Magnetostrophic MRI in the Earth's Outer Core

0.31 MB·English

by  Ludovic Petitdemange

GEOPHYSICALRESEARCHLETTERS,VOL.???, XXXX, DOI:10.1029/,
Magnetostrophic MRI
in the Earth’s Outer Core
1,2 1,3 1,2
Ludovic Petitdemange, Emmanuel Dormy, and Steven A. Balbus
We show that a simple, modified version of the Mag-
netorotationalInstability(MRI)candevelopintheouter
liquidcoreoftheEarth,inthepresenceofabackground
9 shear. It requires either thermal wind, or a primary in-
0 stability, such as convection, to drive a weak differential
0 rotationwithinthecore. TheforcebalanceintheEarth’s
2 core is very unlike classical astrophysical applications of

n theMRI (such as gaseous disks around stars). Here,the
a weak differential rotation in the Earth core yields an in-
J stability byits constructiveinteraction with theplanet’s

9 much larger rotation rate. The resulting destabilising
mechanism is just strong enough to counteract stabiliz-

] ingresistiveeffects,andproducegrowthongeophysically
P
interesting timescales. We give a simple physical expla-
E
nationoftheinstability,andshowthatitreliesonaforce
.
h balance appropriate to the Earth’s core, known as mag-
p netostrophic balance.
-
o
r
t
s 1. Introduction
a
[ TheMagnetorotationalInstability(MRI)isimportant

for differentially rotating astrophysical objects such as
1
gaseous disks, because it forces a breakdown of laminar
v
flowintoturbulence,producingtheenhanceddissipation
7
and transfer of angular momentum necessary for accre-
1
tion of matter onto the central massive object (Balbus
2
andHawley,1991). Diskinvestigatorsbeganwithpurely
1
hydrodynamicalcandidatemechanisms,butnowthecen-
.
1 ter of interest is squarely upon magnetohydrodynamics.
0 Interest in the Earth’s core, by contrast, has been mag-
9 netic almost from the start. The principal problem, of
0 course, has been to understand how fluid motions in
: the core generate a magnetic field. This is often ap-
v
proached via kinematic processes in which a hydrody-
i
X namically turbulent fluid exponentially amplifies a very
weak seed magnetic field. Here, we argue of a version of
r
a theMRI,dynamicallycouplingboththevelocityandthe
magnetic field, can also provide a mechanism for linear
instability in the Earth’s core. While this mechanism is
not meant to serve as the primary dynamo process, it
could be an important source of secondary instabilities
and magnetic secular variation.
1MAG(ENS/IPGP)
LRA,D´epartement dePhysique
EcoleNormaleSup´erieure
24,rueLhomond
75231ParisCedex05,France.
2LERMA,ObservatoiredeParis,CNRS/UMR8112.
3InstitutdePhysiqueduGlobedeParis,
CNRS/UMR7154.
Copyright2009bytheAmericanGeophysical Union.
0094-8276/09/$5.00
1
X - 2 PETITDEMANGEET AL.: MRI INTHE EARTH’SOUTERCORE
We are hardly the first authors to study the effects
of differential rotation on the dynamical stability of the
geodynamo (see e.g. Acheson, 1983; Ogden and Fearn,
1995;Fearnetal. 1997). Butinpreviouscalculationsthe
emphasis has been upon purely azimuthal fields, nonax-
isymmetric disturbances, and magnetic instabilities. In
this work, the dynamical focus is much different. Here
themagneticcouplingistothepoloidalfieldcomponents,
axisymmetric disturbances are front and center, and the
instability, while relying on the presence of a magnetic
field, has its seat of free energy entirely in differential
rotation. The MRI is a somewhat novel concept in this
context,andisworthyofstudyinisolation. Fortunately,
it can be understood in very direct and simple physical
terms.
