 Article
 Open Access
Modelling the Earth’s core magnetic field under flow constraints
 V. Lesur^{1}Email author,
 I. Wardinski^{1},
 S. Asari^{1},
 B. Minchev^{2, 3} and
 M. Mandea^{1, 4}
https://doi.org/10.5047/eps.2010.02.010
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB 2010
 Received: 22 July 2009
 Accepted: 23 February 2010
 Published: 6 August 2010
Abstract
Two recent magnetic field models, GRIMM and xCHAOS, describe core field accelerations with similar behavior up to Spherical Harmonic (SH) degree 5, but which differ significantly for higher degrees. These discrepancies, due to different approaches in smoothing rapid time variations of the core field, have strong implications for the interpretation of the secular variation. Furthermore, the amount of smoothing applied to the highest SH degrees is essentially the modeler’s choice. We therefore investigate new ways of regularizing core magnetic field models. Here we propose to constrain field models to be consistent with the frozen flux induction equation by coestimating a core magnetic field model and a flow model at the top of the outer core. The flow model is required to have smooth spatial and temporal behavior. The implementation of such constraints and their effects on a magnetic field model built from one year of CHAMP satellite and observatory data, are presented. In particular, it is shown that the chosen constraints are efficient and can be used to build reliable core magnetic field secular variation and acceleration model components.
Key words
 Geomagnetism
 core field modeling
 core flow modeling
 frozenflux
1. Introduction
Following the launch of the magnetic survey satellites Oersted in 1999, and CHAMP in 2000, a global set of high quality magnetic vector data is now available. Currently, this set spans nearly ten years and has led to time varying models of the core magnetic field of unprecedented accuracy. One of the major achievements is the modeling of the magnetic field Secular Acceleration (SA) i.e. the time evolution of the magnetic field Secular Variation (SV). Indeed, it is crucial to model as accurately as possible the secular acceleration because it has a profound effect on the SV which in turn affects estimates of the liquid outer core flow, just below the CoreMantle Boundary (CMB). Properly describing the flow at the top of the liquid outer core is essential as it is one key piece of information to understand the dynamics of the core, with implications for other physical process such as long timescale changes in the length of the day.
The secular acceleration is modeled in the available core magnetic field models with time variations described by cubic (or higher order) Bsplines. For example this is the case for the CM4 (Sabaka et al., 2004), GUFM (Jackson et al., 2000) or CALS7K (Korte and Constable, 2004) models. However, these models, that are needed to describe relatively long term variations of the Earth’s core magnetic field, have been built with relatively few data per year, and therefore have been strongly smoothed in time.
In this manuscript we present our approach to model both the core magnetic field and the flow on the core surface simultaneously. More specifically, a core magnetic field model, built to fit a magnetic data set, is coestimated together with a flow model using the radial diffusionless induction equation (hereafter the FFequation). Constraints are applied exclusively on the flow model in order to obtain the best possible core field model. At a glance, it is not obvious why smoothing the flow is preferable to smoothing the field. Both regularization techniques are, however, likely to single out a possible mechanism (i.e. diffusion or advection) for the SV generation in the core. Here, by using the FFequation we favor an advective process, and show that smoothing in time the field is likely to favor diffusion. Indeed, it is wellknown that some diffusion must exist, and therefore an advective process, even if dominant, cannot be the exclusive source of the SV. To avoid this pitfall, we impose the diffusionless hypothesis (hereafter the FFhypothesis) in a weak form, such that the data set can always be properly fitted.
The idea of imposing the FFhypothesis on a core field model has already been used for example by Bloxham and Gubbins (1986) and Jackson et al. (2007). These authors require the magnetic flux to be constant in time over areas on the CMB defined by null flux curves at different epochs. Our approach is different because we coestimate the flow and the field and therefore impose some constraints on the flow. Furthermore, the FFhypothesis is applied continuously in time. Closer to our approach is that of Waddington et al. (1995) where observatory data are fit by parameterizing the flow on the core surface. Their work is sometimes seen as an early attempt of a data assimilation technique where the physical model for the flow evolution is replaced by an hypothesis of steady flow. Beggan and Whaler (2009) also used a steady flow model, combined with Kalman filtering, to forecast change of the magnetic core field. In our case the flow is allowed to vary in time and the difference with assimilation techniques is that evolution equations for the flow are not introduced. On the other hand, the new assimilation techniques recently developed for the magnetic field modeling are not yet based on real data but only on models (Fournier et al., 2007; Liu et al., 2007). We also note that the approach we follow in this study has been independently suggested, in their conclusion, by Whaler and Holme (2007).
Here, our main goal is to investigate how well one can control a core field model by applying constraints on the coestimated flow model. The methodology is tested on a vector CHAMP satellite and observatory data set spanning only one year. By using such a short time span, we make sure that the constraints applied on the flow have an obvious effect on the field model. On the other hand, the resulting core field model cannot be of the same quality as models recently derived from the full set of available satellite and observatory data, such as GRIMM (Lesur et al., 2008) and xCHAOS (Olsen and Mandea, 2008). In particular, as for other models built from short time span data sets (see for example Olsen, 2002), the acceleration cannot be accurately modeled. We also impose some strong restrictions on the flow by first truncating the flow model to a relatively low SH degree, and second, choosing the same temporal representation for the flow coefficients and for the Gauss coefficients. As for the field models, the obtained flow model cannot be of the same quality of recently published models, nevertheless, it is of sufficient quality for the purpose of this study.
The manuscript is organized as follow. The necessary assumptions of the problem and the implementation details are presented in the next section. In the third section the application to the CHAMP satellite and observatory vector data is presented. The fourth section is dedicated to evaluating the effect of the regularization on the field and flow models. Finally, in the fifth section, the obtained magnetic field, the fit to the data, and the flow models are discussed and compared with the GRIMM magnetic field model.
2. Theoretical Background
2.1 The discrete problem
2.2 Solving for the Gauss coefficients

