Originally published as:

Wardinski, I., Lesur, V. (2012): An extended version of the C3FM geomagnetic field model: application of a continuous frozen-flux constraint. - Geophysical Journal International, 189, 3, pp. 1409—1429.

DOI: http://doi.org/10.1111/j.1365-246X.2012.05384.x Geophysical Journal International Geophys. J. Int. (2012) 189, 1409–1429 doi: 10.1111/j.1365-246X.2012.05384.x

An extended version of the C3FM geomagnetic field model: application of a continuous frozen-flux constraint

Ingo Wardinski and Vincent Lesur Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences, Section 2.3: Earth’s Magnetic Field, Germany. E-mail: [email protected]

Accepted 2012 January 11. Received 2012 January 11; in original form 2011 May 3

SUMMARY A recently developed method of constructing core field models that satisfies a frozen-flux constraint is used to built a field model covering 1957–2008. Like the previous model C3FM, we invert observatory secular variation data and adopt satellite-based field models to constrain the field morphology in 1980 and 2004. To derive a frozen-flux field model, we start from a field model that has been derived using ‘classical’ techniques, which is spatially and temporally smooth. This is achieved by using order six B-splines as basis functions for the temporal evolution of the Gauss coefficients and requiring that the model minimizes the integral of the third-time derivative of the field taken over the core surface. That guarantees a robust estimate of the secular acceleration. Comparisons between the ‘classical’ and the frozen-flux field

models are given, and we describe to what extend the frozen-flux constraint is adhered. Our GJI Geomagnetism, rock magnetism and palaeomagnetism models allow the interpretation that magnetic diffusion does not contribute to the observed secular variation. Additionally, the resulting frozen-flux field model shows a more or less constant spatial complexity, so that the spatial complexity of the magnetic field imposed by the 2004 satellite field model is maintained backward in time to the beginning of the model period, 1957. Therefore, we understand the frozen-flux constraint as an instrument that aids the backward projection of high spatial resolution in core field models to earlier times. Key words: Inverse theory; Magnetic anomalies: modelling and interpretation; Rapid time variations.

most powerful tool to diagnose core surface flow. Therefore, it is 1 INTRODUCTION important to testify the validity of this hypothesis. Measurements of the Earth’s magnetic field provide one of the few Bloxham & Gubbins (1986) discussed two categories of testing data sources to investigate the Earth’s deep interior. Changes of the the consistency of the frozen-flux hypothesis and geomagnetic ob- Earth’s magnetic field are of particular importance as they can be servations. The first category of test involves estimations of the used to probe geomagnetically relevant geophysical processes. The temporal changes of flux integrals either taken over individual flux induction equation portrays the processes causing these changes patches or the entire core surface. Consistency with the frozen-flux hypothesis would require that these changes are zero. Though sev- ∂ B =∇×(u × B) + η∇2B. (1) t eral studies have been dedicated to this test (see Jackson & Finlay The first term on the right-hand side describes the advection of 2007, for references), none of them demonstrate that flux changes magnetic field B by the fluid flow u and the second term represents can unambiguously be attributed to diffusion and not to uncertain- the diffusion of the magnetic field subject to the magnetic diffusivity ties in the field models. The second category includes field models η. Estimates of the relative importance of these two terms have that are built to fit observations and are constrained to satisfy the led to the approximation that the advective contribution is prior to frozen-flux condition, such that there is no temporal evolution of magnetic diffusion on a global scale and timescales shorter than a the magnetic flux through individual flux patches at the core–mantle few hundred years. The hypothesis that this approximation is valid boundary (CMB; Chulliat & Olsen 2010), and the preservation of is the so-called frozen-flux hypothesis Roberts & Scott (1965). the flux topology Constable et al. (1993); Jackson et al. (2007). Although, the appropriateness of this hypothesis has been criticized The results of these analyses suggest either conformance or failure Love (1999) and the role of diffusion promoted Amit & Christensen of the frozen-flux hypothesis. However, Backus (1988) showed that (2008), there are dynamical motivations to ignore diffusion on short inferences about the significance of magnetic diffusion at the core timescale Jault (2008). surface cannot be obtained without adequate priors. As a consequence of the frozen-flux hypothesis, the ambiguity We will revisit these two tests later in this paper. in the inversion of the induction equation for core surface flow In a recent study, Wardinski & Holme (2006) presented a contin- is reduced and turns the diffusionless induction equation into the uous model of the main field and its secular variation at the CMB

C 2012 The Authors 1409 Geophysical Journal International C 2012 RAS 1410 I. Wardinski and V.Lesur for 1980–2000. This model, C3FM, was constrained to fit satellite 2DATA field models at its endpoints. In this study, we follow the same ap- Thedatausedinthisworkarebasedonmeasurementstakenfrom proach; however, the developed model will cover a longer period, a network of geomagnetic observatories. Before the year 1957, this from 1957 to 2008, where the starting epoch, 1957, is chosen be- network mainly covered Europe, North America and Japan. Since cause at this time the number of magnetic field observations made then the number of observatories has increased rapidly and extended at geomagnetic observatories drastically increased. Similar to the to other regions of the globe. previous version, the new model is built to fit a MAGSAT satellite field model (Cain et al. 1989) for 1980 and the GRIMM field model Lesur et al. (2008) for 2004. The new model is constructed using order six B-splines to continuously model the temporal evolution 2.1 Observatory data of the main field, secular variation and secular acceleration. This is a conclusion of a comparison between the secular acceleration The main sources of data are a compilation of observatory an- energies of the xCHAOS field model (Olsen & Mandea 2008) and nual means provided by the British Geological Survey (BGS; GRIMM, which showed that the amplitude of the xCHAOS ac- http://www.geomag.bgs.ac.uk/gifs/annual_means.shtml) and the celeration energy tends to reach a maxima at node points and can database of observatory hourly mean values held at the World Data become unrealistically large (see fig. 7 of Lesur et al. 2008). The Centre for Geomagnetism (Edinburgh; http://www.wdc.bgs.ac.uk/). choice of using order six B-splines has implications for our mod- In the first step of the field modelling, three components, north- els of time-dependent core surface flow. So far, the quality of the ward (X), eastward (Y) and downward (Z), were compiled for each secular acceleration has not been discussed in its use for core flow observatory. Where available, monthly means were computed from inversions, even if this is implicitly done when a continuous field hourly means. We define monthly means as the average over all model is inverted for a time-dependent core surface flow. Tempo- days of the month and all times of the day. Otherwise, observatory ral changes in the core surface flow have been related to changes annual means were used. in the Earth’s rotation (Jackson et al. 1993; Jackson 1997) and to Additionally, the monthly means of 20 observatories have been geomagnetic jerks (Bloxham et al. 2002; Wardinski et al. 2008). transcribed from yearbooks of geomagnetic observatories which In Section 2, we describe the data set. Section 3 lays out the are not in the above compilations. Almost all of these are outside classical modelling approach with its treatment of noise related to Europe, Table 1 gives an overview of the locations and data periods external field variation. Section 4 is dedicated to a discussion of the of these observatories. frozen-flux field modelling method, which comprises simultaneous To check for continuity, the secular variation estimates of the inversion for the core field and core surface flow from magnetic field transcribed data were merged with those of the data held at the WDC. observation; its characteristics are specified. In Section 5, we present In one case of the observatory in Teheran, TEH, the differences the results of the two modelling approaches concerning field mor- between the two data sets exceeded a few hundreds nT yr−1 that phology, secular variation patterns and their posteriori variances. lead us to discard data of this observatory between 1970 and 1973. Based on the models posteriori variances, the magnetic flux at the The observatory annual means of the BGS-database form the core surface is discussed. This section ends with a comparison of backbone of our field model computation. However, as we will the two resulting core surface flow models and their fit to observed discuss later, the monthly means are needed to successfully reduce secular variation. We conclude in the last section and the presented contributions to the magnetic field measurements from sources of results are discussed in the light of previous studies. external origin.