TheEarth’s coreis, bycomparison toaccretion disks,
arelatively small object, in which resistiveeffectsareon
anequalfooting withdynamicalprocesses. Therotation
propertiesoftheEarthalsosignificantlydifferfromthose
of an accretion disk. To leading order, they correspond
tosolidbodyrotation,withonlyaweakdifferentialrota-
tion. Atfirstsight,itisfarfromobviousthattheEarth’s
core is a venuefor theMRI.
The purpose of this Letter is to show that, despite its
weakdifferentialrotationandsignificantresistiveeffects,
the Earth’s core can in fact host the MRI. We derive
a WKB dispersion relation relevant to this asymptotic
regime (i.e. magnetostrophic balance), and demonstrate
excellent agreement between this local description and
global numerical simulations in spherical geometry. The
final section speculates on potential applications of our
results.
2. Modelling
ThephysicalparameterregimerelevanttotheEarth’s
outer core dynamics is relatively well constrained. In
order to investigate the MRI in a conducting fluid, it
is also necessary to provide a reasonable approximation
to the basis flow profile over which the instability may
develop. Apart from very restricted components of the
velocity field, imaging of the core flow is limited to sur-
faceflows(e.g. Bloxham&Jackson1991; Amit&Olson,
2004; Eymin & Hulot, 2005). Moreover, while secular
magneticfieldvariationsallowforanorder-of-magnitude
estimate for the zonal flow (Pais and Hulot 2000), it is
not possible at present to achieve a direct, detailed or
reconstructive imaging of the global differential rotation
of the Earth’s core. Asymptotic studies early identified
the essential role of the zonal shear in the geodynamo
(Taylor, 1963), and this has remained central to our un-
derstandingofoutercoredynamics. Numericalmodelsof
convectioninrotatingspheres(e.g. Christensen,2002)or
inconvectivelydrivendynamos(e.g. Aubert,2004), also
observed the presence of a strong zonal shear. Besides
surface flow reconstructions, the only (indirect) observa-
tionalevidenceforsuchshearisinferred from seismolog-
ical data that have been interpreted as rotation of the
solid inner core at a rate of about 0.15o per year rela-
tivetothemantle(Vidale,Dodge&Earle,2000;seealso
Dumberry 2007). But is this change in the angular ve-
locity profile spread across thewhole core radius or,say,
localised in a narrow shear layer? There are no observa-
tional constraints on theway thecorresponding jump in
angular velocity is actually accommodated by theflow.
PETITDEMANGEET AL.: MRI INTHE EARTH’SOUTERCORE X - 3
3. Magnetostrophic-MRIlocaldescription
Considerthestabilityofadifferentially rotatingfluid,
withangularvelocityΩafunctionofsandzinastandard
cylindrical coordinate system (s,φ,z). The density ρ,
pressure P, magnetic field B are functions of s and z
in the equilibrium state, and of course depend on time t
as well when the equilibrium is perturbed. The fluid is
characterizedbyakinematicviscosityν andresistivityη.
We work in theWKB limit, assuming that all linearized
perturbations havethespace-time dependence
Q exp(ik s+ik z+σt) (1)
s z
∝
where Q is an infinitesimal (Eulerian) disturbance in a
fluid quantity. Our starting point is the general disper-
sion relation of a constant density fluid adapted from
Menou, Balbus, & Spruit (2004):
k2 1
k2ση4ν −ση2η s3 D(s4Ω2) −4Ω2(k·VA)2 =0, (2)
z h i
e e
where
k ∂ ∂
= s , (3)
D (cid:16)kz ∂z − ∂s(cid:17)
and
ση2ν = (σ+ηk2)(σ+νk2)+(k·VA)2 , (4)
(cid:2) (cid:3)
e ση2η = (σ+ηk2)2+(k·VA)2 . (5)
The Alfv´en velocit(cid:2)y V is defined as (cid:3)
A
e
B
VA = (6)
√µρ
where µ is the magnetic permitivity. Finally, we note
that Ω is the full angular velocity, i.e. if Ω denotes the
0
Earth rotation rate, and V the velocity in the rotating
frame, then Ω=Ω +V /s .
