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The axial dipole strength and flow in the outer core

K.A. Whaler a,⇑, R. Holme b

a School of GeoSciences, University of Edinburgh, Edinburgh EH9 3JW, UKb School of Environmental Sciences, University of Liverpool, Liverpool L69 3GP, UK

a r t i c l e i n f o

Article history:Received 7 January 2011Received in revised form 6 July 2011Accepted 13 July 2011Available online 23 July 2011Edited by Keke Zhang

Keywords:Core surface flowGeomagnetic intensityDecadal length-of-day variations

a b s t r a c t

From 1590 to 1833, although extensive measurements of the geomagnetic field were made, no directmeasurements of its intensity are available; field models for this period either make use of indirect(palaeomagnetic) intensity determinations or assume a similar change in dipole strength to that seenpresently. Here, we examine the impact of different choices of field model strength on models of thecore-surface flow, derived from the geomagnetic secular variation. We find that for a reasonable rangeof possible changes in the field strength, the variability that this unknown imposes on the flow structureis small compared with other choices in the modelling strategy, such as the degree to which thegeomagnetic models are fit, the smoothness of the flows, and a priori information included to reducenon-uniqueness in the flow calculation. The flows predict variations in core angular momentum ofsimilar magnitude to those seen in recent times. These variations are larger than seen in models fromgeodetic data, but do not seem inconsistent with the geodetic data themselves. We therefore suggest thatit is reasonable to study core flows back at least until 1650, thereby doubling the time interval for whichflows can be considered to constrain the behaviour of the geodynamo.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

The geomagnetic field and its temporal, or secular, variation area powerful probe of the Earth’s core. In particular, they can be usedto model the flow of the iron-rich liquid at the top of the core,which in turn constrains the dynamics and evolution of the coreand geodynamo. Models describing the field and its temporal var-iation are available on a wide range of time scales, with vastly dif-fering resolution, from the recent decade of continuous, highquality satellite data, through to time-averaged models of the pal-aeomagnetic field. Changes over decades to centuries are crucialfor observing and understanding field behaviour, and relating itto core dynamics. The most fruitful data source for investigatingchanges over this time scale, trading off length of availabilityagainst resolution, has been the historical data base covering thelast few centuries, for which gufm (Jackson et al., 2000) is thestate-of-the-art field model. This model covers the time periodfrom 1590 to 1990, giving a high quality, temporally continuousimage of the field at the core surface reflecting the variable quan-tity, distribution and fidelity of the data.

However, the historical era covered by gufm is sharply dividedinto two parts. No measurements of the absolute intensity of thefield were made prior to 1832, when Weber used instrumentationGauss had invented to determine the horizontal intensity of the

magnetic field (Malin, 1982). This leaves a fundamental non-uniqueness in the field determination, characterised fully by (Hulotet al., 1997). The gufm field model allows calculation of field inten-sities prior to 1840, hereafter referred to as the ‘‘pre-Gauss era’’,but, following (Barraclough, 1974), these are based upon anassumed linear axial dipole decay with time. While such a lineardecay provides a good approximation to the field behaviour since1840, that this should have been the case in the past is puresupposition.

Recently, Gubbins et al. (2006) and Finlay’s (2008) haveattempted to constrain the evolution of the field using archaeo-magnetic data, by comparing the intensity predictions of historicalmodels to archaeomagnetic data used in the longer archaeomag-netic time scale models of Korte and co-workers (Korte et al.,2005; Korte and Constable, 2005), since only a few intensity dataare required to solve the ambiguity (Hulot et al., 1997). They findno evidence in the data for the assumed dipole decay rate; in fact,Finlay’s (2008) is unable to rule out a constant axial dipole for theperiod 1590–1840.

The geomagnetic field evolves by the combined action of advec-tion and diffusion, both of which are necessary ingredients ofgeodynamo activity. Their relative importance is measured bythe magnetic Reynolds number, estimates of which suggest thatthe former dominates on the decadal to centuries timescale andthe latter on the much longer timescale associated with fieldexcursions and polarity reversals. Thus over our shorter timescale,we can neglect diffusion and treat the field lines as tied to fluidparcels; in this approximation, the field lines act as tracers of the

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⇑ Corresponding author.E-mail address: [email protected] (K.A. Whaler).

Physics of the Earth and Planetary Interiors 188 (2011) 235–246

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flow (Roberts and Scott, 1965). This enables the flow at the core–mantle boundary (CMB) to be deduced through linear or linearisedinversion (depending on how the problem is posed), albeit withsome inherent ambiguity (Roberts and Scott, 1965; Backus,1968). A number of studies have investigated the timescale overwhich this ‘‘frozen-flux hypothesis’’ is valid, the effect of neglectingdiffusion, the nature of the inherent ambiguity on the flow, the ex-tent to which making additional assumptions can reduce it, exam-ined the practicalities of inverting the field changes for the flow,and analysed the results obtained (see Holme (2007) for a recentreview). Flow modelling has focussed almost entirely on the periodpost 1840, for risk of contamination by unmodelled intensity vari-ations prior to this. Indeed, Finlay’s (2008) has cautioned againstinverting for CMB flows in the pre-Gauss era for this reason. Somepreliminary models have been calculated for earlier epochs (LeHuyet al., 2000a) although their validity has been strongly challenged(Jackson, 2000) (but see also the response by Huy et al., 2000b).