Table 1. The table lists the locations of observatories, components measured and the epochs for which data were transcribed from yearbooks. These yearbooks are part of a Library held at the Niemegk Observatory. Name Code Colatitude East longitude alt. (m) components epochs Teheran TEH 35.730 51.380 1367 DHZ/DHI 1961–1963, 1966, 1968, 1971–1973 Quetta QUE 30.180 66.950 1737 XYZ 1959–1962, 1965–1970, 1973, 1974 Cha Pa CPA 22.350 103.833 1550 DHZ 1964–1967 Muntinlupa MUT 14.370 121.020 62 DHZ 1962 Chichijima CBI 27.093 142.179 154 DHZ 1964–1967, 1989, 1990 Hollandia HNA −2.570 140.520 230 DHZ 1960–1962 Tromso TRO 69.670 18.950 105 DHZ 1961–1963, 1966, 1968–1990 Ankara ANK 39.891 32.764 905 DHZ 1986–1989 Istanbul ISK 41.063 29.062 130 DHZ 1996 Jaipur JAI 26.917 75.800 0 DHZ 1979 Ujjain UJJ 23.183 75.783 0 DHZ 1979 Kodaikanal KOD 10.230 77.470 2323 DHZ 1990–1995 Shillong SHL 25.550 91.880 0 DHZ 1979 Teoloyucan TEO 19.750 260.820 2280 XYZ 1956–1964 Paramaribo PAB 5.820 304.780 2 DHZ 1958–1963, 1966 Tatuoca TTB −1.200 311.480 10 DHZ 1960–1962 Sao Miguel SMG 37.770 334.350 175 DHZ 1973–1977 Luanda Bales LUA −8.920 13.170 53 XYZ 1966, 1967 Lwiro LWI −2.250 28.800 1680 XYZ 1958–1970 Antananarivo TAN −18.920 47.550 1375 XYZ 1967–1970, 1981

C 2012 The Authors, GJI, 189, 1409–1429 Geophysical Journal International C 2012 RAS C3FM2 1411

2.2 Repeat station data estimates of the X-component becomes

The data collected at repeat stations are usually quasi-daily and daily dX/dt|τ = X(τ + 6) − X(τ − 6), (2) mean values which are finally reduced to a common epoch by mak- τ ing use of nearby geomagnetic observatory data or a geomagnetic where denotes a particular month. Likewise, observatory annual field model, such as the International Geomagnetic Reference Field means are treated using (IGRF). This procedure, in turn, may reduce the significance of the dX/dt|t+1/2 = X(t) − X(t − 1), (3) measurements and may bias the measurements towards geomag- netic field models. This should not be a problem in the first place. where t is in years. Secular variation estimates derived by (2) and − However for repeat stations outside Europe, North America and (3) are nT yr 1. These estimates are also computed from the repeat Japan, the nearest observatory is usually hundreds and sometimes station data. To obtain secular variation estimates, data are consid- more than thousand kilometres away. Long distances between obser- ered where the occupations are at least a year but not more than vatories have resulted in regions where the magnetic field is not very 10 yr apart. This approach is crude as it implicitly assumes secu- well known by either observations or field models. Therefore, such lar variation to vary linearly at these stations between two sequent reduction techniques load some uncertainties on the repeat station occupations. For estimates made from occupations that are only data. Our first choice would be to use unreduced repeat station data, separated by a few years this may yield acceptable values, but the but these are not available. Consequently, the repeat station data secular variation estimates are becoming imprecise for longer inter- used in this study are considered as add-ons to the observatory data. vening periods between occupations. The benefit of more sophis- For regions where the coverage with geomagnetic observatories is ticated methods of secular variation estimation from repeat station good, there will be no improvement in the recovering magnetic field data could be investigated in future. changes by adding repeat station measurements. Therefore, only The data transcribed from yearbooks have been processed in repeat station data from regions sparsely covered by geomagnetic the same manner and the standard deviation between the data and observatories were used; neglecting most of the observations made a initial field model are computed. Here, the standard deviation at European repeat stations. Only data collected at the western and is taken as a proxy of the data quality. Table 2 lists the standard eastern European margins are taken into account. This also applies deviation of the secular variation estimates. Only a few observatories to North America (except Alaska) where also a dense network of show extreme deviations (TEH, TEO and KOD), whereas the rest geomagnetic observatories exists. However, there are regions where exhibits minor to moderate deviations from the initial field model. repeat station data are the only source of geomagnetic field mea- Fig. 2, shows the temporal distribution of processed observatory surements, for example, the near East. Here, repeat surveys were and repeat station data. carried out in 1980 and 1983 in Saudi Arabia; in 1956, 1963 and 1966 in Egypt and in the 1990s in Israel. The surveys made in Egypt and Saudi Arabia were conducted by various surveying companies 3CONSTRUCTIONOFTHE to explore natural resources, who mapped the local magnetic field NON-FROZEN-FLUX FIELD MODEL with a high spatial resolution. However, only a few stations have The details of the model parametrization of the non-frozen-flux been visited more than once, from those the data have been ex- field model is largely similar to that used to derive the previous tracted. The data set from Vietnam (where also a dense network model. Not much of the construction has changed from the set up exists) has been treated similarly. Only data from the most northern and southern stations are extracted. Overall, we obtain a total of 522 Table 2. Standard deviation σ of stations with a total of 1058 secular variation triplets (X˙ , Y˙ , Z).˙ In secular variation estimates from Fig. 1, the global distribution of these stations is shown. transcribed observatory with re- spect to an initial field model. Units are in (nT yr−1). 2.3 Data processing Code σ X σ Y σ Z 3 Similar to the previous model, C FM, secular variation estimates TEH 238.741.2 116.8 are computed from observatory monthly and annual means, respec- QUE 16.311.316.6 tively. Then, for example, the computation of the secular variation CPA 16.810.318.0 MUT 15.511.810.7 CBI 14.64.74.3 HNA 31.419.840.1 TRO 14.99.813.6 ANK 11.512.317.5 ISK 12.75.010.4 JAI 27.27.834.2 UJJ 20.313.411.4 KOD 28.227.4 149.9 SHL 20.79.78.0 TEO 37.929.652.5 PAB 13.512.97.3 TTB 33.243.510.0 SMG 44.214.041.8 Figure 1. The locations of repeat stations used in this study. Large black LUA 22.613.523.0 stars mark stations with more than seven occupations, red stars mark sites LWI 17.86.85.7 with more than three and less than seven occupations, green stars mark sites TAN 14.612.112.2 with less than four occupations.

C 2012 The Authors, GJI, 189, 1409–1429 Geophysical Journal International C 2012 RAS 1412 I. Wardinski and V.Lesur

To assess the robustness of the secular acceleration, a second model is derived which minimizes the second-time derivative of the radial component of the field over the core surface    2 4π t2 ∂2B r d dt = mT N −1m . (8) − ∂ 2 T (t2 t1) t1 S(c) t Here, we report solutions that are adjusted to minimize the inte- gral of the horizontal derivative of B  r 4π ∇ 2 = T −1 ( h Br) d m NS m (9) (t2 − t1) S(c) suggested by Shure et al. (1982). In fact, we considered different spatial constraints such as minimizing the heat flux generated by Ohmic dissipation at the CMB Gubbins (1975) and minimizing the observable part of magnetic diffusion at the CMB Gubbins & Figure 2. The number of data with respect to time. The red curve indicates Bloxham (1985); Bloxham (1987). All constraints, when applied, the number of observatory annual means, the black curve shows the number provide similar results, which only differ in some smaller scale of observatory monthly means and the blue curve shows the number of features. repeat station data used in this study. Finally, the model covariance matrix is given by −1 = κ −1 + κ −1 , Cm s NS t NT (10) of the previous model, C3FM. Therefore, we briefly describe the κ κ construction the non-frozen-flux field model and refer to Wardinski where the damping parameters s and t control the spatial and & Holme (2006). temporal smoothness of the model, respectively. The Earth’s magnetic field B is assumed to be the gradient of The rms secular variation misfit is measured by differences be- =−∇ tween model and the observed secular variation, that is scalar potential V: B V and without magnetic sources the  potential has to satisfy Laplace’s equation ∇2V = 0, for which the   Nobs  1 solution for internal sources only in spherical geometry is M = (Obs − Mod)2, (11) N L l obs i=1 V (t) =∼ a (gm (t)cos(mφ) l where N obs is the number of observations. The total residual secular l=1 m=0 variation is (Obs − Mod)2.   + i a l 1 + hm (t) sin(mφ)) Pm (cos θ), (4) l r l 3.1 Solving scheme where a is the Earth’s radius (6371.2 km), (r, θ, φ) the geocen- tric coordinates and Pm (cos θ) are the Schmidt quasi-normalized Solutions are sought in a very similar manner as for the previ- l 3 associated Legendre functions ous model, C FM. We briefly recall the iterative re-weighted least-   squares scheme which is applied to all secular variation estimates 2π π 4π m θ φ 2 θ θ φ = obtained in Section 2.3: (Pl (cos )cos(m )) sin d d (5) φ= θ= 2l + 1 0 0 (1) An initial model is computed by setting the uncertainty of all with degree l and order m. The maximum spherical harmonic degree observatory data to 1 nT yr−1 and to 5 nT yr−1 for repeat station data, = { m , m } of (4) is chosen to be L 14. The Gauss coefficients gl hl are respectively. This down-weights the significance of repeat station expanded in time using order six B-splines M n(t) with respect to observatory data and can be justified by a rather N N crude estimation of the secular variation. m = mn , m = mn . (2) The deviations of the data from the initial model are calculated, gl (t) gl Mn(t) hl (t) hl Mn(t) (6) n=1 n=1 and adopted as new weights for the data. Data with large scatter from Like Lesur et al. (2010b), we seek a reliable estimate of the main the model are therefore down-weighted. field secular acceleration. Therefore, the temporal model constraint (3) A model is derived from the newly (re-)weighted data set. is to minimize the norm of the third-time derivative of the radial (4) Data are rejected with weighted residuals greater than ± −1 component of the field over the core surface 3nTyr . (50) An interim model is derived from this reduced data set.    2 4π t2 ∂3B r d dt = mT N −1m , (7) The next steps in the solving scheme of the previous model (t − t ) ∂t 3 T 2 1 t1 S(c) (C3FM) were a generalization of step 2, where the covariance be- where m the model vector containing the Gauss coefficients, S(c)is tween the different secular variation residuals in the (X,Y,Z) direc- a spherical surface of radius c = 3485 km, the estimated radius of tions at each location were considered. This allowed the inclusion the core, and t1, t2 are the model start and end epoch, respectively. of correlated errors in the inversion scheme. The time integral is computed using a Newton–Cotes formula of a We now alter the scheme. After the interim model is computed closed type, for example, Bode’s rule Abramowitz & Stegun (1973). (point 5) the covariant formalism is replaced by a technique of noise Minimizing the third-time derivative makes it necessary to place removal Wardinski & Holme (2011). The noise is very likely to be further conditions on the second-time derivatives of the radial field related to external field variations and it is found that removal of this at the model endpoints; best results are obtained when these are set signal enhances the resolution of fine-scale detail in secular varia- to zero. tion. In this study, the noise-removal technique has been extended to