0 φ
ToseewhethertheunstableMRImodescanbepresent
in a very simple model, we consider the case of an ini-
tial poloidal magnetic field which is locally vertical and
Ω is a function only of s. (We stress here that we use
a vertical magnetic field for the sake of simplicity of the
presentation, but that the overall conclusion can be eas-
ily extended to the case of more general applied fields.)
Then
dΩ2
(s4Ω2)= 4Ω2+s κ2. (7)
D −(cid:18) ds (cid:19)≡−
For the current application, the shear gradient is very
small compared with 4Ω2, and κ, which is known as the
“epicyclic frequency” in the astrophysical literature, is
very nearly 2Ω. With these assumptions, the dispersion
relation becomes
k2 (σ+ηk2)(σ+νk2)+(k·VA)2 2+4Ω2(σ+ηk2)2+(k·VA)2sdΩ2 =0.
k2 ds
z
(cid:2) (cid:3)
(8)
This is the form that we have used (see below) for com-
parison with numerical simulations.
Ingeodynamoapplications, theviscosity isverysmall
compared with the resistivity (the ratio is 10−5 10−6)
−
and it may bedropped. The instability of interest arises
X - 4 PETITDEMANGEET AL.: MRI INTHE EARTH’SOUTERCORE
from the final term of this dispersion relation, if the an-
gular velocity decreases outwards. This is a small term
in the sense that Ω is regarded as large, and we expect
thatthegrowthrateσ willitselfbeverysmallcompared
withΩ. Wemaythereforedropallσ termsfromourdis-
persionrelation,exceptforthosewhicharemultipliedby
Ω2. (This procedure may be formalized by normalizing
σ with respect to sdΩ/ds and then treating
| |
ε s dΩ 10−6,
≡ Ω ds ∼
asavanishinglysmallexpansionparameter.) Weobtain
k2 dΩ2
(k·VA)4+4Ω2(σ+ηk2)2+(k·VA)2s =0(.9)
k2 ds
z
This relatively simple dispersion relation has a cor-
respondingly simple physical interpretation. Consider
magnetostrophic balance with an axial wave number
(k=kez),
1
2Ω×V = ∇Π+ (B·∇)B, (10)
− µρ
where Π includes the material and magnetic pressures
as well as the centrifugal potential, along with the full
induction equation
∂
+V·∇ B =(B·∇)V +η 2B. (11)
∂t ∇
(cid:16) (cid:17)
The equilibrium profile is to leading order V = s[Ω +
φ 0
(s s0)dΩ/ds], and B = Bez. Consider linear velocity
−
perturbations v and v , and magnetic field perturba-
s φ
tions b and b of the form exp(σt+ikz). In magne-
s φ
tostrophic balance, the radial and azimuthal linearized
equationsare respectively
ikB
2Ωv = 0 b , (12)
− φ µ0ρ s
ikB
2Ωv = 0 b . (13)
s µ ρ φ
0
Thesame components of theinduction equation are
(σ+ηk2)b =ikB v , (14)
s 0 s
dΩ
(σ+ηk2)b =ikB v +s b . (15)
φ 0 φ ds s
Combining (14) and (13) leads to
1
(σ+ηk2)b = (kV )2b , (16)
s −2Ω A φ
whereas (15) and (12) imply
(kV )2 dΩ
(σ+ηk2)b = A +s b . (17)
φ (cid:20) 2Ω ds(cid:21) s
Noticetheabsenceofpressure-likeperturbationsforaxial
wavenumbers. Thelast two equations combine to give
dΩ2
(kV )4+4Ω2(σ+ηk2)2+(kV )2s =0, (18)
A A ds
which is just equation (9) for axial wavenumbers.
