ensemble inversion of time-dependent core flow models

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ensemble inversion of time-dependent core flow models

Nicolas Gillet, Alexandra Pais, Dominique Jault

To cite this version:

Nicolas Gillet, Alexandra Pais, Dominique Jault. ensemble inversion of time-dependent core

flow models. Geochemistry, Geophysics, Geosystems, AGU and the Geochemical Society, 2009,

pp.2008GC002290. �10.1029/2008GC002290�. �hal-00377131�

Niolas Gillet

†∗

, MariaAlexandra Pais

‡

& Dominique Jault

†

AeptedforpubliationatGeohem., Geophy s., Geosyst., April2009

†

LGIT,UniversityJosephFourier,CNRS, BP53,38041GrenobleCedex9,FRANCE.

‡

CFC,DepartmentofPhysis,UniversityofCoimbra,3004-516Coimbra,PORTUGAL.

∗

orrespondingauthor: Niolas.Gilletobs.ujf-grenoble.fr

Résumé

Quasi-geostrophi oreowmodelsarebuilt fromtwoseularvariationmodelsspanning

the periods 19602002 and 19972008. We relyon anensemblemethodto aount for the

ontributionsoftheunresolvedsmall-salemagnetield interatingwithore surfaeows

to the observed magneti eld hanges. The dierent ore ow members of the ensemble

solution agree up tospherial harmoni degree ℓ≃10,and this resolvedomponent varies onlyweaklywithregularization.Takingintoaount theniteorrelationtimeofthesmall-

saleonealedmagnetield,wendthatthetimevariationsofthemagnetieldourring

overshorttime-sales,suhasthegeomagnetijerks,anbeaountedforbytheresolved

largesalepartoftheowtoalargeextent.Residualsfromourowmodelsare30%smaller

for reentepohs,after 1995. Thisresult isattributedto animprovement inthe quality of

geomagneti data.The magnetield models showlittle frozen-uxviolation for the most

reent epohs,withinourestimateoftheapparentmagnetiuxhangesattheore-mantle

boundaryarisingfromspatialresolutionerrors.Weassoiatethemoreimportantuxhanges

deteted atearlier epohswith unertaintiesinthe eld modelsat large harmonidegrees.

Ouroreowmodelsshow,atallepohs,aneentriandplanetarysaleanti-ylonigyre

irlingaroundtheylindrialsurfaetangenttotheinnerore,atapproximately30

◦

and60

◦

latitudeundertheIndianandPaioeans,respetively.Theyaountwellforthehanges

inore angularmomentumforthemostreentepohs.

1 Introdution

Largeandmediumsalesoforesurfaeowshavebeenapturedastheresultoftheontinuous

observation of the large-sale (harmoni degree ℓ ≤ 13^{) Earth's magneti eld from low Earth}

orbiting satellites sine 1999 [ Holme and Olsen, 2006℄. Unfortunately, from the Earth surfae

upwards,thesmall-salemagnetieldB˜ ⁽ℓ >13⁾originatingfromtheoreisdiulttoisolate: itsintensityathigherdegreesbeomesweakerthanthatofthelithospherield. Thesmall-sale

oreeld is muh strongerat theore surfae,where it interats signiantly with themedium

salesoftheowandontributestothelarge-saleseularvariation(SV)[ Hulotetal.,1992℄.This

SV signalorresponds to spatial resolution errors,sometimes alled errors of representativeness

in thedata assimilationommunity [ Kalnay, 2003℄.Both Eymin and Hulot [2005℄and Pais and

Jault [2008℄reliedonastohasti approahto quantify itatdisretetimes.Theyfoundthat the

spatialresolutionerrorsdominatetheerrorbudgetforthelargelengthsaleseularvariation.