C 2012 The Authors, GJI, 189, 1409–1429 Geophysical Journal International C 2012 RAS C3FM2 1413 cope with observatory data, which are discontinuous between 1957 Eq. (1) is applied at the core surface; however, only the radial and 2008 or shorter than the model interval. A minimum length for component of the magnetic field is known to be continuous across continuous data series of 100 months is required. the CMB. Therefore, the equation to derive core surface flow reads

∂B 1 ∂2 r + u ·∇ B + B (∇ · u ) = η (r 2B ) +∇2B . ∂ h h r r h h 2 ∂ 2 r h r 3.2 The role of a priori field models at satellite epochs t r r (13) Some of the spatial characteristics of the resulting field model are This is the radial induction equation, where η is the magnetic diffu- highly determined by the selection of the apriorifield model to sivity and ∇h =∇−∂/∂r r the horizontal gradient. In making the which the sought model should fit at the satellite epoch 1980. This frozen-flux assumption, the diffusive part is ignored (r.h.s.) and the is most notable for the field morphology, which becomes more and radial induction equation, reduces to more dissimilar to the ones of gufm1 Jackson et al. (2000), depend- ∂B ingonthegradetofittheapriorimodel in 1980. For instance, r + ·∇ B + B ∇ · = . ∂ uh h r r( h uh) 0 (14) a reversed flux patch under St Helena appeared around 1920 in t gufm1; however, we find this patch emerging between 1963 and To solve (14), further assumptions have to be made to reduce the 1970 whether the classical field model is constrained to fit the apri- non-uniqueness in the horizontal components of the flow. Among ori model in 1980 or not. This highlights a major difference between the assumptions which can be made (see Holme 2007, for details field models based on main field and secular variation data, such as and references), we assume the flow to be purely toroidal Whaler gufm1 and a model made up from secular variation data only. Ap- (1980), that is, ∇h · uh = 0. A series of studies have shown that parently, gufm1 shows some more spatial details, whereas our model this simplification is a reasonable assumption with an adequate is loosing some of these details. However, both types of models are fit to the observed secular variation (Bloxham 1988a; Lloyd & in good agreement with observed secular variation. However, the Gubbins 1990; Wardinski et al. 2008). Moreover, purely toroidal advantage of using secular variation estimates as model input is that flow has been found to contain a strong geostrophic component, these are not subject to crustal biases and annual external field vari- which explains about 90 per cent of the flow energy (Wardinski ation. As a consequence, to sustain similarity in spatial features with et al. 2008). other field models (e.g. gufm1), the classical field model requires The inversion of the classical field model is very similar to the an apriorisatellite field model in 1980 as fixed reference. description given by Wardinski et al. (2008). The flow solution is However, a problem arises from the application of the apri- sought to represent a large-scale flow and is truncated at spherical ori satellite field model for 1980 which was used in the C3FM. It harmonic degree 14. The flow spherical harmonics are constrained imposes a significant change in the total unsigned flux to converge by this limit. This is ensured by minimizing the norm  of the second spatial derivatives of the flow components Bloxham

U(t) = |Br| d . (12) (1988a). Also temporal smoothness of the flow solution is required S(c) and the knots of the spline interpolation are spaced as for Model This quantity is an important criteria in the verification of the frozen- 1. The appropriate spatial and temporal regularization is chosen by flux hypothesis and the two apriorimodels differ by several hun- considering a trade-off curve between rms secular variation misfit dreds MWb. With that flux change, it would be impossible to built and the mean flow velocity. The solution is chosen to be located at a frozen-flux field model. the ‘knee’ of this curve. The main cause of this problem is the way the apriorifield The flows are discussed in Section 5.7 and the inversion param- model for 1980 was obtained. This model was derived by tapering a eters and flow model characteristics are given in Table 7. MAGSAT field model, which was originally truncated at spherical harmonic degree 66 Cain et al. (1989). The intention of tapering, as described in Wardinski & Holme (2006) and Wardinski et al. 4 FROZEN-FLUX FIELD MODELLING (2008) is to control the uncertainties related to small-scale secular variation beyond spherical harmonic degree 12 or 13. A variety Several studies in the past have been dedicated to verifying the of tapered field models have been derived by applying a range of frozen-flux hypothesis (see Jackson & Finlay 2007, for refer- tapering factors. This in turn leads to different attenuation of the ences). Most recently, Chulliat & Olsen (2010) and Jackson et al. field strength determined by degrees 13 and 14 for 1980. However, (2007) tested this hypothesis. However, these studies differed in the a strong attenuation of the field at these scales leads to a large methodologies deployed. Central to both approaches is to quantify reduction in the total unsigned flux. Therefore, it is a trade-off, the magnetic flux through individual patches at the core surface and and as we are interested in building models which conform to the to constraint the model to have equal fluxes at different epochs. frozen-flux hypothesis, we revoke the idea of tapering and use a Here, we use an alternative approach of constructing frozen-flux truncated and untapered satellite field model for 1980. Then, the field models based on a co-estimation of the field and flow at the core total unsigned flux of the resulting model in 1980 differs by less surface from geomagnetic observations made at the Earth’s surface. than 100 MWb from the model in 2004. The method was previously developed by Lesur et al. (2010a) and is now applied to a longer data series of observatory measurements only. 3.3 Flow inversion To compare the core surface flow of the frozen-flux field model (see 4.1 Model construction next section), the classical field model is also inverted for a flow model. The inversion requires a brief outline of the assumptions For the readers convenience, this section briefly recalls the co- made in this study; we refer to Holme (2007) for a more complete estimation method, but we refer to Lesur et al. (2010a) for a more discussion of core flow inversion techniques. detailed description.

C 2012 The Authors, GJI, 189, 1409–1429 Geophysical Journal International C 2012 RAS 1414 I. Wardinski and V.Lesur

The frozen-flux field model is sought by co-estimating the core be forced to have a convergent spectrum. The flow is then required field and the core surface flow via the linear systems to minimize Bloxham’s ‘strong norm’ (Bloxham 1988b; Jackson 1997) y = A · g + ed , (15)   λ ∇ ∇ · 2 + ∇ ×∇ · 2 . and 2 ( h( h uh)) ( h(rˆ h uh)) d dt (20) T S(c) t m ˙ · = · + = li The damping parameter λ2 controls to what extent the flow follows (t) g Ag(t) f et where f m (16) sli {l,m,i} this constraint. Weexpect that minimizing (20) constrains efficiently are solved simultaneously. y is the observation vector, A the design the secular variation. Secondly, the flow model is chosen such that it varies smoothly in time matrix, g is the field model vector and ed is the uncertainty of the     observation. Further, Ag(t) is the matrix linking the flow model λ ∂ 2 , ˙ · 3 t uh d dt (21) vector f, to the secular variation vector (t) g. The uncertainty et T S(c) is introduced to account for the misrepresentation of the flow and where λ is the associated control parameter of the flow acceleration field temporal variability by polynomials. Furthermore, the flow is 3 ∂ u . We expect such a constraint to efficiently regularize the inverse decomposed into toroidal and poloidal components t h problem as in the limit of a constant flow there is a unique flow u = utor + upol =∇h × T rˆ +∇h rS. (17) solution of the frozen-flux equation (Voorhies & Backus 1985). Finally, it is required that the flow acceleration at the starting and T and S are scalars which are expanded in Schmidt-normalized real ending epochs is minimized by spherical harmonics in space and B-splines of order six in time,      represented by t m , sm . We assume the core surface flow is purely 2  li li λ4 ∂t uh d  , (22) = toroidal (i.e. upol 0), this simplifies the problem and reduces the S(c) t1,t2 computational costs significantly. However, the formalism could be where t1 and t2 are the epochs 1957 and 2008.4, respectively. extended to any assumption about the flow morphology. The set of Gauss coefficients g and flow coefficients f, truncated at spherical harmonic degree 14, are sought by a method of iterative 4.2 Solving scheme least squares. The Gauss and flow coefficients gm , sm , t m are li li li {l,m,i} To find a solution for the joint inversion of geomagnetic data for estimated such that they minimize the cost function the field and flow at the core surface, two strategies are applied.