By restricting the perturbations exclusively to the
PETITDEMANGEET AL.: MRI INTHE EARTH’SOUTERCORE X - 5
magnetic field components, equations (16) and (17) pro-
duce a clear picture of the instability with no “out-of-
phase” terms. Consider an outward radial displacement
and its associated radial magnetic field. If kV is not
A
too large, the radial field is significantly sheared by the
differentialrotationtoproduceanegativeazimuthalelon-
gation of the field line following equation (17). This
results in a positive azimuthal magnetic tension force,
which must be balanced in the magnetostrophic regime
withanopposingCoriolis force, hencewithayetgreater
radially outward displacement and magnetic field (16).
An instability is at hand. This mechanism is illustrated
in Figure1,andseen throughadirect simulation in Fig-
ure 2.
The growing solution for σ is
σ= |k·VA| sdΩ2 k2(k·VA)2 1/2 ηk2.(19)
2Ω (cid:20)(cid:12)(cid:12) ds (cid:12)(cid:12)− kz2 (cid:21) −
(cid:12) (cid:12)
Wewishtofindt(cid:12)hema(cid:12)ximumgrowthrateoftheinsta-
bility, which will be associated with a particularly wave
vector k. The easiest way to proceed is to use the vari-
ables
k2
X =k·VA, Y = , (20)
k2
z
withtheunderstandingthatX >0,andtheparameters
V2 B2 dΩ2
Λ= A = 0 , a= s (21)
2ηΩ 2µ ρηΩ (cid:12) ds (cid:12)
0 0 0 (cid:12) (cid:12)
(cid:12) (cid:12)
ΛistheElsassernumber,whichistypically(cid:12)oford(cid:12)erunity
for theEarth’s core. Then
2Ωσ =X(a X2Y)1/2 X2Y/Λ. (22)
− −
This is a decreasing function of Y everywhere, with a
maximum at the lower boundary Y = 1. This means
that themost rapidly growing wavenumber k lies along
z
therotation axis. Thus
2Ωσ =X(a X2)1/2 X2/Λ. (23)
(max) − −
Requiringthepartial derivative∂/∂X tovanishleads to
thepolynomial
a2
X4 aX2+ =0. (24)
− 4(1+Λ−2)
The physically meaningful wavenumbersolution is
X2 (k V )2 =Ω sdΩ 1 (1+Λ2)−1/2 , (25)
≡ z A (cid:12) ds(cid:12) −
(cid:12) (cid:12)(cid:2) (cid:3)
and corresponds to a gro(cid:12)wth r(cid:12)ate of
dΩ Λ/2
σ= s . (26)
(cid:12) ds(cid:12)1+√1+Λ2
(cid:12) (cid:12)
(cid:12) (cid:12)
It is of interest to examine the range of k in which
z
instability exists. For the case k = k , the dispersion
z
relation(9)admitsgrowingsolutionsforvaluesofk2 less
z
than
sdΩ2/ds
k2 = | | . (27)
z V2+4Ω2η2/V2
A A
X - 6 PETITDEMANGEET AL.: MRI INTHE EARTH’SOUTERCORE
Notice that when V is very large, k is very small be-
A z
cause large k perturbations are stabilized by magnetic
z
tension forces. Conversely, when V is tiny, resistiv-
A
ity damps large wavenumber perturbations. The max-
imum value of k allowing the greatest range of unsta-
z
ble wavenumbers corresponds to Λ=1. Hence, Elsasser
numbers of order unity naturally emerge in an MRI-
influenceddynamo.