Ensemble methodsareroutinelyusedinatmospheriforeastingtoprodueestimatesforthe

omplete probability density of the state variable [ Wunsh, 2000℄. Ensemble foreasting helps

improve the foreastby ensemble averaging, and provides an indiationof thereliability of the

foreast[ Kalnay, 2003, p. 236℄.Weremark that ensemble methods are suitableto quantify the

ontributionoftheonealedB˜ ^{interatingwiththesurfaeoreowto theobservedlarge-sale}

SV.UsinganensemblemethodmakesitstraightforwardtoaountforthetimevariabilityofB(t)˜ ^.

Weantiipatethatimprovingourknowledgeofthetimepropertiesoftheseularvariationsignal

thatresultsfrom theadvetionofB˜ ^{mayhelpalleviateitsimpatonthealulationofthelarge-}

saleoresurfaeow.Typialtimesalesforthemagnetieldstrutureshavebeenderivedfrom

theratiooftheSVandmain eldspetra[ Hulotand Mouël,1994℄.Extrapolatingthosespetra,

as obtainedfrom time-dependent eld models, one nds that B˜ ^{typially has orrelation times}

oftheorder of20yrsand belowforharmonidegreesabove13.It isthusreasonabletosuppose

that its advetion entailsspatial resolution errors that are orrelated in time. We test this idea

andgenerateanensembleoforeowsolutionsfromanensembleofsmall-salemagnetieldsB˜

orrelatedintime.

All disussions on ore dynamis relying on observations of magneti eld time hanges are

potentiallyaeted bythespatial resolutionerrorsthat weseekto quantify.As anexample,the

observedtemporalvariationsof theunsigned magnetiux areoften interpretedas evidene for

magnetidiusion[ GubbinsandBloxham,1985℄.Itisnoteasy,however,todisentangletherespe-

tiveontributionsofmagneti diusion,andthat ofindution involvingtheunresolvedmagneti

eld,tothehangesinmagnetiuxattheoresurfae.Anotherexampleisthesuggestion[ Blox-

hametal.,2002℄thatgeomagnetijerksmayresultfromtheinterationbetweentorsionalAlfvén

wavesandtheradialmagnetield at theoremantleboundary(CMB).Aordingto Bloxham

et al.[2002℄,dierenesin thegeometry oftheradialmagneti eldfrom one plae totheother

explaintheobservationthatgeomagnetijerksareseeninsomeomponentsatsomeobservatories

butnotdetetedinothers.Weshallinvestigaterstwhihomponentofthelargesaleoreow

aountsfor the rapid hanges of themagneti eld, one time orrelation of spatial resolution

errorsistakenintoaount,andseondly whetherspatial resolutionerrorshavemoreimpaton

magnetiseriesinsomeobservatoriesthanin others.

The set-upof ourore ow inverse problem isdetailed in Ÿ2.Wealulate quasi-geostrophi

time-dependentoreowsfromtwoSV modelsoveringannualtodeadaltimesales:theom-

prehensive model CM4 [ Sabaka et al., 2004℄, whih overs 19602002, and the model xCHAOS

[ Olsen and Mandea, 2008℄, derived from satellitedata and annual dierenesof monthly means

overthe period 19972008. Theamplitudeofthe spatialresolution errors,theirtime orrelation

and SV preditionsfrom ourdierent owmodelsat loationof magneti observatoriesaredis-

ussedin Ÿ3. Estimates ofthehangesin themagneti ux throughthemain reverseux path

(beneath SouthAtlanti) assoiatedwith thespatialresolutionerrorsarethen arriedoutinŸ4.

Theomputedoreowsaredesribedin Ÿ5,whereaomparisonwithindependentlengthofday

dataisalsoarriedout.Finally,inŸ6wedisusstheperspetivesfortheoreowinverseproblem.