These observations motivate us to derive a suite of core flowmodels corresponding to a range of dipole decay rates, based onthe gufm model. We seek to determine whether the flow modelsare highly sensitive to the unknown variation in field intensity,or whether model consistency allows us to nonetheless draw con-clusions as to the possible structure of the core surface flow. Notethat the unknown field strength is not the only problem; through-out the period 1590–1990 covered by the gufm model the quality,quantity and distribution of the data increase markedly. Thischange is mapped into gufm, which improves in both reliabilityand spatial resolution. We must therefore be careful that thisdata-distribution effect does not affect the interpretation of ourflow models.

The gufm model expresses the field through spherical harmon-ics in space and cubic B-splines in time, with a spherical harmonictruncation level of 14 and knot points every 2.5 years, and was de-rived from the simultaneous inversion of over 350000 data. To finda model adequately representing the field at the CMB, spatial reg-ularisation was imposed on the inversion, since downward contin-uation preferentially amplifies small-scale features and theiruncertainties. Similarly, without temporal regularisation, the fieldcan change unfeasibly rapidly, given that the signal originatingfrom convection in the liquid outer core propagates through theweakly conducting mantle which preferentially attenuates rapidvariations. gufm was obtained by minimising an objective functionconsisting of three terms: measures of the goodness of fit betweenthe data and model predictions, and of spatial and temporalsmoothness, with damping parameters controlling their relativeimportance. Not all harmonics are resolved at all epochs: themaximum effective harmonic degree increases with time withthe improved data quality, quantity and distribution; unresolvedharmonics are controlled with spatial regularisation. The ambigu-ity in the field model arising from lack of intensity control in thepre-Gauss era was overcome by imposing a constant axial dipoledecay rate of 15 nT/year (Barraclough, 1974).

In the next section, we briefly review the method for invertingthe rate of change of the field, or secular variation (SV), for CMBflows, and explain how the gufm model for the pre-Gauss erawas modified to test the influence of the axial dipole SV on theflows deduced. In section three, we present the results. Followingthat, we discuss their geophysical implications. Our summaryand conclusions are given in the final section.

2. Method

The magnetic field and its temporal variability is parameterisedin the gufm model by the spherical harmonic expansion of themagnetic scalar potential, truncated at degree and order L = 14:

Uðr; h;/; tÞ ¼ aXL

l¼1

Xl

m¼0

ar

� �ðlþ1ÞPm

l ðcos hÞðgml ðtÞ cos m/

þ hml ðtÞ sin m/Þ ð1Þ

The fgml ; h

ml g are the spherical harmonic, or Gauss, coefficients, a is

the Earth’s reference radius, 6371.2 km, and the Pml are Schmidt

quasi-normalised associated Legendre polynomials, with (r, h, /)spherical polar coordinates with origin at the Earth’s centre. The(negative) gradient of the potential gives the magnetic field; com-ponents of the field (e.g., orthogonal north, east and verticallydownwards components traditionally measured by modern instru-mentation) are linearly related to the Gauss coefficients, whereasintensities (horizontal or total) and angular data such as declinationand inclination are non-linearly related, necessitating an iterativeapproach to inversion (Gubbins and Bloxham, 1985).

The CMB flow is deduced from the field and its SV through theradial component of the induction equation in the frozen-flux limit(Roberts and Scott, 1965; Holme, 2007):

_Br þrH:ðvBrÞ ¼ 0 ð2Þ

where Br is the radial magnetic field component, the over-dot (timederivative) indicates SV, the subscript H indicates horizontal com-ponent and v is the velocity. The velocity, assumed to have vanish-ing radial component at the CMB, is expressed through a sphericalharmonic decomposition of the streamfunction, T , and velocity po-tential, S, representing the toroidal and poloidal ingredients of theflow, respectively:

T ¼X1l¼1

Xl

m¼0

Pml ðcos hÞðT m

lc cos m/þ T m

ls sin m/Þ

S ¼X1l¼1

Xl

m¼0

Pml ðcos hÞðSm

lc cos m/þ Sm

ls sin m/Þ ð3Þ

where

vH ¼ r^ ðT rÞ þ rHðrSÞ ð4Þ

In practice, as with the field, the spherical harmonic expansionsfor T and S are truncated, also at degree and order 14, so that weseek the large-scale component of the CMB flow. Substitutingspherical harmonic expansions for Br and _Br , evaluated at theCMB, and v into Eq. (2), and using the orthogonality of the sphericalharmonics, gives a set of linear equations relating the SV coeffi-cients f _gm

l ;_hm

l g to the flow coefficients fT ml ;S

ml g (e.g., Whaler,

1986). However, the resulting equations cannot be straightfor-wardly inverted to obtain the CMB flow. Roberts and Scott(1965) and Backus (1968) showed that there is severe non-unique-ness involved, in that many flows acting on the main field generateno radial SV. Subsequently, a number of non-uniqueness reducingassumptions (that can be tested against the data) have been devel-oped. The most widely used are that the flow is toroidal, tangen-tially geostrophic, or steady, but constraints on its helicity canalso be imposed (see Holme, 2007, for a recent review). ‘‘Snap-shots’’ of toroidal, tangentially geostrophic or helical flow can beobtained directly with the additional constraint imposed, but stea-dy flows must be based on a minimum of three epochs to provide aunique solution (Voorhies and Backus, 1985). As with field model-ling, solutions are usually obtained by regularised inversion, mod-elling the data by a spatially smooth flow; here we minimise anobjective function with a regularisation term that minimises the‘‘strong norm’’ of Bloxham (1988)

ZCMB

r2Hvh

� �2þ r2

Hv/

� �2dS ð5Þ

236 K.A. Whaler, R. Holme / Physics of the Earth and Planetary Interiors 188 (2011) 235–246


To leading order, this damps velocity coefficients of degree l bya factor proportional to l5. The fit to data is determined in the tra-ditional way by minimising the mean square misfit to the secularvariation defined at the Earth’s surface. Such a procedure is appro-priate for historical epochs during which observational error islikely to be responsible for the largest uncertainty in observed sec-ular variation; for more recent epochs, the effects of small-scaleflow (Eymin and Hulot, 2005) or diffusion (Holme and Olsen,2006) would be significant.

