmonicexpansionof the jump of the Le Huy et al. [1998](limited to orderand degree4)
first time derivative of the field from the jump observed compareswell with previousstudies[Malin and Hodat the observatories. This is only a first approxima- der, 1982; Gu.bbins,1984]. Moreover,at eachobserva-
regularity
=1.50
10
.-'
(16)
//
1
(64 S/m)
2
3
4
5
Log2(dilation)
Figure7. Ridge
functions
obtained
fora surface
jerk(corresponding
to c•- 1.5at theCMB)
filteredthrougha conducting
shellwith ro- rc- 2000km andfor er- 1, 4, 16, and64 Sm-1
(seetextfordetails).Thecurvature
of theridges
is morepronounced
forlargerconductivities.
MANDEA
ALEXANDRESCU
ET AL.: MANTLE
:48.•8
ELECTRICAL
'J•'
•
CONDUCTIVITY
17,743
'
--25.5
1
10
100
1000
Electrical conductivity (S/m)
Figure 8. Misfit functionC'(G,rr) (equation(33)) obtainedby fitting syntheticridgefunctions
to the experimental ridge functions of 36 observatories.The symbol m is located at the absolute
minimum
(22.4)whose
coordinates
arec•= 1.$$,andrr-- 8 Sm-1.
tory the directionsqt of the jerk computedusing (29)
gradients, especially toward high conductivities. The
are close to those obtained by the wavelet analysis of
the data. The synthetic ridge functions have been com-
valley widens toward small conductivities with a rather
fiat bottom indicating that the domain of acceptable
puted from syntheticseriesobtainedfrom (24) subject solutions extends itself on the left of the absolute minto the sameprojectionas in (29):
imum. Values of G along the valley axis get smaller
as rr increases,leading to an increasinglymore singular
Z qt' B•(RE,Or,
source:
ß
j (•1G,/•, to)'
ßk,• (t - •) d•,
for G =
i the first
time
derivative
is discon-
tinuous at to, implying an unrealistic discontinuity in
the velocity flow at the CMB responsiblefor the jerk
n
[Le Huy et al., 1_998].If we choosea regularity as high
(30)
as possible, the conductivity rr must be taken smaller
than 10 Sm-•. In any case,it is safeto saythat on
where(0t, qh)are the coordinates
of the/th observatory the basisof the misfit criterion we have adopted, the
and the directions of the unit vectors qt are given in conductivityrr is lessthan a few tensof Sin-•.
We have up to now considereda 2000 km thick conTable i (as noted, they are very closeto thoseof Le
Huy et al. [1998]).
ductiveshell, but an infinite numberof acceptablesoEquation(30) hasbeenusedto computea collection lutionsexistwhichbelongto the classof conductivity
of ridgefunctionsrt,•,,,,(a) for the values(0t,½t)of the distributionshavingthe samekernelsk,• (n = 1,4).
coordinates of each of the 36 observatories and for r•, -
Limiting ourselvesto the subclassof constant conducti-
rc- 2000km,G G [1.0,2.0]
andrr G [1,1024Sm-1]. vity distributions,the curvesshownin Figure 9 give
Then, for each(G,rr) pair belongingto theseintervals, the (to - re, rr) pairsequivalent,from the point of view
a misfit was computed:
of the presentcomputation,to (r•- rc = 2000 km,
cr-- 8 S m-1). Of course,
the computations
couldbe
C(a,rr)- • E [r,(aj)- r,,a,a
(aj)]2
t
(31)
extendedto any regularrr(r) distribution.
j
where the index j runs over the discrete dilations for
whichthe ridgefunctionof a givenobservatoryis avail-
6.
Discussion
The presentstudy is basedon a simplistictwo-shell
able. A contourmap of the misfit function C
model of the Earth, the outer insulating and the inis shownin Figure 8. The absoluteminimum(22.4) ner having a constant conductivity. It is known that
is locatedat the pointc• = 1.55andrr = 8 S m-1. the conductivityof the upper mantle is very low (<
One can observethat the isovaluesof C (G, rr) define 1 S m-1) [Achache
et al., 1981;Constable,
1993;Pea well-marked valley of minime[ surrounded by steep tersonsand Constable,1996], and it is likely that the
17,744
MANDEA
ALEXANDRESCU
ET AL'
MANTLE
ELECTRICAL
}"•
..........
x'.
n=1
.....
-.............n=2
x .' ....
- ....
•.. .......
lOO
CONDUCTIVITY
n=3
n=4
,•,:-..
o
lO
1•0
•00
1000
1000
Thicknessof the conductivelayer (km)
Figure 9. Relationshipbetweenthe thicknessr(,- re of the conductive
layerand its conductivity
rr givingthe sametransferfonctionsk•(t) as thoseobtainedwith ro- rc = 2000km and rr =
8 Sm-•. For example,for n = i the diffusiveeffectsof a layerwith ro- rc = 2000 km and
rr= 8 Sm-• andof a layerwithro- rc = 500km andrr= 108Sm-• areidentical.
conductivity distribution (as a function of depth) of the
The assumptionthat the jerk originated as a singularity
silicatelowermantle is regular [Shanklandet al., 1993]. introduced at the base of the mantle is the simplest
Our upper bound estimate bears in fact on the electromagnetic time constant r of the mantle, transla,ted in
possible.
terms of the conductivity of an inner shell. Of course,
Acknowledgments.
We thank David Barraclough and
Cathy Constable for cmmnents. This study has been partly
funded by the CNRS-INSU progrmn "Intdrieur de la Terre."
most
of the
contribution
to
r could
be concentrated
in a thin layer at the bottom of the ma.ntle (reducing
This
is IPGP
contribution
1599.
(to- r•)""andincreasing
rr). In orderto establish
the
existenceof such a layer, other kinds of evidence have to
be found. Buffett et al. [1990],for example,proposed References
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D. Gibert, and G. Saracco, G•osciences Rennes, Universit• de Rennes 1, Bat.
15 Campus de Beaulieu, 35042
Rennescedex,France. ([email protected];
[email protected])
G. Hulot, J.-L. Le Mou[l, and M. Mandea Alexandrescu,
Institut de Physique du Globe, B.P. 89, 4 Place Jussieu,
75252 Paris cedex 5, France. ([email protected];
[email protected];
[email protected].
fr)
(ReceivedOctober20, 1998; revisedMarch 24, 1999;
acceptedApril 2, 1999.)