= 0 + λ1 1, In the first strategy, the flow and field models are obtained after T −1 numerous iterations from a initial model, where all the flow and 0 = (y − Ag) Ce (y − Ag),  field coefficients are set to 1. The second strategy starts from initial = t · − ˙ · T · g˙ · · − ˙ · , field and flow models which are sequentially obtained by inverting 1 wi (Ag(ti ) f (ti ) g) W (Ag(ti ) f (ti ) g) geomagnetic data for a field model and then by inverting the field ti (18) model for the flow (a two-step inversion). From the combined initial λ where the scalar parameter 1 has to be adjusted. models, the final frozen-flux field model is obtained by iteration.

Herein, the functional 0 defines the weighted misfit between Usually, the iteration converged after 4 sometimes a few more steps data and model. The error covariance matrix Ce accounts for the er- depending on the settings of the damping parameters. The conver- ror in the data. The functional 1 is defined by the condition of min- gence of the first strategy is very slow, but both strategies have been imizing the mean square difference, integrated over the Earth’s core found to provide the same results. surface S(c) and time, between the observed secular variation and Contrary to the solving scheme of the classical field, re-weighting the secular variation generated by the flow—that is, if the elements and de-noising of the data are not applied here, simply due to g˙ g˙ = 4π(l+1)2 of the diagonal weight matrix W are defined by wl (2l+1) ,then computational costs. Therefore, the input data of the frozen-flux the functional 1 is the discrete equivalent of the integral field modelling are those obtained after the de-noising step of the     classical field model. 2 λ1 ∂t Br(t) +∇h · (uhBr) d dt. (19) Contrary to the solving scheme of the classical field model, the T S(c) use of an apriorifield model for 1980 is abandoned. If we used the However, the elements of Wg˙ have a more complex dependence same apriorimodel as in Wardinski & Holme (2006), then no model on the spherical harmonic degree. They account for the modelling could have been found which obeys the frozen-flux constraint. As errors in the induction equation and are given in Lesur et al. (2010a). pointed out already in Section 3.2, there are uncertainties in the In (18), the summation in 1 correspond to the integration over spherical harmonic degrees from 12 to 14 of the apriorifield t model in 1980 that render the estimates of the total unsigned flux time in (19). The sampling points ti and the associated weights wi are those of Gaussian integration rules and are such that the products as vague, that is a significant difference in the total unsigned flux of the different time-dependent terms are integrated exactly over the between 1980 and 2004 is possible as well as a match. Therefore, time span of the core field model T . no such constraint is applied in the solving for the frozen-flux field Because Ag(ti ) in the functional 1 depends on the Gauss coeffi- model to minimize the aprioribias. m Solutions for a range of values of damping parameters (λ , λ , λ ) cients gli and is multiplied by f, this optimization problem is clearly 2 3 4 non-linear and has to be solved iteratively. However, the iterative are explored. The parameter controlling the frozen-flux constraint λ = × −10 process is unlikely to converge unless some constraints are applied is unchanged, that is, 1 1 10 . Table 3 compiles the set of on the flow model because the problem is ill-posed for the flow model parameters and related model characteristics. The ratio    Holme (2007). To obtain the best possible magnetic field model fit- 2 ∂t Br(t) +∇h · (uhBr) d ting the data and simultaneously reducing the null space for the flow, R(t) = S(c)    (23) ∂ 2 two types of constraints are considered. First, the flow model can S(c) t Br(t) d

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Table 3. Specifications of the damping parameter frozen-flux field model. In all inversions, the parameter controlling the frozen-flux constraint is kept 5 RESULTS −10 fixed, that is, λ1 = 1 × 10 . The parameters R and M are defined by (23) and (11), respectively. M is given in nT yr−1. 5.1 The non-frozen-flux field model

Solution λ2 λ3 λ4 R M In section 4, we described our scheme of deriving a non-frozen-flux − − field model. We first want to discuss the effect of applying different 2A 1×10 2 1×102 1×103 0.1469×10 6 12.38 2Ba 1×10−2 1×102 1×10−4 0.1405×10−6 12.32 temporal constraints onto the secular acceleration. Therefore, two model solutions are sought. First, Model 1A is built by minimizing 2C 1×10−2 1×101 1×103 0.1602×10−6 12.23 − − − the third-time derivative of the radial component of the field over the 2D 1×10 2 1×101 1×10 4 0.1505×10 6 12.22 core surface, for example, (7), and secondly Model 1B minimizes × −2 × 0 × −2 . × −6 . 2E 1 10 1 10 1 10 0 1530 10 12 21 the second-time derivative (8). Model 1A is found by varying the − − − 2F 1×10 6 1×101 1×10 4 0.9528×10 0 11.73 spatial damping parameters, where the temporal damping is kept amarks the preferred solution. fixed, but have been adjusted to match major features of the secu- lar variation, such as known geomagnetic jerks. By analysing the trade-off between the model’s spatial complexity and the misfit, the Table 4. Statistics of the non-frozen-flux field models 1A and 1B. preferred solution is taken to be near the knee of the trade-off curve. M is the model misfit. We note that the model is more responsive to the spatial constraint Model 1A Model 1B than it was in the setting of Wardinski & Holme (2006), where the Number of data spatial constraint was basically disregarded. This is likely because retained 220 829 220 809 of the longer model time span and the greater variability in data den- rejected 451 471 sity and quality. Then, Model 1B is derived to same level of noise, total 221 280 221 280 that is, misfit. In Table 4, the models parameters and the misfit are − M (nT yr 1) 11.81 11.79 given. Damping parameters In Fig. 3, the morphology of the radial magnetic field component −2 −12 −10 κs (nT )7.2× 10 2.2 × 10 of the two model solutions at the core surface are displayed. The 3 −2 2 −2 −2 −3 κt [(nT yr ) ] [(nT yr ) ]7.0× 10 9.0 × 10 greatest differences in field morphology occur before 1970. After Model norms 1970, field maps are very similar. The map of Model 1B of 1965 2 12 12 Ns (nT ) 668.9 × 10 678.7 × 10 shows a higher complexity and also is very similar to maps of gufm1 3 2 2 2 × 3 × 3 Nt [(nT yr ) ] [(nT yr ) ] 38.164 10 1766.080 10 around this epoch. A further significant difference between Models 1A and 1B is 0 seen, when the temporal behaviour of g¨1 is compared (Fig. 4). 0 specifies to what amount the frozen-flux constraint is satisfied. Val- Clearly, g¨1 derived from Model 1B shows numerous maxima, which ues similar to 1 and larger mean that this constraint is not fulfilled coincide with the knots of the model B-spline parametrization. by the model, therefore values significantly smaller than 1 indicate These maxima certainly are not realistic but are rather a numerical that the frozen-flux constraint is approximately satisfied. artefact due to the temporal parametrization of the model similar to

Figure 3. The radial component of the magnetic field computed for the epochs 1960 and 1970 based on Model 1A (left-hand panel) and Model 1B (right-hand panel) . The colour-scale is adjusted to fit the maxima of the field at 2006. Zero-contour lines are added to aid the identification of reverse flux patches and the magnetic equator.

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0 Figure 4. The temporal behaviour of g¨1 derived from Models 1A and 1B. the problems pointed out by Lesur et al. (2008). On the other hand, reflect different stages of secular acceleration may be a matter of 0 g¨1 derived from Model 1A tends to show such maxima as well. future analysis. However, these are not as numerous as for Model 1B. Interestingly, In the following, the results derived with the temporal constraint some of the maxima of Model 1A match those of Model 1B (i.e. (7) are discussed. Model 1A therefore represents the non-frozen- around 1979), whereas some others appear to be slightly shifted flux field model and for the remainder of the study is labelled as (i.e. around 1999), and there is a period of no apparent maximum Model 1. Maps of the radial field at the CMB are shown in Fig. 5 0 for g¨1 of Model 1A between 1960 and 1970. Whether or not these and discussed in Section 5.3.

Figure 5. The radial component of the magnetic field computed from Model 1 for the epochs 1960, 1970, 1980, 1990, 2000 and 2008 at the core surface. The colour-scale is adjusted to fit the maxima of the field at 2008.

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Figure 6. The radial component of the magnetic field computed from Model 2 for 1959, 1969, 1979, 1989, 1999 and 2008 at the core surface. The colour-scale is adjusted to fit the maxima of the field at 2008.