We use the steady and tangentially geostrophic flow assump-tions to represent the range of results; previous studies haveshown that the large scale flow obtained depends primarily onthe data and regularisation, with non-uniqueness reducingassumptions having a secondary effect on the appearance of theflow, i.e., in practice, regularisation imposes uniqueness (e.g.,Holme, 2007).

Differences between flows can be gauged by the root-mean-square (RMS) flow speed difference as well as visually. Since(Whaler, 1986)Z

CMBv2dS ¼ 4p

Xl;m

lðlþ 1Þ2lþ 1

ðT ml

cÞ2 þ ðT ml

sÞ2 þ ðSml

cÞ2 þ ðSml

sÞ2� �

ð6Þ

the RMS Dv is given by

Dv2RMS ¼

Xl;m

lðlþ 1Þ2lþ 1

ðDT ml

cÞ2 þ ðDT ml

sÞ2 þ ðDSml

cÞ2 þ ðDSml

sÞ2� �

ð7Þ

where the DT ml and DSm

l are the differences between coefficients oftwo models.

We can also calculate the correlation between flows. Since theyaverage to zero over the CMB, the correlation coefficient betweenflows v1 and v2 is

r ¼R

CMB v1 � v2dSffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRCMB v2

1dSR

CMB v22dS

q ð8Þ

whereZCMB

vi � vjdS ¼ 4pXl;m

lðlþ 1Þ2lþ 1

T ml

ci T

ml

cj þ T

ml

siT

ml

sj þ S

ml

ci S

ml

cj þ S

ml

siS

ml

sj

� �

ð9Þ

The gufm model coefficients were modified as follows to inves-tigate the effect on flow inversion of the unknown field intensity inthe pre-Gauss era. First, the axial dipole coefficient was fixed at its1840 value, the most extreme case proposed by Finlay’s (2008)compared to the �15 nT/year change imposed in the gufm model:

g01ðtÞ ¼ g0

1ð1840Þ ð10Þ

Then, to preserve the values of the declination and inclinationangles (the pre-Gauss era data) predicted by the model, so thatthe fit of the original and modified model to the data is identical,the remaining coefficients must be scaled by the ratio of the axialdipole values:

aðtÞ ¼ g01ð1840Þg0

1ðtÞð11Þ

to give

gnew ¼ goriga ð12Þ

where g denotes a gml or hm

l other than g01. The axial dipole SV coef-

ficient is zero, since g01 is fixed at its 1840 value:

_g01 ¼ 0 ð13Þ

and the values for the other SV coefficients become (differentiatingEq. (12))

_gnew ¼ _gorigaþ gorig _a ð14Þ

To a close approximation, _a is a constant since gufm assumes alinear decay of g0

1 at a rate much smaller than g01 itself. For the lar-

ger (i.e., lower degree) SV coefficients, the effect is approximately alinear scaling since in the right-hand side of Eq. (14) the first termdominates, with the obvious exception of the axial dipole term. Amore general, if less conceptually revealing, treatment can be ob-tained by a full rescaling of gufm: coefficients are calculated as afunction of time, rescaled according to Eq. (11), and new splinecoefficients obtained by a least-squares fit. The results from thetwo methods are not distinguishable.

We also investigated the effect of imposing a constant axial di-pole, and a constant axial dipole decay rate, throughout the fullgufm era, simply to test how sensitive the flows are to its rate ofchange. Rescaled spline coefficients in identical format to that gi-ven for gufm are included as electronic attachments to this paper,for both a constant axial dipole (mmc1.txt), and for constant axialdipole decay rate (mmc2.txt). Since the intensity of the field is con-strained by measurements in the final 150 years of the model,there is no straightforward change that can be made to the othercoefficients that means that the modified model fits the data aswell as the original. The predicted field direction, however, remainsunchanged, and so we altered the coefficients as described above,recognising that this modifies the fit to the post-1840, but not tothe pre-Gauss era, data.

3. Results

The biggest change to the gufm coefficients from assuming(Finlay’s (2008)) most probable axial dipole behaviour, i.e., nochange, occurs at the start of its period of validity. We thereforeinvestigate the effect of the assumed axial dipole strength on theflow for the late 16th and early 17th centuries. A steady flow cal-culated from coefficients for epochs 1590, 1600 and 1610 is shownin Fig. 1(a). We investigated a range of damping parameters con-trolling the relative importance of fitting the data and regularisingthe flow; the flow shown in Fig. 1 is our preferred model. A firstobservation is that this flow (and others with different dampingparameters) looks significantly different from flows deduced from20th century field models (or even directly from data), a point towhich we return later. The modified flow deduced from field andSV coefficients with axial dipole fixed at the 1840 value is shownin Fig. 1(b), for the same damping parameter. We could insteadhave presented flows with the same fit to the data, or the samesolution norm, as Fig. 1(a) but they are virtually identical to thatshown in Fig. 1(b). However, we must bear in mind that afterrescaling the SV and setting _g0