At this point it is helpful to have some explicit num-
bers. WetakeΩ=7.27 10−5 s−1 and sdΩ/ds 10−10
s−1,thelattercorrespon×dingtoalowerboundfor≃thean-
gularvelocity gradientinwhichthecharacteristiclength
scale associated with shear in the core is the outer core
radiusitself (worse case MRI scenario). With Λ=1 and
η 1 m2 s−1, the Alfv´en velocity is 0.012 m s−1. From
≃
equation (25), we find
kV 4.6 10−8 s−1
A
≃ ×
which translates to a wavelength of some 1600 km. The
characteristic growth time from equation (26) is then
about 1500 years. If, on theother hand,we assume that
asimilarjumpinangularvelocityisaccomodatedaccross
anarrowshearlayer(suchastheStewartsonlayer,Stew-
artson1966),thenthecharacteristicgrowthtimecouldbe
as short as a year.
What is the relationship between the unstable mode
consideredhereandthemaintenanceoftheclassicalTay-
lor constraint? It is well known that in the magne-
tostrophic limit, the geostrophic component of the flow
mustadjusttoensureTaylor’sconstraint(Taylor,1964).
Amodificationofthenon-geostrophicflowsimilartothe
one envisioned here could in principle rapidly alter the
zonal shear on which the instability itself relies. How-
ever, in the particular configuration investigated, the
wavenumber is axial. Thus, at least in the linear phase
of the disturbance, the Taylor constraint remains unaf-
fected.
4. Direct numerical simulations
Todemonstratenumericallytheabovemechanism,we
base our study on a very idealized model of the Earth
core, chosen for illustrative purposes. We consider sim-
plesphericalCouetteflowdrivenbyenforcingdifferential
rotationbetweentheinnercoreandthemantle. Theap-
plied magnetic field B is vertical and uniform. In the
0
simpler hydrodynamical case, a strong shear layer will
developonthecylindertangenttotheinnercore(Proud-
man,1956;Stewartson,1966). Withtheparticularchoice
ofaverticalappliedfield,thisshearisonlyslightlymod-
ified by MHD effects (see however Dormy et al., 1998).
In what follows, we will work in time units of (2Ω )−1
0
andspaceunitsofr ,theradiusoftheoutersphere. Itis
o
also convenientto introduce thedimensionless velocity
V
U0 = 0 , (28)
2r Ω
o 0
where V0 is the unperturbed velocity measured in the
frame rotating at Ω . As before, the linear perturbed
0
velocityis v. Fortheperturbedvelocity v and magnetic
field b, we introduce dimensionless variables v′ and b′:
v′ =v/(2r Ω ), b′ =b/B . (29)
o 0 0
PETITDEMANGEET AL.: MRI INTHE EARTH’SOUTERCORE X - 7
The governing fluid equations may then be written
∂∂vt′ +(U0·∇)v′+(v′·∇)U0 =−∇π+E∆v′+ PEmΛ(∇×b′)×ez−ez×v′,
(30)
∂b′ =∇ (v′ ez+U0 b′)+ E ∆b′, (31)
∂t × × × Pm
∇ v′ =∇ b′ =0. (32)
· ·
In addition to the Elsasser number Λ, we have intro-
duced the standard Ekman and magnetic Prandtl num-
bers, which are respectively:
E =ν/(2Ω r2) Pm=ν/η, (33)
0 o
andπistheperturbedvalueofthedimesionlesspressure.
AsamodelfortheEarth’score,weinvestigateaspher-
icalshell. TheangularvelocityΩ isdefinedbytheouter
0
sphere. In this reference frame, the inner sphere rotates
with angular velocity Ω > 0, in order for the differen-
i
tial rotation to decrease outward. We define the Rossby
numberas
Ro=Ω /Ω . (34)
i 0
Let us now compare the local WKB analysis (8) with
the global growth rate obtained numerically. We first
need to compute the dimensionless steady velocity pro-
file U0. If the magnetic field is weak enough, this can
be obtained as a solution of the hydrodynamic problem.
TheresultsaredisplayedinTable1. IftheElsassernum-
ber (21) becomes large, the field obviously affects the
steady solution, and we then perform simulations with
the steady MHD state as initial configuration, see Ta-
ble 2.