First, the flow model can be forced to have a convergent spectrum. The flow is then required to minimize Bloxham’s “strong norm ”(Bloxham, 1988; Jackson, 1997), Another option is to minimize a weaker norm (Gillet et al., 2009), The first and second terms in the integral (22) measure the amount of up/downwelling and the radial vorticity respectively. The damping parameters λ_{2B} and λ_{2W} controls to what extent the flow follows these constraints.

Second the flow model is chosen such that it varies smoothly in time, namely: where λ_{3} is the associated damping parameter and denotes the flow time derivative. We expect such a constraint to efficiently regularize the inverse problem as in the limit of a constant flow there is a unique flow solution of the FFequation (Voorhies and Backus, 1985). We also expect that minimizing Eq. (23) constrains efficiently the secular acceleration.
2.3 Errors and weight matrices
In the inverse problem (17), that consists of estimating a core field model from a data set, the weight matrix W^{ d } depends, at first, on the estimated data accuracy and also on the data density. The matrix is then updated during the iterative least squares inversion process used to derive the model. The data errors are assumed to be uncorrelated so the weight matrix is diagonal.
When the field and the flow are coestimated, the functional Φ_{1} and the diagonal weight matrix W^{ ġ } are introduced (Eq. (18)). The W^{ ġ } matrix elements result from three different contributions, namely: the surface integration weights in Eq. (19), the representation errors e_{ r } and the truncation errors e_{ t } (see Subsection 2.1 for these error definitions).

Observational errors that account for the errors in the Gauss coefficients estimated through the optimization process defined by Eq. (17).

A second type of truncation error to account for the fact that short wavelengths of the magnetic field can interact with the short wavelengths of the flow to generate long wavelength secular variation (Eymin and Hulot, 2005).
In the present work, observational errors are not considered because the field and the flow are coestimated. The second type of truncation errors is also omitted but, in return, we have to impose a rapidly converging spectrum onto the flow. This is not ideal, but it is necessary at this stage, as the field model cannot be efficiently constrained if the flow model has too many degrees of freedom.
3. Application to CHAMP and Observatory Data

Only vector data are used.

The data are selected for quiet magnetic conditions.