1 to zero, flows obtained with thesame damping parameter will be slower (because there is lesspower in the SV) and have a smaller sum of squares of residuals(SSR) between the input and flow predicted SV coefficients(because the modified SV coefficients themselves are in generalsmaller in magnitude). However, we have established that thedifferences between the original and modified flows are notcorrelated with the flows themselves (either on a degree-by-degree basis or overall), indicating that reducing the amount ofSV does not simply weaken the original flow. The original andmodified flows are highly correlated, with a coefficient given byEq. (8) of 0.99. They are also very similar visually, and this isconfirmed when the difference between them is plotted(Fig. 1(c)). The RMS flow speeds are 11.2 km/year (original) and10.4 km/year (modified), and their difference is 1.9 km/year. This

K.A. Whaler, R. Holme / Physics of the Earth and Planetary Interiors 188 (2011) 235–246 237


difference may seem high, but could be substantially reduced by adifferent choice of damping parameter. RMS speeds and othercharacteristics of the flow for all models discussed are give inTable 1.

The significant difference between the flows presented in Fig. 1and those from more recent eras prompted us to test whether the20 year time interval over which the flow was assumed steady wastoo short for changes in the main field to be reliably determinedfrom so few data. We therefore doubled the time period and soughta flow steady over 40 years from coefficients at epochs 1590, 1610and 1630. Again, the flow determined has little in common withmodern era flows, and, as for the shorter time interval, modifyingthe coefficients to match the 1840 axial dipole value makes littledifference to the resulting flow (Table 1).

It is well known that ‘‘end effects’’ can afflict regularised mod-elling of data. For example, gufm is penalised to minimise asquared norm of secular variation. This is balanced by the require-ment also to fit the data, but near the end points there are no datato fit on one side of the interval, so the model minimises theobjective function by tending towards uniform secular variation.This effect is in addition to the fact noted above that data for theearliest epochs are limited, and so gufm may not be as good then.Therefore we limited the earliest date of our next calculation to1630, and calculated a steady flow for 1630–1670, as shown inFig. 2(a). Some flow features are at variance with those obtainedfrom more recent eras, such as strong southward motion alongthe Greenwich meridian and comparatively little westward driftin the hemisphere centred on it. However, several of the flow

20.0km/yr

20.0km/yr

20.0km/yr

Fig. 1. (a) CMB steady flow for 1590–1610 from gufm; (b) after modifying coefficients as described in the text such that the axial dipole is constant; (c) difference betweenflows from original (a) and modified (b) coefficients (continent outlines on this and subsequent plots shown for reference).



Table 1Statistics of flows using the original and (modified) coefficients.‘‘S’’ and ‘‘G’’ in the ‘‘Type’’ column indicate steady and tangentially geostrophic flows, respectively. ‘‘Solution norm’’indicates the value of the square root of the integral given by Eq. (5). ‘‘�v ’’ and ‘‘ �dv ’’ are the root-mean-square flow speeds and flow differences, respectively. r is the correlationcoefficient given by Eq. (8). Note that the SSR for the single epoch tangentially geostrophic flow is from a fit to a factor 3 smaller number of SV coefficients, but that thetangentially geostrophic constraint approximately halves the number of independent model parameters.

Epoch(s) Type Original (modified) Difference

k SSR (nT/year)2 Solution norm (103 km/year/rad2) �v (km/year) �dv (km/year) r

1590 1600 1610 S 2 � 10�4 81.4 (62.4) 203 (127) 11.2 (10.4) 1.9 0.991590 1610 1630 S 2 � 10�4 248 (203) 722 (589) 14.9 (14.2) 2.0 0.991630 1650 1670 S 2 � 10�4 255 (218) 422 (311) 9.2 (8.2) 2.1 0.981630 1650 1670 S 2 � 10�5 88.2 (78.9) 3324 (2778) 19.9 (18.6) 2.6 0.991650 G 2 � 10�4 200 (87.2) 545 (258) 12.7 (10.7) 4.4 0.941630 1650 1670 SG 2 � 10�4 938 (570) 1368 (687) 13.3 (10.8) 5.1 0.931900 1920 1940 S 2 (1.85) � 10�4 412 (298) 529 (530) 14.7 (14.7) 4.1 0.96

20.0km/yr

20.0km/yr

20.0km/yr

Fig. 2. As for Fig. 1 for the period 1630–70.



features recognisable from recent era flows (e.g., Holme, 2007) areapparent, such as considerably slower flows beneath the (north-ern) Pacific, an anti-clockwise gyre beneath the Indian Ocean,and a southward jet from the north pole at around 90�E (here orig-inating from a little further east). This suggests that parts of thelarge-scale flow pattern have changed little over the interveningthree or more centuries, and points to the influence of the lowquantities of and uneven data distribution in the early gufm epochs(though the possibility that the flow did change around that timecannot be ruled out). Once again, a flow deduced from field andSV coefficients changed to match the 1840 axial dipole value is vir-tually identical to the original (Fig. 2(b) and (c); Table 1).

We might expect any differences between the flows deducedfrom the original and modified gufm models to be more apparentwhen fitting the data more closely, so to this end the damping

parameter was reduced, by a factor of 10. The flows, presented inFig. 3, are now considerably more energetic, but again are virtuallyindistinguishable visually, and the RMS difference between them iscomparable to values obtained previously (Table 1). The features ofthe flows are broadly similar to those obtained with a larger damp-ing parameter in Fig. 2 (meaning the flow is robust), though somesmaller scale features are apparent. Note particularly that thechange in the damping parameter has led to much greater changesin the flows than changes in the normalisation for field intensity.