5.2 The frozen-flux field model frozen-flux constraint, a reasonable fit to the data and shows the needed flexibility close to the model endpoints. Maps of the radial Table 3 lists a set of models which are constructed by invoking field of the frozen-flux field model, Model 2, are shown in Fig. 6. the frozen-flux constraint. Their characteristics differ as they are In the following, comparisons of several model characteristics are subject to varying damping parameters. And apparently, there is a examined. hierarchy of the efficiency of the damping parameters in altering the model characteristics. Like in Lesur et al. (2010a), a convergent flow spectrum is required to provide a reasonable fit to the data. 5.3 Field morphology From Table 3, it can be seen that varying the parameter that con- Before we compare the frozen-flux and the non-frozen-flux models, λ λ trols the model’s temporal flexibility (i.e. 3 and 4) has only a minor we discuss the radial magnetic field at the CMB for both models. effect on the misfit and the ratio R. This ratio is close to 1 when Snapshots of the radial magnetic field component of Model 1 at the flow spatial constraint is weakened, that is, (20). In this case, the core surface for different epochs, Fig. 5, shows a temporal the flow becomes more energetic, the fit to the data marginally im- evolution of the field morphology. Maps for the starting epochs proves, but the field model fails to satisfy the frozen-flux constraint. show only a few features, whereas maps for the later epochs are One possible explanation is that a weaker spatial constraint on the richer in distinctive features. This enhancement of spatial resolution flow increases flow complexity. This results in small-scale secular results from a spatial increase in the coverage of the geomagnetic variation which must eventually be balanced by magnetic diffusion. field measurements taken by satellite missions. The highest spatial It is widely accepted that the frozen-flux approximation should be resolution is reached between 2001 and 2008, when the field model valid on a large-scale for timescales considered here but becomes is constrained to fit the satellite magnetic field model GRIMM. less valid towards smaller scales. To this end, a higher flow com- The snapshots at 2000 and 2008 of Model 1 (Fig. 5) and Model 2 plexity, that is, substantial small-scale flow requires a higher spatial (Fig. 6) are very similar, and are entirely determined by the endpoint resolution than included here. constraint, which is the same for both models. The high-resolution Version 2B is our preferred frozen-flux model and is referred satellite data from the epochs 2001 to 2008 allow more small- to thereafter as Model 2. This model has the tightest fit to the scale features at the CMB to be distinguished, such as the reverse

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Figure 7. The spatial complexity of Models 1 and 2 as a function of time.

flux patches under California and Yucatan. However, these reversed ular variation patches of opposite signs under the Indian Ocean. flux patches are not very strong features and accommodate very These features are present throughout the model period and prop- minor magnetic flux of a few MWb. Towards the beginning of the agate westward. A close relationship between these features and model period, the snapshots of the two models are visibly different. a wave train of intense equatorial flux spots seen by an analysis For instance, the map of Model 2 at 1960 shows more structural of satellite data Jackson (2003) may exist, possibly resulting from complexity than Model 1 at the same epoch. This is the case for slow magnetic-Coriolis waves in the Earth’s core (Finlay & Jackson most of the epochs and the general impression that Model 2 has more 2003; Finlay et al. 2010). The existence of such particular structure structural complexity than Model 1 is true. The spatial complexity of secular variation may hold conclusions about the mechanisms of both models are measured by computing involved in producing them.  2 5.5 The posteriori model variance Br d (24) S(c) In general, drawing inferences about Br and its variation at the for given epochs. The results are shown in Fig. 7. Notably, the spatial CMB from magnetic field measurements taken near Earth’s sur- complexity of Model 2 stays more or less constant in time, whereas face requires some aprioribounds on the spherical harmonics of the spatial complexity of Model 1 varies dramatically. It is very high degrees at the CMB. These bounds place margins of con- likely that the spatial complexity of Model 1 is a consequence of fidence, that is, uncertainties concerning the modelled Br at the the varying data density (cf. Fig. 2). CMB. In the following, we attempt to examine the frozen-flux hypothesis. Therefore, it is needed to quantify the uncertainties in the computation of the magnetic flux through the reverse flux 5.4 Regions of intense secular variation patches. For the discussion of the secular variation at the CMB, we adopt By following Bloxham & Gubbins (1986), the related variance a method of mapping which emphasizes regions of higher secu- of the radial magnetic field at a point is given by lar variation Holme et al. (2011). In Fig. 8, maps of normalized T var(Br) = a Ca, (25) secular variation of Model 1 are shown. For each of these maps, + m θ φ the secular variation is normalized by its maximum at the given where a is a vector made of (l 1)Pl (cos )cosm and C is the epoch. Therefore, the scale ranges between ±1. Furthermore, to posteriori covariance matrix. The posteriori covariance matrix of ease the identification of relevant secular variation features, those the model is defined by with magnitudes less than ± 25 per cent of the maximum secu- 2 T −1 −1 C = σ (A Ce A + Cm) (26) lar variation are blanked out. Normalization allows the comparison of maps at different epochs. Most notable are the regions of high and is closely related to the model resolution matrix secular variation under South Africa and their evolution in time. T −1 −1 T −1 R = (A Ce A + Cm) A Ce A (27) Features under Antarctica are well pronounced at the beginning of the model period, getting fainter before they becoming brighter by again towards 2000. Over the model period, a few features can T e Cee be observed in the region under the Pacific Ocean that confirms σ 2 = . (28) N − tr(R) earlier findings of weak secular variation for this area Vestine & Kahle (1966). σ 2 not only depends on the number of data N but also on the trace Similar maps of Model 2 are shown in Fig. 9. These maps show of the resolution matrix tr(R). The reliance of the resolution on nearly no notable secular variation under the Pacific, the North the model covariance matrix Cm and therefore on the damping pa- Atlantic and Antarctica. In comparison to the secular variation maps rameters (sets of κ, λ) itself does not allow a rigorous use of these of Model 1, the structures in Fig. 9 tend to be more meridionally error estimates. In the framework of Bayesian inference (Backus aligned, and in particular this alignment is becoming more devel- 1988), this means that well-resolved model parameters (by means λ oped towards the model end. Perhaps, most interestingly are sec- of decreased damping parameters i) do not necessarily guarantee

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Figure 8. The normalized radial component of the secular variation computed from Model 1 at the same epochs as in Fig. 5. that the model parameters are well-determined. A model parameter spatial constraint and the parameter κs controlling this constraint appears to be resolved if its prior variance is significantly decreased (cf. Section 5.3). by the data, but decreasing the damping will increase the posteriori The magnetic flux through individual reversed flux patches at variance of the model parameter. Because weakening the apriori the CMB derived from Models 1 and 2 is analysed at the epochs beliefs, that is, applying very small damping parameters reduces 1965, 1980, 1985 and 2005, respectively. The results of this analysis the numerical stability of the inversion. This is particularly true for are summarized in Table 5. In total, we examined the flux through the co-estimation of the core field and core surface flow which is 14 patches, these are also defined (Table 5). A number of smaller a non-linear problem. It is found that a convergent flow spectrum patches have been disregarded in the analysis, as these exhibit minor is a fundamental requirement for obtaining a model solution. Tak- magnetic flux (<2 MWb) or exist for only a short time. ing this into account, it could be possible that our chosen damping Some of the analysed patches only exist in Model 1, such as the parameters are too strong, which in turn has lead to a systematic patches 2a and 2b, which then join patch 3. The patch 3 forms a underestimation of posteriori model variances. We regard the esti- gigantic region of reverse flux in the Southern Hemisphere, and mated variances discussed in the following sections as lower bounds may be the primary cause of the South Atlantic anomaly. Another on the model uncertainties. distinctive feature of the two models is the patch near the North Pole. This is present in Model 1 for the first three epochs and split into two patches under Ellesmere Island (patch 11a), and Barents Sea (patch 11b), during the year 2000. In contrast, Model 2 shows 5.6 Magnetic flux at the core surface two individual patches, but not a single patch under Greenland To evaluate to what extent the field models are in agreement with the (patch 11). Table 6 compares the magnetic flux through some of the frozen-flux hypothesis, hence, the success of our strategy to force patches common between this study and those of Chulliat & Olsen Model 2 to follow the frozen-flux constraint, we first compute the (2010) and Jackson et al. (2007). These are six or seven patches, magnetic flux through individual reverse flux patches at the CMB respectively (the patch under Antarctica is not present in the model for given epochs. Thereafter, we discuss the temporal evolution of of Jackson et al. 2007). The comparison is given for the epoch 1980, the absolute unsigned flux integral computed from the two models. and a first inspection makes it obvious that the computed magnetic We note that results for Model 1 depends strongly on the chosen fluxes are quite different.

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Figure 9. The normalized radial component of the of the secular variation computed from Model 2 at the same epochs as in Fig. 5.