Highlatitude three component vector data are selected at all local times.

Mid and lowlatitude data are selected along the X and Y Solar Magnetic (SM) directions during night times.
With such a selection process the data set combines 35975 X and Y (SM) satellite data at mid and low latitudes, 65147 satellite vector data over polar regions, 38954 X and Y (SM) observatory data at mid and low latitudes, and finally 13974 observatory vector data at high latitudes. Gaps in the satellite data distribution are seen at high latitudes due to the blinding of one of the CHAMP star cameras by the sun, whereas at mid and low latitudes they are due to the local time selection applied to minimize the contribution of the ionospheric field in vector data. Observatory data present no gaps over the time period.
The magnetic field contributions modeled are similar to those in GRIMM. The core field model (see Eqs. (5) and (6)) is parameterized using SH up to degree 14 and an order 4 polynomial in time (i.e. this is equivalent to order 5 Bsplines in between two spline knots). A static internal field is modeled up to SH degree 20, which is enough to avoid aliasing effects in the core field model. The large scale external field is modeled only at SH degree 1, but the modeling is robust only in the X and Y SM directions due to the data selection at mid and low latitudes. As in GRIMM, the time variations of this large scale external field are parameterized using a piecewise linear polynomial in time with a node every three months. The rapid time variations are parameterized using the VMD (time series of the estimated disturbances due to large scale external fields (Thomson and Lesur, 2007)). Crustal offsets are coestimated for observatory data in order to account for the unknown contributions from the lithosphere. The high latitude ionosphere field or the toroidal field generated by Field Aligned Currents (FAC) are not considered in the modeling. Again further details can be found in Lesur et al. (2008).

Two further series of models are built using the approach described in the previous section. They differ by the measure used to minimize the flow complexity in space: either Eq. (21) (hereafter the BSNmodel series) or Eq. (22) (hereafter the WSNmodel series). In both models, Eq. (23) is minimized to guarantee temporal smoothness of the flow.
4. Effect of the Regularization on the Coestimated Field and Flow Models
In this section we compare and discuss the choice of regularization for the coestimation of the core magnetic field and flow models. All results presented in this section are for the BSN models and are obtained using an L_{2} measure of the data misfit. This measure does not lead to the best solutions but is sufficient to understand the response of the model solutions to the constraints applied to the inversion process. The results obtained using an L_{1} measure of the misfit require much longer computation time, and are therefore derived only for a limited small set of damping parameters. Such results are presented in the next section.
 1)
The unknown short wavelengths of the flow and SV can interact to contribute to the large wavelength of the SA. The spectrum of the modeled SV is not convergent at the CMB and therefore this truncation error for the background SA is likely to be significant. However, as for the advection of the short wavelengths of the field by the short wavelengths of the flow, we do not expect this effect to be dominant at small SH degrees. Again, this holds because our flow is essentially large scale and has a convergent spectrum.
 2)
The flow resolved by the inversion process is only part of the true flow because when advecting the field lines, part of the flow does not contribute to the SV. This hidden flow can nevertheless contribute to the background SA. If we accept the common consensus that the strength of this hidden flow is at most of the same order of magnitude as the modeled flow (see for example Rau et al. (2000) or Asari et al. (2009)), then we can assume that the interaction of the SV with the hidden flow is not larger than the interaction with the modeled flow. Overall, our estimated background SA may not be accurate, but the order of magnitude is acceptable. The power spectrum of the estimated background SA is shown in Fig. 4.
Mean (M) and root meansquares (SD) misfit values for all data types and core field models, in nT. (SM) stands for SolarMagnetic and H. lat for HighLatitudes.
Data types  Number of data  USN  BSN  WSN  GRIMM  