For all four sets of flows, the differences between them,although small, are most obvious in two regions, beneath thesouthern Atlantic and Indonesia. The differences are systematic –the difference flow is north-pointing. There is also a distinct pat-tern in the differences between the coefficients, illustrated inFig. 4(a). As with the coefficients themselves, the differences

20.0km/yr

20.0km/yr

20.0km/yr

Fig. 3. As for Fig. 2 but with smaller damping parameter.



decrease with increasing harmonic degree. Consistently amongstthe largest differences are to the poloidal coefficients S0

1, S02 and

S04. That S0

2 changes substantially is not surprising: this is the onlyflow coefficient that generates axial dipole secular variation fromthe axial dipole main field, at the CMB still the largest contributionto the main field by an order of magnitude. The other axial poloidalflow term differences presumably compensate for the effect of thischange on other SV coefficients. Changes to the other zero ordercoefficients tend to be smaller, including to T 0

1 and T 03, which

determine predictions of changes in length-of-day arising fromangular momentum exchanges between the core and mantle (seeSection 4).

Hence with steady flows, the influence of changing the rate atwhich the dipole decays over time is small, especially when com-pared with the effect of changing the damping (fit to data). To testwhether the result is dependent on the choice of non-uniquenessreducing assumption, we constructed tangentially geostrophic(TG) flows from the original and modified gufm coefficients for1650. Tangential geostrophy is applied by further flow regularisa-tion (e.g., Holme, 1998). In Fig. 5, we plot the flows and their differ-ences. These flows are even more like those obtained frominverting 20th century SV, with flow in the hemisphere centredon the Greenwich meridian dominated by westward drift, espe-cially south of the equator, and a clockwise gyre beneath the North

Fig. 4. Absolute values of coefficient differences between original and modified flows for toroidal (left) and poloidal (right) parts of the flow, plotted as a function of degree, l,along the ordinate and order, m, along the abscissa, with negative m corresponding to coefficients multiplying sinm/ in Eq. (3). (a) 1630–70 Steady flow (damping parameter2 � 10�4); (b) 1650 TG flow; (c) 1630–70 steady, TG flow; (d) 1900–40 steady flow. Greyscale range is 0–1 in (a) and 0–1.5 in (b)–(d).



Atlantic ocean. However, in this case there are noticable visual dif-ferences between the original and modified flows; the differencesare not confined to particular parts of the core surface, and arenot in the same direction as those for steady flows beneath thesouthern Atlantic and Indonesia (where the differences areamongst the smallest). There is also a larger flow difference,DvRMS = 4.4 km/year, a large (more than a factor of two) changein goodness-of-fit, and a lower correlation coefficient (Table 1).The changes in the flow coefficients differ from the purely steadyflows, not surprisingly because for those flows the greatest changeswere seen in zonal poloidal terms S0

l , which for tangential geostro-phy are required to be identically zero. Changes to the toroidal flowcoefficients are significant to a higher harmonic degree, whereasthose to the poloidal flow coefficients tend to be largest form ’ l (Fig. 4(b)). The result is not sensitive to how strongly the

TG constraint is imposed. However, TG flows provide a poorer fitto the data than steady flows for an equivalent flow strength, con-sistent with previous studies (e.g., Bloxham, 1989). The largechange in the SSR between the original and modified flows arisesfrom the widely differing ability of the two to predict the axial di-pole secular variation. In non-TG flows this is accomplished pri-marily by the action of S0

2 on the axial dipole, but this coefficientvanishes for TG flows, so _g0

1 is generated by flows advecting smalleramplitude field coefficients, and is thus more sensitive to the de-tails of the flow and the field on which it acts. The two flows pre-dict the non-axial dipole coefficients to a similar accuracy.

Steady, TG flows for 1630–1670 from both the original andmodified coefficients look more like the 1650 single epoch TGflows than the 1630–1670 steady flows (Fig. 4(c)), so tangentialgeostrophy, rather than steadiness, appears to be more important

20.0km/yr

20.0km/yr

20.0km/yr

Fig. 5. As for Fig. 1, but for a single-epoch tangentially geostrophic flow for 1650.



for controlling the flow morphology. The differences betweenflows obtained from the original and modified coefficients alsomirror those between their single epoch TG counterparts, i.e.,extending to high degree for the toroidal coefficients and concen-trated in coefficients for which m ’ l for the poloidal ones. TheRMS flow difference, at 5.11 km/year, is the largest of all the flowpairs considered here, and the correlation coefficient, at 0.93, isthe lowest.

To compare with the pre-Gauss era results, we examine the ef-fect on a steady flow of undertaking the same modifications to aperiod in the 20th century when the quantity and quality of datamakes the field and SV, including the axial dipole and its rate ofchange, well constrained. We cannot make the changes as far for-ward as we went back from the time of the first intensity measure-ments to make an approximately equivalent change to the axialdipole. Thus we chose instead a 40-year period centred on 1920when _g0

1 was a maximum and its value before and after highly var-iable (Jackson, 2000), so setting it to zero has the largest impact.The gufm field and SV are altered as for the pre-Gauss era andwe calculate steady flows for 1900–1940. Note in this case weare increasing the magnitude of the field coefficients but stilldecreasing the power in the SV because of the dominance of zero-ing _g0