To best illustrate the above theoretical analysis, we
use numerical values as close as feasible to the regime
described. Simulations use Pm = 0.5, the differen-
tial rotation is weak compared to the planet rotation
(Ro<<1),andtheEkmannumberisdecreased tosmall
values. ToaccommodatesmallEkmannumbers,wehave
implemented an axisymetric version of the Parody code
(Dormy et al., 1998, Christensen et al., 2001, and later
collaborative developments) with radial resolution rang-
ing from 500 to 1000 points in radius and between 200
and 500 harmonics.
Our results are summarized in Table 1. In agree-
mentwiththeoreticalexpectations,theradialwavevector
k varies approximately in proportion to the dominant
s
shear, i.e. E1/4. The value of k is taken from numeri-
s
cal simulations by computing the width of the unstable
mode at mid-intensity, and the value of k is obtained
z
by direct fourier transform. It is found that the most
unstable mode’s half wavelength generally occupies the
full radial extent of the sheared region. The inner core
and the associated equatorial singularity of the Ekman
layer are the natural modal boundaries; the equatorial
singularitydividesthetangentcylinderintoindependent
domains. Finally, to limit the stabilizing effect of diffu-
sion, we must use E much less than unity.
The induced azimuthal magnetic field clearly repro-
duces the mechanism of the magnetostrophic MRI (see
figure1), despitethepresenceof finiteinertialanddiffu-
siveeffects(whichwereneglectedinthetheoreticalcalcu-
lation),andthecomplicationsassociatedwithabounded
spherical domain (also neglected).
SomeofthesecasesinvolvelargevaluesoftheElsasser
X - 8 PETITDEMANGEET AL.: MRI INTHE EARTH’SOUTERCORE
number,one may then worry that they are not fully self
consistent. To check this, we show, in Table 2, the in-
stability parameters of such strong field profiles, which
now also have a (weak) dependence on Λ. For the field
geometry used here, however, both the rotation profile
and the applied field require relatively little adjustment.
Agreementwiththelocal descriptionisalsoobtainedfor
such configurations.
5. Conclusions and Discussion
Attheveryleast, themagnetostrophicMRIdiscussed
here seems to be an efficient regulatory mechanism in
the outer core, providing a back reaction against the
build up of angular velocity at smaller radii. This could
be tested in full scale numerical simulations of spherical
shells whose gross aspect and parameters could be regu-
lated eitherto admit theMRI or not.
More interestingly, might there be traces of the mag-
netostrophicMRIingeomagnetic observations? Because
of the severe limitations in our knowledge of differential
rotationwithintheEarth’score,itisnotpossibletooffer
anykindof definitivetreatment. Itis, however, interest-
ing tonote that some axial variation in thezonal flow is
requiredtoaccountforchangesinthelengthoftheaday
over millenial timescales (Dumberry & Bloxham 2006).
In that respect It is interesting that axial wavenumber
disturbancesgrowparticularlyrapidlyinthemodelstud-
ied here.
Anothertemptativeapplication isbasedontheobser-
vationthatthegrowthrateinquestionisdirectlypropor-
tionaltotheshear,whichcouldbehighlylocalized. Thus,
growth times considerably shorter than 1500 years are
possible: inprinciple,sheartimesoforderyearscouldbe
produced. This suggests a possible connection with the
observations of rapid events of internal origin, known as
“geomagnetic jerks” or “geomagnetic impulses”. These
rapid changes in thesecular variation were detected and
characterized from observatory data (Courtillot et al.,
1978; Le Mou¨el et al., 1982) and found to correspond to
localised patchesofrapidfieldvariationatthesurfaceof
thecore, using global field models (Dormy and Mandea,
2005). Wereemphasize, however,that theestablishment
ofacompellinglinkbetweenMRIunstableconfigurations
in the Earth core and geomagnetic impulses will require
considerablymoreeffort. Nevertheless,theMRIoffersfor
consideration a novel magnetohydrodynamic mechanism
that is able to induce significant field variations over a
range of observationally interesting timescales.