M  SD  M  SD  M  SD  M  SD  
Sat. X (SM)  35975  −0.50  3.49  −0.51  3.50  −0.51  3.50  −0.68  3.60 
Sat. Y (SM)  35975  −0.68  3.71  −0.69  3.71  −0.69  3.71  −0.75  3.75 
Sat. X H. lat  65147  1.36  47.08  1.37  47.08  1.37  47.08  0.30  46.97 
Sat. Y H. lat  65147  −2.68  51.48  −2.67  51.49  −2.67  51.49  −2.71  51.53 
Sat. Z H. lat  65147  −0.95  20.54  −0.93  20.57  −0.94  20.56  −0.74  20.95 
Obs. X (SM)  38954  −0.06  3.25  −0.06  3.25  −0.06  3.24  —  — 
Obs. Y (SM)  38954  −0.08  3.24  −0.10  3.23  −0.10  3.23  —  — 
Obs. X H. lat  13974  −3.72  27.78  −3.91  27.85  −3.89  27.84  —  — 
Obs. Y H. lat  13974  −0.15  13.53  0.17  13.57  0.17  13.56  —  — 
Obs. Z H. lat  13974  −0.59  20.88  0.63  20.94  0.62  20.94  —  — 
By strongly damping the flow time variations one may try to reach the limit where only the background SA is significant. This limit was not reached at low SH degrees even for our largest damping value λ_{3} = 10^{6} (not shown). At higher SH degrees (i.e. from degree 10 and above) the spectra do not change significantly with the damping value and obviously the solutions get closer to this limit. This difference in behavior depending on the SH degree is simply due to the fact that the data set used does not resolve well the acceleration at high SH degrees.
5. Results and Discussions
In this section we present and compare the results obtained using different regularization techniques. All these results were obtained using a L_{1} measure of the misfit. The number of iterations before reaching a stable solution is relatively large for such models and data sets: 20 iterations for the USN model and up to 60 iterations for the BSN or WSN models (for the definition of these models refer to Section 3).
The USNmodel is derived using the regularization parameter λ_{1U} = 10^{−4.5} and λ_{2U} = 10^{−025} in Eq. (30). It is relatively difficult to set these damping parameter values by examining the tradeoff between the fit to the data and the roughness of the solution. We therefore simply set these values such that the power spectra of the model solution, secular variation and acceleration stay reasonably close to those of the GRIMM model. The BSN and WSN models are derived as described in Section 4 and the chosen damping parameter values are λ_{1} = 10^{−3}, λ_{3} = 10^{4.25} for both models and λ_{2B} = 10^{−2.25}, λ_{2W} = 10^{−05} for BSN and WSN respectively (see Eqs. (21), (22), and (23)). The data misfits for these three models are given in Table 1 together with the number of data. For all three models, the fit to the data is good at mid and low latitudes but degrades closer to the poles. This is to be expected, because of the chosen data selection criteria. For comparison Table 1 also gives the fit to the same satellite data set for the GRIMM model with its lithospheric component truncated at SH degree 20. Not surprisingly, at mid and low latitudes the fit for the GRIMM model is slightly worse than for the other three models, because it was built on much larger time span data set.
The GRIMM model presents a SV with slightly less power from SH degrees 6 to 8, explained by the fact that GRIMM is built on a different data set and is smoothed in time. SVBSN and SVWSN models are essentially the same. They have very similar power spectra and the power spectrum of their differences never exceeds 0.08 (nT/yr)^{2}. Similarly, the power spectrum of the differences between the SVBSN (or SVWSN) model and the associated SV estimated through the FFequation (i.e. the SV generated exclusively from the advection of the field lines by the flow) never exceeds 5.0 10^{−10} (nT/yr)^{2} in 2005.4. The ratio ϱ, defined in Eq. (31), is 0.919^{−15} for the BSN model and 0.219^{−14} for the WSN model. Therefore, the models built follow (not exactly but very closely) the FFhypothesis. As expected, due to the regularization technique employed, these two SV models can be downward continued to the CMB without further regularization. The SVUSN model is clearly dominated by the regularization above SH degree 10. Its power drops excessively rapidly from SH degrees 11 to 14.
The SA models power spectra, in Fig. 6, are significantly different. As described above, the damping parameter values of the BSN, USN, WSN models have been adjusted such that all SA power spectra match around SH degree 1. From there, the SABSN and SAWSN power spectra stay more or less constant up to SH degree 3 and then drop regularly down. As discussed in Section 4, the SABSN and SAWSN power spectra presented here for SH degree higher than 8, are the lowest possible for the models to be compatible with the FFhypothesis. The behavior of the SAUSN model is clearly anomalous and it shows that the selected data set over a single year does not resolve well the acceleration. It is also clear that the integrals defined in Eq. (30) impose