Table 5. Comparison of the magnetic flux through reversed flux patches and the absolute magnetic flux through U(t)the entire core surface in MWb, computed from Models 1 and 2. The patches are numbered from South to North, geographical names (e.g. islands) are attached to allow easy identifications on maps. For some epochs, patches did not exist, therefore their fluxes could not be derived. Patch Site Model 1 Model 2

1965 1980 1985 2005 1965 1980 1985 2005 1 Antarctica 49 104 113 84 48 64 69 83 2aSofAfrica500––––––– 2b SW Atlantic 505 – – – –––– 3 S Atlantic – 1313 1388 1377 1169 1208 1211 1212 4 St Helena – 61 87 119 97 94 92 105 5EasterIsl.3 39461816151415 6Mariana4382877347424259 7 Azores – 42 53 28 12 13 14 25 8 Black Sea – 0 3 13 17 15 15 12 9 North Pacific 22 53 60 55 49 49 46 28 10 East Siberia – – 0 15 7 10 14 13 11Greenland5497105––––– 11aBarentsSea– – – 5064595648 11b Ellesmere – – – 90 68 69 74 79 U(t) 37 472.4 39 788.3 40 239.7 40 064.6 39 988.8 40 008.3 40 005.8 40 071.1

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Table 6. Comparison of the magnetic fluxes obtained in this study and those ilar considerations of the model variances allow an uncertainty of by different authors. ±5 MWb to be placed on the magnetic flux of the St Helena patch Chulliat & Jackson throughout the model period. If these uncertainties are treated as Site Model 1 Model 2 Olsen (2010) et al. (2007) lower bounds, then Model 2 conforms well to the frozen-flux as- sumption. 1980 1980 1980 1980 A more general quantity to be considered is the temporal change Antarctica 104 64 79 – of the total unsigned flux (12). The details of its computation are S Atlantic 1313 1208 1416 1318 laid out in the Appendix A. The total unsigned flux through the St Helena 61 94 94 143 core surface is invariant with time, if the relative importance of Easter Isl. 39 15 25 22 magnetic diffusion is negligible Gubbins (1991). Fig. 10 shows this Mariana 82 42 80 28 quantity derived from Models 1 and 2, respectively. The curve of North Pacific 53 49 33 34 N. Pole 97 69 79 53 Model 1 shows a strong variation, the total unsigned flux ranges between 37 215.3 and 40 424.5 MWb, whereas this quantity of Model 2 varies to a significant smaller extent; between 39 841.5 and 40 115.5 MWb. The error bars on these curves reveal that the The temporal variation of the magnetic flux through individual uncertainty in the total unsigned flux of Model 1 is very large patches have been used as an indicator for the failure or confor- towards the beginning of the model period. It could be that the mance with the frozen-flux assumption. And, when the temporal varying data density is affecting the model uncertainty and there- behaviour of the magnetic flux of the individual patches derived fore the spatial constraint is more effective at the beginning of the from Model 1 is considered, then all fluxes vary with time and some model and becomes less effective towards the end. Evidently, the changes are significant, for example, for the patch under St Helena. total unsigned flux uncertainty of Model 2 is less effected by the Flux variability, appearing and disappearing patches, and joining varying data density and stays more or less constant throughout and splitting patches may indicate that the frozen-flux hypothesis is the model period and shows no visible end effects. Our uncertainty not justified for Model 1. To quantify the significance of the mag- estimates are in good agreement with those of Gillet et al. (2009), netic flux variation in Model 1, lower bounds for the error of flux where they attempted to estimate the error on the unsigned flux from computations are derived by using the posteriori model variance the interaction of the flow with unresolved magnetic field at small- (see previous section). From (26), we can assign a variance to the length scales. In their fig. 11 (bottom), the total unsigned flux pre- Gauss coefficients and compute the variance of the radial field at a dicted from the ensemble flow solutions shows a spread of roughly given point on the core surface, which then allows us to place an ±30 MWb. This spread provides a different measure of uncertainty uncertainty on the magnetic flux through individual patches. In case than ours. However, bear in mind that this depends on the choice of the St Helena patch, the uncertainties for the epochs 1965, 1980, of a prior flow spectrum (regularization), and that variable damp- 1985 and 2005 are ±20, ±1 ±6and±5 MWb, respectively. ing parameters may alter the range of the predicted total unsigned Temporal changes of the magnetic flux through patches can also flux spread. be observed in Model 2. However, these changes are less pro- The first-time derivatives of the total unsigned flux for both mod- nounced than the changes derived from Model 1. The largest varia- els are shown in Fig. 11 and it appears that the total unsigned flux tion occurs for patch 1, which changes its flux by 35 MWb (almost change of Model 1 is less than the total unsigned flux posteriori 75 per cent) in this period. However, this region is sparsely covered uncertainty. Which, on the other hand, does not allow us to rate by geomagnetic observatories and the data of the magnetic field Model 1 as a model consistent with frozen flux. Furthermore, the in this region are loaded by external field noise caused by auroral change of the total unsigned flux of Model 2 ranges between −1 − electrojet and field-aligned currents, which also make it difficult to and 5 MWb yr 1 which we consider to be a good implementation of separate core and crustal fields. If at all, this may only indicate a the frozen-flux constraint. To facilitate a more detailed discussion, local failure of the frozen-flux assumption under Antarctica. Sim- the rate of change of the total unsigned flux attributed to individual

Figure 10. Comparison of the evolution of the absolute unsigned flux of the two models: Model 1 and Model 2. The error bars indicate the posteriori uncertainty.

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Figure 13. The rate of change of the total unsigned flux for several trunca- tion degrees derived from Model 2. Figure 11. Comparison of the rate of change of the total unsigned flux. To allow a comparison the curves scale with different measure; the curve Table 7. Damping parameters and diagnostics of the two flow solutions at derived for Model 1 scales with the left y-scale and the curve derived for the epochs shown in Fig. 14. The unit of the mean velocity is (km yr−1), Model 2 scales with the right y-scale. whereas the strength of the components is given as percentage of the total flow energy. Model parameter and diagnostics Flow 1 Flow 2 Damping parameters −1 −2 −6 −2 λs ([km yr ]) 2 × 10 1 × 10 −1 −2 λt ([km yr ]) 10 10 Energy of flow components relative to total flow energy 1969 Mean velocity (km yr−1) 17.1 8.97 Symmetric component 57.88 77.63 Asymmetric component 42.12 22.37 Geostrophic component 86.12 91.99 Ageostrophic component 13.88 8.01 Zonal toroidal component 13.32 79.24 Figure 12. The rate of change of the total unsigned flux for several trun- Non-zonal-toroidal component 86.68 20.76 cation degrees derived from Model 1. Spherical harmonic degrees having 1990 minor contributions to the rate of change of the total unsigned flux are shown −1 in light grey. Mean velocity (km yr ) 12.01 10.9 Symmetric component 66.53 91.58 Asymmetric component 33.47 8.42 Geostrophic component 88.21 91.69 spherical harmonic degree l is computed first by Ageostrophic component 11.79 8.31   Zonal toroidal component 28.53 81.15 Non-zonal-toroidal component 71.47 18.86 U(l) = |Br(l)| d − |Br(l − 1)| d (29) S(c) S(c) 2007.5 −1 for a given time and then it is differentiated in time. It is assumed Mean velocity (km yr ) 13.4 9.69 Symmetric component 74.78 83.79 U = 0 for the spherical harmonic degree l = 0. These quantities Asymmetric component 25.22 16.21 are computed from both models and shown in Figs 12 and 13. Geostrophic component 90.82 87.95 Notably, the curves differ in their amplitude. The curves of Model Ageostrophic component 9.18 12.05 −1 1 range between ±60 MWb yr , whereas the curves of Model 2 Zonal toroidal component 39.26 77.88 − mainly range between ±10 MWb yr 1. The largest flux contributor Non-zonal-toroidal component 60.74 22.12 of Model 2 is the first spherical harmonic degree. This has a similar amplitude to the first spherical harmonic degree flux change of 5.7 The related core surface flow and core angular Model 1, which in turn means that the total unsigned flux change momentum (c.a.m.) changes of degree 1 is equally controlled by the constraints of both models. The effect of model regularization becomes different for higher In this section, we compare the related core surface flow of the two spherical harmonic degrees. The largest flux changes in Model 1 models. By inverting Model 1 for its likely core surface flow, we seek are carried by degrees 11 and 14, whereas Model 2 shows the largest a purely toroidal flow, subject to spatial and temporal constraints flux changes in degrees 1 and 6. This finding differs from results and a adequate fit to the observed secular variation. Table 7 lists the of Gillet et al. (2010) where recent field models for the periods flow model characteristics. 1960 and 2000 show the largest flux changes in degrees 12 and 14. Fig. 14 shows maps of the two models’ core surface flows. The We conclude that the frozen-flux field model efficiently minimizes comparison of the flow reveals apparent differences in spatial details high-degree flux changes. at all epochs (1969, 1990 and 2008), but in particular in 1969,

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Figure 14. Maps of the core surface flow at the epochs 1969, 1990 and 2008 derived from Model 1 (left-hand column) and derived from Model 2 (right-hand column). Note that the scale for the arrows differs between the maps of left- and right-hand columns. where Flow 1 (flow of Model 1) shows similarities with purely toroidal flows of previous studies (e.g. Holme 2007), such as the winding current system in the Southern Hemisphere. Generally, the flow of Model 2 (Flow 2) appears to be much simpler than Flow 1. In Section 5.2, we mentioned that there is a trade-off between flow energy and the fulfilment of the frozen-flux constraint; highly energetic flows tend to be unable to satisfy the frozen-flux constraint. However, both flows commonly have a strong westward-directed motion near the equator. The results of a detailed flow analysis and the relative strength of some fundamental flow components at the three epochs (1969, 1990 and 2008) are given in Table 7. A general feature of the two flows is the large fraction of geostrophic motion, about 90 per cent of the flow energy. Both flows have similar ageostrophic components (mainly, Figure 15. The averaged power spectra of the two flow models. The bars the flow across the equator), but the most significant disagreement display the temporal variability of the flow power. between the two flows is the partitioning between zonal and non- zonal components. Flow 1 tends to have more energy stored in the non-zonal component than in the zonal component. In Flow 2, the zonal component dominates the non-zonal, which causes Flow 2 to A further possible explanation could be that Flow 2 has to meet be more symmetric along the equator than Flow 1. an extra constraint on the flow acceleration at the model endpoints, Another apparent difference is that the mean velocity of Flow 1 whereas there is no such constraint placed for Flow 1. is larger than that of Flow 2, therefore, Flow 1 is more energetic and In this study, order six B-splines are used to represent the core appears to have considerable larger amount of energy in small-scale surface flow and the field coefficients in time. Therefore, robust features than Flow 2. This can be clearly seen in the energy spectra estimates of flow acceleration are obtained, which in turn is of of the flows (Fig. 15). The spectrum of Flow 2 decays much more particular relevance for computations that rely on flow changes. rapidly than the spectrum of Flow 1. Again, this is related to the From these flows, time-series of relative c.a.m. can be derived fact that highly energetic flows violating the frozen-flux constrain. (Jault et al. 1988; Jackson et al. 1993). The derivation involves two

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Figure 16. Predictions of the relative core angular momentum changes from the flow derived from Model 1 (black line) and the flow of the frozen-flux field model (red line). The observed LOD is shown as blue curve. Note that the flow predictions are shifted to allow easy comparison.