1, though in this case only marginally. Again a visual inspec-tion shows little difference between the flows calculated fromthe original and modified coefficients. However, modern field mod-els are more accurate, so a more rigorous comparison of the flowsdeduced from them is required. Hence, rather than simply compar-ing flows obtained with the same damping parameter, we match arelevant flow metric. The sum of squares of residuals to the SVcoefficients is not an appropriate choice, as one would expect abetter fit for the weaker SV when _g0

1 is set to zero, and this is borneout by the results obtained from the pre-Gauss era (see Table 1).Thus we reduce the damping parameter when inverting the mod-ified coefficients to match the solution norm, the quantity mini-mised along with the SV coefficient fit. Despite the significantmodifications made to the field and SV coefficients in this case,the two flows are very similar. The RMS difference between themis 4.1 km/year (compared to RMS flow speeds of almost 15 km/yearfor both flows); this rather large RMS difference arises primarilyfrom changes of 1 km/year or more in coefficients T 0

1, S02 and T 1

2c

(Fig. 4(d)). However, there is very little structure to the pointwisesmall flow differences when they are plotted. The ‘‘westward drift’’coefficient T 0

1 is unusually large (� �10 km/year) for this period(see also Jackson, 1997, Fig. 8(a)). The two solutions are well corre-lated, with a correlation coefficient of 0.96.

The preceding calculations all suggest that CMB flows are insen-sitive to the value and temporal variability of the axial dipole. As afinal test of this, we inverted for two sets of TG flows covering thefull gufm timespan. In the first, the axial dipole was assumed to de-cay at a constant rate (defined by its values in 1840 and 1990); inthe second, the axial dipole value was fixed (again at its 1840 va-lue). In both cases, the other coefficients were scaled as describedabove to preserve the predicted inclinations and declinations. Thecalculations were performed for three damping parameters, givingdifferent fits to the data, with consequent variation in flow magni-tude and complexity. Using the same three choices of dampingparameter throughout does not account for differences in gufmmodel uncertainties and resolution over time, but retains theadvantages of consistency and simplicity. Apart from a short periodduring the early- to mid-twentieth century, when the axial dipolewas very rapidly varying (Jackson, 2000), the models are depen-dent primarily on damping parameter, with relatively small differ-ences depending on the assumption about the axial dipolebehaviour. In all cases, the smooth models are very similar. Themisfit and model solution norm are both generally lower for inver-sions of data modified to have no axial dipole SV (for the same

epoch and damping parameter), implying that it is easier to fitthese modified data. This is consistent with the observations ofHolme and Olsen (2006), and the flows of Jackson reported by(Whaler and Davis, 1997) who noted that the amplitude of _g0

1

tended to be anomalously under-predicted by TG flow models(although its spatial variability is well reproduced), possibly dueto diffusion, and that flow models are only weakly sensitive tothe uncertainty assigned to this coefficient in a weighted (variabledata uncertainties) inversion.

4. Geophysical implications

In the previous section we have demonstrated that the flowsdepend only weakly on assumptions about the rate of dipole decay,and therefore it is appropriate to examine flow models prior to1840 (and the absolute determination of intensity) with only a lit-tle less confidence than flow models from later epochs. In particu-lar, changes to the rate of dipole decay have considerably lessimpact on flow structure than a change in damping parameter,or to the application of additional nonuniqueness reducingconstraints.

Why is this the case? If we consider the frozen-flux inductionEq. (2), and allow for the change in scaling, we obtain

_Br þ Br_aaþrH:ðvBrÞ ¼ 0 ð15Þ

(also given by Jackson (2000) as his Eq. (4)). The new term is thechange to the SV caused by the change in the dipole scaling. Formost low-degree SV coefficients, which dominate the part of theSV the flow must fit, the first term dominates the second. In otherwords, the change to the SV from the rescaling is relatively small,as also noted by Huy et al. (2000b). Hence, although Jackson(2000) was correct to state that calculating flows from pre-Gaussera models is strictly invalid, we have demonstrated that for realis-tic variations in field intensity the flows are insensitive to its precisevalue. To illustrate why this might be the case, consider ratios of Br

and _Br to those of a to _a (defining the relative sizes of the first andsecond terms). Estimating the former in the root-mean-squaresense on a harmonic degree basis as the timescale of that compo-nent gives values in the range 100–200 years for degrees 2–7 (Hulotand Le Mouël, 1994). The latter is just g0

1/ _g01, which even for periods

of rapid change such as around 1920 is 1000 years. In fact, thisunderplays the discrepancy between the two timescales, since thelarger values for the first term come from times when the strengthof a particular degree is low, in which case it is less significant forcore flow modelling – stronger flow will be generated when SVcoefficients are large, and main field coefficients small, i.e., for shorttimescales. At the Earth’s surface, degree 2 SV generally contains asignificant amount of power, and has a timescale of approximatelyan order of magnitude less than the a ratio at all times. For the axialdipole, the first term of Eq. (15) is obviously not dominant as thelarge contribution to Br leads to a sizable change to the SV; however,as has been discussed, this coefficient is notoriously badly fit byflow models. There are good reasons why this might be the case;for example, unmodelled effects of diffusion are likely to be obser-vationally most significant in this component of the field, becauseof its large magnitude (Holme and Olsen, 2006). Further, the influ-ence of small scale flow and field giving rise to large scale SV mayalso increase the uncertainty to which this part of the SV shouldbe fit (Eymin and Hulot, 2005; Pais and Jault, 2008). Finally, themagnitude of the axial dipole field has no effect on the SV predictedby TG flows (LeMouël, 1984).