constraints on the high SH degree of the SAUSN model that are too strong, leading to an unrealistic decrease in the spectrum. Such a model is not compatible with the FFhypothesis (see discussion in Section 4). We see immediately the effect of using the new approach for regularizing the magnetic field inversion process: the time behavior of the field model is consistent with the underlying physical process described by the FFequation and advection is favored as a possible source of the SV. The spectrum of the SA for GRIMM is above the others for degrees 3 to 9. This does not necessarily mean that this spectrum is too high. As stated above, one year of data is not enough to resolve well the SA and by accumulating data over several years, the spectra from both the BSN and WSN model accelerations would possibly rise. However, above SH degree 9 the GRIMM SA is controlled by the applied regularization and probably drops too rapidly to be consistent with the FFhypothesis.
The SA obtained through our inversion process is only valid for the first two or three SH degrees. Most of the SA patterns observed in the GRIMM model correspond to the SH degree 4 or 5. These SH degrees are not resolved here. We observe however, that the BSN and WSN SA models can be downward continued to the CMB without further regularization. There, they are dominated by their short wavelengths, and the patterns, mainly controlled by the FFequation, are associated with strong gradients of the SV model (maps of the SA are not shown here).
6. Conclusions
We derived a core magnetic field model spanning the 2004.87–2005.94 period from CHAMP satellite and observatory data. The field model is coestimated together with a model of the flow at the top of the core and we impose the constraint that the field model closely follows the FFhypothesis continuously in time. Despite the shortness of the data time span, the SV model is surprisingly accurate around 2005.4. Similarly the SA model can be resolved for the first SH degrees. However, our main point in this work has been to investigate how well a core field model inversion process can be regularized by constraints applied on the coestimated flow model. In this respect, the results are very encouraging. First, we have shown that imposing a convergent spectrum on the flow immediately constrains the secular variation such that it can be downward continued without further regularization to the CMB. Second, we have shown that smoothing temporal variations of the flow affects the magnetic field acceleration magnitude.
In addition, we have seen that at the Earth’s surface the observed SA is mainly controlled by the temporal flow variations. We suggest that the background SA, that does not depend directly on the flow variations, defines a lower limit of the acceleration power spectrum for the core field compatible with the FFhypothesis. Then, if the spectrum of a core magnetic field model acceleration falls below this limit, the time behavior of this field model is inconsistent with the FFhypothesis. Deriving flow time variation information from such an anomalous model would be unlikely to lead to acceptable results. It would be interesting to test these hypotheses in a dynamo simulation where the induction equation is solved in a selfconsistent manner.
In the approach used, the core flow model is truncated at relatively small SH degrees and, even if the flow has a rapidly converging spectrum, the interactions between the small scales of the field and the flow could be better accounted for. However, it is not straightforward in the presented framework. This will have to be investigated in a forthcoming study.
Diffusion necessarily exists and one could argue that imposing the FFhypothesis constraint is not a valid approach at these timescales. We think nevertheless, that to impose the constraint is definitively an approach worth studying. Further, we observe that even under the flow constraints presented above, the error in Eq. (15) stays very small. Therefore, it is fairly easy to build a model respecting the FFhypothesis, although that is not a sufficient condition to make the hypothesis valid. Some preliminary work has been done to apply the technique on data sets covering longer time span. There are no apparent serious further difficulties. However, this has to be investigated in detail in future work.
Declarations
Acknowledgements
We would like to acknowledge the work of CHAMP satellite processing team and of the scientists working in magnetic observatories. We would like also to thanks the reviewers for their constructive comments that certainly help in improving this manuscript. I. W. was supported by the European comission under contract No. 026670 (EC research project MAGFLOTOM). IPGP contribution 2611.
Authors’ Affiliations
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