0 0 omitting the re-weighting and de-noising steps in Model 2. The rms zonal toroidal flow coefficients (t1 and t3 )   secular variation misfit of repeat station residuals are clearly larger 12 . . . = . 0 + 0 . than the observatory values. This is likely, because these data were c a m 1 138 t1 t3 (ms) (30) 7 not treated by the de-noising formalism there are no large differ- In Fig. 16, the derived c.a.m. is compared with a rough fit to ences between the rms secular variation misfit of the two models. the observed length-of-day variation (LOD), where signals of Although the benefit from using repeat stations may not be evi- atmospheric and oceanic angular momentum variation have been dent, we believe that these observations may provide a valuable removed, similar to Holme & de Viron (2005). Although Flow 1 data source for geomagnetic field modelling in regions of sparse shows a higher temporal variability, it barely meets the details of observatory coverage. The actual benefit of repeat station data to the observed LOD. In particular, the amplitude of the predicted geomagnetic field modelling should be evaluated in more detail by variation is larger than the observed LOD. Some features of the future studies. Flow 2 prediction and observations correlate well and seemingly It might also be of interest to study the match between Model provide a closer match to the observations, only the amplitude of 2 and the apriorimodel for 1980, which was not used in the the prediction appears to be slightly too large. So far, the cause of setup of Model 2. Fig. 19 shows the spectra of the apriorifield the large amplitude difference of the two predictions is unclear, but model and Model 2 for 1980 at the core surface, (the spectrum from recent computations (Jackson et al. 1993; Whaler & Holme 2011) Model 1 is not shown, as the graph exactly matches the one of the of the c.a.m. show also larger amplitudes than the LOD. apriorifield model). The spectra begin to diverge from spherical harmonic degree 7, but show a close match for degrees 9 and 10. From degree 10 onwards, the differences become larger. The reason a priori 5.8 The fit to observatory data and field models for the spectral differences may result from the different effective In Figs 17 and 18, we examine the fit of the modelled and the minimization of high-degree flux changes of the two modelling observed secular variation at a number of magnetic observatory approaches. sites chosen to represent a broad geographic distribution. The figures Furthermore, the secular variation of Model 2 sometimes tends show also a comparison between the modelled secular variation of to show a higher temporal variability, but this is very marginal Models 1 and 2. The match between the modelled and the observed and cannot be considered as significant. Overall, we conclude that secular variation is, in general, good. This can also be concluded frozen-flux field modelling is capable of successfully accounting from the rms secular variation misfit (11) between observed and for the observed secular variation. modelled secular variation given in Table 8. The largest scatter appears for Xand˙ ZofAlibagwheretheresid-˙ 6 DISCUSSION uals are larger in the earlier epochs (Fig. 17). Also, the modelled secular variation of the two models is in good agreement. In par- When core surface flow and the magnetic field are co-estimated ticular, the geomagnetic jerks which occurred in this period are from geomagnetic observations, there is a marked improvement in recovered by both models equally well. Apparent deviations of the the description of the magnetic field. This is clear when maps of two models are found for Xand˙ Y˙ at Mbour. However, the values in the radial field at the CMB Figs 5 and 6 are compared, especially Table 8 do not indicate major deviations between the models, they if the spatial complexity of maps towards the beginning of the rather suggest that the models fit the data to the same level. model period are examined. The spatial complexity of Model 1 To quantify how well both models fit the data globally, Table 8 varies as time proceeds backwards and is generally poorer than the shows also the global rms secular variation misfit of the observa- spatial complexity of Model 2. Although Model 1 is constrained to tory and repeat station residuals with respect to Models 1 and 2. fit satellite field models for 1980 and 2004, this is not enough to The observatory values differ for all three components by about maintain the spatial complexity towards the beginning of the model 0.5 nT yr−1, but Model 1 shows smaller rms secular variation mis- period. This is because of the lack of data. Conversely for Model 2, fit. It could be that the deviations between the two models are due to the spatial complexity at 1960 is very similar to that satellite field

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Figure 17. Comparison between the observed (grey dots) and the modelled secular variation of Model 1 (black curve) and Model 2 (red curve) in Sitka (SIT), Niemegk (NGK), Kakioka (KAK) and Alibag (ABG)—from top to bottom. The vertical lines, marked with the characters A–H, indicate occurrence times of geomagnetic jerks. (The regular pattern seen in some of the observatory values, that is, dY/dt of Niemegk and Kakioka is due to rounding secular variation values to the nearest integer.)

models. The spatial complexity of the satellite epoch is maintained non-linear, the co-estimation is formulated as a single inverse prob- backwards in time by the frozen-flux modelling approach. This, is lem with one consistent set of uncertainties. Although inferences an important benefit of the frozen-flux modelling procedure. about the core surface flow might be limited, as we require the flow It should be noted that co-estimating core surface flow and the to be large scale and purely toroidal, co-estimation of flow and field magnetic field from one set of data has a different error budget than may provide an alternative to regularized field models. Perspectives the sequential estimation of the core surface flow by first computing on alternative field and flow co-estimation have been considered by a field model from observations and then deriving the core surface Fournier et al. (2010), and for a possible estimation of flow prior flow. The sequential flow estimation requires the assignment of two see Fournier et al. (2011). individual inverse problems. Their solutions are subject to different Our finding that the flow of the frozen-flux model (Model 2) sets of uncertainties (errors) and aprioribeliefs, which are indepen- is much simpler than the flow of Model 1 could be regarded as dent and therefore unlikely to be consistent with each other. Though a consequence of a uniform error budget. It could also be that

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Figure 18. The same comparison as in Fig. 17 for the geomagnetic observatories MBour (MBO), Gnangara (GNA) and Hermanus (HER)—from top to bottom.

Table 8. rms Secular variation misfit (M) at the observatories shown in Figs 17 and 18 for Models 1A, 1B (marked with subscript 1A and 1B) and Model 2 (marked with subscript 2) and the rms secular variation misfit of all observatory and repeat station residuals.

Observatory code M(X˙ 1A) M(X˙ 1B ) M(X˙ 2) M(Y˙ 1A) M(Y˙ 1B ) M(Y˙ 2) M(Y˙ 1A) M(Y˙ 1B ) M(Y˙ 2) SIT 11.10 11.17 10.98 5.58 5.21 5.24 8.55 8.44 8.05 NGK 8.36 8.17 8.21 3.02 2.86 2.93 6.82 6.41 6.57 KAK 12.88 12.72 12.76 1.84 1.73 1.92 4.03 3.92 4.20 ABG 18.60 18.61 18.48 8.09 8.47 8.38 13.92 13.91 12.66 MBO 13.85 14.10 13.94 5.78 5.44 5.68 6.23 6.32 5.63 GNA 10.59 11.82 10.57 4.62 5.69 4.94 9.40 9.99 8.77 HER 11.89 12.23 11.96 4.70 5.45 4.88 5.77 6.05 5.58 Observatories 13.60 13.51 13.93 8.77 8.76 9.23 12.52 12.55 13.01 Repeat stations 18.41 18.38 18.60 18.33 18.39 18.96 18.02 18.09 18.69 inversions of field models for core surface flow in previous studies ation by the interaction of large-scale flow and unknown small-scale systematically overfitted the secular variation and therefore lead to magnetic field leads to uncertainties of about 9 (km yr−1)2 for the more energetic flows. flow and 186 (nT yr−1)2 on average for the secular variation, respec- Limitations and uncertainties in applying our method of frozen- tively. Therefore, truncation of the main field at a given spherical flux field modelling arise as it is based on a spatially low-pass filtered harmonic degree may lead to a significant error in our estimation of core field and secular variation; small scales of the core field and the secular variation within the frozen-flux field model. However, secular variation are shadowed by the crustal field and transient Wardinski et al. (2008, see their fig. 12) showed that flow estima- magnetic fields induced in the Earth by external field variations. In tion was significantly erroneous only in a region under southeast our approach to compute the core surface flow, we assume that the Asia. These uncertainties are similar in magnitude, but are differ- amplitudes of the main field and secular variation above spherical ent in shape compared with those shown in fig. 11 of Eymin & harmonic degree 14 are insignificant and therefore can be disre- Hulot (2005). We note that no match between our maps and those garded. This may be viewed as a crude simplification and in fact of Eymin & Hulot (2005). Our results suggest that our approach of Eymin & Hulot (2005) determined that contributions to secular vari- frozen-flux field modelling approach gives a sufficiently accurate