Consideration of Eq. (15) also explains why the difference plotspresented in Figs. 2 and 3 look similar. The first two terms can beconsidered as two sources of SV; as the flow equation is linear, we



can split the flow into parts which fit each SV term. The flowchanges fit the second (new) term, in which the large contributionfrom the modification to the axial dipole SV is modelled adequatelyeven in the case of strong damping, and so decreasing the dampingfurther does not greatly alter this part of the flow.

Of course, all of our results depend on the variation in fieldintensity being within the range of that observed historically;had we modelled the dipole decay as an order of magnitude greaterthan seen today, substantial flow differences would be expected.However, we have no reason to adopt such extreme models.

What geophysical consequences follow from these observa-tions? The large-scale flow back to the mid-seventeenth centuryshows features that are broadly similar to more recent flows, inparticular circulation under Siberia, westward flow under theAtlantic, and the southern gyre. Indeed, these flows arguably showmore coherence than the field and SV from which they are derivedwhen compared between epochs in the pre- and post-Gauss eras. Apossible explanation for this is that the field model demonstratesclearly the influence of changing data availability (in generalincreasing in complexity with time as the data improve in quality,quantity and distribution); this is an unphysical effect, fortunatelynot modelled by the flow, which instead is dominated by the com-mon large-scale features that do not change greatly with time.These models provide support for the dynamo being in a regimewhich is close to steady, consistent with locking to the lower man-tle (e.g., Willis et al., 2007). In fact, we can trace the development ofthe familiar broad-scale flow pattern through changes in the field.The high latitude flux patches beneath the Americas are evidentfrom the beginning of gufm, but that beneath Siberia only becomesproperly established in around 1680. Its subsequent intensificationdoes not influence the first-order feature of TG flows, which isaround flux patches. Thus, once the flux patch pattern is estab-

lished, the corresponding basic geometry of this part of the globalflow is established. Changes in the gufm field model centred on theregion beneath the southern Indian Ocean give rise to the gyrethere that resolves around the end of the 17th century. Again, thissuggests either that the flow pattern becomes similar to today’saround that time, or more likely that an improvement in field res-olution enables the basic flow pattern that persists throughout tobe recovered by inversion from then on. The comparison breaksdown for the very earliest epochs, as seen in Fig. 1. This may bedue to the geomagnetic data being too limited, or to gufm end ef-fects (see, for example, Fig. 10c in Jackson et al., 2000). The evolu-tion of the field model shows considerable differences in this earlyperiod; as a result, we suggest that prior to 1640, flows generatedfrom it should be treated with caution.

From these flows we can predict core angular momentum andconsequent changes in length-of-day (LOD) over the completetimespan of gufm. A number of studies, beginning with those ofJault et al. (1988) and Jackson et al. (1993), have demonstrated thattime-varying core flows adequately conserve angular momentumin the core–mantle system, assuming that torques on the core ex-cite motions on cylindrical annuli centred on the rotation axis. Thecalculated LOD variations involve only two zonal toroidal coeffi-cients, being given by

dT ¼ 1:138 dT 01 þ

127

dT 03

� �ð16Þ

with variation in length of day dT measured in milliseconds, and theflow coefficients in km/year. In Fig. 6 we present angular momen-tum predictions for the three sets of TG flow models describedearlier: firstly, derived from gufm, secondly from gufm modifiedto have a constant dipole decay rate, and finally from gufm modified

1600 1650 1700 1750 1800 1850 1900 1950Year

−6

−4

−2

0

2

4

6

8

10

12

ΔLO

D (m

s)

Constant dipoleConstant dipole decaygufmObserved LOD

Fig. 6. Predictions of variations in length of day from tangentially geostrophic flows with different assumptions about dipole behaviour (see text for further details). Dampingparameters are 2 � 10�2 (black lines – heavy damping), 2 � 10�3 (red lines – intermediate damping), and 2 � 10�4 (blue lines – as used for the majority of the flows alreadypresented). y-intercept is arbitrary; flow curve predictions are offset from observed LOD changes for easy comparison. (For interpretation of the references to colour in thisfigure legend, the reader is referred to the web version of this article.)



to have constant axial dipole strength over the whole period. Whatcan be concluded from this figure? First, and most clearly, as withthe flow plots, the changes due to different assumptions concerningthe decay of the dipole are smaller than the changes due tovariations in damping parameter – the complexity we choose forthe flows. In particular, the predicitions for the three sets ofsmoothest flows are almost indistinguishable for most of the inter-val. Jackson et al. (2000) noted that there is considerable variationin axial dipole SV between 1840 and 1990; the effect is quantifiedby comparing the curves for flows derived from gufm and aftermodifying it by assuming constant dipole decay post 1840. Differ-ences can be seen, particularly around 1920 when the dipole decayis at its maximum, but nonetheless there is no change in the overallform of the curves. Prior to 1840, the curves derived from a lineardipole decay (i.e., the original gufm model) and from gufm with aconstant dipole do show offests, but these offsets are slowly vary-ing, implying little change in the variation in LOD. Note that thevariations shown for our flows are probably larger than are appro-priate. We minimise mean square misfit to the observed SV atEarth’s surface, which as damping is decreased, results in a veryclose fit to the axial dipole. TG flows do not fit this variation well(see, for example, flows of Jackson in Whaler and Davis, 1997). Ifwe were to allow more realistic error bounds for _g0

1, either due topossible diffusional effects (Holme and Olsen, 2006) or to allowfor the influence of small scale flows on large scale SV (Pais andJault, 2008), these predictions would become closer. Using a null-space approach, Asari et al. (2010) conclude that both diffusional ef-fects and a non-TG flow component are required to fit the variation,although Asari et al. (2009) find that the non-TG component doesnot adversely affect the LOD variation predictions.