2 Methodology

2.1 Formalism and notations

Vetorsm^,y^andx^{storethespherialharmonioeientsforthemainmagnetield,seular}

variation, poloidal and toroidal salarsofthe oresurfaeowmodelrespetively. The`data' y

(andtheirassoiatederrorse^{)arelinkedtotheoreowmodel}x^,ateveryepoht^{,viatheforward}

problem

y(t) =A[m(t)]x(t) +e, ⁽¹⁾

orrespondingtothefrozenux radialindution equationattheCMB[e.g.Holme,2007℄

∂Br

∂t =−∇^h·(uBr). ⁽²⁾

We hooseto trunate the seular variationdata set at harmoni degreeℓy = 13^{. Themain}

eld m= [m,m]˜ ^{isomposed ofalargesalepart}m^{obtainedfrom publishedgeomagnetield}

models(up todegreeℓm= 13^{),andasmall-salepart}m˜ ^(degrees ℓm< ℓ≤ℓm^{)estimatedwitha}

stohastiapproah(Ÿ2.3).Wetrunatetheoreowmodelat ℓx= 26 ^{[ Pais andJault, 2008℄,a}

degreehighenoughtoinludeinterationsbetweenm^andx^{possiblygeneratingseularvariation}

at ℓ ≤ ℓy^{. The trunation level for} m˜ ^{is hosen as} ℓm = 40^{, a degree high enough to inlude}

interationsbetweenm˜ ^andx^{possiblygeneratingseularvariationat}ℓ≤ℓy^{.Thevetorssizesof}

m^, y ^andx ^{at asingleepohare then}Nm =ℓm(ℓm+ 2)^,Ny =ℓy(ℓy+ 2)^and Nx = 2ℓx(ℓx+ 2)^,

respetively.

[ Bakus, 1968℄. For short timesale dynamis, the ratio between Lorentz and Coriolis fores is

alsotheratiobetweenthefrequeniesof Alfvénand inertialwaves,and an beestimated bythe

Lehnert [1954℄numberλ=B/Ωc√ρµ0 ^{[ Jault,2008℄,where} c^{is theouteroreradius,}ρ^theore

density,µ0^thepermeabilityoffreespae,Ω^{therotationrate,and}B^{isanestimateofthemagneti}

eldstrength. B ^{istypiallyofafewmTinside theore,basedonalongtimesalefore balane}

[ Starhenko and Jones, 2002℄, whih givesλ ∼ 10⁻⁴ ≪ 1^{. This motivates the use of the quasi-}

geostrophi (QG) hypothesis, a onstraintwhih allowsa ow invariantparallel to therotation

axis z^{to be desribed everywhere in theouter ore. It implies the tangentialgeostrophy(TG)}

onstraint∇h·(ucosθ) = 0 ^{at theCMB [ Hills, 1979, Le Mouël, 1984℄, plus}non-penetrationat the tangent ylinder(the ylindertangent to theinner ore and alignedwith therotationaxis)

andequatorialsymmetryoutsidethetangentylinder[ Pais andJault,2008℄.Wedeomposethe

oreowas u=ue+uzˆz^{. The}z^{-invariantequatorialomponent}ue ^{of theow an bedened}

withastream funtionψ(s, φ, t)^as

ue(s, φ, t) =∇ ×(zψ). ⁽³⁾

Thedesriptionoftheowisompletedbytheexpressionofitsaxialomponent

uz(s, φ, z, t) = −sz

H(s)²us(s, φ, t), ⁽⁴⁾

whihensuresthenon-penetrationonditionattheCMB,with(s, φ, z)^{theylindrialoordinates}

andH(s) =√

c²−s^{2 the}half-heightofauidolumn.

There exists a basis of ow vetors that automatially satisfy the TG onstraint [ Le Mouël

etal., 1985,Bakusand Le Mouël,1986℄.Theowoeientsw ^{in thatbasisarerelatedto the}

oeientsx^inthetoroidal/poloidalexpansionthroughtheorthogonalmatrixG^:

x(t) =Gw(t). ⁽⁵⁾

Thenumberofunknownsfor asingleepohis thenreduedto Nw=ℓ²_x ^{the sizeof thevetor}w

[ Jakson, 1997℄.We denote H(t) = A(t)G ^{the matrixrelating the ow oeients}w ^{to the SV}

oeientsy^.