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equally well, leading him to conclude that all small-scale patches of reversed flux cannot be resolved with certainty. The finding of O’Brien (1996) agrees with our results in Sec- tion 3.2 that a field model for the epoch 1980 can be built that has the same total unsigned flux as a field model based on CHAMP data for the epoch 2004. The uncertainties of the field at the CMB at spherical harmonic degrees 12–14 allow for this. Moreover, it should be noted that any MAGSAT-based field model is built from data in a distinct local time regime unlike recent satellite field mod- els. More precisely, MAGSAT sampled the field along dawn–dusk orbits, whereas the 2004 satellite field model is exclusively made of quiet night time data. Therefore, it seems very likely that the MAGSAT data carry a stronger contribution from external fields. This contribution is likely to be strongest in regions under the au- roral and equatorial electrojet. Such signals are known to have a non-vanishing strength in dawn time. In particular, the period when MAGSAT was flying, numerous days of enhanced magnetic activ- ity of Kp = 3.0 were recorded. Enhanced magnetic activity raises the amplitude of such external signals. This effect which will sig- Figure 19. Mauersberger–Lowes spectra of the apriorimain field model nificantly alter the physical properties of the models and may also for 1980 (black line) and the Model 2 (red line). have affected the study of Chulliat & Olsen (2010), causing an ap- parent failure of the frozen-flux assumption. This requires further elaboration. It is certain that magnetic diffusion must take place in the liquid outer core. However, we are pessimistic whether its contribution is currently resolvable from observations of the Earth’s magnetic field.

ACKNOWLEDGMENTS We thank the reviewers for their helpful comments. We also would like to acknowledge the CHAMP data centre and those involved in the CHAMP data processing, Furthermore, we like to record our gratitude to the scientists working at magnetic observatories and Figure 20. The absolute of the differences between the secular variation the WDC for Geomagnetism (Edinburgh) for compiling these data derived from Model 2 and the observed secular variation in 2003. These in a database and Anne Froebel for the transcription of monthly signals cannot be explained by the frozen-flux field model. means from geomagnetic observatory yearbooks. We also acknowl- edge the support of the International Space Science Institute (Bern, description of the core field, secular variation and the related core Switzerland). All maps in this paper were generated with a free soft- surface flow. ware, GMT (Wessel & Smith 1991). The work was partly funded by the Deutsche Forschungsgemeinschaft Schwerpunktprojekt 1488. Thus far, we have used the ratio (23) to quantify the failure of the frozen-flux field modelling approach: this gives a global measure. The spatial distribution of the failure is however also of interest. Therefore, the difference between advective secular variation and REFERENCES model secular variation of the frozen-flux field model are com- puted and mapped. In Fig. 20, the absolute values of this difference Abramowitz, M. & Stegun, I.A., 1973. Handbook of Mathematical Func- tions, Dover, New York, NY. are shown. Large differences become visible under South Africa, Amit, H. & Christensen, U.R., 2008. Accounting for magnetic diffusion in Siberia and North America. Regions of large differences appear core flow inversions from geomagnetic secular variation, Geophys. J. Int., in the vicinity of intense flux patches (cf. Fig. 6) and therefore 175, 913–924. a close relationship between these patches and the cause of the Backus, G.E., 1988. Bayesian inference in geomagnetism, Geophys. J. R. violation might be likely. In this context, it should be noted that a astr. Soc., 92, 125–142. previous study found a very similar morphology for the secular vari- Bloxham, J., 1987. Simultaneous stochastic inversion for geomagnetic main ation, which is due to field decay modes (see Wardinski et al. 2008, field and secular variation. I. A large-scale inverse problem, J. geophys. fig. 13). Res., 92, 11 597–11 608. While Chulliat & Olsen (2010) found a local failure of the frozen- Bloxham, J., 1988a. The determination of fluid flow at the core surface flux assumption underneath St Helena in the Mid-Atlantic between from geomagnetic observations, in Mathematical Geophysics, A Sur- vey of Recent Developments in Seismology and Geodynamics, ch. 9, 1980 and 2005, Jackson et al. (2007) could not find observational pp. 189 – 209, eds Vlaar, N.J., Nolet, G., Wortel, M.J.R. & Cloetingh, evidence contradicting the assumption for the period between 1882 S.A.P.L., Reidel, Dordrecht. and 2000. These results seem to be conflicting, but may be explained Bloxham, J., 1988b. The dynamical regime of fluid flow at the core surface, by the different methodologies used. In this context, it appears Geophys. Res. Lett., 15, 585–588. interesting that O’Brien (1996) argued that field models with and Bloxham, J. & Gubbins, D., 1986. Geomagnetic field analysis. IV. Testing without individual flux patches explain magnetic field observations the frozen-flux hypothesis, Geophys.J.R.astr.Soc.,84, 139–152.

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Intense equatorial flux spots on the surface of the Earth’s core, Nature, 424, 760–763. The total unsigned flux is given by Jackson, A. & Finlay, C.C., 2007. Geomagnetic secular variation and  its applications to the core, in Geomagnetism, Treatise on Geophysics U(t) = |Br| d . vol. 5, ch. 5, pp. 148–193, ed. Kono, M., Elsevier, Amsterdam. S(c) Jackson, A., Bloxham, J. & Gubbins, D., 1993. Time-dependent flow at In this study, the total unsigned flux (14) is computed using the the core surface and conservation of angular momentum in the coupled Gauss–Legendre quadrature to numerically calculate the integral core-mantle system, in Dynamics of the Earth’s Deep Interior and Earth for given epoch t Rotation, Geophys. Monogr. Ser. vol. 72, pp. 97–107, eds Le Mouel,¨ J.-L.,  + + Smylie, D.E. & Herring, T., American Geophysical Union, Washington, 1 L 1 2L 1 D. C. |B | d = wL+1 |B (θ ,φ )| (A1) r 2 i r i j Jackson, A., Jonkers, A.R.T. & Walker, M.R., 2000. Four centuries of ge- S(c) i=1 j=1 omagnetic secular variation from historical records, Phil. Trans. R. Soc. with Lond., A, 358, 957–990.   Jackson, A., Constable, C.G., Walker, M.R. & Parker, R.L., 2007. Models L l of Earth’s main magnetic field incorporating flux and radial vorticity |B θ ,φ |= + m φ r( i j )  (l 1) (gl cos(m j ) constraints, Geophys. J. Int., 171, 133–144. = = l 1 m 0  Jault, D., 2008. Axial invariance of rapidly varying diffusionless motions in   +  a l 2  the Earth’s core interior, Phys. Earth planet. Inter., 166, 67–76. + m φ m θ  . hl sin(m j )) Pl (cos i ) (A2) Jault, D., Gire, C. & Le Mouel,¨ J.L., 1988. Westward drift, core motions and r  exchanges of angular momentum between core and mantle, Nature, 333, + × + 353–356. The integrand is sampled at (L 1) (2L 1) nodes in latitu- Lesur, V., 2006. Introducing localized constraints in global geomagnetic dinal and longitudinal direction, L = 14 is the maximum spherical field modelling, Earth Planets Space, 58, 477–483. harmonic degree. The location of the nodes are determined by Lesur, V., Wardinski, I., Rother, M. & Mandea, M., 2008. GRIMM—the θ = = ,..., + , GFZ Reference Internal Magnetic Model based on vector satellite and i arccos(xi ) i 1 L 1 observatory data, Geophys. J. Int., 173, 382–394. 2π j φ = j = 1,...,2L + 1 (A3) Lesur, V., Wardinski, I., Asari, S., Minchev, B. & Mandea, M., 2010a. j (2L + 1)

C 2012 The Authors, GJI, 189, 1409–1429 Geophysical Journal International C 2012 RAS C3FM2 1429 with xi the ith zero of the Legendre polynomials PL+1(x). The nodes Lesur (2006). However, for U(t) this cannot be guaranteed. wL+1 weights i are given by The results of summations are compared to results obtained by a tessellation of the sphere by spherical triangles similar to the method

L+1 2 −2 applied by Jackson et al. (2007). For this method, the number of w = (∂ P + (x )) . (A4) i 2 x L 1 i (1 − xi ) steps in refining an icosahedron is set to seven. A match between the  results of both methods is obtained when 150 points along latitude

The summation would be exact for S(c) Br d with 15 points and 290 points along longitude are chosen. This adds up to 43 500 along latitude and 29 points along longitude giving a total of 435 nodes.

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