As with previous studies (e.g., Ponsar et al., 2003), the matchbetween LOD variations predicted by TG core flows and theobservations (Stephenson and Morrison, 1995) is excellent fromthe mid-1800s. Prior to that, back to around 1680, the patternof the variation is well predicted by the flows, but its amplitudeis much larger than that observed – the observed variations aremuch more subdued than in the post-1850 era (also as seen inthe alternative LOD model of (McCarthy and Babcock, 1986)from the same data), whereas the amount of variation predictedby the flows stays the same. This might be a problem with ourflows, but it could also arise from the LOD data. For the period1620–1800, these are derived mainly from timings of lunar occ-ultations and solar eclipses, depending on time integral of theLOD variations. The scatter in these data is large, as can be seenin Fig. 2 in (Stephenson and Morrison, 1984); if we integrate theLOD variation predictions from our flows, the resulting changesfall well within the scatter of the data. The flow predictions alsohave slightly more structure; for example, the increase in LODfrom 1710 to a maximum in 1760 is predicted by the flows tohave two peaks, at around 1730 and 1770, followed by a reduc-tion until 1790. A similar double peak in the predictions, com-pared to a single peak observed, occurs in the late 1800s. Priorto 1680, there is a large divergence between the observationsand predictions, the two being anti-correlated. As remarkedupon earlier, the core flows in this period do not resemble thoseinferred for later epochs, and we have some doubts about theirveracity.

5. Conclusions

We have shown that the lack of constraint on the fieldstrength in the pre-Gauss era has a relatively small effect onadvective CMB flows deduced from inverting SV coefficients.The flow changes tend to be concentrated beneath the southAtlantic ocean and Indonesia, areas that have experienced highSV. However, even in these areas, the changes are sufficiently

small that we feel able to consider flows calculated for epochsprior to 1840 in the same way as more recent flows. For similarphysical constraints, the main features of the pattern in currentcore flows appear to have been established by at least the mid-dle - end of the 17th century. In constrast, flows at the begin-ning of the period covered by the gufm field model look ratherdifferent from those deduced for recent times; this could reflect‘‘end effects’’ in the gufm model, a paucity of data, or a genuinechange in core flow morphology.

The uncertainty in field strength in the pre-Gauss era is usuallyaccommodated during field modelling by fixing the axial dipole va-lue, and, for temporally varying fields, its rate of change. We havealso demonstrated that the amount of axial dipole SV has noresolvable effect on the flow structure, even when (such as duringthe 20th century) it is the largest component of the SV at theEarth’s surface. This means we are justified in calculating flowsin the pre-Gauss era, comparing them to current flows, and com-paring their predictions of dT to the data. There are surprisinglymany large-scale features that are similar in current and earlierepoch flows (compare those presented here with, e.g., Fig. 1 ofHolme (2007)), for instance, the anti-clockwise gyre beneath thesouthern Indian Ocean, a clockwise counterpart to its east in TGflows, westward drift in a band centred on the equator beneaththe Atlantic hemisphere, weak flows beneath the Pacific in steadyflows, but an equatorial band of eastward flow in TG flows. There isa marked change in amplitude of the dT data in the mid-1800s thatis not reflected in the flow predictions, but the patterns match wellfrom the end of the 17th century.

Jackson (2000) suggested that the most appropriate way to con-strain the intensity prior to the start of measurements would be toensure that the field model obeys frozen-flux. This is undoubtedlytrue, and the variations in axial dipole amplitude we have imposedlead to field changes that are inconsistent with frozen-flux. How-ever, the original gufm is itself not consistent with frozen flux.The advective flows we calculate do not appear sensitive to therescaling of the field strength; therefore, the flow determinationmust be insensitive to any effective diffusion created by thisrescaling.

Most importantly, in this paper we have shown that, given areasonable estimation of how the field strength could have var-ied, the uncertainties in the overall magnitude of the field haveonly a small effect on the models of core flow derived from SV.This effect is much smaller than different choices of appropriatefit to data (through the damping parameter) or non-uniquenessreducing assumption (here steady or TG). Much remains to bedone: in particular, our parameterisation of fit to the SV is verynaive, with the axial dipole considered no worse than other SVcoefficients, when it might instead be reasonable to ignore thiscomponent of the SV entirely. Nevertheless, in showing thatflows can sensibly be calculated before 1840, we have doubledthe length of time available for calculating core surface flowswith which to constrain the evolution of the geodynamo.

Acknowledgements

This study was supported under the NERC GEOSPACE consor-tium, with Grants NER/O/S/2003/00674 and NER/O/S/2003/00675. As part of an undergraduate project, Christian Lynch per-formed some of the early calculations that convinced us this studywas worth undertaking. We thank Richard Stephenson and LeslieMorrison for their LOD time series, and for useful discussions,and Ciaran Beggan for assistance in producing Fig. 4. Themanuscript was improved by helpful comments from Guest EditorGauthier Hulot, referee Vincent Lesur and an anonymous referee.Flow plots were produced with GMT (Wessel and Smith, 1998).



Appendix A. Supplementary data

Gufm spline coefficients re-scaled such that the axial dipole isconstant (mmc1.txt) or decays at a constant rate (mmc2.txt).Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.pepi.2011.07.006.

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