We follow Jakson [1997℄ for the implementation of the time-dependent problem (see also

Bloxham andJakson [1992℄).Theoreowoeientsareexpandedintermsofabasisofubi

B-splines funtions Fp(t) ^{[ Lanaster and} Salkauskas, 1986℄uniformly spanning the timeinterval

[ts, te] ^withknot-spaing∆t^:

w(t) =

P

X

p=1

Fp(t)w^p. ⁽⁶⁾

Wedenote W^thevetor

w¹. . .w^P

andX ^thevetor

x¹. . .x^P

,with x^p=Gw^p.Wesample

the time-span [ts, te] ^{with steps} δt^{. At every epoh} tj ^{we estimate the main eld oeients}

m(tj) = m(tj) + ˜m(tj)^{needed to build the interation matries} A(tj) ^{(see Ÿ2.3),and thedata}

oeients y(tj) = ∂m

∂t (tj)^{. From the latter we generate a data vetor} Y = [y(ts). . .y(te)]^,

a ombined set of SV Gauss oeients alulated at eah time-step, assoiated with the error

vetorE = [e(ts). . .e(te)]^{. The forwardproblem is now written} Y =HW+E^{, where} H ^{is a}

bandedmatrixwithbandwidth4Nw^{,alulatedfromtheinterationmatries}H(tj) =A(tj)G^and

thevalueFp(tj)^{takenbyeahB-spline}Fp ^atepoh tj^.

Thesolutionofourproblemisfoundbyminimizingtheobjetivefuntion

J(W) =kY− HWk²Cy+ξkWk²Qw +µkWk²Pw, ⁽⁷⁾

with thegeneri notation kVk²M =V^TM⁻¹V^{. The rstterm is} χ² = kY− HWk²Cy^{, ameasure}

ofthe misttotheSV data, withthedata ovarianematrixC_y ^{as detailed in Ÿ2.2.Theseond}

termisaspatialregularizationoftheowmodel,withξ^{adampingparametertunedtoadjustthe}

orrespondstotheequatorialsymmetryandnon-penetrationonstraintsatthetangentylinder,

imposedusingaweakformwithµ^{aparameterbigenoughsothattheseonstraintsarepratially}

satised.ThedampingmatrixQ_w^{,together withtheonstraintmatries} P_w^{,aredened in Ÿ2.2.}

Minimizing the ostfuntion J^{, as dened in equation (7), isalinear} optimizationproblem. Its solutionis

W=

H^TC⁻¹_y H+ξQ⁻¹_w +µP⁻¹_w ⁻¹

H^TC⁻¹_y Y. ⁽⁸⁾

Inpratie,δt= 1^yrandP= 24^{,whihorrespondstoaknotspaing}∆t= 2^{years,areusedto}

invertthe CM4SV oeientsspanning [ts, te] = [1960,2002]^{. Convergeneof thesolutionwith} δt^{hasbeenhekedfor. Similarly,}δt= 0.5 ^yearandP = 25 ^{areused toinvertthexCHAOSSV}

oeientsoverthetime-span [ts, te] = [1997,2008] ^{(i.e.aknotspaing} ∆t= 0.5 ^{year). Finally,}

wederivex(t)^fromW^{usingEquations(5)and(6).}

2.2 Regularization, onstraints and error model

Thenormusedthroughoutourstudytoregularizetheproblemis

Q³(u)²= Z

CMB D²+V² ds

= (te−ts)⁻¹kWk²Qw = (te−ts)⁻¹kXk²Qx, ⁽⁹⁾

with the horizontal divergene D= ∇^h·u^{and the radialvortiity} V = ˆr· ∇ ×u^{. Theangular}

braketsdenotethetime-averagingoperator:

h. . .i= 1 te−ts

Z te

ts

. . . dt . ⁽¹⁰⁾

ThematrixQ⁻¹_x ^isblo-diagonal,withelementsvaryingwithharmonidegreeℓ^as[ℓ(ℓ+ 1)]²/(2ℓ+

1)∝ℓ^3,andQ⁻¹_w =G^TQ⁻¹_x G^{.Theextralinearonstraintsin(7),namelytheequatorialsymmetry}

andnon-penetrationatthetangentylinder,arealulatedasinPais andJault [2008℄.Theyan

bewrittenintheformLx=L Gw= 0^,whihyieldsP⁻_w¹=G^TL^TLG^{.Inpratietheparameter}µ

islargeenoughsothat inreasingitdoesnotaetthesolution.