Acknowledgments. We thank M. Dumberry and an
anonymousrefereeforconstructivecriticismsthatgreatlyim-
provedthepresentationofthepaper. Computationsreported
here were carried out on the CEMAG computing center at
LRA/ENS.SABacknowledges supportfromtheFrenchMin-
istryofHigherEducationintheformofaChaired’Excellence
awardwithadditionalfundsfromtheR´egionIledeFrance.
References
Acheson,D.J.(1983),Localanalysisofthermalandmagnetic
instabilitiesinarapidlyrotatingfluid,Geophys.Astrophys.
Fluid Dynam.,27,123–136
Amit,H.andP.Olson(2004),Helicalcoreflowfromgeomag-
neticsecularvariation,Phys. Earth Plan. Int.,147,1–25.
Aubert, J. (2005), Steady zonal flows in spherical shell dy-
namos,J. Fluid Mech.,542,53–67.
PETITDEMANGEET AL.: MRI INTHE EARTH’SOUTERCORE X - 9
Balbus, S.A. and J.F. Hawley (1991) A powerful local shear
instabilityinweaklymagnetizeddisks,Astrophys. J.,376,
214–222.
Bloxham, J.,andA.Jackson (1991), Fluidflow near thesur-
faceofearth’souter core,Rev. Geophys.,29,97-120.
Christensen,U.etal(2001)Anumericaldynamobenchmark,
Phys. Earth Plan. Int.,128,25–34.
Christensen, U. (2002), Zonal flow driven by strongly su-
percritical convection in rotating spherical shells, J. Fluid
Mech.,470,115–133.
Courtillot, V., J. Ducruix, J.L. Le Mou¨el (1978), Sur une
acc´el´eration r´ecente de la variation s´eculaire du champ
magn´etiqueterrestre,Acad.Sci.,Paris,C.R.,D287,1095-
1098.
Dormy, E., P. Cardin and D. Jault (1998), MHD flow in a
slightlydifferentiallyrotatingsphericalshell,withconduct-
inginnercore,inadipolarmagneticfield,EarthandPlan-
etary Science Letters,160,15–30.
Dumberry,M.(2007),Geodynamicsconstraintsonthesteady
and time-dependent inner coreaxial rotation, Geophys. J.
Int.,170,886–895.
Dumberry,M.andJ.Bloxham(2006),Azimuthalflowsinthe
Earth’s core and changes in the length of the day at mil-
lenialtimescales,Geophys. J. Int.,165,32–46.
Dormy,E.andM.Mandea(2005), Trackinggeomagnetic im-
pulses at the core-mantle boundary, Earth Planet. Sci.
Lett.,237,300–309.
Eymin,C.andG.Hulot(2005),Oncoresurfaceflowsinferred
fromsatellitemagnetic data, Phys. Earth Plan. Int.,152,
200–220.
Fearn, D.R., C.J. Lamb, D.R. McLean,R.R. Ogden (1997),
The influence of differential rotation on magnetic insta-
bility, and nonlinear magnetic instability in the magne-
tostrophic limit, Geophys. Astrophys. Fluid Dynam., 86,
173–200.
Le Mou¨el, J.L., J. Ducruix, C. Ha Duyen (1982), The world-
widecharacter of the 1969-1970 impulseof the secular ac-
celerationrate,Phys. Earth Planet. Inter.,28,337-350.
Menou K., S.A. Balbus, and H.C. Spruit (2004), Local ax-
isymmetricdiffusivestabilityofweaklymagnetized, differ-
entiallyrotating,stratifiedfluids,Astrophys. J.,607,564–
574.
Ogden,R.R.,D.R.Fearn(1995),Thedestabilisingroleofdif-
ferentialrotation, Geophys. Astrophys. Fluid Dynam.,81,
215–232.