Otherquadratinormsouldhavebeenused,suhastheminimumenergynorm[ Madden and

LeMouël,1982,Pais etal.,2004℄

Q¹(u)²= Z

CMB

kuk²ds

∝ ℓ(ℓ+ 1)

2ℓ+ 1 ∼ℓ¹, ⁽¹¹⁾

whih gives the r.m.s. veloity Q^{1 and orresponds to a relativelyweaker damping of the high}

harmoni degrees, or the more severe and widely used `strong norm' [ Bloxham, 1988, Jakson

etal.,1993,Jakson,1997℄

Q⁵(u)²= Z

CMB k∇^hDk²+k∇^hVk² ds

∝[ℓ(ℓ+ 1)]³

2ℓ+ 1 ∼ℓ⁵. ⁽¹²⁾

Our norm Q³(u)^{2 sales as} ℓ^{3, as does the strongest omponent of the norm used in Pais and}

Jault [2008℄(symmetripartoftheReynoldstensor).

Weassumethattheerrorsarestationaryandthatthedataovarianematrixisdiagonal.We

denoteσ^d2(ℓ) =Eh e^d_ℓm²i

thedata ovarianes,whihareindependentofthespherialharmoni

orderm^{(isotropierrors).Supposingthat theenergyofthedataerrorspreadsuniformlyoverall}

harmonidegree (i.e.atLowesspetrum),one obtainsa blo-diagonalovarianematrixCy ^of

whihelementsarethevarianesσ^d2(ℓ) =η/(ℓ+1)(2ℓ+1)^{.Theapriorinoiselevelissetat}η= 0.4

(nT/yr)

2

forboththeCM4andxCHAOSmodels.Thishoieissomewhatarbitrary,andabetter

desriptionofboththespatialandtemporalstatistialbehaviourofdataerrorsouldbeusefulin

futurework.First,therearesomehintsthattheoeientsofthedataovarianematrixshould

dependonℓ−m^{to aountforapoorknowledgeof themagnetieldin auroralregions[ Olsen}

and Mandea, 2007℄. Seondly the use of a temporal regularization to generate time-dependent

eld models redues the time variability of the SV oeients at high degrees, penalizing the

instantaneousseularaeleration

∂²Br

∂t^{2 at small sales.There isindeed atrade-o betweenthe}

spatialomplexityandthetemporalvariabilityofamagnetieldmodel.OlsenandMandea[2008℄

gavepreedenetotheformer overthelatter,as notedbyLesur et al.[2008℄whoalsoremarked

thattheseularaelerationpreditedbyxCHAOSisfullyontrolledbytheregularizationproess

abovedegree11.Introduingtimeorrelationinthedataerrors(nondiagonalovarianematrix,

i.e. E

e^d_ℓm(t)e^d_ℓm(t±τ)

6

= 0^{) or a} time-orrelated noise at small sales (e.g. with an ensemble approahforthedataerrors)mightbeawaytoaddressthisissue.

2.3 Small-sale magneti eld with zero mean and Gaussian time or-

relation funtion

An ensemble of K ^matries A^{k is alulated from an ensemble of} K ^{small-sale magneti}

elds m˜^{k. We onsider the small-sale eld as a random noise with a Gaussian entered time}

orrelation. A set of K ^{random small sale main eld models} m˜^k(t) ^{is generated, satisfying :}

∀k∈[1, K]^,∀ℓ∈ ℓm, ℓm

,∀m≤ℓ^,∀(t, t^′)∈[ts, te]^2, E

˜

m^k_ℓm(t),m˜^k_ℓm(t^′)