Proudman,I.(1956),Thealmost-rigidrotationofviscousfluid
between concentricspheres,J. Fluid Mech.,1,505–516.
Stewartson, K. (1966), On almost rigid rotations, Part 2, J.
Fluid Mech.,26,131–144.
Taylor,J.B.(1964),TheMagneto-HydrodynamicsofaRotat-
ing Fluid and the Earth’s Dynamo Problem, Proc. Roy.
Soc. Lond.,274,274–283.
Vidale,J.E.,D.A.Dodge,andP.S.Earle(2000),Slowdiffer-
ential rotation of the Earth’s inner coreindicated by tem-
poralchanges inscattering,Nature,405,445–448.
Ludovic Petitdemange, Emmanuel Dormy, Steven Bal-
bus, LRA, D´epartement de Physique, Ecole Normale
Sup´erieure, 24, rue Lhomond, 75231 Paris Cedex 05, France
([email protected])
X - 10 PETITDEMANGEET AL.: MRI INTHE EARTH’SOUTERCORE
Table 1. Results obtained for the most unstable mode in
a direct integration of the governing equations. We list the
growth rate (σnum) and the wavenumber (ks and kz). We
alsolistthegrowthrate(σth)obtaineddirectlyfromthelocal
dispersion relation (8) with these wavenumber values. The
variablesE,Ro,andΛaredefinedinequations(21),(33)and
(34). Thenumericalagreementisexcellent.
2E Ro 2Λ σth σnum ks kz
2×10−6 0.005 1.0 6×10−4 6.4×10−4 π/0.17 π/0.25
25 1.0×10−3 1.09×10−3 π/0.17 π/0.6
10−6 0.01 5 8.8×10−3 1.16×10−2 π/0.14 π/0.15
0.01 25 1.0×10−2 1.0×10−2 π/0.14 π/0.33
0.005 25 3.6×10−3 3.6×10−3 π/0.14 π/0.4
0.005 5 4.2×10−3 4.8×10−3 π/0.14 π/0.1
0.0025 25 2.0×10−4 2.3×10−4 π/0.14 π/0.6
5×10−7 0.005 50 4.0×10−3 4.4×10−3 π/0.11 π/0.25
0.005 25 6.0×10−3 6.4×10−3 π/0.11 π/0.14
15 7.0×10−3 6.2×10−3 π/0.11 π/0.125
10 7.1×10−3 7.6×10−3 π/0.11 π/0.1
5 6.0×10−3 6.4×10−3 π/0.1 π/0.071
2 3.0×10−3 4.8×10−3 π/0.1 π/0.067
1 2.24×10−3 3.0×10−3 π/0.1 π/0.058
0.5 1.16×10−3 1.36×10−3 π/0.11 π/0.11
0.005 0.3 6.96×10−4 2.03×10−4 π/0.1 π/0.14
0.0025 25 2.0×10−3 1.93×10−3 π/0.1 π/0.25
0.0075 25 6.2×10−3 6.6×10−3 π/0.1 π/0.2
2.5×10−7 0.0025 25 2.0×10−3 3.2×10−3 π/0.08 π/0.15
10−7 0.0025 1 3.0×10−4 2.2×10−4 π/0.06 π/0.04
25 4.0×10−3 4.8×10−3 π/0.06 π/0.083
Table 2. Results obtained using the steady MHD state as
initialconfiguration. ComputationsareperformedwithRo=
5×10−3,Pm=0.5andE=2.5×10−7. AsforTable1,both
thenumericalparametersofthemostunstablemodeandthe
correspondingtheoretical growthratearereported.
2Λ σth σnum kr kz
3 5.06×10−3 5.70×10−3 π/0.92 π/0.056
1 2.06×10−3 1.84×10−3 π/0.95 π/0.056
0.3 8.44×10−4 2.76×10−4 π/0.10 π/0.11