=σ_m²(ℓ)exp

"

−1 2

t−t^′ τm(ℓ)

²#

, ⁽¹³⁾

wherem˜^k_ℓm(t)^{denotesaoeientofdegree}ℓ^andorderm^{ofthespherialharmoniexpansionof}

themagnetieldatepoht^.E[. . .]^{representsthe}mathematialexpetation,τm(ℓ)^isthetypial

orrelationtimeforthemaineld oeientsofdegreeℓ^and σ_m²(ℓ)^{theirvariane.Thehoieof}

astationary Gaussian orrelated stohasti proess is justied by thework of Hulot and Mouël

[1994℄.Itreetsthestatistialbehaviourofboththeobservedhistorialandarheomagnetields

[ Hongreetal.,1998℄.

In order to estimate the varianesσ_m²(ℓ) =E

˜ mℓm(t)²

, we t an exponentialurveto the

Lowesspetrumofthemain eldmodelGRIMM[ Lesuretal.,2008℄fordegreesℓ∈[2,13]^,epoh

2003.5,and extrapolateitfordegreesℓ >13^{.It gives}σ_m²(ℓ) = 1.09×10⁹e^−1.26ℓ/(ℓ+ 1)(2ℓ+ 1)^.

For eah degree ℓ^{, typial orrelationtimes} τm(ℓ) ^{for themain eld are estimated from SV and}

maineld spetra[ HulotandMouël,1994℄,

R(ℓ) =τm(ℓ)⁻²=



 X

m≤ℓ

gℓm² +h²ℓm





−1

X

m≤ℓ

g˙²ℓm+ ˙h²ℓm

, ⁽¹⁴⁾

wheretheupper`dot'denotesthetimederivative.Apowerlawtforharmonidegreesℓ≤11^of

GRIMMat2003.5givesR(ℓ)≃1.47×10⁻⁶ℓ^{2.75yrs−2.Its}extrapolationgivesestimatesfrom22to 5yearsforharmonidegrees14to40ofB˜^{.AsmentionedinŸ2.2,theSVoeientsarelikelytobe}

toomuhorrelatedathighdegrees,duetotemporalregularizationoftime-dependenteldmodels.

ThusthehoieofapowerlawttoR(ℓ)^isquestionable[seee.g.HolmeandOlsen,2006℄,andwe alsotriedtheaseofanexponentialttoR(ℓ)^forℓ∈[2,11]^.WeobtainR(ℓ)≃1.1×10⁻⁵e^0.44ℓ,

that isτm ^{from 14to 0.05yearsforharmonidegrees14to40of} B˜^{.Inthis latterasethesmall}

salesaremuh lessorrelatedin time,asillustratedin Figure1.Thisgurealsoshowsthat our

synthetioeienttime series,omputed bylow-passlteringarandomnoise,haveorrelation

timesverylosetowhatisrequested.Thesetimeseriesarenormalizedinordertohavetherequired

varianeσ_m²(ℓ)^{,andsampledevery}δt^{toproduetheoeients}m˜ℓm(tj)^.

Weonatenatethelarge-salemaineldoeientswiththesmall-salerandomonesintoan

ensembleofK^modelsm^k(t) =

m(t),m˜^k(t)

. Fromthese,wegenerateK ^matries

A^k(t) =A[m^k(t)] =A(t) + ˜A^k(t), ⁽¹⁵⁾

0 5 10 15 20 25 30 35 40 10

10

10

10

10

harm. degree τ

(yrs)

Fig.1Themaineldorrelationtimeτm^{asafuntionofharmonidegree}ℓ^{,estimatedfromthe}

ratioR(ℓ)^{fortheGRIMMmodelatepoh2003.5(green}triangles),itsexponentialandpowerlaw ts(dotted blak),and alulatedfromthesyntheti timeseries (redirles :powerlawt; red

rosses: exponential t).Only harmonidegreesaboveℓ = 13^{(to the rightof the vertialline)}

areusedintheensembleinversion.